Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Hajati, Nasim
Master of Applied Science - MASc
University of British Columbia
Assessing the performance of the cardiovascular system is a crucial part of testing new medications and biomedical devices. To ensure the safety of these new products, it is necessary that their impact on performance factors, such as the strength or elasticity of the heart muscles be evaluated. These tests are typically performed on animals; specifically, rodents have been playing a significant role in such biomedical experimentations. Traditional methods of studying the cardiovascular system involve invasive surgeries to reach the heart and its vessels, to manually manipulate the blood return flow to the left ventricle and monitor the change in the patterns of ventricular or arterial pressure and volume through plotting pressure-volume graphs. These surgical methods often lead to blood loss or damage to vessels and organs of the rodents, which affects the test results, making them less accurate. They are also subject to human error which leads to low repeatability. Considering the issues of the traditional approaches, the literature suggests using a manually-operated Lower Body Negative Pressure (LBNP) chamber as a non-invasive alternative. This approach places sealed chambers around the lower body of the rodents and creates suction inside the chamber to control the blood flow to the heart by drawing the blood toward the lower body as an effect of the negative pressure. To remove direct human intervention, an automated servo-controlled system was built by previous UBC students, which controlled the blood pressure level, to be later plotted against the measured volume. This study aims to fix the shortcomings of that device, including low interface speed and lack of safety features for the electronics. In addition, this study performs system identification on the automated chamber and rodents’ left ventricular pressure models and designs a robust controller for the system based on the derived models.
OPTIMIZATION OF A LOWER BODY NEGATIVE PRESSURE CHAMBER TO CONTROL THE LEFT VENTRICULAR PRESSURE OF RODENTS by Nasim Hajati B.Sc., Sharif University of Technology, 2021 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Electrical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) April 2024 © Nasim Hajati, 2024 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, the thesis entitled: Optimization of a Lower Body Negative Pressure Chamber to Control the Left Ventricular Pressure of Rodents submitted by Nasim Hajati in partial fulfilment of the requirements of the degree of Master of Applied Science . Rudolf Seethaler, School of Engineering Supervisor Klaske Van Heusden, School of Engineering Supervisory Committee Member Ahmad Al-Dabbagh, School of Engineering Supervisory Committee Member Sumi Siddiqua, School of Engineering University Examiner iii Abstract Assessing the performance of the cardiovascular system is a crucial part of testing new medications and biomedical devices. To ensure the safety of these new products, it is necessary that their impact on performance factors, such as the strength or elasticity of the heart muscles be evaluated. These tests are typically performed on animals; specifically, rodents have been playing a significant role in such biomedical experimentations. Traditional methods of studying the cardiovascular system involve invasive surgeries to reach the heart and its vessels, to manually manipulate the blood return flow to the left ventricle and monitor the change in the patterns of ventricular or arterial pressure and volume through plotting pressure-volume graphs. These surgical methods often lead to blood loss or damage to vessels and organs of the rodents, which affects the test results, making them less accurate. They are also subject to human error which leads to low repeatability. Considering the issues of the traditional approaches, the literature suggests using a manually-operated Lower Body Negative Pressure (LBNP) chamber as a non-invasive alternative. This approach places sealed chambers around the lower body of the rodents and creates suction inside the chamber to control the blood flow to the heart by drawing the blood toward the lower body as an effect of the negative pressure. To remove direct human intervention, an automated servo-controlled system was built by previous UBC students, which controlled the blood pressure level, to be later plotted against the measured volume. This study aims to fix the shortcomings of that device, including low interface speed and lack of safety features for the electronics. In addition, this study performs system identification on the automated chamber and rodents’ left ventricular pressure models and designs a robust controller for the system based on the derived models. iv Lay Summary Evaluating the changes in patterns of arterial/left ventricular pressure and volume is a key factor in the assessment of cardiovascular performance. Invasive surgical methods of monitoring these factors are prone to inconsistency, low repeatability, and human error. As a solution, previous UBC students built an automated Lower Body Negative Pressure chamber, based on a patent on the same subject, to control left ventricular pressure in rodents. This study aims to increase the reliability and safety of the chamber. The novelty of the work lies in the use of system identification methods to derive the models of the LBNP chamber and rodents’ cardiovascular system. In addition, it is the first time that a robust control method has been adopted in an LBNP system. This work contributes to the research on the impact of health conditions and injuries, as well as pharmaceutical and biomedical products, on cardiovascular performance. v Preface This work is the implementation of a patent by Christopher West, Rudolf Seethaler, and Aaron Phillips (West et al., 2022). The previous version of the device was built as the requirement of a Capstone project, by Ben Crowther, Dustin French, Aiman Ridwan Mohd Hafash, Justin Thomson, and Grady Whitmore, under the supervision of Dr. Rudolf Seethaler and Dr. Christopher West (Crowther et al., 2021). Tests on the old device performed by Liam Stewart, Mahdi Ahmadian, Liisa Wainman, Jennifer Duffyand, and Dr. Christopher West, showed suboptimal function of the electronics and software design and difficulty of controller tuning, which led to the need to create a second, optimized version of the device. The research question, to rebuild the LBNP chamber with optimal performance, high reliability, and a robust controller was, therefore, created by Dr. Rudolf Seethaler and Dr. Christopher West. The ethics approval from the University of British Columbia Animal Care Committee and the Canadian Council for Animal Care was obtained by Dr. Christopher West’s lab. In Chapter 2, the mechanical structure of the chamber and surgical platform is the same as previously designed in the Capstone project. Electronic circuitry design, simulation and testing, and preparation of the controller box housing were done by me, according to the requirements set by Dr. Christopher West and Kyle Fricke, Transonic Research and Development head of engineering at the time. The interface and microcontroller programming were done by me with the help and supervision of Dr. Seethaler. In vivo experiments were designed by me, Dr. Rudolf Seethaler, Dr. Christopher West, and Dr. Oliver Wearing. The surgical preparation of the experiments was done by Oliver Wearing at the animal research facility of UBC Okanagan, and I vi was present in all experiments to monitor the performance of the device and solve any hardware/software bugs. Analysis of the data, literature review, and implementation of modeling (Chapter 3) and control system design (Chapter 4) were done by me, based on the literature cited in each of the chapters. Chapter 5 shows the results of the experiments and algorithms performed by me, and Chapter 6 concludes our work and elaborates on the future work. Figures 1.1, 1.2 and 1.3 are drawn based on (Goodwill, 2020). The uncertainty boundaries in Figure 4.2 of Chapter 4 and Figure 5.15 of Chapter 5 were plotted using a MATLAB function developed by Dr. Klaske Van Heusden for their paper (van Heusden et al., 2018). Other figures in the document are original unless otherwise mentioned. The instrumentation for catheterization was produced by Transonic Inc. NY, USA. The software used for the simulation and design of the electronics board were OrCAD PSpice, Altium Designer, Texas Instruments Analog Engineer’s Calculator, and Texas Instruments Filter Design Tool. The libraries of the components inside the design were either made manually or taken from SnapEDA free download website. The circuit board was printed by JLCPCB. Graphic User Interface and Microcontroller codes were programmed in Visual Studio and Visual Studio Code, and data analysis, modeling, and controller design were done in MATLAB. Currently, there are no publications on the work. However, there will be two papers written on this project in the future, one on the engineering and design aspects of the device, and one with the focus on physiological testing performed using the device, at Dr. Christopher West’s lab. vii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ........................................................................................................................ vii List of Tables ............................................................................................................................... xii List of Figures ............................................................................................................................. xiii List of Abbreviations ................................................................................................................ xvii Acknowledgements .................................................................................................................... xix Dedication ................................................................................................................................... xxi Chapter 1: Introduction ................................................................................................................1 1.1 Derivation of PV Loops .................................................................................................. 4 1.2 Lower-Body Negative Pressure Chamber ...................................................................... 5 1.3 Automated LBNP............................................................................................................ 7 1.4 Problem Statement .......................................................................................................... 7 Chapter 2: LBNP Architecture.....................................................................................................9 2.1 LBNP as a Closed-Loop Control System ..................................................................... 10 2.2 Old Design .................................................................................................................... 11 2.2.1 Chamber’s Mechanical Design ............................................................................. 12 2.2.2 Hardware Electronics ............................................................................................ 15 2.2.2.1 The Microcontroller Unit (MCU) ..................................................................... 16 2.2.2.2 Catheter Input, Voltage Scaling and ADC ........................................................ 17 viii 2.2.2.3 Box Pressure Sensor ......................................................................................... 18 2.2.2.4 Vacuum Pump and Driver Module ................................................................... 18 2.2.2.5 Cooling System ................................................................................................. 19 2.2.2.6 Power Supply Unit ............................................................................................ 19 2.2.2.7 Electrical Housing and Connections ................................................................. 19 2.2.3 GUI/Microcontroller Software.............................................................................. 21 2.2.3.1 Derivation of the Pressures from Raw Measurement ....................................... 22 2.3 New Design ................................................................................................................... 24 2.3.1 Hardware Electronics ............................................................................................ 24 2.3.1.1 The Microcontroller Unit (MCU) ..................................................................... 25 2.3.1.2 Catheter Input, Voltage Scaling and Anti-Aliasing .......................................... 26 2.3.1.2.1 Voltage Division Stage ............................................................................... 26 2.3.1.2.2 Anti-Aliasing Stage ..................................................................................... 27 2.3.1.2.3 Buffering ..................................................................................................... 29 2.3.1.2.4 Dual Supply ................................................................................................ 30 2.3.1.2.5 Analog-To-Digital Conversion ................................................................... 31 2.3.1.3 Box Pressure Sensor ......................................................................................... 32 2.3.1.4 Analog Outputs ................................................................................................. 32 2.3.1.5 Digital Data Output ........................................................................................... 34 2.3.1.6 OLED Information Display .............................................................................. 34 2.3.1.7 Vacuum Pump Driver ....................................................................................... 35 2.3.1.7.1 Reverse Polarity Protection......................................................................... 36 2.3.1.8 Emergency Stop ................................................................................................ 37 ix 2.3.1.9 Cooling System ................................................................................................. 37 2.3.2 Software ................................................................................................................ 40 2.3.2.1 LBNP Desktop App .......................................................................................... 41 2.3.2.2 Microcontroller Software .................................................................................. 42 2.3.2.2.1 Open-Loop Mode of Operation .................................................................. 43 2.3.2.2.2 Closed-Loop Mode of Operation ................................................................ 44 2.4 Simulation and Circuitry Testing .................................................................................. 45 2.5 In-Vivo Testing and Experiment Design ...................................................................... 47 2.6 Summary ....................................................................................................................... 51 Chapter 3: System Identification and Modeling .......................................................................52 3.1 Open-Loop LNBP ......................................................................................................... 52 3.2 Voltage-Box Pressure Mathematical Model ................................................................. 53 3.3 System Identification Algorithm................................................................................... 57 3.4 Representation of the System........................................................................................ 57 3.5 Parameter Estimation Algorithm .................................................................................. 60 3.5.1 Problem Formulation ............................................................................................ 60 3.5.2 Least Square Fitting: Levenberg-Marquardt Method ........................................... 61 3.5.2.1 Steepest-Descent Method.................................................................................. 62 3.5.2.2 Gaussian-Newton Method ................................................................................ 63 3.5.2.3 Levenberg-Marquardt Method .......................................................................... 63 3.6 Model Assessment ........................................................................................................ 64 3.7 Model Orders ................................................................................................................ 65 3.8 Derivation of the Model Parameters from Rat/Chamber Specifications ...................... 66 x 3.8.1 Linear Regression ................................................................................................. 67 3.8.2 Stepwise Linear Regression .................................................................................. 68 3.9 MATLAB Implementation ........................................................................................... 69 3.10 Summary ....................................................................................................................... 70 Chapter 4: Robust Controller Design ........................................................................................71 4.1 Representation of the Uncertainty ................................................................................ 72 4.2 Visualization of the Uncertainty in Frequency Domain ............................................... 73 4.3 Derivation of the Nominal Model and Uncertainty Region.......................................... 75 4.4 Controller Synthesis ...................................................................................................... 76 4.4.1 Design Criteria ...................................................................................................... 