Robust stability and stabilization studies for uncertain switched systems based on vector norms approach

Abstract

In this paper, we investigate the problems of robust stability and stabilization via a state feedback controller for both continuous and discrete-time switched linear systems with polytopic uncertainties. These considered systems are represented in the state form. Then, a transformation under the arrow form is performed. Thus, based on the construction of a common Lyapunov function associated with the application of the Kotelyanski lemma, the $M$—matrix proprieties and the aggregation techniques, new sufficient conditions for robust stability and stabilization under arbitrary switching laws are established. It should be pointed out that the obtained results are explicit, easy to apply and formulated in terms of the polytopic uncertainty parameters. In addition, this proposed method allows us to avoid the search of a common Lyapunov function which is a difficult matter. For illustration, a DC motor model with separate excitation under variable mechanical loads is used to show the effectiveness and the potential of the proposed techniques.

Appendices

Appendix 1

1.1 Borne–Gentina stability study approach

Let us consider the linear process described in state space by:$\dot{{x}}=Ax$, where $A$ is an $n\times n$ matrix. Hurwitz conditions applied to characteristic polynomial parameters of $A,\,A=\left\{ {a_{i,j} } \right\}$, leads to the asymptotic stability domain in parameters space $a_{i,j} .$ Kotelyanski elaborates a particular lemma, well adapted for stability study when off diagonal elements of matrix $A$ are non-negative.

1.2 Borne–Gentina practical stability criterion [20]:

Let consider the nonlinear continuous process described in state space by: $\dot{{x}}=A(.)x$; $A(.)$ is an $n\times n$ matrix, $A(.)=\left\{ {a_{i,j} } \right\} .$ If the overvaluing matrix $M(A)$ has its non-constant elements isolated in only one row, the verification of the Kotelyanski condition enables to conclude to the stability of the initial system.

As an example, if the non-constant elements are isolated in only one row of $A$, Kotelyanski lemma applied to the overvaluing matrix obtained by the use of the $n$ regular vector norm $p(x)$ with $x=\left[ {x_{1} ,x_{2} ,\ldots ,x_{n} } \right] ^{T}$, such that:

$p(x)=\left[ {\left| {x_{1} } \right| ,\left| {x_{2} } \right| ,\ldots ,\left| {x_{n} } \right| } \right] ^{T},$ leads to the following stability conditions of initial system:

$$\begin{aligned}&\begin{array}{l} a_{1,1} <0, \\ \left| {{\begin{array}{cc} {a_{1,1} } &{} \quad {\left| {a_{1,2} } \right| } \\ {\left| {a_{2,1} } \right| } &{} \quad {a_{2,2} } \\ \end{array} }} \right| >0, \\ \ldots , \\ \end{array}\\&\quad \quad \left( {-1} \right) ^{n}\left| {{\begin{array}{cccc} {a_{1,1} } &{} \quad {\left| {a_{1,2} } \right| } &{} &{} \quad {\left| {a_{1,n} } \right| } \\ {\left| {a_{2,1} } \right| } &{} {a_{2,2} } &{} &{} \quad {\left| {a_{2,n} } \right| } \\ \vdots &{} \vdots &{} \quad {\ldots } &{} \quad \vdots \\ {\left| {a_{n,1} (.)} \right| } &{} \quad {\left| {a_{n,2} (.)} \right| } &{} &{} \quad {a_{n,n} (.)} \\ \end{array} }} \right| >0 \end{aligned}$$

The Borne–Gentina practical criterion applied to continuous systems generalizes the Kotelyanski lemma for nonlinear systems and defines large classes of systems.

