MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation84130310.1155/2009/841303841303Research ArticleMatrix Measures in the Qualitative Analysis of Parametric Uncertain SystemsPastravanuOctavianMatcovschiMihaela-HanakoKalmar-NagyTamasDepartment of Automatic Control and Applied InformaticsTechnical University “Gh. Asachi” of Iasi 700050 IasiRomaniatuiasi.ro20091609200920090712200808062009270720092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The paper considers parametric uncertain systems of the form x˙(t)=Mx(t),M,n×n, where is either a convex hull, or a positive cone of matrices, generated by the set of vertices 𝒱={M1, M2,,MK}n×n. Denote by μ the matrix measure corresponding to a vector norm . When is a convex hull, the condition μ(Mk)r<0, 1kK, 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 μ(Mk)0, 1kK, 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 μ. 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 Mn×n, the matrix norm induced by a generic vector norm      is defined by M=supyn,y0My/y=maxyn,y=  1My, and the corresponding matrix measure (also known as logarithmic norm) is given by μ    (M)=limθ0(I+θM-1)/θ ([1, page 41]). The spectrum of M is denoted by σ(M)={z  det(sI-M)=0} and λi(M)σ(M), i=1,,n, represent its eigenvalues. If Mn×n is a symmetrical matrix, M0 (M0) means that matrix M is negative (positive) definite. If Xn×m, then |X| represents the nonnegative matrix (for m2) or vector (for m=1) defined by taking the absolute values of the entries of X. If X,Yn×m, then “XY”, “X<Y” 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 . Paper  provided bounds for the computer solution and the accumulated truncation error corresponding to the backward Euler method. The work in  gave a characterization of vector norms as Lyapunov functions for time-invariant linear systems. The work in  developed sufficient conditions for the stability of neural networks. The work in  explored contractive invariant sets of time-invariant linear systems. The work in  formulated sufficient conditions for the stability of interval systems. The work in  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 .

The current paper considers parametric uncertain systems of the form

ẋ(t)=Mx(t),M,n×n, where is either a convex hull of matrices,

h={Mn×nM=k=1KγkMk,γk0,k=1Kγk=1}, or a positive cone of matrices,

c={Mn×nM=k=1KγkMk,γk0,M0}, generated by the set of vertex matrices 𝒱={M1,M2,,MK}n×n.

In investigating the evolution of system (1.1) we assume that matrix M 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 M is arbitrarily selected from , the trajectory initialized in x(t0)=x0, namely, x(t)=x(t;t0,x0)=eM(t-t0)  x0, is defined for all t+.

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 h is an interval matrix [7, 1013].

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

Definition 1.1.

(a) The uncertain system (1.1) is called stable if the equilibrium {0} is stable, that is, ε>0,t0+,δ=δ(ε)>0:x0δx(t;t0,x0)ε,tt0 for any solution of (1.1) corresponding to an M.

(b) The uncertain system (1.1) is called exponentially stable if the equilibrium {0} is exponentially stable, that is, r<0:  ε>0,t0+,δ=δ(ε)>0:x0δx(t;t0,x0)εer(t-t0),tt0 for any solution of (1.1) corresponding to an M.

Remark 1.2.

Using the connection between linear system stability and matrix eigenvalue location (e.g. ), we have the following characterizations.

The uncertain system (1.1) is stable if and only if M:σ(M){sRes0}  and if  Reλi0(M)=0  then  λi0(M)  is simple. In this case, the matrix set is said to be quasistable.

The uncertain system (1.1) is exponentially stable if and only if r<0:M:σ(M){sResr}. In this case, the matrix set is said to be Hurwitz stable.

Definition 1.3.

Consider the function V:n+,V(x)=x, and its right Dini derivative, calculated along a solution x(t) of (1.1): D+V(x(t))=limθ0V(x(t+θ))-V(x(t))θ,t+.

V is called a common strong Lyapunov function for the uncertain system (1.1), with the decreasing rate r<0, if for any solution x(t) of (1.1) corresponding to an M, we have t+:D+V(x(t))rV(x(t)).

V is called a common weak Lyapunov function for the uncertain system (1.1), if for any solution x(t) of (1.1) corresponding to an M, we have t+:D+V(x(t))0.

Definition 1.4.