77 4.4.1.1 Stability ............................................................................................................. 77 4.4.1.2 Performance ...................................................................................................... 78 4.4.2 Controller Architecture ......................................................................................... 81 4.4.2.1 Convex-Concave Optimization ......................................................................... 83 4.5 Summary ....................................................................................................................... 85 Chapter 5: Results........................................................................................................................86 5.1 Results of Chapter 3: System Identification ................................................................. 86 5.1.1 Goodness of Fits ................................................................................................... 90 5.1.2 Stepwise Linear Regression .................................................................................. 91 5.2 Results of Chapter 4: Controller Design ....................................................................... 96 5.2.1 Uncertainty Regions and Nominal Values .......................................................... 101 5.2.2 Controller Synthesis ............................................................................................ 102 5.2.3 Closed-Loop Experiments .................................................................................. 106 xi Chapter 6: Conclusion ...............................................................................................................110 6.1 Current Limitations and Future Work......................................................................... 111 Bibliography ...............................................................................................................................114 Appendices ..................................................................................................................................125 Appendix A ............................................................................................................................. 125 A.1 Raspberry Pi Pico IO and Input Headers for Peripherals ....................................... 126 A.2 Catheter Reading Modules ...................................................................................... 127 A.3 Analog Outputs ....................................................................................................... 128 A.4 Vacuum Pump Motor Driver .................................................................................. 129 xii List of Tables Table 2.1 Key Components in the Design of an Automated LBNP ............................................. 15 Table 3.1 Linear Regression Models ............................................................................................ 68 Table 4.1 LBNP Models ............................................................................................................... 71 Table 5.1 Rats' Specifications ....................................................................................................... 92 Table 5.2 V-Box Gain Coefficients .............................................................................................. 93 Table 5.3 V-Box Time Constant Coefficients .............................................................................. 93 Table 5.4 Box-MAP Gain Coefficients ........................................................................................ 94 Table 5.5 Box-MAP Time Constant Coefficients......................................................................... 95 Table 5.6 Closed-Loop Experiments .......................................................................................... 106 xiii List of Figures Figure 1.1 Pressure-Volume Loop (drawn based on (Goodwill, 2020)) ........................................ 2 Figure 1.2 End-Systolic and End-Diastolic Relations (drawn based on (Goodwill, 2020))........... 3 Figure 1.3 (a) ESPVR Change with Contractility, (b) EDPVR Change with Compliance, (drawn based on (Goodwill, 2020)) ............................................................................................................ 4 Figure 2.1 Block Diagram of a Closed-Loop Control System........................................................ 9 Figure 2.2 Overview of LBNP ...................................................................................................... 10 Figure 2.3 LBNP Closed-Loop Block Diagram ........................................................................... 11 Figure 2.4 LBNP Chamber, picture taken from (Crowther et al., 2021) ...................................... 13 Figure 2.5 Operation Platform, picture taken from (Crowther et al., 2021) ................................. 13 Figure 2.6 (a) LNBP Chamber, (b) Chamber and Platform from the Side ................................... 14 Figure 2.7 Old Design's Electronics Modules .............................................................................. 16 Figure 2.8 Old LBNP Board Installation and connections, picture taken from (Crowther et al., 2021) ............................................................................................................................................. 20 Figure 2.9 Schematics of the Old Design, picture taken from (Crowther et al., 2021) ................ 20 Figure 2.10 Graphical User Interface, picture taken from (Crowther et al., 2021) ...................... 22 Figure 2.11 Blood Pressure ........................................................................................................... 23 Figure 2.12 New Design’s Electronic Modules ............................................................................ 25 Figure 2.13 Microcontroller Unit .................................................................................................. 26 Figure 2.14 Anti-Aliasing Filter Order using TI Analog Engineer’s Calculator .......................... 28 Figure 2.15 Two-Stage Anti-Aliasing Filter using TI Filter Design Tool .................................... 28 Figure 2.16 Anti-Aliasing Filter Characteristics derived from TI Filter Design Tool.................. 29 xiv Figure 2.17 Positive Voltage Regulator ........................................................................................ 30 Figure 2.18 Negative Supply Circuitry ......................................................................................... 31 Figure 2.19 Catheter External ADC.............................................................................................. 31 Figure 2.20 Isolated Power for I2C Connections.......................................................................... 33 Figure 2.21 ISO I2C Transfer Circuitry ........................................................................................ 34 Figure 2.22 Half-Bridge Module Schematics ............................................................................... 35 Figure 2.23 Sensorless Detection and Hall Sensor/Encoder Control ........................................... 36 Figure 2.24 Reverse Polarity Protection ....................................................................................... 37 Figure 2.25 3D View of the PCB in Altium Designer .................................................................. 38 Figure 2.26 The Controller Board ................................................................................................. 39 Figure 2.27 (a) Back/Side View, (b) Front View of the Controller Box ...................................... 40 Figure 2.28 LBNP User Interface ................................................................................................. 42 Figure 2.29 Phase Currents (green) and IS Pin Voltages (yellow) of BTS7960, (a) Phase 1, (b) Phase 2 .......................................................................................................................................... 46 Figure 2.30 Phase Voltages of Three Phases at (a) 95% PWM (b) 60% PWM ........................... 46 Figure 2.31 In-Vivo Experimentation at the Animal Facility ....................................................... 49 Figure 2.32 LabChart Graphs ....................................................................................................... 50 Figure 2.33 PV Loops Graph ........................................................................................................ 50 Figure 3.1 Open-Loop LBNP Block Diagram .............................................................................. 52 Figure 3.2 Hydraulic Analogy ...................................................................................................... 54 Figure 3.3 Block Diagram of the Box Pressure System ............................................................... 56 Figure 3.4 Output Error Model Structure ..................................................................................... 59 xv Figure 4.1 Unstructured Uncertainty Disk .................................................................................... 73 Figure 4.2 (a) Average-Based (b) Optimization-Based Models for Diastolic Pressure (Used codes of (van Heusden et al., 2018) to draw the uncertainty bounds in the diagrams) ............... 75 Figure 4.3 Robust Stability Nyquist Visualization ....................................................................... 78 Figure 4.4 Nominal Performance Condition ................................................................................. 80 Figure 4.5 Robust Performance Condition ................................................................................... 81 Figure 5.1 Typical Step Response of a Rats' Blood Pressure ....................................................... 86 Figure 5.2 Multiple Arrhythmias in the Experiment .................................................................... 87 Figure 5.3 Plot of MAP, Not Given Enough Time to Reach S.S. ................................................ 88 Figure 5.4 MAP in Voltage of 2V ................................................................................................ 88 Figure 5.5 Outliers in (a) Gain, (b) Time Constant, of the Voltage-Box Pressure Models .......... 89 Figure 5.6 Histograms of Fit Percentages, (a) V-Box (b) Box-Diastolic, (c) Box-MAP, (d) Box-Systolic .......................................................................................................................................... 90 Figure 5.7 Bode Plot of V-Box ..................................................................................................... 97 Figure 5.8 Nyquist Plot of V-Box ................................................................................................. 97 Figure 5.9 Bode Plot of V-DP....................................................................................................... 98 Figure 5.10 Nyquist Plot of V-DP ................................................................................................ 98 Figure 5.11 Bode Plot of V-MAP ................................................................................................. 99 Figure 5.12 Nyquist Plot of V-MAP ............................................................................................. 99 Figure 5.13 Bode Plot of V-SP ................................................................................................... 100 Figure 5.14 Nyquist Plot of V-SP ............................................................................................... 100 xvi Figure 5.15 Nyquist Uncertainty Regions for (a) V-Box (b) V-Diastolic (c) V-MAP (d) V-Systolic (Used codes of (van Heusden et al., 2018) to draw the uncertainty bounds in the diagrams) .................................................................................................................................... 102 Figure 5.16 V-Box Closed-Loop Step Response ........................................................................ 103 Figure 5.17 V-DP Closed-Loop Step Response ......................................................................... 104 Figure 5.18 V-MAP Closed-Loop Step Response ...................................................................... 104 Figure 5.19 V-SP Closed-Loop Step Response .......................................................................... 105 Figure 5.20 CL V-Box Step Resp. with Arterial Catheter .......................................................... 107 Figure 5.21 CL V-MAP Step Resp. with Arterial Catheter ........................................................ 107 Figure 5.22 CL V-Box Ramp Resp. with Arterial Catheter ....................................................... 107 Figure 5.23 CL V-MAP Ramp Resp. with Arterial Catheter ..................................................... 107 Figure 5.24 CL V-SP Ramp Resp. with Arterial Catheter .......................................................... 108 Figure 5.25 CL V-SP Step Resp. with LV Catheter ................................................................... 108 Figure 5.26 CL V-SP Ramp Resp. with LV Catheter ................................................................. 109 xvii List of Abbreviations ADC Analog-to-Digital Converter BP Blood Pressure CL Closed Loop DAC Digital-to-Analog Converter DP Diastolic Pressure 𝐄𝐞𝐬 End-Systolic Elastance EDPVR End-Diastolic Pressure-Volume Relationship ESPVR End-Systolic Pressure-Volume Relationship GUI Graphic User Interface IAE Integrator Absolute Error IE Integrator Error IO Input/Output LBNP Lower-Body Negative Pressure LTI Linear Time Invariant LV Left Ventricle MAP Mean Arterial Pressure MCU MicroController Unit OE Output Error OL Open Loop Op-Amp Operational Amplifier xviii OS Operating System NP Nominal Performance PCB Printed Circuit Board PI Proportional-Integrator PID Proportional-Integrator-Differentiator PV Pressure-Volume RP Robust Performance RPi Raspberry Pi RS Robust Stability SP Systolic Pressure SQP Sequential Quadratic Programming SS Steady State xix Acknowledgements I would like to begin by thanking my supervisor Dr. Seethaler, from whom I have learned so much, not only in engineering but also in life. Whenever I was worried I was not doing well in my work, he would remind me to look at what was valuable to me, and not society's expectations of me as a graduate student. He taught me to choose my unique path and to enjoy the experiences I gained and the friends I made along the way because that was what mattered the most. Thank you for teaching me to find joy and fun in everything I do! Next, I would like to thank my amazing committee members. Dr. Van Heusden, thank you for sharing your control engineering experiences with me, and being a patient supporter whenever I was lost and confused in my work. And thank you, Dr. Al-Dabbagh for all your insightful feedback and encouragement. Many thanks to Dr. West, Oliver, and Mahdi for their help on the physiology side of the project. I feel honoured to have worked with such a dynamic and smart team. I would also like to thank Dr. Shirazi and Dr. Eisenstein, whom I bothered a lot, but they have been nothing but kind and supportive to me and I have used their valuable advice in all stages of my master's journey. My special thanks go to my loved ones from whom I am separated by immigration. I know that my parents have had teary eyes and sleepless nights, because all they could have of their only daughter was a 6-inch screen calling a few times a week, but they have never stopped being there for me and encouraging me to thrive and progress. My friends, who are now all over the world, xx Iran, Belgium, Greece, Canada (yes, Ontario and Manitoba are still long distance!), and who despite being so far from, my heart feels the closest to; thank you for being with me through all my ups and downs. Another special thank you, to my partner (in crime!), for all his love and help, and for being a patient listener whenever I had a mental breakdown over my controller codes not working for the thousandth time! And finally, many thanks to my wonderful community and friends here, who I was lucky to find, and who have become my new family, far from home. Thanks for supporting me and not letting me feel lonely. xxi Dedication To Iranian women, the brave souls who fight for their freedom and equal rights every day with hearts full of hope and courage.1 Chapter 1: Introduction Measuring heart performance is crucial in testing the effects of new medications, biomedical devices, or health conditions on cardiovascular systems. Due to the dangerous or terminal nature of these tests these experiments are, at least initially, conducted on animals, often rodents. The commonly used factors of heart performance are “heart contractility” and “heart compliance”. When the heart contracts during the time interval between the first and second sounds of the cardiac cycle it is known as Systole, and Contractility is the intrinsic contractional strength of the heart at that time (Calvert & Lefer, 2012). The phase of the heartbeat, when the heart muscles relax and the chambers are filled with blood is known as Diastole, and the heart’s capacity to expand in that time is known as Compliance (Levine, 1972). Cardiac performance can be measured in vivo by recording the blood pressure and blood volume in the left ventricle (LV) and plotting the left ventricular Pressure-Volume (PV) loops (Gaasch et al., 1976; Solaro, 2011), which illustrate the relationship of pressure and volume within the LV over a cardiac cycle (see Figure 1.1). The loop in Figure 1.1 shows the periods of Systole and Diastole, in red and blue sections, respectively. The width of the loop along the volume axis is the Stroke Volume, which is the volume of blood ejected from the heart’s left ventricle during each systolic phase of the cycle (Bruss & Raja, 2024). The point where the ejection is over, and the aortic valves closes is called End-Systole, and the pressure/volume of that point is called End-Systolic pressure/volume. The point after the relaxation of the heart muscles is over and the heart is filled with blood is called End-Diastole, and the pressure/volume at that point is called End-Diastolic pressure/volume. 