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Observability based techniques to analyze and design user-interfaces : situation-awareness and displayed… Eskandari Naddaf, Neda 2015

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Observability Based Techniques toAnalyze and Design User-InterfacesSituation-Awareness and Displayed InformationbyNeda Eskandari NaddafB.Sc., Sharif University of Technology, 2006M.Sc., Sharif University of Technolog, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2015c© Neda Eskandari Naddaf 2015AbstractFor continuous-time LTI systems under human control and under sharedcontrol, this thesis studies techniques to determine whether or not a givenuser-interface provides the information required to accomplish a certain task.It is well known that attaining Situation Awareness (SA) is essential tothe safe operation of the systems involving human-automation interaction.Hence, through two different approaches, the work in this thesis evaluatesand designs user-interfaces based on the satisfaction of SA requirements bythe user.In the first approach, observability-based conditions under which a user-interface provides the user with adequate information to accomplish a giventask are identified. The user is considered to be a special type of ob-server, with capabilities corresponding to different levels of knowledge re-garding the input and output derivatives. Through this approach, the“user-observable/user-predictable” subspaces for systems under shared con-trol are defined and formulated. In addition, state estimation is consideredto incorporate a processing delay. Hence, the “delay-incorporating user-observable/user-predictable” subspaces are formulated and are comparedwith the space spanned by the combination of the states which create thetask. If the task subspace does not lie in the relevant space, then the user-interface is incorrect, meaning that the user cannot accomplish the desiredtask with the given user-interface.In the second approach, to determine the required information to be dis-played, a model of attaining SA for the users is proposed. In this model,the user is modeled as an extended delayed functional estimator. Then, theinformation needed for such an estimator to make correct estimations aswell as the desired expansion of the functional of the states to let the useriiAbstractprecisely reconstruct and accurately predict the desired task is determined.Additionally, it is considered that in practice, to attain the situation aware-ness, the estimation of the task states does not necessarily need to be precisebut can be bounded within certain margins. Hence, the model of the userattaining SA is also modified as a “bounded-error delayed functional obser-vation/prediction”. Such an observer/predictor has to exist for a systemwith a given user-interface, otherwise, the safety of the operation may becompromised.iiiPrefaceMy contributions in this thesis resulted in two conference papers and fivejournal articles.• A version of Chapter 3 is published in:N. Eskandari, and M. Oishi, “Computing observable and predictablesubspaces to evaluate user-interfaces of LTI systems under shared con-trol,” in Proceedings of the IEEE conference on Systems, Man andCybernetics, Alaska, USA, October 2011, pp. 2803–2808. This paperreceived the Best Student Paper Award at this conference.R. Cortez, D. Tolic, I. Palunko, N. Eskandari, R. Fierro, M. Oishi,and J. Wood, A hybrid framework for user-guided prioritized searchand adaptive tracking of maneuvering targets via cooperative UAVs,In Advances in Intelligent and Autonomous Aerospace Systems, in theProgress in Aeronautics and Astronautics Series American Instituteof Aeronautics and Astronautics, Reston, VA, vol. 241, 2012.• The results from Chapter 4 are published in:N. Eskandari, G. A. Dumont, and Z. J. Wang, “Delay-incorporatingobservability and predictability analysis of safety-critical continuous-time systems,” IET Control Theory and Applications, vol. 9, pp. 1692–1699(7), July 2015.• The results from Chapter 5 are published in:N. Eskandari, G. A. Dumont, and Z. J. Wang, “An observer/predictorbased model of the user for attaining situation awareness,” IEEETransactions on Human-Machine Systems, 2015.ivPreface• The results from Chapter 6 were submitted as:Neda Eskandari, Guy Dumont, and Z. Jane Wang. “Bounded-errordelayed functional observer/predictor: Existence and design”.• The results from Appendix A are published in:N. Eskandari, M. M. Oishi, and Z. J. Wang, “Observability analysisof continuous-time LTI systems with limited derivative data,” AsianJournal of Control, vol. 16, no. 2, pp. 623–627, 2014.• The results from Appendix B are published in:N. Eskandari, Z. J. Wang, and G. Dumont, “Modeling the user as anobserver to determine display information requirements,” in In Pro-ceedings of IEEE International Conference on Systems, Man, and Cy-bernetics, Manchester, UK, October 2013, pp. 267–272.N. Eskandari, Z. J. Wang, and G. A. Dumont, “On the existenceand design of functional observers for LTI systems, with applicationto user modeling,” Asian Journal of Control, 2014.I, hereby, declare that I am the first author of this thesis. I conductedthe literature survey, identified the problem to solve, formalized the theo-retical solution and performed the analysis, and finally implemented all thesimulations. In addition, I am responsible for writing all of the drafts of thisthesis and the resulted papers. My supervisors Professor Guy A. Dumontand Z. Jane Wang guided my research and provided me with complete sup-port including validating the methodologies and editing all the manuscriptsco-authored by them. Part of the thesis is the result of research collabora-tion with an additional contributor, Dr. Meeko Oishi for her technical andediting feedbacks.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations and Symbols . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Situation awareness and display design . . . . . . . . . . . . 31.3 Research objectives . . . . . . . . . . . . . . . . . . . . . . . 51.4 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.1 Observability and predictability . . . . . . . . . . . . 71.4.2 Observer and predictor design . . . . . . . . . . . . . 81.5 Summary of contributions . . . . . . . . . . . . . . . . . . . 102 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . 122.2 A framework for human-automation interaction . . . . . . . 142.2.1 Schematic framework . . . . . . . . . . . . . . . . . . 142.2.2 Mathematical framework . . . . . . . . . . . . . . . . 16viTable of Contents2.2.3 Task formulation . . . . . . . . . . . . . . . . . . . . 172.3 Approaches taken in the thesis . . . . . . . . . . . . . . . . . 172.3.1 Subspace analysis . . . . . . . . . . . . . . . . . . . . 172.3.2 The modeling technique . . . . . . . . . . . . . . . . . 192.4 List of the assumptions . . . . . . . . . . . . . . . . . . . . . 213 Novel observability/predictability subspaces . . . . . . . . . 233.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 233.2 User-observable subspace . . . . . . . . . . . . . . . . . . . . 253.3 User-predictable subspace . . . . . . . . . . . . . . . . . . . . 293.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.1 Nagoya A300 Accident, 1994 . . . . . . . . . . . . . . 323.4.2 A remotely controlled fleet of UAVs . . . . . . . . . . 343.5 Discussion and an alternative presentation of the system . . 353.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Novel observability/predictability subspaces (long term anddelayed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 384.2 Delay-incorporating user-observable subspace . . . . . . . . . 394.3 Delay-incorporating user-predictable subspace . . . . . . . . 444.3.1 Validation of the displayed information . . . . . . . . 464.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4.1 delay-incorporating user-observable subspace . . . . . 474.4.2 Delay-incorporating user-predictable subspace . . . . 484.4.3 Task accomplishment . . . . . . . . . . . . . . . . . . 494.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . 505 User-interface analysis through modeling the user as an es-timator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 515.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.1 Existence conditions for an extended functional esti-mator . . . . . . . . . . . . . . . . . . . . . . . . . . . 57viiTable of Contents5.2.2 Model of the user as an estimator . . . . . . . . . . . 645.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.4 Towards display design . . . . . . . . . . . . . . . . . 705.3 Application example . . . . . . . . . . . . . . . . . . . . . . . 735.4 Summary and discussion . . . . . . . . . . . . . . . . . . . . 756 User-interface analysis through modeling the user as an es-timator with bounded error . . . . . . . . . . . . . . . . . . . 776.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 776.2 Bounded-error estimator . . . . . . . . . . . . . . . . . . . . 786.3 Application example . . . . . . . . . . . . . . . . . . . . . . . 836.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 866.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 896.4 Summary and discussion . . . . . . . . . . . . . . . . . . . . 897 Thesis summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.1 Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . 917.2 Possible future directions . . . . . . . . . . . . . . . . . . . . 93Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96AppendicesA On the effect of higher derivatives on the observability sub-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109A.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 109A.2 Unknown-input observability subspace . . . . . . . . . . . . . 110A.3 Observability subspace with limited information about theinput and output derivatives . . . . . . . . . . . . . . . . . . 113A.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117B On the effect of higher derivatives on the existence of anddesign of functional observers . . . . . . . . . . . . . . . . . . 120B.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 120viiiTable of ContentsB.2 Existence conditions of a functional observer . . . . . . . . . 125B.3 Estimating the order of the functional observer . . . . . . . . 135B.3.1 Design procedure of the functional observer (B.6) . . 139B.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140B.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . 140B.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . 142C List of publications . . . . . . . . . . . . . . . . . . . . . . . . . 146ixList of Tables3.1 The observable and predictable subspaces for different mea-surements of a leader-follower formation . . . . . . . . . . . . 355.1 PKPD coefficient for a 21 years old 100 kg patient . . . . . . 746.1 Patients’ parameters from [1] and [2] . . . . . . . . . . . . . . 856.2 Percentage effectiveness of the estimator designed for nominalmodel P2 on estimating the task for other patients . . . . . . 876.3 Percentage effectiveness of the estimator designed for nominalmodel P7 on estimating the task for other patients . . . . . . 87xList of Figures1.1 Model of SA, from [3]. As is shown in the figure, SA whichconsists of three levels of processing the information is nec-essary for the user to make correct decisions on the controlaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 A simple schematic model of a human controlling a system.The user gets information about the automation from thedisplay and applies control action through the controls. . . . 152.2 The required relationship between the elements of a system.The user have to be provided with information regarding thetask in order to accomplish it. . . . . . . . . . . . . . . . . . . 182.3 Block diagram presentation of the system and the user. . . . 215.1 Non-delayed observation of the desired functional, Tx, of thestates of the system (5.39). . . . . . . . . . . . . . . . . . . . 685.2 Using a non-delayed observer matrices for predicting Tx, whilethe actual observer has also internal delay. . . . . . . . . . . . 695.3 Delayed prediction of the desired functional, Tx, of the statesof the system (5.39), with τ = 0.2 and τ1 = 0.3. . . . . . . . . 70A.1 The effect of providing information about input and its deriva-tives on the observability subspace. We could show thatadding information about input and its derivatives can re-sult in larger observability space. The dashed-dotted linesrepresent containment. . . . . . . . . . . . . . . . . . . . . . . 116xiList of FiguresB.1 The error of estimating the functional z0(t) of (B.62) with theobserver (B.63) . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.2 The error of estimating the functionals of system (B.64) withthe observer (B.67) . . . . . . . . . . . . . . . . . . . . . . . . 144xiiList of Abbreviations andSymbolsAcronymsDOA Depth of AnesthesiaGPS Global Positioning SystemLTI Linear Time InvariantOCM Optimal Control modelPD PharmacodynamicPK PharmacokineticSA Situation AwarenessSGA Situation Goal OutputTHS Trimmable Horizontal stabilizerNotations and OperatorsMany notations are introduced in the body of this thesis with their descrip-tion provided in the respective sections. Following, we provide the mainimportant notations and operators that are used more frequently through-out this document:OH User-observable subspacePH User-predictable subspaceO∗H Delay-incorporating user-observable subspaceP∗H Delay-incorporating user-predictable subspacexiiiNotations and OperatorsOUI Unknown-input observability subspaceOH,τ1 Delayed observable spaceT Task spaceC Controllability subspaceO Observability subspaceλ Available number of derivatives of inputγ Available number of derivatives of outputτ Prediction horizonτ1 Processing delayR(.) Column spaceN (.) Null space(.)⊥ Orthogonal complementrank(.) Rank of a matrixR Real numbersIN N ×N identity matrixdiag{·} (Block) Diagonal matrix⊕ Subspace summation∩ IntersectionxivAcknowledgmentsI am using this opportunity to express my gratitude to my supervisors Pro-fessor Guy A. Dumont and Professor Z. Jane Wang without whom thisresearch would not be possible. Their continuous understanding, support,and patience helped me to overcome a series of difficulties and conduct thepresented research. I would also like to thank my supervisors for all thetechnical guidance that they provided me with.I also wish to deeply thank my loved ones who have supported methroughout the entire process. Many thanks to my dear husband, Hossein,who has been incredibly supportive of me and my work since the very firstday of our marriage, more than eleven years ago! His selfless patience andhis faith in me throughout the difficult phases of my Ph.D. was one of themain reasons that this work could come to existence. Words cannot expresshow grateful I am to my lovely parents for all of the sacrifices that they havemade for my success. Their consistent support and also prayers for me waswhat sustained me thus far. I would also like to thank my brother and mysister-in-law who have always been available for me during this Ph.D..I am grateful to my friends and colleagues for helping me throughoutthis process.xvChapter 1Introduction1.1 MotivationIn complex cyber-physical systems, correct human interaction with the sys-tem is key for effective operation. Oftentimes, such systems are very largeand simple intuition is not enough to determine whether a user-interface, adevice through which the human applies a control input and through whichthe system provides the human with information about the output, displaysthe required information. Not only the representation of information, butalso the information content of the display, play important role in havinga good interface. While too much information can overwhelm the user,presenting too little information can result in non-determinism in the user-interface from the user’s point of view [4]. Mathematical tools and methodsto determine whether a given user-interface allows accomplishment of a fea-sible task can help prevent non-determinism and other inconsistencies thatcould arise through incorrect user-interface design.Consider an example about 1994 Nagoya Airbus A300 incident [5]. Forthis flight, an automatic go-around maneuver was inadvertently triggeredduring a manual approach to landing. While the flight crew applied a pitch-down command to achieve their desired path, automation applied a pitch-up command to reach a reference altitude rapidly. Unaware of the effectof their input, cabin crew continued to descend along the glideslope. Theautomation adjusted the trimmable horizontal stabilizers (THS) to makethe aircraft ascend, while the flight crew acted to counter the automationthrough the elevators. Meanwhile, the THS gradually moved from −5.3◦to −12.3◦, producing an orientation very close to the nose-up limit. Theflight crew eventually disengaged the autopilot and decided to abort the11.1. Motivationlanding. A pitch-up command to the elevators, in combination with theTHS orientation, generated a stall condition which resulted in a crash [6].Although it might not seem obvious that under given situation, the task oflanding was not a safe task, mathematical tools can be developed to helpthe designers come up with such a conclusion.Researchers have previously identified potential sources of mode confu-sion [7] in discrete user-interfaces for aircraft flight management systemsmodeled as discrete-event systems [4, 8–10] and as discrete abstractions ofhybrid systems [6, 11]. Methods based on model-checking evaluate a dis-crete event system that represents the underlying system dynamics anduser-interface for deadlock and other problematic states. Methods basedon composition of finite state machines compare the evolution of trajecto-ries in a discrete event system that represents the underlying system, toanother discrete event system that maps the states to a known output,essentially providing a simplified representation of the information on theuser-interface that evolves according to the system dynamics [12]. How-ever, while continuous components of the interface are quite common (e.g.,altimeter, speedometer, others), little work has been done on evaluatingthe correctness of user-interfaces with continuous components that have notbeen abstracted to discrete states [4, 13].As interfaces and the underlying systems become more complex, infor-mation beyond what is contained in the interface may not be accessible.We aim to identify tools that assess the correctness of a user-interface fora given task, an especially relevant problem in systems for which intuitionand simulation may not be enough to assure that an interface is effective.While in many systems, such as a human-driven car, the user has accessto information beyond what is simply contained in the interface, we focushere solely on information contained in the interface, e.g., a remote operatorcontrolling a fleet of UAVs or a pilot performing a task in high altitude.Using the mathematical tools that we develop, we can demonstrate thatfor the aircraft in the above example under shared control, the display didnot provide the user with sufficient information to perform the task of land-ing safely. More importantly, by making further investigations we can show21.2. Situation awareness and display designthat even with further information in the display e.g. about the automa-tion desired trajectory, accomplishing landing is not a safe task to be doneunder shared control. So given these results, we can suggest either provid-ing the user with a predicted information about the process of landing ordisengaging the autopilot during the phase of landing.1.2 Situation awareness and display designDespite successful efforts to increase the autonomy of systems and devicesused daily, many systems function under the shared control of the human andthe computer. Indeed many fully automated systems need to be supervisedand monitored by a human operator. Because correct interaction betweenthe user and the plant is crucial, it is essential for the user to understandwhat the system is doing and what it is intending to do in a near future.For this, the system should provide the required information for the userto achieve complete Situation Awareness (SA), i.e. to ”keep the operatortightly coupled to the dynamics of the environment” [14].Endsley formally defines SA as ”the perception of the elements in theenvironment within a volume of time and space, the comprehension of theirmeaning and the projection of their status in the near future” [3]. Theschematic framework for SA that was suggested by Endsley in [3] is shownin Figure 1.1.From Figure 1.1 and the above definition, three components of SA areas follows [3, 15]:• Level 1: Perception. Level 1 of SA is about being aware of the existenceof the elements in the environment, the dynamics of the system, andmore. Lack of correct perception can increase the chances of wrongunderstanding about the situation.• Level 2: Comprehension. Based on the particular goal of the user,each part of the perceived information in Level 1 can be of certainimportance. Comprehension of information is about integrating and31.2. Situation awareness and display designFigure 1.1: Model of SA, from [3]. As is shown in the figure, SA whichconsists of three levels of processing the information is necessary for theuser to make correct decisions on the control action.synthesizing this information to determine the significance of each por-tion of the information and to attain further and possibly high levelunderstanding about the environment.