The time-dependent set Xrε(t;t0)={xnxεer(t-t0)},t,t0+,tt0,ε>0,r0 is called invariant with respect to the uncertain system (1.1) if for any solution x(t) of (1.1) corresponding to an M, we have t0+,x0n,x0εx(t;t0,x0)εer(t-t0),tt0, meaning that any trajectory initiated inside the set Xrε(t0;t0) will never leave Xrε(t;t0).

A set of the form (1.12) with r<0 is said to be exponentially contractive.

A set of the form (1.12) with r=0 is said to be constant.

This paper proves that matrix-measure-based inequalities applied to the vertices Mk𝒱, 1kK, 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 Mk𝒱, 1kK, satisfy some supplementary hypotheses, a unique test matrix M* can be found such that a single inequality using μ    (M*) 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 Results2.1. Uncertain System Defined by a Convex Hull of MatricesTheorem 2.1.

Consider the uncertain system (1.1) with =h, the convex hull defined by (1.2), which, in the sequel, is referred to as the uncertain system (1.1) and (1.2). Let μ     be the matrix measure corresponding to the vector norm     , and r<0 a constant. The following statements are equivalent.

The vertices of the convex hull h fulfill the inequalities Mk𝒱:μ    (Mk)r.

The function V defined by (1.8) is a common strong Lyapunov function for the uncertain system (1.1) and (1.2) with the decreasing rate r.

For any ε>0, the exponentially contractive set Xrε(t;t0) 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.

Inequalities (2.1) are equivalent to M:μ    (M)r.

Inequality (1.10) is equivalent to t0+,x0n,x(t;t0,x0)er(t-t0)x0,tt0.

The matrix measure μ     fulfills the equality

Mn×n:μ    (M)=limθ0eMθ-1θ. 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 Mkh, for k=1,,K. Conversely, if (2.1) is true, then, from the convexity of the matrix measure, we get Mh,M=k=1KγkMk:μ    (M)k=1Kγkμ    (Mk)k=1K(γkr)=(k=1Kγk)r=r.

(R2) If inequality (2.3) is true, then, for any solution x(s) of (1.1) and (1.2) with initial condition set at s0=t0 as x(s0)=x0, we have D+V(x(s0))=limθ0x(s0+θ;s0,x0)-x0θ(limθ0erθ-1θ)x0=rV(x(s0)). Conversely, let t00 and x0n. If inequality (1.10) holds for x(t)=x(t;t0,x0), consider the differential equation ẏ(t)=ry(t) with the initial condition y(t0)=V(x(t0))=V(x0). Then, according to [14, Theorem 4.2.11], V(x(t))y(t)=er(t-t0)y(t0)=er(t-t0)V(x0), for all tt0.

(R3) For Mh and θ>0, we have eMθ=I+θM+θO(θ), with limθ0O(θ)=0. The triangle inequality I+θM-θO(θ)I+θM+θO(θ)I+θM+θO(θ) leads to (I+θM-1)/θ-O(θ)(eMθ-1)/θ(I+θM-1)/θ+O(θ). By taking limθ0, we finally obtain the equality (2.4).

Proof of Part II. (i)(ii) For any solution x(t) to (1.1) and (1.2) corresponding to an Mh, we get t+:D+V(x(t))=D+  x(t)=limθ0  x(t+θ)-x(t)θ=limθ0eMθx(t)-x(t)θ[limθ0eMθ-1θ]x(t)=(R3)μ    (M)x(t)(R1)rx(t).

(ii)(i) For all t0,θ+,eMθ=supx00eMθx0/x0=supx00x(t0+θ;t0,x0)/x0(R2)erθ. Hence, we have μ    (M)=(R3)limθ0(eMθ-1)/θlimθ0(erθ-1)/θ=r.

(ii)(iii) By contradiction, assume that there exists ε*>0 such that the exponentially contractive set Xrε*(t;t0) is not invariant with respect to the uncertain system (1.1) and (1.2). Then there exists a trajectory x̃(t) of (1.1) and (1.2) for which condition (1.13) is violated, meaning that we can find t*,t**+, t**>t*t0, so that x̃(t*)ε*er(t*-t0) and x̃(t**)>ε*er(t**-t0). This leads to er(t**-t*)x̃(t*)<x̃(t**), which contradicts (2.3). As a result, according to (R2), we contradict (ii).

(iii)(ii) For arbitrary tt0, by taking ε=x0 in (1.13), we get (2.3) that is equivalent to (1.10), via (R2).

Remark 2.2.

