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Multivariable Extremum Seeking Control for Dynamic Maps through Sliding Modes and Periodic Switching Function

Nerito Oliveira Aminde, Tiago Roux Oliveira, Liu Hsu

TL;DR

The work addresses real-time optimization of uncertain multivariable dynamic systems by steering the output of a nonlinear map y = h(z) toward a unique maximizer y^* = h(z^*). It introduces a multivariable extremum seeking controller based on periodic switching, sliding modes, and time-scaling to handle arbitrary relative degree, with a cyclic directional search and a ramp-based reference. A modulation function and a robustness-focused modulation design guarantee finite-time convergence of the sliding surface and convergence to within O(sqrt(η) + ε) of the optimum, under a set of assumptions (H1–H6). An illustrative two-input, one-output example demonstrates rapid convergence to y^* and bounded closed-loop signals, validating the practical applicability of the approach for real-time optimization of dynamic maps.

Abstract

This paper presents the design of an extremum seeking controller based on sliding modes and cyclic search for real-time optimization of non-linear multivariable dynamic systems. These systems have arbitrary relative degree, compensated by the technique of time-scaling. The resulting approach guarantees global convergence of the system output to a small neighborhood of the optimum point. To corroborate with the theoretical results, numerical simulations are presented considering a system with two inputs and one output, which rapidly converges to the optimal parameters of the objective function.

Multivariable Extremum Seeking Control for Dynamic Maps through Sliding Modes and Periodic Switching Function

TL;DR

The work addresses real-time optimization of uncertain multivariable dynamic systems by steering the output of a nonlinear map y = h(z) toward a unique maximizer y^* = h(z^*). It introduces a multivariable extremum seeking controller based on periodic switching, sliding modes, and time-scaling to handle arbitrary relative degree, with a cyclic directional search and a ramp-based reference. A modulation function and a robustness-focused modulation design guarantee finite-time convergence of the sliding surface and convergence to within O(sqrt(η) + ε) of the optimum, under a set of assumptions (H1–H6). An illustrative two-input, one-output example demonstrates rapid convergence to y^* and bounded closed-loop signals, validating the practical applicability of the approach for real-time optimization of dynamic maps.

Abstract

This paper presents the design of an extremum seeking controller based on sliding modes and cyclic search for real-time optimization of non-linear multivariable dynamic systems. These systems have arbitrary relative degree, compensated by the technique of time-scaling. The resulting approach guarantees global convergence of the system output to a small neighborhood of the optimum point. To corroborate with the theoretical results, numerical simulations are presented considering a system with two inputs and one output, which rapidly converges to the optimal parameters of the objective function.
Paper Structure (13 sections, 2 theorems, 29 equations, 5 figures)

This paper contains 13 sections, 2 theorems, 29 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Statement
  3. Control objective
  4. Analysis by Singular Perturbation
  5. Redesign of the controller via time-scaling
  6. Multivariable extremum seeking controller via periodic switching function
  7. Design of the cyclic directional search
  8. The singular case Lg
  9. Design of the modulation function for Lg
  10. Modulation function design for Lg
  11. Global Convergence
  12. Illustrative example
  13. Conclusions

Key Result

Proposition 1

Consider the systems (sistinteg)--(saidamensura), search direction (dirbusca), reference trajectory (modref) and control law (eqdrakunov). Outside the regions $\mathcal{D}_{\Delta}$ and $\mathcal{D}_{\Delta_i}$, if the modulation function $\rho$ in (eqdrakunov) is designed as satisfying (funcmodgen2), while $z \notin \mathcal{D}_{\Delta_i}$, one has: (a) the sliding mode $s=k\varepsilon$ is reach

Figures (5)

  • Figure 1: Multivariable extremum seeking control via periodic switching function and cyclic search for dynamic maps.
  • Figure 2: Vector parameters $z$ converge to $(0,0)$ starting from the initial condition $z(0)=(-2,4)$ and plant output converges to $y^*=2$.
  • Figure 3: Phase portrait shows the convergence to the equilibrium point marked with a red asterisk), starting from the initial condition $z_0=(-2, 4)$.
  • Figure 4: (a) control signals $u_1$ and (b) the cyclical search direction $\sigma_1$ and $\sigma_2$, with period $T_s=5 s$ scaled by $\eta=0.01$.
  • Figure 5: Output trajectories going to the optimal point $y^*=2$, from two initial conditions $z_0=~(-2, 4)$ and $z_0=~(0, 5)$.

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • proof