2 Figure 1.1 Pressure-Volume Loop (drawn based on (Goodwill, 2020)) When the cardiac muscle cells stretch initially, before contraction, it is known as preload. To determine the effect of a medication, biomedical device, or health condition on heart performance, one could test the subject before and after that condition to compare the effect. For that purpose, one can manipulate the preload on the left ventricle by reducing the return of venous blood to the heart. By doing so, a series of PV loops will be generated over a range of decreasing ventricular preloads. While restricting return of blood to the LV, each subsequent PV loop reduces in size as End-Diastolic (i.e. filling) volume is reduced, causing each successive beat to produce weaker contractions, generating lower Systolic pressures, and ejecting smaller Stroke Volumes (see Figure 1.2). 3 Figure 1.2 End-Systolic and End-Diastolic Relations (drawn based on (Goodwill, 2020)) By plotting a line through the ESPVR (End-Systolic Pressure-Volume Relationship) points over this series of beats over a broad range of loading (i.e. preload) conditions and calculating its slope, one can calculate the LV End-Systolic Elastance (Ees); a ‘load-independent’ index of LV contractility. Similarly, plotting a curvilinear line through the EDPVR (End-Diastolic Pressure-Volume Relationship) points of these loops can determine a load-independent index of LV compliance (Knaapen et al., 2007). The impact of the conditions, such as medications, will change contractility and compliance functions and the effect of that condition on the heart performance of the subject will be evident through the changes in the ESPVR and EDPVR points (see Figure 1.3). The research on the cardiovascular performance, therefore, seeks to plot the subjects’ PV loops under normal and affected circumstances and measure the change in the two graphs. 4 (a) (b) Figure 1.3 (a) ESPVR Change with Contractility, (b) EDPVR Change with Compliance, (drawn based on (Goodwill, 2020)) 1.1 Derivation of PV Loops As mentioned in the previous section, to derive the PV loops for an animal one should change the preload gradually by manipulating the return flow of blood to the heart, and then by measuring the left ventricular pressure and volume and plotting them, performance factors can be assessed. Measuring the left ventricular pressure and volume in animals can be done by cardiac catheterization (Ahmed & Hajouli, 2024), which is an invasive procedure in which a catheter, containing a pressure sensor and a sensor capable of estimating the volume, is surgically placed within the subject’s left ventricle. The idea of catheterization was first used in the 17th century by Stephen Hales (Hall, 1987) and has evolved over the last four centuries. In contrast, cardiac imaging is part of the non-invasive approaches, and includes methods such as Echocardiography (Todd et al., 2022), Sonomicrometry (Heppelmann et al., 2015), and MRI (Wu, 2021). Due to the 5 costly and labour-intensive nature of such methods, catheterization is still the preferred method of pressure-volume measurement (Wei et al., 2014). Manipulating LV preload requires methods of controlling the venous return of blood to the heart. Traditional approaches of restricting the venous return were performed using invasive surgical techniques, such as hemorrhage (Mobin et al., 2023; Stonko et al., 2022) or occlusion (Ahmadian et al., 2022; Wainman et al., 2021). In hemorrhage, the venous blood in the rat’s body is gradually drawn so that less can return to the heart and fill the LV, reducing preload. Occlusion is the method of restricting the blood flow through the major vein that returns blood to the heart, the Vena Cava, by adopting a constricting ligature or applying pressure with a cotton swab. Since in these methods there is a need to access the Vena Cava, there is a chance of organ and vessel damage to the subject or excessive blood loss that affects the pressure and volume measurements or causes adverse reactions in the subject. In addition, direct human intervention in these methods makes the experiments prone to human error which could affect the study results and the repeatability of the experiments. Therefore, there is a need to find a less invasive method with more repeatable and accurate measurements. As a solution to this problem, (West et al., 2022) invented and patented a method of using an automated Lower-Body Negative Pressure (LBNP) Chamber to get more consistent measurements of heart performance non-invasively. 1.2 Lower-Body Negative Pressure Chamber LBNP is a technique for redistributing blood in the upper body by causing blood to pool in the lower body through the regional application of negative pressure. As a result, LBNP reduces the flow of venous blood back to the heart (Goswami et al., 2019). In this approach, the subject is 6 placed inside a chamber, which is sealed around the lower part of the body, and a negative pressure is created inside the chamber using a vacuum pump. The idea of LBNP was developed over 60 years ago as a means of inducing hypotensive anesthesia (Crystal & Salem, 2015). Later on, the approach found more applications in cardiovascular research (Stevens & Lamb, 1965), orthostatic1 simulation (Brown et al., 1966), and cardiovascular deconditioning2 prevention (Stevens, 1966). The National Aeronautics and Space Administration (NASA) conducted various experimentations on this technique at the Skylab program, studying the cardiovascular effects and useability of this method to reduce the adverse effects of space flights and microgravity conditions on the heart (Charles & Lathers, 1994; Hoffler & Johnson, 1975; Johnson et al., 1975, 1977). The use of LBNP on animals also dates back to the same time and has been used to study the cardiovascular system; examples of these works are research on rabbits to study the cardiovascular reflex to an LBNP (Bennett et al., 1976), monkeys to find their cardiovascular responses to a simulated space flight (McCutcheon et al., 1979) and rats to recognize vasodilators (de Metz & van Zwieten, 1979) and study the effect of training on their blood pressure changes (Tipton et al., 1982). In recent studies, LBNP has been used to study cardiac performance under physiological stress (Williams, 2016) and to assess the effects of spinal cord injury on heart function (Hayes et al., 2021). The LBNP methods in the literature are controlled manually, and although the method is non-invasive, there is still a potential for human error involved in the manual control of the vacuum pump and, therefore, in the negative pressure created around the lower body of the subject. 1 Decrease in blood pressure after standing or sitting up (Campos Munoz et al., 2024) 2 Changes in function/structure of the cardiovascular system caused by reduced gravity conditions, such as space flight (Stenger et al., 2018) 7 1.3 Automated LBNP Following the idea later patented by (West et al., 2022) and in an attempt to increase the repeatability of the experiments by removing direct human intervention, (Crowther et al., 2021) built an automated LBNP chamber for rodents, which was used in the work of (Stewart, 2022). The LBNP system in that study was a servo-controlled automated chamber, receiving feedback from an arterial pressure catheter and a barometric pressure sensor inside the lower-body chamber and running a heuristically-tuned control algorithm to regulate the vacuum pump attached to the chamber, automatically. The automated LBNP system designed by (Crowther et al., 2021) had issues on both the hardware and software sides and lacked safety measures for the hardware electronics (e.g., no emergency shut-down button, no reverse polarity protection for the voltage supply, loose connections), and slow-running software which was not capable of plotting output signals in real-time. These issues with the device could harm the subject since, for instance, the only means of shutting down the experiment was through a software button which had a delay to operate. In case of an adverse reaction in a rat or drop of blood pressure below the safe threshold, the rat would become damaged irreversibly. In addition, the control algorithm running on the system was tuned heuristically without any knowledge of the system model or prior studies on the stability of the controlled LBNP. 1.4 Problem Statement This thesis aims to rebuild the rodents’ automated LBNP chamber with the following objectives: 8 Obj 1. Redesigning the electronics hardware to ensure suitable data reading and transfer, ensure the safety of the subject in emergencies as well as reliable vacuum pump regulation. Obj 2. Performing code optimization to speed up the performance of the device, as well as adjusting the graphic user interface for real-time monitoring of the signals. Obj 3. Identification of the LBNP chamber and rats’ arterial blood pressure models to use in the control design. Obj 4. Design a robust controller to ensure the stability of the system, and achieve good performance in terms of disturbance rejection, fast response, and low overshoot. Chapter 2 discusses the device architecture and the changes, and advancements of the new design are compared to the one in (Crowther et al., 2021). Chapter 3 uses system identification methods to find models for box and arterial pressures as a function of the vacuum pump voltage. Finally, in Chapter 4, we discuss the uncertainty of the system caused by variation in rats and experiment results, use robust control design rules to find an optimal robust controller, and compare the controller gain values with the results of the closed-loop experiments performed on chamber pressure and arterial pressures (Diastolic, Systolic, and Mean Arterial Pressure) as well as the results on maximal LV pressure. 9 Chapter 2: LBNP Architecture This chapter discusses the technical design considerations of the device. The previous design done by the students for the capstone project (Crowther et al., 2021) will be shown and compared to the new system, highlighting the parts that have remained the same and changes that have been made to improve the device's safety, reliability, and user-friendliness. This chapter will cover an overview of the hardware and software modules, as well as tests done at the engineering lab and in-vivo experimentation at the animal facility. From a control engineering point of view, the LBNP system is a closed-loop system with sensor feedback from the chamber and the rat. The components of a closed-loop control system are as shown in the block diagram of Figure 2.1. Figure 2.1 Block Diagram of a Closed-Loop Control System In Figure 2.1, the Reference block sets the desired values for the output signal and the Controller block receives the difference between the reference and the measured output (or the Feedback Signal) to create a control output which sets the Actuator’s parameters. The Actuator is the block that runs and makes changes in the Plant (the open-loop system that needs to be controlled, e.g., 10 the chamber pressure or the diastolic pressure). Finally, the Sensor block measures the Plant outputs and transfers them back into the Controller. 2.1 LBNP as a Closed-Loop Control System An overview of the LBNP system can be found in Figure 2.2. Figure 2.2 Overview of LBNP Our LNBP setup is a closed-loop system designed to control arterial/left ventricular blood pressure, as well as the pressure inside the chamber. As seen in Figure 2.2, the Reference block (see Figure 2.1) is the User Interface running on a desktop computer, where the user can enter desired pressures and control parameters. The controller is the LBNP Controller Box, which has a microcontroller and peripheral hardware running inside to control the system. The controller box outputs a voltage as the control output, which runs a vacuum pump (actuator). The vacuum creates User Interface Controller Box Vacuum Pump LBNP Chamber Box Pressure Sensor Catheter 11 a negative pressure inside the LBNP chamber, which draws the blood circulation toward the lower body of the rat (Plant/Process) to make the pressure reach the reference value set by the user. There are two sensors in the system, one inside the chamber that measures the barometric pressure and the other one in the form of a catheter inside the rat’s artery or left ventricle to measure blood pressure. Finally, the pressure values are returned to the user interface for plotting and monitoring. The system’s closed-loop block diagram is drawn as in Figure 2.3. Figure 2.3 LBNP Closed-Loop Block Diagram 2.2 Old Design The initial design for the capstone project (Crowther et al., 2021) was a full closed-loop system, with the mechanical design of the chamber, sensors, controller box, and a vacuum pump. The vacuum chamber in the new design is the same as the original design, without any alterations. The electronics design and the software of the interface and microcontroller have changed, as discussed in section 2.3, to improve reliability and efficiency. In the following section, the components of the previous design are discussed. 12 2.2.1 Chamber’s Mechanical Design The objective for the chamber’s design is to have a box of such length that a typical young adult rat would fit inside comfortably, with the maximum seal possible to allow minimal air leakage. There should be an opening to allow the portion of the rat’s body above the thorax3 to stick out from the LBNP chamber, another opening the size of the vacuum hose to evacuate the air, and finally, a small third opening for the box sensor wire to go through. To reach these objectives and to create a durable chamber for high testing volume, the Capstone team made a transparent acrylic box as shown in Figure 2.4. This design included a bottomless box with two removable front plates of sizes 2.29” and 2.63” to accommodate different-sized rats. The front plates can be fixed to the rest of the box using three clamps on each of the chamber’s walls. There is always a small space between the rat’s body and the opening of the front plate, which may cause air leakage, but all previous efforts to create a seal around the rat had caused stress to the subject’s body and had interfered with the measurements. Therefore, instead of creating a perfect seal, the team had chosen a suitable vacuum pump to compensate for the leakage and create a box pressure drop of at least 20 mmHg. The box was designed to be bottomless so that it could be placed on top of the subject without the need to move it so that there was minimal stress to the rat or interruptions of catheter measurements. A fixed platform, as demonstrated in Figure 2.5, was designed to be the rat’s operation table. This design also allowed for putting a heating pad underneath the rat’s resting area 3 Thorax: area of the rat’s body where the heart lies, separated from the abdomen by the diaphragm (Lőw et al., 2016) 13 to ensure the rat’s body temperature remained constant during the tests. The full LBNP chamber used in an experiment is shown in Figure 2.6. Figure 2.4 LBNP Chamber, picture taken from (Crowther et al., 2021) Figure 2.5 Operation Platform, picture taken from (Crowther et al., 2021) 14 (a) (b) Figure 2.6 (a) LNBP Chamber, (b) Chamber and Platform from the Side Vacuum Hose Opening Box Sensor Opening Surgery Platform Front Plate Heating Pad 15 2.2.2 Hardware Electronics In both the old and the new designs, there are key components that the system needs to have in order to function effectively. These components must be present in all the designs; however, there is a degree of freedom in the choice of electronics and the interfaces between these components. These key elements are shown in Table 2.1. Table 2.1 Key Components in the Design of an Automated LBNP Module Function Box Sensor Measures the chamber pressure and feeds it back into the system. Catheter Measures the blood pressure and feeds it back into the controller. Vacuum Pump Pumps the air out of the chamber to create a negative pressure in the room surrounding the lower body of the rat. Microcontroller Unit (MCU) Receives and processes the data from the box and catheter sensors, controls the vacuum pump based on the control algorithm and user commands, and transfers the pressure results back for monitoring. Communication Protocol with the Graphical User Interface (GUI) A means of communication to connect the MCU and GUI to send commands from the GUI to the MCU and transfer pressure measurements and reference data from the MCU to the GUI. 16 In this part, the design choices of the previous LBNP by (Crowther et al., 2021) are introduced as context to understand the advancements and improvements in the new design in section 2.3. In the following, the overall architecture and components of the old system are described. In addition, the specific limitations and deficiencies of that design are detailed, highlighting the drawbacks and required improvements that the new design aims to address. The overall design of the previous version can be seen in the flowchart of Figure 2.7. Figure 2.7 Old Design's Electronics Modules 2.2.2.1 The Microcontroller Unit (MCU) The microcontroller unit is the central computing unit of the controller box. It reads and processes sensors and user commands, runs the controller algorithm, and outputs voltages in the form of Pulse Width Modulation (PWM) signals (Bolic, 2023) to manage the input of the vacuum. In the old version, a Raspberry Pi 4.0 was used as the development board for the MCU. 17 The Raspberry Pi 4.0 is a complete computer with a Linux Operating System running on it. It has designated inputs for peripherals including a mouse and keyboard and an HDMI cable to connect to a display monitor. The GUI that had been developed for this version could only run on the Raspberry Pi computer, and therefore, the MCU unit also served as a user interface. A drawback of this design was that it would limit the amount of manipulation the user could do with the data since it could not run on a desktop computer. Furthermore, it was space consuming, as it required its own monitor, mouse, and keyboard. Also, the Linux OS running on it is a powerful tool for programmers, but most non-programmer users, who are the main target audience of the device, are unfamiliar with it; therefore, problems were difficult to fix for the user. A picture of the user interface built for this version can be found in Figure 2.10. 