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Optimum settings for PID control of nonlinear systems

Robert Vrabel

TL;DR

This work addresses the challenge of controlling nonlinear systems of the form $y^{(n)} + f(y, y', \dots, y^{(n-1)}) = u(t)$ by introducing a piecewise affine approximation Lin$_h$ over an $n$-simplex partition of a compact domain. It proves existence, uniqueness, and convergence of the piecewise-linear model to the original nonlinear system using Banach and Gronwall tools, and defines a transfer-function-like description in the linearized regime with $G_h(s) = 1/(s^n + a_h\cdot(1,s,\dots,s^{n-1}))$ converging to $G(s) = 1/(s^n + \nabla f(y)\cdot(1,s,\dots,s^{n-1}))$ as $h\to\infty$. The authors couple this framework with PID control, optimizing $(K_p,K_i,K_d)$ via Particle Swarm Optimization to minimize ITAE and ISO criteria, demonstrated on linear and nonlinear first-order examples. The approach offers a tractable path to design and analyze nonlinear controllers using familiar linear tools, while controlling approximation error through refinement (larger $h$) and balancing computational cost. Overall, it provides theoretical guarantees of convergence and practical, PSO-driven PID tuning for nonlinear systems, with clear guidelines on when increased piecewise resolution yields meaningful performance gains.

Abstract

This paper investigates the application of piecewise linear approximation for the control of nonlinear systems, particularly focusing on the effective linearization of systems modeled by the differential equation y^(n) + f(y,y',...,y^(n-1)) = u(t). We explore the use of PID controllers in conjunction with piecewise linearization, tackling the challenges of nonlinear dynamics by dividing the function f into smaller regions or simplices over a predefined compact set D subset of R^n. The parameter h determines the number of linear pieces and influences the approximation accuracy. By increasing h, the system's behavior increasingly mirrors that of the original nonlinear system, leading to enhanced performance of the control system. The paper further discusses the optimization of PID controller parameters through the Particle Swarm Optimization method, with a focus on minimizing the ITAE and ISO criteria. Numerical simulations demonstrate the efficacy of the approach, highlighting the trade-off between computational complexity and approximation accuracy in the context of control system design.

Optimum settings for PID control of nonlinear systems

TL;DR

This work addresses the challenge of controlling nonlinear systems of the form by introducing a piecewise affine approximation Lin over an -simplex partition of a compact domain. It proves existence, uniqueness, and convergence of the piecewise-linear model to the original nonlinear system using Banach and Gronwall tools, and defines a transfer-function-like description in the linearized regime with converging to as . The authors couple this framework with PID control, optimizing via Particle Swarm Optimization to minimize ITAE and ISO criteria, demonstrated on linear and nonlinear first-order examples. The approach offers a tractable path to design and analyze nonlinear controllers using familiar linear tools, while controlling approximation error through refinement (larger ) and balancing computational cost. Overall, it provides theoretical guarantees of convergence and practical, PSO-driven PID tuning for nonlinear systems, with clear guidelines on when increased piecewise resolution yields meaningful performance gains.

Abstract

This paper investigates the application of piecewise linear approximation for the control of nonlinear systems, particularly focusing on the effective linearization of systems modeled by the differential equation y^(n) + f(y,y',...,y^(n-1)) = u(t). We explore the use of PID controllers in conjunction with piecewise linearization, tackling the challenges of nonlinear dynamics by dividing the function f into smaller regions or simplices over a predefined compact set D subset of R^n. The parameter h determines the number of linear pieces and influences the approximation accuracy. By increasing h, the system's behavior increasingly mirrors that of the original nonlinear system, leading to enhanced performance of the control system. The paper further discusses the optimization of PID controller parameters through the Particle Swarm Optimization method, with a focus on minimizing the ITAE and ISO criteria. Numerical simulations demonstrate the efficacy of the approach, highlighting the trade-off between computational complexity and approximation accuracy in the context of control system design.
Paper Structure (11 sections, 5 theorems, 137 equations, 8 figures)

This paper contains 11 sections, 5 theorems, 137 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Banach Contraction Principle and Gronwall Inequality
  3. Linear interpolation of $n-$dimensional function $f$ over an $n-$simplex
  4. Lipschitz Constant of the Piecewise Linear Appro-xi-ma-tion $\text{Lin}_h(y)$
  5. Existence and uniqueness of a solution for nonlinear system
  6. Convergence of the solution of the piecewise linear approximation to the solution of the original nonlinear system
  7. Definition of transfer function for nonlinear system
  8. Numerical simulations
  9. Example 1: PID control for a linear first-order system
  10. Example 2: PID control for a nonlinear first-order system
  11. Conclusions

Key Result

Theorem 1

Let $(X, d)$ be a complete metric space, and let $T: X \mapsto X$ be a contraction mapping, meaning there exists a constant $c \in [0, 1)$ such that Then, there exists a unique point $x^* \in X$ such that $T(x^*) = x^*$.

Figures (8)

  • Figure 1: Feedback control system with piecewise linear approximation of the nonlinear system.
  • Figure 2: Convergence of piecewise linear approximation to the original nonlinear control system.
  • Figure 3: Dirac delta with $\sigma=0.01$
  • Figure 4: Dirac doublet with $\sigma=0.01$
  • Figure 5: $C(s) = 2.4 + \frac{4.0}{s} + 0.25s$. PID-Controlled Output (closed-loop tuning), ITAE+ISO criterion ($\alpha =2000$): error 3.3411
  • ...and 3 more figures

Theorems & Definitions (6)

  • Theorem 1: Banach Contraction Principle
  • Theorem 2
  • Definition 3
  • Lemma 4
  • Theorem 5
  • Theorem 6