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Modeling and stability analysis of structurally-varying dynamic systems with application to mechanical… Peng, Jie 1993

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MODELING AND STABILITY ANALYSIS OF STRUCTURALLY-VARYINGDYNAMIC SYSTEMS WITH APPLICATION TO MECHANICALPROCESSINGByJie PengB.S. (Electrical Engineering), Shanghai Jiao Tong University, 1982M.A.Sc. (Mechanical Engineering), University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1993© Jie Peng, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of Mect) cm■ cal Ev13'1 ek,f iv\`The University of British ColumbiaVancouver, CanadaDate  0 -& heYB, t993DE-6 (2/88)AbstractWith the ever-increasing sophistication of engineering practices, more time-variant andstructurally-varying dynamic systems are required to accomplish various demandingtasks. The research on various aspects of the dynamics of these complex systems istherefore becoming increasingly active. The current project, which is the modeling andstability analysis of structurally-varying dynamic system, is initiated in such a context.A structurally-varying system is defined as a dynamic system that consists of a number ofstructural subsystems and are interconnected together through a finite set of constraints,which are time varying. The dynamics of the overall system depends on not only the dy-namics of the subsystems but also the interconnections between them. The focus of thisresearch is the stability analysis of linear, structurally-varying dynamic systems. Sincethe system has a time-varying structure, the stability condition of the system is gener-ally changing with time (or more accurately with the constraints between subsystems).For different situations, test criteria for evaluating the stability of structurally-varyingsystems are developed. The relationship between the system stability and time-varyingconstraints is investigated. The relationship of the subsystem dynamics and the overallsystem dynamics is also studied. The main purposes of this study can be summarized asfollows: first is to evaluate the stability of the system under certain constraints, and sec-ond is to deliberately change these constraints according to a set of desired criteria (suchas change the constraints of the system in order to stabilize it) to maintain the systemwithin a desired operating region, which may serve design and development purposes.iiTable of ContentsAbstract^ iiTable of Contents^ iiiList of Figures viAcknowledgement^ viii1 Introduction 11.1 Literature Review ^ 11.1.1^Modeling of Large-Scale Dynamic Systems ^ 11.1.2^Stability and Stabilization of Dynamic Systems 21.2 Objectives of the Proposed Research ^ 71.3 Motivations for the Proposed Research 92 Basic Concepts and Definitions 112.1 Structurally-Varying Systems ^ 112.2 Review of Concepts of Stability of Dynamic Systems ^ 152.3 Definition of Concepts of Stability for An SVS 172.4 Summary ^ 233 Stability of Structurally-Varying Systems With Fixed Order 243.1 Introduction ^ 243.2 Modeling of Switching Instants ^ 24iii3.3 State Space Approach ^ 273.3.1^Recursive State Space Model of a Structurally-Varying System 273.3.2^Analysis of Static Stability of an SVS ^ 353.3.3^Analysis of Dynamic Stability Via State Space Model ^ 423.3.4^Recursive Algorithm for Estimation of -y i (Ai, At) 453.4 Energy Function Approach ^ 493.4.1^Analysis of Dynamic Stability Via Energy Function ^ 493.5 Summary 544 Stability of Structurally-Varying Systems With Time-Varying Order 564.1 Introduction ^ 564.2 Energy Function Approach ^ 574.3 Modeling of Structural Perturbations ^ 594.4 Modeling of Switching Instants 654.4.1^Modeling of Switching Instants by Process Compatibility ^ 664.4.2^Modeling of Switching Instants by Motion Compatibility ^ 734.5 Analysis of Dynamic Stability Using Process Compatibility ^ 774.5.1^Perturbation on Kinetic Energy Function ^ 854.5.2^Perturbation on Potential Energy Function 874.5.3^Experimental Study of Dynamic Stability Using Process Compat-ibility ^ 954.6 Analysis of Dynamic Stability Using Motion Compatibility ^ 1014.7 Issues of Dynamic Control of the SVS ^ 1074.8 Summary ^ 1115 Concluding Remarks 112ivNomenclature^ 115Bibliography 117Appendices^ 122A Norms and Inner Product^ 122B The Solution of X(i) = Ax(t) + Bu(t)^ 124C Matrix Measure^ 125D Properties of Symmetric Matrices^ 127vList of Figures2.1 An Example of a Fixed-Order SVS ^  132.2 An Example of a Varying-Order SVS  142.3 Diagram of Structure of the SVS ^  152.4 Example of Dynamic Stability and Instability of an SVS ^ 223.1 Illustration of Switching Instants  ^263.2 Example of a fixed order SVS ^  314.1 Energy Function ^  594.2 Example of A Varying-Order SVS ^  624.3 Structural Variation of the Example System ^  634.4 Illustration of A Breaking Instant (1)  674.5 Schematic Diagram of the Order-Varying SVS ^  714.6 Schematics of the SVS at the Breaking Instance  724.7 Illustration of A Breaking Instant (2) ^  744.8 Definition of the Coordinates  794.9 Example System ^  904.10 Result of Numerical Simulation: A Dynamic Stable Case Using ProcessCompatibility ^  964.11 Result of Numerical Simulation: A Dynamic Unstable Case Using ProcessCompatibility ^  974.12 A Moving Mechanical Device ^  984.13 Image Processing System  99vi4.14 Experimental Result: Position Profile ^  1014.15 Experimental Result: Velocity Profile  1024.16 Experimental Result: Energy Profile ^  1034.17 Result of Numerical Simulation: A Dynamic Stable Case using MotionCompatibility ^  106viiAcknowledgementI am grateful to a number of people for their assistance during the whole period of theresearch and the writing of this thesis. First and foremost I must thank my supervisorDr. C.W. de Silva for his continuous support and advice during my graduate studiesat UBC. Also, I would like to thank the research supervisory committee members, Dr.Cherchas, Dr. Dunwoody in the mechanical engineering department and Dr. Ma in theelectrical engineering department. Special thanks goes to my wife Wei for her unreservedsupport, understanding and encouragement throughout these years. Without them, it isimpossible to finish this work.This work is supported by grants to Dr. C.W. de Silva from the Natural Sciences andEngineering Research Council for a NSERC Research Chair in Industrial Automationand a research grant (No. 91-5867).viiiChapter 1Introduction1.1 Literature Review1.1.1 Modeling of Large-Scale Dynamic SystemsSince the early 1960s, the field of dynamic modeling and analysis of large complex struc-tures has been a very active research area [Hurty, 1971], [Meirovitch, 1980]. A number ofimportant techniques, such as component-mode synthesis (CMS) [Hurty, 1965], branch-mode analysis (BMA) [Gladwel1,1964], and component-mode substitution [Hurty, 1971],have been developed for that purpose and gradually improved over the past severaldecades. These techniques, in common, make use of the information collected from thesubstructure (sometimes called component) analysis to study the overall structure andthey are closely related to experimental modal analysis described in [de Silva, 1984]. Thebasic idea is to treat a complex structure as an assemblage of connected substructures, orcomponents. Each subsystem is analyzed or treated separately to derive an appropriatedynamic model. Then the dynamic model for the overall connected structure is formu-lated on the basis of the individual substructure models using the constraint conditionswhich are derived from the connections among the substructures. A considerable amountof literature on these techniques is available, but they have almost exclusively consideredsystems with time-invariant constraints, due to the fact that the techniques were orig-inally developed for the modal analysis of large systems with time-invariant structures[Gladwel1,1964], [Hurty, 1965], [Smet et al, 1989]. Lately, Peng and de Silva extended1Chapter 1. Introduction^ 2the CMS technique to systems with time-varying constraints [Peng and de Silva,1991].This decomposition-aggregation approach is also widely used by electrical engineers[Bondi etc., 1980], [Hsu etc., 1980], [Vidyasagar, 1981] in the study of the theory of dy-namic systems. Instead of direct analysis of the whole system, analysis is carried out on anaggregate model which consists of subsystems and interconnections [Michailesco, 1980],[Lunze, 1985]. This actually brings about conceptual simplifications because the dynamicanalysis of the subsystems, which usually have lower order, is often simpler. Compre-hensive study of the dynamics of large scale, interconnected systems can be found in[Siljak, 1978], [Vidyasagar, 1981] and [GrujiC,1987].1.1.2 Stability and Stabilization of Dynamic SystemsAs a very important characteristic of dynamic systems, stability has been another ac-tive research area for centuries. Early work of the stability of mechanical systems datedback to the eighteenth century when Euler considered the eigenvalue problem of col-umn buckling. Since this pioneering groundwork, a great number of mathematicians andengineers have been working on this fascinating subject for generations. The stabilitytheory, as Leipholz pointed out, has experienced a dramatic development toward a cer-tain degree of perfection [Leipholz, 1987]. Numerous approaches have been developed forthe determination of stability of various dynamic systems. Through the investigation ofvibration of a dynamic system, in 1788 Lagrange demonstrated a theory of stability forvibratory motion of mechanical systems about an equilibrium position [Lagrange, 1788].Energy criteria were used to determine the stability of the equilibrium positions. Routhlater extended this method to the stability analysis of perturbed motions. He tried toapply the energy criteria to the investigation of the stability of states of motion. Themost prominent theory of stability in the nineteenth century was developed by A.M. Lia-punov in his famous doctoral dissertation published in 1892 [Liapunov, 1949]. LiapunovChapter 1. Introduction^ 3attempted to establish a stability theory for general motions. Among the numerouscontributions, the most notable one is his second method, which is also called the Li-apunov Direct Method [LaSalle and Lefschetz, 1961]. It provides a method to solve thestability problem of state differential equation of a system without actually solving theequation. Information about the stability property of the system is deduced directlyfrom its model, although the stability definition is phrased in terms of the system mo-tions. Inspired by Liapunov's work in the late nineteenth century, Poincare startedqualitative analysis of nonlinear differential equations as a result of his investigation oforbital stability. Many good books are available today on the classic theory of stabil-ity such as [Bellman, 1953], [LaSalle and Lefschetz, 1961], [Hahn, 1967], [Leipholz, 1987]and [Vidyasagar, 1993], which provide a thorough, comprehensive study of stability ofdynamic systems.For the analysis of the stability of dynamic systems, there are several methods whichare often used. The Liapunov function method [LaSalle and Lefschetz, 1961] is one of themost widely used methods of stability analysis of dynamic systems. The broadness ofthis principle constitutes a difficult, often impossible task: finding of a Liapunov func-tion. The method itself does not suggest a way to construct the Liapunov function.One has to analyze a variety of trial functions for the kind of dynamic systems thatare under investigation in order to determine a Liapunov function, which is one of thedrawbacks of the Liapunov method. In spite of the drawbacks, this method is a verypopular approach for the stability analysis of linear, time-varying systems. New theo-ries for stability analysis of time-varying systems are actively being pursued by manyresearchers [Ljung, 1982], [Kosut and Anderson, 1985], [Wittenmark, 1990]. The Com-parison method [Grujie,1987] is a similar method and is basically an extension of theLiapunov function method. A comparison function has to be constructed. Although itis generally not easy to find a comparison function, there are several common ways toChapter 1. Introduction^ 4initiate an attempt. After a comparison function is constructed, the stability analysis iscarried out based on it. From the analysis of the comparison function, a sufficient con-dition for stability of the original system can be determined. Also as a general theorem,small gain theory gives sufficient conditions of bounded-input-bounded-output (BIBO)stability of dynamic systems [Desoer and Vidyasagar, 1975]. The Perturbation method isanother widely used method. The stability of a perturbed system is based on the stabilityof its unperturbed counterpart and perturbation properties.The recent trend of stability analysis has been on finding a qualitative measure of asystem stability property for various dynamic systems [Lunze, 1988]. Topological andfunctional analytical methods for the treatment of the operator equation have beenused particularly in the development of stability theories, such as L p-space method[Vidyasagar, 1981], in which the formation of the dynamic system description and thedefinition of system stability are based on the concepts of operator and mapping in thelinear space theory. Very recently, a new algebraic factorization approach is innovated byYoula [Youla, 1976] and Vidyasagar [Vidyasagar, 1985]. The latest Ho. optimal controltheory is also developed based on the concept of factorization [Francis, 1987]. The centralidea of the so-called "factorization" approach is that of "factorizing" the transfer matrixof a system as the "ratio" of two stable rational matrices. Based on this factored systemtransfer matrix, a simple parameterization of all compensators that stabilize a given plantcan be obtained. One could then, in principle, choose the best compensator for variousapplications [Youla, 1976]. The factorization approach is a very general framework, whichencompasses continuous-time systems as well as discrete-time systems, lumped as well asdistributed systems, one-dimensional as well as multidimensional systems. However, thefactorization approach is a computation-intensive approach. For systems which have atime-varying structure, required "refactorizing" in real-time may limit its application.Robust control is another new approach for stabilization and control of uncertainChapter 1. Introduction^ 5systems with unknown, sometimes time-varying uncertainty. To date, the field of robustcontrol is still in a stage of intensive research [Lunze, 1988]. Numerous criteria have beenderived to characterize the uncertainty such that the stability is guaranteed if the criteriaare satisfied. A robust controller can sometimes be designed to stabilize the system fora given uncertainty bound [Jabbari, 1991]. The literature concerning this problem isquite extensive [Lunze, 1988], [Qu and Dorsey, 1991], [Olas, 1991], [Haddad etc., 1992],and [Bauer, 1992]. Jabbari [Jabbari, 1991] developed a state feedback controller based onthe Lyapunov technique. The time domain framework is preserved along with the abilityto readily incorporate the time-varying uncertainty. The uncertainty, which is describedas a perturbation to the state space model of the system, is assumed to satisfy certainmatching conditions. Robust control without the matching conditions has been studiedrecently by Chen [Chen, 1990], Qu and Dorsey [Qu and Dorsey, 1991]. It is shown in theirwork that a general control law can be designed to guarantee the stability of the uncertainsystem if the nominal system can be stabilized with an arbitrarily large convergence rate.The Riccati approach is another widely used method in robust control [Petersen, 1986],[Schmitendorf, 1988], in which the bound of the uncertainties does not enter explicitlyinto the control scheme but appears implicitly in an associated Riccati equation forsolution of the feedback control gain. Instead of the matching conditions, the uncertaintyfunctions are assumed to be linear combinations of unknown parameter variations withconstant bounds and weighting matrices. In the development of a state feedback controllaw, prior knowledge of the structure of the uncertainties is used. The Liapunov andRiccati equation approaches have been shown to be very effective in analysis and synthesisof the systems. Much research has been done via these tools especially for robust stabilityand stabilization for finite dimensional time invariant systems.For linear systems with varying structure (which may include variations of both sys-tem parameters and system order), the usual way to estimate the parameters is throughChapter 1. Introduction^ 6some kind of estimation algorithm. Usually a certain form of the system model is as-sumed [Niu etc, 1982], [Guo etc., 1982]. If information is not sufficient in order to assumea reasonably good model, an artificially selected black-box parametrization for a systemis sometimes used [Ljung, 1982]. In this case, the system is parameterized according toinput-output properties instead of physical insight [Guo etc., 1982]. In some sense, it isa pure mathematical practice and system physics is either completely neglected, or notused. In some applications, not only the system parameters, but also the system orderhave to be identified simultaneously [Niu etc, 1982]. Usually a great deal of computa-tion time is required. Recently, a different approach is adopted by Peng and de Silvain the stability analysis of systems with time-varying structure [Peng and de