Robust stability of hybrid Roesser models against parametric uncertainty: a general approach

Abstract

This paper aims at proposing a general framework for the establishement of LMI conditions to analyse the robust stability of a singular hybrid Roesser model subject to parametric uncertainties. The uncertain parameters are involved through implicit Linear Fractional Representations (LFR). Special focus is put on the influence of the number of uncertain parameters and the dimensionality of the model. More precisely it is shown that each dimension can nearly be regarded as an uncertain parameter and the other way around. Therefore, their influence on the conservatism of the obtained condition is very similar.

Appendices

A Some useful properties of implicit LFR

In this appendix, we focus a little on the properties of implicit LFRs. We only give the three properties that are useful in the body of the paper.

Let three implicit LFRs be given:

$$\begin{aligned} M_1&= \Delta _1\star \left[\begin{array}{c@{\quad }c} A_1&B_1\\ C_1&D_1\\ E_1&F_1 \end{array}\right],\end{aligned}$$

(45)

$$\begin{aligned} M_2&= \Delta _2\star \left[\begin{array}{c@{\quad }c} A_2&B_2\\ C_2&D_2\\ E_2&F_2 \end{array}\right].\end{aligned}$$

(46)

$$\begin{aligned} M&= \bar{\Delta }\star \left[\begin{array}{c@{\quad }c} \bar{A}&\bar{B}\\ C&D\\ \bar{E}&\bar{F}\\ \end{array}\right], \end{aligned}$$

(47)

with $\bar{\Delta }$ a block diagonal matrix which satisfies

$$\begin{aligned} \bar{\Delta }=T^{-1}\Delta T, \end{aligned}$$

(48)

where $\Delta$ is also a block diagonal matrix and where $T$ is a matrix which enables the permuation of the blocks in $\Delta$ to get $\bar{\Delta }$. Then it comes:

Summation:

$$\begin{aligned} M_1+M_2=\left[\begin{array}{cc} \Delta _1&0\\ 0&\Delta _2 \end{array}\right] \star \left[\begin{array}{c@{\quad }c|c} A_1&0&B_1\\ 0&A_2&B_2\\ \hline C_1&C_2&D_1+D_2\\ \hline E_1&0&F_1\\ 0&E_2&F_2 \end{array}\right]. \end{aligned}$$

(49)

Multiplication:

$$\begin{aligned} M_1M_2=\left[\begin{array}{cc} \Delta _1&0\\ 0&\Delta _2 \end{array}\right] \star \left[\begin{array}{c@{\quad }c|c} A_1&B_1C_2&B_1D_2\\ 0&A_2&B_2\\ \hline C_1&D_1C_2&D_1D_2\\ \hline E_1&F_1C_2&F_1D_2\\ 0&E_2&F_2 \end{array}\right]. \end{aligned}$$

(50)

Block permutation:

$$\begin{aligned} M=\Delta \star \left[\begin{array}{c@{\quad }c} A&B\\ C&D\\ E&F\\ \end{array}\right], \end{aligned}$$

(51)

with

$$\begin{aligned} \left[\begin{array}{c@{\quad }c} A&B\\ E&F \end{array}\right]= \left[\begin{array}{c@{\quad }c} T&0\\ 0&T \end{array}\right] \left[\begin{array}{c@{\quad }c} \bar{A}&\bar{B}\\ \bar{E}&\bar{F} \end{array}\right] \end{aligned}$$

(52)

Descriptor concrete S-procedure

In this appendix, we first recall the abstract full block S-procedure (Yakubovich 1971) formulated about in the same way as in Scherer (2001) and then we make it more concrete by specializing it to the case where implicit LFRs are involved.

Theorem 3

(Scherer 2001) Let the following mathematical objects be introduced :

• $\mathbb{\nabla }$, a compact set of complex matrices ${\Delta }$;

• An Hermitian matrix $\Theta$;

• A matrix $V\in \mathbb{R }^{l\times n}$;

• ${\fancyscript{S}}({\Delta })$, a family of subspaces $\mathbb C ^l$ continuously depending on ${\Delta }$ over $\mathbb{\nabla }$;

• ${\fancyscript{B}}({\Delta })=\displaystyle \{x\in \mathbb C ^n : Vx\in {\fancyscript{S}}({\Delta })\},\,\,{\Delta }\in \mathbb{\nabla }$.