2.2.2.2 Catheter Input, Voltage Scaling and ADC A catheter sensor was used to measure the rat's blood pressure. The model of sensor used in the old version is the SciSense Admittance Pressure-Volume (PV) catheter (used with an ADV500 PV system, Transonic, NY, USA). The sensor outputs signals in the range of -8V to 8V through a BNC cable to the control box. To be able to read the signal inside the Raspberry Pi, this value needed to be scaled to a 0 to 5V range, for which an LM358-N Operational Amplifier in a Non-Inverting configuration (Analog Devices, n.d.) was used. The scaled-down voltage was then fed into a 10-bit Analog-to-Digital Converter (ADC) of model MCP3008 after which it was fed to the MCU. As seen in Figure 2.9, in the old design, the V- lower supply pin on the LM358-N was connected to ground; in other words, the Op-Amp was used in single-supply mode. This mode is typically 18 avoided with negative inputs because “Op-Amp inputs are highly susceptible to reverse voltage breakdown” (Macini, 1999). Furthermore, since the common ground of the Op-Amp is not at a zero reference in this mode, extra components are required to create an artificial reference node (Kitchin, 2001), which cannot be seen in the previous version of the device. Another issue with the circuit in Figure 2.9 is that no Anti-Aliasing filtering units (Analog Devices, 2002) were included; also, no buffers are present to prevent input impedances in different stages to load the prior ones, which may cause loss of transfer signal (Ultimate Electronics, 2021). 2.2.2.3 Box Pressure Sensor An HSCDRNN030PAAA3 was used in the old design to measure the pressure inside the chamber, which was a 12-bit, 3.3V, 30psi analog sensor. This sensor is highly reliable and accurate, making it a common choice for medical purposes. Currently, the cost of this sensor in the Canadian electronic components market is in the range of 80-143 Canadian Dollars. From a marketing point of view, this sensor might need more financial justification for this product since sufficient accuracies are achievable using sensors with a price range of less than or equal to 20 CAD, one example of which is used in the new design. 2.2.2.4 Vacuum Pump and Driver Module A 12V, 10A conventional car vacuum was used in the design of (Crowther et al., 2021) to pump the air out of the chamber. It had enough power to supply the required pressure drop, so we continued to use this device in the new design. 19 To drive the vacuum’s motor, the old design had implemented a BTS7960 dual-half bridge module with a 5.5V-27V, 42A output. The Infineon Technologies BTS7960 chip used in this module is discontinued, and other H-bridges should be used for new prototypes. In addition, our lab testing (see section 2.4) showed that this driver’s performance is unreliable. 2.2.2.5 Cooling System A heatsink attached to the back of the driver module was used to prevent overheating. In addition, a small 12V(DC) cooling fan was also implemented to cool down the board, as seen in Figure 2.9. 2.2.2.6 Power Supply Unit A 12V, 10A variable DC power supply was used to supply the required power to run the vacuum. The same supply unit was also used in the new design. 2.2.2.7 Electrical Housing and Connections As can be seen in Figure 2.8, the circuitry of the old device was installed on a copper breadboard. The Raspberry Pi module was installed separately at the bottom of the control box, and connections to the main board were made using breadboard cables. These connections were problematic, as the wires would get loose and cut off the connections. The other issue was a lack of an emergency power-off button. The only shut-off button was on the user interface, which had delays in turning off the system and could damage the animal in emergency cases. 20 In the new design, all the circuitry is implemented on a single PCB board, which was designed in Altium Designer software, and an emergency button is installed to improve safety. Figure 2.8 Old LBNP Board Installation and connections, picture taken from (Crowther et al., 2021) Figure 2.9 Schematics of the Old Design, picture taken from (Crowther et al., 2021) 21 2.2.3 GUI/Microcontroller Software Similar to hardware, there are main programming modules that should be present for the LBNP chamber to operate. These modules are as follows: 1. A Graphic User Interface (GUI), where the user interacts with the computer application of the device, gives different commands, and observes signals. 2. Hardware interactions, where software works with hardware components such as the box and catheter sensors and vacuum pump, get their states and set values for them. 3. Trajectory generation and control algorithms to ensure the pressure reaches the desired value set by the user. As mentioned in section 2.2.2.1, since the microcontroller used for the old design was Raspberry Pi 4 running a Linux OS, all three modules above were done by the RPi. The codes were written in Python, and the GUI was made using the “Tk” Interface package. Figure 2.10 shows the old device’s user interface. The interface had a Start/Stop button, slots for setting the desired pressure and hold times, options to enter desired control parameters, and the option to save the session to file. On the right part was a plot showing box pressure, desired blood pressure, and the measured blood pressure. However, the plots were drawn only after the experiment was over and were not real-time. A Proportional-Integrator (PI) controller (Astrom & Hagglund, 1995) code was implemented to control the pressure. 22 Figure 2.10 Graphical User Interface, picture taken from (Crowther et al., 2021) 2.2.3.1 Derivation of the Pressures from Raw Measurement To derive the values for systolic, diastolic, and mean arterial pressures, certain computations need to be performed on the raw catheter input. The raw catheter signal is a combination of two sine waves representing the cardiac pressure in each heartbeat, plus the breathing cycles, and is in the form seen in Figure 2.11. 23 Figure 2.11 Blood Pressure Systolic and diastolic pressures are the maximum and minimum blood pressure peaks in each heartbeat. To derive them, a peak detection algorithm was implemented. Mean arterial pressure is the average of the blood pressure in one cardiac cycle and is derived by the following formula (DeMers & Wachs, 2024), which was also implemented in the codes. 𝑀𝐴𝑃 = 𝐷𝑃 +13(𝑆𝑃 − 𝐷𝑃) (2.1) In (2.1) DP, SP, and MAP are Diastolic, Systolic and Mean Arterial pressures, respectively. The same methods of deriving the pressure measurements were also used in the new LBNP. The issues with the software that required to be addressed in the new design were as follows: 24 1. The user interface needed to run on any desktop computer as the user might need to monitor or manipulate the data on the same lab computer as their other applications are running, without using extra peripherals just for this device. 2. The plots needed to be monitored in real time so that if there were any problems, the user could immediately shut down the experiment. 3. Python is an interpreted language, and while it is the perfect language for machine learning and offline applications, it typically is not ideal for real-time applications since it has a slower performance than compiled languages and it cannot guarantee that the control loop is timed accurately. C/C++/C# family is a better substitute for these applications (GeeksforGeeks, 2020). 2.3 New Design As mentioned in section 2.2.1, the mechanical design of the chamber was acceptable for LBNP. The power supply and the vacuum pump model in the new architecture are also the same as those in the old one. This section will discuss how the new hardware, electronics, and software on the MCU and GUI resolved/improved the issues with the old device. 2.3.1 Hardware Electronics The flowchart in Figure 2.12 shows the modules in the new system. The details of the new system and its advancements compared to the old one are discussed in the following subsections. 25 Figure 2.12 New Design’s Electronic Modules 2.3.1.1 The Microcontroller Unit (MCU) To separate the GUI and MCU, the microcontroller board is changed to a Raspberry Pi Pico. Raspberry Pi Pico is a development board running a dual-core 133MHz ARM Cortex M0+ processor with 2 SPI, 2 I2C, 2 UART pins, and 16 controllable PWM channels. It does not run an OS inside and can connect to desktop computers via one of the UART pins. This change resolves the issue of the MCU requiring separate peripherals to run, and as the GUI can run on any computer, data can be saved and manipulated more efficiently. Figure 2.13 shows the Raspberry 26 Pi Pico’s Input/Output (IO) pins. The push button is implemented to reset the microcontroller when programming, without the need to unplug it from the computer. Figure 2.13 Microcontroller Unit 2.3.1.2 Catheter Input, Voltage Scaling and Anti-Aliasing The catheter reading module in the new design addresses a few considerations, as described in the subsections below. 2.3.1.2.1 Voltage Division Stage In the new design, similar to the one in (Crowther et al., 2021), the same model SciSense PV catheter is used to read the rat's blood pressure. Therefore, the voltage needed to be scaled before 27 being fed into the Raspberry Pi Pico, which operates at a 3.3V logic level. This scaling was achieved using a potentiometer acting as two voltage divider resistor parts. 2.3.1.2.2 Anti-Aliasing Stage Before feeding the analog signal from the catheter to the ADC, anti-aliasing considerations should be addressed. A typical breathing frequency for a rodent is assumed to be less than 85 breaths per minute. The heartbeat is approximately five times the breathing cycle, making it 425 beats per minute or 7.08Hz. Thus, we design the anti-aliasing filter for a signal frequency of up to approximately 10Hz and 5V amplitude. We set the sampling rate of the MCU to be 100 times faster than the signal, so the rate 𝑓𝑠 would be 1 kHz. Therefore, according to the Nyquist-Shannon sampling theorem (Tan & Jiang, 2019), Nyquist frequency 𝑓𝑁 (stop-band) is equal to: 𝑓𝑁 =𝑓𝑠2= 500 𝐻𝑧 (2.2) and the cut-off frequency 𝑓𝑐 (pass-band) is approximately: 𝑓𝑐 =𝑓𝑁10= 50 𝐻𝑧 (2.3) Knowing the stop-band and pass-band frequencies, Texas Instruments Tools can be used to design the filter (Karki, 2023). Using Texas Instrument’s Analog Engineer’s Calculator, a third order Bessel filter is found suitable for our application, as seen in Figure 2.14. 28 Figure 2.14 Anti-Aliasing Filter Order using TI Analog Engineer’s Calculator With the information gained through the TI Analog Engineer Calculator, we then used their Filter Design Tool (TI, n.d.) to find the architecture of the filter, as shown in Figure 2.15. Figure 2.15 Two-Stage Anti-Aliasing Filter using TI Filter Design Tool The characteristics of the designed filter are shown in Figure 2.16. The order of the filter for the optimal condition should be 3; although a second-order filter would have less phase shift, it would take longer to settle, and its group delay would be greater than that of the third order filter; also, 29 the passband frequency would have to be around 33Hz in the second order case. A fourth-order filter would have a quicker response, but the phase shift and the group delay would be higher. Figure 2.16 Anti-Aliasing Filter Characteristics derived from TI Filter Design Tool 2.3.1.2.3 Buffering Between each stage of this module a buffer is implemented to convert high-impedance signals to low impedance and, as a result, restore the signal flow to be the strongest it can be. The new design has one impedance-matching buffer between the catheter input and the voltage division unit, one between the voltage division and the anti-aliasing filter, and one before the ADC to speed up digitization. 30 2.3.1.2.4 Dual Supply Since the input voltage to the Op-Amps is -8V to 8V, we implemented a dual supply system for the circuit. A low-current 12V power supply is installed to supply this part of the system to avoid noise interference between the motor supply section of the board and the other parts. To design the supply circuitry for this section, the Op-Amp’s requirements must be met to drive the signals as large as ±8V. According to the TL97x Op-Amp family datasheet the supply voltage should be at least 1.5V - 2V higher than the maximum input voltage. To have a reliable positive supply that is sufficient for the Op-Amps an NCP3334DADJR2G positive adjustable voltage regulator chip is used to create a positive voltage of just above 10.2V (with standard resistors in the market we were able to create a voltage of approximately 10.2V), as shown in Figure 2.17. Figure 2.17 Positive Voltage Regulator 31 A negative charge pump and a negative voltage regulator were used to create the negative voltage. The charge pump chip TC1044SCOA creates a negative supply equal to its input. The negative voltage signal was then fed to the MIC5271YM5-TR chip to provide a reliable -10V signal to serve as a negative supply for the Op-Amps. The negative supply circuitry can be seen in Figure 2.18. Figure 2.18 Negative Supply Circuitry 2.3.1.2.5 Analog-To-Digital Conversion Raspberry Pi Pico has 3 internal ADC modules, which we used for the current sensing of the motor driver (see section 2.3.1.7). For the catheter reading, an ADC121S051CIMF chip was used to convert the analog signal to digital and to feed it through one of the digital pins of Pico (see Figure 2.19). Figure 2.19 Catheter External ADC 32 2.3.1.3 Box Pressure Sensor To measure the chamber’s pressure, an Adafruit BMP390 module was used, which operates with a low-noise 24-bit absolute barometric sensor Bosch BMP390 chip. The module has wirings for both SPI and I2C modes, but we used I2C here. Unlike the pricey analog sensor in the old design, the price of this module is currently about 15 CAD. It serves the resolution needed in this project, and has in-built ADCs, which removes the need to add any external ADC modules. 2.3.1.4 Analog Outputs To check for any disparities between the calculations in our system and the ADV500 software running on the lab’s data acquisition computer, also for debugging purposes, the reference pressure and sensor measurements are sent via BNC cables to be read by the lab’s data acquisition system. We needed to output these signals using a Digital-to-Analog Converter. For a high-speed, low-noise transfer of the data, an isolated I2C bus was implemented. Usage of the isolation ICs separates electrical circuits so that signals can pass through, whereas ground loops, high common-mode transients, or stray currents are blocked. First, an isolated power supply needed to be set up. For that, we used an ADUM5010ARSZ-RL7 integrated DC-to-DC converter chip, as shown in Figure 2.20. 33 Figure 2.20 Isolated Power for I2C Connections Secondly, we set up the data transfer system from the microcontroller to the Digital-to-Analog Converter (DAC). For this purpose, the Si8606AD I2C isolator with two unidirectional digital channels was chosen, and for the DAC, the AD5694RBRUZ-RL7 chip was used, which is a quad 16-/14-/12-bit string DAC with an I2C interface. Finally, a reference chip was used to ensure the output was bound to the range of 0-3V. The chip adopted for this purpose is a REF193GSZ. The schematic of this part is shown in Figure 2.21. Since the AD5694RBRUZ-RL7 has an internal buffer implementation at the output, there is no need to add another buffer before the BNC connector. The four jumpers in Figure 2.21 are present to determine the I2C line address, set outputs to zero-scale or mid-scale, and determine the span of the DAC based on the datasheet. 34 Figure 2.21 ISO I2C Transfer Circuitry 2.3.1.5 Digital Data Output In case some data is required to be read digitally, an FTDI module is used to transfer the data from the controller board to the computer through a serial connection. For this purpose, we used the SparkFun DEV-14050 module in 3.3V logic mode. Currently, this function is not being used, but for the user’s future needs, the controller board can be programmed to output desired data through it. 2.3.1.6 OLED Information Display As another means of data monitoring, an Adafruit Monochrome 0.96” 128x64 OLED Graphic Display is installed on the board in SPI mode. Currently, this module is not being used, but in the 35 future, it can be used to display reference pressures, modes of control, or other data requested by the user, directly on top of the control box. 2.3.1.7 Vacuum Pump Driver The vacuum pump used in the new design is the same 12V, 10A one as the old device. The new driver module was designed based on the Infineon Technologies BLDCSHIELDIFX007TTOBO1 module to solve the reliability problems with the old driver. The driver system has been implemented on the same PCB board as the other components to minimize the usage of wired connections. The module comprises three IFX007TAUMA1 8V-40V, 55A half-bridges, enabling it to drive two-phase or three-phase vacuum motors, thus giving the user the freedom to pick different models of vacuums. Each half-bridge, as seen in Figure 2.22, has a “Current Sense” pin to check the phase currents for debugging purposes. Figure 2.22 Half-Bridge Module Schematics 36 To accommodate different modes of driving three-phase motors, circuitry for both “Sensorless Detection” and “Hall/Single-Ended Encoder” is implemented (see Figure 2.23). By default, the Sensorless Detection mode is used, but one can change the mode by cutting off the 0Ohm jumpers R131, R134, and R137 in Figure 2.23, soldering them to the slots R147, R148, and R149, and connecting an encoder or hall sensor to Header J1. Figure 2.23 Sensorless Detection and Hall Sensor/Encoder Control 2.3.1.7.1 Reverse Polarity Protection The input of the motor driver module is a high-current 12V input. The connector used for this purpose is an SAE connector (similar to the old device) whose polarity is visible. Nonetheless, for extra levels of protection, in case the type of the connector in a future design is different, a reverse polarity protection circuitry is implemented before the motor driver circuit that cuts off the current if the polarity is mistakenly reversed (see Figure 2.24). 