Silva,1993],[Peng and de Silva,1992]. Based on known subsystem models and constraints betweenthem, the model of the overall system can be updated in real-time by a recursive algo-rithm. The stability of the system can then be determined. General discussion on dy-namics and stability analysis of adaptive control systems can be found in [Egardt, 1979],[Anderson etc., 1986] and [Astrom and Wittenmark, 1990].In the stability analysis of mechanical systems, Walker and Schmitendorf proposedan approach to evaluate the stability of a linear, time-invariant system without actuallysolving the equation of motion [Walker and Schmitendorf, 1973]. The asymptotic stabilityof a mechanical system is determined by evaluating the rank of a special evaluationmatrix constructed from the parameter matrices of the system. The stability of systemswith uncertain, linear and time-varying parameter perturbations was studied by Chenand Hsu [Chen, 1988]. Sufficient stability conditions for such systems are derived byusing the possible bound of the perturbation in conjunction with the classical Liapunovapproach. More recently, Lin [Lin et al., 1991] studied the stability of a system subjectedto parameter perturbations and model uncertainties. Asymptotic stability and bounded-input-bounded-output (BIBO) stability for a class of lumped-parameter systems underChapter 1. Introduction^ 7nonlinear time-varying perturbations are analyzed. The stability analysis is carried outbased on the analysis of time domain response. The final stability criterion is stated interms of a perturbation bound and several matrix norms.It has to be pointed out that the multilateral meanings of the concept of stabilityhave led to various methods for stability analysis which have been formulated separatelyaccording to different stability definitions. However, there are some common features ofall stability definitions and the associated analysis methods. In general, the most crucialissue in the stability analysis of dynamic systems is to determine the characteristics usedto define the stability of a system. Certain quantities, such as norms of the state vector,are sometimes emphasized, and used to characterize the system state response at anydesired time. Other methods include total energy function or trajectories in phase spaceof the system. Although there has been a desire and effort to unify these concepts,apparently none has been satisfactory.1.2 Objectives of the Proposed ResearchWith the increasing complexity of process control problems, more sophisticated and effi-cient control strategies and theories are required in order to manipulate the operation ofthe process effectively and economically. This research is initiated under such a situation.The main objective of the work is to develop modeling methodology and an approach forstability analysis for a class of time-varying dynamic systems that are termed structurally-varying systems, (SVS for short). A majority of the system analysis and control theoryprocedures developed to date is limited to linear and time-invariant systems or structure-fixed systems, which constitute only a small portion of real systems. For the analysis andcontrol of more complicated time-varying or structure-varying systems, new approachesare needed.Chapter 1. Introduction^ 8In the work to be presented, the dynamic modeling and stability analysis of an SVSwill be investigated. We assume that an SVS consists of a number of subsystems, whichare connected together. By subsystems, we mean physical entities to be identified by asuitable partitioning method. Some knowledge of the dynamics of the subsystems, whichmay be linear, is assumed. It is believed that the behavior of the overall system can bepredicted from the dynamics of the subsystems and the constraint conditions among thesubsystems. This idea is based on the hypothesis that the dynamic characteristics of asystem are solely determined by its own structure. If the subsystems are known, and alsothe constraints among them are known, it can be said that the overall system will beknown. Hence, the dynamics of the overall system can be synthesized from that of thesubsystems and the constraint conditions among the subsystems. Besides, the stabilitycondition of an SVS will usually change if the structure of the SVS changes due tothe variations of constraint condition between subsystems. The relationship between thesystem characteristics, particularly the time-varying model and the stability of the overallsystem, and its structural variations will also be studied.The proposed research has a variety of practical applications. One of them can bein the building and deployment of a space station. In the mission of building a spacestation, all materials have to be moved out into space by a space shuttle. The spacestation may be assembled piece by piece by either astronauts or robots controlled fromthe space shuttle. The structure of the space station being built keeps varying, which hasto be maintained stable at any instant of time. The widely used pick-and-place operationscarried out by industrial robots in factories are another example of an SVS. A directapplication of the proposed research will be in the design of a robotic fish processingworkcell. During the overall working period, the architecture of the workcell may varyat different stages of operation. Hence a proper control strategy has to be developed todeal with the variations of the system structure. In our research, we will concentrate onChapter 1. Introduction^ 9the theoretical aspects, especially the modeling and the stability analysis of such typesof systems. Some illustrative examples will also be given in the process of developmentof the theory in order to demonstrate the effectiveness of the theory and the procedureto apply the theory.1.3 Motivations for the Proposed ResearchFactory automation is widely recognized as an important goal for remaining competitivein the manufacturing sector both nationally and internationally. Robotics is one of themajor research areas in manufacturing automation, which has been motivated both byeconomic objectives (i.e., enhances productivity, profitability, and quality) and sociolog-ical objectives (i.e., a desire to improve the quality of human life by releasing humansfrom repetitive, hazardous, or strenuous tasks).The fish processing industry is an old one. The technology used in current fish pro-cessing is rather outdated. The majority of the work is done manually. With today'soutdated methods of fish processing, considerable wastage is inevitable. Upgrading thefish processing technology will result in improved raw product recovery; it is estimatedthat recovering an additional one per cent of the raw product through improved process-ing would result in as much as $5-million annual savings for the Canadian fish processingindustry [de Silva, 1990]. With modern robot technology, we could even go beyond thatgoal. There is a further promise of recovering anywhere from three-to-five per cent ofthe raw product, and furthermore, productivity can be increased by speeding up thewhole plant process. On the other hand, fish cutting is a boring and tedious job. Thesharp blade of a cutter is a potential danger to the workers, especially in long workshiftsand considering the fact that the environment is very unpleasant and slippery. Develop-ing fish processing technology relieves the humans of such hazardous work, can enhanceChapter 1. Introduction^ 10productivity and keep fish processing economically viable.This thesis is organized into five chapters. In Chapter two, we will propose some newdefinitions of stability for an SVS. The basic concepts associated with the definitionsand terminology to be used in the later study will also be presented. Chapter three andChapter four concentrate on the development of dynamic models and stability analysis foran SVS. Major results and contributions of the research will be summarized in Chapterfive.Chapter 2Basic Concepts and DefinitionsIt has been known that although the research on stability of dynamic systems has been anactive area for centuries, there is hardly a universal definition of this important concept[Leipholz, 1987].  This is due to the fact that the types of dynamic systems consideredvary and also the performance requirements can be specified in many different ways.However, this fact has not prevented the theory on stability of dynamic systems fromevolving. In fact it has provided a fertile subject for analytical research. The usualpractice of the studies of stability has been that for the kind of dynamic systems whichare of interest to us, the definition of stability is first tailored to the particular needs ofthe problem, and then the relevant stability theory in that particular sense is developed.In this chapter, we will first discuss the dynamic system which we are interested in,specifically a structurally-varying system or SVS, and then we will provide an appropriatedefinition of stability for the SVS.2.1 Structurally-Varying SystemsFor the purposes of the present development, an SVS is assumed to be composed of anumber of linear, deterministic and lumped-parameter subsystems. Lumped-parametersystems are those for which all energy storage or dissipation can be lumped into a finitenumber of discrete spatial locations. They are described by ordinary differential equa-tions. The way these subsystems are structurally integrated is time-variant. But thedynamics of the subsystems are assumed to be time-invariant.11Chapter 2. Basic Concepts and Definitions^ 12A simple example of an SVS is presented in Figure 2.1. The system consists of twosubsystems. There is a dynamic constraint or dynamic connection between them whichis assumed to be time-varying. mi , ki and ci are the subsystem parameters, and kciand cci are parameters of the dynamic connection. In the real-time operation, the twoconstraints are released sequentially. From the subsystem point of view, the boundarycondition is time-varying. On the other hand, the structure of the overall system is alsotime-varying due to the structural perturbation. Part (a) shows the fully constrainedsystem configuration and part (b) shows the system configuration after the system iscompletely disintegrated into two subsystems. It is not difficult to see that the systemorder is a constant in this type of SVS. The variation of the structure will only changethe parameters of the overall system.Figure 2.2 provides an example of another type of SVS. Two subsystems are con-nected with each other through two mass nodes m 1 and m2 , which may be called a rigidconstraint or rigid connection. Each of them can be considered as a combination of twosmaller masses, m11, m12 and m21 , m22 respectively. A rigid connection is assumed to bein one of the two states, either connected or disconnected (binary constraint model). Thistwo-state constraint model can also be called the static constraint model. The meaningof the term static can be interpreted as the dynamics of the constraint being negligible.In this case, the connection between two subsystems is rigid, or in other words, each con-straint has infinite stiffness. In this type of SVS, the system order will change when theconstraint condition between the two subsystems changes. The system can be consideredto be growing bigger in the sense that the order of the overall system increases when theconstraints of the subsystems are being released. On the other hand, the system canbe considered to be shrinking when a new constraint is applied to the subsystems sincethe order of the overall system will decrease. These two types of SVS will be studied inChapter 3 and Chapter 4 respectively.Chapter 2. Basic Concepts and Definitions^ 13subsystem #1 subsystem #2(a) Before Disintegrationsubsystem #1 subsystem #2(b) After DisintegrationFigure 2.1: An Example of a Fixed-Order SVSChapter 2. Basic Concepts and Definitions^ 14subsystem #1 subsystem #2(a) Before Disintegrationsubsystem #1 subsystem #2(b) After DisintegrationFigure 2.2: An Example of a Varying-Order SVSChapter 2. Basic Concepts and Definitions^ 15Configuration 0^Configuration 1^Configuration 1-2^Configuration 1-1^Configulation 1 - - ►Wv.\'‘W- - ►At1-1^Ott^ 4    I •^• 4^t1^t1-2^t1-1^t 1 istots -t o ► 1Figure 2.3: Diagram of Structure of the SVSIn general, any SVS can be modeled by s system configurations and s switchinginstants over a period of time, which is of interest to us. Figure 2.3 describes thismodel. An SVS has an invariant or fixed system structure between any two instantsof structural variations, which are called the structural switching instants. The systemstructure takes a new configuration after a structural variation. It can be seen that thestructural variation of the SVS is of a discrete nature. The configurations of the SVSare connected to each other through the structural switching instants, and conversely,the structural switching instants are related to each other through system configurations.2.2 Review of Concepts of Stability of Dynamic SystemsBefore we start to discuss the concepts of stability for an SVS, some conventional defi-nitions of stability are reviewed. Although there are a variety of definitions of stability,they can in general be grouped into two categories, i.e., perturbation definition and re-sponse definition. A system is said to be stable if when a small disturbance is applied,the motion of the system will return to its initial equilibrium point after a period ofChapter 2. Basic Concepts and Definitions^ 16time. If the system is not able to return to its initial equilibrium point under a smalldisturbance, the system is said to be unstable. This definition can be considered as aperturbation definition of stability. The stability of a system can also be defined fromits response performance. If a well-behaved excitation produces a desired response over atime interval, the system can be considered stable. If a well-behaved excitation does notproduce a desired response, the system is considered unstable. Here, by "well-behaved"we mean the excitation is applied within a certain range. The definition of the range isdetermined by the particular problem we are facing. In addition, by "desired" we meanthat the response of the system is what we actually want and this response meets thespecial requirements of the particular task. These two categories can be unified if we lookat them from another perspective. They all use system response over a period of timeas a measuring variable or evaluation function. If the response of the system satisfiescertain requirements, the system is said to be stable. Otherwise, the system is said to beunstable. Usually, the requirements include convergence rather than divergence of theresponse of the system over a certain period of time.Generally, the definitions of stability of dynamic systems consist of four elements:convergence, bounds, time interval and the input. From the definitions of stability, it isusually possible to relate certain system dynamics to its stability. The stability conditionof a system can then be expressed in terms of the particular dynamic characteristics ofthe system, for instance, the eigenvalues of the system. In order to examine the stabilityof dynamic systems, a measuring variable or evaluation function has to be selected whichallows us to examine the dynamic characteristics of the system. This measuring variableor evaluation function carries the information of the dynamic characteristics of the systemfrom which the stability can be determined. For instance, given a linear system= Ax+BuChapter 2. Basic Concepts and Definitions^ 17y = Cxwith u(t) as the input vector, y(t) as an output vector and x(t) as a state vector, wehave the following definition of stability:• A system is said to be bounded-input-bounded-output (BIBO) stable if for eachadmissible bounded input u(t), the output y(t) is also bounded.This definition is based on the system response, which is the measuring variable. Thestability of this system can also be restated in terms of system eigenvalues. If all eigen-values of the system have negative real parts, the system is said to be stable. In the latercase, however, the eigenvalues are the parameters which describe the special dynamiccharacteristics of the system.2.3 Definition of Concepts of Stability for An SVSOne reason to discuss the concept of stability for an SVS is that an SVS has somespecial dynamic features, which can make most of the popular stability definitions nei-ther applicable nor appropriate in the stability analysis of this class of systems. Mostof the research on stability study of a dynamic system has focused upon the dynamicperformance of the system over an infinite -time period and the criteria are consequentlyinfinite-time ones. It is unrealistic to ascertain the stability condition of a system duringthe structure-varying period merely from an infinite-time criterion since we are interestednot only in the system stability after the system operates for a long period of time, whichin other words, can be mathematically interpreted as the system stability as time co,but also in the system stability in a relatively short period of time. Also, the time depen-dence of the structure of an SVS is of discrete nature. Each system configuration can beconsidered as a time-invariant system. Its stability is also of interest to us. Practically,Chapter 2. Basic Concepts and Definitions^ 18it is often true that the stability during a finite period of operation would be of interestto us. Most important of all, the influence of continuous