• Level 3: Prediction. The highest stage of SA is projection which isabout using all information obtained in Level 1 and 2 to make pre-dictions about the future states of the system and the elements of theenvironment.As is clear from Figure 1.1, having SA about the desired task is necessaryfor the user to be able to make correct decisions and finally to choose thereasonable and safe control action. Therefore, it is of interest to developtechniques to evaluate and determine the minimum information requiredfor the user to achieve SA.The tools that we develop for analyzing available displays as well asdesigning better displays are based on the theory of SA. For instance, forthe aircraft incident described in Section 1.1, we will later show that theavailable display of A300 did not provide the pilot with adequate informationto achieve SA about the task of landing which finally led to a crash.41.3. Research objectives1.3 Research objectivesAs has already been discussed, it is well-known from the literature thatattaining SA is key for the user to have correct interaction with the au-tomation [3, 16]. Not having SA may lead to wrong decision-making andpossibly a faulty control action by the operator. Attaining SA necessitatesthree stages of processing the information, i) perception of the information,ii) comprehension of the information, and iii) projection or prediction of theinformation [3, 16].With the purpose of analyzing and modeling the process of attainingSA and its relationship with displayed information, this thesis considers thecases for which the only source of information for the user is the display.Therefore, the display has to be designed to allow the user to perceive, com-prehend, and predict the desired information to attain correct SA. Morespecifically, the display design requires a careful selection and clear presen-tation of the information for the user to perform those tasks properly.To achieve the main objective of this thesis which is determining therequired displayed information, observability-based conditions under whicha user-interface provides the user with adequate information to accomplisha given task are identified. In addition, a model of attaining SA for the usersis proposed.1.4 Related workCognitive modeling has long been a topic of interest for human-factors re-searchers, engineers, clinical psychologists, and scientists in any other fieldwhich is creating a link between human behavior and mathematical con-cepts [17–19]. As opposed to conceptual models which discuss and evaluatethe specifications of the system and the interactions between the user andthe plant in verbal terms [20], cognitive models use analytical tools andtechniques to provide more precise and valid results and predictions of thebehavior of such systems [21]. The main difference between a cognitivemodel and a statistical or a generic mathematical model is that in a cogni-51.4. Related worktive model certain specifications of the human and cognitive parameters areconsidered to customize the analytical model and to address the behavior ofthe human.Although, in general, mostly the statistical based models of human areconsidered and named as cognitive models [22–24], we also can consider themodels of human suggested in other types of papers as cognitive models. Forinstance, over the past few decades, much work has been done on modelingthe human operator as a controller [25–28]. Each of these models includesvarious parts representing different specifications of the human, such as,neuromuscular term, proprioceptive feedback, and delay and in each paperthe results are compared with the real-life results. In more recent papers,uncertainties were also incorporated in the human models [29, 30].Because SA is essential for the user to properly accomplish a task, it isnecessary to either 1) analyze the processing of the information by the userto evaluate whether and/or how the user is capable of attaining SA aboutthe important states of the system or 2) to extend the human model to alsoinclude obtaining SA before acting as a controller.Several researchers have investigated conceptual models of human infor-mation processing for attaining SA in various applications [31–33]. Howevernone of these papers quantified this process with a rigorous analytical model.To our knowledge, the optimal control model (OCM) [34, 35] was thefirst model to mathematically capture the ability of the user as an estimatorin addition to a controller. In general, in the OCM, the user estimates thestates of the system and acts on the system based on these estimations.This model states that the human attains delayed noisy information fromthe display then makes further estimations about the states of the system.The estimator block in the OCM is considered to be a Kalman Filter. Apredictor block is also included in this model which simply compensates forthe inherent delay of the user. This model had aimed at capturing somelimitations of the human, including the time delay, neuro-motor dynamics,and some controller specifications. However, given that the time delay isconsidered to be canceled out by the predictor and no further effort is madeto customize the Kalman filter based on the limitation of the user, the esti-61.4. Related workmation part of this model is more of a mathematical model than a cognitivemodel.In addition to the above models, other researchers also modeled thehuman as an observer-based fault detector with the focus on modeling thedecision-making process [36, 37].1.4.1 Observability and predictabilityObservability analysis and observer design are our main tools through thisthesis.The concept of observability was first introduced by Luenberger [38, 39]for single and multi-variable systems. Later, through using a geometricalapproach based on the controlled and conditioned invariant spaces, Basileand Marro [40] could formulate the observability subspace for systems withunknown inputBy taking derivatives from the output equation inx˙ = Ax+Buy = Cx(1.1)where R(.) shows the column space and based on definition, we haveDefinition 1. The least (A,F)-conditioned invariant containing a space Xis the subspaceJm = Yn−1where Yn−1 is defined by a recursive relationshipY0 = X , Yi = X ⊕A(Yi−1 ∩ F), i = 1, · · · , n− 1.In Chapters 3 and 4, we draw mainly on control and observability con-cepts by Kalman [41] and Luenberger[38, 39], as well as the work by Basileand Marro [40] in unknown-input observability to obtain the user-observablesubspace – that is, a subspace which is observable to the user with additionallimitations.71.4. Related work1.4.2 Observer and predictor designIn Chapters 5 and 6, we model the user as a functional observer under a setof given assumptions about the actual behavior of the user.For many applications, such as implementing a control law, monitoringthe behavior of an automated system, or for fault detection, one needs tohave access to all or a portion of the state space. If the desired portion ofthe state space is not directly measured, an observer is required to estimatethe desired states based on the measured states. Over the past few decades,numerous research papers have been published on observing the states oflinear systems. Luenberger initially came up with the ideas of designingfull-state observers to estimate all states of a deterministic linear systemand reduced-order observers to reconstruct the unmeasured states of a de-terministic linear system [38, 39]. In many applications, however, there is noneed for reconstructing all unmeasured states, since only a specific portionof the state space is important to the users/designers (e.g., for state feed-back control). Therefore, to reduce the cost of reconstructing unnecessarystates, functional observers were introduced.Functional observers are a specific type of observers which can recon-struct a desired functional without reconstructing all unmeasured states.The first functional observer, which was capable of reconstructing a scalarlinear functional of the states of a linear system, was presented in [39]. Thistechnique was followed by other efforts [42, 43] and was later extended forobserving multi-functionals of LTI systems [42, 44, 45].While designing observers for the reconstruction of linear functionals re-mains a popular topic, researchers are also interested in investigating meth-ods for reducing the order of the functional observers [46] and for evaluat-ing the minimum required order [47, 48]. An excellent piece of work [49]presents a necessary and sufficient condition for the existence of a “sameorder functional observer” (i.e., a functional observer with the same orderas the desired functional). Later [50], [51], and [52] extended [49] to evaluatethe existence of “potentially higher order observers” that can reconstruct thedesired functional in cases that the observer in [49] does not exist. Another81.4. Related workapproach for assessing the existence of functional observers is the eigenspaceanalysis [53].Functional observers are of great importance, not only for systems withknown input, but also for systems with unknown input. Researchers havemade efforts to design functional observers for systems with unknown input[54–57]. As in his work on systems with known input and no uncertainty[49], in [58] Darouach defined necessary and sufficient conditions for the exis-tence of a “same order functional observer” for systems containing unknowninputs. Further, by modifying [58], more general existence conditions forunknown input functional observers were reported in [59].Researchers studied the existence of functional observers for the recon-struction of desired functionals for different types of systems, including lineardescriptor systems [52, 60], time delay systems [61–65], and two dimensionalsystems [66]. Also functional observers are designed for the purposes suchas fault detection [67, 68]. There are also works on the existence and designof common/simultaneous functional observers [69, 70]. The references men-tioned here are only a portion of the vast body of literature on functionalobservers for linear systems.In most of the above papers on the existence and design of functionalobservers, the overall procedure for designing the functional observer for anLTI system is as follows:• Assume to have a parametrized observer in an LTI form. Based on theavailability of the input, the input will be introduced to the dynamicsof the observer.• Formulate the error of estimation to be the difference between theestimated functional and the actual functional of the states.• Find the dynamics of the error by taking derivative from the estimationerror we formulated before.• Formulating the error dynamics as e˙ = C1e+C2 where C1 and C2 arefunctions of the observer matrices.91.5. Summary of contributions• To make the error asymptotically approach zero, C1 should be designedto be stable and C2 have to be zero.The effect of the availability of the delayed output on the size of thedesired functional has also been evaluated [71–73]. A rather comprehensivediscussion of functional observers for systems with known and unknowninputs is available in [74].1.5 Summary of contributionsChapters 3 through 6 cover techniques for the analysis and syntheses of userinterfaces to satisfy the requirements of SA theory.In Chapter 3, we identify observability-based conditions under whichuser-interfaces provide the user with information to accomplish a given task,formulated as a subset of the state-space. We introduce the “user-observablesubspace” and the “user-predictable subspace” which are subspaces basedon observability and predictability requirements of the user and limitationsof the user regarding input signals.The main contribution of Chapter 4 is to modify the results of Chapter3 by incorporating the delay of estimation. Therefore, the notions of delay-incorporating user-observable and delay-incorporating user-predictable spacesare defined and are formulated in this chapter. The delay that we considerthroughout this thesis is the information processing delay of the user. Onthe other hand, our system of interest is delay free.In Chapter 5, we model SA with specific consideration of the capabil-ities of the user. We, thus, evaluate the existence of the novel delayedobservers/predictors and then design them. We, also, suggest a techniquefor determining the required displayed information.The work in Chapter 6 comes up with an estimator model that makesbounded estimations of the desired states of a deterministic LTI system.Therefore, a novel technique to check the boundedness of the estimationand the prediction errors of a desired functional, with an estimator whichdynamics is delayed, is suggested.101.5. Summary of contributionsIn the appendices, the effect of providing higher derivatives of the inputand/or the output signal on the observability subspaces and on the orderand the structure of the functional observer is investigated.11Chapter 2Problem definitionIncorrect human-automation interaction can be due to reasons such as non-deterministic plant behavior or the user having inadequate understandingof the plant dynamics. In addition, inadequate and improper display ofinformation could be hazardous to the safety of the system.2.1 Problem statementConsider a user interacting with an automated system through the user-interface. Building a reliable interaction between this user and the auto-mated plant requires providing the user with necessary information aboutthe states of the plant and to set up a proper task for him/her to accom-plish. Through processing the available information, the user can achievesituation awareness and then use it to accomplish a desired set of tasks.Various researchers have examined human information processing [3, 75–77]. Parasuraman et. al. [76] introduced a four-stage model of humaninformation processing which includes information acquisition, informationanalysis, decision and action selection, and action implementation. Accord-ing to the situation awareness model of Endsley [3], the user first attainssituation awareness. Decision making and action implementation are twostages that follow situation awareness. Sherry’s SGA model of human infor-mation processing, which was later used by Sherry et.al. [77] to demonstrateone of the reasons for users’ confusion about the behavior of automation,also consists of attaining information about the situation, re-scheduling thegoals, and finally action.Essentially, three common fundamental stages of human information pro-cessing have been pointed out by the above mentioned researchers, including122.1. Problem statementi) understanding the situation, ii) decision making, and iii) action implemen-tation. In this work, we focus on the first stage of information processing –that is, understanding the situation. Our main goal here is to evaluate thedisplayed information and obtain the required information which has to bedisplayed so that the user can accomplish a desired task.Analyzing the available information has been discussed in detail by Ends-ley and it was specifically termed as Situation Awareness (SA) [3]. AttainingSA which necessitates perception, comprehension, and prediction of the in-formation, can be done using working memory or long term memory. Asstated by Cowan [78], “long-term memory is a vast store of knowledge anda record of prior events” and the working-memory is the memory which“is used to plan and carry out the behavior”. Other researchers also men-tioned that understanding and predicting the required information is vitalfor having an effective human-automation interaction and that problemswith predictability can result in false expectations [7, 79].Factors such as learning and attention can clearly affect the perceptionof the information (the first stage of attaining SA). Here, as we make As-sumption 1; presented at the end of this chapter; we focus on the latter twostages of attaining SA to help us evaluate and determine the informationcontent of the display.Note that insufficient understanding about the situation is not solelydue to lack of information in the user-interface, but it can also be the resultof having an indeterministic automation or a user who is unfamiliar withsystem dynamics, e.g., as in category II of Pilot Induced Oscillations (PIO)[80]. In this thesis we particularly consider the case where the user is fullyfamiliar with the deterministic dynamics of the system and our goal is toanswer what information has to be presented on the display to achieve thedesired task and whether it is displayed.132.2. A framework for human-automation interaction2.2 A framework for human-automationinteractionSo far, various frameworks for analyzing human-automation interaction havebeen developed. Jamieson and Vicente [81] suggested a feedback loop modelin which feeding back important signals to the user could help them identifyand localize the source of failure in the elements of the system. Sheridansuggested a set of frameworks for systems with different levels of autonomy[82]. In [83], using a framework for supervisory control, Cummings showedthat it is necessary for the user of some systems to act in collaboration withthe plant, rather than exclusively acting as a supervisor.2.2.1 Schematic frameworkFor this thesis, we consider a delay-free system whose evolution is modeledby an LTI model.In Figure 2.1, we consider the user to be a function that maps the dis-played information to the user input. This mapping involves different stagesof information processing. In our framework, we similarly consider the userto first obtain situation awareness, then decide on the required action, andfinally act on the system. As mentioned earlier, our focus is on the stage ofobtaining situation awareness and our purpose is to evaluate whether or notthe task is defined correctly and whether the user has access to the necessaryinformation. We also want to determine what information is required forthe user to attain SA.We consider the user to be highly trained and experienced with thedynamics of the automation. The terms trained, experienced, and noviceare frequently used in the literature to define the level of proficiency of theuser, however, defining these terms formally and precisely is not an easytask. A comparison between commercial pilots, who always need to havehigh levels of training and experience, and licensed drivers, who are notrequired to have high level of proficiency, can clarify the difference betweenlevels of training and experience for users of various systems [31]. Here by the142.2. A framework for human-automation interactionFigure 2.1: A simple schematic model of a human controlling a system. Theuser gets information about the automation from the display and appliescontrol action through the controls.terms trained and experienced, we specifically mean that the user is capableof perceiving information on the inputs, outputs, and their derivatives. Wetherefore ignore the information-acquisition delay of the user. The usercan then make use of the perceived information to reconstruct and predictimportant states of the system. Recall that reconstruction (comprehension)and prediction are two stages of attaining situation awareness.For a fully experienced and trained user with a well-developed mentalmodel about the behavior of the system, comprehension and prediction areachieved using long term memory. This process of pattern-matching whichtakes place without loading the working memory is almost instantaneous.152.2. A framework for human-automation interactionIn this thesis, we assume that the only role of the mental model is to providethe user with adequate understanding about the system’s dynamics.For the purpose of Chapter 3 we ignore the effect of any delay, however,for Chapters 4-6 we assume that the comprehension of the non-measuredpart of the task functional and also the prediction the task functional haveto take place within the working memory which is associated with certainamount of processing delay [84]. By focusing on the processing delay andassuming the perception delay to be negligible, we build our technique onAssumption 7 for Chapters 4- Mathematical frameworkIn the body of this thesis, we consider the system as a delay-free systemwhose evolution of the states is modeled by the LTI modelx˙(t) = Ax(t) +Buh(t) + Fra (2.1)where x(t) ∈ Rn is the state vector. The inputs in (2.1) can be categorized asthe known input which is the low-level human input, uh(t) ∈ Rmb , controlledby the user and the input which is the time-invariant reference trajectory,ra ∈ Rmf , tracked by the automation. Here, we consider the referencetrajectories to be unknown, unless they are measured in the display. In(2.1), the matrices A, B, and F have compatible dimensions. Other thanChapter 3, in all other chapters we also assume that no poles of the systemare on the imaginary axis.Based on our specific application of user-interface design, we assume thateach displayed measurement is either a reference trajectory or a combinationof the states. Also, the output consists of two sets of measurementsy(t) = Cx(t),yr(t) = Dra,(2.2)where y ∈ Rpx is the set of measured combinations of states and yr ∈ Rpr isthe set of measured reference trajectories in the display.The models of the system used in the appendices are slightly different162.3. Approaches taken in the thesisfrom the main model above. We will present those models in their respectivechapters.2.2.3 Task formulationTasks are often specified in terms of conditions that must always be met, ormust be eventually met. For example, potential tasks for a remotely drivencar could be Always travel under the speed limit, or At some time, stop atthe stop sign. Hence we formulate the task as a function f : Rl → Rs withs subtasks, such thatF = {x | f(Tx) ≥ 0}, T ∈ Rl×n (2.3)with task matrix T comprised of l linear combinations of the state. ForAlways F , the state trajectory must lie in F for all time in order for thetask to be successfully completed. For Eventually F , when the state entersthe set F at some finite time, the task has been successfully completed.For example, for a point-mass car with position x and speed v, successfulcompletion of the task Always travel under the speed limit is indicated forstates that remain in F = {x | vmax − v ≥ 0} for all time. Successfulcompletion of the task Eventually stop at the stop sign is indicated for statesthat reach F = {x | x = xstop ∧ v = 0} at some finite time.Denote the task space T by the row space of T , that is, T = R(T T ).2.3 Approaches taken in the thesisIn this thesis we perform two types of analysis to determine the correctnessof the user interface content. In Chapters 3 and 4 we analyze the correctnessof this information via subspace analysis and in Chapters 5 and 6 we comeup with a model of the user attaining SA.2.3.1 Subspace analysisDegani and Heymann [79], introduced a schematic diagram to define the de-sired relationship between the elements of system which are the user-model,172.3. Approaches taken in the thesisthe task, and the user-interface during human-automation interaction. Us-ing their suggested model, they could describe the interrelation betweenthese elements of the system [79]. When user’s capabilities, informationfrom the display, and task requirements are aligned, the correct interactionbetween the elements of the system is possible. As an extension to theirmodel, we suggest Figure 2.2 which presents the necessary relationship be-tween the elements of the system.Figure 2.2: The required relationship between the elements of a system.The user