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Covariance Control and its Relationship to 17 Other Control Problems Robert Skelton Department of Aerospace Engineering Department of Aerospace Engineering 1 17 Different Control Design Problem Continuous-Time Case Discrete-Time Case 1.

  1. Covariance Control and its Relationship to 17 Other Control Problems Robert Skelton Department of Aerospace Engineering Department of Aerospace Engineering 1

  2. 17 Different Control Design Problem Continuous-Time Case Discrete-Time Case 1. Stabilizing Control 1. Stabilizing Control 2. Covariance Upper Bound Control 2. Covariance Upper Bound Control 3. Linear Quadratic Regulator 3. Linear Quadratic Regulator 4. L∞ Control 4. l∞ Control 5. H∞ Control 5. H∞ Control 6. Positive Real Control 6. Robust H 2 Control 7. Robust H 2 Control 7. Robust l∞ Control 8. Robust L∞ Control 8. Robust H∞ Control 9. Robust H∞ Control Skelton RE, Iwasaki T, Grigoriadis K (1998): A Unified Algebraic Approach to Control Design. Taylor & Francis, London. Department of Aerospace Engineering 2

  3. Linear Matrix Equalities and Inequalities • Existence Condition • Solutions for G Department of Aerospace Engineering 3

  4. Different Interpretations of the Lyapunov Equation • Stability Condition • Controllability Gramian Department of Aerospace Engineering 4

  5. Different Interpretations of the Lyapunov Equation • Stochastic Interpretations zero mean white noise with unit intensity Steady-state covariance matrix (Linear Matrix Inequality) Upper Bound Output Covariance Matrix Department of Aerospace Engineering 5

  6. Different Interpretations of the Lyapunov Equation • Deterministic Interpretations Department of Aerospace Engineering 6

  7. Control Design Problem Consider the LTI system and a dynamic controller Closed-loop system dynamics form Skelton RE, Iwasaki T, Grigoriadis K (1998): A Unified Algebraic Approach to Control Design. Taylor & Francis, London. Department of Aerospace Engineering 7

  8. 17 Different Control Design Problem Continuous-Time Case Discrete-Time Case 1. Stabilizing Control 1. Stabilizing Control 2. Covariance Upper Bound Control 2. Covariance Upper Bound Control 3. Linear Quadratic Regulator 3. Linear Quadratic Regulator 4. L∞ Control 4. l∞ Control 5. H∞ Control 5. H∞ Control 6. Positive Real Control 6. Robust H 2 Control 7. Robust H 2 Control 7. Robust l∞ Control 8. Robust L∞ Control 8. Robust H∞ Control 9. Robust H∞ Control Skelton RE, Iwasaki T, Grigoriadis K (1998): A Unified Algebraic Approach to Control Design. Taylor & Francis, London. Department of Aerospace Engineering 8

  9. Stabilizing Control Skelton RE, Iwasaki T, Grigoriadis K (1998): A Unified Algebraic Approach to Control Design. Taylor & Francis, London. Department of Aerospace Engineering 9

  10. Covariance Upper Bound Control Upper bounds on the output covariances Skelton RE, Iwasaki T, Grigoriadis K (1998): A Unified Algebraic Approach to Control Design. Taylor & Francis, London. Department of Aerospace Engineering 10

  11. Linear Quadratic Regulator Skelton RE, Iwasaki T, Grigoriadis K (1998): A Unified Algebraic Approach to Control Design. Taylor & Francis, London. Department of Aerospace Engineering 11

  12. Skelton RE, Iwasaki T, Grigoriadis K (1998): A Unified Algebraic Approach to Control Design. Taylor & Francis, London. Department of Aerospace Engineering 12

  13. Skelton RE, Iwasaki T, Grigoriadis K (1998): A Unified Algebraic Approach to Control Design. Taylor & Francis, London. Department of Aerospace Engineering 13

  14. Integrating Information Architecture and Control Actuator Plant Sensor Controller Convex Problem Faming Li, Mauricio C. de Oliveira, and Robert E. Skelton. “Integrating Information Architecture and Control or Estimation Design”. SICE Journal of Control, Measurement, and System Integration, Vol.1(No.2), March 2008. Department of Aerospace Engineering 14

  15. Motivation • A true systems design theory would include plant design, appropriate modeling, sensor/actuator selection and control design in a cohesive effort • A theory such as this may be impossible to develop but there are steps in that direction that are achievable • Most control problems are defined AFTER sensor and actuator location and precision has been decided • Defining INFORMATION ARCHITECHTURE (IA) as the selection of instrument type (sensor/actuator), instrument precision (SNR), instrument location, and the control or estimation algorithm, the problem of finding the best IA to meet certain customer requirements can be solved Department of Aerospace Engineering 15