structural perturbations of anSVS cannot be studied by the conventional infinite-time criteria. Hence the concepts ofconventional stability have to be modified in order to accommodate these special featuresof an SVS.The stability of an SVS can be studied in terms of fixed-structure stability i.e., staticstability or varying-structure stability, i.e, dynamic stability. Static stability describes thesystem stability condition at each fixed configuration, and dynamic stability describesthe variation of system stability condition when the system structure changes due toperturbations.By fixed-structure stability or static stability, we mean the stability of a particularsystem configuration. We look at the stability of each system configuration individually.When the time interval between two switching instants of an SVS is large enough, theconventional stability theory can be adopted to study the static stability of the SVS inthat time interval. The static stability is appropriate and it has some physical meaningin this situation. However, if the time interval between two switching instants of an SVSis not large enough, the stability analysis results using the conventional stability theorywould be inappropriate. Since each system configuration is of time-invariant structure,the conventional stability definitions and theories, such as BIBO stability, can be applieddirectly to its stability analysis. The analysis of static stability of an SVS will revealinformation on the stability of individual system configurations. Basically, each systemconfiguration is placed and analyzed on an infinite time scale, as the system is a time-invariant one. The time scale is stretched from a finite period of time to an infiniteperiod of time. Analysis of the static stability is in fact no different to stability analysisof ordinary dynamic systems. The definition of static stability could be considered asone of the conventional definitions of stability.Chapter 2. Basic Concepts and Definitions^ 19On the other hand, the dynamic stability of an SVS is meant to represent the changeof the stability condition from one system configuration to the next due to perturbations,which could be structural or state-variable-related. All together, they can be calledsystem perturbations. The system configurations are linked to each other through thesystem perturbations. We mainly look at the change of the evaluation function of stabilityfrom one system configuration to the next rather than the stability condition of individualconfigurations. Therefore, the influence of the structural variation on the stability of anSVS is important. It is evident that the dynamic stability of an SVS investigatesthe system stability on a finite time interval only. The differentiation or the changeis emphasized. At different system configurations, the dynamic stability of an SVS isgenerally different. Therefore, the dynamic stability for every configuration has to bestudied in order to determine the dynamic stability of an SVS. This issue will be themajor topic of the present research.Based on our previous discussion, the definition of dynamic stability of an SVS isgiven here:1. Definition of Dynamic Stability: The time span which we are interested in isdivided into a number of equal segments. Each segment (such as the time interval[ti, ti+1 ]) corresponds to a configuration of the SVS and ti and ti+i are the timeinstants when the structural changes of the SVS occur. The SVS is said to bedynamically stable in the time interval [t i , ti+1 ] ifAxraz xrax _ xraix <where Di is the change of the maximum state response (in the sense of a suit-able norm) in two consecutive system configurations. einam maxIII x(t) t E[ti, ti+1 ]}, x:ny = max{11 x(t) II; t E [ti_ i , t i ]l, and II II represents a suitable norm.Chapter 2. Basic Concepts and Definitions^ 20In this definition, the state response of the SVS is used as the evaluation function.The dynamic stability of an SVS can also be defined by using an energy functionas the evaluation function.2. Definition of Dynamic Stability: The time span which we are interested in isdivided into a number of equal segments. Each segment (such as the time interval[ti , t i+1 ]) corresponds to a configuration of the SVS and ti and 4 +1 are the timeinstants when the structural changes of the SVS occur. The SVS is said to bedynamically stable in the time interval [ti, ti+i ] ifDi = AEr" = Er" — Era,- < 0where Di is the change of the maximum value of an energy function in two consecu-tive system configurations. Er' = max{E(t); t E [4,4+1]}, E.17217 = max{E(t); t E[4-1, ti]}.It is not difficult to see that the concept of dynamic instability of an SVS can alsobe defined by using either system state response or system energy function as theevaluation function.3. Definition of Dynamic Instability: The time span which we are interested in isdivided into a number of equal segments. Each segment (such as the time interval[ti , ti+1 ]) corresponds to a configuration of the SVS and t i and ti+i are the timeinstants when the structural changes of the SVS occur. The SVS is said to bedynamically unstable in the time interval [ti, ti +d ifA = Asnaz = xr" — xmx > 0where Di is defined in (1).Chapter 2. Basic Concepts and Definitions^ 214. Definition of Dynamic Instability: The time span which we are interested in isdivided into a number of equal segments. Each segment (such as the time interval[ti, ti+1 ]) corresponds to a configuration of the SVS and t i and ti+i are the timeinstants when the structural changes of the SVS occur. The SVS is said to bedynamically unstable in the time interval [ti, ti +i ] if= .6.Er" = Er" — Err > 0where Di is defined in (2).In figure 2.4, two examples are provided which illustrate the concepts of dynamicstability and instability of an SVS.The above concepts define the stability (and instability) of an SVS at a particulartime interval. Since an SVS is classified as a time-varying system, its stability conditionis generally changing with time. The system stability over a period of time T can beknown if the stability condition of the SVS over every time interval (such as [ti, t i+1 ])is known. It can be observed that the excitation to the SVS is not included in thedefinitions of the stability. However, we assume that there exists an external force whichis applied at the structural switching instant and causes the variation of the constraintbetween subsystems.It should be noted that the definition of dynamic stability is designed for investigationof system dynamic performance of an SVS, either state response or energy value, duringa finite period of time. It is different from conventional definitions of stability suchas Liapunov stability or asymptotic stability which consider the dynamic response of asystem in an infinite time scale.Also, it has to be pointed out that the theories of static and dynamic stability dealwith different dynamic aspects of an SVS. They are independent of each other. StaticChapter 2. Basic Concepts and Definitions^ 22EvaluationFunction11X11 or E^ATime(a) Dynamic StabilityEvaluationFunctionil)(11 or E^A^■Time(b) Dynamic InstabilityFigure 2.4: Example of Dynamic Stability and Instability of an SVSChapter 2. Basic Concepts and Definitions^ 23stability of an SVS does not assure the dynamic stability of the SVS and vice versa. Inother words, even when every configuration is statically stable, it is still possible that theSVS is dynamically unstable. On the other hand, a dynamically stable SVS may havestatically unstable configurations. Therefore, in order to determine the stability of anSVS comprehensively, the analysis of both static and dynamic stability has to be carriedout.2.4 SummarySince the dynamics of an SVS has some special features, the concepts of static anddynamic stability are introduced and defined in order to study the stability of the SVScomprehensively. They are modifications of some of the conventional stability conceptsand designed particularly for the special dynamic characteristics of the SVS. Based onthese new concepts, the stability of the SVS can be analyzed more thoroughly and theinfluence of the discrete structural variations on the stability of the SVS can be studied.Chapter 3Stability of Structurally-Varying Systems With Fixed Order3.1 IntroductionIt has been known that there are two types of SVS, an SVS with fixed order and an SVSwith varying order. In this chapter, the stability analysis for an SVS with fixed order iscarried out. The SVS to be studied is assumed to have only dynamic connections betweensubsystems, which implies that the connection between any two subsystems only consistsof a spring with finite stiffness and a damper, as is shown in figure (2.1). There is nomass coupling between subsystems. This type of SVS has a constant order throughoutthe entire time period of operation regardless of the perturbations on the stiffness anddamping matrices of the system. There is no perturbation on the mass matrix of thesystem. Two evaluation functions, state response function and energy function will beemployed to carry out the stability analysis. A number of criteria for the evaluation ofstability of an SVS will be derived for both static stability and dynamic stability.3.2 Modeling of Switching InstantsIt is known from Chapter 2 that an SVS can be modeled by a series of configurations andswitching instants. Each configuration between two switching instants can be consideredas a time-invariant system and the structural variation occurs only at the switchinginstant. In this chapter, we will study an SVS which has only flexible connectionsbetween subsystems, which means a connection is composed of either a spring or a24Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^25damper. There is no mass coupling between subsystems. As a result, the total numberof the mass nodes is a constant over the period of time of interest. Hence, the systemorder is maintained.A schematic diagram of the switching instant is given in figure (3.1). There are twoboundary mass nodes, m 1 and m 2 . Dotted lines represent the connections of these twomass nodes to other parts of the systems. The connection between these two mass nodesconsists of a spring and a damper. The connection between subsystems is emulated bya switch. At the time of structural change, the switch is turned off instantly so thatthe two mass nodes are disconnected. The connection components can be consideredremoved from the system.Since the forces applied on the mass nodes by the spring and the damper are finite,the application or removal of them will change the system structure only and will notcause any sudden change of motion of the mass nodes, which means neither displacementnor velocity vector has a sudden change at the switching instant. If we defined = dl(t)d2 (t)we can haved(tt) = d(C)anda ( i-sE ) a(c)where t = t i is the instant of a structural variation. In other words, there is no pertur-bation on displacement and velocity due to the structural variation. It has been shownpreviously that the dimension of the displacement and velocity vectors will not changeeither. Therefore, we havedl (t)d2(t)d,+1(ti)^di ( ti )^ai(ti)^ (3.4)Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^26Figure 3.1: Illustration of Switching Instantsat the structural switching instants. The change of the system dynamics can then bedetermined based on the structural perturbation only.However, it should be pointed out that this model of the switching instant is anideal one. It assumes that the connection components, which could be either a springor a damper, are massless. In reality, this assumption may not be right. There maybe some mass and energy associated with the connection components. When they aredisconnected from the system, the mass and energy may go away with them. The dy-namic model of the switching instant may have to be modified if this factor is taken intoconsideration.Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^273.3 State Space ApproachWe first use state response as the evaluation function. The state space model of an SVSwill be developed and then the stability analysis will be carried out based on the statespace model.3.3.1 Recursive State Space Model of a Structurally -Varying SystemAs has been discussed in Chapter 2, the systems being studied are confined to a certainclass of dynamic systems. The system is composed of a number of smaller systems, orsubsystems. Each of them is modeled byMj + K, 0 (3.5)where Mj , C; and Ki are the mass, damping and stiffness matrices, respectively, forthe jth subsystem, and dj is the displacement vector for the jth subsystem. The overallsystem model can generally be assembled from the subsystem modelsM(t) d + C ( t ) K(t) d = 0 (3.6)where M(t), C(t) and K(t) are the mass matrix, damping matrix and the stiffness matrix,respectively, for the overall system. d is the displacement vector for the overall system.All three parameter matrices are composed of the corresponding parameter matrices ofsubsystems and constraint parameter matricesM(t) M° Mc(t)C(t) = C° Cc(t) (3.7)K(t)^K° Kc(t)where M°^C° = diag{q} and K° = diag{Kj}. Superscript o denotesoriginal and c denotes constraint. Subscript j denotes the subsystem number. M i , CiChapter 3. Stability of Structurally-Varying Systems With Fixed Order^28and Ki are the parameter matrices of the jth subsystem. Mc(t), Cc(t) and K°(t) are con-straint parameter matrices, which describe the dynamic connections between subsystems,andd(t) =d i (t)d2 (t) dm(t)The symmetry of parameter matrices M(t), C(t) and K(t) are ensured by Maxwell'sreciprocity theorem if all subsystem parameter matrices and coupling constraint matricesMc(t), 1<c(t) and Cc(t) are symmetric [Meirovitch, 1986].Since we assume that the subsystems are time-invariant and the constraints whichconnect the subsystems are also time-varying, the configurations of an SVS in two sep-arate constraint conditions are generally different. For any configuration i, the systemmodel can be written asMia+Cia+Kid=0 (3.8)To derive the state space model of the SVS, we assumex=then,=Sincedd(3.9)(3. 10)(ci ci + Ki d)Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^29d[ Ki —1V1;-1 Cid[ Ki Ci x (3.11)and----^[ci1=[0^x (3.12)Combining equations (3.11) and (3.12), we obtain the state space model for the configu-ration i,X = Ai x^and^x(ti) = xi^ (3.13)wherex= [ a ]d0AiKi^C.and xi is the initial condition of configuration i. Similarly, we can have the model forconfiguration i + 1Mi+i d + C i+1 d + KJ+, d = 0^ (3.14)Therefore, the state space model for configuration i 1 can be written asx = Ai+1 x,^and^x(ti+i) =^ (3.15)where0—1\441.1 Ki+1 _mi-+11 ci+iChapter 3. Stability of Structurally-Varying Systems With Fixed Order^30The change of parameter matrices due to the variation of the system constraint con-dition at time t = ti+i is modeled byAMi+i = Mi+1 — Mi=^— Ci^ (3.16)= K j+1 - KiSince we assume that the system only has a spring-damper connection, we haveMi = Mi+1, i.e., AMi = 0, which implies that the variations of constraints do not incurany perturbation on the mass matrix. Substituting equation (3.16) into equation (3.15),we get=—NW (Ki^—Mi1 (Ci ACi+i )0— AVAKi+ i^— Mi 10Ci+10^I^0^0—NW Ki •— l\V C i^—A4z4 AKi+i^ACi+ithat isAi+1 = Ai + AAi+iwhere0^IKi^Ci0^0-1\4 -i- 1 A/Ci.o . -Mi 1 LOC4+1Ai ==(3.17)mlklsubsystem #1kcl(t)kc2(t)connection subsystem #2Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^31d2r-•Figure 3.2: Example of a fixed order SVSAi+1 is separated into two terms. The first term A i is the model of the previousconfiguration, and the second term is the model perturbation due to the variation ofconstraints at time ti+i • Equation (3.17) is the recursive state space model of the SVS.• Example: Consider the system described in figure 3.2. The initial connectionbetween two subsystems consists of a spring with stiffness Ica . We assume that thesystem stiffness increases at time t i by the value Ica . Using the precedure developedpreviously, we can haved = d1d2M(t) = M° =Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^32K(t) = K° Kc(t) =^0^Kc(t)0 k2When t <K`(t) =^kci —kciandK l = K(t) =lc,.^0^4_0 k2ka —Icak2 kaWhen t > t i ,Kc(t) =^+ Ica^— kc2— Icc2^kc2Hence,K2 = K(t) =^kl 0^kc —kc^0 k2^—kc kc— ka — kcz— kc2^k2 + kc, + kc2AK 2 = K 2 —^=[ki + kci. + ka — kci. — kc2—Ica — Ica k2 + ka + kc2[kl + ka —Ica  = [ kc2—ka k2 + ka —kc2—Icc2kczChapter 3. Stability of Structurally-Varying Systems With Fixed Order^33The state space model of the overall system can then be derived;X = A ix,^0 <t < t1X^A2x ,^t1 < t < oowherex= ad0 0 IA l = —(k1 + lcci)/mi kcl/ml 0__(A40)-1 K1 0Ica /m2 —(k2^ko.)/m2 00 0 0AA 2 = —Icc2/mi kca/mi 0ka/m2 —kc2/m2 001A2 =^AA2 =_(M°) -1 K2 00^0^I— (1c1 +^+ Icc2)/mi^(ka + kc2)/mi^0(kci kc2)/m2^—(k2^Icc2)/m2 0In order to investigate the relation between overall system stability and subsystemstability, we rewrite equation (3.6) asM° a + C° cl + K° d Mc(t) a + cc(i) a + Kc(i) d = 0^(3.18)Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^34where M°, C° and K° are assembled from the dynamic models of the subsystems andare time-invariant. Mc(t), Cc(t) and Kc(t) describe the coupling constraints betweensubsystems and are determined solely by the constraint conditions at time t.The model for configuration i can then be written asmo d + co ci + K° d Mat) d + q(t) d + Kat) d = 0The system matrix can be rewritten asAi =where0As—(M°) -1 K° —(M°) -1 C°0^0As^—(M°)-1 Kf(i) _(M°)-1 cf(i)Since k is time-invariant, the subscript i can be dropped. SoAi = A° +Since the recursive constraint model for the SVS isM7 = WK-1=^AqKf^Kf_1(3. 