#### Abstract

The paper considers *parametric uncertain systems* of the form ,
where is either a *convex hull*, or a *positive cone* of matrices, generated by the
set of vertices . Denote by the matrix measure corresponding to a vector norm . When is a convex hull, the condition
,
, is necessary and sufficient for the existence of *common strong Lyapunov functions* and *exponentially contractive invariant sets* with respect to the trajectories of the uncertain system. When is a positive cone, the condition ,
, is necessary and sufficient for the existence of *common weak Lyapunov functions* and *constant invariant sets* with respect to the trajectories of the uncertain system. Both Lyapunov functions and invariant sets are described in terms of the vector norm used for defining the matrix measure . Numerical examples illustrate the applicability of our results.

#### 1. Introduction

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

For a square matrix , the *matrix norm* induced by a generic *vector norm * is defined by , and the *corresponding matrix measure* (also known as *logarithmic norm*) is given by ([1, page 41]). The spectrum of is denoted by and , , represent its eigenvalues. If is a symmetrical matrix, () means that matrix is negative (positive) definite. If , then represents the nonnegative matrix (for ) or vector (for ) defined by taking the absolute values of the entries of . If , then āā, āā mean componentwise inequalities.

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 [2]. Paper [3] provided bounds for the computer solution and the accumulated truncation error corresponding to the backward Euler method. The work in [4] gave a characterization of vector norms as Lyapunov functions for time-invariant linear systems. The work in [5] developed sufficient conditions for the stability of neural networks. The work in [6] explored contractive invariant sets of time-invariant linear systems. The work in [7] formulated sufficient conditions for the stability of interval systems. The work in [8] presented a necessary and sufficient condition for componentwise stability of time-invariant linear systems.

A compact survey on the history of matrix measures and the modern developments originating from this notion can be found in [9].

The current paper considers *parametric uncertain systems* of the form

where is either a *convex hull *of matrices,

or a *positive cone* of matrices,

generated by the set of vertex matrices

In investigating the evolution of system (1.1) we assume that matrix is fixed, but arbitrarily taken from the matrix set defined by (1.2) or (1.3). Thus, the parameters of system (1.1) are not time-varying. Consequently, once is arbitrarily selected from , the trajectory initialized in , namely, , is defined for all

The literature of control engineering contains many papers that explore the stability robustness by considering systems of form (1.1), a great interest focusing on the case when the convex hull is an *interval matrix* [7, 10ā13].

For system (1.1) we define the following properties, in accordance with the definitions presented in [14ā16] for a dynamical system.

*Definition 1.1. *(a) The uncertain system (1.1) is called *stable* if the equilibrium is stable, that is,
for any solution of (1.1) corresponding to an .

(b) The uncertain system (1.1) is called *exponentially stable* if the equilibrium is exponentially stable, that is,
for any solution of (1.1) corresponding to an .

*Remark 1.2. *Using the connection between linear system stability and matrix eigenvalue location (e.g.ā[15]), we have the following characterizations.(a)The uncertain system (1.1) is stable if and only if
In this case, the matrix set is said to be *quasistable*.(b)The uncertain system (1.1) is exponentially stable if and only if
In this case, the matrix set is said to be *Hurwitz stable*.

*Definition 1.3. *Consider the function
and its right Dini derivative, calculated along a solution of (1.1):
(a) is called a *common strong Lyapunov function* for the uncertain system (1.1), *with the decreasing rate *, if for any solution of (1.1) corresponding to an , we have
(b) is called a *common weak Lyapunov function* for the uncertain system (1.1), if for any solution of (1.1) corresponding to an , we have

*Definition 1.4. *The time-dependent set
is called *invariant *with respect to the uncertain system (1.1) if for any solution of (1.1) corresponding to an , we have
meaning that any trajectory initiated inside the set will never leave .(a)A set of the form (1.12) with is said to be *exponentially contractive*.(b)A set of the form (1.12) with is said to be* constant*.

