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Adaptive optimal $\ell_\infty$-induced robust stabilization of minimum phase SISO plant under bounded disturbance and coprime factor perturbations

Victor F. Sokolov

TL;DR

This work tackles robust stabilization of a discrete-time SISO minimum-phase plant with unknown nominal coefficients, bounded disturbances, and bounded coprime-factor perturbations within the $\ell_1$ framework. It develops an adaptive scheme that uses set-membership (polyhedral) estimates and treats the control objective $J(\theta)$ as an identification criterion, transforming a nonconvex online problem into a linear-fractional program via a change of variables to $\zeta=(\xi,\delta^w,\delta)^T$. A dead-zone updating rule yields finite-time convergence of online estimates, and the method includes online verification of both estimates and a priori assumptions. The adaptive controller is shown to achieve, with prescribed accuracy, the same steady-state upper bound on the output as an optimal controller with known parameters, while enabling online testing of model validity and robustness. Simulations on a 10-parameter plant demonstrate improved robustness and computational tractability relative to RLS-based approaches, confirming the practicality of the proposed methodology in the presence of structured disturbances and perturbations.

Abstract

This paper addresses the problem of optimal robust stabilization of a discrete-time minimum-phase plant in the framework of robust control theory in the $\ell_1$ setup and under poor a priori information. Coefficients of the transfer function of the plant nominal model with stable zeros are unknown and belong to a known bounded polyhedron in the space of coefficients. The gains of coprime factor perturbations of the plant and the upper bound of external disturbance are also unknown. The problem under consideration is to design adaptive controller that minimizes, with the prescribed accuracy, the worst-case asymptotic upper bound of the output. Solution of the problem is based on set-membership estimation of unknown parameters and treating the control criterion as the identification criterion. A hard nonconvex problem of on-line computation of optimal estimates is reduced, under additional nonrestrictive assumption, to a linear-fractional programming via a nonlinear transformation of estimated parameters. Despite the non-identifiability of the unknown parameters, the proposed adaptive controller guarantees, with the prescribed accuracy, the same optimal asymptotic upper bound of the output of adaptive system as the optimal controller for the plant with known parameters. In addition to the optimality of adaptive control, the proposed solution provides on-line verification/validation of current estimates and a priori assumptions.

Adaptive optimal $\ell_\infty$-induced robust stabilization of minimum phase SISO plant under bounded disturbance and coprime factor perturbations

TL;DR

This work tackles robust stabilization of a discrete-time SISO minimum-phase plant with unknown nominal coefficients, bounded disturbances, and bounded coprime-factor perturbations within the framework. It develops an adaptive scheme that uses set-membership (polyhedral) estimates and treats the control objective as an identification criterion, transforming a nonconvex online problem into a linear-fractional program via a change of variables to . A dead-zone updating rule yields finite-time convergence of online estimates, and the method includes online verification of both estimates and a priori assumptions. The adaptive controller is shown to achieve, with prescribed accuracy, the same steady-state upper bound on the output as an optimal controller with known parameters, while enabling online testing of model validity and robustness. Simulations on a 10-parameter plant demonstrate improved robustness and computational tractability relative to RLS-based approaches, confirming the practicality of the proposed methodology in the presence of structured disturbances and perturbations.

Abstract

This paper addresses the problem of optimal robust stabilization of a discrete-time minimum-phase plant in the framework of robust control theory in the setup and under poor a priori information. Coefficients of the transfer function of the plant nominal model with stable zeros are unknown and belong to a known bounded polyhedron in the space of coefficients. The gains of coprime factor perturbations of the plant and the upper bound of external disturbance are also unknown. The problem under consideration is to design adaptive controller that minimizes, with the prescribed accuracy, the worst-case asymptotic upper bound of the output. Solution of the problem is based on set-membership estimation of unknown parameters and treating the control criterion as the identification criterion. A hard nonconvex problem of on-line computation of optimal estimates is reduced, under additional nonrestrictive assumption, to a linear-fractional programming via a nonlinear transformation of estimated parameters. Despite the non-identifiability of the unknown parameters, the proposed adaptive controller guarantees, with the prescribed accuracy, the same optimal asymptotic upper bound of the output of adaptive system as the optimal controller for the plant with known parameters. In addition to the optimality of adaptive control, the proposed solution provides on-line verification/validation of current estimates and a priori assumptions.
Paper Structure (13 sections, 5 theorems, 87 equations, 3 figures)

This paper contains 13 sections, 5 theorems, 87 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. System description
  4. Robust perfomance of optimal closed loop system with known parameters
  5. Problem formulation
  6. Nonidentifiability
  7. Optimal estimation
  8. Adaptive optimal stabilization
  9. Model verification in closed loop
  10. Model verification under Assumption A4
  11. Model verification under Assumption $J(\theta) \le J_*$
  12. Simulations
  13. Conclusion

Key Result

Theorem 1

Following statements hold for the optimal closed loop system (plant) and (optcont). 1. The optimal closed loop system (plant) and (optcont) with the perturbations memory $\mu=+\infty$ is robustly stable if and only if and for the system with the zero initial data $y_{1-n}^0$. 2. For the system (plant) and (optcont) with bounded memory perturbations ($\mu<+\infty$) and arbitrary initial data $y_{

Figures (3)

  • Figure 1: Graphs of $y_t$ for the RLS algorithms (\ref{['rls']}) (left) and the optimal algorithm (\ref{['zetat1']})--(\ref{['zetat3']}) (right), $\pm J(\theta)$ -- red dash lines.
  • Figure 2: Graphs of $y_t$ for the RLS algorithms (\ref{['rls']}) (left) and the optimal algorithm (\ref{['zetat1']})--(\ref{['zetat3']}) (right), $\pm J(\theta)$ -- red dash lines.
  • Figure 3: Left - the graphs of $u_t$ (blue line) and $\pm\bar{u}_t$ (red lines); right - the graph of $I(\zeta_t)$ (blue line) and $J(\theta)$ (red line).

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2: Non identifiability of $\xi$
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof