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Theory and Design of Extended PID Control for Stochastic Systems with Structural Uncertainties

Baoyou Qu, Cheng Zhao

TL;DR

This paper extends PID control to a broad class of nonlinear stochastic systems with arbitrary relative degree n by introducing an extended PID controller. Using Lyapunov-based analysis and a constructive parameter-design framework, the authors prove global mean-square stability and provide explicit tracking bounds that depend on the noise intensity and chosen gains. The results yield an (n+1)-dimensional unbounded set of admissible controller parameters and show that the steady-state error can be made arbitrarily small with large gains, even under structural uncertainties. Simulations on a third-order stochastic system corroborate the theory and demonstrate robustness to parameter uncertainties and noise. Together, these contributions broaden the applicability of PID-style control in uncertain, noisy nonlinear settings and offer practical guidelines for parameter selection.

Abstract

Since the classical proportional-integral-derivative (PID) controller has continued to be the most widely used feedback methods in engineering systems by far, it is crucial to investigate the working mechanism of PID in dealing with nonlinearity, uncertainty and random noises. Recently, Zhao and Guo (2022) has established the global stability of PID control for a class of uncertain nonlinear control systems with relative degree two without random perturbations. In this paper, we will consider a more general class of nonlinear stochastic systems with an arbitrary relative degree $n$, and discuss the stability and design of extended PID controller (a natural extension of PID). We demonstrate that, the closed-loop control systems will be globally stable in mean square with bounded tracking errors provided the extended PID parameters are selected from an $(n+1)$-dimensional unbounded set, even if both the system nonlinear drift and diffusion terms contain a wide range of structural uncertainties. Moreover, the steady-state tracking error is proved to be proportional to the noise intensity at the setpoint, which can also be made arbitrarily small by choosing the controller parameters suitably large.

Theory and Design of Extended PID Control for Stochastic Systems with Structural Uncertainties

TL;DR

This paper extends PID control to a broad class of nonlinear stochastic systems with arbitrary relative degree n by introducing an extended PID controller. Using Lyapunov-based analysis and a constructive parameter-design framework, the authors prove global mean-square stability and provide explicit tracking bounds that depend on the noise intensity and chosen gains. The results yield an (n+1)-dimensional unbounded set of admissible controller parameters and show that the steady-state error can be made arbitrarily small with large gains, even under structural uncertainties. Simulations on a third-order stochastic system corroborate the theory and demonstrate robustness to parameter uncertainties and noise. Together, these contributions broaden the applicability of PID-style control in uncertain, noisy nonlinear settings and offer practical guidelines for parameter selection.

Abstract

Since the classical proportional-integral-derivative (PID) controller has continued to be the most widely used feedback methods in engineering systems by far, it is crucial to investigate the working mechanism of PID in dealing with nonlinearity, uncertainty and random noises. Recently, Zhao and Guo (2022) has established the global stability of PID control for a class of uncertain nonlinear control systems with relative degree two without random perturbations. In this paper, we will consider a more general class of nonlinear stochastic systems with an arbitrary relative degree , and discuss the stability and design of extended PID controller (a natural extension of PID). We demonstrate that, the closed-loop control systems will be globally stable in mean square with bounded tracking errors provided the extended PID parameters are selected from an -dimensional unbounded set, even if both the system nonlinear drift and diffusion terms contain a wide range of structural uncertainties. Moreover, the steady-state tracking error is proved to be proportional to the noise intensity at the setpoint, which can also be made arbitrarily small by choosing the controller parameters suitably large.

Paper Structure

This paper contains 10 sections, 10 theorems, 210 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Notations
  4. The Control System
  5. Assumptions
  6. Global Mean Square Stability
  7. Parameter Design and Performance Analysis
  8. Proof of The Main Results
  9. Simulations
  10. Conclusion

Key Result

Proposition 1

Suppose Assumptions A 1 and A 2 hold, and the $(n+1)$-parameters $k_0$, $k_1$, $\cdots,$$k_n$ are admissible for the two-tuples $(L,M)$. Then, the closed-loop system (n-order control system)-(extended pid controller) will be globally stable. Moreover, there exist some positive constants $C_1$, $C_2$ where $z^*$ and $u^*$ are defined in (z*) and (unique u^* w.r.t y^*) respectively.

Figures (3)

  • Figure 1: Curves of $\mathbf{E}|e(t)|^2$ under different system parameters. The system initial state $(x_1(0),x_2(0),x_3(0))$ is $(0.5,0.5,0.3)$.
  • Figure 2: Curves of $\mathbf{E}|e(t)|^2$ under different $\sigma$. The system parameters $(a,b,c,d,\mu)=(0.4,-0.3,0.5,6,5.2)$, and the initial state $(x_1(0),x_2(0),x_3(0))$ is $(0.9,0,0.1)$.
  • Figure 3: Curves of $\mathbf{E}|u(t)|^2$ and $\text{Var}(u(t))$ under different $\sigma$. The system parameters $(a,b,c,d,\mu)=(0.4,-0.3,0.5,6,5.2)$, and the initial state $(x_1(0),x_2(0),x_3(0))$ is $(1.3,0,0.1)$.

Theorems & Definitions (29)

  • Remark 1
  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 19 more