This paper proves that matrix-measure-based inequalities applied to the vertices , , provide necessary and sufficient conditions for the properties of the uncertain system (1.1) formulated by Definitions 1.3 and 1.4. The cases when the matrix set is defined by the convex hull (1.2) and by the positive cone (1.3) are separately addressed. When is a symmetric gauge function or an absolute vector norm and the vertices , , satisfy some supplementary hypotheses, a unique test matrix can be found such that a single inequality using implies or is equivalent to the group of inequalities written for all vertices. Some numerical examples illustrate the applicability of the proposed theoretical framework.

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.

#### 2. Main Results

##### 2.1. Uncertain System Defined by a Convex Hull of Matrices

Theorem 2.1. *Consider the uncertain system (1.1) with , the convex hull defined by (1.2), which, in the sequel, is referred to as the uncertain system (1.1) and (1.2). Let be the matrix measure corresponding to the vector norm , and a constant. The following statements are equivalent.*(i)* The vertices of the convex hull fulfill the inequalities
*(ii)* The function defined by (1.8) is a common strong Lyapunov function for the uncertain system (1.1) and (1.2) with the decreasing rate .*(iii)* For any , the exponentially contractive set defined by (1.12) is invariant with respect to the uncertain system (1.1) and (1.2).*

*Proof. *We organize the proof in two parts. Part I proves the following results.(R1)Inequalities (2.1) are equivalent to
(R2) Inequality (1.10) is equivalent to
(R3)The matrix measure fulfills the equality
Part II uses (R1), (R2), and (R3) to show that (i), (ii), and (iii) are equivalent.*Proof of Part I*. (R1) If (2.2) is true, then (2.1) is true, since , for . Conversely, if (2.1) is true, then, from the convexity of the matrix measure, we get

() If inequality (2.3) is true, then, for any solution of (1.1) and (1.2) with initial condition set at as we have
Conversely, let and . If inequality (1.10) holds for , consider the differential equation with the initial condition . Then, according to [14, Theoremā4.2.11], , for all

(R3) For and , we have , with . The triangle inequality leads to . By taking , we finally obtain the equality (2.4).*Proof of Part II*. (i)(ii) For any solution to (1.1) and (1.2) corresponding to an , we get

()(i) For all . Hence, we have .

(ii)(iii) By contradiction, assume that there exists such that the exponentially contractive set is not invariant with respect to the uncertain system (1.1) and (1.2). Then there exists a trajectory of (1.1) and (1.2) for which condition (1.13) is violated, meaning that we can find , , so that and . This leads to , which contradicts (2.3). As a result, according to (R2), we contradict (ii).

(iii)(ii) For arbitrary , by taking in (1.13), we get (2.3) that is equivalent to (1.10), via (R2).

*Remark 2.2. *The equivalent conditions (i)ā(iii) of Theorem 2.1 imply the exponential stability of the uncertain system (1.1) and (1.2). Indeed, if the flow invariance condition (1.13) from Definition 1.4 is satisfied, then condition (1.5) from Definition 1.1, for exponential stability, is satisfied with . Conversely, if (1.5) is true for a certain , but not for , then condition (1.13) is not met. In other words the uncertain system (1.1) and (1.2) may be exponentially stable without satisfying the equivalent conditions (i)ā(iii) of Theorem 2.1.

*Remark 2.3. *Theoremā2.2 in [7] shows that condition (i) in Theorem 2.1 is *sufficient *for the Hurwitz stability of the convex hull of matrices defined by (1.2). According to Remark 2.2, the uncertain system (1.1) and (1.2) is exponentially stable. The fact that condition (i) in Theorem 2.1 is *necessary and sufficient* for stronger properties of the uncertain system (1.1) and (1.2) remained hidden for the investigations developed by [7].

*Remark 2.4. *Theorem 2.1 offers a high degree of generality for the qualitative analysis of uncertain system (1.1) and (1.2). Thus, from Theorem 2.1 particularized to the vector norm , , (where is a nonsingular matrix) and the corresponding matrix measure , we get the following well-known characterization of the quadratic stability of uncertain system (1.1) and (1.2), (e.g., [16, page 213]). Indeed, according to ([1, page 41]) inequality (2.1) in Theorem 2.1 means , which is equivalent to the condition , with , for all . In other words, Theorem 2.1 provides a comprehensive scenario that naturally accommodates results already available in particular forms for uncertain system (1.1) and (1.2).

