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Optimal Infinite-Horizon Mixed $\mathit{H}_2/\mathit{H}_\infty$ Control

Vikrant Malik, Taylan Kargin, Joudi Hajar, Babak Hassibi

Abstract

We study the problem of mixed $\mathit{H}_2/\mathit{H}_\infty$ control in the infinite-horizon setting. We identify the optimal causal controller that minimizes the $\mathit{H}_2$ cost of the closed-loop system subject to an $\mathit{H}_\infty$ constraint. Megretski proved that the optimal mixed $\mathit{H}_2/\mathit{H}_\infty$ controller is non-rational whenever the constraint is active without giving an explicit construction of the controller. In this work, we provide the first exact closed-form solution to the infinite-horizon mixed $\mathit{H}_2/\mathit{H}_\infty$ control in the frequency domain. While the optimal controller is non-rational, our formulation provides a finite-dimensional parameterization of the optimal controller. Leveraging this fact, we introduce an efficient iterative algorithm that finds the optimal causal controller in the frequency domain. We show that this algorithm is convergent when the system is scalar and present numerical evidence for exponential convergence of the proposed algorithm. Finally, we show how to find the best (in $\mathit{H}_\infty$ norm) fixed-order rational approximations of the optimal mixed $\mathit{H}_2/\mathit{H}_\infty$ controller and study its performance.

Optimal Infinite-Horizon Mixed $\mathit{H}_2/\mathit{H}_\infty$ Control

Abstract

We study the problem of mixed control in the infinite-horizon setting. We identify the optimal causal controller that minimizes the cost of the closed-loop system subject to an constraint. Megretski proved that the optimal mixed controller is non-rational whenever the constraint is active without giving an explicit construction of the controller. In this work, we provide the first exact closed-form solution to the infinite-horizon mixed control in the frequency domain. While the optimal controller is non-rational, our formulation provides a finite-dimensional parameterization of the optimal controller. Leveraging this fact, we introduce an efficient iterative algorithm that finds the optimal causal controller in the frequency domain. We show that this algorithm is convergent when the system is scalar and present numerical evidence for exponential convergence of the proposed algorithm. Finally, we show how to find the best (in norm) fixed-order rational approximations of the optimal mixed controller and study its performance.
Paper Structure (16 sections, 11 theorems, 46 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 16 sections, 11 theorems, 46 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prior Works
  3. Contributions
  4. Preliminaries
  5. Linear-Quadratic Control
  6. Input-Output Formalism
  7. Controller
  8. Mixed $\mathit{H}_2/\mathit{H}_\infty$ Control
  9. Main Results
  10. A Fixed-Point Iteration Algorithm
  11. Finite-Dimensional Parameterization of the Optimal Controller
  12. Algorithm
  13. Convergence Analysis
  14. Rational Approximation
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $\gamma \in (\gamma_{\infty}, \gamma_{2})$ be an admissible $\mathit{H}_\infty$ norm bound. The infinite-horizon mixed $\mathit{H}_2/\mathit{H}_\infty$ control problem (prob:MHH) is equivalent to the following max-min problem: where the dual variable $\Uplambda \in \mathscr{L}_1^{d_w\times d_w}$ is a positive definite, self-adjoint, $d_w\times d_w$-block Laurent operator.

Figures (5)

  • Figure 1: The spectral norm, $\overline{\sigma}(T_K^\ast(\mathrm{e}^{\jmath\omega})T_K(\mathrm{e}^{\jmath\omega}))$ of the mixed $\mathit{H}_2/\mathit{H}_\infty$ controller for $\gamma \in \{60, 68, 75\}$ at different frequency values, for the system [AC17]. The cost of the mixed $\mathit{H}_2/\mathit{H}_\infty$ controller follows $H_2$ and clips at the threshold $\gamma$ for some frequencies.
  • Figure 2: The spectral norm, $\overline{\sigma}(T_K^\ast(\mathrm{e}^{\jmath\omega})T_K(\mathrm{e}^{\jmath\omega}))$ of the mixed $\mathit{H}_2/\mathit{H}_\infty$ controller ($\gamma = 60$) and a $6^{th}$ order rational approximation at different frequency values, for the system [AC17]. The cost of the rational controller closely follows the optimal mixed $\mathit{H}_2/\mathit{H}_\infty$ controller.
  • Figure 3: The spectral norm, $\overline{\sigma}(T_K^\ast(\mathrm{e}^{\jmath\omega})T_K(\mathrm{e}^{\jmath\omega}))$ of the mixed $\mathit{H}_2/\mathit{H}_\infty$ controller for $\gamma \in \{10, 11, 12\}$ at different frequency values, for system [REA4]. The cost of the mixed $\mathit{H}_2/\mathit{H}_\infty$ controller follows $H_2$ and clips at the threshold $\gamma$ for some frequencies.
  • Figure 4: The spectral norm, $\overline{\sigma}(T_K^\ast(\mathrm{e}^{\jmath\omega})T_K(\mathrm{e}^{\jmath\omega}))$ of the mixed $\mathit{H}_2/\mathit{H}_\infty$ controller ($\gamma = 10$) and a $4^{th}$ order rational approximation at different frequency values, for system [REA4]. The cost of the rational controller closely follows the optimal mixed $\mathit{H}_2/\mathit{H}_\infty$ controller.
  • Figure 5: The variation of $r_d(\gamma)$ (defined in \ref{['eq:ratio']}) with $\gamma$ for the [REA4] system. The plot indicates different $\mathcal{N}_1, \mathcal{N}_2$ in \ref{['eq:ratio']} chosen at random. Note that the contraction ratio is always less than $1$ and decreases with an increase in $\gamma$.

Theorems & Definitions (14)

  • Theorem 3.1: Strong Duality
  • Lemma 3.2: Wiener-Hopf Method
  • Remark 3.3
  • Theorem 3.4: Saddle Point
  • Corollary 3.5
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3: Fixed-Point Solution
  • Lemma 5.1: Monotonicity
  • Theorem 5.2
  • ...and 4 more