19)(3.20)(3.21)where AMf = 0 is used, the constraint matrix Af can be further expressed in a recursiveformDA;^ (3.22)[ —(M°) -1 K__ 1 (t) —(M°) -1 q_ 1 (t)A7_ 1 =0^0Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^35whereoAJK = _(M°)_1 AKf: _(1\40)-1 AqFor various constraint conditions, we have different 1<c(t) and Cc(0. In general, thecharacteristics of the Kc(t) and Cc(t) are dependent on the physical properties of theconnection between subsystems. Af_ 1 is the constraint matrix for configuration i —1 andDAs is the constraint variation matrix at time t = ti. Substituting equation (3.22) intoequation (3.20) yieldsAi = A° +^+= A° + E AA;i=o(3.23)It can be seen that A° is determined solely by the dynamics of the original subsystem,which is the time-invariant part of Ai. Af, on the other hand, is determined by couplingconstraints between subsystems, which is the time-varying part of Ai . The model forconfiguration i of the SVS can then be expressed asX(t) = Ai(t) x(t)= (A° + E DAB) x(t)^ (3.24)j=0where each AA; describes a structural variation of the SVS.3.3.2 Analysis of Static Stability of an SVSFrom the definition given in Chapter 2, we know that the static stability can be consideredas an extension of the conventional stability concepts to an SVS. For each of the systemChapter 3. Stability of Structurally-Varying Systems With Fixed Order^36configurations, we can find out if the configuration is stable (in the conventional sense,such as bounded-input-bounded-state (BIBS) stable or asymptotically stable) when itsmodel is known. In this section, we are not studying the static stability of each systemconfiguration separately. Plenty of work has been done in this area before. We areinvestigating how the static stability of an SVS changes from one system configurationto the next due to the structural variation. To a degree, this problem is similar to therobustness problem of dynamic systems [Haddad etc., 1992].It has to be pointed out that although the stability analysis is carried out based onthe state space model of the SVS which is composed of subsystems modeled by thesecond-order-matrix-equation, the results derived here are not limited to the system ofthis category. The theory to be developed can be applied to any dynamic system as longas its state space model is available.The solution of equation (3.15) can be written asx(t) = i+1 (,6,t) x(ti+i) t E [ti+i, 4+2] (3.25)where ski+i (At) is the state transition matrix for configuration i, i = 0,1,2, • • •,m and4,i+1(At) eA i+ , At, At = t — ti+1 . This group of equations determines the time historyof the state response at any time instant for the SVS. Substituting equation (3.17) into413 i+1 (At) yields44-1-1(At) e(At+AAi+i) At= eAi At eAAi+ 1 At= 41)i(At) • AC■i+i (At)^(3.26)whereck,(At) = eA sChapter 3. Stability of Structurally-Varying Systems With Fixed Order^37AC+1(At) ebiti+1 Atand it can be observed that Ili(At) is the state transition matrix of configuration i, whichdetermines the system stability at configuration i.The condition of the static stability of an SVS can then be derived according toequation (3.26). They are presented in the following theorems.Theorem 3.1: If the system configuration i is statically stable in the sense of BIBSfor the time period [t i , oo), the system configuration i + 1 will also be static stable in thesense of BIBS if II 6,4loi+1 (At) II is bounded.Proof: Suppose the configuration i is stable for [ti, oo). Since II la i (At) II is bounded,E^oo)where pi E R. The notation II cbi(At) II refers to the norm of the linear transformationx C(At)x, x E Rn, which is induced by the standard Euclidean norm on Rn. Weknow thatC+1 (At) = c(zsa). Ac +1 (At)Therefore11 cbi-Fi(A t ) 11=11 4.i(At) Ac+1(At) 11_11 cbi( At ) II • 11 L\ci-Fi(At) 11If II 6.434 1 (At) II is bounded11 AC-pi(At) Il< 196,where 1)4, E R, thenII< Pi PA = Pi+ 1and pi+i E R. Therefore, the system configuration i + 1 is BIBS stable, i.e., staticallystable. This concludes the proof of Theorem 3.1.Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^38Now we take a further look at the condition that II AC(At) 11 is bounded. It isknown thatAC(At) = CAA` AtThe main concern here is to find out what condition AAi has to satisfy in order tokeep the system statically stable. The result of the study is summaried in the followingtheorem.Theorem 3.2: The 11 Ack i (At) 11 will be bounded if and only if the eigenvaluesof AA,: have negative real parts. The proof for a similar theorem can be found in[Chen, 1988].Theorem 3.1 and Theorem 3.2 present the sufficient conditions of the static sta-bility of a configuration. The following theorem gives a condition to make a staticallyunstable configuration statically stable after a structural perturbation.Theorem 3.3: If the system configuration i is statically unstable in the sense ofBIBS for the time period [t i , oo), the system configuration i 1 will be statically stablein the sense of BIBS if II A■foi+i (At) 0.Proof: Suppose the configuration i is unstable for [ti, oo), which implies thatII 41i(Ai) II> M for any M E R. SinceII C+1(At) II=11^Ai.fi(At)^C(At)II II Acki+i(At)If 11 AC+1(At) 11= 0, we haveII C-Fi(At) 115_ 0 <where pi+i E R. Hence, the configuration i 1 is BIBS stable, i.e., statically stable inthe sense of BIBS.As a theoretical result, theorem 3.3 provides the design method to stabilize a stati-cally unstable configuration through modifying its constraint condition. However, it hasChapter 3. Stability of Structurally-Varying Systems With Fixed Order^39to be modified slightly in order to be applied in engineering practice because the condi-tion II 6.1, i (At) 11= 0 hardly has any practical meaning. This situation results from theassumption that a statically unstable system has the transition matrix 11 4 ,i(At) 11-4 oo,which is also of little practical significance. In reality, the system would be consideredunstable if its output exceeds a certain accepted level and tends to diverge. The outputwill usually saturate at the physical limits of the system rather than reach infinity. Thephysical limit of the system is usually the maximum output the system could reach.The previous study has focused on the relation between two consecutive configurationsof the SVS. The following theorem provides the criterion for the evaluation of staticstability of the overall system based on the stability of subsystems.Theorem 3.4: If all unconstrained subsystems are BIBS stable, the constrainedsystem will be statically stable in the sense of BIBS if every constraint applied in thetime interval of interest is stable.For all constraints to be stable, we mean II AckaAt)^pi , A E R and for every i,i = 1,2,• • •,m.Proof: Suppose the unconstrained subsystems are BIBS stable. We know fromequation (3.20)Ai = A° + .A4Hence, the solution to equation (3.24) can be written asexp{(A° E LIADAt} x(ti)exp{A'At} exp{(E ADAt} x(ti)(1,i(At) x(t i )4,0 (6.0 41(6d) x(t i )^t E [ti, ti+1]^ (3.27)Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^40where43°(L t)^exp{A°}a,a,t)^A41(6,0J.0Since all unconstrained subsystems are BIBS stable, we haveII l'o(At)11< Po,^t E [ti, oo)where Po E R. We also know40 i(At) = 4.°(At). (1; (At)ThereforeII Iii(At)II = II °(i t) '6■('At) II41'°(At) II • II 6A'aAt) II4'° (At) II • II II '61 '.;(At)i=oIf every II .6.4) .7(A t) II is bounded, i.e.,4,;(At) < pi < Pmaxwhere Amax = max{Phi = 0,1,2, • • •, m}. We haveII^ i(At) II^Po PnntaxTherefore, the overall constrained system is BIBS stable, i.e., statically stable in thesense of BIBS.• Example: Assume that a system has the model= A°xChapter 3. Stability of Structurally-Varying Systems With Fixed Order^41with0^1A° =—2 —3Its eigenvalues are a i = —1, A2 = —2. It can be shown that it is BIBS stable. If atthe time t = t i a coupling constraint is applied to the system, which has the modelAc^0 00 —5Since11 4)°(At) 11 2 =11 exp{^At} 11 2 < 10 —°5The constraint is stable. Hence, according to Theorem 3.4, we know that theoverall constrained system is statically stable, which can be verified. The model ofthe overall constrained system isic = AxwithA = + A.' =—2 —8Its eigenvalues are A i = —7.74, A = —0.26 and it is BIBS stable, i.e., staticallystable in the sense of BIBS.Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^423.3.3 Analysis of Dynamic Stability Via State Space ModelAlthough an SVS can be considered as a special type of time-varying system, the timedependency of its system structure is not continuous in general. In order to examine thestability of this type of system, we select an evaluation function which is associated withthe system configuration. When the system model changes, the value of this functionwill also change. In this section, the state response of an SVS is selected as the evalu-ation function first. The change of the maximum state response of consecutive systemconfigurations is used to determine the stability of an SVS.The concept of dynamic stability has been discussed in Chapter 2. It has been shownthat dynamic stability studies a different aspect of stability of an SVS from what staticstability does. It is determined by comparison of the values of evaluation functions ofdifferent configurations of the SVS. It is not closely related to the static stability ofthe SVS. A system configuration can be dynamically stable even when it is staticallyunstable.We start the analysis from equation (3.13). The solution to equation (3.13) isx(t) = exp[Ai(At)] x(ti) (3.28)= i(At) x(ti) t E [ti, ti-Fl] (3.29)where obi is the state transition matrix for each configuration, i = 1, 2, • • •,m and At =t — ti . This group of equations determines the time history of the state response at anytime instant for the SVS.It is also known that the change of the constraint condition at the time t = ti canbe modeled by a perturbation on system matrix Ai_ 1 . Therefore, a recursive state spacemodel can be determined for the system. We can writeAi = Ai-1 AAi^ (3.30)Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^43Then we can determine a recursive relation for the state transition matrix asC(At) = exp[(Ai_ i AAi) At]floi_ i (At) • AC(At)^(3.31)whereC- 1 (At) = exp[Ai_ i (At)]AC(At) = exp[AA i (At)]The derivation of stability criteria is based on the recursive state space model. It isknown [Vidyasagar, 1978] thatII x(t)II_< exp(ft:A[Ai]dT) II x(ti)exp{it[A i] At} II x(ti) II= -yi(Ai, At) II x(ti)II(3.32)t E [ti, ti+i ]where -yi(Ai, At) = exp{p[Ai] At}, p[Ai] is the matrix measure of Ai, At = t — ti andII x(t) II is a suitable norm of the state vector x(t). The computation of -yi(Ai, At) consistsof algebraic calculation only and is usually very simple. This feature distinguishes itselfand makes this approach very suitable for real-time applications. The definitions of themathematical concepts and their properties are given in the Appendix. The followingexample illustrates how to calculate the matrix measure of a matrix.• Example: For a system= Axwith10^1 1x(0) ^, A =4 0 1Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^44The matrix measure of A isico,„[A] = maxifaii E aid I}i#i= 2Hence,-yi(A, t) = exp[10 2dr] = exp(2t)SoII x^-ri(A,i) II x(0) II.exp(2t) II x(0) II.= 10 exp(2t)which gives the upper limit of II x II. at any time t.Theorem 3.5: An SVS is dynamically stable during the time interval [ti,ti+i]At) < 1.Proof: From equation (3.32), we haveII x(t) II^< 'y (Ai, At)^t E^ti+1]x(ii)If -yi(Ai , At) < 1, we havex(t)  <1II x(ti) II —Equivalently,II x(t) II^II^(ti) IIOn the other hand,II x(ti)^fir_T^t E [ti-1)ti]ifChapter 3. Stability of Structurally-Varying Systems With Fixed Order^45where ;77 is the maximum value of II x(t) II in the time interval [4_ 1 ,4 Accordingly,x'nar = II x (t)^x:nrwhich proves that the system is dynamically stable in time interval [t i , ti+1 ].3.3.4 Recursive Algorithm for Estimation of 71 (A,, At)Since -yi(Ai, At) is different for every configuration, an efficient algorithm is needed if-yi (A i , At) is used to evaluate the system stability. We defineAt) =-^(Ai+i , At) —^At)^ (3.33)By substituting equation (3.30) into (3.32) we obtainAt)exp{ft[Ai.fi ]At} — exp{A[AdAt}expfp[Ai AAi4.1]At} — exp{A[Ai]At}(exp{/2[A.A.i.4. 1 ]At} — 1) • exp{it[AdAt}= i3i+1 7i(Ai, At)where-yi (A i , At) = exp{p[AdAt}1^41 = exp{p[AAi+dAt} — 1The recursive algorithm for estimating -yi +1 (A,4 1 , At) can be obtained as(3.34)(3.35)At) =^At) +^At)5_ (1 +^At)=^At)^(3.36)withx(0)^10 ,0Ao =—1 00 —1Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^46where cxi 4.1 = 1 -I- i3i 4.1 conveys the information of current structural perturbation andis determined by AA i+i only. Using this recursive algorithm, yi +i(Ai+i , At) can beestimated recursively. The dynamic stability of the SVS can then be determined byTheorem 3.5.• Example: For a system= AxWe assume that the system model varies by AA 1 at time tAA, =0 0which gives—2 1A l =0 —1Sincepc.3 [A 1 ] = —1Hence,71(Ai, At) = exp(—.6a) < 1andAccording to Theorem 3.5, we know that the system is dynamically stable in thetime interval (t 1 , oo).Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^47In Theorem 3.5, the stability condition has been stated through -yi(Ai, Lit), whichis an overall system parameter and the subsystem dynamics is not reflected in it. Fromequation (3.20), we have(1>i(6 .t) = exp[(A° An At]4°(At) • 4; (At) (3.37)where°(L t) = exp[A° (At)]1:07(A t) = exp[A. (At)]It can be seen that the state transition matrix 40 i is determined by (1) the dynam-ics of unconstrained subsystems which is described by time-invariant 40 (At), and (2)constraint-dynamic characteristics which is described by 4i (At).If an artificial system which has the system matrix A° is created, the system statetransition matrix would be ck°(At) and the stability condition of 40°(At) does not changebecause A° is time-invariant. In the analysis of the SVS, this is due to the fact that thestability condition of 4°(L t) is determined solely by unconstrained subsystems, whichare assumed time-invariant. Their stability can be studied separately using conventionalanalysis tools, such as modal analysis, at the subsystem level. Based on that, the relationbetween the stability of the overall system and that of unconstrained subsystems can bedetermined.Theorem 3.6: An SVS is dynamically stable if-yf(A., At)< -y°(A°, At)wheree(A°, At) = exp{A[A°] At}1Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^48-yf (117, At)^exp{E µ[DAB] At}i=o-y°(A°, At) is solely dependent on the subsystem dynamics and it is independent of thesystem configuration. However, -yf(Af, At) is determined by the constraint dynamics andit is dependent on the system configuration.Proof: It is known that -yi(Ai, At) = explp[Ai] At}. By substituting equation (3.23)into -yi(A i , At) we get-yi(Ai, At)= -yi(A°, A7, At)exp{A[A° E A.nAt}i=o< exp{.L[AlAt} • explE p[AgAt}j=0It follows that,II x(t) II exp{p[A.1 At} • explE I.L[A.k] Atl• II x(ti) IIJ=o= -y°(A°, At) • -1(14, At). IIx(ti) 11If-r(A°, At) • -yafq, At) < 1we get41"^x(t)^x(ti) IINote thatII x(ti)^x(t) II= fincrHencet E [ti,t E [ti_ l , ti]max =II x(t i )^x721.Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^49andDi < 0Therefore, the system is dynamically stable during the time interval [ti, ti +d. FromTheorem 3.6 the stability of the SVS can be analyzed from its subsystem stability andits constraint conditions.3.4 Energy Function ApproachIn this section, the dynamic stability of the SVS is studied by using the energy functionas the evaluation function. This approach is especially appropriate when the systemmodel is given in the form of a second-order-matrix-equation since the energy functionis readily available in this case. The dynamic stability criteria are given based on thechange of the energy function due to the structural perturbation.3.4.1 Analysis of Dynamic Stability Via Energy FunctionIt is known from section 3.3.1 that the models of the configuration i and i +1 of the SVScan be written respectively asMilt) di + ci(t) di + Ki(t) di = 0andmi+i(t) ai+, + ci+i(t)^+ Ki+l(t) di+1 0The corresponding energy functions are1 •Ei (t) = —2 dTMieli + 2—1 dtKidi1 AT iv A\^TEi+i (t) = — La • 1^Lti+i^22 j+ Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^50respectively. At the instant of a structural variation, we haveMi^= MiCi Ci+ 1^Ci ACi+1Ki Ki+1 =^AKi+ianddi -4 di+iSubstituting these relation into Ei +i (t) expression, we haveOrEi+ i (t) = 2dTiMi ;i+a -Fi12dT (K AK )(I1 „,+ -dT AK • d •2 I^ 2 2+1^2+1 2+1Ei+i (t) = Et(t)d- AEi(t)where1 • T^•^1 Tgc (t)1 , T^,AZ4.1(t) = -a • a.n.i+l a2 "4It has to be pointed that although El'(t) and Ei(t) have the same form, they arenot equal over the time interval [4+1,4+2] because in general di+ i(t)^di(t) for t E[4+1 ,4+2]. However, at the instant of the structural variation t^we do have= di(ti+1). Therefore(ti+i) = E:(ti+i) AEi+i(ti+i)= Ei(ti+i ) AEJ-1-1(4+1)Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^51It is known that when the damping matrix Ci is positive definite, the system willdissipate energy. The total system energy will decrease with time [Meirovitch, 1980].Hence, max{Ei(t)} = Ei(ti), t E [ti, ti+d and the final value E i (t i+i ) will be the minimumone. If we can calculate the max{Ei(t)} from the available information, we will be ableto find the difference of maximum energy of two consecutive configurations. Therefore,the stability in the sense of energy change can be determined.It is known that the derivative of Ei(t) can be computed as1 ••^•^1 • ,ti(t) = —2[d: Midi + , Midi] + —2[d: Kidi <KA]2[dTM1^+1 •i cocijai 2^+Kidd^(3.38)Since Mi, Ki, Ci are symmetric, we can havemi di + Ki di =^ (3.39)and(mi di + Ki di)T = ar Mi + dT Ki^Ci^ (3.40)HenceE^HaT ciiai +(3.41)It can be seen that the rate of energy decay is determined by the damping matrix. Accord-ing to Liapunov's stability theorem, we know that the static stability of the configurationis assured when Ci is positive definite.Considering the fact thatmax{Ei+i (t)} =^)gr(ti+i ) 1E,4 1 (ti+1 )=^) AEi+i (ti+i )^[ti+i, ti+2 ]^(3.42)Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^52The energy change of the system due to the structural variation is fully described byAEi+i (t). If AEi+i(t) > 0, we have Ei d_ 1 (t) > Ei (t). It implies that the structuralvariation increases the overall system energy level. Hence, the system is dynamicallyunstable in the time period [ti, according to the definition given in Chapter 2. Onthe other hand, if AEi+i (t) < 0, we have Ei+i (t) < E=(t). It can be shown that thestructural variation causes the overall system energy level to drop, which implies thatthe system is dynamically stable in the time period [ti, t;.0 ]. Sincemin{E1(t)} = Ei (ti+i )ti+,= max{Ei(t)} — ft ^Ciaidt^ (3.43)Substituting equation (3.43) into equation (3.42), we havemax{Ei+i (t)}.