##### 2.2. Uncertain System Defined by a Positive Cone of Matrices

Theorem 2.5. *Consider the uncertain system (1.1) with , the positive cone defined by (1.3), which, in the sequel, is referred to as the uncertain system (1.1) and (1.3). Let be the matrix measure corresponding to the vector norm . The following statements are equivalent.*(i)*The vertices of the positive cone fulfill the inequalities
*(ii)* The function defined by (1.8) is a common weak Lyapunov function for the uncertain system (1.1) and (1.3).*(iii)* For any , the constant set defined by (1.12) is invariant with respect to the uncertain system (1.1) and (1.3).*

*Proof. *It is similar to the proof of Theorem 2.1 where we take *. *

*Remark 2.6. *The equivalent conditions (i)ā(iii) of Theorem 2.5 imply the stability of the uncertain system (1.1) and (1.3). Indeed, if the flow invariance condition (1.13) from Definition 1.4 is satisfied, then condition (1.4) for stability from Definition 1.1 is satisfied with Conversely, if (1.4) is true for a certain but not for then condition (1.13) is not met. In other words the uncertain system (1.1) and (1.3) may be stable without satisfying the equivalent conditions (i)ā(iii) of Theorem 2.5.

*Remark 2.7. *Theoremā2.3 in [7] claims that for (i.e., condition (i) in Theorem 2.5 with strict inequalities) is *sufficient *for the Hurwitz stability of the positive cone of matrices (1.3). However this is not true. Inequalities , , imply , which, together with , for example, [1], yield . Thus, condition (1.7) for the Hurwitz stability of the positive cone of matrices defined by (1.3) may be not satisfied. Although the hypothesis of [7, Theoremāā2.3] is stronger than condition (i) in our Theorem 2.5, this hypothesis can guarantee only the stability (but not the exponential stability) of the uncertain system (1.1) and (1.3). Moreover, as already mentioned in Remark 2.3 for the matrix set defined by (1.2), [7] does not discuss the necessity parts of the results.

#### 3. Usage of a Single Test Matrix for Checking Condition (i) of Theorems 2.1 and 2.5

Condition (i) of both Theorems 2.1 and 2.5 represents inequalities of the form

which involve all the vertices of the matrix sets defined by (1.2) or (1.3), respectively. We are going to show that, in some particular cases, one can find a single test matrix such that the satisfaction of inequality

guarantees the fulfillment of (3.1).

Given a real matrix , let us define its *comparison matrix * by

Proposition 3.1. *(a) If the following hypotheses (H1), (H2) are satisfied, then inequality (3.2) is a sufficient condition for inequalities (3.1).*(H1)*The vector norm is a symmetric gauge function ([17, page 438]) (i.e., it is an absolute vector norm that is a permutation invariant function of the entries of its argument) and is the corresponding matrix measure.*(H2)*Matrix satisfies the componentwise inequalities
for some permutation matrices , .**(b) If the above hypotheses (H1), (H2) are satisfied and there exists such that , then inequality (3.2) is a necessary and sufficient condition for inequalities (3.1).*

*Proof. *(a) We organize the proof in two parts. Part I proves the following results.(R1) If is a permutation matrix, then
(R2)Given , if the componentwise inequality
is fulfilled for a permutation matrix then
Part II uses (R2) to show that (3.2) implies (3.1).*Proof of Part I*. (R1) From the definition of the matrix norm, there exists , , such that . Since the considered vector norm is permutation invariant, we have for a permutation matrix . This leads to . Let us prove that the strict inequality does not hold. Assume, by contradiction, that . Then, there exists , , such that . Hence, with satisfies that contradicts the definition of . Consequently, .

Similarly we prove that , yielding . Thus, we get and, consequently, = . By taking we obtain equality (3.5).

(R2) First, we exploit the componentwise matrix inequality (3.6). For small , we get that leads to the following componentwise vector inequality , with .

Since is a symmetric gauge function, it is also an absolute vector norm, and, equivalently, a monotonic vector norm [17, Theoremāā5.5.10]. Consequently, that implies Thus, we get and . By taking we obtain the inequality .

Similarly, the componentwise matrix inequality leads to . Finally, we have .*Proof of Part II*. From (3.4), according to (R2) we get , , which together with (3.2) lead to (3.1).