Appendix 2

1.1 Proof of Theorem 4

The matrix $M_{C}$ has its off-diagonal elements positive and the ones non constant are isolated in the last row. Thus, by referring to results obtained in [39], the conditions of Theorem 4 can be deduced from the Kotelyanski conditions [20]. These conditions require having the principal minors of alternated signs, the ${\upalpha }_{j}$ are chosen all negative:

$$\begin{aligned} \begin{array}{l} {\upalpha }_{1} <0,\, \\ {\upalpha }_{1} {\upalpha }_{2} >0, \\ \,\ldots ,\, \\ \left( {-1} \right) ^{n-1}\prod \limits _{j=1}^{n-1} {{\upalpha }_{j} } >0,\, \\ \left( {-1} \right) ^{n}\det \left( {M_{C} } \right) \,>0 \\ \end{array} \end{aligned}$$

The $n-1$ first conditions are checked because the ${\upalpha }_{j}$are negative, however the last condition yields to: $\left( {-1} \right) ^{n}\det \,\left( {M_{C} } \right) =\left( {-1} \right) ^{n}\left| {{\begin{array}{ccccc} {{\upalpha }_{1} } &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad {\left| {{\upbeta }_{1} } \right| } \\ 0 &{} \quad \ddots &{} \quad \ddots &{} \quad \vdots &{} \quad \vdots \\ \vdots &{} \quad \ddots &{} \quad \ddots &{} \quad 0 &{} \quad \vdots \\ 0 &{} \quad \cdots &{} \quad 0 &{} \quad {{\upalpha }_{n-1} } &{} \quad {\left| {{\upbeta }_{n-1} } \right| } \\ {\overline{{t}}^{1}} &{} \quad \cdots &{} \quad \cdots &{} \quad {\overline{{t}}^{n-1}} &{} \quad {\overline{{t}}^{n}} \\ \end{array} }} \right|$

$$\begin{aligned} =\left( {-1} \right) ^{n}\left[ {\overline{{t}}^{n}\prod \limits _{q=1}^{n-1} {{\upalpha }_{q} } -\sum \limits _{\mathrm{j=1}}^{n-1} {\left( {\overline{{t}}^{j}\left| {{\upbeta }_{j} } \right| \prod \limits _{\begin{array}{l} j=1 \\ j\ne q \\ \end{array}}^{n-1} {{\upalpha }_{j} } } \right) } } \right] >0 \end{aligned}$$

Then the theorem is obtained by dividing this condition by $\left( {\left( {-1} \right) ^{n-1}\prod \nolimits _{q=1}^{n-1} {{\upalpha }_{q} } } \right)$ such that: $-\overline{{t}}^{n}+\sum \nolimits _{j=1}^{n-1} {\overline{{t}}^{j}\left| {{\upbeta } _{j} } \right| {\upalpha }_{j}^{-1}>0}$

Appendix 3

1.1 Proof of corollary 1

If there exist It is clear that, for$j=1,\ldots ,n-1$, such that:

${\upbeta }_{j} \sum \nolimits _{l=1}^{N_{l} } {{\upmu }_{il} } \left( t \right) P_{il} \left( {\upalpha } \right) =-{\upbeta }_{j} \sum \nolimits _{l=1}^{N_{l} } {{\upmu }_{il} \left( t \right) } {\upgamma }_{il}^{j} <0$ and the comparison system can be chosen identically to [39]:

$$\begin{aligned} M=\left[ {{\begin{array}{ccccc} {{\upalpha }_{1} } &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad {{\upbeta }_{1} }\\ 0 &{} \quad \ddots &{} \quad \ddots &{} \quad \vdots &{} \quad \vdots \\ \vdots &{} \quad \ddots &{} \quad \ddots &{} \quad 0 &{} \quad \vdots \\ 0 &{} \quad \cdots &{} \quad 0 &{} {{\upalpha }_{n-1} } &{} \quad {{\upbeta } _{n-1} } \\ {\mathop {\max }\limits _{i\in I} \left( {\sum \limits _{l=1}^{N_{l} } {{\upmu }_{il} \left( t \right) } {\upgamma }_{il}^{1} } \right) } &{} \quad \cdots &{} \quad \cdots &{} \quad {\mathop {\max }\limits _{i\in I} \left( {\sum \limits _{l=1}^{N_{l} } {{\upmu }_{il} \left( t \right) } {\upgamma }_{il}^{n-1} } \right) } &{} \quad {\mathop {\max }\limits _{i\in I} \left( {\sum \limits _{l=1}^{N_{l} } {{\upmu }_{il} \left( t \right) } {\upgamma }_{il}^{n} } \right) } \\ \end{array} }} \right] \end{aligned}$$