= max{Ei (t)} —^Ciaidt AEi+i(ii+1)[t1+1,ti+2]Hence,Amax{Ei+i}^max{Ei+i(t)} — max{Ei(t)}ti+ , .= AEi+i(ti+i)—^clT Ciaidtt i + l^ ti+i .idZ-Virti+1Cii+1 — cl; Cictidtt i(3.44)(3.45)It can be seen that Amax{Ei+i } consists of two terms which have the quadratic form.The sign of the Amax{Ei+1 } is determined by the positive-definiteness of the matricesAKi+i and Ci. It is usually true that Ci is positive definite. It is evident that if AKi+iis negative-definite, Amax{Ei+1 } < 0. Therefore, the dynamic stability criterion can bestated in the following theorem.Chapter 3. Stability of Structurally-Varying Systems With Fixed Order^53Theorem 3.7: For any SVS, if (1) AKi +i is negative semi-definite, (2) Ci is positivedefinite, the system will be dynamically stable during the time period [4, 4 +1 ].The proof of the Theorem 3.7 is straightforward based on the derivation processdone before. Hence, it is not rewritten.By Theorem 3.7, the dynamic stability of an SVS during a certain time intervalcan be examined using the variational parameter matrices and the damping matrix.Especially, if the system is a conservative one, the condition provided in the Theorem3.7 becomes a necessary one.Theorem 3.8: For any conservative SVS, iff 6.Ki +i is positive semi-definite, thesystem will be dynamically stable during the time period [4, 4-E1]•proof: Since the system is conservative, C i = 0 for i = 1,2, • • -,m. Therefore,equation (3.45) can be rewritten asAmax{Ei+1 } = max{Ei44 (t)} — max{Ei(t)}6.Ei+1 (4+1)1—2 diT+1 6,Ki+idi-Ei (3.46)If AKi+i is negative semi-definite, we will have dr+1 0Ki+i di+1 < 0. Therefore,Amax{Ei+1} < 0 and the system is dynamically stable. On the other hand, if thesystem is dynamically stable, we must have Amax{Ei+i } < 0, which implies thatd,T+1 6,Ki+i di+i < 0. Therefore, L‘Ki+i has to be negative semi-definite. This provesthe sufficient and necessary conditions of the dynamic stability of the SVS.• Example: From the previous example shown in figure 3.2, we know that initiallyIcc = /ca . If the system stiffness is increased by Ica at time t = t iAK^lca —IcaChapter 3. Stability of Structurally-Varying Systems With Fixed Order^54SinceI — AK I= A —^Ica= A 2 — 2kc2 A = 01Cc2^kc2The eigenvalues of AK are A i = 0, as = 2k2c . AK is positive semi-definite.According to Theorem 3.8, we know that the system is dynamically unstableafter the structural variation.On the other hand, if initially kc^+ ka , and the system stiffness is decreasedby Ica at time t = t 1 , we haveAK +^—ka^[ki + Ice —Iceka + k2 kc[—ka kawhere Icc = ko. + Ica. SinceA +^—IcaI AI — AK I= = A 2 + 2kc2 A = 0—Ica A +The eigenvalues of AK are A i = 0, A2^—2kzc . If k2 > 0, we have A2 < 0.Therefore, AK is negative semi-definite. According to Theorem 3.8, we knowthat the system is dynamically stable after the structural variation.3.5 SummaryIn this chapter, both static and dynamic stability of the fixed order SVS has beenstudied. System state response and energy function have been employed respectivelyas the evaluation functions. The stability analysis has been carried out based on theChapter 3. Stability of Structurally-Varying Systems With Fixed Order^55recursive state space model developed for the fixed order SVS. A number of criteria forevaluating the static and dynamic stability have been derived. In particular, the relationbetween the subsystem stability and that of the overall system has been studied.Chapter 4Stability of Structurally-Varying Systems With Time-Varying Order4.1 IntroductionIn this chapter, we study the stability of an SVS which has a time-varying order. Theorder of a system of this type will change when a variation of constraints of the systemoccurs. The system can be considered growing when the order of the system increases, orshrinking when the order of the system decreases. A new approach has to be developedin order to accommodate the order variation of the SVS.In the previous study, we have assumed that the subsystems are connected to eachother through springs and dampers and there is no mass coupling between the subsystems.In that situation, the structural variations of the SVS can be characterized analytically bythe structural perturbations or the change of the system stiffness and damping matricesalone. The system mass matrix remains virtually unchanged. Most important of all,the dimensions of parameter matrices are kept unchanged for every system configurationregardless of constraint conditions among subsystems. Therefore, comparison of theparameter matrices and the state variables, which is a crucial step in predicting thechange of the system dynamics, can be made. If this assumption is dropped, i.e., theconnection between subsystems is composed of not only springs and dampers but alsomass elements, the order of the overall system will consequently change whenever theconstraint condition among subsystems changes, as will the dimensions of the parametermatrices. Hence, simple direct comparison or algebraic operation of any of the parameter56Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^57matrices becomes unfeasible due to the incompatibility of their dimensions. Therefore,the stability analysis of an order-varying SVS becomes more difficult than that of a fixedorder SVS.The approach adopted to attack this problem is to find a descriptive scalar variable,which should carry the stability information and be determined by structural properties ofthe SVS. This variable has to be easily computable and physically meaningful consideringthe fact that the algorithm for the computation of the stability condition may be usedin real-time applications. From this variable, the system stability can be predicted.4.2 Energy Function ApproachIt has been known that in the analysis of the dynamic stability of the SVS, the key issueis to look at how the evaluation function would change, if it does, with the variationof the system structure. In the previous stability study of the fixed order SVS, thestate response was used as the evaluation function. Basically, we carried out analysis onthe state transition matrix. In the case of order-varying SVS, this approach becomesdifficult to apply since the state transition matrices for different system configurationshave different dimensions, which make the comparison of the system matrices of differentconfigurations impossible in a meaningful way. Also, the dimension of the system statevariable will change when an SVS moves from one configuration to another, i.e., thedimension of the system state variable will either grow or shrink. Therefore, the approachto use the state response as the evaluation function becomes inappropriate in the presentcase.In such a context, we start by considering using the energy function as the evalua-tion function in the stability analysis of an order-varying SVS. It is known (from thedefinitions given in Chapter 2) that if the energy in a system grows over a significantChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^58time interval, the system may be considered unstable. On the other hand, if the energyremains unchanged or is even diminishing in a system, the system may be consideredstable. In the stability analysis of the order-varying SVS, we will be focusing on thechange of the energy function due to the structural perturbations, i.e., dynamic stabilityrather than static stability since the static stability is assured by the positive-definitenessof the damping matrix C. In other words, we will study how the energy function varieswhen the system takes a new configuration instead of looking at the changing rate of theenergy function within a fixed configuration.The approach we adopted is to calculate and compare the maximum value of theenergy functions max{Ei(t)} and max{Ei+i (t)} of two consecutive configurations ofthe SVS. Then, we use the difference A{Emax } = max{Ei+i (t)} — max{Ei(t)} as themeasuring variable to determine the dynamic stability of the SVS over this time interval.For the kind of systems we are studying, it is known that every system configuration isstatically stable, i.e., for any system configuration i, we have ti(t) < 0 over the timeperiod [ti, ti+d, which has been proven in the previous chapter. In other words, thesystem is dissipating energy during each time interval in which the system structure isfixed. Therefore, max{Ei(t)} = Ei(ti), [ti, ti +d, which implies that at the initial instantof each system configuration, the energy function Ei(t) assumes its highest value overthe period. Then the energy keeps dissipating, as shown in Figure 4.1.The criteria of the dynamic stability can then be developed by comparing the initialvalues of energy function of consecutive system configurations. In general, we havemax{Ei (t)} = max{Ei (tm) : ti (t,72 ) = 0; Ei (ti ); Ei(ti+i ); t Edepending on the characteristics of the energy function.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^59E(ti) Configuration i ti ti Configuration i+1 6+ 1ti++ 1Figure 4.1: Energy Function4.3 Modeling of Structural PerturbationsIn the analysis of the dynamic stability of an SVS, the key issue is to look at howthe energy function would change, if it does at all, with the variations of the systemstructure. It is known that the order of the overall system will change if the constraintcondition between subsystems varies. In the situation when the system order is varying,we no longer have dim{d i(t)} = dim{di+i (t)}. The parameter matrices of differentsystem configurations cannot be directly compared to each other due to the fact thatdim{Mi}. Hence, the theorems for stability evaluation of an SVS asderived in Chapter 3 become invalid.In order to study the stability of the order-varying SVS, the structural perturbationhas to be modeled first. Considering the fact that the orders of two consecutive systemChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^60configurations (ith and i lth for instance) are not equal, we assumedim{^= nidimIdi+11^ni-Fiandni+i = Ani+iAni+i more (or less) degrees of freedom are added to (or subtracted from) the previoussystem configuration i after the system structure varies. Hence, Ani +i more coordinatesare required in order to fully define the dynamics of the new system configuration i 1.We selectd i+1 = [ dAi+1 (i)as(t) (4.47)where the symbol "," is used to indicate that this part of di+1 is inherited from thecoordinate di(t) of the previous configuration. clAi+i (t) is the new coordinates added tothe new system configuration. Then, we partition the parameter matrices accordinglymf+1Mi+i(mf+i)TKfA44,41k.7+1(4.48)Ki+1= ( 1c41)T K i14 1where MI: and IV: are of the same dimensions as Mi and Ki . Mpi+1 and KAi+1 can beconsidered as parameter matrices describing the newly-created part of the SVS due tothe structural perturbation. m4 1 and 1c4 1 can be considered as parameter matrices ofthe connection between the original part and the newly-created part of the SVS.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^61Since MY, KY have the same dimension as Mi and Ki, we can define the perturbationof parameter matrices on the previous system configuration as^AMY = MY — Mi^ (4.49)AKY = KY — Ki (4.50)where AMY and AK': are the perturbations of parameter matrices on the previous con-figuration. On the other hand, we can defineMiMi 0 0^0(4.51)KiKi 00^0where the dimensions of the Mi and Ki are compatible with those of Mi +1 andTherefore, we haveAMi+1 = Mi+i — (4.52)AKi+i = Ki+1 — KiSubstituting equation (4.48) and (4.51) into equation (4.52), we obtainAMi+ iOMp^m,+1(m41)T mAi+i (4.53)AK i+iOKp^1c7+1(1cf4.1)T^Icfri+1An+1 and AK i+1 can be considered as generalized parameter perturbation matrices forconfiguration i+1. They describe the change of the system model due to the structuralvariation occurring at time instant t = ti.• Example: Consider the system described in figure 4.2. We assume that initiallythe two switches are both on.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^62Subsystem #1d11^d12K11^K12K21^K22Hd21^d22Subsystem #2Figure 4.2: Example of A Varying-Order SVSThe system is actually the same as the one shown in (a) of figure 4.3. At timet = t i , switch #2 is turned off so that the system takes a new configuration, as isshown in (b) of figure 4.3. The parameter matrices for system configuration #1 areM1 =7111 0 k2 —k2[kiK1 =0 M2 —k2 k2At t = t i , the system configuration changes. Its order increases by 1.The parameter matrices for the system configuration #2 areM2 =m1 0 0 lc;0 m2 0 K2 —1c; k; 00 0 M3 —14 0 Ici3Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^63ml'(b)Figure 4.3: Structural Variation of the Example SystemChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^64We partition the parameter matrices of M2 and K2 asMT 0M2 —0 M;K1K2 =KwhereMi = Mil^0 KT. = k;^—k;0^m2 k;—14ka2 =0Hence, the perturbation of the parameter matrices on configuration #1 can bedetermined,AMT = — M i =1/^0 M1 00 M2^0 M20^00 m2 —m2[ ki, -k12^ki + k2 -k2_^-4 k;^—k2^k2. [—k2^—(4 - k2 )-(4 - k2) k2 - k2where m il = m1 and kli = ki have been used. Also by equation (4.51), we haveM1 01v11 =0 0ml 0 00 m2 00 0 0 AKT = Ki — K i =Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^65Ki =K1 00^0k2—k2—k2k2000 0 0Hence,mi 0 0AM2 = M2 — M 1 = 0 m2 00 0 m3m1 0 00 m2 00 0 00^0^00 m; - M2 00^0^m3k; -k; —k;AK 2 = K2 — K1 = —k; k; 0—k3 0^k31 _—k2^—(4 - k2 ) 14=^-(14 - k2 ) 14 - k2^014^0^14—k2 0k2 00 0k2—k20which are the generalized parameter perturbation matrices of configuration #2.4.4 Modeling of Switching InstantsAs has been discussed in Chapter 2, any SVS can been modeled by a series of config-urations and switching instants over a period of time. Each configuration of the SVScan be treated as a time-invariant system and the structural change of the SVS occursat the switching instant. In this section, the switching instant will be analyzed and itsChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^66dynamic model will be developed. It has to be pointed out that since the model of theswitching instant is developed for the analysis of the stability of the SVS, the emphasisis placed on the dynamic characteristics rather than the physical characteristics of theswitching instant. More detailed analysis of the physical characteristics is being carriedout in our Industrial Automation Laboratory in another project.4.4.1 Modeling of Switching Instants by Process CompatibilityIn order to illustrate the idea of process compatibility, we start with a simple system whichinitially consists of one mass node with one degree of freedom (d.o.f). The schematicdiagram of the system is shown in Figure 4.4. An external impulsive force is applied onthe mass node at the time t. As a result, the mass is broken into two smaller mass nodesand each of these smaller masses will have one d.o.f.At the instant of the break-up, the system momentum may change due to the appli-cation of the external impulsive force. After the separation, the system total energy mayalso change If it goes up, we say the variation of the structure makes the system unsta-ble. In other words, the system has the trend of increasing kinetic energy and therefore issaid to be dynamically unstable. On the other hand, if the kinetic energy level remainsor even decreases, the system is said to be dynamically stable. The key issue here is todetermine the change of the energy of the system and find its varying trend.In order to calculate the energy of the system, the velocities of the mass nodes haveto be determined first. It is known thatF = ppoat — ppre M1V1 M2V2 — my (4.54)where ppost and ppre are the momenta of the system before and after the break-up. F isthe external impulsive force applied on the initial mass node. v1 and v2 are the velocitiesof the two mass nodes after the separation. v is the velocity of the mass node beforeFdt- tChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^67mi •Tel • rn2 t—tTd2Figure 4.4: Illustration of A Breaking Instant (1)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^68separation. For simplicity, we assume that the mass node is broken into two equal parts,i.e., m 1 = m2 = mp = m/2, which is called an equal split. The equal split will lead tov1 = v2 = vp . ThereforeF = ppoid — ppr e = 2 mpvp — my= mvp — my = m(vp — v)^(4.55)F= vP — V = —m (4.56)OrFV = —P m(4.57)It can be seen that the velocity of post-separation v p is determined by the velocity ofpre-separation v and the external impulsive