The equivalent conditions (i)–(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 δ(ε)=ε. Conversely, if (1.5) is true for a certain δ(ε)>ε, but not for δ(ε)=ε, 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)–(iii) of Theorem 2.1.

Remark 2.3.

Theorem 2.2 in  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 .

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 x2H=Hx2=(xTHTHx)1/2, xn, (where H is a nonsingular matrix) and the corresponding matrix measure μ    2H, we get the following well-known characterization of the quadratic stability of uncertain system (1.1) and (1.2),     𝒬0:for all  Mk𝒱:𝒬Mk+MkT𝒬0 (e.g., [16, page 213]). Indeed, according to ([1, page 41]) inequality (2.1) in Theorem 2.1 means λmax((1/2)HMkH-1+(1/2)(HMkH-1)T)=μ    2H(Mk)r<0, which is equivalent to the condition 𝒬Mk+MkT𝒬0, with 𝒬=HTH, for all Mk𝒱. 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 MatricesTheorem 2.5.

Consider the uncertain system (1.1) with =c, the positive cone defined by (1.3), which, in the sequel, is referred to as the uncertain system (1.1) and (1.3). Let μ     be the matrix measure corresponding to the vector norm     . The following statements are equivalent.

The vertices of the positive cone c fulfill the inequalities Mk𝒱:μ    (Mk)0.

The function V defined by (1.8) is a common weak Lyapunov function for the uncertain system (1.1) and (1.3).

For any ε>0, the constant set X0ε(t;t0) 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 r=0.

Remark 2.6.

The equivalent conditions (i)–(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 δ(ε)=ε. Conversely, if (1.4) is true for a certain δ(ε)>ε, but not for δ(ε)=ε, 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)–(iii) of Theorem 2.5.

Remark 2.7.

Theorem 2.3 in  claims that μ    (Mk)<0 for k=1,,K (i.e., condition (i) in Theorem 2.5 with strict inequalities) is sufficient for the Hurwitz stability of the positive cone of matrices c (1.3). However this is not true. Inequalities μ    (Mk)<0, k=1,,K, imply supMcμ(M)=0, which, together with for all  Mc:maxi  =  1,,n{Re{λi(M)}μ    (M), for example, , yield supMcmaxi=1,,n{Reλi(M)}0. Thus, condition (1.7) for the Hurwitz stability of the positive cone of matrices c defined by (1.3) may be not satisfied. Although the hypothesis of [7, Theorem  2.3] 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 h defined by (1.2),  does not discuss the necessity parts of the results.

3. Usage of a Single Test Matrix for Checking Condition (i) of Theorems <xref ref-type="statement" rid="thm1">2.1</xref> and <xref ref-type="statement" rid="thm2">2.5</xref>

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

μ    (Mk)r,k=1,,K, which involve all the vertices 𝒱={M1,M2,,MK}n×n 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 M*n×n such that the satisfaction of inequality

μ    (M*)r guarantees the fulfillment of (3.1).

Given a real matrix A=(aij)n×n, let us define its comparison matrix A̅=(a̅ij)n×n by

a̅ii=aii,i=1,,n;a̅ij=|aij|,i,j=1,,n,  ij.

Proposition 3.1.

(a) If the following hypotheses (H1), (H2) are satisfied, then inequality (3.2) is a sufficient condition for inequalities (3.1).

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 μ     is the corresponding matrix measure.

Matrix M*n×n satisfies the componentwise inequalities PkTMk¯PkM*,k=1,,K, for some permutation matrices Pkn×n, k=1,,K.

(b) If the above hypotheses (H1), (H2) are satisfied and there exists M**𝒱 such that μ    (M*)=μ    (M**), then inequality (3.2) is a necessary and sufficient condition for inequalities (3.1).

Proof.

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

If Pn×n is a permutation matrix, then An×n:μ    (PTAP)=μ    (A).

Given An×n, if the componentwise inequality PTA¯PM* is fulfilled for a permutation matrix Pn×n, then μ    (A)μ    (M*). 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 x*n×n, x*=1, such that A=Ax*. Since the considered vector norm      is permutation invariant, we have   x*=PTx*=1 for a permutation matrix Pn×n. This leads to A=Ax*=(AP)(PTx*)APPTx*=AP. Let us prove that the strict inequality A<AP does not hold. Assume, by contradiction, that A<AP. Then, there exists y*n×n, y*=1, such that APy*=AP>A. Hence, x**=Py*n×n with x**=Py*=y*=1 satisfies A<Ax**  that contradicts the definition of A. Consequently, A=AP.