the $n$th $M$ principal minor $\Delta _{_{n} } =\sum \nolimits _{l=1}^{N_{l} } {{\upmu }_{il} } \left( t \right) P_{il} \left( 0 \right) >0\,i=1,\ldots ,N$ and $l=1,2,\ldots ,N_{l}$

$$\begin{aligned}&-\sum \limits _{l=1}^{N_{l} } {{\upmu }_{il} \left( t \right) } {\upgamma }_{il}^{n} (.)+\sum \limits _{j=1}^{n-1} {\left( {{\upalpha }_{j} } \right) ^{-1}\sum \limits _{l=1}^{N_{l} } {{\upmu }_{il} \left( t \right) } {\upgamma } _{il}^{j} (.)} {\upbeta }_{j} \\&\quad =\prod \limits _{j=1}^{n-1} {\left( {{\upalpha }_{j} } \right) ^{-1}\sum \limits _{l=1}^{N_{l} } {{\upmu }_{il} } \left( t \right) P_{il} \left( 0 \right) } \end{aligned}$$

Appendix 4

1.1 Proof of Theorem 8

The matrix $M_{D}$ with all elements positive and the ones non constant are isolated in the last row. Thus, the conditions of Theorem 8 can be deduced from the Kotelyanski conditions in the discrete case applied to matrix $\left( {I_{n} -M_{D} } \right)$ [32].

The $n-1$ first conditions are checked because $\left| {{\upalpha }_{j} } \right| \in ] {0,1} ]j=1,\ldots ,n-1$, however the last condition yields to:

$$\begin{aligned}&\det \,\left( {I_{n} -M_{D} } \right) =\left| {{\begin{array}{ccccc} {1-\left| {{\upalpha }_{1} } \right| } &{} \quad 0 &{} \quad \cdots &{} \quad 0 &{} \quad {-\left| {{\upbeta }_{1} } \right| } \\ 0 &{} \quad \ddots &{} \quad \ddots &{} \quad \vdots &{} \quad \vdots \\ \vdots &{} \quad \ddots &{} \quad \ddots &{} \quad 0 &{} \quad \vdots \\ 0 &{} \quad \cdots &{} \quad 0 &{} \quad {1-\left| {{\upalpha }_{n-1} } \right| } &{} \quad {-\left| {{\upbeta }_{n-1} } \right| } \\ {-\overline{{t}}_{D}^{1}} &{} \quad \cdots &{} \quad \cdots &{} \quad {-\overline{{t}}_{D}^{n-1}} &{} \quad {1-\overline{{t}}_{D}^{n}} \\ \end{array} }} \right| >0\\{}&=\left[ {\left( {1-\overline{{t}}_{D}^{n}} \right) \prod \limits _{q=1}^{n-1} {\left( {1-\left| {{\upalpha }_{q} } \right| } \right) } -\sum \limits _{j=1}^{n-1} {\left( {\overline{{t}}_{D}^{j}\left| {{\upbeta } _{j} } \right| \mathop {\mathop {\prod }\limits _{j=1}}\limits _j\ne q ^{n-1} {\left( {1-\left| {{\upalpha }_{j} } \right| } \right) } } \right) } } \right] >0, \end{aligned}$$

Then the Theorem 8 is obtained by dividing this condition by $\bigg ( {\prod \nolimits _{q=1}^{n\!-\!1} {\bigg ( {1\!-\!\bigg | {{\upalpha } _{q} } \bigg |} \bigg )} } \bigg )$ such that: