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Energy-Gain Control of Time-Varying Systems: Receding Horizon Approximation

Jintao Sun, Michael Cantoni

Abstract

Standard formulations of prescribed worst-case disturbance energy-gain control policies for linear time-varying systems depend on all forward model data. In discrete time, this dependence arises through a backward Riccati recursion. This article is about the infinite-horizon $\ell_2$ gain performance of state feedback policies with only finite receding-horizon preview of the model parameters. The proposed synthesis of controllers subject to such a constraint leverages the strict contraction of lifted Riccati operators under uniform controllability and observability. The main approximation result is a sufficient number of preview steps for the incurred performance loss to remain below any set tolerance, relative to the baseline gain bound of the associated infinite-preview controller. Aspects of the result are explored in a numerical example.

Energy-Gain Control of Time-Varying Systems: Receding Horizon Approximation

Abstract

Standard formulations of prescribed worst-case disturbance energy-gain control policies for linear time-varying systems depend on all forward model data. In discrete time, this dependence arises through a backward Riccati recursion. This article is about the infinite-horizon gain performance of state feedback policies with only finite receding-horizon preview of the model parameters. The proposed synthesis of controllers subject to such a constraint leverages the strict contraction of lifted Riccati operators under uniform controllability and observability. The main approximation result is a sufficient number of preview steps for the incurred performance loss to remain below any set tolerance, relative to the baseline gain bound of the associated infinite-preview controller. Aspects of the result are explored in a numerical example.
Paper Structure (13 sections, 17 theorems, 109 equations, 4 figures)

This paper contains 13 sections, 17 theorems, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic notation and problem formulation
  3. Main contributions
  4. Related work
  5. Organization
  6. Main result
  7. Proof of the main result
  8. Lifted Riccati operator contraction
  9. Riccati operator lifting
  10. Continuity of closed-loop l2 gain performance
  11. Proof of Theorem \ref{['theorem:h_inf_T_lb']}
  12. Numerical Example
  13. Conclusion

Key Result

Theorem 1

Given $\gamma \in \mathbb{R}_{>0}$, suppose the sequence $(P_t)_{t\in\mathbb{N}_0}\subset\mathbb{S}_+^n$satisfies the following: where the $\gamma$-dependent time-varying $\ell_2$ gain Riccati operator for the system eq:ltv_sys_w--eq:output_z_def is given by with Further, in the system eq:ltv_sys_w--eq:output_z_def, let where Then, with $J_\gamma$ as per eq:J_def,$(\forall \, w \in \ell_2)~ J_

Figures (4)

  • Figure 1: Nominal trajectory.
  • Figure 2: $\underline{\kappa}$ (black), $0.1\overline{\delta}$ (red), and $\overline{\rho}$ (blue), in Theorem \ref{['theorem:h_inf_T_lb']} for lifting steps $d\in[10:40]$.
  • Figure 3: Preview steps $d(\lceil\underline{T}\rceil+1)$ sufficient according to Theorem \ref{['theorem:h_inf_T_lb']} for different performance loss bounds $\beta$relative to baseline gain bound $\gamma=125$ ($1-50\%$), and lifting steps $d\in[20:40]$.
  • Figure 4: $\delta(X_{t+1},P_{t+1})$ for: (i) $T=1$, $d=40$${\color{black} \implies} \overline{\rho}^T\overline{\delta}=2.90$ (blue); and (ii) $T=2$, $d=40$${\color{black} \implies} \overline{\rho}^T\overline{\delta}=1.08$ (red).

Theorems & Definitions (36)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 3
  • Theorem 4
  • ...and 26 more