37 Figure 2.24 Reverse Polarity Protection 2.3.1.8 Emergency Stop To stop the operation of the device in emergencies, e.g., in case the pressure drop in the subject goes above the safe limit or the subject's health factors are compromised, a GPB527D202BB 14V(DC), 20A(DC) pushbutton was installed after the SAE voltage input connector. When pushed, it turns off the vacuum immediately, allowing the pressure inside the chamber to return. However, the other parts of the board, including the microcontroller and analog/digital data transfers, will continue their operation, monitoring the data to check for potential causes of that emergency situation. 2.3.1.9 Cooling System To cool the driver system, three adhesive heatsinks were installed on the half-bridge chips. Also, similar to the previous design, a 12V(DC) fan is connected to the side of the device, which operates as soon as the driver part is turned on. 38 Complete design schematics of the new device can be found in Appendix A. A three-dimensional view of the PCB is shown in Figure 2.25, and pictures of the new box are seen in Figure 2.26 and Figure 2.27. Due to the shortage of time and the need to prepare the device in time for the physiological experiments, the housing used was a generic one bought from DigiKey, whose dimensions were larger than the board. This problem forced us to use wired connections between the board, the output BNCs, and the push button; we also had to drill imperfect holes and slots into the box, and did not have a proper housing for the fan. In the future, a proper box that is compliant with the board dimensions and specifications should be 3D-printed to resolve this issue. Figure 2.25 3D View of the PCB in Altium Designer Low-Current Supply Vacuum Supply Emergency Stop Cooling Fan Vacuum Output OLED Display Measurement Output Reference Output Catheter Input RPi Pico FTDBox Sensor Input 39 Figure 2.26 The Controller Board (a) Fan 40 (b) Figure 2.27 (a) Back/Side View, (b) Front View of the Controller Box 2.3.2 Software In the new design, the GUI is a .exe computer application written in C#, which can be opened and run on any desktop. Hardware interactions and control algorithms are written in C and run on the Raspberry Pi Pico on the control board. The GUI and the microcontroller are connected with serial communication through Pico’s main serial port, which exchanges user commands and reference values, as well as transferring the measured sensor values by the hardware to the GUI for monitoring. Box In Pico In Low-Curr Pwr High-Curr Pwr 41 2.3.2.1 LBNP Desktop App The GUI was designed using Visual Studio, and its Windows app can be seen in Figure 2.28. At the top right corner of the app, two modes can be defined for the system. The Voltage Control mode is an open-loop operation, which was used to gather data for our modeling and system identification, as will be described in Chapter 3. In this mode, the user enters voltage commands for the vacuum pump and records the system's behaviour in response to that input. The Pressure Control mode is a closed-loop system with a PI controller that uses a reference type (i.e., Box Pressure, MAP, DP or SP), desired pressure drops, and rise and hold times, then controls the selected pressure to match the reference. The real-time plots in the interface use Visual Studio's ScottPlot library to plot the raw catheter input, reference pressure, control pressure, and PWM values with a sample time of 100 Hz (10 times slower than the control loop) as the experiment is running. Other fields in the interface are used to select the communication port with the microcontroller, set controller parameters, and calibrate the catheter sensor to match the LBNP with the ADV500 system. There is also an option to save the session to a file. 42 Figure 2.28 LBNP User Interface 2.3.2.2 Microcontroller Software The codes for the microcontroller were developed using Visual Studio Code in C language. The control loop is written in an Interrupt that runs at a speed of 1kHz to make the controller’s integral calculations precise. An algorithm to derive BP, SP, and MAP is implemented, similar to what was discussed in section 2.2.3.1. A second-order digital Bessel lowpass filter (Pashtoon, 1987) was also implemented to smoothen the data prior to the control phase. The continuous form of such a lowpass filter is as follows (Roberts, 2001): 𝐹(𝑠) =𝜔2𝑠2 + 2𝜁𝜔𝑠 + 𝜔2 (2.4) where 𝜔 is the cut-off frequency in rad/s and 𝜁 is the damping factor. A Backward Euler discretization approximation (2.5) can be used with a sampling time 𝑇𝑠 = 10−3𝑠 to discretize the above filter: 43 𝑠 =1 − 𝑧−1𝑇𝑠 (2.5) Using (2.4) and (2.5), the digital lowpass filter would have the following format: 𝐹(𝑧−1) =𝑇𝑠2𝜔2𝑧−2 − 2(1 + 𝜁𝜔𝑇𝑠) + (1 + 𝜔𝑇𝑠(2𝜁 + 𝜔𝑇𝑠)) (2.6) Therefore, the recursive implementation of this filter is as follows: 𝑎0𝑦𝑖 + 𝑎1𝑦𝑖−1 + 𝑎2𝑦𝑖−2 = 𝑏0𝑢𝑖 (2.7) where 𝑢𝑖 is the 𝑖-th raw measurement, 𝑦𝑖, 𝑦𝑖−1 and 𝑦𝑖−2 are filtered values at times 𝑖, 𝑖 − 1 and 𝑖 − 2, respectively, and the filter coefficient are as follows: 𝑎0 = 1 + 𝜔𝑛𝑇(2𝜁 + 𝜔𝑛𝑇), 𝑎1 = −2(1 + 𝜁𝜔𝑛𝑇), 𝑎2 = 1, 𝑏0 = 𝜔𝑛2𝑇2 (2.8) 2.3.2.2.1 Open-Loop Mode of Operation In Open-Loop mode or “Voltage Control”, the user enters their desired voltage, the rise time (0 for step input, other values for ramp), and the time to hold this value. The microcontroller translates these values to PWM signals at each sample time and outputs this value to the vacuum pump. For a ramp input command of a maximum voltage 𝑉𝑚𝑎𝑥, rise time 𝑇𝑟 and hold time 𝑇𝐻 the value of voltage 𝑉𝑖 at each sample 𝑡 = 𝑖 is calculated using the below algorithm: 44 𝑰𝒇 (𝒊 < 𝑻𝑹): 𝑉𝑖 = 𝑉𝑖−1 +𝑉𝑚𝑎𝑥𝑇𝑅𝑇 𝑬𝒍𝒔𝒆 𝑰𝒇 (𝑻𝑹 < 𝒊 < 𝑻𝑹 + 𝑻𝑯): 𝑉𝑖 = 𝑉𝑚𝑎𝑥 𝑬𝒍𝒔𝒆: 𝑉𝑖 = 0 𝑬𝒏𝒅 and the PWM values for the motor become: 𝑃𝑊𝑀𝑖 =𝑉𝑖𝑉𝑚𝑎𝑥× 𝑃𝑊𝑀𝑚𝑎𝑥 (2.9) where for an RPi Pico for a sampling frequency of 1kHz, 𝑃𝑊𝑀𝑚𝑎𝑥 = 62499 (Raspberry Pi Ltd, 2024). The pressure change according to the voltages is then monitored and saved to files. 2.3.2.2.2 Closed-Loop Mode of Operation In closed-loop mode or “Pressure Control”, the program gets reference pressure type and values, the duration in which the signal should linearly increase to reach the final value (rise time), and the time the reference holds its value (hold time) as input and matches the pressure to the reference. The user can select a sequence of a maximum of four pressure control “sessions” with different desired pressures and rise and hold times as seen in Figure 2.28 (P1 to P4). The trajectories made for the pressure at each session are computed linearly, similar to the voltage control case in the algorithm defined at the top of this page. The PI controller then compares the reference with the sensor feedback and changes the PWM commands accordingly. The values for the controller parameters were picked heuristically (as explained in section 5.2.3) in the closed-loop experiments 45 performed in the lab. It is important to note that the PI controller applies on PWM, and therefore, the parameters which will be derived in Chapter 4, should be multiplied by a factor of 𝑃𝑊𝑀𝑚𝑎𝑥100 (%) or 624.99 to be used in the interface. 2.4 Simulation and Circuitry Testing The operation of each module on the board was verified by running simulations in OrCAD PSpice software, building each module individually on a breadboard, and checking signals with an oscilloscope and multimeter. Also, the hardware interaction parts of the code were tested with each component. We verified the OLED function with RPi Pico using Adafruit’s library (Adafruit/Adafruit_SSD1306, 2010/2024). The FTDI board was tested by transferring strings of characters using a simple serial connection code, and the output was monitored in the serial terminal software PuTTY. Output DACs were tested by programming the Pico to send out certain levels of digital signals (constant and square pulse) and checking the BNC output on an oscilloscope. Motor drivers were tested by plotting the phase voltages and currents of each phase on the oscilloscope. Through these tests, we found the unreliability of the BTS7960 module. By comparing the BTS7960 chips’ voltages and currents as two/three phases of a motor driver, significant mismatches were found between different manufactured chips. As an example of these issues, Figure 2.29 shows phase currents (green graphs) and the current sensing IS pin voltages (yellow graphs) of two half-bridges in no-load conditions; in identical conditions for both chips, 46 IS voltage pin amplitudes are not matched (Note that the oscilloscope’s voltage division for Figure 2.29 (b) is twice (a)). (a) (b) Figure 2.29 Phase Currents (green) and IS Pin Voltages (yellow) of BTS7960, (a) Phase 1, (b) Phase 2 Furthermore, Figure 2.30 compares the phase voltages of three half-bridge chips in a 95% PWM (a) and a 60% PWM (b). As the PWM decreases, the voltages on each phase become noisier. (a) (b) Figure 2.30 Phase Voltages of Three Phases at (a) 95% PWM (b) 60% PWM 47 BMP390 sensor readings were verified using the Adafruit library (Adafruit/Adafruit_BMP3XX, 2018/2024) as well as by creating negative pressures inside the chamber with the vacuum and reading the outputs of BMP390. As for the catheter input, since we were running the tests without a rat, we used a function generator to create a periodic signal with the typical breathing cycle of a rat (between 60-85Hz) and tested each stage of the module, which were assembled on the breadboard, using an oscilloscope. Also, before ordering the components and assembling the catheter reading module, the voltage-scaling and anti-aliasing system was simulated and verified in PSpice. Once the operation of the individual modules were verified and the software was developed for the device, we tested the completed system with the user interface. We gave pressure commands through the app to control the chamber pressure, ran the system with the vacuum pump, and monitored the graphs on the interface. Once the device succeeded in all the tests, and objectives 1 and 2 of section 1.4 (hardware and software optimization) were achieved, we proceeded to test the device at the animal facility, performing in-vivo experiments. 2.5 In-Vivo Testing and Experiment Design In-vivo experiments were designed to test the device’s performance on rats and collect data for modeling (Chapter 3) and controller design (Chapter 4). The device was tested on 10 rats in open-loop mode. Since the left ventricular pressure acts as a driving force for the arterial pressure, arterial pressure and LV pressure have high similarities in their behaviors; plus, since the artery is easier of the two to access, it tends to be more commonly used in experiments. Therefore, for our 48 open-loop experimentations, an arterial catheter was used. The blood volume was recorded by the ADV500 system to plot PV loops and was not an input to our LBNP system. The procedure for the open-loop experiments was as follows. A Step input was given to the system with a voltage starting from 2V and a hold time of 10-12 seconds. Then, the Step value was increased 1-2 volts at a time until the rat’s pressure was at the edge of the safety thresholds (rat’s blood pressure drop of 30mmHg, equivalent to 6-7 volts of vacuum input). Each experiment was repeated 3-5 times, and between each repetition, the rats were given enough time to recover their blood pressure. In addition to the open-loop experiments, closed-loop tests were performed on rat number 10 to have a basis for a comparison between the heuristics-based controller and the robust controller in Chapter 4 (see the results in section 5.2.3). Closed-loop experiments were performed using readings from the box pressure and both arterial and left ventricular blood pressures, as well as different values of controller gain, and pressure references of type Step and Ramp. The experimental results were monitored and recorded on LBNP’s GUI. The ADV500 system’s data plotter LabChart also recorded data channels, including the box pressure sent out by LBNP’s BNC cable. Figure 2.31 shows the experimental setup, the interface on the laptop (it could also run alongside LabChart on the lab's computer, but to have a better view of all the graphs at the same time, we ran the interface on a separate laptop), LabChart plotting different channels and the PV loops in real-time. The LBNP chamber with the rat and the vacuum pump is also shown in Figure 2.31. 49 Figure 2.31 In-Vivo Experimentation at the Animal Facility Figure 2.32 Gives a better view of the graphs being monitored. The trend of changes in MAP versus box pressure (LBNP Pressure channel in LabChart) can be seen in this image. Finally, Figure 2.33 shows the change in the PV loops. The ESPVR and EDPVR can be seen in the PV graph for this subject. 50 Figure 2.32 LabChart Graphs Figure 2.33 PV Loops Graph 51 2.6 Summary Chapter 2 discussed how the design of the new LBNP device addressed the issues of the old device made by (Crowther et al., 2021). Improvements in both electronics and software were made and tested for performance. Section 2.5 explained the in-vivo experiment to assess the system’s performance in the presence of the subjects. Once the data from in-vivo experiments are collected, modeling and control algorithms can be completed. The next two chapters discuss the methods and algorithms adopted for these purposes. 52 Chapter 3: System Identification and Modeling In the previous chapter, the design of the LBNP system was discussed, and objectives 1 and 2 of the thesis (hardware and software optimization) were achieved. It was also shown that in-vivo experimentations were conducted at the animal facility to create a dataset for system identification and control system design. This chapter will explain the system identification algorithms and techniques used to find the models for the chamber pressure, as well as the rat’s Systolic (SP), Diastolic (DP), and Mean Arterial pressures (MAP). The results of this system identification process can be seen in Chapter 5. 3.1 Open-Loop LNBP The LBNP system, without feedback or controller present, is shown in Figure 3.1. Figure 3.1 Open-Loop LBNP Block Diagram In the open-loop model of Figure 3.1, two models can be seen. The vacuum’s voltage causes the box pressure to change, and this change in the box pressure affects the value of DP, SP, and MAP. The goal of this chapter is to identify the three model sets below: 1. From vacuum voltage to box pressure 2. From box pressure to DP/SP/MAP 53 3. The model of vacuum, chamber and rat as one whole system: the model from vacuum voltage to DP/SP/MAP As it will be discussed in the next section, a grey-box model can be derived for the first model based on the estimations of fluid dynamics equations. However, there has been no literature on mathematical modeling of a rat’s arterial blood pressure based on an LBNP’s vacuum pump voltage or chamber pressure. Therefore, the “Rat” system model is a black box. 3.2 Voltage-Box Pressure Mathematical Model The LBNP system uses a vacuum that evacuates the air from the chamber to create a desired negative pressure. Parameters that play a role in this system are the leakages from the LBNP chamber and the vacuum, and vacuum motor specifications. The input to the system is the voltage that controls the motor’s rotational speed, and the output is the pressure inside the chamber. Figure 3.2 shows a hydraulic block diagram of the chamber and pump. The vacuum pump provides the necessary airflow to compensate for the total leakage 𝑄. In the DC circuit model, it is shown as a current source. 𝑄 is dependent on the vacuum motor’s rotational speed 𝜔𝑚, and therefore, we can write: 𝑄 = 𝜆𝜔𝑚 (3.1) where 𝜆 is a coefficient describing the relationship between the motor’s speed and the airflow. There are two sources of leakage: 𝑄𝑝 represents leakage losses in the pump, and 𝑄𝑏 is the leakage between the rat’s torso and the opening for the rat in the chamber. 54 We assume that these leakages are laminar flow and thus, are related to a pressure drop by a hydraulic resistance (Van de Vegte, 1993): 𝑄𝑏 =𝑃𝐵𝑜𝑥𝑅𝑏 (3.2) where 𝑃𝐵𝑜𝑥 is the pressure difference and 𝑅𝑏 is the hydraulic resistance between the inside and outside of the chamber. For the leakage air past the vanes of the vacuum pump, with a flow resistance of 𝑅𝑝 we can write: 𝑄𝑝 =𝑃𝐵𝑜𝑥𝑅𝑝 (3.3) The two leakages add up to the total flow created by the vacuum pump: 𝑄 =𝑃𝐵𝑜𝑥𝑅𝑝+𝑃𝐵𝑜𝑥𝑅𝑏 (3.4) Figure 3.2 Hydraulic Analogy To simplify, we use 𝑅𝑒𝑞 as the equivalent of the 𝑅𝑝 and 𝑅𝑏 in parallel, and use (3.1) to find a substitute for 𝑄. Q Qb Qp Rb Rp 55 𝑄 =𝑃𝐵𝑜𝑥𝑅𝑒𝑞⟹ 𝑃𝐵𝑜𝑥 = 𝑅𝑒𝑞𝑄 = 𝑅𝑒𝑞𝜆𝜔𝑚 (3.5) Power in hydraulic systems is defined by pressure multiplied by flow; therefore, if we write the energy balance between the vacuum motor power and the hydraulic system: 𝑇𝑙𝑜𝑎𝑑 ×𝜔𝑚 = 𝑄𝑃𝐵𝑜𝑥 (3.6) where 𝑇𝑙𝑜𝑎𝑑 is the total torque load that the vacuum motor needs to overcome. From equations (3.1), (3.5) and (3.6), we have: 𝑇𝑙𝑜𝑎𝑑 =𝑅𝑒𝑞𝑄2𝜔𝑚= 𝜆2𝑅𝑒𝑞𝜔𝑚 (3.7) The vacuum pump has a Brushless DC (BLDC) motor, therefore, the below equations can be written, assuming the motor has a torque constant 𝐾𝑡, inertia 𝐽𝑚, and damping constant of 𝐵𝑚 (Abdolhosseini & Abdollahi, 2023): 𝐿𝑎𝑑𝑖𝑎𝑑𝑡+ 𝑅𝑎𝑖𝑎 = 𝑉𝑖𝑛 − 𝐾𝑎𝜔𝑚 (3.8 a) 𝐾𝑡𝑖𝑎 = 𝐽𝑚𝑑𝜔𝑚𝑑𝑡+ 𝐵𝑚𝜔𝑚 + 𝑇𝑙𝑜𝑎𝑑 (3.8 b) In equations (3.8 a) and (3.8 b), 𝐾𝑎 is the armature constant, 𝑅𝑎 is the armature resistance and 𝐿𝑎 is the armature inductance. If we assume that the motor is fast enough that the time constant of motor, which is 𝜏 = 𝐿𝑎/𝑅𝑎, is negligible, we can rewrite equation (3.8 a) in the form below: 𝑖𝑎 =𝑉𝑖𝑛𝑅𝑎−𝐾𝑎𝑅𝑎𝜔𝑚 (3.9) By replacing 𝑖𝑎 in equation (3.8 b) with the value derived in (3.9), we have: 𝐾𝑡𝑅𝑎(𝑉𝑖𝑛 − 𝐾𝑎𝜔𝑚) = 𝐽𝑚𝑑𝜔𝑚𝑑𝑡+ 𝐵𝑚𝜔𝑚 + 𝑇𝑙𝑜𝑎𝑑 (3.10) 56 After substituting 𝑇𝑙𝑜𝑎𝑑 in (3.10) by its value derived in (3.7), and applying a Laplace transform, equation (3.10) will become: 𝐾𝑡𝑅𝑎(𝑉𝑖𝑛 − 𝐾𝑎Ω𝑚) = 𝐽𝑚𝑠Ω𝑚 + (𝐵𝑚 + 𝜆2𝑅𝑒𝑞)Ω𝑚 (3.11) Equations (3.5) and (3.11) can be used to draw a block diagram from voltage to box pressure, as seen in Figure 3.3: Figure 3.3 Block Diagram of the Box Pressure System where 𝐵𝑒𝑞 = 𝐵𝑚 + 𝜆2𝑅𝑒𝑞. Based on the diagram in Figure 3.3, the transfer function from 𝑉𝑖𝑛 to 𝑃 is given by: 𝑃𝐵𝑜𝑥𝑉𝑖𝑛= 𝜆𝑅𝑒𝑞 ×Ω𝑚𝑉𝑖𝑛= 𝜆𝑅𝑒𝑞 ×𝐾0𝑇𝑠 + 1 (3.12) where the steady-state gain from 𝑉𝑖𝑛 to 𝑃 is: 𝐾 = 𝜆𝑅𝑒𝑞 × 𝐾0 = 𝜆𝑅𝑒𝑞𝐾𝑡/𝑅𝑎𝐵𝑒𝑞 + 𝐾𝑡𝐾𝑎/𝑅𝑎 (3.13) and the time constant 𝑇 is: 𝑇 =𝐽𝑚𝐵𝑒𝑞 + 𝐾𝑡𝐾𝑎/𝑅𝑎 (3.14) Therefore, the model we expect to see in the data is a first order lag from 𝑉𝑖𝑛 to 𝑃 with gains and time constants in (3.13) and (3.14). 