force . F which is applied on the mass nodeat the switching instant. Equation (4.56) determines the change of the velocity of theexample system. Since the displacement of the system cannot change instantly, we haved = d1 = d2 (4.58)where the definitions of d, d1 and d2 can be found in Figure 4.4. If the switching instantof the system can be modeled by process compatibility relations such as equation (4.57)and (4.58), we say that the system is in process compatibility at the switching instant.For the system shown, the kinetic energy function for the initial configuration is1^2Ep,.e = 2my (4.59)where m is the mass and v is the velocity of the mass node. After the system is brokenup, the kinetic energy function becomes1^2Epost = 21 2m2v 2Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^69l rnpv2 17npvp22^P 21rn v2 = —97/V 2P P 2 P(4.60)where mp is the mass of two smaller mass nodes and vp is the velocity of two mass nodesafter equal split. Based on two energy functions, we can find the varying trend of thekinetic energy due to the structural variation.AE Epost Epre1^1= — 7711) 2 — —777. V 22^P 22m (v2P _ u_2) (4.61)It can be seen that the sign of LE is dependent on the difference of vp and v. IfV 2 > V 2thenAE > 0The system would be said to be dynamically unstable after the break-up. On the otherhand, ifV 2P < v2thenAE < 0and the system would be said to be dynamically stable after the break-up.Substituting equation (4.57) into equation (4.61), we have1AE = 2m[(v + ;1-) 2 — v2 ]2-1 m [v 2 + 2—Fv + ( 1 )2 — v 2 ]  m21^F[2Fv (4.62)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^70It can be seen that the stability condition of the system can then be stated in terms ofthe direction of the externally applied impulsive force F. As long as F is applied in thesame direction as v, we would haveAE > 0Therefore we know the system energy is going to increase after the break-up and con-sequently the system is dynamically unstable. For the system to be dynamically stable,we must have AE < 0, which meansF22Fv — <0mSince F has to be applied in the opposite direction to the v, we have—2111•Iv1+  F12mOr0-2 I v I ^ <FinallyIF15 2 1v 1mThis result indicates that the external impulsive force has to be applied in the oppositedirection of the velocity of the mass node and its magnitude must lie within certain rangeif we don't want to increase the energy of the system. In other words, if the mass node ispushed forward, or pushed backward too hard, the system would be dynamically unstable.F has to be applied in a certain direction and stay within a certain range in order forthe system to be dynamically stable.In order to incorporate this break-up model into the analysis of stability of generalorder-varying SVS, we study the multi-boundary-node breaking process. Looking at theChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^71Figure 4.5, we can see there are # boundary nodes which are to be split into two separatesmaller mass nodes at time t = ti. After the break-up, there would be 2/3 mass nodesproduced from the original # boundary nodes.boundarysubsystem #1Figure 4.5: Schematic Diagram of the Order-Varying SVSAlthough these boundary nodes are connected to both subsystems, we assume thatat the instance of structural variation, the forces applied on boundary nodes from otherinternal parts of the system are negligible compared to the externally applied impulsiveforce F. Therefore, at the switching instant, the system boundary can be considered asa group of isolated mass nodes as is shown in Figure 4.6.vp2vpivpi•v2vifl•vlvplf2 ir * vp2Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^72* vpl^ fiFigure 4.6: Schematics of the SVS at the Breaking InstanceBased on previous analysis, it is not difficult to obtainvpi = vimifor each of the # boundary nodes. Writing them in vector form, we haveV = V + MbnldF(4.63)(4.64)OrV = V + AV^ (4.65)where vi, = {vpi , vp2 , • • •, vpo}T , v = {vi , v2 , • • •, V iii}T Mbnd = diag {m i , m2 , • • • , ms},F = f2, • • •, MT, and Ay = mb:-- dr. The physical meaning of these variables isclearly shown in figure (4.6). v is the pre-breaking velocity vector of the /3 boundarynodes. vp is the post-breaking velocity vector of the 2/3 newly-created mass nodes. M bndis the mass matrix of the (3 boundary nodes before break-up. F is the vector of theexternally applied impulsive force at the breaking instance. From equation (4.65), thevelocities of the pre-breaking boundary nodes and post-breaking mass nodes are relatedthrough the impulsive force F.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^734.4.2 Modeling of Switching Instants by Motion CompatibilityIn the previous section, the process compatibility has been used in the analysis of theswitching instants of the SVS. It has been seen that the velocities of the boundarynodes of the SVS change instantly at the switching instant due to the externally appliedimpulsive forces. The important point is that the external impulsive force is applied inthe same d.o.f. of the boundary nodes. Hence, the velocities of the post-breaking massnodes change after the switching instant. If the impulsive force which causes the breakingof the boundary mass nodes is applied in a slightly different way, the pre-breaking velocityand post-breaking velocity of the boundary mass nodes will be exactly the same. In thissituation, motion compatibility occurs, and will be used to analyze the switching instants.As has been discussed before, the boundary mass nodes can be modeled as a group ofisolated mass nodes without any connection to any subsystems at the switching instants.To demonstrate the idea of motion compatibility, we present the model for each of theboundary nodes in Figure 4.7. The boundary node can be thought of being composed oftwo equal smaller mass nodes. The external impulsive force is applied on the boundarynode at time t, which will be broken into two mass nodes and each of them has one d.o.f..In this case, the external impulsive force is applied in the d.o.f. in which the mass isconstrained.If the equal split is assumed, we will have m 1 = m2 = m/2. Hence,1Ppost =^m2v2 = m(vi + v2)ppre = MV(4.66)(4.67)where ppost and pp, are the momenta of the system before and after the break-up. v isthe velocity of the mass node before separation. v1 and v2 are the velocities of the twomass nodes after the separation. It is not difficult to obtain v 1 = v2 vp if m 1 = m2.Fm2dChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^74Figure 4.7: Illustration of A Breaking Instant (2)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^75According the law of conservation of momentum, we havePpre = Ppost^ (4.68)1my = —2myup vp) = myTherefore,(4.69)= VP^ (4.70)It can be seen that the velocity of post-separation yp is the same as that of pre-separationv. Since the displacement of the system cannot change instantly, we haved = d1 = d2^ (4.71)where the definitions of d, d1 and d2 can be found in figure (4.7). If the switching instantof the system can be modeled by equation (4.70) and (4.71), we say that the system isof motion compatibility at the switching instant.The kinetic energy function of the system for the initial configuration isEpre^1 my2^(4.72)After the system is broken up, the kinetic energy function becomes1^2 ^2—2 miv i + —2 m2Y 2Epost2 2( P^P^P2m)(v 2 + v 2 ) = mv 21 V 2= —2 mTherefore, we have(4.73)Epre = Epost^ (4.74)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^76It can be seen that the energy of the boundary nodes remains unchanged at the switchinginstant.By incorporating this break-up process into the stability analysis of multi-boundary-node order-varying SVS, we haveVpi = Vi^ (4.75)for each of the 13 boundary nodes. Writing them in the vector form, we havevp =^ (4.76)where vp {vp1 , vp2 , • • •, vo }T is the the post-breaking velocity vector of the 2/3 newly-created mass nodes. v = {v1 , v2 , • •, vo}T is the pre-breaking velocity vector of theboundary nodes.Comparing equations (4.57), (4.58) with equations (4.70), (4.71), we observe thatthe motion compatibility is actually a special case of the process compatibility withF = 0. When the external impulsive force which causes the break-up of the boundarynode is applied ih the d.o.f. in which the boundary node is constrained, we will haveF = 0. There is no instantaneous change in the velocity of the boundary node. Hence,the switching instant of the SVS can be analyzed by using motion compatibility. Onthe other hand, when the external impulsive force is applied in the same d.o.f. as theboundary node, F 0. There is an instantaneous change in the velocity of the boundarynode. Therefore, the switching instant of the SVS has to be analyzed by using processcompatibility.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^774.5 Analysis of Dynamic Stability Using Process CompatibilityTo evaluate the dynamic stability, the energy functions for two consecutive system con-figurations have to be calculated1 •Ei(t) = 2^2^ (4.77)Ei+i (t) = + (4.78)Substituting parameter matrix equation (4.52) into equation (4.78) yieldsEi+ I (fit) •^1= . 1--c1T±1 (tt) Mi+iiii+i(tt) + id 1 (tt) Ki+idi+i (tt)2a • — 1 T=-. f:141M (14i + AMi+l)ili-F1K) + id;+1 ( 18-i + AKi+l)di+11 •^• 1= idiFi (tt) g4iiii+1(tt) + 2d41(til) kid41(tt)1 • 1 T--Pic14 1 (tt) AMi-Fiai+i(tt) + -2-di+i (tt) AKi_Fi di+i (tt)= ei(tt) + Aei+i (tt) (4.79)whereei (t-iF )^4+1(it )^dr+i(tt) Kidi FlT1 ^2d 2+^Aei.+1(tt) = 2 2+—d• (V- ) + ^ ?' 1 (e ) AKi_F ldi+ 1 (tp )Substituting equation (4.47) and equation (4.51) into equation (4.80), we haveei(tt)= -2 [di (tt) Mi (EL(ti ) C17(tt)Ki Cli(tt)]1 .1i(tt) T Mi 0 .1i(tn—^3( .db,i+i (tt) 0^0 ilAi+1(tt)TKi 00^0CVOclAi+i (tn(4.80)(4.81)(4.82)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^78After the energy function is determined, the stability of the order-varying SVS can bestudied. From equation (4.65), the initial velocity of a configuration can be establishedin terms of the final velocity of the last configuration at a structural switching instant,aii-1(t,t)==Ai(ti-)+Aa,+1(ti) (4.83)where di+i(tt) is the initial velocity of the system configuration i 1, di(ti ) is the finalvelocity of the system configuration i with the dimension adjusted to the Cli+i(tt) andAili+i (ti) is the velocity perturbation introduced by process compatibility. di +i(tt) canbe divided into three elements,- Lint -di (t-1-)bnd d (it ) (4.84)a:n°(0where superscript int denotes internal nodes, superscript bnd denotes the boundary nodesand superscript new denotes newly-created nodes in the new system configuration. Asbefore, the symbol ",,," denotes inherited from the coordinate of the previous systemconfiguration. The physical meaning of every element of the coordinate can be seen inFigure 4.8.hintHence, we know that the first element d i (tt) is inherited from the coordinates ofthe internal nodes of the previous system configuration i. Since these internal nodes arenot changed at the instant of the structural variation, we have• intdi (ti = dint(q) (4.85)where Clnt(q) is the real coordinate of the internal nodes of the system configuration i.h bndFor the second element d i (ft), it is inherited from the coordinates of the boundarynodes of the previous system configuration and the mass nodes they correspond to areErn _pewU 1+1boundary mass nodes perturbed original^newly-createdboundary mass nodes mass nodesChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^79internal mass nodes^original internal mass nodesFigure 4.8: Definition of the Coordinatessplit at the instance of structural variation. Therefore, it can be related to ilEind(q) byequation (4.65), i.e.,bnd di (CiF) = dbnd(q) ACI!'nd(ti)^ (4.86)where dIrd(q) is the real coordinate of the boundary nodes of the system configurationi and .Ailnd(ti) is the velocity perturbation on the d/rd(ti)•The third element ainzit ) corresponds to the newly-created mass nodes, which donot exist in the previous system configuration. However, since they are produced fromthe boundary nodes of the previous system configuration, the following relation appliesto it if the equal split is assumed,bndCrilTeiu (tn = d i (tn= Cl!nd(g) Ail Ird(t i )^ (4.87)It has been shown from the previous analysis that the velocity perturbation at thetime t = ti is the function of external impulsive force F, i.e.,Aant i) (Mrd ) -1 F^ (4.88)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^80where Wfrd = diag{mrd, mr, • • •,mbond}, F =^f2, • • •, MT and p is the number ofthe boundary nodes of the previous system configuration i. Substituting equations (4.85),(4.86), (4.87) and (4.88) into equation (4.84) yields-^ ntLi^-di (tn^aiint )6nddi (in^ivrd(q) + Okra (ti)_ d,+1(is) _^at:4(in + Aat:nd(ti) _0amtn + Aar(t i )ab,nd(ti-) _^_ Add(ti)aint^) 0iltmtn + (Mbnd)i 1 Faknd(in ^(Mend) s 1 FComparing to equations (4.83), we haveci!nt(q)^Ai(q) =^aird(q)knd(q)0^Aai+l(ti) =^(mtind)-1 F (4.89)(4.90)(4.91)(mlind)-1 FAt the switching instant, there is a sudden change in velocity, which is determined bythe externally applied impulsive force F.Using this model of break-up, the dynamic stability of the order-varying SVS can beanalyzed by process compatibility. As previously derived in equation (4.79), we haveEi+i (tt) = ei(tt) Aei+i (tt)ai+i(tt) =Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^81withei(i)^1441(tt) miai+i(tn + -2-1 cq+i (tnftidi+iAei+i(i) = -2-1(141(tt) 6,1(41441(ts)We further assumeei(tt) = Ei(tT) AeF(ti )^(4.92)where AeF(ti) is a small perturbation term which is caused by the velocity perturbation,or more directly by the externally applied impulsive force F. Substituting equation (4.92)into equation (4.79) yieldsEi+i (tt)^Ei(tT) AEi(ti)^= Ei(tT) AeF(ti)^(4.93)where Ei(tT) is the energy value just before the structural variation. AEi (ti ) is the energyperturbation occurring at the instant of structural variation. This perturbation termconsists of two terms, one is the structural perturbation term Aei+i (tt) and the otheris the state variable perturbation term AeF(ti). In order for a system to be dynamicallystable, we must haveAEi(ti ) = Ei+i (tt) —^) < 0AeF(ti) Aei4. 1 (fil") < 0Since the condition of ei+i (tt) < 0 is easy to find, the condition of AEi(ti) < 0 can bedetermined if we can find the condition of eF(ti) < 0.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^82In order to determine AeF(ti), we substitute equation (4.83) into equation (4.80),ei(tt)^--14:141(tt)^+^kidi+i= 2[(a=(ti)2[( micAicin+ Aai+ictim + 2diT+1 (02 •[u-Ii(tnT Midi(tn 20Cli+1 (ti )T g/IiCli(Cr)IC/IjAili+1 (ti )]^1 d4 1 (tt)^ (4.94)Since _ dint(q)Cli(t i7) = Cl Eind(q)dLind(tz)anddint(ti)ai(tn = d ird(q)dtpd(c)we haveai(tn[ at.,,nd(tz )di (q )d!nd(q)(4.95)(4.96) T^= ICliK)^[Mi^d Erd(tr)^0 o^knd(q)= dROMiiii(tn•,T^„di (ti) Midi(q) (4.97)(4.98)(4.99)(4.100)anddT+1Tdi(tn^[Ki 0^di(q)cl /rd(q)^0 0^dIrd(ti- )= dT(q)K idi(g)Hence, we can rearrange equation (4.94) asei(tn = -j2 [CliT(q)^2AiliT+1(ti)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^83where+AC1T+i(ti)^2diT+1(tt) kidi+i1^1= 2 jT (ti- )MiCli(tT) 2 c1T(t -i- )Kidi(tT)+.64_ 1 (toicil ia i (t-i-)+-1-?A iT+1 (ii ) KLAa i+1 (ti )Ei(q) Aep(ti) (4.101)Ej(C) =^ CIT(tT)MiCli(tT)^. 4:1T(tT)Kidi(ti . )^ (4.102)AeF(ti)^ACIT1(ti) Midi(ti)T) + AilTF4 (ti) 1C4jAili+i (ti)^(4.103)The energy perturbation AeF.(ti) due to the velocity perturbation is then determined.We defineai[F, di(q)] = ACIT+i(ti) Midi(ti)^(4.104)then,Aep(ti )^ai[F, di(ti)]^ 6,(1T+. 1 (ti )^ (4. 1 05 )ai(F, di(C)) can be considered as a control variable that provides the constraint conditionon the externally applied impulsive force F. If F satisfies the condition, the system wouldbe dynamically stable. Based on equation (4.105), the criteria for evaluating the dynamicstability of the order-varying SVS using process compatibility can be derived.Theorem 4.1 Assume that dynamics of the structural variation is dominated by theprocess compatibility. An order-varying SVS would be dynamically stable if AeF (ti )Aei+I (tt) < 0.Theorem 4.2 Assume that dynamics of the structural variation is dominated by theprocess compatibility. An order-varying SVS would be dynamically stable if1. Both AM i+1 and AKi+i are negative semi-definite,Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^842. 2cxj[F, d i (tn]^g/liAili+1(ti) < o.Proof: Since both AMi+1 and AKi+i are negative semi-definite,AMi+iiii-f-i(in dT-Fi(tt) AKi+idi+i(tt) < 0Since 2ai [F,^)] c14 1 (tt) AKi+idi+i (tn < 0, we haveai[F, di(tn] ACIT+. 1 (ti) 'miAaj+1 (ti )+ (V4. 1 (tt)^+ dr+1(4- ) AKi+idi+i(i) < 0i.e., AeF(ti)+6,ei+i (in < 0. According to the definition, we know that the configurationi 1 of the SVS is dynamically stable.Theorem 4.3 Assume that dynamics of the structural variation is dominated by theprocess compatibility. A conservative order-varying SVS would be dynamically unstableiff AeF(ti) > 0.More specifically, we haveTheorem 4.4 Assume that dynamics of the structural variation is dominated by theprocess compatibility. A conservative order-varying SVS would be dynamically unstableif1. Both AMi+i and AKi+ i are positive semi-definite,2. ai[F,di(tn] > 0Proof: Since both AMi+i and AKi+i are positive semi-definite,iiiT+1 (0 AA/1414441(o + dr+i (tt) AK i+lai+1 (in > 0Considering ai [F, di (tn] > 0 and g/li is positive semi-definite, we have[F, d i (tT)]^AaT+i (ti )+ 2 tzliT+ i (tt)^+ -21 d iT+1(ti+ )^> 0Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^85i.e., AeF(ti)+ ei+i (tt) > 0. According to the definition, we know that the configurationi 1 of the SVS is dynamically unstable.4.5.1 Perturbation on Kinetic Energy FunctionIt has been shown that the energy function changes suddenly at the structural switchinginstant. At the structural switching instant, the forces applied on mass nodes from otherinternal parts of the system can be considered relatively small compared to the externallyapplied impulsive force and therefore are neglected. The mass nodes can then be treatedas a group of isolated ones and the mass matrix of the system at the switching instantcan be expressed by two diagonal matrices,Ming^0Matut(q ) -7 (4.106)0^MondMint^0^0(tt) = 0^M ^0 (4.107)0^0^Mnewwhere M.