Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear Controller Design for Chance Constrained Systems

Georg Schildbach, Paul Goulart, Manfred Morari

TL;DR

The paper addresses constrained linear control for discrete-time LTI systems with additive disturbances by focusing on the stationary regime, where the state $x$ and input $u=Kx$ converge to zero-mean distributions with covariance $ar{X}$ and $Kar{X}K^T$ respectively.It formulates the controller design as a convex SDP that jointly optimizes the gain $K$ and the stationary covariance via the Lyapunov constraint $ar{X}-(A+BK)ar{X}(A+BK)^T-W=0$, using the change of variables $Y=KX$ to obtain LMIs for tractable optimization.Chance constraints on $x$ and $u$ are handled through SCCs and JCCs, with bounds derived from Chebyshev inequalities for WSS disturbances and normal-tail/chi-squared bounds for Gaussian disturbances, allowing either single or joint constraints to be incorporated.The approach extends to output feedback by incorporating a Kalman filter and redefining decision variables to accommodate estimator dynamics, maintaining a separation principle in the NRMs case and preserving convexity of the CSP.An illustrative satellite example demonstrates substantial reductions in constraint violations under CC-LQR compared to unconstrained LQR, validating the method’s practical effectiveness and its potential for systematic, offline design.

Abstract

This paper is concerned with the design of a linear control law for linear systems with stationary additive disturbances. The objective is to find a state feedback gain that minimizes a quadratic stage cost function, while observing chance constraints on the input and/or the state. Unlike most of the previous literature, the chance constraints (and the stage cost) are not considered on each input/state of the transient response. Instead, they refer to the input/state of the closed-loop system in its stationary mode of operation. Hence the control is optimized for a long-run, rather than a finite-horizon operation. The controller synthesis can be cast as a convex semi-definite program (SDP). The chance constraints appear as linear matrix inequalities. Both single chance constraints (SCCs) and joint chance constraints (JCCs) on the input and/or the state can be included. If the disturbance is Gaussian, additionally to WSS, this information can be used to improve the controller design. The presented approach can also be extended to the case of output feedback. The entire design procedure is flexible and easy to implement, as demonstrated on a short illustrative example.

Linear Controller Design for Chance Constrained Systems

TL;DR

The paper addresses constrained linear control for discrete-time LTI systems with additive disturbances by focusing on the stationary regime, where the state $x$ and input $u=Kx$ converge to zero-mean distributions with covariance $ar{X}$ and $Kar{X}K^T$ respectively.It formulates the controller design as a convex SDP that jointly optimizes the gain $K$ and the stationary covariance via the Lyapunov constraint $ar{X}-(A+BK)ar{X}(A+BK)^T-W=0$, using the change of variables $Y=KX$ to obtain LMIs for tractable optimization.Chance constraints on $x$ and $u$ are handled through SCCs and JCCs, with bounds derived from Chebyshev inequalities for WSS disturbances and normal-tail/chi-squared bounds for Gaussian disturbances, allowing either single or joint constraints to be incorporated.The approach extends to output feedback by incorporating a Kalman filter and redefining decision variables to accommodate estimator dynamics, maintaining a separation principle in the NRMs case and preserving convexity of the CSP.An illustrative satellite example demonstrates substantial reductions in constraint violations under CC-LQR compared to unconstrained LQR, validating the method’s practical effectiveness and its potential for systematic, offline design.

Abstract

This paper is concerned with the design of a linear control law for linear systems with stationary additive disturbances. The objective is to find a state feedback gain that minimizes a quadratic stage cost function, while observing chance constraints on the input and/or the state. Unlike most of the previous literature, the chance constraints (and the stage cost) are not considered on each input/state of the transient response. Instead, they refer to the input/state of the closed-loop system in its stationary mode of operation. Hence the control is optimized for a long-run, rather than a finite-horizon operation. The controller synthesis can be cast as a convex semi-definite program (SDP). The chance constraints appear as linear matrix inequalities. Both single chance constraints (SCCs) and joint chance constraints (JCCs) on the input and/or the state can be included. If the disturbance is Gaussian, additionally to WSS, this information can be used to improve the controller design. The presented approach can also be extended to the case of output feedback. The entire design procedure is flexible and easy to implement, as demonstrated on a short illustrative example.

Paper Structure

This paper contains 16 sections, 10 theorems, 55 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Stationary State and Input
  4. Optimal Controller Design
  5. Paper Outline
  6. Stationary Distribution
  7. Single Chance Constraints (SCCs)
  8. Joint Chance Constraints (JCCs)
  9. Controller Synthesis Problem
  10. Feasibility of the CSP
  11. Optimality of the CSP
  12. Output Feedback
  13. Kalman Filter (KF)
  14. Stationarity Conditions
  15. Example
  16. ...and 1 more sections

Key Result

Lemma 2

Let $K$ be strictly stabilizing for $(A,B)$ and $\{w_{t}\}_{t\in\mathbb{N}}$ be WSS or NRM. The covariance of the closed-loop states $\{x_{t}\}_{t\in\mathbb{N}}$ converge to a unique stationary value $\bar{X}\succ 0$ that satisfies the discrete time Lyapunov equation

Figures (3)

  • Figure 1: Comparison of SCC bounds for WSS (solid line) and NRM (dashed line) disturbances, according to Table \ref{['Tab:SCC']}.
  • Figure 2: Comparison of JCC bounds for WSS (solid line) and NRM (dashed line) disturbances, as defined in Equation \ref{['Equ:CompJCCBound']}.
  • Figure 3: Model of a spinning satellite.

Theorems & Definitions (16)

  • Lemma 2: Stationary Variance
  • Remark 3: Notation
  • Lemma 4: Stationarity Condition
  • Remark 5: Partial Ordering
  • Lemma 6: SCC on the State
  • Lemma 7: SCC on the Input
  • Remark 8: Polytopic Constraints
  • Lemma 9: JCC on the State
  • Lemma 10: JCC on the Input
  • Theorem 11: Minimum Probability Level
  • ...and 6 more