have to be provided with information regarding the task in orderto accomplish it.From Figure 2.2, the human-automation interaction is not correct for alltasks unless the task requirements is entirely included in the intersection ofuser-model and user-interface – that is, to have a correct human-automationinteraction, it is necessary for the user-interface to provide the user withinformation related to any feasible task.Having developed a framework for shared control systems and havingformulated the task physically and mathematically, we are ready to come-up with a mathematical criterion for evaluating the information content of182.3. Approaches taken in the thesisthe display for the system presented in Figure 2.1 to meet the requirementsof a task formulated in (2.3). Hence, we should evaluate whether the user hasaccess to the required information to attain situation awareness regardingthe task. We therefore impose Assumption 5; presented at the end of thischapter.2.3.2 The modeling techniqueTo have a successful HAI, both the data-driven (bottom-up) informationprocessing and the goal-directed (top-down) processing for SA are consideredto be vital [85, 86]. The emphasis of the goal-driven processing is on payingdirect attention to and then processing the most important informationrelated to the goal. Literature reviews on the relationship between attentionand working memory have been provided in [87, 88]. In their discussions, [87,88] brought evidence on how the attention acts to filter out the unnecessaryinformation at both the early stages of perception and the late stage ofprocessing the information in working memory. Overall, one main role ofattention during the post perceptual stage of processing information is toreduce or cancel out the distractions while comprehending the target [89].The adverse effect of irrelevant information on the ability of the userto understand and perform a desired task has been investigated extensivelyin the literature. Through an experiment performed over two years on agroup of elementary students, [90] concluded that “ the problem-solvingability is related to the ability of reducing the memory accessibility of non-target and irrelevant information”. Similar comments were made in [91, 92],e.g., lack of the capability of selecting information relevant to the task andsuppressing irrelevant information will result in poor performance of theworking memory and poor comprehension. It was shown that the abilityto suppress irrelevant information declines for elderly adults [93]. It wasalso mentioned that, for more challenging tasks, during the processing andmanipulating of information in the executive working memory, the userswill have higher focus on the actual goal and irrelevant task informationwill have less effect [88].192.3. Approaches taken in the thesisThe above discussions suggest that it is desirable for a good user (agood problem solver) to concentrate on the target and suppress irrelevantinformation. Intuitively, a trained and experienced user generally will notperform unnecessary processing of information. Therefore, it is not realisticto assume that the user reconstructs all observable and predictable statesto only make use of the desired functional of the states of the system. Forinstance, let us use an aircraft with eight lateral and longitudinal states asan example. If the desired purpose for the pilot is to keep the angle of attackbounded, intuitively s/he will not aim to reconstruct and predict all otherunmeasured states of the system, e.g., the yaw rate is of no interest here.We therefore have Assumption 6; presented in Section 2.4.Based on Assumption 6 and since we consider the comprehension andprediction of the unmeasured information to be a challenging task that canoccupy the executive working memory of the user, we consider the user tobehave as a functional estimator in order to process the information. Webelieve that this model is a reasonable model to start with, although it issimplistic as it cannot capture some common conditions such as mind wan-dering - that is, the decoupling of information processing from the primarytask [94, 95]. The simplified model is in fact addressing perfect users whoperceive all information provided in the display and can fully concentrateon what they are asked to do.The block diagram presented in figure 2.3 contains the model of thesystem that we deal with and the model of the human that we suggest. Thehuman is modeled as an observer/predictor that acts upon the informationprovided in the display. In this diagram, the dashed dotted blocks are theimportant parts of the human model which in this thesis we do not focus on(i.e. the direct effect of mental model as well as the effect of the informationfrom the environment).As is shown in figure 2.3, we consider the users to have access to theinformation about input, output, and some of their derivatives. They mightalso have access to the reference trajectory from the automation. This useris assumed to make delayed estimations of desired functionals of the statesof the system, yet the directly measured information and its derivatives are202.4. List of the assumptionsFigure 2.3: Block diagram presentation of the system and the user.not delayed. While the input and output signals are often noisy, in thismodel we have not considered any noise. In addition, in real world, theremight exist parametric uncertainties in the dynamics of the system whichaffect A, B, F , and C.2.4 List of the assumptionsFor the purpose of this thesis and based on the above discussion in thecurrent chapter, we consider the following general assumptions:Assumption 1. The user can perceive all the displayed information.212.4. List of the assumptionsAssumption 2. The user is provided with no additional information beyondwhat is on the display.Assumption 3. Depending on the application, the users might have knowl-edge about the derivatives of their own inputs, up to λ derivatives and alsohave knowledge about the derivatives of the outputs, up to γ derivatives.Regarding the task we assume :Assumption 4. To assure feasibility of the task, let T ⊆ C, the controllablesubspace of (2.1).To evaluate the correctness of the displayed information in Chapters 3and 4, we consider:Assumption 5. In order to accomplish or monitor a desired task, all rowsof Tx should be mathematically observable and predictable by the user.and for Chapters 5 and 6, we assume:Assumption 6. For the purpose of reconstructing the desired functional,the user does not estimate all observable states unless reconstructing all ob-servable states is feasible and necessary for the estimation of the desiredfunctional.In addition, for Chapters 4-6, we also consider:Assumption 7. Comprehension of the directly measured combination ofthe states of the system does not incorporate delay. However, prediction ofall combinations of the states and also comprehension of the non-measuredfunctionals of the states are associated with a delay, τ1.Assumption 8. Matrix A has no eigenvalues on the imaginary axis.22Chapter 3Novel observability/predictability subspaces toanalyze user-interfacesIn this chapter, we identify basic observability-based conditions under whicha user-interface provides the user with adequate information to accomplisha given task, formulated as a subset of the state-space.We, thus, formulate the user-observable subspace in Section 3.2, andthe user-predictable subspace in Section 3.3. We evaluate the mentionedsubspaces for the case that the users do not know the reference trajectory,ra, and the case that they know the pattern of changes of the referencetrajectory, i.e., the reference trajectory and its derivatives are known.3.1 Problem formulationAs has been discussed in Section 2.2.2, we model our system as in equation(2.1) such that the evolution of the states is a function of the human lowlevel input and the automation reference trajectory. In this chapter wenarrow down our investigations to the cases for which no derivative of thehuman input is known to the user and the pattern of changes of the outputis entirely known by the user, i.e. λ = 0 and γ = ∞; this assumption willbe relaxed in the next chapters.In the formalism of Assumption 5, we say that the task should be user-observable and user-predictable, i.e., the user should be able to reconstructthe desired combination of the state of the system at the current and the233.1. Problem formulationnext instant in time. Mathematically, in this chapter, we define a state tobe user-predictable if and only if the user can reconstruct both the currentvalue of the state and its higher derivatives, up to a pre-specified degree.For the purpose of formulating the user-observable/predictable spaces,in chapters 3 and 4 we extensively use projection matrices. Hence, it isworthwhile to first introduce the concept of orthogonal projection onto aspace through an example.By definition, the projection matrix, P , onto the subspace X △= span(X)is a symmetric matrix which can be computed asP = X(XTX)−1XT . (3.1)Lets consider the space V with the basis vector v such thatv =1 00 00 1.From (3.1), the orthogonal projection onto V can be obtained asP =1 0 00 0 00 0 1(3.2)and the orthogonal projection of any vector u on the space V is Pu.We can also formulate the projection onto the null space of vector v byconsidering X =[0 1 0]. In this case the projection matrix will beP =0 0 00 1 00 0 0. (3.3)Now, for a system of form (2.1), if we project the state vector ontothe left null space of matrix Bh – that is, N (BhT ), the result will showthe combinations of the states which are not affected by the input u(t).243.2. User-observable subspaceSimilarly, projection onto N (F T ) states that the projected vector is free ofthe effect of ra.3.2 User-observable subspaceConsider system (2.1) and- the output equation in 2.2. We first assume thatthe automation reference trajectory is not shown to the user, i.e. in 2.2 wehave D = 0 .The first derivative of the output equation isy˙ = CAx+ CBuh + CFra, (3.4)with Buh is known to the user.Consider an orthogonal projection matrix Pr,1 ∈ Rp×p onto N ((CF )T ) .Hence, this matrixPr,1y˙ = Pr,1CAx+ Pr,1CBuh (3.5)removes the unknown reference trajectory from the first derivative of theoutput equation. For the special case in which m ≥ p and rank(CF ) = p,Pr,1 = 0.Now consider an orthogonal projection Ph,1 ∈ Rp×p onto N ((Pr,1CB)T )and an orthogonal projection Pr,2 ∈ Rp×p onto N ((Ph,1Pr,1CAB)T ). Thematrices Ph,1 and Pr,2 remove the unknown Bu˙h and the unknown Fra fromthe second derivative of the output equation respectively, therefore,M2y¨ = M2CA2x +M2CABuh. (3.6)with Mi = Pr,i+11∏k=iPh,kPr,k for i = 2, · · · , n.Continuing in a similar fashion with higher derivatives,Miy(i) = MiCAix+MiCA(i−1)Buh, (3.7)and the projection matrices are selected such that they remove the unknown253.2. User-observable subspacevalues from the derivatives of output equation. Hence,1∏k=i−1(Ph,kPr,k)CAi−2B = 0MiCAi−1F = 0.(3.8)Note that the projection matrices are chosen to remove the unknownvalues from the derivatives of the output equation, hence, depending on theavailability of different types of inputs, the analytical expressions of thesematrices may change, e.g. in Corollaries 1 and 2.Combining the output equation, (3.5), (3.6), and (3.7), we obtainyPr,1y˙M2y¨...Mn−1y(n)= Opx+0Pr,1CBM1CAB...MnCAn−1Buh (3.9)withOp =CPr,1CAM2CA2...MnCAn. (3.10)Theorem 1. Under Assumption 3 for λ = 0 and γ = ∞, the user-observablesubspace of system (2.1) with an unknown reference trajectory isOH , R(CT )⊕R((Pr,1CA)T )⊕n−1∑i=2R((MiCAi)T ), (3.11)where Mi = Pr,i+11∏k=iPh,kPr,k and the projection matrices can be obtained263.2. User-observable subspacerecursively asPr,i = p(N ((1∏k=i−1(Ph,kPr,k)CAi−1F )T )),Ph,i = p(N (Pr,i1∏k=i−1(Ph,kPr,k)CAi−1B)T )),Pr,1 = p(N ((CF )T )),Ph,1 = p(N (Pr,1CB)T )),(3.12)where p(M) means the projection onto the space M.Proof. Given that the only unknown values in (3.9) is the state vector x,the combinations of the state which span the row space of of Op can bereconstructed by the user. Hence,R(OpT ) = R(CT )⊕R((Pr,1CA)T )⊕R((M2CA2)T )⊕ · · · ⊕ R((MnCAn)T ).(3.13)Equation (3.13) and equivalently the right hand side of (3.11) representa subspace that can be reconstructed by the user for λ = 0 and γ = ∞ –that is, Op , OH.Note that a user-observable system is a system for which OH = Rn.Corollary 1. Under Assumption 3, the user-observable subspace of system(2.1) with a known reference trajectory isOH , R(CT )⊕n−1∑i=1R((1∏k=iPh,kCAi)T ), (3.14)wherePh,i , p(N ((∏1k=i−1 Ph,kCAi−1B)T )),Ph,1 , p(N ((CB)T )).(3.15)273.2. User-observable subspaceProof. Similar to the proof of Theorem 1, we now consider that ra and itsderivatives are available, therefore, there is no need for projection matrices toremove the reference trajectory from the derivatives of states equations.Corollary 2. Under Assumption 3 for λ = 0 and γ = ∞, the user-observable subspace of• the human-driven system isOH , R(CT )⊕n−1∑i=1R((1∏k=iPh,kCAi)T ), (3.16)wherePh,i , p(N((∏1k=i−1 Ph,kCAi−1B)T )),Ph,1 , p(N ((CB)T )).(3.17)• the systems under reference tracking is equivalent to the observablesubspace of those systems.OH , R(CT )⊕n−1∑i=1R((1∏k=iPr,iCAi)T ), (3.18)wherePr,i , p(N((∏1k=i−1 Pr,kCAi−1F )T )),Pr,1 , p(N ((CF )T )).(3.19)Proof. Consider that for a human-driven system F = 0. In addition, weassume that the reference trajectory is unknown in the case of monitoringthe states of an automated system, but, for reference tracking it is assumedto be known. Hence, the user-observable subspace of different paradigmsof human-automation interaction can be easily obtained from (3.11) and(3.14).283.3. User-predictable subspace3.3 User-predictable subspaceTo obtain the user-predictable space, we define a transformation matrixS ,EOHTE⊥OHT , (3.20)with EOH the basis of the user-observable subspace of (2.1) and E⊥OH anyorthogonal complement of EOH . The space EOH and therefore E⊥OH canbe easily obtained from the results of Theorem 1. Using x¯ , Sx the statevector breaks into user-observable states, x¯OH , and user-unobservable states,x¯UOH . The transformed system is˙¯xOH˙¯xUOH = A¯x¯OHx¯UOH+ B¯uh + F¯ ra (3.21)with A¯ = SAS−1, B¯ = SB, F¯ = SF , C¯ = CS−1, and D¯ = DS−1.A¯ =A¯gOH A¯g12A¯g21 A¯gUOH ,F¯ =B¯gOHB¯gUOH , B¯ =B¯HOHB¯HUOH .(3.22)From (3.21), the derivatives of the user-observable states can be calcu-latedEOHT x˙ =[A¯gOH A¯g12]x¯OHx¯UOH+B¯HOHuh + EOHTFra(3.23)with x¯UOH and the unknown reference trajectory preventing the user fromreconstructing the derivative of user-observable states.Consider two projection matrices Px,1 and Pr,1 with the property to re-293.3. User-predictable subspacemove the unknown values from (3.23) – that is, Px,1 is orthogonal projectiononto N (A¯g1,2) and Pr,1 is an orthogonal projection on toN ((Px,1EOHTF )T ).Hence, fromPr,1Px,1EOHT x˙ = Pr,1Px,1A¯gOH x¯OH + Pr,1Px,1B¯HOHuh, (3.24)the first derivative of the states which spans R((Pr,1Px,1EOHT )T ) can bereconstructed.Continue in a similar fashion and take higher derivatives from the stateequation. At each stage introduce three projection matrices to remove theunknown values, ra, x¯UOH , and B¯HOH u˙h. Therefore, withNi =2∏k=i(Ph,k−1Pr,kPx,k)Pr,1Px,1,the ith derivative of state can be obtained fromNiEOHTx(i) = NiA¯giOH x¯OH +NiA¯gi−1OHB¯HOHuh. (3.25)Theorem 2. For the LTI system (2.1) with an unknown reference trajectoryand under Assumption 3 for λ = 0 and γ = ∞, the user-predictable subspaceisPH ,np−1⋂i=1R((NiEOHT )T ) (3.26)where np shows the degree of derivatives of the states which need to be avail-able for their predictability, Ni =2∏k=i(Ph,k−1Pr,kPx,k)Pr,1Px,1, andPx,i = p(N ((Ni−1A¯gi−1OHA¯g1,2)T )),Pr,i = p(N ((Px,iNi−1A¯gi−1OHEOHTF )T )),Ph,i = p(N ((Pr,iPx,iNi−1A¯gi−1OHETOHB)T ))(3.27)303.3. User-predictable subspaceandPx,1 = p(N (A¯gT1,2)),Pr,1 = p(N ((Px,1EOHTF )T )),N1 = Pr,1Px,1,(3.28)such that k is the highest derivative which should be observable by the userand p(M) means the projection onto the space M.Proof. From (3.24), the user can reconstruct the derivative of those com-binations of states which span R(EOHP Tx,1P Tr,1). Similarly, from (3.25), theuser can reconstruct the ith derivative of those combinations of states whichspan R(EOHNiT ). Therefore, in order to reconstruct higher derivatives ofstate vector (up to the kth derivative), the user predictable subspace can beobtained as equation (3.26) and Theorem 2 is proved.Note that a user-predictable system is the one that satisfies PH = Rn.Corollary 3. Under Assumptions 3 for λ = 0 and γ = ∞, the user-predictable subspace of system (2.1) with a known reference trajectory isPH ,np−1⋂i=1R((NiEOHT )T ) (3.29)where Ni =2∏k=i(Ph,k−1Px,k)Px,1 andPx,i = p(N ((Ni−1A¯gi−1OHA¯g1,2)T )),Ph,i = p(N ((Px,iNi−1A¯gi−1OHETOHB)T ))(3.30)andPx,1 = p(N (A¯gT1,2)),N1 = Px,1,(3.31)such that k the highest derivative which should be observable by the user.Proof. The proof is trivial. Consider that when the reference trajectory andits derivatives are all available, there is no need for an orthogonal projection313.4. Examplesmatrix to remove them from the derivatives of the output equation.3.4 Examples3.4.1 Nagoya A300 Accident, 1994For the Airbus accident of Section 1.1, we model the aircraft prior to stallas an LTI system under shared control, and aim to determine whether thepilots had access to the required information about the aircraft to assurepredictability of the states relevant to the desired task.Consider the linearized longitudinal dynamics of an aircraft in trimmedlevel flight, with state x = [V, α, q, θ] consisting of total velocity V , angle ofattack α, pitch rate q and pitch angle θ and state matrixA =−0.0414 10.9259 0 −32.8000−0.0013 −0.5017 1.0000 00 −0.4184 0 00 0 1.0000 0, (3.32)with the numerical values for stability derivatives taken from data for theBoeing 747 [96]. The equations of the motion of the aircraft from whichmatrix A is achieved are provided in [97].In order to achieve and keep a desired altitude, we consider that theautomation is following a glide slope which approaches zero, i.e. Cd =[1 0 − 1 0]. Hence, both the pilot and the autopilot were trying to controlthe flight path angle, one in order to descend and the other one in order toascend the aircraft. The accident investigation report [5] states that the pilotand the co-pilot expected their control input to override the autopilot controlaction, which in fact was not a valid expectation, therefore, we consider thereference trajectory to be unknown to the user. In Airbus A300, the pilotscontrol the flight path angle using the elevators,B =[0 −0.0305 −0.4039 0]T(3.33)and the autopilots control it through the angle of the horizontal tail (THS),323.4. Examplesthus, the automation affects the states throughBa =[0 −0.0636 −0.8656 0]T, (3.34)The input δe and δih represent elevator deflection and THS deflection re-spectively. Having Ba from (3.34), it is straight forward to obtain F .Velocity, pitch angle, and flight path angle are all available in standardA300-600 pilot displays [98]. Hence, for the augmented systemC =1 0 0 00 0 0 10 1 0 −1. (3.35)We define the desired task of the pilot to be descending on the desiredglideslope, γ = γdes. Since γ = α− θ, by (2.3), we obtainT = span([0 1 0 −1]T). (3.36)By applying Theorem 1, we obtain the user-observable subspace OH =span(e1, · · · , e4) which shows that by measuring the mentioned states of thesystem, the pilot would be able to reconstruct the unknown pitch rate.We assume that the first two derivatives of user-observable states shouldbe observable in order to achieve a reliable prediction of their values. From(3.26), the user predictable subspace isPH = span100−0.8251,01−0.0735−0.9644. (3.37)Since T * PH, our method demonstrates that the modeled user-interfacedoes not provide the necessary information for the user to predict the sys-tem’s behavior.If we assume that the automation’s desired trajectory is known to theuser, still the flight path angle and, as a result, the angle of attack will not333.4. Examplesbe user-predictable. Hence, using our method could be helpful in the earlystages of the design of flight management systems by showing the designersthat in some specific modes of the flight, flying under shared control, e.g.,shared control on the flight-path angle, can be hazardous even for pilots whoare aware of the autopilot’s goal.3.4.2 A remotely controlled fleet of UAVsSeveral researchers have developed methods for stabilization and controlof the formation of a group of UAVs for both continuous-time and discrete-event systems [99–101]. We consider a fleet of two point mass vehicles, flyingin a leader-follower formation and following a real-time reference trajectorydefined by a remote operator. We model each vehicle as a double integrator,Ai =[0 10 0], Bi =[01]. (3.38)In the leader-follower formation, each vehicle only has access to the infor-mation about the position of its leader and the main leader is following areference trajectory.We model the formation such that the position of each vehicle is thereference trajectory for its direct follower and the overall trajectory of theformation is the one which is defined by the user to the leader of the forma-tion. Hence,A =[Ai 0B Ai],B =[BiT 0]T.