(b) The sufficiency is proved by (a). The necessity is ensured by the equality and the inequality (resulting from ).

Proposition 3.2. *(a) If the following hypotheses (H1), (H2) are satisfied, then inequality (3.2) is a sufficient condition for inequalities (3.1).*(H1)*The vector norm is an absolute vector norm and denotes the corresponding matrix measure.*(H2)*Matrix satisfies the componentwise inequalities
**(b) If the above hypotheses (H1), (H2) are satisfied and there exists such that , then inequality (3.2) is a necessary and sufficient condition for inequalities (3.1).*

*Proof. *(a) We use the same technique as in the proof of Proposition 3.1 to show that, for a given satisfying the componentwise inequality , the monotonicity of implies .

(b) The proof of necessity is identical to Theorem 2.1.

*Remark 3.3. *Proposition 3.2 allows one to show that the characterization of the componentwise exponential asymptotic stability (abbreviated CWEAS) of interval systems given by our previous work [12] represents a particular case of Theorem 2.1 applied for an absolute vector norm.

Indeed, assume that parametric uncertain system (1.1) and (1.2) is an interval system; that is, the convex hull of matrices has the particular form . This system is said to be CWEAS if there exist *, *, and such that *, **: **, **,* where , denote the components of the initial condition and of the corresponding solution , respectively. According to [12] the interval system is CWEAS if and only if , where and the matrix is built from the entries of the matrices and by , and , *, *. On the other hand, Theorem 2.1 characterizes CWEAS if applied for the vector norm , with . At the same time, we can use Proposition 3.2(b), since (3.8) is satisfied with , is an absolute vector norm, and there exist belonging to the set of vertices of such that , , , , , which implies . Thus is a necessary and sufficient condition for the CWEAS of the interval system. Finally we notice that is equivalent to , , showing that the CWEAS characterization derived in [12] for interval systems is incorporated into the current approach to parametric uncertain systems.

*Remark 3.4. *Propositions 3.1 and 3.2 can be stated in a more general form, by using, instead of a single test matrix , several test matrices , with being significantly smaller than . Each , will have to satisfy inequality (3.4) in Proposition 3.1 or inequality (3.8) in Proposition 3.2, for some vertex matrices in , such that .

#### 4. Illustrative Examples

This section illustrates the applicability of our results to three examples. Examples 4.1 and 4.2 refer to case studies presented by literature of control engineering, in [18, 19], respectively. Example 4.3 aims to develop a relevant intuitive support for invariant sets with respect to the dynamics of a mechanical system with two uncertain parameters.

*Example 4.1. *Let us consider the set of matrices [18]:
Paper [18] shows that matrices *, * and have the following common quadratic Lyapunov function:
since , .

We define the convex hull of matrices having the set of vertices (4.1), that is,

and the positive cone of matrices having the set of vertices (4.1), that is,
In we define the vector norm
where , in accordance with Remark 2.4, and consider the corresponding matrix measure . For the vertex-matrices in (4.1) simple computations give , , and .(i)Theorem 2.1 applied to the qualitative analysis of uncertain system (1.1) and (4.3) reveals the following properties.(a)The function
is a common strong Lyapunov function for the uncertain system (1.1) and (4.3) with the decreasing rate . (b)Any exponentially contractive set of the form
is invariant with respect to the uncertain system (1.1) and (4.3).(ii)Theorem 2.5 applied to the qualitative analysis of uncertain system (1.1) and (4.4) reveals the following properties.(a)The function defined by (4.5) is a common weak Lyapunov function for the uncertain system (1.1) and (4.4).(b)Any constant set of the form
is invariant with respect to the uncertain system (1.1) and (4.4).

*Example 4.2. *Let us consider the interval matrix [19]:
Obviously, the set can be regarded as a convex hull with vertices
The comparison matrices *, **,* of the vertices of (4.10) are built in accordance with (3.3) yielding *, * (since all are essentially nonnegative matrices). The dominant vertex satisfies inequalities (3.8), meaning that also satisfies inequalities (3.4) with all the permutation matrices equal to the unity matrix, . Therefore we can apply both Propositions 3.1 and 3.2.