57 3.3 System Identification Algorithm In order to find the models mentioned in 3.1, a system identification process is required. The three main components of a system identification procedure are as follows: 1. Choosing a dataset with inputs that are persistently exciting and maximally informative (Ljung, 1999). 2. Picking a set of candidate model structures based on a priori knowledge and engineering intuition. 3. Choosing suitable validation methods for the models. A dataset of Step responses of the open-loop model is gathered from in-vivo experiments, as discussed in section 2.5. This section discusses the model structures, fitting algorithms, and assessment methods used in the thesis. The goal is to find the simplest models that represent the data well enough to use linear controller design rules. Also, the choice of high-order models seems unreasonable because the system in question is the internal function of a rat's body, which may vary in different subjects. Compensations for uncertainties in the model and neglected model dynamics will be addressed in Chapter 4. 3.4 Representation of the System To find a set of candidate models, the system needs to be represented in a mathematical form. We assume that our models in Figure 3.1 are Linear Time-Invariant (LTI). The models can be represented, as described in (Ljung, 1999), with the following structure: 𝑤(𝑡) + 𝑓1𝑤(𝑡 − 1) + ⋯+ 𝑓𝑛𝑓𝑤(𝑡 − 𝑛𝑓) = 𝑏1𝑢(𝑡 − 1) + ⋯+ 𝑏𝑛𝑏𝑢(𝑡 − 𝑛𝑏) (3.15 a) 58 𝑦(𝑡) = 𝑤(𝑡) + 𝑒(𝑡) (3.15 b) where 𝑢(𝑡 − 𝑘) and 𝑤(𝑡 − 1) are the input and undisturbed output of time 𝑡 − 𝑘, 𝑒(𝑡) is a white gaussian noise disturbance, and 𝑦(𝑡) is the real measurement of the system’s output at time 𝑡. We define a shift operator 𝑞 as follows: 𝑞−𝑘𝑥(𝑡) = 𝑥(𝑡 − 𝑘) (3.16) and using this operator, rewrite (3.15 a) as seen below: 𝑤(𝑡) + 𝑓1𝑞−1𝑤(𝑡) + ⋯+ 𝑓𝑛𝑓𝑞−𝑛𝑓𝑤(𝑡) = 𝑏1𝑞−1𝑢(𝑡) +⋯+ 𝑏𝑛𝑏𝑞−𝑛𝑏𝑢(𝑡) (3.17) By defining 𝐵(𝑞) and 𝐹(𝑞) in the following format: 𝐹(𝑞) = 1 + 𝑓1𝑞−1 +⋯+ 𝑓𝑛𝑓𝑞−𝑛𝑓 (3.18 a) 𝐵(𝑞) = 1 + 𝑏1𝑞−1 +⋯+ 𝑏𝑛𝑏𝑞−𝑛𝑏 (3.18 b) equation (3.17) can be written in a transfer function form: 𝑦(𝑡) =𝐵(𝑞)𝐹(𝑞)𝑢(𝑡) + 𝑒(𝑡) (3.19) representing the block diagram in Figure 3.4. This representation of the system is called Output Error Model (Ljung, 1999). 59 Figure 3.4 Output Error Model Structure In order to derive the models of the system, parameters 𝑓1,…, 𝑓𝑛𝑓 and 𝑏1,…, 𝑏𝑛𝑏 should be estimated. The estimation parameters vector is, therefore, defined as follows: 𝜃 = [𝑓1, … , 𝑓𝑛𝑓 , 𝑏1, … , 𝑏𝑛𝑏] (3.20) Also, since in equation (3.15 a), 𝑤(𝑡 − 𝑘) is never observed or measured directly, it carries a 𝜃 in the notation, meaning it can change based on the parameter estimations. Hence, the set of models to be estimated is as follows: 𝑤(𝑡, 𝜃) + 𝑓1𝑤(𝑡 − 1, 𝜃) + ⋯+ 𝑓𝑛𝑓𝑤(𝑡 − 𝑛𝑓 , 𝜃)= 𝑏1𝑢(𝑡 − 1) + ⋯+ 𝑏𝑛𝑏𝑢(𝑡 − 𝑛𝑏) (3.21) or, in a transfer function format: ?̂?(𝑡|𝜃) = 𝑊(𝑡|𝜃) =𝐵(𝑞)𝐹(𝑞)𝑢(𝑡) (3.22) where ?̂?(𝑡|𝜃) is introduced to conform with the output notation 𝑦(𝑡) and is defined as the predicted undisturbed output, constructed with the parameter vector 𝜃 and past inputs. 60 3.5 Parameter Estimation Algorithm In the previous section, the system was represented as an LTI model, the Output Error (OE) structure was introduced, and the estimation parameter vector and estimation model were defined in (3.20) and (3.22), respectively. This section discusses the algorithms to find the optimal values for the estimation problem. To find the implementation of these algorithms using the MATLAB function oe the reader can refer to section 3.9. 3.5.1 Problem Formulation To solve the estimation problem, the formulation of the problem should first be defined. Suppose that we found a candidate model with a parameter vector 𝜃∗. To evaluate this model, the prediction error is defined as follows, using the measured output 𝑦(𝑡) and the predicted output ?̂?(𝑡|𝜃): 𝐸(𝑡, 𝜃∗) = 𝑦(𝑡) − ?̂?(𝑡|𝜃∗) (3.23) The best model is one that can minimize the error between the measured output and the predicted output. The size of the prediction error vector is determined using a scalar-valued norm, which should be minimized. A commonly used norm for such problems is the two-norm. The quadratic norm parameter evaluation function is defined as below: 𝑉𝑁(𝜃, 𝑍𝑁) =∑||𝐸(𝑡, 𝜃)||22 𝑁𝑖=1 (3.24) where 𝑁 is the number of datapoints in the dataset and 𝑍𝑁 is the input-output information derived from the dataset, defined as follows: 𝑍𝑁 = [𝑦(1), 𝑢(1), 𝑦(2), 𝑢(2),… , 𝑦(𝑁 ), 𝑢(𝑁)] (3.25) 61 The best model is found by minimizing 𝑉𝑁(𝜃, 𝑍𝑁) over the set of all possible models 𝐷𝑀: 𝜃𝑁 = argmin𝜃∈𝐷𝑀𝑉𝑁(𝜃, 𝑍𝑁) (3.26) 3.5.2 Least Square Fitting: Levenberg-Marquardt Method The problem defined in equation (3.26) is a Least Squares problem (Dennis Jr. & Schnabel, 1996), where a vector 𝜃 should be found, such that it is a local minimizer to the function of the sum of squares of the error’s two-norm. Since our LBNP models are Single-Input-Single-Output (SISO), the matrix form 𝐸 in equation (3.24) can be replaced with a scalar value 𝜖 and the norm can be written in the form below: 𝑉𝑁(𝜃, 𝑍𝑁) =∑𝜖2(𝑡, 𝜃)𝑁𝑖=1 (3.27) A suitable numerical method should be adopted to find the minimum of (3.27). The family of Newton algorithms are commonly used for this purpose, where the value of the function, its gradients and Hessians are used to find the search direction 𝑓𝑖 in the general form below (Ljung, 1999): 𝜃(𝑖+1) = 𝜃(𝑖) + 𝛼𝑓(𝑖) (3.28) In (3.28), 𝛼 is a coefficient used to reach a suitable decrease in the value of 𝑉𝑁(𝜃, 𝑍𝑁). A generally used formulation of the equation (3.28) is as follows: 62 𝜃𝑁(𝑖+1)= 𝜃𝑁(𝑖)− 𝛼𝑁(𝑖)[𝑅𝑁(𝑖)]−1𝑉𝑁′ (?̂?𝑁(𝑖), 𝑍𝑁) (3.29) where the subscript (𝑖) is the 𝑖-th iteration and 𝑅𝑁(𝑖) is a 𝑑 × 𝑑 matrix, 𝑑 being the dimension of the parameters vector, that modifies the search direction. 𝛼𝑁(𝑖) is the step size that should be chosen such that: 𝑉𝑁(𝜃𝑁(𝑖+1), 𝑍𝑁) < 𝑉𝑁(𝜃𝑁(𝑖), 𝑍𝑁) (3.30) In equation (3.29), 𝑉𝑁′ (𝜃𝑁(𝑖), 𝑍𝑁) is the gradient of 𝑉𝑁 (see equation (3.27)) and can be defined as follows: 𝑉𝑁′ (𝜃, 𝑍𝑁) = −2∑𝜓(𝑡, 𝜃)𝜖(𝑡, 𝜃)𝑁𝑖=1 (3.31) where 𝜓 is the gradient matrix of ?̂?(𝑡|𝜃∗) with respect to 𝜃 (Ljung, 1999). By varying the values of 𝛼𝑁(𝑖) and 𝑅𝑁(𝑖) in equation (3.29) three algorithms are achievable, which are explained in the following subsections. 3.5.2.1 Steepest-Descent Method This method picks the identity matrix as the parameter and relies solely on the step size and the gradient. Step size in this method can be fixed or chosen using backtracking or exact line search (Gordon & Tibshirani, 2012). This method loses its efficiency close to the minimum and, therefore, it is not ideal for our work. 63 3.5.2.2 Gaussian-Newton Method This method picks the step size 𝛼𝑁(𝑖) to be 1, and uses an approximation of the Hessian of the function 𝑉𝑁(𝜃, 𝑍𝑁) for 𝑅𝑁(𝑖) (Wills & Ninness, 2008). From equation (3.31) the Hessian can be calculated as follows: 𝑉𝑁′′(𝜃, 𝑍𝑁) ≅ 2∑𝜓(𝑡, 𝜃)𝜓𝑇(𝑡, 𝜃)𝑁𝑖=1≜ 𝐻𝑁(𝜃) (3.32) In cases where the dataset is not informative enough or the model is over-parametrized, this method will fail since the Hessian matrix will become singular/close to singular. The next method introduced, which is the method used in this thesis, will overcome this issue. 3.5.2.3 Levenberg-Marquardt Method In order to solve the issue raised in the Gaussian-Newton method, (Levenberg, 1944) and (Marquardt, 1963) suggested these choices for the step size and the search direction modifier term: 𝜶𝑵(𝒊) = 𝟏 (3.33 a) 𝑹𝑵(𝒊)(𝝀) = 𝑯𝑵(?̂?𝑵𝒊 ) + 𝝀𝑰 (3.33 b) They used a modifier scalar term 𝜆 to fix the singularity issue. This method is a balance between Steepest-Descent and Gaussian-Newton methods, since as 𝜆 goes toward zero, the method will become close to Gaussian-Newton, and as it goes toward infinity, it behaves like Steepest-Descent. The parameter 𝜆 is chosen by the iterative process below. Assume that we are at the step 𝑖 in the iterative process and we know the values of 𝑉𝑁(𝜃𝑁(𝑖), 𝑍𝑁) and the 𝜆(𝑖−1) that was used to derive 64 this value. The question is to choose a suitable 𝜆(𝑖) to keep the condition 𝑉𝑁(𝜃𝑁(𝑖+1), 𝑍𝑁) <𝑉𝑁(𝜃𝑁(𝑖), 𝑍𝑁) true and to create the value 𝑉𝑁(𝜃𝑁(𝑖+1), 𝑍𝑁) for the next iteration. 𝑺𝒕𝒆𝒑 𝟎: Start with 𝜆0 = 10−2. Pick a coefficient 𝜈 > 1 (typically it is chosen to be 10) and repeat this iteration until the stop condition is met. 𝑺𝒕𝒆𝒑 𝒊: Use 𝜆(𝑖−1) from the previous iteration to calculate 𝑉𝑁(𝜃𝑁(𝑖+1), 𝑍𝑁 , 𝜆(𝑖−1)) and 𝑉𝑁 (𝜃𝑁(𝑖+1), 𝑍𝑁 ,𝜆(𝑖−1)𝜈): • 𝑰𝒇: 𝑽𝑵 (?̂?𝑵(𝒊+𝟏), 𝒁𝑵,𝝀(𝒊−𝟏)𝝂) ≤ 𝑽𝑵(?̂?𝑵(𝒊), 𝒁𝑵) 𝜆(𝑖) =𝜆(𝑖−1)𝜈 • 𝑰𝒇: 𝑽𝑵 (?̂?𝑵(𝒊+𝟏), 𝒁𝑵,𝝀(𝒊−𝟏)𝝂) > 𝑽𝑵(?̂?𝑵(𝒊), 𝒁𝑵) & 𝑽𝑵(?̂?𝑵(𝒊+𝟏), 𝒁𝑵, 𝝀𝒊−𝟏) ≤ 𝑽𝑵(?̂?𝑵(𝒊), 𝒁𝑵) 𝜆(𝑖) = 𝜆(𝑖−1) • 𝑰𝒇: 𝑽𝑵 (?̂?𝑵(𝒊+𝟏), 𝒁𝑵,𝝀(𝒊−𝟏)𝝂) > 𝑽𝑵(?̂?𝑵(𝒊), 𝒁𝑵) & 𝑽𝑵(?̂?𝑵(𝒊+𝟏), 𝒁𝑵, 𝝀𝒊−𝟏) > 𝑽𝑵(?̂?𝑵(𝒊), 𝒁𝑵) Successively multiply 𝜆 by 𝜈 until for the smallest 𝐾, we have: 𝑉𝑁(𝜃𝑁(𝑖+1), 𝑍𝑁 , 𝜆𝑖−1𝜈𝐾) ≤ 𝑉𝑁(𝜃𝑁(𝑖), 𝑍𝑁) 𝑬𝒏𝒅: when the infinity norm of the gradient is less than the defined tolerance (typically 𝑡𝑜𝑙 = 10−4): ||𝑉𝑁′ (𝜃𝑁(𝑖), 𝑍𝑁)||∞ ≤ 𝑡𝑜𝑙 3.6 Model Assessment For evaluating the predicted models, the criteria Fit Percentage is used (Zavala-Yoe & Galván Duque, 2010) , which uses Normalized Root Mean Squared Error (NRMSE) (Eikeland et al., 2021) as a measure of the similarity between the model and the real output, as follows: 65 𝑭𝒊𝒕 𝑷𝒆𝒓𝒄𝒆𝒏𝒕𝒂𝒈𝒆 = (𝟏 − 𝑵𝑹𝑴𝑺𝑬) × 𝟏𝟎𝟎 = (𝟏 −∑ (𝒀𝒎𝒐𝒅𝒆𝒍 − 𝒀𝒎𝒆𝒂𝒔)𝟐𝑵𝒊=𝟏∑ 𝒀𝒎𝒆𝒂𝒔𝟐𝑵𝒊=𝟏) × 𝟏𝟎𝟎 (3.34) 3.7 Model Orders The models for V-Box Pressure and Box Pressure-Blood Pressure systems that were fit to the dataset are first order models of the form below: 𝑮(𝒛) =𝒃𝟎𝟏 − 𝒂𝟏𝒛−𝟏 (3.35) which is a discrete system with a sampling frequency of 100Hz. Assuming a high value of 85 breaths per minute for the rats, and cardiac cycles with a frequency of 5 per each breathing cycle, the cardiac cycles happen at a speed of 425 cycles per minute or 7.083Hz. If we approximate this value by an upper bound of 10Hz, the sampling rate is still at least 10 times faster than each cycle. This fast sampling frequency records enough datapoints in each cycle to allow for a continuous approximation of the system. Therefore, since continuous transfer functions are generally easier to comprehend and analyze, the model in (3.35) is transformed into the continuous form below using a Zero-Order Hold: 𝑮(𝒔) =𝑲(𝒔 + 𝒛𝟎)𝒔 + 𝒑𝟎 (3.36) where 𝑧0 and 𝑝0 are the zero and the pole of the system, respectively. In our experiment repetitions, the derived zeros were all at least 100 times faster than the poles, therefore this form is further simplified to: 𝐺(𝑠) =𝐾𝑠𝑠𝜏𝑠 + 1 (3.37) with 𝐾𝑠𝑠 and 𝜏 being the steady-state gain and the time constant, respectively. The third sets of models in section 3.1 (from voltage to DP/SP/MAP) can be found by multiplying models 1 and 2: 66 𝑃𝐵𝑉=𝑃𝐵𝑃𝐵𝑜𝑥×𝑃𝐵𝑜𝑥𝑉 (3.38) where 𝑃𝐵 represents the three blood pressure factors DP, SP and MAP. Dominant pole approximation states that if one pole is at least 5 times faster than the other in a second-order system, the system behaves like a first-order system with the slower pole. The average time constants of the models in the LNBP system, for 𝑃𝐵𝑜𝑥𝑉 is 0.63s, and for 𝑃𝐵𝑃𝐵𝑜𝑥 is 1.41s. As the response times of the systems are close, dominant pole approximation cannot be used and 𝑃𝐵𝑉 must remain a second-order system in the calculations. The model for this system is as follows: 𝑃𝐵𝑉=𝐾𝑠𝑠1𝐾𝑠𝑠2(𝜏1𝑠 + 1)(𝜏2𝑠 + 1) (3.39) Where 𝐾𝑠𝑠1 and 𝐾𝑠𝑠2 are the steady-state gains for 𝑃𝐵𝑜𝑥𝑉 and 𝑃𝐵𝑃𝐵𝑜𝑥, and 𝜏1 and 𝜏2 are their time constants, respectively. 3.8 Derivation of the Model Parameters from Rat/Chamber Specifications In Chapter 4, we will find a robust controller based on different models’ variations. This controller can serve as a stable starting point for the LBNP. The controller is chosen to be as optimal as possible, but there is a trade-off between the uncertainty of the system and its performance. Therefore, a question that arises is: Is it possible to target one rat and derive a controller for that subject instead of accounting for the variation of all the models in the dataset? 67 In in-vivo experiments, the mass, chest size and age of the rats are measured and known before preparing the subject for the tests. The size of the front plate’s opening is also known. To answer the above question, we hypothesized that the parameters in models from voltage to box pressure, and from box pressure to Diastolic, Systolic and Mean Arterial Pressures can be derived from the subjects’ specifications. In other words, the hypothesis is, in models of form (3.37), can the steady-state gain 𝐾𝑠𝑠 and time constant 𝜏 be derived as a function ℱ of the area of unsealed plate opening around the rat 𝐶, mass of the subject 𝑀, and their age 𝐴: [𝐾𝑠𝑠, 𝜏] = ℱ(𝑀, 𝐶, 𝐴) (3.40) 3.8.1 Linear Regression To find the function ℱ we perform a linear regression between the model parameters and the rat’s/chamber’s specifications. In this process, to ensure the validity of the linear regression models, we assessed the goodness of the models, as well as the significance of each of the three parameters 𝑀,𝐶 and 𝐴. The goal is to find the simplest model that is significant enough to explain the variability of the models. Therefore, we need to test for constant, linear, interactions, pure-quadratic, quadratic, and polynomial relations. As the order of the model increases, there is a risk of overfitting while not having any advantage in predicting new data. Therefore, we start from the lowest order, i.e., constant, and increase the complexity step by step. 68 The formula for each model is as seen in the table below. In the table, 𝑛 is the number of exploratory variables put inside an 𝑛 × 1 vector 𝑋 and 𝑋𝑖 is the 𝑖-th element of the vector X. 𝐴0, 𝐴1, … are the coefficients matrices and the added indices 𝐴2,𝑖 𝑎𝑛𝑑 𝐴2,𝑖𝑗 are scalar values. Table 3.1 Linear Regression Models Model Formula Explanation Constant 𝐴0 A constant intercept value 𝐴0 Linear 𝐴0 + 𝐴1𝑋 Intercept and linear terms for each variable Interactions 𝐴0 + 𝐴1𝑋 + ∑ 𝐴2,𝑖𝑗𝑋𝑖𝑋𝑗𝑛𝑖,𝑗=1,𝑖≠𝑗 Intercept, linear forms and product of distinct variables Pure-Quadratic 𝐴0 + 𝐴1𝑋 +∑𝐴2,𝑖𝑋𝑖2𝑛𝑖=1 Intercept, linear, and squares of the variables Quadratic 𝐴0 + 𝐴1𝑋 + ∑ 𝐴2,𝑖𝑗𝑋𝑖𝑋𝑗𝑛𝑖,𝑗=1 Intercept, linear, distinct variables’ products and squares of the variables Polynomial - Polynomials with terms up to the specified degrees. 3.8.2 Stepwise Linear Regression A stepwise linear regression method is used to test the validity of the models and find the best fit. Stepwise regression balances the complexity of the model with the predictability (the amount of variance that the model describes) (Li et al., 2023). 69 The Forward Stepwise Linear Regression algorithm starts with the simplest model, which is constant. It adds variables to the regression one by one, performs an ANOVA test (Sharma, 2020), and repeats the process until the criterion specified for it is met. The chosen criterion is: an F-test is performed on the Sum of Squared Error (SSE) resulting from adding a term, and the p-Value of the F-test is compared to the defined threshold. If the p-Value is less than the defined critical value, the term will be added. It will be shown in section 5.1.2 of Chapter 5 (Results) that there are simple models to predict the parameters of the Box pressure model and the MAP model from measurements of the plate size, and the subject’s mass, chest size, and age. These could be used in future work to design a controller, based on the methods discussed in Chapter 4. 3.9 MATLAB Implementation The implementation of the Output Error method in MATLAB can be done by the command oe. The reason for using an Output Error Method (OE), rather than methods such as Order Estimation or Autoregressive models, is that OE is a nonlinear approach, which is widely used in cases where the structure of the model is unknown or in case the model contain nonlinearities such as delays (Zhang, 2004). The Levenberg-Marquardt method is the most common solver used with the OE method for reasons explained in 3.5.2.3 and can be defined with the lm prompt for the MATLAB solver in oeOptions.SearchMethod. The default for MATLAB’s OE solver is auto which switches automatically between different solvers. Using this option might be beneficial, as nonlinear structures in OE do not necessarily result in convex least squares problems and might lead to local optima rather than global; thus, choosing the correct initial value is of high significance. By 70 switching between different approaches, the sensitivity to initial conditions is reduced. In our study, lm and auto resulted in the same models. The Fit Percentage criteria can be seen in the reports generated by the oe function. The command for Stepwise Linear Regression is stepwiselm and for ANOVA, the anova command can be used. 3.10 Summary This chapter discussed the approaches used for finding models for our system, including the four subsystems: Voltage-Box Pressure and Box Pressure-DP/SP/MAP. A physical first-order model was found for our Voltage-Box Pressure system based on the hydraulic equations; for the Box Pressure-DP/SP/MAP, we could not find a model based on literature, so a black-box modeling approach was adopted. The model might not be the true and accurate system representation, but as (Box, 1979) mentions, “All models are wrong, but some are useful.”, we aim for an approximate model simple enough to allow us to use classical controller design rules. In the next chapter, we will address any uncertainty in the model and data and design a robust controller for the device. The modeling was done with the “Output Error Method” black-box representation of the system, a least-square problem was solved with the “Levenberg-Marquardt” approach to solve the least square fitting problem, and the “Fit Percentage” criterion validated the models. Finally, to find models using measurements of the chamber and rats, a “Stepwise Linear Regression” was performed to derive the parameters of our first-order models based on the chamber’s opening size and the rats’ mass, chest size, and age. 71 Chapter 4: Robust Controller Design Chapter 3 explained the methods for finding models for box pressure and blood pressure, with the vacuum's voltage as input. The results of the System Identification algorithms are demonstrated in Chapter 5. Our goal in this chapter is to find a robust controller to match the pressures to the desired values. For that purpose, the following derived models should be studied: Table 4.1 LBNP Models System Model Name Formula Vacuum Voltage – Box (Chamber) Pressure 𝐺𝑉𝐶 𝐾𝑠𝑠1𝜏1𝑠 + 1 Box Pressure – Blood Pressure (DP/SP/MAP) 𝐺𝐶𝐵 𝐾𝑠𝑠2𝜏2𝑠 + 1 Vacuum Voltage – Blood Pressure 𝐺𝑉𝐵 𝐾𝑠𝑠1𝐾𝑠𝑠2(𝜏1𝑠 + 1)(𝜏2𝑠 + 1) As the user only has access to the vacuum voltage, the target models for controller design are 𝐺𝑉𝐶 and 𝐺𝑉𝐵. In this chapter, first, the uncertainty in our models is represented, and the variability of the system is identified. After that, robust control methods are used to find the best controllers for our two models in question, within our stability and performance conditions. 