(3.39)Under Assumption 3 and based on the availability of human referencetrajectory we can evaluate the user-observable and the user-predictable sub-spaces for different measurements from Corollary 2 and 3. Table 3.1 showshow the various measurements can affect the mentioned subspaces.As opposed to what one might expect, Table 3.1 demonstrates that pro-viding information about the position of the leader does not let the user343.5. Discussion and an alternative presentation of the systemMeasurement C O, OH, and PHLeader position e1T span(e1, e2)Follower position e3T R4Table 3.1: The observable and predictable subspaces for different measure-ments of a leader-follower formationcomprehend and predict the position of the follower. Essentially, by mea-suring the states of the leader the user may only be capable of accomplishingtasks related to the states of the leader. The reason lies behind the fact thatin the leader-follower formation the states of the leader are not affected byits follower – that is, changes in the states of the leader are only made bythe operator’s command.The states of the follower are on the other hand affected by the states ofits leader. From Table 3.1, in order to accomplish any possible set of tasksin a leader-follower formation, i.e., in order to satisfy the inclusion T ⊆ PHfor all possible sets of tasks, the position of the follower has to be measuredin the user interface.3.5 Discussion and an alternative presentation ofthe systemIn this chapter we modeled the evolution of the system as a function ofthe low-level human input and the automation’s reference trajectory. Analternative presentation of the system with the relevant analysis is providedin our conference paper [102] in which we consider the states of the systemto be a function of low-level inputs from both the user and the automation.In the alternative presentation of the system, we consider the system to be353.6. Summarya non delayed continuous time LTI system whose evolution is modeled asx˙ = Ax+[Bh Ba Bboth]uhuaubothy = Cx(3.40)with A ∈ Rn×n, Bh ∈ Rn×mh , Ba ∈ Rn×ma , Bboth ∈ Rn×mb , C ∈ Rp×n,state vector x(t) ∈ Rn, output y(t) ∈ Rp, human input uh(t) ∈ Rmh , au-tomation input ua(t) ∈ Rma , and merged input uboth(t) ∈ Rmb . We assumeuh, ua and uboth to be exhaustive and mutually exclusive inputs to thesystem, which model the effect of different actuators. In contrast to thestandard LTI model [103], the input is categorized in one of three ways: asan input uh controlled solely by the user, an input ua controlled solely bythe automation, or as an input uboth which controlled by both the user andautomation. Hence, Bh models how the human input affects the system,Ba models how the automation input affects the system, and Bboth modelshow the merged control input, in which commands from the human and theautomation are combined in some fashion (not specified here), affects thesystem. Since the automation input and intent are unknown to the user, weconsidered ua, uboth, and their derivatives, to be unknown.Detailed discussion on how we obtained the user-observable and the user-predictable subspaces of such system is provided in [102].3.6 SummaryThis chapter presented necessary conditions for evaluating the informationcontent of a user-interface for an LTI system under shared control. Two sub-spaces, the user-observable subspace OH and the user-predictable subspacePH, were formulated. The user-predictable subspace is compared to the tasksubspace. If the task subspace does not lie in the user-predictable subspace,then the user-interface is not correct, meaning that it is not possible for thehuman to accomplish the desired task with the information provided on thegiven user interface.363.6. SummaryThe results of this chapter are acceptable for the systems in which asmall amount of delay in comprehension and prediction of the task vectordoes not affect the safety of the system. However, for safety-critical systemswhich have to follow a precise trajectory of the functionals of the states,these results are not precise enough. In addition, here we have assumedthat a subsequent value of a functional of the states is known if the currentfunctional and its derivative are known. By relaxing these assumptions, wewill be able to modify our tool so that it can be used for the analyses ofthe displayed information in safety-critical systems. In the next chapter, wemodify the above observability and predictability subspaces by consideringthe perception delay as well as by looking at longer term predictions ratherthan the instantaneous prediction.37Chapter 4Novel observability/predictability subspacesconsidering the delay andlong term predictionIn this chapter, for a system of the form (2.1), we modify the results ofChapter 3 by taking into account the processing delay of the estimation andthe prediction which we simply ignored in the previous chapter. In addition,in Chapter 3 we assumed that a subsequent value of a functional of the statesis known if the current functional and its derivatives are known. Here, werelax this assumption and consider the actual evolution of the states ofthe system which depends on the current functional, the inputs, and inputderivatives. By relaxing these assumptions, we will be able to modify ourtool so that it can be used for the analyses of the displayed information insafety-critical systems.We determine formulas for the delay-incorporating user-observable sub-space, and the delay-incorporating user-predictable subspace of shared con-trol systems in Section 4.2 and 4.3 respectively. An example on a remotelydriven car is provided in Section Problem formulationUnder Assumption 5, the user-interface of a safety-critical system undershared control must provide the user with information that results in a delay-384.2. Delay-incorporating user-observable subspaceincorporating user-observable and a delay-incorporating user-predictable sub-spaces which we define them as below.Definition 2. The delay-incorporating user-observable subspace, O∗H, is aspace which is spanned by the combination of current states, x(t), which areknown to the user at current time, t.Definition 3. The delay-incorporating user-predictable subspace, P∗H, is aspace which is spanned by the combination of upcoming states, x(t + τ),which are known to the user at current time, t.To make this more clear, consider a user who aims to reconstruct a com-bination of the states x(t) given uh(t) and y(t). Although this reconstructionmight be feasible for the user, due to the processing delay, the desired com-bination of x(t) will become available to the user at time t+ τ1. For a safetycritical system, this late understanding about the states of the system canbe unacceptable.Under Definitions 2 and 3, for system (2.1), we formulate the delay-incorporating user-observable subspace, and the delay-incorporating user-predictable subspace.From Assumption (7), we can write,O∗H = OH,y ⊕OH,τ1 (4.1)where O∗H is the delay-incorporating user-observable space, OH,y△= R(CT )is obtained from the directly measured combinations of the states, y(t) =Cx(t), and OH,τ1 is the delayed observable space which is the space spannedby the non-measured functional of states which can be reconstructed givenτ1 ≥ 0 delay.4.2 Delay-incorporating user-observable subspaceIn this section, we determine the combination of the states at time t, i.e.x(t), which can be reconstructed by time t. Mathematically, this means394.2. Delay-incorporating user-observable subspaceto obtain what combination of x(t), can be reconstructed given y(t − τ1),uh(t− τ1), and some of their derivatives.Consider the output equation and its ith derivative for i ∈ {1, · · · , γ} attime t− τ1,y(t− τ1) = Cx(t− τ1),y(i)(t− τ1) = CA(i)x(t− τ1) + CAi−1Buh(t− τ1)+· · · + CBuh(i−1)(t− τ1) + CAi−1Fra.(4.2)putting all the derivatives of the output equation together in a matrixform we obtainY0:γ(t− τ1) = Ox(t− τ1) +HxU0:γ(t− τ1) +Hrra, (4.3)withY0:γ(t− τ1) △=y(t− τ1)y˙(t− τ1)...y(γ)(t− τ1),U0:γ(t− τ1) △=uh(t− τ1)u˙h(t− τ1)...uh(γ)(t− τ1).(4.4)In (4.3), O is the observability matrix and Hx is the Toeplitz matrixobtained from (4.2), in addition,Hr△=0CAF...CAγ−1F. (4.5)In equation (4.3), the delayed-states of the system – that is, x(t − τ1),are formulated as a function of the input, output, their derivatives up to the404.2. Delay-incorporating user-observable subspaceγth derivative, and the reference trajectory at time t− τ1.From Assumption 3, only λ derivatives of the input is known to theuser. Hence, it is not possible to directly obtain x(t − τ1) from (4.3). We,therefore, define P0 = I andPi =min(i−1,λ+1)∏j=0Pi,jPi,r. (4.6)If we select Pi,r and Pi,j as follows, pre-multiplication of the ith derivativeof the output equation by ∏ik=0 Pi will remove the unknown values from theith derivative of output.• The matrix Pi,r is the projection matrix onto the left null-space ofDc(∏i−1k=0 PkCAi−1F ), where Dc ∈ {0, I} is the complement of D –that is, Dc +D = I all the time.• For j ≤ λ, Pi,j = I and for j = λ+1 where i > j, Pi,j is the projectiononto the left null-space of (Pi,r∏i−1k=0 PkCAi−j−1B).Now, consider a matrixM △=P0 0 0 00 P1P0 0 00 0 . . . 00 0 0 ∏γi=0 Pi, (4.7)and pre-multiply it in (4.3) to eliminates the unknown values of the inputderivatives and the reference trajectory. Hence,MY (t− τ1) = MOx(t− τ1) +MHxU0:λ(t− τ1) +MHrra (4.8)in which the only unknown value is x(t− τ1).414.2. Delay-incorporating user-observable subspaceTheorem 3. In a system of form (2.1) and under the Assumption 3, thedelay-incorporating user-observable subspace is of the formO∗H△= R(CT )⊕R((e−Aτ1)T (MO)TP Tτ1), (4.9)where M is from (4.7) and Pτ1 is from (4.15).Proof. Since in general the delay is small, we consider τ21 to be negligible,hence,uh(t) = uh(t− τ1) + τ1u˙h(t− τ1). (4.10)Note that to make the results more precise, it is straight forward tomodel the current state as a larger series of previous states and modify therest of the results as per need.Under (4.10), the states of a continuous time system (2.1) evolve asx(t) = eAτ1x(t− τ1) + c, (4.11)wherec =∫ tt−τ1eA(t−T )[B F ][uh(T )ra]dT. (4.12)By introducing the variablesθ0 , (eAτ1 − I)A−1B,θ1 , ((eAτ1 − I)A−1 − τ1I)A−1B,θ2 , (eAτ1 − I)A−1F.(4.13)From (4.10) - (4.13), we can writex(t− τ1) = e−Aτ1x(t)− e−Aτ11∏k=0θkuh(k)(t− τ1)− e−Aτ1θ2ra. (4.14)We can combine (4.8) and (4.14) to formulate x(t) as a function of theinput, output and their derivatives at time t− τ1. As, new unknown uh(t−424.2. Delay-incorporating user-observable subspaceτ1), u˙h(t− τ1), and ra may arise, we introducePτ1△=1∏k=0Pτ1,kPτ1,r (4.15)where Pτ1,0, Pτ1,1, and Pτ1,r are defined as follows.• Thematrix Pτ1,r is a projection onto the left null-space ofDc(MOe−Aτ1θ1).• For j ≤ λ, Pτ1,j is an identity matrix and it is a projection onto theleft null-space of (∏j−1k=0 Pτ1,kPτ1,rMOe−Aτ1θk) otherwise.From (4.14) and (4.15) we can rewrite (4.8) asPτ1MY0:γ(t− τ1) = Pτ1MOe−Aτ1x(t)− Cknown (4.16)whereCknown = Pτ1e−Aτ1θ0uh(t− τ1) + Pτ1e−Aτ1θ1u˙h(t− τ1)+Pτ1e−Aτ1θ2ra − Pτ1MHxU0:λ(t)− Pτ1MHrra.(4.17)Hence, the combination of x(t) which spans R((Pτ1MOe−Aτ1)T ) can bereconstructed from Y0:γ(t − τ1) and U0:λ(t − τ1). Therefore, from (4.1),Theorem (3) is proved.Procedure. The following steps are required to calculate the delay-incorporatinguser-observable subspace, O∗H:• Determine matrices O, Hx, and Hr and calculate the value of θ0-θ2from (4.13).• Obtain Pi from (4.6) and Pτ1 from (4.15).• Determine M from (4.7).• Determine the delay-incorporating user-observable subspace from (4.9).434.3. Delay-incorporating user-predictable subspaceCorollary 4. A delay-incorporating user-observable space is also user-observable.Proof. By definition, the user-observable space is the space which is spannedby the combination of the current states which are known to the user at thecurrent time, for τ1 = 0.From (4.8), the user-observable subspace can be formulated asOH △= R((MO)T ). (4.18)We also have the equation of O∗H from (4.9).Under Assumption (8), the matrix A is of full rank, henceR((e−Aτ1)T (MO)TP Tτ1) = R((MO)TP Tτ1).In addition, for random matrices N and Q of compatible dimensions, wehave R(NQ) ⊆ R(N). Hence,O∗H△= R((e−Aτ1)T (MO)TP Tτ1)= R((MO)TP Tτ1)⊆ R((MO)T ),(4.19)which proves that O∗H ⊆ OH.4.3 Delay-incorporating user-predictablesubspaceFrom Definition (3), the delay-incorporating user-predictable space is thespace which can be spanned at time t based on the information available onx(t+ τ). As in Section 4.2, we can write the upcoming states as a functionof Y0:γ(t− τ1) and U0:λ(t− τ1).Theorem 4. In a system of form (2.1) and under the Assumption 3, thedelay-incorporating user-predictable subspace is of the formP∗H△= R((e−A(τ+τ1))T (MO)TP Tτ1P Tτ ), (4.20)444.3. Delay-incorporating user-predictable subspacewhere Pτ is from (4.24), Pτ1 is from (4.15), and M is from (4.7).Proof. From (4.16) andx(t) = e−Aτx(t+ τ)− e−Aτ1∑i=0δiuh(t)(i) − e−Aτδ2ra (4.21)where τ is the required prediction horizon andδ0△= (eAτ − I)A−1B,δ1△= ((eAτ − I)A−1 − τI)A−1B,δ2△= (eAτ − I)A−1F,(4.22)we can writePτ1MY0:γ(t− τ1) = Pτ1MOe−A(τ1+τ)x(t + τ)− Pτ1MOe−Aτ δ1uh(t− τ1)−Pτ1MOe−Aτ (δ2 + τ1δ1)u˙h(t− τ1)− Pτ1MOe−Aτδ3ra − Cknown(4.23)where Cknown is from (4.17).By pre-multiplying (4.23) byPτ△=1∏k=0Pτ,jPτ,r (4.24)we can remove all unknown values from it. In (4.24),• Thematrix Pτ,r is a projection onto the left null-space of Dc(Pτ1MOe−A(τ1+τ)δ2).• For j ≤ λ, Pτ,j is an identity matrix and it is a projection onto theleft null-space of (∏j−1k=0 Pτ,jPτ,rPτ1MOe−A(τ1+τ)δk) otherwise.Hence, the functional of the upcoming states of the system, x(t + τ),which span the row space of PτPτ1MOe−A(τ1+τ) can be reconstructed bythe user by time t.Procedure. The steps that are required to calculate the delay-incorporatinguser-predictable subspace, P∗H are as follows:454.3. Delay-incorporating user-predictable subspace• Obtain all the required matrices to determine O∗H.• Calculate δ0 − δ2 from (4.22).• Obtain Pτ from (4.24).• Determine the delay-incorporating user-predictable subspace from (4.20).Corollary 5. A delay-incorporating user-predictable space is also delay-incorporating user-observable.Proof. As in the proof in Corollary 4, if is straight forward to show thatwith A being a full rank matrixP∗H△= R((e−A(τ+τ1))T (MO)TP Tτ1P Tτ )= R((e−Aτ1)T (MO)TP Tτ1P Tτ )⊆ R((e−Aτ1)T (MO)TP Tτ1),which proves that P∗H ⊆ O∗H.Corollary 6. A delay-incorporating user-predictable space is also user-predictable.Proof. Consider τ1 = 0, hence Pτ1 = I and from (4.20), PH can be formu-lated asPH△= R((e−A(τ))T (MO)TP Tτ ). (4.25)As in Corollary 4 and 5, it is trivial to show that P∗H ⊆ PH4.3.1 Validation of the displayed informationIn Section 4.1 we stated that for safety-critical systems under human orshared control, the user-interface must provide the user with information464.4. Examplesthat results in a delay-incorporating user-observable and a delay-incorporatinguser-predictable task. Hence, we can introduce the following proposition.Proposition 1. In order for a user to be able to accomplish a desired taskin a safety-critical condition for a system of form (2.1) and under the as-sumptions 7-4, the following inclusion is necessaryT ⊆ P∗H, (4.26)where P∗H is the delay-incorporating user-predictable subspace, formulated in(4.20).4.4 ExamplesWe consider a remotely driven point mass car modeled as a double integratorand stabilized to have poles on −2 and −3. The system matrices areA =[0 1−2 −3], B =[01]. (4.27)Our goal is to evaluate whether for such a system the displays (whichmeasure the position or the velocity of the car) are effective to accomplisha desired task, when the processing delay is τ1 = 0.2.We consider two cases, 1) a user who controls the system via a knownforce with known constant rate – that is, all derivatives of the input areknown and 2) a user whose input to the system is complicated and random,thus, has no knowledge about input derivatives – that is , λ = 0. For bothcases, we consider γ = 1.Our desired task is stopping at a stop sign, hence, we can define the taskspace as T = R([10],[01])which spans R2.4.4.1 delay-incorporating user-observable subspaceIt is first required to calculate the delay-incorporating user-observable sub-space for different measurements of states available to the user.474.4. ExamplesFrom (4.9), for λ = ∞, we can obtain M = I and Pτ1 = I. Hence,O∗H|λ=∞ = R(CT )⊕R((e−Aτ1)TOT ). (4.28)Also, for λ = 0, we can obtain M = I therefore,O∗H|λ=0 = R(CT )⊕R((e−Aτ1)TOTP Tτ1). (4.29)From (4.28) and (4.29), for either of the two different displays includinga GPS with C = [1 0] and a speedometer with C = [0 1], we can show thatO∗H|λ=∞ spans R2.In addition, when λ = 0, with either of the measurements in the displaywe can obtainPτ1 =[0.9919 0.08990.0899 0.0081],hence, O∗H|λ=0 = R2.The results state that for such a system, regardless of the type of themeasurements and the complexity of users’ input, the user can reconstructboth states of the system.4.4.2 Delay-incorporating user-predictable subspaceFor τ = 0.1, we calculate the delayed-incorporates user-predictable subspaceof this system for different measurements of states provided in the display.From (4.20), for λ = ∞, we can obtain Pτ = I, henceP∗H|λ=∞ = R((e−A(τ+τ1))TOT ) (4.30)Also, for λ = 0 and from (4.20)P∗H|λ=0 = R((e−A(τ+τ1))TOTP Tτ1P Tτ ) (4.31)From (4.30), we can obtain P∗H|λ=∞ = R2 with either the GPS or thespeedometer.484.4. ExamplesWhen λ = 0, for either of the displays,Pτ =[0.0081 −0.0899−0.0899 0.9919],hence, P∗H|λ=0 = ∅. This states that, with a complicated input, the user canreconstruct no combination of the states x(t+ τ) at time t.The results above can help the reader understand Corollary 5 better,as it is clear that in all of the above cases, the delay-incorporating user-predictable space is a subset of the delayed incorporated user-observablespace.We now consider λ = 0 for the case of having no processing delay, τ1 = 0.We thus can obtain the user-predictable subspace to bePH = R([1.0000−0.0526])for either of the displays. Based on Corollary 6, the delay-incorporatinguser-predictable space is always a subset of the user-predictable space whichwe also have shown it to be the case in this example.The result of having no delay shows that not considering the processingdelay can result in a larger user-predictable subspace. Overlooking this delaycan mislead the designer to misjudge the capability of the user to accomplisha task; i.e., by ignoring the delay, the designer might find the user capable ofaccomplishing the task, while, due to the existence of the processing delay,the space which is predictable by the user may not include the task spaceor even is empty. Thus, for safety critical systems, it is not safe to simplyignore this value as it may result in hazardous outcomes.4.4.3 Task accomplishmentWith any of the suggested displays, when λ = ∞, both the delay-incorporatinguser-observable and the delay-incorporating user-predictable spaces span R2.Hence, T ⊂ PH ∗|λ=∞. This means, regardless of the displayed information,if the pattern of changes of the user’s input is all clear to the user, it might494.5. Summary and conclusionbe possible for the user to stop at a stop sign.On the other hand, it is clear that T * PH ∗ |λ=0 for either of thedisplays. Hence, under the Assumption 5, for a complicated input to thesystem, neither the GPS nor the speedometer are effective for a human tocontrol the velocity of the mass. Hence, for such an input, regardless of thetype of the measurements, there is always a chance that the user cannotestimate and predict the task precisely.For this example, although not having a processing delay is helpful inexpanding the user predictable space, it still does not help with task accom-plishment.4.5 Summary and conclusionIn this chapter, for safety-critical LTI systems, two novel subspaces, thedelay-incorporating user-observable subspace, O∗H, and the delay-incorporatinguser-predictable subspace P∗H, with possibly longer term predictions, wereformulated. As in Chapter 3, these subspaces were compared to the taskspace for a feasible task. If the task space does not lie in the relevant space,then the user-interface of a safety-critical system is incorrect, meaning thatin such a system there exists a possibility that the user cannot accomplishthe desired task with the given user-interface.In the next two chapters, we suggest models for the process of attainingSA by the user. These models let us evaluate the correctness of a givendisplay. For cases with unchanging operating conditions, with the aid of thementioned models, the information which is required to be included in thedisplay for the safety of the task can also be determined.50Chapter 5User-interface analysisthrough modeling the user asan estimatorHaving a system as in (2.1), our goal in this chapter is to introduce a detailedmodel of a human attaining SA.In Sections 5.2.1 and 5.2.2, we present the existence conditions and thedesign procedure of an extended delayed functional observer/predictor con-sidered to be the model of attaining SA by the user. We follow by anexample on the existence and design of the delayed/non-delayed functionalobserver/predictor. In section 5.2.4, we suggest a technique to determine therequired information to be displayed. Finally, in Section 5.3, we investigatea safety critical application, prediction of the depth of anesthesia duringsurgery.5.1 Problem formulationTo model the process of attaining SA by the user, we take into account theusers’ limitations and capabilities regarding the information presented tothem and estimated by them. The process of attaining SA includes observa-tions as well as predictions by the user. Processing of information generallyintroduces a delay [104–106]. Assuming the derivatives of the inputs andthe outputs might be available, we design (and evaluate the existence of) anovel estimator for LTI systems generating delayed estimates of the currentand upcoming desired functional of states. Since we consider the user to i)515.1. Problem formulationonly reconstruct and predict the desired set of states rather than the entirestate space, ii) make delayed estimations, and iii) possibly have knowledge ofthe derivatives of the inputs and outputs, we model the process of attainingSA as an extended delayed functional observation/prediction.Since, in this chapter, reconstruction and prediction of the desired statesof the system are considered to be delayed, we model the user as a delayedobserver/predictor for the functional:z0(t+ τ) = Tx(t+ τ), 0 ≤ τ, (5.1)where τ defines the prediction horizon. In (5.1), the task matrix T ∈ Rl×nis defined in (2.3).In some cases, it is not possible to estimate the functional z0(t + τ)directly and it is necessary for the user to also estimate the functional Rx(t+τ) such that R ∈ RX×n. We select the rows of R to be linearly independentfrom the rows of T . Hence, we introduce the extended functional asz(t+ τ) =[TR]x(t+ τ), (5.2)where R is selected such that L = [T T , RT ]T is of full row rank. For casesthat Tx(t+ τ) can be estimated directly, we have R = ∅.We introduce a theorem providing conditions needed for an estimatorwith human specifications to exist, i.e., the required conditions on the sys-tem, the display, and the task so that the user can attain SA toward specificgoals. If these conditions are not satisfied for a triplet of dynamics, measure-ments, and task, then it is not possible for the user to attain SA regardingthe specific task through the available information. Possible solutions tosuch a problem could be i) modifying the content of the display, and/or ii)expanding the task. In some systems, however, none of the above modifica-tions help in attaining SA. This non-existence of the observer/predictor canitself be informative to the system designer, e.g., for making better decisionson how to provide the required information to the user.The analyses in this chapter are under Assumption 3 for γ ∈ {0, 1} and525.2. Methodologyλ ∈ {0, 1}. It is worth mentioning that employing a technique similar to theone in this chapter, it is straightforward to design a non-delayed/delayedfunctional estimator for 1 ≤ γ and 1 ≤ λ.5.2 MethodologyBased on Assumption 3, we introduce the extended output vector and theextended input vector asY0:γ(t) = [y1T (t), y˙T1 (t), · · · , y1(γ)T (t)]T ,U0:λ(t) = [uhT (t), u˙Th (t), · · · , uh(λ)T (t)]T ,(5.3)hence, as we consider γ ∈ {0, 1} and λ ∈ {0, 1}, only two cases for theextended output vector and two cases for the extended input vector mayexist. Analytically, the extended output vector can be written asY0:γ(t) = Oγx(t) +M1,0:γU0:γ(t) +M2,γra, (5.4)where for γ ∈ {0, 1}, the observability matrix Oγ ∈ R(γ+1)px×n , a Toeplitzmatrix M1,0:γ ∈ R(γ+1)px×(γ+1)px , and matrices M2,γ ∈ R(γ+1)px×pr andU0:γ(t) ∈ R(γ+1)px are defined as follows,O0 = C , O1 = [CT , ATCT ]TM1,0:0 = 0 , M1,0:1 =[0 0CB 0]M2,0 = 0 , M2,1 =[0CF]U0:0(t) = uh , U0:0(t) = [uhT (t), u˙Th (t)]T .