Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards stochastic realization theory for Generalized Linear Switched Systems with inputs: decomposition into stochastic and deterministic components and existence and uniqueness of innovation form

Elie Rouphael, Manas Mejari, Mihaly Petreczky, Lotfi Belkoura

Abstract

In this paper, we study a class of stochastic Generalized Linear Switched System (GLSS), which includes subclasses of jump-Markov, piecewide-linear and Linear Parameter-Varying (LPV) systems. We prove that the output of such systems can be decomposed into deterministic and stochastic components. Using this decomposition, we show existence of state-space representation in innovation form, and we provide sufficient conditions for such representations to be minimal and unique up to isomorphism.

Towards stochastic realization theory for Generalized Linear Switched Systems with inputs: decomposition into stochastic and deterministic components and existence and uniqueness of innovation form

Abstract

In this paper, we study a class of stochastic Generalized Linear Switched System (GLSS), which includes subclasses of jump-Markov, piecewide-linear and Linear Parameter-Varying (LPV) systems. We prove that the output of such systems can be decomposed into deterministic and stochastic components. Using this decomposition, we show existence of state-space representation in innovation form, and we provide sufficient conditions for such representations to be minimal and unique up to isomorphism.
Paper Structure (7 sections, 11 theorems, 17 equations)

This paper contains 7 sections, 11 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result
  4. Conclusion
  5. Appendix: Proofs of Theorems \ref{['decomp:lemma']} and \ref{['decomp:lemma:inv']}
  6. Proof of Theorem \ref{['decomp:lemma']}
  7. Proof of Theorem \ref{['decomp:lemma:inv']}

Key Result

Theorem 1

For a sGLSS of the form eqn:LPV_SSA, $\mathbf{S}_d=(\{A_{\sigma},B_{\sigma}\}_{\sigma=1}^{n_{\mu}},C,D,\textbf{u})$ is an asGLSS of $(\textbf{y}^d,\bm{\mu})$ and $\mathbf{S}_s=(\{A_{\sigma},K_{\sigma}\}_{\sigma=1}^{n_{\mu}},C,F,\mathbf{v})$ is an asGLSS of $(\textbf{y}^s,\bm{\mu})$.

Theorems & Definitions (33)

  • Definition 1: Admissible process, PetreczkyBilinear
  • Example 1: White noise
  • Example 2: Discrete valued i.i.d process
  • Example 3: Markov chain
  • Remark 1: LPV systems
  • Definition 2: ZMWSSI, PetreczkyBilinear
  • Definition 3: SII process, PetreczkyBilinear
  • Definition 4: White noise w.r.t. $\bm{\mu}$
  • Definition 5: Stationary GLSS
  • Theorem 1
  • ...and 23 more