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Second Moment Polytopic Systems: Generalization of Uncertain Stochastic Linear Dynamics

Yuji Ito, Kenji Fujimoto

Abstract

This paper presents a new paradigm to stabilize uncertain stochastic linear systems. Herein, second moment polytopic (SMP) systems are proposed that generalize systems with both uncertainty and randomness. The SMP systems are characterized by second moments of the stochastic system matrices and the uncertain parameters. Further, a fundamental theory for guaranteeing stability of the SMP systems is established. It is challenging to analyze the SMP systems owing to both the uncertainty and randomness. An idea to overcome this difficulty is to expand the SMP systems and exclude the randomness. Because the expanded systems contain only the uncertainty, their stability can be analyzed via robust stability theory. The stability of the expanded systems is equivalent to statistical stability of the SMP systems. These facts provide sufficient conditions for the stability of the SMP systems as linear matrix inequalities (MIs). In controller design for the SMP systems, the linear MIs reduce to cubic MIs whose solutions correspond to feedback gains. The cubic MIs are transformed into simpler quadratic MIs that can be solved using optimization techniques. Moreover, solving such non-convex MIs is relaxed into the iteration of a convex optimization. Solutions to the iterative optimization provide feedback gains that stabilize the SMP systems. As demonstrated here, the SMP systems represent linear dynamics with uncertain mean and covariance and other existing systems such as independently identically distributed dynamics and random polytopes. Finally, a numerical simulation shows the effectiveness of the proposed method.

Second Moment Polytopic Systems: Generalization of Uncertain Stochastic Linear Dynamics

Abstract

This paper presents a new paradigm to stabilize uncertain stochastic linear systems. Herein, second moment polytopic (SMP) systems are proposed that generalize systems with both uncertainty and randomness. The SMP systems are characterized by second moments of the stochastic system matrices and the uncertain parameters. Further, a fundamental theory for guaranteeing stability of the SMP systems is established. It is challenging to analyze the SMP systems owing to both the uncertainty and randomness. An idea to overcome this difficulty is to expand the SMP systems and exclude the randomness. Because the expanded systems contain only the uncertainty, their stability can be analyzed via robust stability theory. The stability of the expanded systems is equivalent to statistical stability of the SMP systems. These facts provide sufficient conditions for the stability of the SMP systems as linear matrix inequalities (MIs). In controller design for the SMP systems, the linear MIs reduce to cubic MIs whose solutions correspond to feedback gains. The cubic MIs are transformed into simpler quadratic MIs that can be solved using optimization techniques. Moreover, solving such non-convex MIs is relaxed into the iteration of a convex optimization. Solutions to the iterative optimization provide feedback gains that stabilize the SMP systems. As demonstrated here, the SMP systems represent linear dynamics with uncertain mean and covariance and other existing systems such as independently identically distributed dynamics and random polytopes. Finally, a numerical simulation shows the effectiveness of the proposed method.
Paper Structure (28 sections, 16 theorems, 87 equations, 4 figures, 1 algorithm)

This paper contains 28 sections, 16 theorems, 87 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Second moment polytopic systems with problem settings
  4. Target systems described by second moment polytopes
  5. Problem statements
  6. Proposed method
  7. Overview
  8. Key idea: Development of expanded systems
  9. Solutions to Problem 1: Stability analysis
  10. Solutions to Problem 2: QMI-based controller design
  11. Solutions to Problem 2: Convex controller design
  12. Demonstration with a numerical simulation
  13. Variety of second moment polytopes
  14. Simulation settings
  15. Simulation results
  16. ...and 13 more sections

Key Result

Theorem 1

For any ${\boldsymbol{x}_{0}} \in \mathbb{R}^{ n }$ and any ${\boldsymbol{\theta}_{t}} \in \mathbb{S}_{\boldsymbol{\theta}}$ for $t \in \{0,1,2,\dots\}$, suppose the following relations: Then, the following property holds for all $t \in \{0,1,2,\dots\}$:

Figures (4)

  • Figure 1: Overview of the proposed method.
  • Figure 2: Results of solving the iterative SDP. The symbols $\blacksquare$ and $\bullet$ denote the maximum eigenvalue ${\lambda_{1} (\boldsymbol{Z}^{(\ell)})}$ and the absolute sum of the other eigenvalues ${\varepsilon( \boldsymbol{Z}^{(\ell)} ) } = \sum_{i=2}^{ n ( n + m )} |{\lambda_{i} (\boldsymbol{Z}^{(\ell)})}|$, respectively, where ${\lambda_{1} (\boldsymbol{Z}^{(\ell)})} + {\varepsilon( \boldsymbol{Z}^{(\ell)} ) } \leq Z_{\mathrm{ub}} = 10$.
  • Figure 3: Without control
  • Figure 4: With the designed controller

Theorems & Definitions (48)

  • Definition 1: Second moment polytope
  • Definition 2: Time-invariant/varying SMP
  • Example 1: Simple example of SMP systems
  • Definition 3: Robust mean-square stability
  • Definition 4: Exponential robust mean-square stability
  • Definition 5: Compression operator $\mathcal{C}$
  • Remark 1: Details of $\mathcal{C}$
  • Definition 6: Expanded system
  • Definition 7: Time-invariant/varying expanded systems
  • Example 2: Demonstration of an expanded system
  • ...and 38 more