The paper considers

First let us present the notations and nomenclature used in our paper.

For a square matrix

Matrix measures were used in the qualitative analysis of various types of differential systems, as briefly pointed out below, besides their applications in numerical analysis. Monograph ([1, pages 58-59]) derived upper and lower bounds for the norms of the solution vector and proposed stability criteria for time-variant linear systems. Further properties of matrix measures were revealed in [

A compact survey on the history of matrix measures and the modern developments originating from this notion can be found in [

The current paper considers

In investigating the evolution of system (

The literature of control engineering contains many papers that explore the stability robustness by considering systems of form (

For system (

(a) The uncertain system (

(b) The uncertain system (

Using the connection between linear system stability and matrix eigenvalue location (e.g. [

The uncertain system (

The uncertain system (

Consider the function

The time-dependent set

A set of the form (

A set of the form (

This paper proves that matrix-measure-based inequalities applied to the vertices

Our results are extremely useful for refining the dynamics analysis of many classes of engineering processes modeled by linear differential systems with parametric uncertainties. Relying on necessary and sufficient conditions formulated in terms of matrix measures, we get more detailed information about the system trajectories than offered by the standard investigation of equilibrium stability.

Consider the uncertain system (

The vertices of the convex hull

The function

For any

We organize the proof in two parts. Part I proves the following results.

Inequalities (

Inequality (

The matrix measure

(

(R3) For

(

(ii)

(iii)

The equivalent conditions (i)–(iii) of Theorem

Theorem 2.2 in [

Theorem

Consider the uncertain system (

The vertices of the positive cone

The function

For any

It is similar to the proof of Theorem

The equivalent conditions (i)–(iii) of Theorem

Theorem 2.3 in [

Condition (i) of both Theorems

Given a real matrix

(a) If the following hypotheses (H1), (H2) are satisfied, then inequality (

The vector norm

Matrix

We organize the proof in two parts. Part I proves the following results.

If

Given

Similarly we prove that

(R2) First, we exploit the componentwise matrix inequality (

Since

Similarly, the componentwise matrix inequality

(b) The sufficiency is proved by (a). The necessity is ensured by the equality

(a) If the following hypotheses (H1), (H2) are satisfied, then inequality (

The vector norm

Matrix

(a) We use the same technique as in the proof of Proposition

(b) The proof of necessity is identical to Theorem

Proposition

Indeed, assume that parametric uncertain system (

Propositions

This section illustrates the applicability of our results to three examples. Examples

Let us consider the set of matrices [

We define the convex hull of matrices

Theorem

The function

Any exponentially contractive set

Theorem

The function

Any constant set of the form

Let us consider the interval matrix [

First, we apply Proposition

The function

Any exponentially contractive set of the form

The dominant vertex

The function

Any exponentially contractive set of the form

is invariant with respect to the uncertain system (

Note that all the conclusions regarding the qualitative analysis of the uncertain system (

Let us consider the translation of the mechanical system in Figure

The system dynamics in form (

The viscous friction coefficients have unique values, namely,

For the initial conditions

The problem can be approached in terms of Theorem

Obviously, condition (

The graphical plots in Figures

As a general remark, it is worth mentioning that the problem considered above is far from triviality. If, instead of condition (

The request

The mechanical system used in Example

Time-evolution of the state-space trajectories corresponding to four initial conditions.

(a) 3D representation of the exponentially contractive set

Many engineering processes can be modeled by linear differential systems with uncertain parameters. Our paper considers two important classes of such models, namely, those defined by convex hulls of matrices and by positive cones of matrices. We provide new results for the qualitative analysis which are able to characterize, by necessary and sufficient conditions, the existence of common Lyapunov functions and of invariant sets. These conditions are formulated in terms of matrix measures that are evaluated for the vertices of the convex hull or positive cone describing the system uncertainties. Although matrix measures are stronger instruments than the eigenvalue location, their usage as necessary and sufficient conditions is explained by the fact that set invariance is a stronger property than stability. We also discuss some particular cases when the matrix-measure-based test can be applied to a single matrix, instead of all vertices. The usage of the theoretical concepts and results is illustrated by three examples that outline both computational and physical aspects.

The authors are grateful for the support of CNMP Grant 12100/1.10.2008 - SICONA.