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Stability Analysis of Adaptive Model Predictive Control Using the Circle and Tsypkin Criteria

Juan A. Paredes, Dennis S. Bernstein

TL;DR

This work analyzes the stability of predictive cost adaptive control (PCAC) for discrete-time Lur'e systems with self-excited oscillations. By treating the adaptive PCAC as an instantaneous LTI in the closed loop, the authors apply the discrete-time circle criterion and Tsypkin criterion to assess absolute stability of the evolving system, with global asymptotic stability guaranteed for the asymptotic dynamics once the controller converges. The approach combines online RLS-based identification with a Block Observable Canonical Form, and a backward-propagating Riccati equation to enable receding-horizon control, yielding an output-feedback dynamic compensator. A numerical example demonstrates that, under sufficient excitation, the closed-loop system satisfies the stability criteria and stabilizes the self-excited oscillations, providing a practical framework for stability guarantees of adaptive MPC in nonlinear SES contexts. The results highlight the importance of excitation and reveal the heuristic nature of instantaneous tests on time-varying dynamics, suggesting directions for deeper theoretical grounding and robustness analyses.

Abstract

Absolute stability is a technique for analyzing the stability of Lur'e systems, which arise in diverse applications, such as oscillators with nonlinear damping or nonlinear stiffness. A special class of Lur'e systems consists of self-excited systems (SES), in which bounded oscillations arise from constant inputs. In many cases, SES can be stabilized by linear controllers, which motivates the present work, where the goal is to evaluate the effectiveness of adaptive model predictive control for Lur'e systems. In particular, the present paper considers predictive cost adaptive control (PCAC), which is equivalent to a linear, time-variant (LTV) controller. A closed-loop Lur'e system comprised of the positive feedback interconnection of the Lur'e system and the PCAC-based controller can thus be derived at each step. In this work, the circle and Tsypkin criteria are used to evaluate the absolute stability of the closed-loop Lur'e system, where the adaptive controller is viewed as instantaneously linear time-invariant. When the controller converges, the absolute stability criteria guarantee global asymptotic stability of the asymptotic closed-loop dynamics.

Stability Analysis of Adaptive Model Predictive Control Using the Circle and Tsypkin Criteria

TL;DR

This work analyzes the stability of predictive cost adaptive control (PCAC) for discrete-time Lur'e systems with self-excited oscillations. By treating the adaptive PCAC as an instantaneous LTI in the closed loop, the authors apply the discrete-time circle criterion and Tsypkin criterion to assess absolute stability of the evolving system, with global asymptotic stability guaranteed for the asymptotic dynamics once the controller converges. The approach combines online RLS-based identification with a Block Observable Canonical Form, and a backward-propagating Riccati equation to enable receding-horizon control, yielding an output-feedback dynamic compensator. A numerical example demonstrates that, under sufficient excitation, the closed-loop system satisfies the stability criteria and stabilizes the self-excited oscillations, providing a practical framework for stability guarantees of adaptive MPC in nonlinear SES contexts. The results highlight the importance of excitation and reveal the heuristic nature of instantaneous tests on time-varying dynamics, suggesting directions for deeper theoretical grounding and robustness analyses.

Abstract

Absolute stability is a technique for analyzing the stability of Lur'e systems, which arise in diverse applications, such as oscillators with nonlinear damping or nonlinear stiffness. A special class of Lur'e systems consists of self-excited systems (SES), in which bounded oscillations arise from constant inputs. In many cases, SES can be stabilized by linear controllers, which motivates the present work, where the goal is to evaluate the effectiveness of adaptive model predictive control for Lur'e systems. In particular, the present paper considers predictive cost adaptive control (PCAC), which is equivalent to a linear, time-variant (LTV) controller. A closed-loop Lur'e system comprised of the positive feedback interconnection of the Lur'e system and the PCAC-based controller can thus be derived at each step. In this work, the circle and Tsypkin criteria are used to evaluate the absolute stability of the closed-loop Lur'e system, where the adaptive controller is viewed as instantaneously linear time-invariant. When the controller converges, the absolute stability criteria guarantee global asymptotic stability of the asymptotic closed-loop dynamics.
Paper Structure (9 sections, 2 theorems, 46 equations, 13 figures)