First, we apply Proposition 3.1 for the usual HĆ¶lder norms , with , which are symmetric gauge functions. We calculate the matrix measures for and obtain the values *, **, *. Consequently all the vertex matrices in (4.10) satisfy the inequalities

Thus, for the qualitative analysis of uncertain system (1.1) and (4.9) we can employ Theorem 2.1 that reveals the following properties.(a)The function
is a common strong Lyapunov function for the uncertain system (1.1) and (4.9) with the decreasing rate *, *.(b)Any exponentially contractive set of the form
is invariant with respect to the uncertain system (1.1) and (4.9).Next, we show that Proposition 3.2 allows refining the properties discussed above of the uncertain system (1.1) and (4.9). The refinement will consist in finding common strong Lyapunov functions and exponentially contractive sets with faster decreasing rates than presented above for *. *

The dominant vertex is an essentially positive matrix (all off-diagonal entries are positive) and we can use the Perron Theorem, in accordance with [20]. Denote by the Perron eigenvalue. From the left and right Perron eigenvectors of we can construct the diagonal matrices , and , , such that, for the vector norms defined in by we have , . These vector norms are absolute without being permutation invariant; hence they are not symmetric gauge functions. Nonetheless, for these norms we may apply Proposition 3.2 with proving that the vertex matrices in (4.10) satisfy the inequalities

Thus, for the qualitative analysis of uncertain system (1.1) and (4.9) we can employ Theorem 2.1 that reveals the following properties.(a)The function
is a common strong Lyapunov function for the uncertain system (1.1) and (4.9) with the decreasing rate .(b)Any exponentially contractive set of the form

is invariant with respect to the uncertain system (1.1) and (4.9).

Note that all the conclusions regarding the qualitative analysis of the uncertain system (1.1) and (4.9) remain valid in the case when we consider the modified interval matrix
which has the same dominant vertex as the original interval matrix (4.9).

*Example 4.3. *Let us consider the translation of the mechanical system in Figure 1. A coupling device CD (with negligible mass) connects, in parallel, the following components: a cart (with mass ) in series with a damper (with viscous friction coefficient ) and a spring (with spring constant ) in series with a damper (with viscous friction coefficient ).

The system dynamics in form (1.1) is described by
where the state variables are the spring force and the cart velocity . We consider when the spring is elongated and when it is compressed as well as when the cart moves to the left and when it moves to the right.

The viscous friction coefficients have unique values, namely, āNs/mm, āNs/mm, whereas the cart mass and the spring constant have uncertain values belonging to the intervals , . Therefore we introduce the notation that allows describing the set of system matrices as the convex hull of form (1.2) defined by the vertices

For the initial conditions *, *, we analyze the free response of the system. We want to see if there exists such that

The problem can be approached in terms of Theorem 2.1, by considering in the vector norm , with , and the exponentially contractive set

Obviously, condition (4.20) is equivalent with the invariance of the set (4.21) with respect to the uncertain system (4.18). By calculating the matrix measures for the vertices *, *, in (4.19), we show that condition (2.1) in Theorem 2.1 is satisfied for . Hence, the set (4.21) is invariant with respect to the uncertain system, and condition (4.20) is fulfilled regardless of the concrete values of , , 1.5ākg 2ākg, .

The graphical plots in Figures 2 and 3 present the simulation results for a system belonging to the considered family, that corresponds to the concrete values , . We take four distinct initial conditions given by the combinations of , at . Figure 2 exhibits the evolution of and , as 2D plots (function values versus time). The dotted lines mark the bounds , as used in condition (4.20), with . Figure 3 offers a 3D representation of the exponentially contractive set defined by (4.21), with , as well as a state-space portrait, presenting the same four trajectories as in Figure 2.

As a general remark, it is worth mentioning that the problem considered above is far from triviality. If, instead of condition (4.20), we use the more general form
then Theorem 2.1 shows that (4.22) can be satisfied if and only if ; if this condition is fulfilled, then (4.22) is satisfied for

The request has a simple motivation even from the operation of the system. Assume that and , . Immediately after , the elongation of the spring will increase (since the damper with moves slower than the damper with ). Thus, at the first moments after , we will have and condition (4.22) is violated.

**(a)**

**(b)**

**(a)**

**(b)**

#### 5. Conclusions

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.

#### Acknowledgment

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