(5.5)In addition to the inputs, outputs, and their derivatives, we also give theuser the ability to incorporate the measured trajectories in estimating thedesired states. We therefore aim to model the user as an estimator of theformω˙(t) = Nω(t) + J1Y0:γ(t) + J2y2(t) +HU0:λ(t)zˆ(t) = ω(t− τ1) + EY0:γ(t)(5.6)535.2. Methodologywhich produces delayed or non-delayed estimates of current or upcomingvalues of a desired functional of states. In (5.6), ω(t) ∈ Rl+X is the state ofthe estimator and τ1 is the estimation delay. It is desirable to determine astable matrix N and matrices J1, J2, H, and E with compatible dimensionsto make the estimation error asymptotically approach zero. From (5.6), itis clear that we only apply the delay term on the desired states which needto be processed and estimated in the working memory – that is, the set ofdesired states which are not directly available to the user.Having the estimator (5.6) to estimate the functional (5.2) of the system(2.1), the prediction error ise(t) = zˆ(t)− z(t + τ)= ω(t− τ1) + EY0:γ(t)− Lx(t+ τ)= ω(t− τ1) + EOγx(t) + EM1,0:γU0:γ(t)+EM2,γra − Lx(t+ τ)(5.7)with the error dynamicse˙(t) = Nω(t− τ1) + J1Oγx(t− τ1) + J1M1,0:γU0:γ(t− τ1) + J1M2,γra+J2Dra +HU0:λ(t− τ1) + EOγAx(t) + EOγBuh(t) + EOγFra+EM1,0:γU1:γ+1(t)− LAx(t + τ)− LBuh(t + τ)− LFra.(5.8)Note that in (5.8), by setting τ = 0 we obtain the error dynamics for theobservation and by setting τ > 0 we obtain the error dynamics for theprediction.Since in general the delay and the value of prediction horizon are small,we can writeuh(t) = uh(t− τ1) + τ1u˙h(t− τ1),uh(t+ τ) = uh(t− τ1) + (τ + τ1)u˙h(t− τ1) + ττ1u¨(t− τ1).(5.9)The states of the continuous time system (2.1) evolve asx(t+ τ) = eAτx(t) + c,x(t) = eAτ1x(t− τ1) + c1,(5.10)545.2. Methodologywherec =∫ t+τt eA(t+τ−T )[B F ][uh(T )ra]dT,c1 =∫ tt−τ1 eA(t−T )[B F ][uh(T )ra]dT.(5.11)By introducing the variablesδ1 , (eAτ − I)A−1B,δ2 , ((eAτ − I)A−1 − τI)A−1B,δ3 , (eAτ − I)A−1F,θ1 , (eAτ1 − I)A−1B,θ2 , ((eAτ1 − I)A−1 − τ1I)A−1B,θ3 , (eAτ1 − I)A−1F,η1 , eA(τ+τ1),η2 , δ3 + eAτθ3,η3 , δ1 + eAτθ1,η4 , δ2 + τ1δ1 + eAτθ2,η5 , τ1δ2,(5.12)and from (5.9) and (5.10), we can writex(t) = eAτ1x(t− τ1) + θ1u(t− τ1) + θ2u˙(t− τ1) + θ3ra,x(t+ τ) = η1x(t− τ1) + η2u(t− τ1) + η3u˙(t− τ1) + η4u¨(t− τ1) + η5ra.(5.13)Hence, under the assumption that γ and λ are selected from {0, 1}, the555.2. Methodologyerror dynamics can be written ase˙(t) = Ne(t) + (NLη1 − LAη1 + [E J1 K J2 H]Q1,1)x(t− τ1)+(NLη2 − LAη2 + [E J1 K J2 H]Q1,2 − LF )ra+(NLη3 − LAη3 + [E J1 K J2 H]Q1,3 − LB)uh(t− τ1)+(NLη4 − LAη4 + [E J1 K J2 H]Q1,4 − LB(τ + τ1))u˙h(t− τ1)+(NLη5 − LAη5 + [E J1 K J2 H]Q1,5 − LBττ1)u¨h(t− τ1),(5.14)where K , J1 −NE and Q1,i are defined in (5.15) for i ∈ {1, · · · , 5}.Q1 ,[Q1,1 Q1,2 Q1,3 Q1,4 Q1,5],OγAeAτ1 Oγ(Aθ3 + F ) Oγ(Aθ1 + B) Oγ(Aθ2 + τ1B) +M1,γ τ1M1,γOγ(I − eAτ1) −Oγθ3 −Oγθ1 −Oγθ2 − τ1M1,γ 0OγeAτ1 Oγθ3 +M2,γ Oγθ1 +M1,γ Oγθ2 + τ1M1,γ 00 D 0 0 00 0 I 0 00 0 0 λI 0(5.15)In (5.15), M1,0 = 0px×mb and M1,1 =[0CB].From (5.14), a τ1-delayed estimator exists and can be designed to es-timate the desired functional z(t + τ) if and only if there exists a set(E, J1, N, J2,H) , where H , [Ha Hb], with a stable N to always satisfyNLη1 + [E J1 K J2 H]Q1,1 = Q2,1,NLη2 + [E J1 K J2 H]Q1,2 = Q2,2,NLη3 + [E J1 K J2 H]Q1,3 = Q2,3,NLη4 + [E J1 K J2 H]Q1,4 = Q2,4,NLη5 + [E J1 K J2 H]Q1,5 = Q2,5,(5.16)565.2. Methodologywith Q2,i are defined in (5.17) for i ∈ {1, · · · , 5}.Q2 ,[Q2,1 Q2,2 Q2,3 Q2,4 Q2,5],[LAη1 L(Aη2 + F ) L(Aη3 +B) L(Aη4 +B(τ + τ1)) L(Aη5 +Bττ1)](5.17)In summary, using the above conditions, we can formulate the problemas follows. We seek to:• evaluate the satisfaction of (5.16) for a desired task T , a given delayτ1, and a given amount of prediction horizon τ to determine whetherit is possible for the user to attain SA regarding the desired task andthus make correct decisions toward its accomplishment.• obtain the model of the user by solving (5.16) forN and[E J1 K J2 H](which also satisfy K = J1 −NE).• seek the triplet (C,D,R) (if there exists any), with a minimum cardi-nality of (C,D), which satisfies conditions in (5.16) to determine therequired information to be displayed. Note that, we define the cardi-nality of (C,D) as rank(C) + rank(D).5.2.1 Existence conditions for an extended functionalestimatorFor LTI systems under shared-control and assuming availability of the deriva-tives of the inputs and outputs, we obtain the necessary and sufficient con-ditions for the existence of a delayed/non-delayed functional observer andpredictor.Recall that we consider a full row rank functional Lx(t+ τ), with Lx(t+τ) =[TR]x(t+τ), whose components are Tx(t+τ) and Rx(t+τ). There-fore reconstructing Lx(t+τ) is sufficient for the reconstruction of the desiredtask, Tx(t + τ). Our goal is to investigate the existence of and then design575.2. Methodologyan observer of form (5.6) to reconstruct the functional Lx(t + τ). Mathe-matically this is equivalent to finding a solution for (5.16).Lemma 1. There exists a solution for (5.16) iff the following two conditionsare simultaneously satisfied:•[E J1 K J2 H]T1 = T2, (5.18)where T1 = Q1ME and T2 = Q2ME and ME is from (5.20).•N = Q2,iHi −[E J1 K J2 H]Q1,iHi,for i ∈ {1, · · · , 5}.(5.19)where Hi is such that LηiHi = I, for i ∈ {1, · · · , 5}.Proof. By selecting Eis to satisfy LηiEi = 0 and His defined earlier, we candefine a full row-rank matrixS1 =[MH ME], (5.20)whereME =E1 0 0 0 00 E2 0 0 00 0 E3 0 00 0 0 E4 00 0 0 0 E5, MH =H1 0 0 0 00 H2 0 0 00 0 H3 0 00 0 0 H4 00 0 0 0 H5.(5.21)Given that S1 is of full row rank, post multiplication of S1 in (5.16) willnot change the results. As a result of this post multiplication, (5.18) and(5.19) are obtained to be an equivalent expression to (5.16).In order for a stable solution for (5.16) to exist, a stable matrix N andmatrices [E J1 K J2 H] have to exist to satisfy both (5.18) and (5.19).585.2. MethodologyClearly, there exists a solution for (5.18) iff span(T2T ) ⊆ R(T1T ) – thatis,rank[T1T2]= rank[T1]. (5.22)Proposition 2. The condition (5.18) is satisfied iffrank(LHS1) = rank(RHS) (5.23)whereRHS ,[Q1Lη1 Lη2 Lη3 Lη4 Lη5], (5.24)andLHS1 ,[Q2RHS]. (5.25)Proof. We can post-multiply S1 from (5.20) in (5.24) and (5.25) to obtainrank(RHS) = rank(RHS × S1)= rank(L) + rank(T1)(5.26)andrank(LHS1) = rank(LHS1 × S1)= rank(L) + rank([T1T2]) (5.27)respectively. From (5.26) and (5.27), we can show that rank(RHS) =rank(LHS1) iff rank[T1T2]= rank[T1]. Thus, (5.23) is the necessaryand sufficient condition for the existence of the solution to (5.18).Proposition 3. The condition in (5.19) is satisfied, with a stable N , iff thefollowing conditions are simultaneously satisfied.1. For all s ∈ C,rank(LHS2,i) = rank(RHS) (5.28)where LHS2,i is formulated in (5.29). In (5.29), ME,i:j is a blockdiagonal portion of ME, defined in (5.21), which only contains Ek on595.2. Methodologyits diagonal where k ∈ {1, · · · , j}. In (5.28), i ∈ {1, · · · , 5} whenτ 6= 0 and τ1 6= 0, i = 1 when τ = 0 and τ1 = 0, and i ∈ {1, · · · , 4}when τ = 0 and τ1 6= 0.LHS2,1 ,sLη1 − LAη1 −Q2,2 −Q2,3 −Q2,4 −Q2,5Q1C2,1,LHS2,2 ,−Q2,1 sLη2 − LAη2 −Q2,3 −Q2,4 −Q2,5Q1C2,2,LHS2,3 ,−Q2,1 −Q2,2 sLη3 − LAη3 −Q2,4 −Q2,5Q1C2,3,LHS2,4 ,−Q2,1 −Q2,2 −Q2,3 sLη4 − LAη4 −Q2,5Q1C2,4,LHS2,5 ,−Q2,1 −Q2,2 −Q2,3 −Q2,4 sLη4 − LAη4Q1C2,5.(5.29)whereC2,1 ,[I 00 ME,2:5], C2,2 ,E1 0 00 I 00 0 ME,3:5,C2,3 ,ME,1:2 0 00 I 00 0 ME,4:5, C2,4 ,ME,1:3 0 00 I 00 0 E5,C2,5 ,[ME,1:4 00 I].2. When τ1 6= 0 and/or τ 6= 0, there exists a Z for which (Λ1−ZΓ1) has605.2. Methodologynegative eigenvalues with acceptable magnitude(Λ1 − Λi)− Z(Γ1 − Γi) = 0 (5.30)for i ∈ {2, · · · , 5} when τ1 6= 0 and for i ∈ {2, · · · , 4} when τ = 0.In (5.30), Γi and Λi areΛi = Q2,iHi − T2T1+Q1,iHi,Γi = (I − T1T1+)Q1,iHi.Proof. The solution to (5.18) has the form of[E J1 K J2 H]= T2T1+ + Z(I − T1T1+) (5.31)for an arbitrary matrix Z with compatible dimension.1. Proof of (5.28).From (5.19) and (5.31), N can be written as N = Λi − ZΓi whereΛi = Q2,iHi − T2T1+Q1,iHi,Γi = (I − T1T1+)Q1,iHi.(5.32)The eigenvalues of N can be placed at any desired values iff rank[sI − ΛiΓi]=l + X , ∀s ∈ C.We first introduce some required matrices to complete the proof. Choose615.2. Methodologyfull-row rank matricesSa,1 =[H1 E1 00 0 I],Sa,i =0 I 0 0Hi 0 Ei 00 0 0 I, for i ∈ {2, 3, 4}Sa,5 =[0 0 IH1 E1 0],(5.33)a full column rank matrixSb =I T2T1+0 (I − T1T1+)0 T1T1+, (5.34)and a full row rank matrixSc,i =[I 0T1+Q1,iHi I]for i ∈ {1, · · · , 5}. (5.35)For each feasible value of i, by first post-multiplying Sa,i in LHS2,ifrom (5.29) and then pre-multiplying Sb and post-multiplying Sc,i inthe resulted matrix, the rank does not change and we haverank(LHS2,i) = rank[sI − ΛiΓi]+ rank[T1]. (5.36)Comparing (5.36) and (5.26) and having rank(L) = l + X , (5.28) issatisfied iff rank[sI − ΛiΓi]= l + X , ∀s ∈ C.2. Proof of (5.30).In proof of (5.28) we showed that, for each feasible value of i, the eigen-values of matrix N can be selected to have a desired values depending625.2. Methodologyon a matrix Z if rank(LHS2,i) = rank(RHS).After satisfaction of (5.28), according to N = Λi − ZΓi, required isa common pair (N,Z) with an stable N which can satisfy the aboveequation for all feasible values of i. Thus, for any feasible pair of (i, j),it is necessary to have Λi−ZΓi = Λj−ZΓj, which is the proof to (5.30).From Lemma 1 and Propositions 2-3, we can introduce Theorem 5 onthe existence of a stable delayed functional estimator to estimate the currentand upcoming desired functional of the states of a system of interest.Theorem 5. For a system of the form (2.1), with γ ∈ {0, 1} availablederivatives of the outputs and λ ∈ {0, 1} available derivatives of the inputs,there exists an estimator of the form (5.6) to make• non-delayed observations, with τ1 = 0 and τ = 0,• delayed observations, with τ1 6= 0 and τ = 0,• non-delayed predictions, with τ1 = 0 and τ 6= 0,• delayed predictions, with τ1 6= 0 and τ 6= 0,iff, a functional Rx exists to extend the desired task functional; as is definedin (5.2); to satisfy the condition (5.23) in Proposition 2 and conditions(5.28) and (5.30) in Proposition 3 and also to satisfy the following condition:• When τ1 6= 0, there exists a pair (N,Z), with Z satisfying (5.30) andN being a stable matrix, to holdN(Ta,1 + ZTb,1) = (Ta,2 − Ta,3) + Z(Tb,2 − Tb,3), (5.37)where[Ta,1 Ta,2 Ta,3 Ta,4], T2T1+,[Tb,1 Tb,2 Tb,3 Tb,4], (I − T1T1+),(5.38)635.2. Methodologyand have compatible dimensions (Ta,4 and Tb,4 have (pr + (λ + 1)B)columns, and Ta,i and Tb,i have equal number of columns for i ∈{1, 2, 3}).In (5.38), T1 = Q1ME and T2 = Q2ME where Q1 and Q2 are obtainedfrom (5.15) and (5.17). In (5.38), T1+ is the pseudo-inverse of T1.Proof. The necessary and sufficient condition for the existence of a delayed/non-delayed estimator of form (5.6) to reconstruct the functional Tx(t+τ) of thesystem (2.1) is the existence of a matrix R and a stable matrix N to satisfyequations (5.16). This problem can be considered as two subproblems: i)existence of a stable solution for (5.16), and ii) satisfaction of the conditionJ1 = K +NE.In the proofs of Propositions 2 and 3, we have already showed that thesatisfaction of (5.23) is necessary and sufficient for the existence of solution to(5.18) and that satisfaction of (5.28) and (5.30) are necessary and sufficientfor the existence of the solution to (5.19).In addition to the existence of the solution for (5.18) and (5.19) and alsothe stability of matrix N , it is required to choose N to satisfy J1 , K+NE.When τ1 = 0, from (5.12), θi = 0 which will result in Q1,2 = 0. Thus,J1 can be selected arbitrarily and it will be straight forward to obtain N tosatisfy J1 , K +NE.On the other hand, when τ1 6= 0, J1 cannot be selected arbitrarily.Therefore, we need to select the pair (N,Z) such that N is stable andJ1 = K + NE. Recall that, from (5.31), the selection of Z will affect thevalues of [E J1 K J2 H]. Equation (5.37) is obtained by plugging in (5.31)to J1 = K +NE.5.2.2 Model of the user as an estimatorAssuming the existence of the estimator (5.6) to reconstruct the functionalTx(t + τ) of the system (2.1), we can determine the related estimator ma-trices N , J1, J2, H, and E for delayed/non-delayed observation/predictionas follows:645.2. Methodology• To design a non-delayed observer (i.e. for τ = 0 and τ1 = 0),– Find a matrix R to satisfy (5.23) and (5.28), then form the matrixL.– Choose H1 and E1 such that LH1 = I and LE1 = 0 respectively.– Calculate Λ1 and Γ1 from (5.32).– Choose Z to have a stable N = Λ1 − ZΓ1 and obtain J2, E, K,and H from (5.31).– Based on N , E and K, calculate J1 from J1 = K +NE.• To design a delayed observer (i.e. for τ = 0 and τ1 6= 0),– Find matrices R, N , and Z to simultaneously satisfy (5.23),(5.28), (5.30), and (5.37), then form the matrix L.– Based on Z, obtain J1, J2, E, K, and H from (5.31).• To design a non-delayed predictor (i.e. for τ 6= 0 and τ1 = 0),– Find matrices R, N , and Z to simultaneously satisfy (5.23),(5.28), and (5.30), then form the matrix L.– Determine J2, E, K, and H from (5.31).– Based on N , E and K, calculate J1 from J1 = K +NE.• To design a delayed predictor (i.e. for τ 6= 0 and τ1 6= 0),– Find matrices R, N , and Z to simultaneously satisfy (5.23),(5.28), (5.30), and (5.37), then form the matrix L.– Based on Z, obtain J1, J2, E,K, and H from (5.31).5.2.3 ExampleIn this example, we validate our method for designing a delayed/non-delayedobserver/predictor to estimate a desired functional. We assume γ = 1 (i.e.,having access to the first derivative of the outputs) and λ = 0 (i.e., noderivative of the low-level input). Our goal is to evaluate the existence of655.2. Methodologya delayed-observer and a delayed-predictor to reconstruct Tx(t+ τ), whereT =[0 1 1]Tand then design such an estimator.Consider a system of the form (2.1) with the following system matrices,A =0 1 −10−2 −3 −12 0 −2,B =[0 0 1]T,F =[1 0 0]T,(5.39)and C = I3×3 – that is, all states are measured. Besides, the referencetrajectory is assumed to be available in the user interface, D = 1. Giventhe system dynamics in (5.39), with all states being measured, the systemis observable and predictable. However, based on our earlier discussion, thestandard observability and predictability of the system is not enough for ahuman operator to accomplish the desired task. Through the conditions ofTheorem 5, we can evaluate whether the human can attain SA about thetask Tx.For the system (5.39), the conditions in Theorem 5 for the existence ofthe delayed and non-delayed estimator where τ ∈ {0, 0.2} and τ1 ∈ {0, 0.3}are satisfied. Therefore, based on the technique suggested in Section 5.2.2,we can design delayed/non-delayed observer/predictor for this system toreconstruct Tx(t+ τ).• Non-delayed observer (τ = 0 and τ1 = 0):Using the design procedure suggested in Section 5.2.2, we can design665.2. Methodologya non-delayed observer as is illustrated in (5.40).NJ2H=−4.22040.05461.2624, J1 =−0.8084−0.2323−4.5347−0.1637−0.96320.3564T,E =0.25150.26051.07780.13420.3544−0.0000T, K =0.25300.86720.01390.40280.53260.3564T.(5.40)Figure 5.1 shows the effectiveness of using the designed observer intracking the desired functional of the states of system (5.39) while theobserver has no delay.We now use the designed observer (same matrices as above) to predictour desired functional while the actual observer is delayed (τ1 = 0.3and τ = 0.2). The results are available in Figure 5.2.From Figure 5.2, it can be seen that the non-delayed observer matricesare not effective for precisely predicting a desired functional while thestructure of the actual observer is delayed too. Hence, a new estimatorhave to be designed to provide us with our desired results.• Delayed-predictor (τ = 0.2 and τ1 = 0.3):From the algorithm provided in Section 5.2.2, we can design a delayedpredictor for the reconstruction of the functional Tx. The predictor675.2. Methodology0 1 2 3 4 5 6 7 8−20246Non−delayed observationtime (sec.)  Actual stateEstimated state0 1 2 3 4 5 6 7 80123Non−delayed observation errortime (sec.)Figure 5.1: Non-delayed observation of the desired functional, Tx, of thestates of the system (5.39).structure is provided in 5.41.NJ2H=−8.47190.18001.8711, J1 =0.65410.20900.3986−0.1792−0.4676−1.1273T, (5.41)685.2. Methodology0 1 2 3 4 5 6 7 8−20246Delayed Prediction(Non−delayed Observer)time (sec.)  Actual statePredicted state0 1 2 3 4 5 6 7 8−1012Delayed prediction error(Non−delayed Observer)time (sec.)Figure 5.2: Using a non-delayed observer matrices for predicting Tx, whilethe actual observer has also internal delay.E =−0.21000.13480.0114−0.07300.10400.2905T,K =−1.12501.35100.4956−0.79780.41361.3335T.Figure 5.3 illustrates the simulation results of such a predictor. Fromthis figure, it is possible to design a delayed estimator for system (5.39)to predict the desired functional, Tx, which confirms the results ofTheorem 5.Since the prediction starts when t ≥ τ , it will result in the discontinuityobserved in Figure 5.3.695.2. Methodology0 1 2 3 4 5 6 7 8−4−20246Delayed Predictiontime (sec.)  Actual statePredicted state0 1 2 3 4 5 6 7 8−505Delayed prediction errortime (sec.)Figure 5.3: Delayed prediction of the desired functional, Tx, of the statesof the system (5.39), with τ = 0.2 and τ1 = Towards display designIn Section 5.2.1, we assumed the availability of certain measurements of thestates and also the automation’s desired trajectory and evaluated whetherthe human can make delayed/non-delayed estimations based on the availabledata. We also showed how the user can obtain the desired estimations.Our main goal in this section is to discuss the stages required for de-signing a display with minimal cardinality to allow the user to accomplish adesired task. For this purpose, we have to find matrices C andD of minimumrank summation to satisfy the conditions in Theorem 5, which are requiredfor the existence of a delayed predictor as well as a delayed observer (i.e.,for prediction, τ = τa and τ1 = τ1,a with τa the prediction horizon amountand τ1,a the information processing delay; and for observation, τ = 0 andτ1 = τ1,a).Clearly, desired for most designers is the real-time determination of the705.2. Methodologydisplayed information such that depending on the operating condition andthe task, the display information can be updated. Analytically, our goal isto solve the problemminC,Drank(C) + rank(D)subject to (5.23) for τ = 0 & τ = τa,(5.28), (5.30), and (5.37).(5.42)Remark. Since some measurements might be easier than others for the userto perceive and process, in some cases, the cardinality of the displayed in-formation is not as important as the nature of the displayed information.Hence, in practice, it might be preferable for a designer to investigate sev-eral valid designs with low cardinalities and select the most suitable one. Inaddition, as the rank of matrix R is directly related to the required orderof the functional estimator, having this matrix with low rank is also veryimportant. Therefore, the reader may even consider selecting a feasible pairof (C,D) with their corresponding matrix R obtained to be of low or evenminimum rank – that is, rather than (5.42), solving an optimization prob-lem with rank(R) being its cost function and the objective function as in(5.42).Although there are techniques to formulate the rank condition in a con-vex form, with the sophisticated constraints in (5.42), solving the mentionedproblem and designing a display of minimum information for a generic caseof online determination of the information is very complicated topic thatshould be a subject to extensive future research.On the other hand, for cases where the operating conditions do notchange and the task is pre-specified, the real-time determination of the dis-play is not necessary and it is possible to simply design the displayed infor-mation in advance and apply it as is during the running of the process. Wecall this technique the off-line determination of the displayed information.For such simpler cases, it is possible to determine the required displayedinformation by manipulating the matrices C and D and verifying the satis-715.2. Methodologyfaction of the conditions in Theorem 5, heuristically.For our specific application of user-interface design, C represents a setof the measured states and D represents the set of the automation’s desiredtrajectories available on the user interface. Thus, we can consider the ma-trices C and D to be diagonal with the elements on the diagonal being zeroor one – that is, no linear combinations of states and no linear combinationsof the trajectories is presented in the interface. Clearly, if a specific linearcombination of the states has a physical meaning to the user, that vectorcan also be added to the set of feasible measurements (e.g. the flight pathangle of an aircraft dynamics which is a linear combination of the statespitch angle and the angle of attack).Considering the processing delay, our goal is to determine the displayedinformation with minimum cardinality that lets us observe/predict the taskfunctional. The required steps to determine the correct displayed informa-tion are as follows:• Initialize by considering certain available displayed information, e.g,for initialization we suggest to have rank(C) = 1 and rank(D) = 0.• For the available display, check the conditions in Theorem 5 (for τ1 = 0and τ1 = τ1,a).• If either the delayed functional observer or the delayed functional pre-dictor does not exist, change the display information or iterativelyincrease the number of the measurements (including the measuredstates and the measured trajectories) and re-investigate the conditionsin Theorem 5.