Similarly we prove that AP=PTAP, yielding A=PTAP. Thus, we get I+θA=PT(I+θA)P=I+θPTAP and, consequently, (I+θA-1)/θ = (I+θPTAP-1)/θ. By taking limθ0 we obtain equality (3.5).

(R2) First, we exploit the componentwise matrix inequality (3.6). For small θ>0, we get 0I+θPTA̅PI+θM* that leads to the following componentwise vector inequality |(I+θPTA̅P)y||(I+θPTA̅P)|y|(I+θM*)|y, with yn.

Since      is a symmetric gauge function, it is also an absolute vector norm, and, equivalently, a monotonic vector norm [17, Theorem  5.5.10]. Consequently, (I+θPTA̅P)y(I+θPTA̅P)|y|(I+θM*)|y|, that implies (I+θPTA̅P)yI+θM*  |y|=I+θM*y. Thus, we get (I+θPTA̅P)=maxy=1(I+θPTA̅P)yI+θM* and ((I+θPTA̅P)-1)/θ(I+θM*-1)/θ. By taking limθ0 we obtain the inequality μ    (PTA̅P)μ    (M*).

Similarly, the componentwise matrix inequality AA¯ leads to μ    (PTAP)μ    (PTA̅P). Finally, we have μ    (A)=μ    (PTAP)μ    (PTA̅P)μ    (M*).

Proof of Part II. From (3.4), according to (R2) we get μ    (Mk)μ    (M*), k=1,,K, which together with (3.2) lead to (3.1).

(b) The sufficiency is proved by (a). The necessity is ensured by the equality μ    (M*)=μ    (M**) and the inequality μ    (M**)r (resulting from M**𝒱).

Proposition 3.2.

(a) If the following hypotheses (H1), (H2) are satisfied, then inequality (3.2) is a sufficient condition for inequalities (3.1).

The vector norm      is an absolute vector norm and μ     denotes the corresponding matrix measure.

Matrix M*n×n satisfies the componentwise inequalities Mk¯M*,k=1,,K.

(b) If the above hypotheses (H1), (H2) are satisfied and there exists M**𝒱 such that μ    (M*)=μ    (M**), 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 An×n satisfying the componentwise inequality AA¯M*, the monotonicity of      implies μ    (A)μ    (A̅)μ    (M*).

(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  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 I={Mn×nM-MM+}. This system is said to be CWEAS if there exist di>0, i=1,,n, and r<0 such that for all  t,t0+, tt0: -dixi(t0)di-dier(t-t0)xi(t)dier(t-t0), i=1,,n, where xi(t0), xi(t) denote the components of the initial condition x(t0) and of the corresponding solution x(t), respectively. According to  the interval system is CWEAS if and only if M̃drd, where d=[d1dn]Tn and the matrix M̃=(m̃ij) is built from the entries of the matrices M-=(mij-) and M+=(mij+) by m̃ii=mii+,i=1,,n, and m̃ij=max{|mij-|,|mij+|}, ij, i,j=1,,n. On the other hand, Theorem 2.1 characterizes CWEAS if applied for the vector norm xD=D  x=maxi=1,,n{xi/di}, with D=diag{1/d1,,1/dn}. At the same time, we can use Proposition 3.2(b), since (3.8) is satisfied with M*=M̃,     D is an absolute vector norm, and there exist M**=(mij**) belonging to the set of vertices of I such that mii**=m̃ii, i=1,,n, |mij**|=m̃ij, ij, i,j=1,,n, which implies μ    D(M̃)=maxi=1,,n{m̃ii+j=1,jinm̃ij(dj/di)}=maxi=1,,n{mii**+j=1,jin|mij**|(dj/di)}=μ    D(M**). Thus μ    D(M̃)r is a necessary and sufficient condition for the CWEAS of the interval system. Finally we notice that μ    D(M̃)r is equivalent to m̃ii(di/di)+j=1,jinm̃ij(dj/di)r, i=1,,n, showing that the CWEAS characterization M̃drd derived in  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 M*, several test matrices M1*,M2*,,ML*, with L being significantly smaller than K. Each M*,  =1,,L, will have to satisfy inequality (3.4) in Proposition 3.1 or inequality (3.8) in Proposition 3.2, for some vertex matrices in 𝒱𝒱, such that =1L𝒱=𝒱.