„„ t (t i- ) is the mass matrix pre-structural switching and M s.t (tt) is the massmatrix post-structural switching. M—int Mbnd ' Kid and Mneu, represent system internal,boundary, perturbed boundary and newly-created mass nodes. They are all diagonal.Since no mass is added to or removed from the system, the system overall mass is aconstant., Mne„, is separated from Mbnd. Therefore,1\46nd = 41)nd — M..^ (4.108)Using the velocity relation given in equation (4.89), we can write the kinetic energyfunction at the switching instant,E7;:e(C) =^Matut(q) CVOChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^86T1 aint(q^[Mint^0 1^Clint(C )2 Gird )^o Mbnd^ard(o[aint(tn]T mint ipi.nt(q) —1 [1:16ind(07 Mbnd ilbnd(q)2^ 2and1 •E;:st(in =^Mewt(tn aid-i(tnint - T -di (ti)^Mint^0^0bnddi (ti)^0 M^0bndnew (41 0^0 Mnew- h int^-di (tibnddi (tianT(tt)1 int^,7 int^1 • bnd^• bnd=^(ti )iT Mint di (ti)^(t-iF)JT Mbnd d i (tif)-F 2 [d7_7(tn7 Mnewwhere Epkee (tn is the kinetic energy function just before the structural variation andEptest (tn is the kinetic energy function just after the structural variation. The change ofthe kinetic energy function at the switching instant can then be expressed asAEke(ti) = EI:it(tt) E i;:e(tn1 • int^• int^1i [di (01 1 Mint di (ti) — Ant (c1 z bnd^T ^bnd^1 •^)1 Mint Cl int (t -i- )=-+[di (ti )]T Mbnd di (tn — i[cl"d(tn]T Mbnd a!nd(tn-F[irs+1u(it)] 7' Mnew am(ti)Substituting equation (4.85) and equation (4.108) into equation (4.109) yieldsAEke(ti) =1 bnd^ • bnd2- [die ^(Mbnd — Mnew) d (riF)1 t— [dr " (tn]T Mbnd^) 1^(iniT Mnew^(rif )1 • bnd^bnd h bnd2 [d; (ti^Mbnd d i (tn — —2 [clbndi (tt)]T Mnew di (tt)1 .:. (4.109)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^872_1 1rat,nd(r. iT mbnd abnd (tnIm 2 Las+.1^/J(t)iT Nine. azie_7(tt) (4.110)Also substituting equation (4.86) and equation (4.87) into equation (4.110), we can have1 •AEke(ti) =^[clt:snd(ti") Aar(ti)iT Mid [ard(tz) + Altd(toi[iirc 1 )IT Mond alimd(q )1^•2_[Ad Ird(ti )]T mbnd alrd(r )- •1^•+ 2 [Ad d(ii)]T Mond Dm(ti)1 •+i[dr(ti Mbnd Aar(ti)Since Mid is diagonal,AEke(t i )^[Aard(ii)]T mbnd ard(c- )1^•+ _2 [,AdIrd(to]T Mid Aed(ii)^(4.111)It can be seen that the perturbation on kinetic energy is determined by (1) the param-eter matrix of the boundary mass, (2) the velocity perturbation due to the externallyapplied impulsive force which causes the system structural variation and (3) the time theimpulsive force is applied.4.5.2 Perturbation on Potential Energy FunctionThe perturbation on the potential energy function of an SVS can be derived as followsTEig(q =^(ti ) Kau,t(q) di(q)= 2clr(q) Ki di(C)andEr,:: t (t -n — 2ciT+1(4)Ksuit(tp) di+1(0= 2 4:1171_1(ft) Ki+iChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^88where K„„t (q) is actually the stiffness matrix of configuration i and K oeut (tt) is thestiffness matrix of configuration i + 1. Ep ,, (q) is the potential energy function justbefore the structural variation and EpPot(tt) is the potential energy function just afterthe structural variation. The change of the potential energy function at the switchinginstant can then be expressed asEgott (tt) — Eret (tT)1^ 1= ic14(tt) Ki+1 di+i(tt) —^Ki di(q)1 1= 2d T ,(t+) (K • + AKi+i ) di+i (tt) — icIT(ti) Ki di(g)2+-Considering equation (4.99), we have1 m1AEP°t(ti) = 2^(t7r) AKi+iH-(4.112)The perturbation on the system potential energy is determined by the generalized stiffnessperturbation matrix.Therefore, the perturbation on the energy function at the switching instant isAE(ti )^AEke(ti) AEP°t (ti)= [Aa Ird (ti)]T mt.nd abnd(tz) + {Aipind(ti )jr Mond Aabind(to1+-2_dr+i (in AKi+1 di+i (it)abnd[F,aird(ini + abnd[F] + a[AKi+i )wherebnd [F abnd (0] = [Aierd(ti)iT Mend at:nd(ti-)^abnd [F]^- {,Aatrd(toiT mb„d AC117'd (ti)^a[AKi+i j^-2cir+1(tn AKi+i di+i (tt)The dynamic stability of the SVS can then be restated as follows:(4. 113).6,EPc' t (ti ) =Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^89Theorem 4.5 Assume that dynamics of the structural variation is dominated by theprocess compatibility. An order-varying SVS would be dynamically stable if1. AKi+i is negative semi-definite,2. abnd[F, ClIrd(tZ)]^—abnd[11.Theorem 4.6 Assume that dynamics of the structural variation is dominated by theprocess compatibility. A conservative order-varying SVS would be dynamically unstableif1. AKi+i is positive semi-definite,2. abnd^abind(tf )] > — abnd [F] .The proof of Theorem 4.5 and Theorem 4.6 is straightforward. The negativesemidefiniteness of the AKi +1 would lead to«[AKi+i ] = diT+1 (tt) AKi+i di+i (tn < 0If a bnd [F , Cl /rd(t;- )]^—abnd [F] ,AE(ti) < 0and the SVS will be dynamically stable. Condition #1 is determined by system struc-tural variation and condition #2 is determined by the way the externally applied impul-sive force is applied on the boundary mass nodes. In general, if the impulsive force isapplied oppositely to the moving direction of the boundary mass nodes within a certainrange, the system dynamic stability condition would be satisfied. The proof of Theorem4.6 can be carried out in exactly the same way as that of Theorem 4.5.In the following example, the application of Theorem 4.5 and Theorem 4.6 willbe demonstrated.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^90fd(a)dlH(b)Figure 4.9: Example SystemChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^91• Example: Consider the system described in (a) of Figure 4.9.The equation of motion is given byand kd 0, d(0) = doandM 1 =- Mbnd m, K 1 = kAt time t t i , an external impulsive force f (the horizontal one) is applied on themass node and the mass is divided into two pieces instantly. The system takes anew configuration, which is shown in (b) of the Figure 4.9. The equations of motionarekdi = 0m2(12 = 0We haveM2 =ml^0 K2 = [lc 0which gives0^M2 0^0AK2 = K2^klOk^0° Ok^0° = 0The velocity perturbation Ad due to f can be calculated as followsfChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^92Hence,1^f^pab"d [F] = m ( 771 ) 2 =2^ 2mf^:abnd [F,Cerd(tn] = 7.77, m d(t 17) = fSince LK 2 = 0, the system dynamic stability is determined by the relation betweenabnd [F] and abnd [F, Cir(tn] only. If we assume a bnd [F, Cl ird(g)] < —abnd [F], wehaveab"d [F, ard(til] = f d(ti ) C —abnd [F] = — f 22mIt is not difficult to see that the external impulsive force f has to be applied in theopposite direction of the d(t), in order for the system to be dynamically stable, i.e.,abnd [F,ard(q)]= —f d(ti) < ^= —Finallyf < 2mci(q)According to Theorem 4.5, the system will be dynamically stable if f is appliedin the opposite direction of the d(t) and f < 2mcktn.On the other hand, the system would be dynamically unstable ifabnd [F,^= f d(ti) > —abnd[F] = If f is applied in the same direction as a(q ), the above equation would be satisfied.According to Theorem 4.6, the system would be dynamically unstable.f22mChapter 4. Stability of. Structurally-Varying Systems With Time-Varying Order^93The energy function before the structural variation is1Ei (t) = E(tT) = -2 m d2 (ti") + -2 k d2 (tT) = -2 k d2 (0)The energy function after the structural variation isE2 (t) E(4 ) = 2 mi di(it) + ma di(ti)^k di(ti)Since^4(4) =^+ Acii (t 1 ) = di (ti ) +^ci2(in =^= di (ti ) +=we have; E2 = 1i mi [di(ti)+ f+ f;1 2 + 1i m2 [ai(ti) + -i-rf-] 2 + 12 k di (C)^1^:^f^1= i m [cii (tT) + 7-71, ] 2 + i k di(ti)1^. 1= 2 m [CC) + (— ) 2 + 2 — d1(r)] + —2 k di (tT)1^m^m 11^.^1 1 f2 = -- rn d?(C)d- i k di (C)-1- -- ( -77-n- ) + fcti (tT)1 f2= Ei + i (-71 ) + feii (q)Therefore,1 f 2AE2 = E2^= -2 (—m) fai(ti7)1 f2AE2 E2 El = -2 ( m— ) fd(tilChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^94If f is applied in the opposite direction to d(C), we have1 12^•AE2 =^) — f d(tT)If f < 2md(q) (i.e., f /2m —^< 0), we will have1 f2^ff^f^di(tT)] 06.E2 < 0which proves that the system is dynamically stable.On the other hand, if f is applied in the same direction as d(tT), it is not difficultto see that1 f2AE2 = -2 ( —m ) fd(iT)— (I!2^) f (kr) > 0m^1Therefore, the system is dynamically unstable. It can be seen that the previouslyderived analytical results have been verified.The numerical simulation results are shown in Figure 4.10 and Figure 4.11 for dy-namically stable and dynamically unstable cases respectively. The initial conditionsof the system d(0) = 10, d(0) = 0 are assumed. The following parameters are usedin the numerical simulation,m = 0.05kg,^ml = m2 = 0.025kg,^k = 6N/mThe impulsive forces f = 1.0N and f = —1.5N are used in stable and unstablecases respectively. The symbols are defined as follows,Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^95—d(t): position of the mass node before structural variation; di (t), d2 (t), posi-tions of the mass nodes after structural variation.—v(t): velocity of the mass node before structural variation; v i (t), v2 (t), veloc-ities of the mass nodes after structural variation.—E(t): energy of the mass node before structural variation; E 1 (t), E2 (t), energyof the mass nodes after structural variation.It can be seen from (a), (b) of Figure 4.10 that although the change of the systemposition is continuous, there is a sudden change in the system velocity. The changeis caused by the external impulsive force f. In this case, f is applied in the samedirection as d(ti) and also f < 2md(C). The system energy level decreases afterthe structural variation, which can be observed from (c) of Figure 4.10. Hence, thesystem is dynamically stable. In another case shown in Figure 4.11, f is applied inthe same direction as d(tT). There is also a sudden jump in the velocity, which canbe observed from (b) of Figure 4.11. From (c) of Figure 4.11, we can see that thesystem energy level increases after the structural variation. Hence, the system isdynamically unstable.4.5.3 Experimental Study of Dynamic Stability Using Process CompatibilityIn order to further illustrate the application of analytical results developed in this researchand verify the numerical simulation results, an experiment is carried out.The experimental setup consists of three parts, a mechanical moving device, an EKTA1000 motion analyser (i.e., high-speed camera) and an image acquisition and processingsystem. The mechanical moving device, which is shown in Figure 4.12, is composedof four SPB 8 super pillow blocks with linear bearings and a tubular solenoid. The0.4000.200^0.300TIME (SECOND)350 (c)300250 E(t)200>-0150El (t)+E2(t)E2(t)El (t)d(t)• • *dl (t). ......• * _ ......^•0.000 0.1000.000 0.100 0.200^0.300 0.400(a)+^d2(t)+.^+ ++2010-20TIME (SECOND)1507100 750OL>1-50 - v(t)-100(b)* * * • * • * *v1 (t)^*+ + + + + + + + + + + + + + + + + + + + + + + + +v2(t)X X XX XX X X XX X X X X X X X X XX XXX XX X++++++++++++++++++++++++++0.000^0.100^0.200^0.300^0.400TIME (SECOND)z1-1-1^10050Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^96Figure 4.10: Result of Numerical Simulation: A Dynamic Stable Case Using ProcessCompatibility750 (c)X X X X X X X X XX X X X X X X X X XX XX X X X XEl (t)+E2(t)+++++++++++++++++++++E(t)500>-0wzw250 -0.4000.000.^.^.0.100 0.200^0.30020(a)0.100 0.4000.200^0.300TIME (SECOND)0.000*d (t)•+ •+ 4... .dl (t). . • • • *• •-1000i=co^-100-20++ +^d2(t)+ +t +150100500:^-Ow>-50-100_^v( t)•-150* * • • * *vl (t)^•v2(t)+++++++++++++++++++++++++TIME (SECOND)(b)•• * •E2(t)0.000^0.100^0.200^0.300^0.400TIME (SECOND)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^97Figure 4.11: Result of Numerical Simulation: A Dynamic Unstable Case Using ProcessCompatibilityChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^98Figure 4.12: A Moving Mechanical DeviceChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order^99Figure 4.13: Image Processing SystemChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 100blocks are used to emulate two mass nodes and the tubular solenoid is used to emulatea switchable connection between the mass nodes. The tubular solenoid is controlled byan electrical signal. The mass of each of the two mass nodes is 0.52kg. One of themass nodes is connected to a spring. The stiffness of the spring is 70N/m. The otherend of the spring is fixed to the supporting structure. The external impulsive force f isgenerated by a 100psi air jet, which can be turned on and off by a switch. The EKTA1000 motion analyser is used to collect and store the experimental data. The samplingrate is set at 500 frames/second. The position is measured by using the EKTA 1000motion analyser. The velocity and energy are calculated from the measured positiondata. After the position data is collected, it is then sent to a PC-based image processingsystem, which is shown in Figure 4.13, and the data is then processed there.Initially, two mass nodes are connected through the tubular solenoid. An initialposition is given to the mass nodes. When they are released, they start to move. Ata point, an air jet is applied to the mass nodes in the same direction as the velocity ofthe mass nodes and the tubular solenoid is activated so that the two mass nodes areseparated. From previous analysis, we know that the air jet will cause the system to bedynamically unstable.The experimental results are shown in Figure 4.14 through Figure 4.16. Figure 4.14shows the position profile of the system. Two mass nodes are separated and go in differentways after the solenoid is activated. Figure 4.15 shows the velocity profiles. The velocitydata is calculated from the experimental position data. It can be seen that it is verynoisy. In order to eliminate the nosie, a low-pass filter with a cutoff frequency of 8 hzis designed using MATLAB. The velocity signal is filtered and the true velocity signalcan then be obtained. The change of the velocity at switching instant can be seen inthe Figure 4.15. The sudden change of the velocity is caused by the air jet. Since thesystem energy is increased, which can be seen in Figure 4.16, we know that the systemd2(t)100E 500co0 d1(t)0.00^0.10^020^0.30^0.40^0.50^0.60TIME (SECOND)-50d(t)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 101Figure 4.14: Experimental Result: Position Profileis unstable after the change of the structure takes place.This experiment further illustrates the concept of dynamic stability of the SVS (anddynamic instability of the SVS) and shows the application of previously derived theo-retical results.4.6 Analysis of Dynamic Stability Using Motion CompatibilityIn the previous section, the stability of the order-varying SVS has been studied using theprocess compatibility. In this section, the stability of the order-varying SVS is analyzedusing the motion compatibility. By motion compatibility, we have at the instant ofstructural variation,di(C) = a,(0a,(q) = a, (0According to equation (4.111), we haveAEke(ti) = [Aa ird (ti)]T Mbnd alrd(t -i- )+ 1 [,a,ard(ti)]T Mbnd Aknd (ti)(4.114)di (t)-0.50-0.55 di (t)-0.60-0.65d*0.300 0.325 0.350 0.375d2(t)0.50.00^0.10^0.20^0.30^0.40^0.50^0.60TIME (SECOND)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 102Figure 4.15: Experimental Result: Velocity ProfileE (t)0.40=3"^0.30>-0w 0.20zw0.100.00^0.10^0.20^0.30^0.40TIME (SECOND)0.50^0.60Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 103Figure 4.16: Experimental Result: Energy ProfileThe motion compatibility (described in equation (4.114)) leads to Astnd(t i) = 0 at theswitching instant. Hence,L\Eke(ti) = 0^ (4.115)which means that there is no change of kinetic energy at the instant of structural vari-ation. It is not difficult to see that motion compatibility is actually a special case ofprocess compatibility. The perturbation on the energy function is solely determined bythe potential energy function, i.e.,1AE(ti) = AEP°t(ti) = idT1-1(in AKi+i di+i(in^ (4.116)The stability theorem can then be stated according to the generalized stiffness perturba-tion matrix AICiA-1.Theorem 4.7: Assume that dynamics of the structural variation is dominated bymotion compatibility. If AKi+i is negative semi-definite, the configuration i+1 will bedynamically stable.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 104Proof: As is shown in equation (4.116), AE(ti) is of quadratic form. If AKi +i isnegative semi-definite, we will have21—cIT1^AKi+idi+i (tt) < 0t+no matter what the values the di +i(tt) takes. Therefore, AE(ti) < 0. According tothe definition, we know the configuration i+1 of the SVS is dynamically stable, whichconcludes the proof.Theorem 4.8: Assume that dynamics of the structural variation is dominated bymotion compatibility. For any conservative SVS, if AKi +i is positive definite, the con-figuration i+1 will be dynamically unstable.Proof: Since the SVS is a conservative system, we havemax{Ei(t)} = Eicmax{Ei+i (t)} =where Eic and Ei+ ic are constant over the time period [ti_ 1 , ti] and [ti, ti+i ] respectively.If AKi+i is positive definite, AE(ti) — Eic > 0, i.e., Ei+ic > Eic. There is asudden jump of energy at the instant t = ti , which makes the configuration i+1 of thesystem dynamically unstable.Theorem 4.7 and Theorem 4.8 provide criteria for evaluation of the dynamic sta-bility and instability of the order-varying SVS for the motion compatibility case. Usingthese two theorems, the dynamic stability of the order-varying SVS can be predictedbased on given structural perturbation, which is described by AKi+1.