Then the next two statements are equivalent :

• a)

  $$\begin{aligned} x^{\prime }\Theta x<0\quad \forall x\in {\fancyscript{B}}(\Delta )\backslash \{0\},\quad \forall {\Delta }\in \mathbb{\nabla }. \end{aligned}$$

  (53)

• b)

  $$\begin{aligned} \exists {X}:\displaystyle \left\{ \begin{array}{lcr} V^{\prime }{X}V+\Theta <0\\ ~\\ z^{\prime }{X}z\ge 0&\,&\forall z\in {\fancyscript{S}}({\Delta }),\forall {\Delta }\in \mathbb{\nabla } \end{array}\right. \end{aligned}$$

  (54)

Corollary 1

Let $\mathbb{X }$ be set of multipliers $X$ which enables the characaterization of a compact set of matrices $\Delta$ defined as follows:

$$\begin{aligned} {\nabla }=\displaystyle \left\{ {\Delta }\,:\, \left[\begin{array}{c} {\Delta }\\ I \end{array}\right]^{\prime }{X}\left[\begin{array}{c} {\Delta }\\ I \end{array}\right]\ge 0,\,\,\forall {X}\in \mathbb{X }. \right\} \end{aligned}$$

(55)

Then the two following statements are equivalent:

• i) $\forall \Delta \in \mathbb{\nabla },$

  $$\begin{aligned} \left[\begin{array}{c} (E-{\Delta }A)^{-1}\quad ({\Delta }B-F)\\ I \end{array}\right]^{\prime }\Theta \left[\begin{array}{c} (E-{\Delta }A)^{-1}\quad ({\Delta }B-F)\\ I \end{array}\right]<0, \end{aligned}$$

  (56)

• ii)

  $$\begin{aligned} \exists X\in \mathbb{X }: \left[\begin{array}{c@{\quad }c} E&F\\ A&B \end{array}\right]^{\prime }{X}\left[\begin{array}{c@{\quad }c} E&F\\ A&B \end{array}\right]+\Theta <0. \end{aligned}$$

  (57)

Proof

This is an application of Theorem 3. Indeed Assume that the subspace ${\fancyscript{S}}(\Delta )$ is defined as

$$\begin{aligned} {\fancyscript{S}}(\Delta )=\text{ Ker}\left(\left[\begin{array}{cc} I&-\Delta \end{array}\right]\right), \end{aligned}$$

(58)

where $\text{ Ker}(.)$ denotes the right nullspace of a matrix. The elements of ${\fancyscript{S}}(\Delta )$ can be characterized by

$$\begin{aligned} z=\left[\begin{array}{c} \Delta \\ I \end{array}\right]q,\quad q\in \mathbb{C }^{2\mu }. \end{aligned}$$

(59)

Therefore, with such a characterization, the definition of ${\nabla }$ in (55) is such that it makes the second inequality in (54) verified. Indeed, it becomes the definition of $\nabla$ itself. Moreover, the matrix $V$ is chosen as follows:

$$\begin{aligned} V=\left[\begin{array}{c@{\quad }c} E&F\\ A&B \end{array}\right]. \end{aligned}$$

(60)

Then for any vector $x$ of the form

$$\begin{aligned} x=\left[\begin{array}{c} (E-{\Delta }A)^{-1}\quad ({\Delta }B-F)\\ I \end{array}\right]y, \end{aligned}$$

(61)

ones gets

$$\begin{aligned} \left[\begin{array}{cc} I&-\Delta \end{array}\right]Vx=0, \end{aligned}$$

(62)

which proves that $Vx$ belongs to ${\fancyscript{S}}(\Delta )$ i.e. that $x$ belongs to ${\fancyscript{B}}(\Delta )$ as defined in Theorem 3. Thus, applying Theorem 3 amounts to proving the equivalence between statements i) and ii). $\square$