72 4.1 Representation of the Uncertainty The variation in the models derived from the in-vivo experiments dataset is an indication of how uncertain the models are. Uncertainty in systems is typically in one of the below categories (Skogestad & Postlethwaite, 2001): 1. Parametric uncertainty, where the model dynamics are known, but the parameters of the model have ranges of uncertainty. 2. Unmodelled or neglected dynamics, which is caused by unknown physical dynamics or deliberately neglected ones to make the models simpler. 3. Lumped uncertainty, which is a combination of all the sources of uncertainty. In this study, the uncertainty is assumed to be lumped. Since this kind of uncertainty is unstructured, frequency domain methods show the best performance in representing them (Skogestad & Postlethwaite, 2001). The unstructured uncertainty is represented by the following notation: 𝐺𝑝(𝑠) = 𝐺0(𝑠) + 𝑤1(𝑠)Δ(𝑠)𝑤2(𝑠); ||Δ(𝑗𝜔)||2 ≤ 1 ∀𝜔 ∈ Ω (4.1) where 𝐺𝑝(𝑠) is all the uncertain models with an open-loop stable nominal model 𝐺0(𝑠). Δ(𝑠) is any stable transfer function with a magnitude of less than one for all frequencies, chosen to represent any dynamic uncertainty in the model, and 𝑤1(𝑠) and 𝑤2(𝑠) are appropriate uncertainty weights (Hindi et al., 2002). Ω = {ω1, … , 𝜔𝑚} is a span of frequencies chosen to represent transfer functions. The format of representing uncertainty in (4.1) is called “additive uncertainty”. 73 4.2 Visualization of the Uncertainty in Frequency Domain The additive uncertainty in (4.1) can be visualized on a Nyquist diagram, as 𝐺0(𝑠) on the nominal model’s Nyquist plot, and a circle around that point with a radius as follows (see Figure 4.1): 𝜌 = max ||Δ||2≤1||𝑤1(𝑗𝜔) Δ 𝑤2(𝑗𝜔)||2 = ||𝑤1(𝑗𝜔)||2||𝑤2(𝑗𝜔)||2 (4.2) Figure 4.1 Unstructured Uncertainty Disk Our aim is to find fast controllers with good disturbance rejection and low overshoot, that make the voltage to box pressure and voltage to blood pressure systems stable; in other words, the problem is to find a balance between performance and uncertainty. In such problems, finding the “tightest” uncertainty region possible is essential. Finding this region is highly dependent on picking a suitable nominal model. One common way of picking the nominal model is to choose the model with the average of the parameter values. The issue with the average-based uncertainty model is that this method yields the biggest uncertainty region. 𝐺0 𝜌 Re Im 74 To derive the nominal model that yields the smallest circles, an optimization problem to minimize 𝜌(𝜔𝑗) should be solved, to find the pair [𝐺0, 𝜌] at each frequency. A comparison of the average-based uncertain model with the radius-optimized uncertain model is shown in Figure 4.2 (a) and (b). In both figures, black lines show the models derived from system identification. The red starred plots show the nominal models, and the red bounds around the models are the uncertainty regions found in each method. Note that the uncertainty regions are only shown for the lower half of the Nyquist diagrams due to their symmetric nature. (a) 75 (b) Figure 4.2 (a) Average-Based (b) Optimization-Based Models for Diastolic Pressure (Used codes of (van Heusden et al., 2018) to draw the uncertainty bounds in the diagrams) 4.3 Derivation of the Nominal Model and Uncertainty Region The optimization problem to derive the tightest uncertainties, as discussed in section 4.2, is indicated below. This optimization problem minimizes the radii of uncertainty disks around the nominal model 𝐺0 at each frequency. minimize 𝐺𝑜,𝜌 𝜌 subject to |𝐺𝑝 − 𝐺0| ≤ 𝜌 This problem translates into a semi-definite programming convex optimization (Hindi et al., 2002), as follows: 76 minimize 𝑡,𝐺𝑜,𝑇1,𝑇2 𝑡 subject to: 𝑇1 ≤ 𝑡, 𝑇2 ≤ 𝑡, [𝑇1 𝐺𝑝𝑖 − 𝐺0(𝐺𝑝𝑖 − 𝐺0)∗𝑇2] ≽ 0, ∀𝑖 In the above optimization, 𝐺𝑝𝑖 indicates the 𝑖-th uncertain model, the syntax (𝐴)∗ means the complex conjugate transpose of 𝐴, and [𝐴] ≽ 0 denotes a positive semi-definite matrix. The weights 𝑤1 and 𝑤2 in (4.2) will be 𝑇11/2 and 𝑇21/2 respectively; thus, the optimal radius at each frequency will be equal to 𝑡. This optimization problem can be solved with CVX software in MATLAB. 4.4 Controller Synthesis In section 4.3, we quantified the uncertainty of our models, accounting for variability in the models’ gain and time constants, as well as uncertainty in their dynamics. In this section, robust control design rules are used to find the “best” controller for all the rats in our population. This population consists of a wide range of variations in rats and experiment recordings, therefore the controller derived in this part can be a starting point to use with the interface, to ensure a stable closed-loop response. One can later adjust the controller parameters further to improve the performance based on the specific subject. 77 4.4.1 Design Criteria To design the optimized robust controller for the device, we must consider both the desired performance of the device and the amount of uncertainty in the models to ensure stability. Since the form of variability was chosen to be unstructured so as to include all kinds of uncertainty, we continue to work in the frequency domain when designing the controller. The following sections show how to represent and ensure acceptable performance and stability in the frequency domain. 4.4.1.1 Stability To ensure closed-loop stability, we use the Nyquist criterion (Desoer & Wang, 1980; Khong, 2023). The loop transfer function of the uncertain system 𝐿𝑝 with a controller 𝐶 can be defined with the nominal model 𝐿0 and uncertainty disks with radii 𝜌𝐿 around the nominal models in the Nyquist diagram: 𝐿𝑝(𝑠) = 𝐺𝑝𝐶 = 𝐶(𝐺0(𝑠) + 𝑤1(𝑠)Δ(𝑠)𝑤2(𝑠)) = 𝐿0(𝑠) + Δ𝐿(𝑠); ||Δ𝐿(𝑗𝜔)||2 ≤ 𝜌𝐿(𝑗𝜔), ∀𝜔 ∈ Ω (4.3) To achieve stability in a model with unstructured uncertainty, no model 𝐿𝑝 in the range of our uncertain models should encircle −1. The graphical visualization of this condition is that no uncertainty disk around 𝐿0 should cover the point −1 (see Figure 4.4). 78 Figure 4.3 Robust Stability Nyquist Visualization From this condition, Robust Stability 𝑅𝑆 is defined as follows: 𝑅𝑆 ⟺ 𝜌𝐿 − |1 + 𝐿0(𝑗𝜔)| ≤ 0 ∀𝜔 ∈ Ω (4.4) 4.4.1.2 Performance Besides stability, we must ensure that the system meets sufficient performance criteria. We mainly focus on disturbance rejection for our criterion since reference tracking can be improved by feeding setpoints through ramps (Astrom & Hagglund, 1995) in the user interface. To examine the performance, the sensitivity function 𝑆 can be analyzed: 𝑆 =11 + 𝐺𝐶=11 + 𝐿 (4.5) 𝐿0 𝜌𝐿 −1 Im Re 79 𝑆 in (4.5) represents the closed-loop transfer function from the output disturbances to the outputs. Hence, it can demonstrate the relative sensitivity of the closed-loop system to the relative plant model error. The peak value of the sensitivity function is denoted by 𝑀𝑠. To achieve good disturbance rejection, 𝑀𝑠 should be low; a value of 𝑀𝑠 ≤ 2 is, therefore, typically used. Other performance factors, such as Gain Margin 𝐺𝑀 and Phase Margin 𝑃𝑀 can also be determined from 𝑀𝑠 (Skogestad & Postlethwaite, 2001): 𝐺𝑀 ≥𝑀𝑠𝑀𝑠 − 1 , 𝑃𝑀 ≥ 2arcsin (12𝑀𝑠) ≥1𝑀𝑠 [𝑟𝑎𝑑] (4.6) The condition of nominal performance (performance for the nominal model) is that for all frequencies and for a given peak value 𝑀𝑠∗ we have: |𝑆| ≤ 𝑀𝑠∗ (4.7) Substituting equation (4.5) into equation (4.7) yields: |1 + 𝐿| ≥1𝑀𝑠∗ (4.8) Graphically, the condition of (4.8) can be visualized as Figure 4.5. For the Nominal Performance 𝑁𝑃 condition to be true, the Nyquist plot of the loop transfer function 𝐿 must avoid a circle around −1 with a radius of 1𝑀𝑠∗: 𝑁𝑃 ⟺1𝑀𝑠∗− |1 + 𝐿(𝑗𝜔)| ≤ 0, ∀𝜔 ∈ Ω (4.9) 80 Figure 4.4 Nominal Performance Condition In an uncertain system, the condition of (4.9) must be true for all models 𝐿𝑝 in the model set. The Robust Performance 𝑅𝑃, defined in (4.10), should therefore be true. 𝑅𝑃 ⟺1𝑀𝑠∗− |1 + 𝐿𝑝(𝑗𝜔)| ≤ 0, ∀𝜔 ∈ Ω (4.10) On the Nyquist diagram, condition (4.10) implies that all points inside the uncertainty circle around the nominal model 𝐿0(𝑗𝜔) with radius 𝜌𝐿 must avoid the worst-case performance circle of radius 1𝑀𝑠∗ (see Figure 4.6). 𝑅𝑃 condition can be written in an Uncertain-Model format (Soltesz et al., 2016) in terms of closed-loop uncertainty circles, as shown below: 𝑅𝑃 ⟺1𝑀𝑠∗+ |𝐶(𝑗𝜔)|𝜌(𝑗𝜔) − |1 + 𝐿0(𝑗𝜔)| ≤ 0, ∀𝜔 ∈ Ω (4.11) 1/𝑀𝑠∗ 𝐿(𝑗𝜔) −1 Im Re 81 Figure 4.5 Robust Performance Condition For the controller to be optimally stable, it has to satisfy condition (4.11). The structure of the controller and how to design said controller will be discussed in the next section. 4.4.2 Controller Architecture The controller structure chosen for this study is a Proportional-Integrator (PI). Since a differentiator can amplify the sensor noises (Shi et al., 2020), we have foregone the D-term in PID. To see the effects of the disturbance, 𝑒 is defined as the error caused by a unit step disturbance at the input. A criterion that is typically used to characterize the error is Integral Absolute Error (IAE): 1/𝑀𝑠∗ Im Re −1 𝜌𝐿 𝐿0 |1 + 𝐿0| 82 𝐼𝐴𝐸 = ∫ |𝑒(𝑡)|𝑑𝑡∞0 (4.12) Computing an infinite integral with sufficient accuracy requires significant computation time and power. As (Åström & Hägglund, 2006) suggest, we can instead, use Integral Error, as defined below, which, for non-oscillatory or well-damped systems is a good approximation of 𝐼𝐴𝐸: 𝐼𝐸 = ∫ 𝑒(𝑡)𝑑𝑡∞0 (4.13) The benefit of using 𝐼𝐸 over 𝐼𝐴𝐸 is that there is no need to solve the infinite integral to compute its value, and its value can be directly obtained by looking at the controller parameters. In our case, for a PI controller where 𝑢 is the control output, the control law is: 𝑢(𝑡) = 𝑘𝑝𝑒(𝑡) + 𝑘𝑖∫ 𝑒(𝑡)𝑑𝑡𝑡0 (4.14) Assuming that a disturbance in the form of a step signal is applied to the system’s input, and the error at time 𝑡 = 0 is zero, since we have an integrator, the steady-state error becomes zero, therefore at 𝑡 → ∞ we can write: 𝑢(∞) − 𝑢(0) = 𝑘𝑖∫ 𝑒(𝑡)𝑑𝑡∞0 (4.15) Since the disturbance is applied at the system’s input, 𝑢(∞) − 𝑢(0) is equal to the change of the disturbance, and therefore from (4.13) and (4.15), we have: 𝐼𝐸 =1𝑘𝑖 (4.16) Equality (4.16) is only valid in well-damped systems, with one integrator, and otherwise might get a zero value and become misleading. 83 To have a good disturbance rejection, we aim to minimize the Integral Error 𝐼𝐸, which from (4.16) implies maximizing the integral gain 𝑘𝑖. With this objective, and the 𝑅𝑃 condition in (4.11), the below optimization problem can be solved to obtain the optimal robust controller for our system: maximize 𝑘𝑝,𝑘𝑖 𝑘𝑖 subject to ∀𝜔∈Ω: 1𝑀𝑠∗+ |𝐶(𝑗𝜔)|𝜌(𝑗𝜔) − |1 + 𝐿0(𝑗𝜔)| ≤ 0 This optimization problem is not a convex problem since it has a {𝑐𝑜𝑛𝑣𝑒𝑥} − {𝑐𝑜𝑛𝑣𝑒𝑥} term. Therefore, it cannot be solved using conventional methods and solvers. We need to use the method described in the next section to overcome this issue. 4.4.2.1 Convex-Concave Optimization The procedure to solve problems like (3.20) is called Convex-Concave Optimization (Yuille & Rangarajan, 2003). In such problems, the optimization objectives and constraints are written as a difference between two convex functions. To solve this issue, this approach replaces the concave functions with a linearization around the solution point of the current iteration 𝑥𝑘: 𝑓(𝑥) − 𝑔(𝑥) ≅ 𝑓(𝑥) − 𝑔(𝑥𝑘) − ∇𝑔(𝑥𝑘)𝑇(𝑥 − 𝑥𝑘) (4.17) The approximation (4.17) is more conservative than the original function as it is an upper bound on it. The problem turns into Sequential Programming (Boyd, 2019), where at each iteration, the optimization is done with the linear approximation (4.17) around 𝑥𝑘 in order to find a new feasible point 𝑥𝑘+1. There is no guarantee that this method will converge to a global minimum, but 84 experience has proven the local minima or saddle points that the method converges to are satisfactory solutions (Hast et al., 2013). In our problem, the −|1 + 𝐿0(𝑗𝜔)| concave part can be approximated around 𝐿0𝑘 (at each iteration 𝑘) by the following function: −|1 + 𝐿0| = −ℜ((1 + 𝐿0𝑘)∗|1 + 𝐿0𝑘|(1 + 𝐿0)) (4.18) where ℜ denotes the real part and (1 + 𝐿0𝑘)∗ is the complex conjugate of the term 1 + 𝐿0𝑘 . Hence, our Sequential Programming Problem is solved by the following algorithm. We initialize our controllers with 𝑘𝑝0 = 0 and 𝑘𝑖0 = 0 for an asymptotically stable system, and then, for an 𝑛 number of iterations, we have: 𝒇𝒐𝒓 𝒌 = 𝟏 𝒕𝒐 𝒏: maximize 𝑘𝑝,𝑘𝑖 𝑘𝑖 subject to: 𝒇𝒐𝒓 𝝎𝝐𝛀: 1𝑀𝑠∗ + |𝑘𝑝 +𝑘𝑖𝑗𝜔| 𝜌 − ℜ( (1 + (𝑘𝑝𝑘 +𝑘𝑖𝑘𝑗𝜔)𝐺0)∗|1 + (𝑘𝑝𝑘 +𝑘𝑖𝑘𝑗𝜔)𝐺0|(1 + (𝑘𝑝 +𝑘𝑖𝑗𝜔)𝐺0)) ≤ 0 𝑬𝒏𝒅 𝑬𝒏𝒅 The above optimization can be solved in CVX software in MATLAB. 85 4.5 Summary In this chapter, we represented the variability of our system in the form of an unstructured uncertainty to account for parameter deviations as well as uncertain dynamics. It was shown that the nominal model and the uncertainty regions in the frequency domain could be found by solving an optimization problem to obtain the smallest disks around the nominal models in the Nyquist diagram. We then proceeded to design a PI controller that stabilized the system for all our experimental data from all the rats while meeting the performance criteria. Finally, it was shown that the optimal robust controller can be found by solving a convex-concave optimization problem that maximizes the integrator gain while avoiding the worst-case performance circle around −1 in the Nyquist diagram for each point in the uncertainty regions. In the Results section, we will compare and validate the robust optimal controllers derived from our optimization procedure with the collected closed-loop experimental data. 86 Chapter 5: Results In Chapter 2, the architectural design of the LBNP system was shown. In Chapter 3 and 4, modeling and controller design were performed on the open-loop data gathered from the in-vivo experiments. This chapter shows the results of the methods and algorithms in the two previous chapters. 5.1 Results of Chapter 3: System Identification To collect an informative dataset, 205 Step experiments were conducted on a total of 10 rats. Out of these, 98 were used for system identification. A typical trend in the pressure drops when a step input is applied, is seen in Figure 5.1. Figure 5.1 Typical Step Response of a Rats' Blood Pressure There were four reasons that would disqualify a dataset from being used: 87 1. The rat underwent multiple arrhythmias, which interrupted the normal pressure change trend of Figure 5.1. In terms of the controller design, arrhythmia does not cause stability issues since a sudden drop is equivalent to a control error of a small value or a negative error. The former makes the controller less aggressive, and the latter shuts down the system letting the air flow back inside the chamber; therefore, in both cases, stability is ensured. Nevertheless, since the arrhythmic periods do not represent the behaviour of the system, they were not used in modeling. An example of this case is seen in Figure 5.2. Figure 5.2 Multiple Arrhythmias in the Experiment 2. The experiment was not given enough time for the pressure trend to reach a steady state (see MAP plot in Figure 5.3). 88 Figure 5.3 Plot of MAP, Not Given Enough Time to Reach S.S. 3. In voltages less than 3-4 volts the vacuum power did not fully compensate for the leakage, and the pressure drop was negligible. The case for a 2V input can be seen in Figure 5.4. Figure 5.4 MAP in Voltage of 2V 89 4. The poles or gains of their calculated models were outliers with quartile criterion (values in the 1.5 interquartile ranges above the upper quartile or below the lower quartile). As an example of the outliers, poles and gains of the Voltage-Box Pressure system are brought in the box chart format in Figure 5.5. Each chart in Figure 5.5 belongs to one rat in the experiments and the dots in the graphs show the outliers. (a) (b) Figure 5.5 Outliers in (a) Gain, (b) Time Constant, of the Voltage-Box Pressure Models 90 5.1.1 Goodness of Fits As discussed in section 3.6, the models were evaluated with the Fit Percentage criteria. The Fit Percentages of the first-order models from voltage to box pressure, and from box pressure to Diastolic, Systolic and MAP are illustrated in the four histograms of Figure 5.6. (a) (b) (c) (d) Figure 5.6 Histograms of Fit Percentages, (a) V-Box (b) Box-Diastolic, (c) Box-MAP, (d) Box-Systolic 91 In histograms of Figure 5.6 the first model (V-Box Pressure) has significantly higher fit percentages compared to the other three. The reason for the low fit percentages in the other three models is that they represent the rats’ internal bodily functions and are subject to more variations and disturbances, such as arrhythmia or breathing cycles. As discussed in section 5.1, non-informative data were removed from the dataset, but the ones with fair behavior were accepted, despite showing few arrhythmias and undesirable fluctuations. 5.1.2 Stepwise Linear Regression In section 3.8.2 we hypothesized that the parameters of the model can be found as a function of the plate size and rat’s mass, chest size and age, and discussed the Stepwise Linear Regression to find these relationships. Also, in section 3.2, a model of vacuum voltage to box pressure was derived based on the vacuum specifications and the leakages. Since the leakage is dependent on the unsealed opening around the rat’s body, we expect that the parameters of the 𝑃𝐵𝑜𝑥𝑉 model be dependent on this area, or: [𝐾𝑠𝑠1, 𝜏1] ∝ 𝑃𝑙𝑎𝑡𝑒 𝑆𝑖𝑧𝑒 − 𝑅𝑎𝑡′𝑠 𝐶ℎ𝑒𝑠𝑡 𝑆𝑖𝑧𝑒 (5.1) and since the plate size used for all the rats are the same size, it can be expected that the gain 𝐾𝑠𝑠1 and time constant 𝜏1 be inversely proportional to the rat’s chest size. The measure of the “chest size” used here is the cross-sectional area of the rat’s chest, since the larger this area, the better the seal would be, and from intuition, one can assume that for rats with larger chests, the model would have a lower gain and/or a faster time constant. 