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Multivariable simultaneous stabilization: A modified Riccati approach

Yufang Cui, Anders Lindquist

Abstract

Simultaneous stabilization problem arises in various systems and control applications. This paper introduces a new approach to addressing this problem in the multivariable scenario, building upon our previous findings in the scalar case. The method utilizes a Riccati-type matrix equation known as the Covariance Extension Equation, which yields all solutions parameterized in terms of a matrix polynomial. The procedure is demonstrated through specific examples.

Multivariable simultaneous stabilization: A modified Riccati approach

Abstract

Simultaneous stabilization problem arises in various systems and control applications. This paper introduces a new approach to addressing this problem in the multivariable scenario, building upon our previous findings in the scalar case. The method utilizes a Riccati-type matrix equation known as the Covariance Extension Equation, which yields all solutions parameterized in terms of a matrix polynomial. The procedure is demonstrated through specific examples.
Paper Structure (7 sections, 3 theorems, 78 equations, 5 figures)

This paper contains 7 sections, 3 theorems, 78 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The simultaneous stabilization problem
  3. The analytic interpolation problem
  4. Computational examples
  5. Example 1
  6. Example 2
  7. conclusion

Key Result

Proposition 1

The two plants $P_0,P_1$ can be simultaneously stabilized by a proper compensator if and only if there exists $\Delta_i(s) \in \mathbb{H}^{m \times m}$, $\operatorname{det} \Delta_i(s) \in \mathbb{J}, i=0, 1$ , such that if $s_{1},s_2$, $\cdots, s_{t}$ are the zeros of $\det(M)$ in $\mathbb{C_+}$, t at $s_{1},s_2$, $\cdots, s_{t}$.

Figures (5)

  • Figure 1: The poles of $P_\lambda$ before stabilization
  • Figure 2: The poles of $P_\lambda$ after stabilization
  • Figure 3: The poles of the stabilized system with diferent $\Sigma$
  • Figure 4: The poles of $P_\lambda$ before stabilization
  • Figure 5: The poles of $P_\lambda$ after stabilization

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3