• Among the valid displays which satisfy all the existence conditions,select those of interest, either for lower rank, or for any application-specific reason which may be important to the designer.As has already been mentioned, for most of the applications, the sug-gested method is necessary but not sufficient to design a good display. Hence,in order to design a display that is compatible with any operating conditionand task, further research is required.725.3. Application example5.3 Application exampleAs for the users of any system with a human controller or under humansupervision/monitoring, attaining SA is indispensable for an anesthetist tomaintain the safety of the anesthetized patient. This importance has beeninvestigated by several researchers [31, 107, 108]. In a recent paper, Fiora-tou et al. [107] discussed SA in the framework of anesthesiology. Accordingto [107], after perceiving the available displayed information and the in-formation from the environment, the anesthetist has to integrate all theavailable data for the identification of the current and the future desiredpatient states. The estimation of the current states of the system is impor-tant for goal accomplishment and for fault detection. As is mentioned in[107], task prediction is also extremely important for the anesthetist to beproactive rather than just being reactive.In order to model a patient under anesthesia, understanding the relation-ship between the dose of the drug and its pharmacological effect is necessary.This model consists of two sub-models, the pharmacokinetic (PK) and thepharmacodynamic (PD) models. The PK model, demonstrates the effect ofthe administered drug on the drug plasma concentration and the PD model,models the relationship between the drug concentration in the effect site andthe observed effect of the drug.We consider a simplified version of the PKPD model described in [109]and [110] to model the effect of propofol administration on the depth ofhypnosis. The PK model in [110] is the well-known 3-compartment modeldeveloped in [111] to evaluate the effect of propofol on the drug concen-tration in different compartments. Pharmacokinetically, a compartment isconsidered to be a group of tissues which have similar kinetic characteristics.A 3-compartment model has three states, i.e. concentrations in i) the bloodand highly perfused tissues (e.g. brain and liver), ii) the muscles and viscera,and iii) fat and bones. The PD model presented in [110] consists of threestates, two of which are associated with the dynamics of the monitor. We,however, only consider the effect site concentration of the drug and linearizethe Hill equation to obtain the depth of hypnosis based on this state of the735.3. Application examplek10 0.0524 k12 0.2359 k13 0.0162k21 0.0892 k31 0.0022 kd 0.1335V1 0.3593 EC50 3.2 γh 4.7Table 5.1: PKPD coefficient for a 21 years old 100 kg patientsystem.Considering no transport delay, the PKPD model for evaluating thedepth of anesthesia is as (2.1) withA =[Apk 0kd 0 0 −kd],B =[Bpk0], F =[04×1],(5.43)whereApk =−(k10 + k12 + k13) k12 k13k21 −k21 0k31 0 −k31,Bpk =V −1100.(5.44)In (5.43) and (5.44), kij and kd are rate constants and V1 is the volumeof the plasma compartment.The desired task which is controlling the depth of anesthesia can bedefined as Tx =[0 0 0 γh(4EC50)−1]x, where EC50 is the 50% effectconcentration and γh is the cooperativity coefficient. The values that we areusing are presented in Table 5.1.From Theorem 5, no delayed predictor for system (5.43) can reconstructthe desired functional Tx(t+ τ) when the measurement in the display is re-stricted to the depth of hypnosis, C =[0 0 0 γh(4EC50)−1]. However,based on the discussion in Section 5.2.4, it can be seen that for an estimatorof the form (5.6) with τ1 = 0.3sec estimation delay, it is necessary and suf-ficient to measure the blood-plasma drug concentration in addition to the745.4. Summary and discussiondepth of hypnosis in order to make correct and precise observations and pre-dictions of the desired functional – that is, C =[1 0 0 00 0 0 γh(4EC50)−1].Hence, from the above analysis we can conclude that it is not possible forthe anesthetist to precisely predict the effect of the drug administration onthe depth of hypnosis unless they are provided with both the depth of hypno-sis and the plasma concentration through the display. Unfortunately, plasmaconcentration measurement is beyond current state of technology and thuscannot be provided. In practice, open-loop population-based PKPD modelsare used to estimate and display both Cp and Ce. However, due to significantinter-patient variability, these estimates come with such large uncertaintiesthat they are not likely to substitute for real measurements, hence hinderingthe ability of the anesthetist to accurately predict the depth of hypnosis.5.4 Summary and discussionThe focus of this chapter was on mathematical modeling of the processof attaining SA for the user. The user was considered to be a functionalobserver and a functional predictor whose estimations of the states of thesystem are delayed. It was also assumed that the user may have knowledgeabout the derivatives of their own inputs and of the outputs.For a system that is controlled by the user and that tracks a desiredreference trajectory with the aid of a computer, we presented a techniqueto evaluate whether it is possible to reconstruct and predict (with delay)a desired set of states given a set of displayed measurements. In additionto obtaining the existence conditions for the extended delayed functionalobserver/predictor with the availability of higher-order derivatives of theinputs and the outputs, we presented a procedure to design such an esti-mator. We also presented a method to determine the minimum informationrequired to display so that the user can accomplish a desired task.The work of this chapter focused on precise comprehension and predic-tion of the task by the user. In most of the systems, however, the users donot require to precisely comprehend and predict the task, but, it is enough,755.4. Summary and discussionyet necessary, for them to make these estimations within a specific bound.That is the inspiration for the next chapter in which the process of attain-ing SA by the user will be modeled as a bounded-error delayed functionalestimator.76Chapter 6User-interface analysisthrough modeling the user asan estimator with boundederrorIn this chapter, for a system modeled in (2.1), we model the process ofattaining SA by the user as a bounded-error delayed functional estimator.In Section 6.2 we will introduce a Theorem and Corollary on the existenceof and the type of estimator that we are looking for. In 6.3 we talk aboutthe anesthesia example in detail and use our tool to make some analysis onsuch systems.6.1 Problem formulationIn the previous chapter, we investigated the correctness of and then designedthe displayed information for safety critical systems in which for attainingSA, the precise comprehension and prediction of the information is necessaryfor their user. The previous results, however, were too restrictive for themajority of the systems in the real world. In most of the systems, theusers do not require to precisely comprehend and predict the task, but, it isenough, yet necessary, for them to make these estimations within a specificbound. Consider a driver trying to maintain the speed of a car within thespeed limits. This speed limit prevents the user from exceeding a specificspeed while the user should also not drive too slowly. Hence, it is important776.2. Bounded-error estimatorfor the driver to be capable of keeping the speed within the pre-specifiedbounds.To model the user as a bounded-error delayed observer/predictor, weobtain the error dynamics as in Chapter 5. We then evaluate the conditionsunder which the error remains bounded for all feasible situations.6.2 Bounded-error estimatorHaving the error dynamics (5.8), our goal is to design the estimator matricesin (5.6) such that the steady-state error remains bounded for all feasiblecombinations of the inputs and the initial states.Under the assumption that γ and λ are selected from {0, 1}, the errordynamics can be written ase˙(t) = Ne(t) + (NLη + [E J1 K J2 H ]Q1 −Q2)x(t− τ1)rauh(t− τ1)u˙h(t− τ1)u¨h(t− τ1), (6.1)where η = [η1 η2 η3 η4 η5], K , J1 − NE and for i ∈ {1, · · · , 5}, Q1,i andQ2,i are defined in (5.15) and (5.17) respectively.We define a matrix XU ∈ Rn×i which columns are selected to be all ifeasible combinations of states, inputs, and input derivatives of a system,and formulateC1 , e¯1 + LηXUC2 , e¯2 + LηXUT1 , Q1XUT2 , Q2XU.with e¯1 = {e1, e1, · · · e1} and e¯2 = {e2, e2, · · · e2} have i columns.Proposition 4. For a system of form (2.1) with a given feasible combi-nations of state-input, XU , there exists a bounded error estimator of form786.2. Bounded-error estimator(5.6) to reconstruct the functional z(t + τ) iff there exists a random vectorV , a stable N , and a matrix Z, to satisfy•T2−NC2 ≤ V T1 ≤ T2−NC1. (6.2)• when τ1 6= 0,N(ZTa,1 + V Tb,1) + Z(Ta,2 − Ta,3) + V (Tb,2 − Tb,3) = 0 (6.3)where[Ta,1 Ta,2 Ta,3 Ta,4] , (I − T1T1+),[Tb,1 Tb,2 Tb,3 Tb,4] , T1T1+and Ta,i and Tb,i are of compatible dimensions.Proof. We need to find the estimator matrices to always keep the steadystate error bounded within the desired values. With the error dynamicsprovided in (5.8) and since we consider the state-input vector to be constant(a discussion will be provided later in this section) the error evolves ase(t) = eNte(0) + Fr (6.4)where the forced response, Fr, isFr = (eNt − I)N−1(NLη + [E J1 K J2 H]Q1 −Q2)x(t− τ1)rauh(t− τ1)u˙h(t− τ1)u¨h(t− τ1).Mathematically, we want the steady-state error to be bounded withinpre-specified values e1 and e2 all the time. Considering that N will be796.2. Bounded-error estimatordesigned to be stable, the boundedness can be formulated asNe1 ≤ (Q2 − [E J1 K J2 H]Q1 −NLη)x(t− τ1)rauh(t− τ1)u˙h(t− τ1)u¨h(t− τ1)≤ Ne2, (6.5)where ≤ shows the element-wise inequality.From (6.5) we can writeNC1 ≤ CT ≤ NC2 (6.6)whereCT , T2 − [E J1 K J2 H]T1. (6.7)• Proof of (6.2): As [E J1 K J2 H]T1 = T2 − CT , the solution existsfor [E J1 K J2 H] if and only ifrank[T1] = rank[T2 − CTT1], (6.8)which is equivalent to saying that for the existence of a solution to[E J1 K J2 H] it is necessary and sufficient that there exists a CTsuch that T2 − CT be a linear combination of the rows of T1. Hence,[E J1 K J2 H] has infinite number of solutions for arbitrary values ofV whereCT = T2 − V T1. (6.9)From (6.6) and (6.9), Condition (6.2) is proved and for the stability ofthe observer, a stable N have to exist.806.2. Bounded-error estimator• Proof of (6.3): In addition to satisfaction of (6.2) and stability ofmatrix N , it is required to choose N to satisfy J1 , K +NE.When τ1 = 0, J1 can be selected arbitrarily. However, when τ1 6= 0, J1is not an arbitrary matrix and we need to select the triplet (N,Z, V )such that N is stable and J1 = K+NE. The solution to [E J1 K J2 H]is as follows[E J1 K J2 H] = (T2 −CT )T1+ + Z(I − T1T1+)= Z(I − T1T1+) + V T1T1+.(6.10)From (6.10) and the condition J1 = K +NE, (6.3) can be achieved.The steady-state error of the estimator will be maximized when theforced response of the error is maximum. This happens at a specific combi-nation of N , [E J1 K J2 H], states, inputs, and input derivatives. The goalhere is to keep the maximum steady state error bounded.Definition 4. The direction of xumax – that is, the (input,state) vectorthat maximizes the error, is that of the singular-vector corresponding to themaximum singular-value of G, where G is the input-output transfer matrixof the error dynamics at a desired frequency [112].From (5.8), we introduceAe , N∗Be , N∗Lη + [E∗ J∗1 K∗ J∗2 H∗]Q1 −Q2Ce , I,(6.11)where N∗ and [E∗ J∗1 K∗ J∗2 H∗] are a feasible solution of the estimatormatrices. Define G to be the transfer matrix representation of (Ae, Be, Ce, 0)at a desired frequency. From Definition 4, xumax can be obtained as thesingular-vector corresponding to the maximum-singular value of G. We canthen define the following Theorem.816.2. Bounded-error estimatorTheorem 6. For a system of form (2.1), if there exists an estimator of form(5.6) to make bounded-error estimations of the current and/or upcomingdesired functionals of the states, then the conditions in Proposition 4 aresatisfied and for the calculated N∗, [E∗ J1∗ K∗ J2∗ H∗], and xumax, a pair(CT , Z) exist to satisfy the following conditions1.Z(I−T1,∗T1,∗+)−CTT1,∗++T2,∗T1,∗+−[E∗ J1∗ K∗ J2∗ H∗] = 0 (6.12)2.N∗C1,∗ ≤ CT ≤ N∗C2,∗ (6.13)In (6.12) and (6.13)C1,∗ , e1 + LηxumaxC2,∗ , e2 + LηxumaxT1,∗ , Q1xumaxT2,∗ , Q2xumax.Proof. As we obtained xumax to be the error-maximizing vector of a systemwith [E J1 K J2 H] = [E∗ J1∗ K∗ J2∗ H∗] and N = N∗, Theorem 6 showsthat with the obtained [E J1 K J2 H] and N and with xumax, still theestimation error can remain bounded (within the pre-specified values).Note that satisfaction of (6.12) and (6.13) are not necessary for the ex-istence of the bounded error observer due to the fact that the designedestimator from Proposition 4 is not unique. Hence, if the designed esti-mator in Proposition 4 does not satisfy the conditions in Theorem 6, stillseveral other estimators may exist which can bound the maximum error ofestimation.The sufficient condition in Theorem 6 will become a necessary and suf-ficient condition by applying recursion, such that the calculated xumax be826.3. Application exampleadded to the vector XU at each stage and a new estimator be designed tillall conditions in Theorem 6 are satisfied. The convergence of such a recur-sion can however remain an open issue. It is, therefore, easier to solve theconditions in the Proposition 4 and Theorem 6 simultaneously to obtain thenecessary and sufficient condition.Corollary 7. Having matrix XUr with columns randomly selected to befeasible (input,state) vectors of the process, there exists a bounded error es-timator of form (5.6) to keep the estimation error within the pre-specifiedbounds iff the following conditions are simultaneously satisfied.• There exists a random vector V , a stable N , and a matrix Z, to satisfy(6.2) and (6.3) for XU = XUr.• For the corresponding N∗, [E∗ J1∗ K∗ J2∗ H∗], and the boundedxumax, there exists a pair (CT , Zn) to satisfy (6.12) and (6.13), forZ = Zn.6.3 Application exampleWe consider the PKPD model described in Section 5.3, equation 5.43. Thematrix F of (2.1) can be designed to make the output follow the referencetrajectory.In (5.43) and (5.44), kij and kd are rate constants and V1 is the volumeof the plasma compartment.The desired task which is controlling the depth of anesthesia can bedefined as Tx =[0 0 0 γh(4EC50)−1]x, where EC50 is the 50% effectconcentration and γh is the cooperativity coefficient.In Chapter 5, we could show that by only measuring the depth of hyp-nosis, it would not be possible for the anesthetist to precisely predict thedepth of anesthetia (DOA). This states that the SA cannot be preciselyachieved by the anesthetist to perform a task on DOA while only having ac-cess to information about DOA. However, adding more information to theuser interface could help to attain SA.836.3. Application exampleWe modify our previous analysis in two directions:1. In the previous chapter, we investigated the possibility that the anes-thetist could estimate the current and future DOA with complete pre-cision. However, intuitively, the anesthetist does not require to pre-cisely reconstruct and predict the task (i.e. reconstruction and predic-tion of the DOA within acceptable ranges would be sufficient). Hence,we use our new technique for this analysis.2. In the previous chapter, we assumed the anesthetist knew the pre-cise dynamics of each patient. This, however, does not sound like areasonable assumption. The internal estimator of the anesthetist canbe considered to be formed based on a model of a nominal patient.This nominal model is the understanding of the anesthetist about anaverage patient in a specific category (e.g. children or adults) and iscreated from the real responses of various patients. In this chapter,we model the human as an estimator designed based on a nominalsystem, we then evaluate whether the obtained model can be used toreconstruct and predict the DOA of each patient within the desiredbounds.For the patients with average response to the administered drug, thePKPD coefficients are presented in Table 6.1. The PK parameters are esti-mated from [1] and the PD values are from [2]. We randomly select two setsof coefficients in Table 6.1 to form the nominal model of an average-patientthat the anesthetist knows internally. We then investigate the chancesthat with such an internal understanding about an average patient, theanesthetist can attain SA regarding the desired task within the acceptablebounds for each patient.846.3.ApplicationexamplePatient k10 k12 k13 k21 k31 V1 kd EC50 E0 γ1 0.0068 0.0019 0.0007 0.0009 0.0001 20.164 1.15 3.95 93.11 1.742 0.0062 0.0019 0.0007 0.0009 0.0001 11.5058 1.34 4.24 92.46 1.903 0.0062 0.0019 0.0007 0.0009 0.0001 10.0848 10.71 5.77 91.47 1.564 0.0061 0.0019 0.0007 0.0009 0.0001 12.331 1.12 4.84 91.6 1.555 0.0065 0.0019 0.0007 0.0009 0.0001 28.0782 3.84 3.97 92.91 1.626 0.0062 0.0019 0.0007 0.0009 0.0001 10.7266 1.89 3.57 94.58 1.577 0.0062 0.0019 0.0007 0.0009 0.0001 10.5432 4.55 4.81 92.89 1.558 0.0059 0.0019 0.0007 0.0009 0.0001 26.8164 1.46 3.71 91.68 1.759 0.0062 0.0019 0.0007 0.0009 0.0001 11.5975 1.16 5.44 90.30 1.5210 0.0063 0.0019 0.0007 0.0009 0.0001 22.44 7.41 3.60 91.38 1.82Table 6.1: Patients’ parameters from [1] and [2]856.3. Application exampleFor our analysis, we consider two type of measurements which include1) only the DOA (rank(C) = 1 and rank(D) = 0), 2) the DOA, theplasma concentration; although measuring the plasma concentration is be-yond current state of technology; and the automation’s desired trajectory(rank(C) = 2 and rank(D) = 1). In addition, we consider three valuesfor the prediction horizon, τ , to analyze the capability of the anesthetist toperform shorter and longer term predictions.Our approach is to consider two of the patients as the nominal models– that is, the average-patient model that the anesthetist knows internally.We then investigate the chances that with such an internal understandingabout an average patient, the anesthetist can attain SA regarding the de-sired task within the acceptable bounds for each patient. Note that if theconditions in Corollary 7 are all satisfied, the estimator that gives boundeddelayed estimations and predictions of the desired task exists and is notnecessarily unique. Hence, among all estimators that may exist based onthe nominal model, some may let the anesthetist attain SA about DOA ofother patients and some may not. It is, however, not clear which of themany estimators that exist is the closest model to the internal estimator ofthe anesthetist. We, therefore, perform a statistical analysis to determinethe chances that the anesthetist can attain SA about various patients basedon different internal estimators.6.3.1 ResultsFor each nominal model and each combination of the measurements and τ ,we design fifty estimators; if there exists any; and then evaluate whether thedesigned estimator is effective to attain SA about other patients.The results are provided in Tables 6.2 and 6.3. Each table shows thechances (percentage) that the estimators designed based on a given nominalmodel are effective to make bounded-error estimations for other patients.As it is expected, the estimators designed to make bounded error estima-tions for the nominal models 2 and 7 are effective to make correct estimationsfor Patients 2 and 7 (respectively) all the times.866.3.Applicationexampleτ (sec) rank(C) rank(D) P1 P2 P3 P4 P5 P6 P7 P8 P9 P100.5 1 0 100 100 24 100 62 90 70 98 100 302 1 5 100 2.5 52.5 5 70 25 7.5 42.5 7.55 1 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A2 1 6 100 6 4 0 4 0 6 4 020 1 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A2 1 0 100 0 0 0 0 0 0 0 0Table 6.2: Percentage effectiveness of the estimator designed for nominal model P2 on estimating the task forother patientsτ (sec) rank(C) rank(D) P1 P2 P3 P4 P5 P6 P7 P8 P9 P100.5 1 0 94 100 74 98 98 98 100 98 100 562 1 0 17.5 25 17.5 0 52.5 100 0 25 05 1 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A2 1 0 8 2 2 4 4 100 2 2 020 1 0 N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A2 1 0 0 0 0 0 0 100 0 0 0Table 6.3: Percentage effectiveness of the estimator designed for nominal model P7 on estimating the task forother patients876.3. Application exampleTo discuss the results of Tables 6.2 and 6.3 in detail, we first clarifythe difference between the selected values of the prediction horizon, τ =0.5, τ = 5, and τ = 20. The very short prediction horizon, τ = 0.5,means that predicting a very short step ahead is desired and the explicitprediction of the states is not required for attaining SA. On the other hand,by τ = 5 and τ = 20 we mean that in order to attain situation awareness,the anesthetist is required to make explicit predictions 5 and 20 seconds inadvance, respectively.From the results obtained for τ = 0.5, we can see that the estimatordesigned based on the nominal internal model of the anesthetist, is notnecessarily capable of reconstructing and predicting the task for each indi-vidual. However, we need to notice that the possibility of making correctbounded estimations depends on the similarity between the actual and thenominal PKPD models. In addition, it can be seen that when the amountof measured information in the display is increased, it becomes less possiblefor the anesthetist to make correct estimations on various individuals basedon the internal model. This can be due to the fact that by introducing ad-ditional measurements the internal estimator of the anesthetist is designedmore specifically for the available internal nominal model.For τ = 5 and τ = 20, when only the DOA is measured in the dis-play, the conditions in Corollary 7 are not all satisfied. So, irrespective ofthe internal model of the anesthetist, the anesthetist cannot make correctbounded estimations of the task. In other words, it is never possible forthe anesthetist to attain SA, even about the patient with the model beingthat of the internal nominal model. By increasing the information in thedisplay, the internal estimators for the anesthetist can be designed to makecorrect estimations on the nominal model. For τ = 5, in the majority of thecases, these internal estimators are not capable to let the anesthetist attainSA about other patients. When longer term prediction is required – that is,τ = 20, the anesthetist can only attain SA about the DOA of the