• Example: Consider the system described in (a) of Figure 4.9 again and assume thef is applied perpendicularly to the direction of motion of the mass node, which willlead to the process of the structural variation dominated by motion compatibility.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 105Then the velocity perturbation is Ad = 0. Therefore,abnd [F , abind(q)] = 0abnd [1] = 0It is known that AK 2 0, i.e., a[AKi+i] = 0. According to Theorem 4.7, weknow that the system is dynamically stable. In fact, the system energy remainsunchanged in this case although the system structure has changed.The numerical simulation results are shown in figure (4.17). Parameters used arem = 0.05kg,^m1 = m2 = 0.025kg,^k = 6N/mThe symbols are defined as follows,— d(t): position of the mass node before structural variation; d 1 (t), d2 (t), posi-tions of the mass nodes after structural variation.—v(t): velocity of the mass node before structural variation; v i (t), v2 (t), veloc-ities of the mass nodes after structural variation.—E(t): energy of the mass node before structural variation; E1 (t), E2 (t), energyof the mass nodes after structural variation.It can be seen from (a) of Figure 4.17 that after the two mass nodes separate,they take different trajectories. There is no sudden change in either the positionor velocity profiles, which can be observed from (a) and (b) of Figure (4.17) . Theenergy remains unchanged after the structural variation. Therefore, the system isdynamically stable.(a)d(t)0.100 0.200^0.300 0.400201000co^-10 -Oa_-200.000dl(t)^• • • •• *+d2(t)+ + + ++ +TIME (SECOND)(b) * * * ** •* •^vl (t) *• •v(t)v2(t)0.000 0.100 0.200^0.300 0.4001501005000-50-100TIME (SECOND)_ - + + + + + + + + + + + + + + + + + + + + + + + + + +E(t)400350300250=-3X XX XXX X X X X X X X X X X X X X X XX X X X X(c)E1 (t)+E2(t)0.4000.000^0.100 0.200^0.300E1 (t)++++++++++++++++++++++++++E2(t)0 200U-1^15010050TIME (SECOND)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 106Figure 4.17: Result of Numerical Simulation: A Dynamic Stable Case using MotionCompatibilityChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 1074.7 Issues of Dynamic Control of the SVSIn this section, the control strategies and their implementation will be discussed. Ithas to be pointed out that since this research project originated from interest in theanalysis of SVS dynamics and its stability, the previously developed analysis approachis mainly for that purpose. It is not our attempt here to present an exhaustive or eventhe representative picture of the general control theory of time-varying dynamic systems.The objective instead is to demonstrate what control strategies are applicable in thestabilization and control of SVS and how these control strategies can be implemented.Since the systems we have investigated are of matrix-second-order form, the controlproblem will be discussed in "mechanical" or physical coordinates.The systems considered have the formm(t) + c(t) + K(t) d f^ (4.117)where M(t), C(t) and K(t) are the mass matrix, damping and stiffness matrices for theoverall system, d is the displacement vector for the overall system, and f is the controlforce generated from n(t) force actuators which can be further represented byf = B(t)u^ (4.118)where f denotes the control force, u denotes the n(t) control inputs, one for each controldevice (actuator), and B(t) can be considered as actuator gain matrix. In general, itsdimensions are determined by (1) the dimension of the system mass matrix M(t) and(2) the dimension of the control input u, which is n(t) x 1. It can be observed that sincethe dimension of the system mass matrix is not constant, the dimension of the controlforce could also vary over the time period of operation.In order to feedback the position and velocity signals, we have to have sensor outputy Cp(t)d Cy (t)il^ (4.119)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 108where y is the sensor output and C p (t) and Cv (t) are position and velocity sensor gainmatrices, which could include the mathematical model of the transducers and possiblythe signal processing units used in measuring of the system response. If we assume thecontrol strategy is modeled by G c (t), the control input can be represented by^u = —Gc(t)y —Gc(t)Cp(t)d — G c (t)C„(t)CI^ (4.120)Hence, the control force can be written asf = —B(t)G c(t)y—B(t)G c(t)Cp(t)d — B(t)G c(t)Cv(t)il^ (4.121)Substituting equation (4.121) into equation (4.117) yieldsM(t) d + C(t) d + K(t) d = —B(t)G c(t)C„(t)CI — B(t)Gc(t)Cp(t)dor simplyM(t);:i C(t) d + K(t) d —ACctri(t)d — AK ctrt(t)dwhereAKcfri(t) = B(t)G c(t)Cp (t)ACctri(t) = B(t)Gc(t)C„(t)(4.1 22)Usually, B(t) and Cp(t), Cv (t) are constant matrices for a particular configuration of theSVS. G.c (t) can then be designed to make AlCctri(t) and ACctri(t) satisfy the stabilityrequirement. .6,1( cot(t) and ACctri(t) can be considered as special perturbations of thesystem. Rearranging equation (4.122), we haveM(t) d + [C(t) + Ac ctri (t)] d + [K(t) .6.1(ctri(t)] d = 0^(4.123)Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 109orm(t) + c/(t) + KV) d =0^(4.124)whereCV) = C(t) + ACctrt(t)W(t) = K(t) + AK ch./(t)The implementation of control strategies can be discussed respectively in two cases.• Original System Contains No ControllerIf the system expressed in equation (4.117) does not contain a controller, the controlstrategy can be designed and implemeted by going through the procedure decribedin this section from equation (4.118) to equation (4.124). Besides general engi-neering concerns, such as the place and the number of sensors, the place and thenumber of actuators, the symmetry of the system parameter matrices has to becarefully maintained in order to apply the previously developed approaches for thestability analysis of the system, which implies that AC ctr/(t) and AK ctri(t) haveto be symmetric. Since in general the SVS is a time-varying system, AC ct,./ (t)and Al( ct,./(t) should accommodate this feature, which means the controller for theSVS should have the ability to adapt to the ever-changing system dynamics. Inparticular, if the system order changes, it may be necessary to add (or remove)some of the sensors or actuators to (or from) the system. In general, not only theparameters of ACctrt(t) and Al(col(t) have to adapt to the different system config-urations, but also the orders of the AC ctri(t) and AlCct,./(t) have to adapt as well.From the perturbation point of view, ACctrt(t) and AlCart(t) can be consideredas special perturbations superimposed on the structural perturbations so that thesystem dynamics are properly controlled.Chapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 110• Original System Contains A ControllerIf the system expressed in equation (4.117) contains a controller, the implementa-tion of control strategies would be slightly different. It is known from chapter 3that the parameter matrices of equation (4.117) can be further expressed asM(t) = M° Mc(t)C(t) = C° Cc(t)K(t) = K° 1<c(t)where M°, C° and K° are determined by subsystem parameter matrices. Mc(t),Cc(t) and Kc(t) are connection parameter matrices. Since it is assumed that thesubsystems are time-invariant, only connection matrices can be modified to changethe dynamics of the overall system. Therefore, the control strategies can only beimplemented through connection matrices.It is not difficult to obtainM. d + C° d + K° d = —Mc(t) d — Cc(t) d — Kc(t) dFor simplicity of illustration, we assume Mc(t) = 0. Thenmo a + co + K° d = _cc(t) _ Kc(t) dIf we separate Cc(t) and Kc(t) asCc(t) = C cs (t) ACctri(t)Kc(t) = Kca(t) AKctr/(t)(4.125)(4. 126)where Cc8(t), K cs (t) model the system structural perturbations, and ACct,./(t),AKciri(t) model the dynamics of the controller. Then, equation (4.126) can beChapter 4. Stability of Structurally-Varying Systems With Time-Varying Order 111rewritten asM° d + [C° + cca(t)] d + [K° + Kcs (t)] d= —.O.Cdri (t) d — AKdri(t) dOrM° d + C(i) d + K(t) d = —ACctrt(t) d _ AKctri(t) d (4.127)Comparing equation (4.127) and equation (4.122), we see that both equations es-sentially have the same form. The design and implementation of AC ct,./(t) andAl(ctri (t) have been discussed before. It is not difficult to realize them throughproper selection of sensors, actuators and control parameters by using the proce-dures decribed previously. However, the symmetry of ACctri(t) and AKctri(t) hasto be maintained in order to use the previously developed approach to analyze thestability of the system.4.8 SummaryThe stability of order-varying SVS has been studied in this chapter by using the energyfunction as the evaluation function. Both process compatibility and motion compatibilityhave been studied. A number of criteria for evaluation of the dynamic stability of theorder-varying SVS have been derived. The control strategy and the implementation ofthe control strategy have also be discussed for the SVS.Chapter 5Concluding RemarksThe stability theory of dynamic systems has been constantly evolving during the past twocenturies. Various approaches have been developed for the stability analysis of differentdynamic systems. These approaches are loosely related to each other and are usuallyapplicable only in the stability analysis of particular kinds of dynamic systems. Althoughthere has long been an effort to unify the stability theory for all branches of mechanics,significant results have rarely been achieved.The proposed research is on the modeling and stability analysis of a special subsetof time-varying dynamic systems, called structurally-varying systems or SVS. The mainfeature of the SVS is that it consists of a number of subsystems which are connectedtogether through a group of time-varying constraints. The dynamic model and the sta-bility condition of the system will usually change if the constraint condition changes. Inreal applications; it is always desirable to predict the change of the stability conditiondue to the variation of the constraint condition, or in other words, the structural pertur-bation. The real-time application sometimes even demands speed in the algorithm forthe evaluation of the change of the stability condition.In order to meet these requirements, new concepts of the stability have been designedand new approaches have been developed to analyze the stability of the SVS. We haveused both the state response and the energy function as the evaluation function in thestability analysis of the SVS. The static and dynamic stability of the SVS have beenthoroughly studied. The major contributions of the work can be summarized as follows:112Chapter 5. Concluding Remarks^ 113• Based on the comprehensive study of various stability theories, new concepts ofstability have been proposed for the special dynamic systems, which are calledstructurally-varying systems. The new concepts lay the foundation for the stabilityanalysis of the SVS.• The recursive state space model of the SVS has been developed so that the dynamicmodel for any system configuration can be derived in two ways, (1) using a modelof its previous configuration and the current structural perturbation, (2) using theunconstrained subsystem models and constraint matrix which provides the mostcurrent system constraint information.• A qualitative measure of the stability for the SVS has been established. A recur-sive estimation algorithm (ry-approach) has been developed for the evaluation ofthe stability of the SVS. By the -y function, the stability of the fixed order SVScan be evaluated. In particular, the algorithm has the features of simplicity andrecursiveness. Therefore it is appropriate for real-time applications.• Thorough analysis has been carried out on the process of structural switching in-stants. Motion compatibility and process compatibility have been proposed andapplied to the stability analysis of the SVS. Different dynamic performances havebeen revealed and their influences on the stability of the SVS have been inves-tigated. A number of criteria for evaluating the stability of the SVS have beenderived. The applications of the analytical results have been illustrated computa-tionally and experimentally.• Using both state response and system energy functions as the evaluation function,a new method for analyzing the stability of the SVS has been developed. Criteriabased on two evaluation functions for predicting the static and dynamic stabilityChapter 5. Concluding Remarks^ 114of the SVS have been derived.NomenclatureSymbol^description SVS structurally-varying systemm, M^mass and mass matrixC damping matrixk, K^stiffness and stiffness matrixd(t) displacement vectorx^ state vectorA, B system matrix and input matrixtf•^state transition matrixmatrix measureexp^exponentialO originalc^ constrainedA changeE, e^energydim dimensionn degree of freedomp^ perturbedT transposed^ velocityF, f impulsive force vectorint^internal115Nomenclature^ 116bnd^boundarynew newly-createdBibliography[Anderson etc., 1986] Anderson, B.D.O., et al, "Stability of Adaptive Systems: Passivityand Averaging Analysis", MIT Press, (1986).[Astrom and Wittenmark, 1990] Astrom, K.J. and Wittenmark, B., "Computer-Controlled Systems: Theory and Design", Prentice Hall, Englewood Cliffs, N. 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Auto. Control, AC-21, pp.319-338, June (1976).Appendix ANorms and Inner ProductSome mathematical definitions of norm and inner product are reviewed in this section.The materials come mainly from [Chen, 1988, DeCarlo, 1989]Definition 1: Let x = [x1,x2 ,• • xn ] T E R. The norm of x, which is a function:Rn^R, can be defined by one of the following expressions:nII X 111 =-11 E xi InII X II2=A (E I Xi 1 2 )112i=1x 110. max I xi Ior in general,nII x IIP =6-- (E I xi IP) 11Pi=1where p ranges between 1 and oo. In particular, the norm II x 11 2 is called the Euclideannorm or 12 norm on Rn.Each of the norms defined here has the following properties:1. II x II> 0 and II x II= 0 iff = O.2. ax 11=1 a x II for all a E R.3. II^+ x2^xi II + II x2 II.122Appendix A. Norms and Inner Product^ 123These properties are easy to verify from the definition.Definition 2: Let A = (a1) E Rnxn. The norm of A is defined as:II A IA sup{  IIIIAxx1111^x^0}= sup{II Ax^II x 11= 1}An immediate consequence of the definition of II A II is that for any x E Rn ,II Ax11s1IA II^II x IISince the norm of II A II is defined through the norm of II x II, it is called an inducednorm. For different 11 x II, we have different II A II.1. For II x 111:^II A Ik= max(E I^= 1,2,2. For II x 11 2 :^II A 112= {Amax (ATA)} 1 /2 .n3.^For II x1100:^II A H oc = ma,x(E I aii 1),^i = 1,2, • • •.j=1The norm of a matrix has the following properties:1. II A II= 0 iff A = O.2. II aA 11=1 a III A II for all a E R.3. II A1 + A2 115_11 A1 II + II A2 II.Appendix BThe Solution of i(t) = Ax(t) Bu(t)For the state-space model of a system,*(t) = A(t)x(t) B(t)u(t),^x(to) = X0we have the solutionx( t ) =^to )xo + J t 4.(t, T) • B(T) • u(T)drtowhere1. 11.(t, to )xo is the zero-input state response,2. j .b(t, T) • B(T) • u(T)dT is the zero-state response.and in particular, in time-invariant cases,4(t, to ) = 1(t — to , 0) = exp[A(t — to)]Hence, the complete solution for a time-invariant syetem isx(t) = exp[A(t — to )]xo^exp[A(t — r)] • B • u(T)dTexp[A(t — to )]xo exp(At)^exp(—AT) • B • u(T)dTAppendix CMatrix MeasureDefinition: Let II • II be an induced matrix norm on Rnxn. The matrix measure isdefined as a function A: Rnxn --+ RIII + EA —1ii[A] = lime,o+From a purely mathematical point of view, the measure /4A] of a matrix A can bethought of as the directional derivative of the norm function II • II, as evaluated at I in thedirection A. The matrix measure has some useful properties, which are provided here.1. — II A II<^A[A] <II A II,^V A E Rnxn .2. p[aA] = ap[A],^V a> 0 and V A E Rnxn .3. max -WA] — 1.4-13], —1.1[—A] ji[B]l < fi[A. B] < p[A] p[B]•4. —p[—A] 5_ Re(Ai) < p,[A] whenever Ai is an eigenvalue of A.The proof of the properties can be found in [Vidyasagar, 1978]. Using the matrix measure,we present a useful theorem.Theorem Cl: Consider the differential equation x = A(t) x, t > 0, where x E B.',A(t) E Rnxn, AO is piecewise-continuous. Let II • II be an norm on Rn, and A[A] denotethe the matrix measure on Rnxn. Then, whenever t > t o > 0, we haveII x(to) II exp{fto --/./[—A(r)]th- }^x(t)^x(to) explito p[A(r)]c/T}125Appendix C. Matrix Measure^ 126andexp.( Ito —A[—A(r)]ch-} <11 4, (t, to) 11.  exP{ Ito it[A(T)]th- }This theorem gives both upper and lower bounds of state variables x and state transitionmatrix 40(t, to ). Proof of this theorem can be found in [Vidyasagar, 1978]. Using thistheorem, the stability of structurally-varying systems can be studied based on the conceptof matrix measure.The calculation of the matrix measure on different norm is provided.1. For 11 x 11.,0 =maxl xi 1, p co [A] = maxi{aii Ejoi I ail 1}.2. For 11 x Il i = EZ=1. I xi I, p i [A] = maxj faii Eioj I3. For 11 x 112= (E7=1 1 xi (2)1/2, /12[A] = Amax {(A*+ A) f*.Appendix DProperties of Symmetric MatricesTheorem D.1: If an n x n matrix A is real and symmetric, its eigenvalues are all real.Theorem D.2: If an n x n matrix A is real and symmetric, then all its eigenvalues are1. positive if A is positive definite.2. nonnegative if A is positive semidefinite.3. negative if A is negative definite.4. nonpositive if A is negative semidefinite.The proof of these theorem can be found in [Orteg, 1987].127

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