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 : 𝒱={M1,M2,M3}3×3,M1=[-3.4610.951-0.410-0.480-2.725-0.172252.903-2.504-1.014],M2=[-3.6900.136-1.144-0.648-2.437-0.2732.314-0.282-0.0734],M3=[-4.800-4.574-0.324-0.386-6.355-0.1893.8663.611-2.046]. Paper  shows that matrices M1, M2, and M3 have the following common quadratic Lyapunov function: VC𝒬LF(x)=xT𝒬x,𝒬=[12.6-5.705.70-5.707.50-2.405.70-2.403.12],𝒬0, since 𝒬Mk+MkT𝒬0, k=1,3¯.

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

h={M3×3M=k=13γkMk,γk0,k=13γk=1},Mk𝒱, and the positive cone of matrices c having the set of vertices 𝒱 (4.1), that is, c={M3×3M=k=13γkMk,γk0,M0},Mk𝒱. In 3 we define the vector norm x2H=Hx2,with  H=[3.2079-0.91731.2116-0.91732.5580-0.33931.2116-0.33931.2397], where H2=HTH=𝒬, in accordance with Remark 2.4, and consider the corresponding matrix measure μ    2H(M)=μ    2(HMH-1). For the vertex-matrices in 𝒱 (4.1) simple computations give μ    2H(M1)=-1.5915, μ    2H(M2)=-0.5271, and μ    2H(M3)=-0.1999.

Theorem 2.1 applied to the qualitative analysis of uncertain system (1.1) and (4.3) reveals the following properties.

The function V:3+,V(x)=x2H is a common strong Lyapunov function for the uncertain system (1.1) and (4.3) with the decreasing rate r=-0.1999.

Any exponentially contractive set X-0.1999ε(t;t0) of the form X-0.1999ε(t;t0)={x3x2Hεe-0.1999    (t-t0)},t,t0+,tt0,ε>0 is invariant with respect to the uncertain system (1.1) and (4.3).

Theorem 2.5 applied to the qualitative analysis of uncertain system (1.1) and (4.4) reveals the following properties.

The function V defined by (4.5) is a common weak Lyapunov function for the uncertain system (1.1) and (4.4).

Any constant set of the form X0ε(t;t0)={x3x2Hε},t,t0+,tt0,ε>0 is invariant with respect to the uncertain system (1.1) and (4.4).

Example 4.2.

Let us consider the interval matrix : I={M2×2M=M0+,||R},M0=[-3.81.60.6-4.2],R=0.17. Obviously, the set I can be regarded as a convex hull with K=24 vertices 𝒱={Mk=M0+k,k=1,,16},where  k=0.17    [±1±1±1±1]. The comparison matrices Mk¯, k=1,,16, of the vertices of 𝒱 (4.10) are built in accordance with (3.3) yielding Mk¯=Mk, k=1,,16 (since all Mk are essentially nonnegative matrices). The dominant vertex M*=M0+R=[-3.631.770.77-4.03]𝒱 satisfies inequalities (3.8), meaning that M* also satisfies inequalities (3.4) with all the permutation matrices equal to the unity matrix, Pk=I. Therefore we can apply both Propositions 3.1 and 3.2.

First, we apply Proposition 3.1 for the usual Hölder norms     p, with p{1,2,}, which are symmetric gauge functions. We calculate the matrix measures rp=μ    p(M*) for p{1,2,} and obtain the values r1=-2.26, r2=-2.5443, r=-1.86. Consequently all the vertex matrices in 𝒱 (4.10) satisfy the inequalities

μ    p(Mk)rp,k=1,,16,p{1,2,}. Thus, for the qualitative analysis of uncertain system (1.1) and (4.9) we can employ Theorem 2.1 that reveals the following properties.

The function V:2+,V(x)=xp,p{1,2,} is a common strong Lyapunov function for the uncertain system (1.1) and (4.9) with the decreasing rate rp, p{1,2,}.

Any exponentially contractive set of the form Xrpε(t;t0)={x2xpεerp(t-t0)},t,t0+,tt0,ε>0,p{1,2,} 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 p{1,2,}.