This paper contains 9 sections, 2 theorems, 46 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Statement of the Control Problem
  3. Predictive Cost Adaptive Control
  4. Online Identification Using Recursive Least Squares with Variable-Rate Forgetting Based on the F-Test
  5. Input-Output Model and the Block Observable Canonical Form
  6. Receding-Horizon Control with Backward-Propagating Riccati Equation (BPRE)
  7. Closed-loop Lur'e system for absolute stability analysis
  8. Numerical Example
  9. Conclusions

Key Result

Theorem A.2

Assume that $G$ is strictly proper, let $M_1 \in {\mathbb R}^{m \times p}$ and $M_2 \in {\mathbb R}^{m \times p}$ be such that $M_2 - M_1$ is positive definite, assume that $\gamma$ is SB with sector bound [$M_1, M_2$], define $H({\bf q}) \stackrel{\triangle}{=} [I_m - M_2 G({\bf q})] [I_m - M_1 G({ Then, the zero solution of xLureA, yLureA is GAS.

Figures (13)

  • Figure 1: Closed-loop LTV control of the discrete-time Lur'e (DTL) system. The LTV controller $G_{{\rm c}, k}$ is applied to the DTL system consisting of the linear system $G$ and feedback nonlinearity $\gamma.$ The linear dynamics of the closed-loop Lur'e system are given by $\tilde{G}_k \stackrel{\triangle}{=} G (I_m -G_{{\rm c},k} G)^{-1},$ which is the LTV system arising from the positive feedback interconnection of $G$ and $G_{{\rm c}, k}.$
  • Figure 2: Example \ref{['ex_1']}: Open-loop response of the DTL system in Example \ref{['ex_1']} for $k\in[100, 150]$. $\gamma$ is SB and DISB with sector bound $[0, 1],$ as indicated by the red, dashed line segments and the green-shaded region.
  • Figure 3: Example \ref{['ex_1']}: DTL system output $y$ and the adaptive controller coefficients $\theta_k$ for $x_0 = 1000B$ and $v_k = 0$. The simulation transitions from open-loop operation to closed-loop operation at the step indicated by the vertical, dashed red line.
  • Figure 4: Example \ref{['ex_1']}: Evaluation of (CC1) and (CC2) for $x_0 = 1000B$ and $v_k = 0.$ Values in red correspond to steps at which $1 - \alpha_{{\rm CC}, k}$ and $\beta_{{\rm CC}, k}$ are nonpositive. Values in blue correspond to steps at which $1 - \alpha_{{\rm CC}, k}$ and $\beta_{{\rm CC}, k}$ are positive. The closed-loop system satisfies (CC1) and (CC2) when the values in both plots are green. For this case, the circle criterion is not satisfied at most steps.
  • Figure 5: Example \ref{['ex_1']}: Evaluation of the Tsypkin criterion for $x_0 = 1000B,$ and $v_k = 0$ with $N = 0.08.$ Values not in black correspond to steps at which $\zeta_{1,k} \neq 0,$$\zeta_{2,N,k} = n + \hat{n}p,$ and $\zeta_{3,N,k} > 0.$ Values in red correspond to steps at which $1 - \alpha_{{\rm TC}, N, k}$ and $\beta_{{\rm TC}, N, k}$ are nonpositive. Values in blue correspond to steps at which $1 - \alpha_{{\rm TC}, N, k}$ and $\beta_{{\rm TC}, N, k}$ are positive. The closed-loop system satisfies (TC1), (TC2) and (TC3) when the values in both plots are green. For this case, the Tsypkin criterion is not satisfied at most steps.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Example V.1
  • Definition A.1
  • Theorem A.2
  • Definition A.3
  • Theorem A.4