patient ifs/he knows the precise model of the patient.886.4. Summary and discussion6.3.2 DiscussionFrom the results obtained in Section 6.3.1, regardless of the nominal model,the type of the measurements, or the definition of the prediction for SA,there is always a chance that the anesthetist cannot predict the task stateswithin the desired bounds. Hence, a hazardous situation may occur at somepoint. The error due to the lack of SA can be as minor as putting thepatient in a slightly lighter or deeper anesthesia than what is desired. It canalso be very serious with the patient being put in too deep of an anesthesia.Obviously, in real world applications, where the anesthetist has access tofurther information about the patient through the environment, the chancesthat s/he cannot control and/or monitor the DOA can be much slimmerthan what we have obtained here.Due to the importance of the concept of SA in the safety of operations,and based on the results that show the existence of the cases that the anes-thetist may have lack of SA about the DOA, we need to seek a way thatguarantees the existence of SA for the anesthetist all the time. The so-lution could be providing the anesthetist with SA through a CDSS whichpresents predicted effect of the anesthetic drug on the patients. Two suchsystems are Navigator Applications Suite by GE or the SmartPilot View byDrager [113]. It is still an open issue to investigate whether with the existinguncertainties and with the differences between the PKPD models used tobuild these devices and the actual PKPD values of each patient, the finalprediction remains in the safe bound or not.6.4 Summary and discussionIn this chapter, the user was modeled as a bounded-error delayed functionalestimator. For accomplishing a desired task safely, this estimator have toexist to reconstruct and predict a specific functional of the states of thesystem within pre-specified bounds.Our method was used to investigate the important problem of safetyof an anesthetized patient. Considering the anesthetist to have an internal896.4. Summary and discussionnominal understanding about the patients, the chances that the anesthetistwill be able to attain SA about the DOA of each patient during surgerywas evaluated. We could show that when the available information is re-stricted to the displayed information, there always exists a possibility thatthe anesthetist cannot attain SA about the patient’s DOA – that is, the un-derstanding of the anesthetists about the depth of anesthesia of the patientis not necessarily correct. This led us to suggest incorporating automateddevices which could provide the current and the predicted values of the DOAdirectly to the doctor.90Chapter 7Thesis summary7.1 Thesis contributionsThe contributions in this thesis are twofold. In the body of the thesis,the effectiveness of the displayed information was investigated. In order tomaintain the coherency of the techniques provided in the body, a part of thecontributions, which is purely analytical, is provided in the appendices.The body of this thesis considered the problem of evaluating the dis-played information and also designing good user-interfaces for LTI systemsunder human or shared control. The theory of situation awareness statesthe importance of the comprehension and prediction of certain informationbefore attempting to perform a task. Based on this theory, we consideredthe user to be a specific type of observer/predictor and evaluated the in-formation which is required for this estimator to make correct estimationsof the desired functionals of the states of the system. We introduced twomain approaches for making such an evaluation of the displayed information.These approaches are applicable to systems of different orders i.e. small sizesystems as well as large systems with many states.The first approach was based on subspace analysis, such that, we eval-uated whether the space spanned with the combination of the states whichare involved in the task can be observed and predicted by the user. Forthis purpose we needed to formulate two main spaces, the user-observableand the user-predictable subspaces and then investigate whether the taskspace is contained in these two subspaces. To determine the user-observableand the user-predictable subspaces, we considered certain limitations for theuser. With such limitations, the space which is spanned by the functionalof the states whose current and future values can be estimated by the user917.1. Thesis contributionsdiffers from the standard observable and predictable subspace.In the second approach, we modeled the user as a specific type of ob-server and predictor. To create this model, we also considered certain user’sspecifications and limitations, including the information processing delay ofthe human , having access to higher derivatives of the measured states, andfocusing only on the desired task. We first modeled the user as a type ofobserver whose goal is to make precise estimations and predictions of thetask functional. We also modeled the user as an estimator whose estimationerror is bounded rather than being precisely zero. Through modeling theuser, we could then evaluate the existing user interfaces as well as determin-ing the required information to be included in the display for the cases thatoff-line design of the displayed information is valid over the entire process.In addition to the results on the display design, in the Appendices, weachieved some novel analytical results on observability subspaces and ob-server design. We evaluated the effect of higher derivatives on 1) the observ-ability subspaces and 2) the existence of, order, and the design of functionalobservers.The contributions of this thesis can be summarized as follows:• Formulating the novel user-observable and user-predictable spaces [102,114] (and also the delay-incorporating user-observable and the delay-incorporating user-predictable) and using them to evaluate the cor-rectness of the displayed information [115].• Modeling the user as a specific type of observer and predictor (for boththe precise estimation [116, 117] and bounded-error estimation [118]of the task functional). Then using this model to evaluate the existingdisplay or to design a new display.• Determining the effect of the availability of higher derivatives of theinput and the output on the observability subspace [119] and on theexistence and order of the functional observer[120].927.2. Possible future directions7.2 Possible future directionsSince the presented research topic in this thesis is very novel and not muchwork has been done on the subject, there are many directions that canbe taken to make current results stronger and more suitable for real worldapplications.1. Incorporating nonlinearities as well as uncertainties and noise:In this thesis we have evaluated the existing display content and havesuggested a design technique for designing the displays for LTI systemswith no noise and uncertainty. This assumption of having a non-noisyand deterministic LTI system is, however, too simplistic when it comesto real world applications.First, throughout the thesis we considered the possible availability ofthe derivatives of input and output signals under the assumption ofhaving non-noisy input and output signals. However, this assumptionhave to be relaxed at some point to give us more realistic results. Forthe case of having noisy signals, the techniques have to be extensivelymodified since it is not straight forward to deal with the derivatives ofthe noise.Besides, consider the PKPD model of the patient we discussed in Sec-tions 5.3 and 6.3. Not only this model is affected by external noiseterms such as measurement noise, the system matrices also are notprecise and have a degree of uncertainty associated with them. Thismay affect the obtained results to a great deal such that while we be-lieve that the estimation error remains bounded during the operation,it may grow beyond the acceptable bounds.In addition to the uncertainties and noises, considering the system tobe a LTI is rather simplistic. On one hand, the linearized systemsare not the most realistic presentation of each system. On the otherhand, in addition to all different sources of noise, the linearizationitself introduces more uncertainties to the model.937.2. Possible future directionsHence, robust and non-linear analysis of the user as an observer/predictoris required to provide us with a guaranteed or at least a more realisticbound of error of estimation. The main benefit of modeling the user asa specific type of robust observer is that this observer can be designedto be robust to the effect of bounded parametric uncertainties as wellas the noise. This is, in fact, how a trained and experienced user wouldbehave.2. Relaxing some assumptions:Several assumptions were made in the thesis to help with obtainingthe initial model of the user.For instance, so far, we considered a specific category of systems inwhich the user-interface was the only source of information for the user.Incorporating the effect of information from the environment on therequired information content of the user-interface is an important workwhich is necessary in order to make our framework more applicable torealistic cases.In addition we made some fundamental assumptions including the lim-ited role of mental model on simply understanding the dynamics andalso non-existence of mind wandering. We believe that at furtherstages of this project, the mentioned simplifications have to be relaxedto provide us with a more comprehensive framework.3. Coming up with a rigorous technique to determine the re-quired information to be displayed:In Chapter 5 we suggested a heuristic algorithm to determine the dis-play content. This algorithm, however, is only valid if the operatingconditions and task are non-changing during the entire process. If theoperating conditions and/or the desired task change at some point,a real time estimation of the displayed information is necessary. Forsuch a real time determination of information, we require an analyticalformulation of an effective display that can be easily updated during947.2. Possible future directionsthe process. In addition, a numerical solution with fast convergencemay be helpful to determine the displayed information in real-time.As has been discussed in Section 5.2.4, a solution to the optimizationproblem (5.42) gives us an analytical tool to determine the informationcontent of the display. Other techniques may also exist and might bemore suitable to determine the required information, either analyti-cally or numerically.4. 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We then evaluate how the availability of thederivatives of input and output signals may affect the results in Section A.3.A.1 Problem formulationConsider a continuous-time LTI system of the formx˙(t) = Ax(t) +Bw(t)y(t) = Cx(t),(A.1)with state vector x(t) ∈ Rn, continuous-time output y(t) ∈ Rp, and coeffi-cient matrices A,B, and C with compatible dimensions. In Section A.2 weconsider w(t) ∈ Rm to be a continuous-time unknown input and in SectionA.3, w(t) is a known input with derivatives which can be unknown. Forsimplicity, in the rest of the chapter, we omit the index (t) from x(t), y(t),w(t), and their derivatives.109A.2. Unknown-input observability subspaceIn [40] the authors show that the observability subspace of (2.1) (with un-known input) is the least (AT ,F⊥)-conditioned invariant containing R(CT )where F △= N (BT )⊥ (i.e. the column space of B). In this chapter we eval-uate the effect of lacking information about some derivatives of input andoutput, rather than unknown input, on the observability subspace of thesystem.A.2 Unknown-input observability subspaceIn a slightly different approach from [40], we re-formulate the unknown-input observability subspace of (2.1) and obtain an equation which is easierto solve and extend. To do so, we introduce the orthogonal projection,P1 ∈ Rp×p, onto the left null space of CB. With pre-multiplying P1 in thefirst derivative of the output equation, we obtainP1y˙ = P1(CAx+ CBw)= P1CAx,(A.2)which leads to the set of states which can be reconstructed regardless of thevalues of unknown inputs. Clearly, when CB is of full column rank, P1 is azero matrix.Continuing in a similar fashion with higher derivatives, and introducingfurther projection matrices, Pk ∀k ∈ {2, · · · , n − 1}, to remove the inputfrom up to the kth derivative of the output equation (i.e. Pk is an orthogonalprojection onto the space N (BT (AT )k−1CT ∏k−1i=1 PkT )), we can have1∏k=iPky(i) =1∏k=iPkCAix, (A.3)where ∏1k=i Pk△= PiPi−1 · · ·P1. Putting the output equation and equation110A.2. Unknown-input observability subspace(A.3) (for i ≤ (n− 1)) together, we haveyP1y˙P2P1y¨...1∏k=n−1Pky(n−1)= Opx (A.4)withOp△=CP1CAP2P1CA2...1∏k=n−1PkCAn−1. (A.5)Theorem 7. The unknown-input observability subspace of (2.1), which itselfis a subset of the observability space of the mentioned system, is a time-invariant spaceOUI△= R(CT )⊕n−1∑i=1R((AT )iCTi∏k=1PkT), (A.6)wherePk = p(N (BT (AT )k−1CTk−1∏i=1PiT ))and p(M) means the projection onto the space M.Proof. By definition, OUI is the largest subspace in which states that span itcan be reconstructed without the knowledge of the input and its derivatives.From (A.4), the states which span the row space of Op can be reconstructed,irrespective of the unknown inputs. Therefore, the unknown-input observ-111A.2. Unknown-input observability subspaceability subspace is the row space of Op,R(OTp)= R(CT)⊕R(ATCTP1T)⊕R((AT )2CTP1TP2T)⊕ · · · ⊕R((AT )n−1CTn−1∏k=1PkT)= R(CT )⊕n−1∑i=1R((AT )iCTi∏k=1PkT).(A.7)Hence, (A.6) is obtained.Since, for any two matrices A andB with compatible dimensions, we haveR(AB) = AR(B) and R(AB) ⊆ R(A), and also from1∏k=iPkCAi−1B = 0,we can obtainR((AT )iCTi∏k=1PkT)⊆[AT (R((AT )i−1CT ) ∩ N (BT ))]. (A.8)From (A.7) and (A.8), we haveR(OpT ) ⊆ R(CT )⊕AT (R(CT ) ∩ N (BT ))⊕AT(R(ATCT ) ∩ N (BT ))⊕ · · · ⊕AT (R((AT )n−2CT ) ∩ N (BT )),(A.9)therefore,OUI ⊆ R(CT )⊕ATn−2∑i=0(R((AT )iCT ) ∩ N (BT ))⊆ R(CT )⊕AT (O ∩N (BT )),(A.10)where O represents the observability subspace of (2.1).As a result of (A.10), we can provide the following remark.112A.3. Observability subspace with limited information about the input and output derivativesRemark . For an unknown-input observable system for which OUI = Rn,R(CT )⊕AT (O ∩N (BT )) = Rn. (A.11)A.3 Observability subspace with limitedinformation about the input and outputderivativesIn this section we obtain an equation that determines how providing moreinformation about input and output derivatives, affects the respective ob-servability subspace of the system. For this purpose w(t) is considered tobe known but with some unknown derivatives.We first obtain the relationship between two observability subspaces, 1)Oi−1: The observability subspace with up to the (i− 1)th input derivativesavailable and 2) Oi: The observability subspace with up to the ith inputderivatives available.Assume that we have information about input and up to its (i − 1)thderivative. So, for the (i + 1)th derivative of the output equation and theprojection matrix P1, we can show thatP1y(i+1) = P1CAi+1x+ P1CAiBw + · · · + P1CABw(i−1). (A.12)Proceeding with higher derivatives, we pre-multiply each equation witha projection matrix (as in the proof of Theorem 7), thenY = Opdx+HW, (A.13)113A.3. Observability subspace with limited information about the input and output derivativeswhere H is a lower-triangular Toeplitz matrices of compatible size andOpd△=[C CA · · · P1CAi+1 · · ·1∏k=n−1−iPkCAn−1]T. (A.14)In (A.13),Y △=[y y˙ · · · P1y(i+1) · · ·1∏k=n−1−iPky(n−1)]TW △=[w w˙ · · · w(i−1)]T.(A.15)The states which span the row-space of Opd can be reconstructed ir-respective of unknown values of input derivatives. Hence, with Oi−1 =R(OpdT ),Oi−1 = R(CT )⊕i∑j=1R((AT )jCT)⊕n−1∑j=i+1R((AT )jCTn−i∏k=1PkT). (A.16)Theorem 8. For (2.1), the observability subspace with up to the ith inputderivatives available isOi = R(CT )⊕ATOi−1O0△= R(CT )⊕ATOUI,(A.17)where O0 is the observability subspace with only 0th input derivative avail-able.Proof. Pre-multiplying P1 in the (i+2)th derivative of the output, eliminatesthe unknown derivative of input. Taking more derivatives and introducingmore projection matrices to remove the unknown values and putting the114A.3. Observability subspace with limited information about the input and output derivativesresults together, we obtain (A.13) withOpd△=CCA...P1CAi+2...1∏k=n−2−iPkCAn−1.(A.18)ThereforeOi = R(CT )⊕i+1∑j=1R((AT )jCT)⊕n−1∑j=i+2R((AT )jCTn−i∏k=1PkT)= R(CT )⊕AT (R(CT )⊕i∑j=1R((AT )jCT)⊕n−1∑j=i+1R((AT )jCTn−i∏k=1PkT))= R(CT )⊕ATOi−1(A.19)and note that the 0th derivative case reduces to the result in Theorem 7.Hence,O0△= R(CT )⊕ATOUI. (A.20)In Figure A.1 we show how providing information about input and itsderivatives can affect the observability subspace of the system. Note thatthe two dimensional representation of subspaces is just a simplification. Ashas been mathematically shown in Theorem 8, by providing informationabout the input and its derivatives, a larger part of the state space can bereconstructed. Note that the largest subspace that can be reconstructed isthe observability subspace, O, for which all information about input, output,115A.3. Observability subspace with limited information about the input and output derivativesFigure A.1: The effect of providing information about input and its deriva-tives on the observability subspace. We could show that adding informationabout input and its derivatives can result in larger observability space. Thedashed-dotted lines represent containment.and their derivatives up to the (n− 1)th derivatives is available.If in addition to limited information about higher input derivatives, theinformation about output derivatives is also limited, a smaller part of thestate space can be reconstructed.Corollary 8. For (2.1), the observability subspace with up to the ith inputderivatives and up to the jth output derivatives available, (with j > i), isOi,j = R(CT )⊕ATOi−1,jO0,j △= R(CT )⊕ATOUI ,j,(A.21)with116A.4. ExampleOUI ,j = R(CT )⊕ATj−1∑i=0(R((AT )iCT ) ∩ N (BT )).Proof. Having limited information about output derivatives will affect (A.6)by changing the upper margin of the summation. On the other hand, therecursive part of equation (A.17) which shows the effect of providing furtherinformation about input derivatives will not be affected. Hence, (A.21) isobtained.Since the ith derivative of input will not show up until taking (i + 1)derivatives from the output equation, it is necessary to have j > i. Other-wise, every i in (A.21) should be replaced by (j − 1).Remark. It is straight forward to show that Theorems 7 and 8 and alsoCorollary 8 will remain the same for a system model (2.1) with both anunknown input (or an input with unknown derivatives) and a known inputwhose derivatives are entirely known.A.4 ExampleConsider the linearized longitudinal dynamics of a Boeing 747 in trimmedlevel flight [121], with state x = [q, V, α, θ, h] consisting of pitch rate q,airspeed V , angle of attack α, pitch angle θ, and altitude h. We consideru(t) = 0 and assume that the user applies the input w(t) = δih(t) whichrepresents the deflection of the horizontal tail. System matrices are providedin (A.24).Consider the case in which the measurement is limited to informationabout the pitch angle, therefore, C = [0 0 0 1 0]. The observable subspaceof this system spans R5, hence, all states are observable. From Theorem 7,the unknown-input observable subspace (A.6) isOUI = span (e1, e4) . (A.22)117A.4. ExampleWe now evaluate how having information about the input and its first twoderivatives but not having information about the third and forth derivativesof input can affect the observable subspace. So, from (A.17)O0 = span (e1, e3, e4)O1 = span (e1, e3, e2 − 0.0086e5, e4)(A.23)and O2 = R5 spans the entire state-space.Now consider a case that the information about the derivatives of outputis also limited. From (A.21), OUI ,0 = OUI. Since from Corollary 8 it isrequired to have j > i, the new observable subspace will span R5 for j ≥ 3.Essentially, when measuring the pitch angle of an aircraft with dynamicsprovided in (A.24), the first two derivatives of input and the first threederivatives of output are enough to reconstruct the entire state space andhigher derivatives of these signals do not provide additional informationabout the states of the system.118A.4.ExampleA =−6.6926 × 10−1 −8.6× 10−6 −8.856 × 10−1 0 −3.45× 10−6−1.6179 × 10−1 −7.588 × 10−3 4.9965 −9.8 4.59 × 10−51.0084 −1.0036 × 10−3 −6.735 × 10−1 0 5.9× 10−61 0 0 0 00 0 −1.338 × 102 1.338 × 102 0,B =[−4.5944 × 10−2 0 −1.912 × 10−3 0 0]T(A.24)119Appendix BOn the effect of higherderivatives on the existenceof and design of functionalobserversIn this appendix we focus on investigating the effects of availability of theextended input and output signals (i.e., higher derivatives of input and out-put signals in this chapter) on the existence of a generic functional observerfor LTI systems.In Section B.2, we derive the existence conditions of a generic functionalobserver of form (B.6) for system (B.1). We provide an estimate of therequired order of the functional observer in B.3, and we suggest a designprocedure for such an observer in Section B.3.1.B.1 Problem formulationThe results provided in this chapter are a specific case of those in Chapter5. Here, we focus specifically on an extension of the model used in [58],x˙(t) = Ax(t) +Bu(t) + αFd(t)y(t) = Cx(t) +G1u(t) + αG2d(t),(B.1)where x(t) ∈ Rn is the state vector, u(t) ∈ Rmu is the known input, d(t) ∈Rmd is the unknown input or the disturbance, y(t) ∈ Rp is the output vectorand the matrices A, B, F , C, G1, and G2 have compatible dimensions. In120B.1. Problem formulation(B.1), α ∈ {0, 1}, where α = 0 for having no unknown inputs and α = 1 forhaving unknown inputs. Instead of introducing the parameter α, F can bealternatively considered as being a zero matrix for systems without unknowninputs.Assuming that we have information about the inputs and the outputs aswell as their derivatives, our goal is to evaluate whether or not there existsa functional observer to reconstruct the linear functionalz0(t) = L0x(t), z0(t) ∈ Rr, (B.2)where L0 ∈ Rr×n.To reduce the required order of the functional observer, we introduce thedesign parameters γ and λ. For the system with γ derivatives of outputsavailable, we redefine the set of outputs as Y0:γ = [yT (t), y˙T (t), · · · , y(γ)T (t)]T ,thusY0:γ(t) = Oγx(t) +M1,0:γU0:γ(t) + αM2,0:γD0:γ(t) (B.3)where Oγ is the observability matrix and M1,0:γ and M2,0:γ are Toeplitzmatrices as follows,Oγ = [CT , ATCT , · · · , AγTCT ]T ,M1,0:γ =G1 0 · · · 0CB G1 · · · 0CAB CB · · · 0... ... . . . ...CAγ−1B CAγ−2B · · · G1,(B.4)M2,0:γ =G2 0 · · · 0CF G2 · · · 0CAF CF · · · G2... ... . . . ...CAγ−1F CAγ−2F · · · G2,U0:γ(t) = [uT (t), u˙T (t), · · · , u(γ)T (t)]T ,D0:γ(t) = [dT (t), d˙T (t), · · · , d(γ)T (t)]T ,121B.1. Problem formulationClearly, for all i ∈ N, we have M1,i:γ+i = M1,0:γ and M2,i:γ+i = M2,0:γ . Wealso can write these matrices in recursive form asM1,0:γ+1 =[G1 0OγB M1,0:γ],M2,0:γ+1 =[G2 0OγF M2,0:γ],(B.5)andU0:γ+1(t) = [UT 0:γ(t), u(γ+1)T (t)]T ,D0:γ+1(t) = [DT 0:γ(t), d(γ+1)T (t)]T ,Based on the extended outputs, Y0:γ , and the extended inputs, U0:λ =[uT (t), u˙T (t), · · · , u(λ)T (t)]T , our goal is to design a functional observer toreconstruct the desired functional, z0(t). Therefore the observer dynamicsisw˙(t) = Nw(t) + JY0:γ +HU0:λ,zˆ(t) = w(t) + EY0:γ ,(B.6)where w(t) ∈ Rr+X . Note that, designing a stable observer is equivalent todetermining matrices J , H, E, and a stable N with compatible dimensions.Similar to what we did in Chapter 5, we also assume that in order toreconstruct the functional z0(t) = L0x(t), it is also necessary to reconstructthe functional Rx(t) where R ∈ RX×n. Hence, we introduce the extendedfunctional asz(t) =[L0R]x(t), (B.7)such that LT = [L0T , RT ]T is of full row rank. For cases that L0x(t) can bereconstructed directly, we have R = ∅.122B.1. Problem formulationChoosing Q , EOγ −L and K = J −NE, we have the estimation errore(t) = zˆ(t)− z(t)= w(t) +EY0:γ(t)− Lx(t)= w(t) +Qx(t) + EM1,0:γU0:γ + αEM2,0:γD0:γ(B.8)with the following dynamicse˙(t) = Ne(t) + (QA+ JOγ −NQ)x(t)+KM1,0:γU0:γ(t) + EM1,1:γ+1U1:γ+1+HU0:λ +QBu(t) + αKM2,0:γD0:γ+EM2,1:γ+1D1:γ+1 + αQFd(t).