The dominant vertex M* is an essentially positive matrix (all off-diagonal entries are positive) and we can use the Perron Theorem, in accordance with . Denote by λmax(M*)=-2.6456 the Perron eigenvalue. From the left and right Perron eigenvectors of M* we can construct the diagonal matrices D1=diag{0.7882,1}, and D2=diag{0.6596,1}, D=diag{0.5562,1}, such that, for the vector norms defined in 2 by xpDp=Dpxp we have μ    pDp(M*)=λmax(M*), p{1,2,}. 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 r=λmax(M*)=-2.6456 proving that the vertex matrices in 𝒱 (4.10) satisfy the inequalities

μ    pDp(Mk)r,k=1,,16,p{1,2,}. Thus, for the qualitative analysis of uncertain system (1.1) and (4.9) we can employ Theorem 2.1 that reveals the following properties.

The function V:2+,V(x)=Dpxp,p{1,2,} is a common strong Lyapunov function for the uncertain system (1.1) and (4.9) with the decreasing rate r=-2.6456.

Any exponentially contractive set of the form X-2.6456ε(t;t0)={x2Dpxpεe-2.6456(t-t0)},t,t0+,tt0,ε>0,p{1,2,}

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 I={M2×2M-MM+},M+=[-3.631.770.77-4.03],M-=[abcd],a-3.63,|b|1.77,|c|0.77,d-4.03, which has the same dominant vertex M* as the original interval matrix I (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 m) in series with a damper (with viscous friction coefficient γ1) and a spring (with spring constant k) in series with a damper (with viscous friction coefficient γ2).

The system dynamics in form (1.1) is described by [Ḟ(t)v̇(t)]=[-kγ2k-1m-γ1m][F(t)v(t)], where the state variables are the spring force F(t) and the cart velocity v(t). We consider F>0 when the spring is elongated and F<0 when it is compressed as well as v>0 when the cart moves to the left and v<0 when it moves to the right.

The viscous friction coefficients have unique values, namely, γ1=2 Ns/mm, γ2=0.5 Ns/mm, whereas the cart mass and the spring constant have uncertain values belonging to the intervals 1.5kgm2kg, 3N/mmk4N/mm. Therefore we introduce the notation M(m;k)=[-2kk-1/m-2/m] that allows describing the set of system matrices ={M(m;k)1.5m2,3k4} as the convex hull of form (1.2) defined by the vertices

𝒱={M1,M2,M3,M4},whereM1=M(1.5;3),M2=M(1.5;4),      M3=M(2;3),      M4=M(2;4).

For the initial conditions F(t0)=F0, v(t0)=v0, we analyze the free response of the system. We want to see if there exists r<0 such that

F0,|F0|3N,v0,|v0|4mm/s|F(t)|3er(t-t0),|v(t)|4er(t-t0),tt0.

The problem can be approached in terms of Theorem 2.1, by considering in 2 the vector norm xD=Dx, with D=diag{1/3,1/4}, and the exponentially contractive set

Xr1(t;t0)={x2xDer(t-t0)},t,t0+,  tt0.

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 μ    D(Mk)=μ    (DMkD-1) for the vertices Mk, k=1,,4, in 𝒱 (4.19), we show that condition (2.1) in Theorem 2.1 is satisfied for r=-0.625  s-1. 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 m, k, 1.5 kg m 2 kg, 3N/mmk4N/mm.

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 m=1.75  kg, k=3.5  N/mm. We take four distinct initial conditions given by the combinations of F0=±3, v0=±4 at t0=0. Figure 2 exhibits the evolution of F(t) and v(t), as 2D plots (function values versus time). The dotted lines mark the bounds ±3ert, ±4ert as used in condition (4.20), with t0=0. Figure 3 offers a 3D representation of the exponentially contractive set Xr1(t;0) defined by (4.21), with t0=0, 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 F0,|F0|F*,v0,|v0|v*|F(t)|F*er*(t-t0),|v(t)|v*er*(t-t0),tt0, then Theorem 2.1 shows that (4.22) can be satisfied if and only if γ2=0.5<(F*/v*)<γ1=2; if this condition is fulfilled, then (4.22) is satisfied for r*=max[3(v*F*-2)k1,0.5(F*v*-2)].

The request γ2<γ1 has a simple motivation even from the operation of the system. Assume that γ1<γ2 and F0=F*, v0>0. Immediately after t0>0, the elongation of the spring will increase (since the damper with γ2 moves slower than the damper with γ1). Thus, at the first moments after t0>0, we will have F(t)>F* and condition (4.22) is violated.

The mechanical system used in Example 4.3.

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

(a) 3D representation of the exponentially contractive set Xr1(t;0). (b) State-space portrait. The same four trajectories as in Figure 2.

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.

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