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Spectrum Assignment of Stochastic Systems with Multiplicative Noise

Xiaomin Xue, Juanjuan Xu, Huanshui Zhang

TL;DR

This work addresses spectrum (pole) placement for stochastic systems with multiplicative noise by introducing an $\alpha$-spectrum framework, where feasible placement satisfies $\\mu^* = \lambda_i\\\lambda_j + \alpha^2$ and $K_u=\alpha I$. For the discrete-time and continuous-time cases, the authors derive feasibility conditions under complete controllability of $(G,F)$ with $G=H+\\alpha L$ and provide constructive feedback laws $K_v$ (and $T_v$ in the continuous-time case) of the form $K_v = a D + b G^n$, guided by Ackermann-like polynomial design. When system matrices are unknown, stochastic approximation algorithms are proposed to learn the feedback gains, using observations of one-step or short-time state transitions and updating rules that converge to the target gains under standard assumptions. Numerical examples validate the approach, showing accurate spectrum placement with errors below $10^{-2}$ and demonstrating rapid convergence (typical $p$ in the hundreds to low thousands, depending on the example). Overall, the paper extends pole assignment to stochastic multiplicative-noise systems and provides practical, provably convergent learning-based methods for spectrum control in both discrete and continuous time.

Abstract

This paper studies the spectrum assignment of a class of stochastic systems with multiplicative noise. A novel $α$-spectrum assignment is proposed for discrete-time and continuous-time stochastic systems with multiplicative noise. In particular, $0$-spectrum assignment is equivalent to the pole assignment for the deterministic systems. The main contribution is two-fold: On the one hand, we present the conditions for $α$-spectrum assignment and the design of feedback controllers based on the system parameters. On the other hand, when the system parameters are unknown, we present a stochastic approximation algorithm to learn the feedback gains which guarantee the spectrum of the stochastic systems to achieve the predetermined value. Numerical examples are provided to demonstrate the effectiveness of the proposed algorithms.

Spectrum Assignment of Stochastic Systems with Multiplicative Noise

TL;DR

This work addresses spectrum (pole) placement for stochastic systems with multiplicative noise by introducing an -spectrum framework, where feasible placement satisfies and . For the discrete-time and continuous-time cases, the authors derive feasibility conditions under complete controllability of with and provide constructive feedback laws (and in the continuous-time case) of the form , guided by Ackermann-like polynomial design. When system matrices are unknown, stochastic approximation algorithms are proposed to learn the feedback gains, using observations of one-step or short-time state transitions and updating rules that converge to the target gains under standard assumptions. Numerical examples validate the approach, showing accurate spectrum placement with errors below and demonstrating rapid convergence (typical in the hundreds to low thousands, depending on the example). Overall, the paper extends pole assignment to stochastic multiplicative-noise systems and provides practical, provably convergent learning-based methods for spectrum control in both discrete and continuous time.

Abstract

This paper studies the spectrum assignment of a class of stochastic systems with multiplicative noise. A novel -spectrum assignment is proposed for discrete-time and continuous-time stochastic systems with multiplicative noise. In particular, -spectrum assignment is equivalent to the pole assignment for the deterministic systems. The main contribution is two-fold: On the one hand, we present the conditions for -spectrum assignment and the design of feedback controllers based on the system parameters. On the other hand, when the system parameters are unknown, we present a stochastic approximation algorithm to learn the feedback gains which guarantee the spectrum of the stochastic systems to achieve the predetermined value. Numerical examples are provided to demonstrate the effectiveness of the proposed algorithms.

Paper Structure

This paper contains 15 sections, 6 theorems, 63 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries
  4. Definition of spectrum assignment
  5. Main Result
  6. Spectrum Assignment
  7. Learning Algorithm
  8. Spectrum Assignment for Continuous-time Systems
  9. Continuous-time Stochastic Systems
  10. Spectrum Assignment
  11. Learning Algorithm
  12. Numerical Examples
  13. Example 1
  14. Example 2
  15. Conclusion

Key Result

Theorem 1

Under Assumption Ass1, the spectrum 9 generated by system 1 can be assigned to the desired set $\{ \mu_1^*, \dots, \mu_\frac{n(n+1)}{2}^*\}$ according to Definition def2. In this case, the feedback gain $K_v$ is given by where $a=$ denotes the coefficient of the following polynomial: and

Figures (1)

  • Figure 1: Communication network with local controller and remote controller

Theorems & Definitions (23)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • ...and 13 more