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Optimal Control for Discrete-Time Systems under Bounded Disturbances

Egor Dogadin, Alexey Peregudin, Dmitriy Shirokih

Abstract

This paper introduces a novel approach to the optimal control of linear discrete-time systems subject to bounded disturbances. Our approach is based on the newly established duality between ellipsoidal approximations of reachable and hardly observable sets. We provide exact solutions for state-feedback control and filtering problems, aligning with existing methods while offering improved computational efficiency. Moreover, our main contribution is the optimal solution to the output-feedback control problem for discrete-time systems which was not known before. Numerical simulations demonstrate the superiority of this result over previous sub-optimal ones.

Optimal Control for Discrete-Time Systems under Bounded Disturbances

Abstract

This paper introduces a novel approach to the optimal control of linear discrete-time systems subject to bounded disturbances. Our approach is based on the newly established duality between ellipsoidal approximations of reachable and hardly observable sets. We provide exact solutions for state-feedback control and filtering problems, aligning with existing methods while offering improved computational efficiency. Moreover, our main contribution is the optimal solution to the output-feedback control problem for discrete-time systems which was not known before. Numerical simulations demonstrate the superiority of this result over previous sub-optimal ones.
Paper Structure (15 sections, 7 theorems, 49 equations, 2 figures, 1 table)

This paper contains 15 sections, 7 theorems, 49 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The -norm for Discrete Systems
  3. Main Results
  4. State-Feedback. -Optimal Controller
  5. Filtering. -Optimal Observer
  6. Output-Feedback. -Optimal Controller
  7. Comparison with known results
  8. Conclusion
  9. Proof of Theorem 1
  10. Proof of Theorem 2
  11. Proof of Theorem 3
  12. Proof of Theorem 4
  13. Proof of Theorem 5
  14. Lemma 1
  15. Proof of Theorem 6

Key Result

Theorem 1

If $\alpha \in \left(\rho^2 \! \left( A \right), 1 \right)$ and $P_\alpha \succ 0$ is a solution of the discrete Lyapunov equation then $\mathcal{R}_\infty \subset \left\{ x \mid x^\top P_\alpha^{-1} x \leq 1 \right\}$.

Figures (2)

  • Figure 1: A comparison of of the closed-loop system's $\varepsilon(\alpha)$-norm between the sub-optimal methods b2-topunov, Khlebnikov2011 and the optimal controller proposed in Theorem \ref{['thm-KL']}.
  • Figure 2: A comparison of the closed-loop system's invariant ellipsoids (in 2D-projection) and sample state trajectories obtained with sub-optimal methods b2-topunov, Khlebnikov2011 and the optimal method proposed in Theorem \ref{['thm-KL']}.

Theorems & Definitions (17)

  • Theorem 1: b1-nazinKhlebnikov2011
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1
  • Theorem 5
  • Remark 2
  • Theorem 6
  • Remark 3
  • proof
  • ...and 7 more