(B.9)From (B.5), (B.9) can be written ase˙(t) = Ne(t) + (QA+ JOγ −NQ)x(t)+(K[G1Oγ−1B]+QB +Ha)u(t)+(KMa + EM1,0:γ)U1:γ+1 +HbU1:λ+α(K[G2Oγ−1F]+QF )d(t)+α(KMb + EM2,0:γ)D1:γ+1,(B.10)whereMa ,[0 0p×muM1,0:γ−1 0],Mb ,[0 0p×mdM2,0:γ−1 0],(B.11)andHa , H λ = 0,[Ha Hb], H λ > 0. (B.12)123B.1. Problem formulationBy choosing β = max(λ, γ + 1), we can write[M1,1 M1,2], M1,0:γ ,[M2,1 M2,2], Ma,(B.13)where M1,2 ∈ R(γ×p)×((β−λ)×mu) and M2,2 ∈ R(γ×p)×((β−λ)×mu). Note thatfor γ ≤ λ, we have M1,1 , M1,0:γ and M2,1 , Ma. Also, for λ = 0,M1,2 , M1,0:γ and M2,2 , Ma.To have an asymptotically stable observer, from (B.9) and (B.13), thematrix N has to be stable and we need to satisfyEOγA+KOγ = LA−NL,α(EOγF +K[G2Oγ−1F]) = αLF,α(EM2,0:γ +KMb) = 0,(B.14)andEOγB +K[G1Oγ−1B]+Ha = LB,EM1,1 +KM2,1 +Hb = 0, λ > 0.(B.15)In addition to (B.14) and (B.15), for γ > λ, we also need to satisfyEM1,2 +KM2,2 = 0. (B.16)In summary, to design a stable functional observer of form (B.6) for thereconstruction of the functional z0(t) in system (B.1), we do the followingtwo steps:1. We evaluate the existence of such a functional observer by checkingwhether there exists a stable matrix N to satisfy (B.14), (B.15), and(B.16).2. After estimating the required order of the observer, we design thefunctional observer by determining the observer matrices to satisfy124B.2. Existence conditions of a functional observer(B.14), (B.15), and (B.16) for an stable N .It is worth mentioning that the process of taking derivatives from theinput and the output signals can amplify the high frequency componentswhich is undesirable. To reduce the adverse effect of these noisy componentsof the extended signals, low-pass filtering might be needed. As our focus hereis on non-noisy systems and non-noisy derivatives of the signals, determiningan appropriate filtering technique is out of the scope of the current chapter(e.g., in our main application of human-automation interaction, we assumethat the user is capable of evaluating the rate of change of the availablesignals).B.2 Existence conditions of a functional observerFor LTI systems with known and/or unknown inputs and with availablederivatives of inputs and outputs, we obtain the necessary and sufficientconditions for the existence of a functional observer.Recall that we have considered having a full row rank functional Lx(t),with Lx(t) =[L0R]x(t), whose components are L0x(t) and Rx(t). There-fore reconstructing Lx(t) results in the reconstruction of L0x(t).With a similar approach to that used in [58], we can write (B.14) and(B.16) as[E K]T1 = T2, (B.17)whereT1 =OγAE1 αOγF αM2,0:γ M1,2OγE1 α[G2Oγ−1F]αMb M2,2andT2 =[LAE1 αLF 0 0],125B.2. Existence conditions of a functional observerwith E1 being selected so that LE1 = 0.Considering (B.17), this equation has a solution if and only if span(T2T ) ⊆R(T1T ) or equivalentlyrank[T2T1]= rank[T1]. (B.18)Theorem 9. An observer with order (r+X ) exists to reconstruct the func-tional z0 = L0x(t) of the system (B.1) (with γ and λ being the design pa-rameters) iff there exists a matrix R ∈ RX×n for which•rank(LHS1) = rank(RHS) (B.19)• For all s ∈ C,rank(LHS2) = rank(RHS) (B.20)whereLHS1 ,L0A αL0F 0 0RA αRF 0 0OγA αOγF αM2,0:γ M1,2Oγ α[G2Oγ−1F]αMb M2,2L0 0 0R 0 0 0, (B.21)LHS2 ,sL0 − L0A −αL0F 0 0sR−RA −αRF 0 0OγA αOγF αM2,0:γ M1,2Oγ α[G2Oγ−1F]αMb M2,2, (B.22)126B.2. Existence conditions of a functional observerRHS ,OγA αOγF αM2,0:γ M1,2Oγ α[G2Oγ−1F]αMb M2,2L0 0 0 0R 0 0 0. (B.23)Proof. • Proof of (B.19).By selecting H1 to satisfy LH1 = I and post multiplying the full rowrank matrixS1 =H1 E1 0 0 00 0 I 0 00 0 0 I 00 0 0 0 Iin (B.21) and (B.23), we can show that satisfaction of (B.19) for a ma-trix R of rank X is equivalent to satisfaction of (B.18) and guaranteesexistence of solution for (B.14) and (B.16).• Proof of (B.20).Assume that condition (B.19) is satisfied, then[E K]= T2T1+ + Z(I − T1T1+) (B.24)are the solutions of (B.17) for arbitrary matrix Z with compatibledimensions. Note that when (I − T1T1+) = 0, matrices E and K arenot affected by Z and can be uniquely obtained as[E K]= T2T1+.It is also required to satisfy the stability of the observer (B.6). We canrewrite the first equation in (B.14) as follows:N = LAH1 −[E K][OγAOγ]H1. (B.25)127B.2. Existence conditions of a functional observerFrom (B.24) and (B.25), we can have N = Λ− ZΓ whereΛ = LAH1 − T2T1+[OγAOγ]H1,Γ = (I − T1T1+)[OγAOγ]H1.(B.26)Clearly the eigenvalues of N can be placed at any desired values iffrank[sI − ΛΓ]= r + X , ∀s ∈ C.By post-multiplying S1 in (B.22), we havesI − LAH1 −T2[OγAH1OγH1]T1. (B.27)Now choose a S2 with full column rankS2 =I T2T1+0 (I − T1T1+)0 T1T1+(B.28)and a S3 with full row rankS3 =I 0−T1+[OγAOγ]H1 I. (B.29)By pre-multiplying S2 and post-multiplying S3 in (B.27), the rank128B.2. Existence conditions of a functional observerdoes not change and we can finally obtain rank(LHS2) , Ψ1 to beranksL0 − L0A −αL0F 0 0sR−RA −αRF 0 0OγA αOγF αM2,0:γ M1,2Oγ α[G2Oγ−1F]αMb M2,2, (B.30)whereΨ1 , rank[sI − ΛΓ]+ rank(T1). (B.31)Similar to the proof of (B.19), if we post-multiply S1 in (B.23), we canshow that its rank isrankOγA αOγF αM2,0:γ M1,2Oγ α[G2Oγ−1F]αMb M2,2L0 0 0 0R 0 0 0= Ψ2, (B.32)whereΨ2 , r + X + rank(T1). (B.33)From (B.30) and (B.32) we have rank[sI − ΛΓ]= r + X iff (B.20)is satisfied.We can show that the results of Theorems in [49] and [58] can be easilyobtained from Theorem 9 by having α = 0 and α = 1 respectively, withγ = 0 and λ = 0.129B.2. Existence conditions of a functional observerTheorem 9 requires finding a matrix R with a low rank (with unknownnumber of rows) which satisfies (B.19) and (B.20) simultaneously. The ma-trix R of lowest rank can be determined through an iterative algorithm, aspresented in Algorithm 1.Algorithm 1 Determining matrix R1: if (B.19) and (B.20) are satisfied with R = 0¯ then2: R = 0¯3: else4: DefineR = {{ei}|i ∈ {1, ..., n} & span(ei) * span(L0T )}and j = 1.5: Rj = {v|span(v) ⊆ span(R) & rank(v) = j}.6: Obtain R such that(L0, R) = {(L0, w)|w ∈ Rj & (L0, w) satisfy (B.19)&(B.20)}.7: if R = ∅ and j < n then8: j = j + 19: go to 5.10: end if11: end ifAs computing R for systems of high order can be numerically hard, wesuggest a sufficient (but not necessary) condition in Proposition 5 belowfor the existence of a functional observer for system (B.1) which does notdepend on the unknown matrix R.To obtain Proposition 5, we first explain a few notations and terms. Wehave[Y1 Y2], with Y1 having n columns, to be the transpose of the basisof Nα[FOγF]T.We calculate YN and YM such that the rows of [YN |YM ] are selected from130B.2. Existence conditions of a functional observerthe rows of [Y1|Y2] andspan(YNT ) ⊆ N([L0POγ]),span(YMT ) ⊆ N ([αM2,0:γ M1,2]T),(B.34)where P = P2P1. Matrices P1 and P2 are projection matrices onto theleft null space of M1,0:γ and αP1M2,0:γ respectively. Note that PY0:γ(t) =POγx(t) is the combination of outputs with no dependency on the inputs.Also we choose M such thatspan(MT ) ⊆ N ([αM2,0:γ M1,2]T)and to satisfyα(L0F +MOγF ) = 0. (B.35)We choose R = KYN , where K is an identity matrix with compatibledimension. Equation (B.34) shows that rows of R are selected to be linearlyindependent from the functional L0x(t) and from POγ – that is, the availablestates which have no dependency on inputs. From (B.35), the rank of (B.21)can be written asrank(LHS1) = rankL0A+MOγAKYNA+KYMOγA0OγAOγTL0KYN0, (B.36)131B.2. Existence conditions of a functional observerwhereT =αOγF αM2,0:γ M1,2α[G2Oγ−1F]αMb M2,2. (B.37)We choose v = [v1 v2]T to be a full row rank matrix such that v1T hasfull row rank equal to the rank of T and v2T = 0 (e.g. v1 = T+ whereT+ is the pseudo-inverse of matrix T and span(v2) = N (T T )). Hence, from(B.36),rank(LHS1) = rankv1[OγAOγ]v1Tv2[OγAOγ]L0A+MOγAKYNA+KYMOγAL0KYN0= q1 + rank(T ),(B.38)whereq1 = rankv2[OγAOγ]L0A+MOγAKYNA+KYMOγAL0KYN. (B.39)132B.2. Existence conditions of a functional observerHaving similar v1 and v2, for (B.23) we haverank(RHS) = rankv1[OγAOγ]v1Tv2[OγAOγ]L0KYN0= q2 + rank(T ), (B.40)whereq2 = rankv2[OγAOγ]L0KYN. (B.41)Proposition 5. For system (B.1), a functional observer of form (B.6) existsto reconstruct the functional L0x(t) if• Condition 1:rankL0A+MOγAYNA+ YMOγAv2[OγAOγ]L0YN= rankv2[OγAOγ]L0YN, (B.42)133B.2. Existence conditions of a functional observer• Condition 2: For all s ∈ C,rankv2[OγAOγ]L0(sI −A)−MOγAYN (sI −A)− YMOγA= rankv2[OγAOγ]L0YN. (B.43)with v1, v2, P , M , YN , and YM already defined.Proof. • Proof of (B.42).Earlier we have proved (B.19). We have also shown that for R = KYN ,equation (B.38) is an equivalent to the rank of (B.21) and equation(B.40) is equivalent to the rank of (B.23). Hence, for the above R,(B.19) and the condition q1 = q2 are equivalent.Through proof with contrapositive, we now assume that there existsno K such that (B.19) can be satisfied when R = KYN , therefore,q1 6= q2|∀K . However, we can clearly see that for K = I, q1 = q2.Thus, there exists a matrix R, not exclusively R = YN , that satisfies(B.19) if (B.42) is satisfied.• Proof of (B.43).The rank of (B.22) can be written asrankv1[OγAOγ]v1Tv2[OγAOγ]L0(sI −A)−MOγAKYN (sI −A)−KYMOγA0, (B.44)134B.3. Estimating the order of the functional observerhence,rank(LHS2) = q3 + rank(T ),∀s ∈ C,(B.45)whereq3 = rankv2[OγAOγ]L0(sI −A)−MOγAKYN(sI −A)−KYMOγA. (B.46)We have already obtained (B.23) to be equal to (B.40). Hence, from(B.40) and (B.45), finding a matrix R to satisfy (B.20) is equivalent tofinding a K for which (q3 = q2) for all (s ∈ C). Satisfaction of (B.43)is essentially equivalent to having (q3 = q2)|K=I for all (s ∈ C).From both parts of the proof, we can show that (B.42) and (B.43) are suf-ficient for the existence of a functional observer (B.6) to reconstruct L0x(t)in system (B.1).Remark. Note that, if YN is a zero matrix, there exists no linearly indepen-dent combinations of the states of the system to be added to L0. Therefore,the existence problem of a state reconstructor is reduced to existence of areconstructor with the same order as the desired functional z0(t).B.3 Estimating the order of the functionalobserverIt is clear that the rank of (B.23) is always smaller than or equal to therank of (B.21), i.e., q2 ≤ q1 for all K. To estimate the required order of theunknown-input functional observer, we first need to determine a matrix Kasuch that, for K = Ka, the rank of (B.23) is equal to q1|K=Ka + rank(T ),where q1|K=Ka + rank(T ) is the rank of (B.21) for K = Ka. Equivalently,135B.3. Estimating the order of the functional observerfrom (B.41) and the desired condition q1 = q2, we need to find a matrix Kasuch thatrankv2[OγAOγ]L0KaYN= q1|K=Ka. (B.47)In order to simplify (B.47), we choose Ka such thatYNTR(KaT ) ⊆ N (v2[OγAOγ]). (B.48)Thus, we haveXa , rank(KaYN )= q1|K=Ka + rank(T )− rank(v2[OγAOγ]).(B.49)As q1 ≤ q for q , (q1|K=I), to satisfy q1 = q2, we need to have Ka ∈RXa×nY N , where nY N is the number of rows of YN , such thatXa ≤ q + rank(T )− rank(v2[OγAOγ]). (B.50)Equation (B.50) provides an upper bound for the number of rows tobe added to the desired functional so that the extended functional satisfies(B.19).On the other hand, if we find a matrix Ka for which R = KaYN satisfiesq1 = q2, we still need to satisfy q2 = q3 . For this purpose, we need to havea matrix Kb ∈ RXb×n whereXb , q2|K=Ka,b + rank(T )− mins∈eigΛ (rank(Cp))≤ q + rank(T )− mins∈eigΛ(rank(Cp)) ,(B.51)136B.3. Estimating the order of the functional observerwhereKa,b =[KaKb](B.52)andCp =sL0 − L0A αL0F 0 0sKaYN −KaYNA αKaYNF 0 0OγA αOγF αM2,0:γ M1,2Oγ α[G2Oγ−1F]αMb M2,2. (B.53)To estimate the minimum order of the unknown-input functional ob-server, required for the reconstruction of L0x(t), we suggest the followingsteps:1. From (B.50), obtain Xa and define L1 to be a set of all feasible (L1 ,KaYN ) ∈ R(r+Xa)×n whose corresponding Ka’s satisfy q1 = q2.2. FindMm = maxL1∈L1(mins∈eigΛ(rank(Cp)))(B.54)and denote the corresponding optimal solution as Lm.3. The required order of the observer is thereforeX = Xa +Xm, (B.55)with Xm ≤ q + rank(T )−Mm.4. The functional to be reconstructed isL =[LmKmYN], (B.56)137B.3. Estimating the order of the functional observerwith Km ∈ RXm×rows of YN being selected so that K =[KaKm]sat-isfies q1 = q3.We can now introduce Proposition 6 as an alternative of Theorem 9 insubspace form.Proposition 6. An observer with order (r+X ) exists to reconstruct L0x(t)in (B.1) (with γ and λ being the design parameters) iff• a matrix Ka exists to satisfyr1TR(K1T ) ∩ N (Pc) = rLTR(K1T ) ∩ N (Pc) (B.57)whereK1 =[I 00 Ka], Ka ∈ RXa×n (B.58)and• for all s ∈ C, a matrix Kb exists to satisfyr2TR(K2T ) ∩N (Pc) = rLTR(K2T ) ∩ N (Pc), (B.59)whereK2 =[K1 00 Kb], Kb ∈ RXb×n, (B.60)138B.3. Estimating the order of the functional observerandr1 =[L0A+MOγAYNA+ YMOγA],r2 =[L0(sI −A) +MOγAYN (sI −A) + YMOγA],rL =[L0YN],Pc = v2[OγAOγ].Proof. We first consider R = KaYN and obtain Ka to satisfy q1 = q2, from(B.39) and (B.41). Then, we modify R =[KaT KbT]Tto also satisfythe other condition – that is, q1 = q3, from (B.39) and (B.46). The rest ofthe proof is straightforward, therefore, the details are omitted.B.3.1 Design procedure of the functional observer (B.6)Considering the system (B.1) and the observer (B.6), we can determine therelated observer matrices N , J , H, and E as follows:• Estimate L from (B.56).• Choose H1 and E1 such that LH1 = I and LE1 = 0 respectively.• Calculate Λ and Γ from (B.26).• Choose Z to result in a stable N = Λ−ZΓ and obtain E and K from(B.24).• Based on N , E, and K, calculate J from J = K −NE.• Calculate H from (B.15).139B.4. ExamplesB.4 ExamplesWe consider two examples. In the first example, we use the proposed methodto evaluate the existence condition and then design a “0-th derivative avail-able” functional observer for a third-order system. In the second example,we show how the order of a functional observer (designed for the recon-struction of a common functional) decreases when higher derivatives of theinput and/or the output signals are available. We then discuss a real-worldapplication of such a functional observer.B.4.1 Example 1Consider a third order continuous-time system with the following dynamics,A =1 1 01 1 00 1 1, F =001C =[1 0 0], L0 =[0 1 0],(B.61)and assume that the system has an unknown input (i.e. α = 1) and noinformation about the derivatives of the inputs and outputs is available –that is, γ = 0 and λ = 0. Hence, M1,0:γ , M2,0:γ are zero matrices, P is anidentity matrix, and Oγ = C.For the above system, Theorem 1 in [58] is satisfied. While withrankCA CFC 0L0 0Y1 0= rank1 1 0 01 0 0 00 1 0 01 0 0 00 1 0 0= 2, (B.62)since, n+ rank(CF ) = 3, the first Theorem in [59] (i.e. Theorem 4.6) is notsatisfied. Therefore, based on [59], there does not exist a functional observerfor this system to reconstruct L0x(t).However, based on Proposition 5 and with having YNx(t) to be an empty140B.4. Examplesset, we can show that there exists an unknown-input functional observer withorder 1 that can reconstruct L0x(t).According to Section B.3.1, we can obtain Λ = −0.5 and Γ =[0.50.5].Hence, choosing Z = [6 1] will result in a 1st-order observerw˙ = −4w − 8y + u,zˆ = w + 3y.(B.63)Note that the matrix Z is not unique.0 1 2 3 4 5 6 7 8− vs. Estimated functional(a) The actual functional presented by the solid line vs. the estimated functionalpresented by the dashed-dotted line0 1 2 3 4 5 6 7 800. error(b) Estimation error of the desired functionalFigure B.1: The error of estimating the functional z0(t) of (B.62) with theobserver (B.63)141B.4. ExamplesFigure B.1 presents the performance of the designed unknown-inputfunctional observer in reconstructing the desired functional, L0x(t). It isclear that the designed first-order observer (B.63) is capable of estimatingL0x(t) with zero steady-state error.B.4.2 Example 2As in [59], we consider the unstable continuous-time unknown-input system,α = 1, with dynamics provided in (B.64).A =−0.0226 −36.617 −18.897 −32.09 3.2509 −0.76260.0001 −1.8997 0.9831 −0.0007 −0.1708 −0.0050.0123 11.72 −2.6316 0.0009 −31.604 22.3960 0 1 0 0 00 0 0 0 −30 00 0 0 0 0 −30,B =[0 0 0 0 0 300 0 0 0 30 0]T, F =[0 0 0 0 1 −1]T(B.64)The output of the system isC =[0 1 0 0 0 00 0 0 1 0 0](B.65)and the goal is to reconstruct the functional L0x =[0 1 1 1 1 1]x(t).Fernando et.al. [59] showed that there exists a third order observer to re-construct the desired functional of the states of the mentioned system. Ourgoal is to investigate whether having a observer with access to the deriva-tives of the input and output signals can be helpful to reduce the requiredorder of the observer. We, hence, consider an observer with access to thefirst derivative of the output with γ = 1 and λ = 0.142B.4. ExamplesTo check the conditions in Proposition 5, we obtainYN =[1 0 0 0 0.0006 −0.0006],YM =[0 0 0 0],M =[0 0 11.3766 0],v2 =[e1T e2T e4T e5T e6T e7T e8T]T.(B.66)It can be seen that the conditions in Proposition 5 are satisfied. Theright hand side and the left hand side of (B.42) are both of rank 6. Besides,(B.43) is not rank deficient for s being selected to be the eigenvalues ofΛ|L=L0 . Hence, there exists a functional observer with access to the firstderivative of the output to reconstruct L0x(t) for this system.On the other hand, it is straight forward to show that Theorem 9 is notsatisfied for R = ∅, as (B.20) is rank deficient for s being selected to bethe eigenvalues of Λ|L=L0 . Hence, having information about derivatives ofoutputs, still the desired functional cannot be estimated directly.To obtain the required order of the functional observer, from (B.50),we obtain Xa = 2 (i.e., at most two additional rows are needed to satisfy(B.19)). However, with YN having only one row, the only extension tothe desired functional can be L =[L0YN]and we can show that for theextended functional, conditions in Theorem 9 are satisfied. Hence, a secondorder functional observer with access to the first derivative of the outputexists to reconstruct the functional L0x(t) of the system (B.64).Using the design procedure provided in B.3.1, we can design a stablefunctional observer with poles being placed at −3 and −1. We therefore143B.4. Examples0 1 2 3 4 5 6 7 8−50510Estimation error of L0xTime (sec.)(a) Estimation error of the desired functional, L0x(t)0 1 2 3 4 5 6 7 8−600−400−2000Estimation error of RxTime (sec.)(b) Estimation error of the complementary functional, Rx(t)Figure B.2: The error of estimating the functionals of system (B.64) withthe observer (B.67)obtainN =[−4.2103 0.0125−311.9616 0.2103],E =[−0.0061 −0.0029 −0.0114 0.0111−2.1213 2.2491 −0.0000 0.0130]× 103,J =[0.0320 0.0445 0.0418 −0.0496−6.4680 1.6618 1.4284 −1.2149]× 103,H =[−28.3051 28.30510.0167 −0.0167].(B.67)Having the above observer, the error dynamics in equation (B.9) are144B.4. Examplesmodeled. Figures B.2 shows that the estimation error of the desired func-tional, L0x(t), and the estimation error of Rx|R=YN asymptotically approachzero, hence, the observer makes correct estimations.145Appendix CList of publicationsJournal articlesN. Eskandari, G. A. Dumont, and Z. J. Wang, “Delay-incorporating ob-servability and predictability analysis of safety-critical continuous-time sys-tems,” IET Control Theory and Applications, vol. 9, pp. 1692–1699(7), July2015.N. Eskandari, G. A. Dumont, and Z. J. Wang, “An observer/predictorbased model of the user for attaining situation awareness,” IEEE Transac-tions on Human-Machine Systems, 2015.N. Eskandari, Z. J. Wang, and G. A. Dumont, “On the existence anddesign of functional observers for LTI systems, with application to usermodeling,” Asian Journal of Control, 2014.N. Eskandari, M. M. Oishi, and Z. J. Wang, “Observability analysis ofcontinuous-time LTI systems with limited derivative data,” Asian Journalof Control, vol. 16, no. 2, pp. 623–627, 2014.N. Eskandari, G. A. Dumont, and Z. J. Wang. ‘Bounded-error delayedfunctional observer/predictor: Existence and design,” submitted.Conference papersN. Eskandari, Z. J. Wang, and G. Dumont, “Modeling the user as an ob-server to determine display information requirements,” in In Proceedings ofIEEE International Conference on Systems, Man, and Cybernetics, Manch-ester, UK, October 2013, pp. 267–272.N. Eskandari, and M. Oishi, “Computing observable and predictable sub-spaces to evaluate user-interfaces of LTI systems under shared control,”146Book Chaptersin Proceedings of the IEEE conference on Systems, Man and Cybernetics,Alaska, USA, October 2011, pp. 2803–2808.Book ChaptersR. Cortez, D. Tolic, I. Palunko, N. Eskandari, R. Fierro, M. Oishi, andJ. Wood, A hybrid framework for user-guided prioritized search and adaptivetracking of maneuvering targets via cooperative UAVs, In Advances in Intel-ligent and Autonomous Aerospace Systems, in the Progress in Aeronauticsand Astronautics Series American Institute of Aeronautics and Astronau-tics, Reston, VA, vol. 241, 2012147


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