  16. Problem Statement • A continuous linear time-invariant system representation: • Noises are modeled as independent zero mean white noises • Inverse of noises is defined as precision Sensor precision Actuator precision Department of Aerospace Engineering 16

  17. Problem Statement Total cost for actuators and sensors: Department of Aerospace Engineering 17

  18. Integrating Information Architecture and Control : Final Result Department of Aerospace Engineering 18

  19. Integrating Information Architecture and Control : Final Result Which produces the control given in the paper: Department of Aerospace Engineering 19

  20. Integrated Plant, Sensor/Actuator and Control Design Actuator Plant Sensor Controller Non - Convex Problem New Contribution - Jointly optimize controller, Sensor/Actuator Design and Plant Parameters in an LMI framework to meet some performance criteria. Department of Aerospace Engineering 20

  21. Integrated Plant, Sensor/Actuator and Control Design • Existing Theory – Integrated Structure and control design – ISCD Paper* • Fix structure parameter then design controller • Fix Controller and then redesign structure – Optimize sensor/actuator precision jointly with control design – IA Paper** • New Contribution - Controller does not need to be fixed in the structure redesign step - LMI framework - Ability to optimize the mass matrix – Jointly optimize controller, Sensor/Actuator Design and Plant Parameters in an LMI framework to meet some performance criteria. *K. M. Grigoriadis and R. E. Skelton. Integrated structural and control design for vector second-order systems via LMIs. In Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207), volume 3, pages 1625–1629 vol.3, June 1998. **Faming Li, Mauricio C. de Oliveira, and Robert E. Skelton. “Integrating Information Architecture and Control or Estimation Design”. SICE Journal of Control, Measurement, and System Integration, Vol.1(No.2), March 2008. Department of Aerospace Engineering 21

  22. Given Data • A continuous linear time-invariant system in descriptor state-space representation: (Plant) (Output) (Measurement) All these matrices are affine in parameters a • • Noises are modeled as independent zero mean white noises Goyal R, Skelton RE,(2019) Joint Optimization of Plant, Controller, and Sensor/Actuator Design. In: Proceedings of the 2019 American control conference, Philadelphia. Department of Aerospace Engineering 22

  23. Given Data • Actuator and sensor precisions are defined to be inversely proportional to the respective noise intensities • Assuming p a , p s and p a are vectors containing the price per unit of actuator precision, sensor precision and structure parameter • The total design price : Department of Aerospace Engineering 23

  24. Information Architecture System Design Problem Statement Department of Aerospace Engineering 24

  25. Closed Loop System • Define the closed-loop state and noise vectors as • The closed-loop system is given by • All the matrices can be expanded as Department of Aerospace Engineering 25

  26. Control Design Problem • The above closed loop system is stable if and only if there exists a X > 0 such • that: • Applying Schur’s Compliment and defining Not an LMI (Non-Convex Constraints) Department of Aerospace Engineering 26

  27. Information Architecture System Design Existence Condition Existence Theorem:- If there exists a matrix X that satisfies all these equations, then the design specifications can be met, and the closed loop system will be stable Not an LMI Design Specifications Department of Aerospace Engineering 27

  28. Convexifying Algorithm Lemma Juan F. Camino, M. C. de Oliveira, and R. E. Skelton, ‘‘Convexifying’’ Linear Matrix Inequality Methods for Integrating Structure and Control Design, J. Struct. Eng., 2003, 129(7): 978-988 Department of Aerospace Engineering 28

  29. Convexifying the Problem • To use the previous Lemma, let us define the matrix G as: • Also define the convexifying potential function as: • LMI (Convex Constraints) Department of Aerospace Engineering 29

  30. Control Design Problem If there exists a matrix Q such that the iteration on the following LMIs converges, then all the design objectives can be met, and the closed loop system will be stable Design Specifications Goyal R, Skelton RE,(2019) Joint Optimization of Plant, Controller, and Sensor/Actuator Design. In: Proceedings of the 2019 American control conference, Philadelphia. Department of Aerospace Engineering 30

  31. Optimization Versions of the Design Problem Goyal R, Skelton RE,(2019) Joint Optimization of Plant, Controller, and Sensor/Actuator Design. In: Proceedings of the 2019 American control conference, Philadelphia. Department of Aerospace Engineering 31

  32. Thank You! https://bobskelton.github.io/index.html

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