92 Regarding the models of the blood pressure, since we did not have any applicable literature or mathematical models, we could only hypothesize that the rat’s internal blood pressure regulation/compensation system was a function of its mass, age and size. The rat’s specifications are as seen in Table 5.1. Table 5.1 Rats' Specifications Rat no. R.1 R.2 R.3 R.4 R.5 R.6 R.7 R.8 R.9 R.10 Chest Cross-Sectional Area (× 𝟏𝟎𝟑 𝒄𝒎𝟐) 1.79 2.01 1.69 1.77 1.91 1.86 2.09 1.91 2.30 2.01 Mass (𝒈) 368 363 363 352 352 367 433 338 418 393 Age (𝒘𝒆𝒆𝒌𝒔) 9.7 9.9 9.9 10.0 10.0 10.6 10.6 10.7 10.7 10.9 To test the hypotheses stated above, the Stepwise Linear Regression algorithm was implemented with 𝑆𝑆𝐸 criterion and a critical p-Value of the F-test is 0.06. The results for 𝑃𝐵𝑜𝑥𝑉 and 𝑃𝑀𝐴𝑃𝑃𝐵𝑜𝑥 are as seen in Tables 5.2 to 5.5. In the tables, C, M, and A are the chest cross-sectional area, mass, and age, respectively. Estimate is the estimation of the intercept or coefficient of each parameter, SE is the Standard Error, tStat or t-statistic is the measure of the significance of the coefficients (Estimate and SE) and pValue is the p-Value of the t-statistic. Below each table, the Root Mean Squared Error RMSE is shown as an indication of the goodness of fit of the derived model. In addition, in non-constant models, Adjusted R2, F-statistics, and the p-Value of the F-statistics is shown, to demonstrate the significance of the model with added terms, compared to the constant model. 93 1. Voltage-Box Pressure Model: • Gain: Table 5.2 V-Box Gain Coefficients Variables Estimate SE tStat pValue Intercept 0.067 0.002 31.023 7.598 × 10−51 RMSE: 0.021 Final Formula: 𝐾𝑠𝑠1 = 0.067 • Time Constant: Table 5.3 V-Box Time Constant Coefficients Variables Estimate SE tStat pValue Intercept 2.103 0.274 7.664 1.786 × 10−11 M −0.004 0.0007 −5.386 5.492 × 10−7 RMSE: 0.021 Adjusted R-Squared: 0.231 F-stat vs. constant model: 29 p-Value of the F-test: 5.46 × 10−7 Final Formula: 𝜏1 = 2.103 − 0.004 ×𝑀 94 The results above indicate that in the 𝑃𝐵𝑜𝑥𝑉 model, gain is independent of the measured variables, and the time constant depends on the rat’s mass. From intuition and for extreme cases, one can assume that with a perfect seal, certain power from the vacuum can cause high values of negative pressure, and in a no seal condition, the gain becomes minimal. Hence, the reason behind the gain being independent might be that the variation of plate/rat sizes or the number of experiments in the dataset was not sufficient for the algorithm to find a meaningful relationship. As for the time constant, the mass of the rat can be a measure of how much air the animal replaces, therefore it plays a more significant role than only the cross-sectional area of the chest. 2. Box Pressure-Mean Arterial Pressure Model: • Gain: Table 5.4 Box-MAP Gain Coefficients Variables Estimate SE tStat pValue Intercept 938.524 200.168 4.689 9.984 × 10−6 C −0.134 0.031 −4.402 2.997 × 10−5 M −2.588 0.546 −4.742 8.097 × 10−6 A −63.538 18.330 −3.466 8.180 × 10−4 C×M 3.436 × 10−4 7.973 × 10−5 4.310 4.239 × 10−5 M×A 0.181 0.0502 3.618 4.954 × 10−4 95 RMSE: 2.01 Adjusted R-Squared: 0.424 F-stat vs. constant model: 14.7 p-Value of the F-test: 1.91 × 10−10 Final Formula: 𝐾𝑠𝑠2𝑀𝐴𝑃 = 938.52 − 0.13 𝐶 − 2.58 𝑀 − 63.54 𝐴 + 0.00034 𝐶 𝑀 + 0.18 𝑀 𝐴 • Time Constant: Table 5.5 Box-MAP Time Constant Coefficients Variables Estimate SE tStat pValue Intercept 74.118 93.311 0.754 0.453 C −0.134 0.0264 −5.0970 1.956 × 10−6 M 0.508 0.258 1.972 0.0517 A −8.109 9.297 −0.872 0.385 C×A 0.0134 0.00258 5.191 1.334 × 10−6 M×A −0.0483 0.0244 −1.979 0.0509 RMSE: 0.96 Adjusted R-Squared: 0.375 F-stat vs. constant model: .212 p-Value of the F-test: 5.81 × 10−9 Final Formula: 𝐾𝑠𝑠2𝑀𝐴𝑃 = 74.12 − 0.13 𝐶 − 0.51 𝑀 − 8.11 𝐴 + 0.0134 𝐶 𝐴 − 0.048 𝑀 𝐴 96 The results for the 𝑃𝑀𝐴𝑃𝑃𝐵𝑜𝑥 model show that the gain and time constant are dependent on the chest cross-sectional area, mass and age of the rats, as well as their interactions. The Adjusted R-Squared value for the gains and time constants is near 0.4, indicating that the goodness of the fit in these models are relatively high. The results of this part can be an indication that a relationship can be found between the rat’s V-Blood Pressure model and its measurements; it can act as a starting point for future investigations to find more accurate relationships between the rat’s blood pressure compensation system and the measurements. Derivation of these relationships is significant in controller design as well. To find a higher performance controller (depending on the user’s needs, e.g., faster response, less overshoot, more disturbance rejection) than what we achieved using robust control rules on the complete dataset, one could use the measurements of the chamber and the rat, derive an estimate of the model gain and time constants as well as their standard deviation, and use that narrowed uncertainty to design a faster controller. 5.2 Results of Chapter 4: Controller Design It was discussed in Chapter 4 that four models in the LBNP system required controlling; from an input of vacuum voltage (or PWM percentage), to outputs of the box, Diastolic, Systolic and Mean Arterial pressures. A robust controller approach was applied to the dataset of open-loop experiments with the arterial catheter. The results of the control design steps are shown in this section. In section 5.2.3, the controller will be compared to the closed-loop experiment results with arterial and left ventricular catheters. 97 The frequency responses of the open-loop systems are found in the Bode and Nyquist diagrams of Figures 5.7 to 5.14. Different colours represent each of the 10 rats. 1. Voltage (PWM Percentage)-Box Pressure Model: Figure 5.7 Bode Plot of V-Box Figure 5.8 Nyquist Plot of V-Box 98 2. Voltage (PWM Percentage)-Diastolic Pressure Model: Figure 5.9 Bode Plot of V-DP Figure 5.10 Nyquist Plot of V-DP 99 3. Voltage (PWM Percentage)-Mean Arterial Pressure Model: Figure 5.11 Bode Plot of V-MAP Figure 5.12 Nyquist Plot of V-MAP 100 4. Voltage (PWM Percentage)-Systolic Pressure Model: Figure 5.13 Bode Plot of V-SP Figure 5.14 Nyquist Plot of V-SP 101 The diagrams above illustrate the similarities between the behaviours of the models in different rats. The uncertainty of the models is also evident in the diagrams. In the following section, the nominal models and uncertainty regions are shown, and the controllers designed based on them are demonstrated. 5.2.1 Uncertainty Regions and Nominal Values The representations of the uncertainty, according to the algorithms in section 4.3, are seen in the Nyquist diagrams of Figure 5.15. The starred diagrams in red represent the nominal models, and the boundaries shown in red on the lower half of the diagrams demonstrate the tightest uncertainty regions in the models. (a) (b) 102 (c) (d) Figure 5.15 Nyquist Uncertainty Regions for (a) V-Box (b) V-Diastolic (c) V-MAP (d) V-Systolic (Used codes of (van Heusden et al., 2018) to draw the uncertainty bounds in the diagrams) 5.2.2 Controller Synthesis By solving the optimization problem in section 4.4.2.1, controller gains for the PI controllers in each of the four systems are derived. We used an upper bound of 𝑘𝑝, 𝑘𝑖 ≤ 15 to bind the control output. The results will be converted to values that can be used in the interface (𝐾𝑝, 𝐾𝑖), as shown in (5.2 a) and (5.2 b). The step responses of the closed-loop systems with the derived control parameters can be seen in Figures 5.16 to 5.19. Each colour in the plots represents a different subject. 𝐾𝑝 = 𝑘𝑝 ×𝑃𝑊𝑀𝑡𝑜𝑝100= 𝑘𝑝 × 624.99 (5.2 a) 𝐾𝑖 = 𝑘𝑖 ×𝑃𝑊𝑀𝑡𝑜𝑝100= 𝑘𝑖 × 624.99 (5.2 b) 103 1. Voltage (PWM Percentage)-Box Pressure Model: • Derived controller gains from the optimization: 𝑘𝑝 = 14.085, 𝑘𝑖 = 14.999 • Translated interface controller gains: 𝐾𝑝 = 8803.1, 𝐾𝑖 = 9374.8 Figure 5.16 V-Box Closed-Loop Step Response 2. Voltage (PWM Percentage)-Diastolic Pressure Model: • Derived controller gains from the optimization: 𝑘𝑝 = 4.639, 𝑘𝑖 = 1.108 • Translated interface controller gains: 𝐾𝑝 = 2899.3, 𝐾𝑖 = 692.5 104 Figure 5.17 V-DP Closed-Loop Step Response 3. Voltage (PWM Percentage)-Mean Arterial Pressure Model: • Derived controller gains from the optimization: 𝑘𝑝 = 3.372, 𝑘𝑖 = 1.074 • Translated interface controller gains: 𝐾𝑝 = 2107.7, 𝐾𝑖 = 671.3 Figure 5.18 V-MAP Closed-Loop Step Response 105 4. Voltage (PWM Percentage)-Systolic Pressure Model: • Derived controller gains from the optimization: 𝑘𝑝 = 2.728, 𝑘𝑖 = 0.522 • Translated interface controller gains: 𝐾𝑝 = 1705.2, 𝐾𝑖 = 326.6 Figure 5.19 V-SP Closed-Loop Step Response The controller gains of the 𝑃𝐵𝑜𝑥𝑉 system are approximately equal to the determined upper bounds, and the slowest time constant of the system is below 3s. The optimal integral gain for DP and MAP in the interface is approximately 600, and for SP this value is just above 300. The range of time constants of the blood pressure models vary between approximately 4s to 25s. Therefore, with the same components used in this LNBP system, a response time of at least 4s should be expected and considered when choosing reference values on the interface. 106 5.2.3 Closed-Loop Experiments As discussed in section 2.5, in addition to the open-loop experiments, closed-loop experiments were performed on subject number 10 with both arterial and LV catheters. The experiments were conducted with Step and Ramp reference trajectories. The total number of experiments conducted can be seen in Table 5.6. Table 5.6 Closed-Loop Experiments Box P. Diastolic P. Mean Arterial P. Systolic P. Arterial Catheter Step:7, Ramp:2 - Step: 3, Ramp:15 Ramp:3 LV Catheter - - - Step:14, Ramp:15 The controllers for these experiments were chosen heuristically; the process started with a low integral gain and zero proportional gain. The integral gain was gradually increased until a large overshoot or stable oscillatory response was detected. Then the integral gain was fixed, and the proportional gain was gradually increased until the response had an overshoot of less than 20%. The range of the controller gains in the experiments were 𝐾𝑝 ∈ [0, 1200], 𝐾𝑖 ∈ [1000, 8000] for the voltage to box pressure models and 𝐾𝑝 ∈ [0, 1200], 𝐾𝑖 ∈ [200, 600] for the voltage to blood pressure models. The values tested that were closest to the optimal controller gains of section 5.2.2 are 𝐾𝑝 = 1000, 𝐾𝑖 = 8000 for voltage to box pressure and 𝐾𝑝 = 1200, 𝐾𝑖 = 600 for voltage to blood pressures (DP/SP/MAP). Examples of the closed-loop responses with these gains can be found in Figures 5.20 to 5.26. The oscillations in the DP/SP/MAP are caused by the breathing cycle of the subjects. 107 1. Arterial Catheter: • Step Input: Figure 5.20 CL V-Box Step Resp. with Arterial Catheter Figure 5.21 CL V-MAP Step Resp. with Arterial Catheter • Ramp Input: Figure 5.22 CL V-Box Ramp Resp. with Arterial Catheter Figure 5.23 CL V-MAP Ramp Resp. with Arterial Catheter 108 Figure 5.24 CL V-SP Ramp Resp. with Arterial Catheter 2. LV Catheter: • Step Input: Figure 5.25 CL V-SP Step Resp. with LV Catheter 109 • Ramp Input: Figure 5.26 CL V-SP Ramp Resp. with LV Catheter The controllers used in the in-vivo closed-loop experiments were tuned heuristically to bind the overshoot while achieving a fast response. The goal of Chapter 4 was to design controllers that ensure stability for at least all of the rats in the experiments’ population. By comparing the derived robust controllers with the heuristically tuned ones and noting the similarity between their P and I gains, the validity of the robust control results can be shown. 110 Chapter 6: Conclusion The current research aimed to develop an automated Lower-Body Negative Pressure chamber to control the left ventricular pressure of rodents. This work enhanced the first version of this device, which was built by UBC students for their Capstone project (Crowther et al., 2021) based on the method later patented by (West et al., 2022), and achieved the objectives below: 1. The electronics hardware was redesigned to ensure the safety of the subject, and achieve reliable vacuum motor driving, and data acquisition and transfer. 2. A new user interface was designed to increase user-friendliness and allow real-time monitoring of the data, and a more optimized code was programmed in C for the microcontroller. 3. Models for the chamber pressure and rats’ blood pressures as a function of the vacuum pump voltage were created and used in the control system design. 4. Optimized robust controllers were designed for the LBNP system to ensure stability while achieving the desired performance of the system. A new, safer, and more reliable hardware and software were developed and tested in both the electrical engineering lab and the animal facility. System identification methods were applied to the data collected from the in-vivo experiments, creating models for controlling the pressure inside the chamber, as well as the subjects’ diastolic, systolic, and mean arterial pressures. A stepwise linear regression was used to derive the model parameters based on the size of the chamber’s opening and the rat’s size, mass, and age. In the end, robust control approaches were utilised to 111 create an optimized stable controller for the defined range of uncertainty, including the population in the experiments. The performances of the derived controllers were compared to the heuristically-tuned controllers with which the closed-loop in vivo experiments were conducted. 6.1 Current Limitations and Future Work Although the device was successful in the experiments, there is room for future improvement in the hardware and software design of the device, as well as the algorithms implemented on the data. There is a need to design and 3D print a suitable housing for the controller box to remove the current wired connections. There have been a few bugs on the software side; the box pressure sensor has drifts, and after two to three hours of experimentation, its initial value would drop by approximately 5 mmHg. This does not interfere with the operation of the device, as the internal algorithms were built to operate on the pressure drops rather than the absolute values; however, for monitoring, it is more reasonable to show an accurate value for the box pressure. Another bug also occurred with the automatic stop; the interface is built to shut down the device after the operating periods are over, yet there were a few times where the device did not stop the experiment and restarted the experiment instead. We stopped those experiments manually with the help of the software interface stop button, but this issue needs to be resolved in future iterations. Regarding the system identification and control, robust controllers were developed based on the variation of all of the data in the dataset. However, this research showed that models of both box pressure and the rat’s blood pressure can be derived from the chamber’s and rat’s specifications; one can use the derived relationships to find models with less uncertainty for each subject and design a better-performing controller. The relationship was found using linear regression. In the 112 future, further research can be done to find more accurate models for the chamber and blood pressures. Another study that can be performed in the future is to further investigate the response of the baroreflex4 in different rats to pressure drops, how this information can be included in the system identification, and what the best control approach would be based on the required outcome. In this version of the system, tuning of the controller gains is done offline, and the values derived are entered in the interface. In future iterations, the tuning algorithm of the controller could be performed automatically on the device. The benefit of this change would be that the target users of the chamber are typically unfamiliar with controller tuning techniques, and with an auto-tuning system present, they can indicate their desired speed/robustness on the interface and the robust optimization techniques can find the best feasible controller for them. To achieve that, a study should be conducted on the computational power of the microcontroller versus that of the desktop computers to decide whether the auto-tuning should be implemented in the desktop app or on the microcontroller inside the control box. Finally, improvements can be made to the instrumentation in the future. Currently, the blood pressure is measured through a cabled catheter. Working with this long cable required caution, attention to details, and multiple adjustments to ensure it stayed in place. In the future, we hope that wireless telemetry instruments can be manufactured so they can replace the catheters and resolve such issues. 4 the blood pressure compensation mechanism that maintains the blood pressure at a nearly constant value 113 This work, which has novelty in the combination of automation and control system engineering with physiological studies to build an automated LBNP, can be used in research related to the effects of new pharmaceutical/biomedical products on the cardiovascular system, as well as in studies on the impact of injuries and health conditions on the baroreflex. We hope that in the future, this automated, non-invasive system can help facilitate more accurate and reliable. 114 Bibliography Abdolhosseini, M., & Abdollahi, R. (2023). Performance analysis of PID controller-based metaheuristic optimisation algorithms for BLDC motor. 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IFAC Proceedings Volumes, 37(13), 501–506. https://doi.org/10.1016/S1474-6670(17)31273-9 125 Appendices Appendix A The schematics of the LBNP controller box can be found in Appendix A. 126 A.1 Raspberry Pi Pico IO and Input Headers for Peripherals 127 A.2 Catheter Reading Modules 128 A.3 Analog Outputs 129 A.4 Vacuum Pump Motor Driver
Thesis/Dissertation
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Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
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Seethaler, Rudolf
Optimization of a lower body negative pressure chamber to control the left ventricular pressure of rodents
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