Covariance Control and its Relationship to 17 Other Control - - PowerPoint PPT Presentation

▶
covariance control and its relationship to 17 other
SMART_READER_LITE
LIVE PREVIEW

Covariance Control and its Relationship to 17 Other Control - - PowerPoint PPT Presentation

Mar 25, 2023 119 likes •447 views

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.

slide-1
SLIDE 1

1 Department of Aerospace Engineering

Robert Skelton

Department of Aerospace Engineering

Covariance Control and its Relationship to 17 Other Control Problems

slide-2
SLIDE 2

2 Department of Aerospace Engineering

17 Different Control Design Problem

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

Continuous-Time Case

  • 1. Stabilizing Control
  • 2. Covariance Upper Bound Control
  • 3. Linear Quadratic Regulator
  • 4. L∞ Control
  • 5. H∞ Control
  • 6. Positive Real Control
  • 7. Robust H2 Control
  • 8. Robust L∞ Control
  • 9. Robust H∞ Control

Discrete-Time Case

  • 1. Stabilizing Control
  • 2. Covariance Upper Bound Control
  • 3. Linear Quadratic Regulator
  • 4. l∞ Control
  • 5. H∞ Control
  • 6. Robust H2 Control
  • 7. Robust l∞ Control
  • 8. Robust H∞ Control
slide-3
SLIDE 3

3 Department of Aerospace Engineering

Linear Matrix Equalities and Inequalities

  • Existence Condition
  • Solutions for G
slide-4
SLIDE 4

4 Department of Aerospace Engineering

Different Interpretations of the Lyapunov Equation

  • Stability Condition
  • Controllability Gramian
slide-5
SLIDE 5

5 Department of Aerospace Engineering

Different Interpretations of the Lyapunov Equation

  • Stochastic Interpretations

zero mean white noise with unit intensity

Steady-state covariance matrix Upper Bound Output Covariance Matrix

(Linear Matrix Inequality)

slide-6
SLIDE 6

6 Department of Aerospace Engineering

Different Interpretations of the Lyapunov Equation

  • Deterministic Interpretations
slide-7
SLIDE 7

7 Department of Aerospace Engineering

Control Design Problem

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

Consider the LTI system and a dynamic controller

Closed-loop system dynamics form

slide-8
SLIDE 8

8 Department of Aerospace Engineering

17 Different Control Design Problem

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

Continuous-Time Case

  • 1. Stabilizing Control
  • 2. Covariance Upper Bound Control
  • 3. Linear Quadratic Regulator
  • 4. L∞ Control
  • 5. H∞ Control
  • 6. Positive Real Control
  • 7. Robust H2 Control
  • 8. Robust L∞ Control
  • 9. Robust H∞ Control

Discrete-Time Case

  • 1. Stabilizing Control
  • 2. Covariance Upper Bound Control
  • 3. Linear Quadratic Regulator
  • 4. l∞ Control
  • 5. H∞ Control
  • 6. Robust H2 Control
  • 7. Robust l∞ Control
  • 8. Robust H∞ Control
slide-9
SLIDE 9

9 Department of Aerospace Engineering

Stabilizing Control

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

slide-10
SLIDE 10

10 Department of Aerospace Engineering

Covariance Upper Bound Control

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

Upper bounds on the output covariances

slide-11
SLIDE 11

11 Department of Aerospace Engineering

Linear Quadratic Regulator

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

slide-12
SLIDE 12

12 Department of Aerospace Engineering

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

slide-13
SLIDE 13

13 Department of Aerospace Engineering

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

slide-14
SLIDE 14

14 Department of Aerospace Engineering

Integrating Information Architecture and Control

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.

Actuator Plant Sensor Controller Convex Problem

slide-15
SLIDE 15

15 Department of Aerospace Engineering

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

slide-16
SLIDE 16

16 Department of Aerospace Engineering

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

Actuator precision Sensor precision

slide-17
SLIDE 17

17 Department of Aerospace Engineering

Problem Statement

Total cost for actuators and sensors:

slide-18
SLIDE 18

18 Department of Aerospace Engineering

Integrating Information Architecture and Control : Final Result

slide-19
SLIDE 19

19 Department of Aerospace Engineering

Which produces the control given in the paper:

Integrating Information Architecture and Control : Final Result

slide-20
SLIDE 20

20 Department of Aerospace Engineering

Actuator Plant Sensor Controller Non - Convex Problem

Integrated Plant, Sensor/Actuator and Control Design

New Contribution - Jointly optimize controller, Sensor/Actuator Design and Plant

Parameters in an LMI framework to meet some performance criteria.

slide-21
SLIDE 21

21 Department of Aerospace Engineering

  • 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.

Integrated Plant, Sensor/Actuator and Control Design

slide-22
SLIDE 22

22 Department of Aerospace Engineering

Given Data

  • A continuous linear time-invariant system in descriptor state-space

representation:

(Plant) (Measurement) (Output)

  • 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.

slide-23
SLIDE 23

23 Department of Aerospace Engineering

  • Assuming pa , ps and pa are vectors containing the price per unit of

actuator precision, sensor precision and structure parameter

  • The total design price :
  • Actuator and sensor precisions are defined to be inversely proportional

to the respective noise intensities

Given Data

slide-24
SLIDE 24

24 Department of Aerospace Engineering

Information Architecture System Design

Problem Statement

slide-25
SLIDE 25

25 Department of Aerospace Engineering

Closed Loop System

  • The closed-loop system is given by
  • All the matrices can be expanded as
  • Define the closed-loop state and noise vectors as
slide-26
SLIDE 26

26 Department of Aerospace Engineering

Control Design Problem

Not an LMI

  • Applying Schur’s Compliment and defining

(Non-Convex Constraints)

  • The above closed loop system is stable if and only if there exists a X > 0 such

that:

slide-27
SLIDE 27

27 Department of Aerospace Engineering

Information Architecture System Design

Existence Condition 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

Existence Theorem:-

Design Specifications

slide-28
SLIDE 28

28 Department of Aerospace Engineering

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

slide-29
SLIDE 29

29 Department of Aerospace Engineering

Convexifying the Problem

  • To use the previous Lemma, let us define the matrix G as:

LMI

  • Also define the convexifying potential function as:
  • (Convex

Constraints)

slide-30
SLIDE 30

30 Department of Aerospace Engineering

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.

slide-31
SLIDE 31

31 Department of Aerospace Engineering

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.

slide-32
SLIDE 32

Thank You!

https://bobskelton.github.io/index.html