# Matrix Measures in the Qualitative Analysis of Parametric Uncertain Systems

Authors: Octavian Pastravanu; Mihaela-Hanako Matcovschi
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

The paper considersparametric uncertain systems of the form x˙(t)=Mx(t),M∈ℳ,ℳ⊂ℝn×n,
where ℳ is either a convex hull, or a positive cone of matrices, generated by the
set of vertices 𝒱={M1, M2,…,MK}⊂ℝn×n. Denote by μ∥∥ the matrix measure corresponding to a vector norm ∥∥. When ℳ is a convex hull, the condition
μ∥∥(Mk)≤r<0,
1≤k≤K, 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 ℳ is a positive cone, the condition μ∥∥(Mk)≤0,
1≤k≤K, 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.


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## Body

## 1. Introduction

First let us present the notations and nomenclature used in our paper.For a square matrixM∈ℝn×n, the matrix norm induced by a generic vector norm∥∥ is defined by ∥M∥=supy∈ℝn,y≠0∥My∥/∥y∥=maxy∈ℝn,∥y∥=1∥My∥, and the corresponding matrix measure (also known as logarithmic norm) is given by μ∥∥(M)=limθ↓0(∥I+θM∥-1)/θ ([1, page 41]). The spectrum of M is denoted by σ(M)={z∈ℂ∣det(sI-M)=0} and λi(M)∈σ(M), i=1,…,n, represent its eigenvalues. If M∈ℝn×n is a symmetrical matrix, M≺0 (M≻0) means that matrix M is negative (positive) definite. If X∈ℝn×m, then |X| represents the nonnegative matrix (for m≥2) or vector (for m=1) defined by taking the absolute values of the entries of X. If X,Y∈ℝn×m, then “X≤Y”, “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 [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 considersparametric uncertain systems of the form(1.1)ẋ(t)=Mx(t),M∈ℳ,ℳ⊂ℝn×n,
where ℳ is either a convex hullof matrices,(1.2)ℳh={M∈ℝn×n∣M=∑k=1KγkMk,γk≥0,∑k=1Kγk=1},
or a positive cone of matrices,(1.3)ℳc={M∈ℝn×n∣M=∑k=1KγkMk,γk≥0,M≠0},
generated by the set of vertex matrices 𝒱={M1,M2,…,MK}⊂ℝn×n.In investigating the evolution of system (1.1) we assume that matrix M is fixed, but arbitrarily taken from the matrix set ℳ 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 ℳ, the trajectory initialized in x(t0)=x0, namely, x(t)=x(t;t0,x0)=eM(t-t0)x0, is defined for all t∈ℝ+.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 ℳh is an interval matrix [7, 10–13].For system (1.1) we define the following properties, in accordance with the definitions presented in [14–16] for a dynamical system.Definition 1.1.
(a) The uncertain system (1.1) is called stable if the equilibrium {0} is stable, that is,
(1.4)∀ε>0,∀t0∈ℝ+,∃δ=δ(ε)>0:∥x0∥≤δ⇒∥x(t;t0,x0)∥≤ε,∀t≥t0
for any solution of (1.1) corresponding to an M∈ℳ.
(b) The uncertain system (1.1) is called exponentially stable if the equilibrium {0} is exponentially stable, that is,
(1.5)∃r<0:∀ε>0,∀t0∈ℝ+,∃δ=δ(ε)>0:∥x0∥≤δ⇒∥x(t;t0,x0)∥≤εer(t-t0),t≥t0
for any solution of (1.1) corresponding to an M∈ℳ.Remark 1.2.
Using the connection between linear system stability and matrix eigenvalue location (e.g. [15]), we have the following characterizations.(a)
The uncertain system (1.1) is stable if and only if
(1.6)∀M∈ℳ:σ(M)⊂{s∈ℂ∣Res≤0}and ifReλi0(M)=0thenλi0(M)is simple.
In this case, the matrix set ℳ is said to be quasistable.(b)
The uncertain system (1.1) is exponentially stable if and only if
(1.7)∃r<0:∀M∈ℳ:σ(M)⊂{s∈ℂ∣Res≤r}.
In this case, the matrix set ℳ is said to be Hurwitz stable.Definition 1.3.
Consider the function(1.8)V:ℝn→ℝ+,V(x)=∥x∥,
and its right Dini derivative, calculated along a solution x(t) of (1.1):
(1.9)D+V(x(t))=limθ↓0V(x(t+θ))-V(x(t))θ,t∈ℝ+.(a)
V is called a common strong Lyapunov function for the uncertain system (1.1), with the decreasing rater<0, if for any solution x(t) of (1.1) corresponding to an M∈ℳ, we have
(1.10)∀t∈ℝ+:D+V(x(t))≤rV(x(t)).(b)
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∈ℳ, we have
(1.11)∀t∈ℝ+:D+V(x(t))≤0.Definition 1.4.
The time-dependent set(1.12)Xrε(t;t0)={x∈ℝn∣∥x∥≤εer(t-t0)},t,t0∈ℝ+,t≥t0,ε>0,r≤0
is called invariantwith respect to the uncertain system (1.1) if for any solution x(t) of (1.1) corresponding to an M∈ℳ, we have
(1.13)∀t0∈ℝ+,∀x0∈ℝn,∥x0∥≤ε⇒∥x(t;t0,x0)∥≤εer(t-t0),∀t≥t0,
meaning that any trajectory initiated inside the set Xrε(t0;t0) will never leave Xrε(t;t0).(a)
A set of the form (1.12) with r<0 is said to be exponentially  contractive.(b)
A set of the form (1.12) with r=0 is said to beconstant.This paper proves that matrix-measure-based inequalities applied to the verticesMk∈𝒱, 1≤k≤K, 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 ℳ 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 Mk∈𝒱, 1≤k≤K, 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 Results

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

Theorem 2.1.
Consider the uncertain system (1.1) with ℳ=ℳh, 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.(i)
The vertices of the convex hullℳh fulfill the inequalities
(2.1)∀Mk∈𝒱:μ∥∥(Mk)≤r.(ii)
The functionV defined by (1.8) is a common strong Lyapunov function for the uncertain system (1.1) and (1.2) with the decreasing rate r.(iii)
For anyε>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.(R1)
Inequalities (2.1) are equivalent to
(2.2)∀M∈ℳ:μ∥∥(M)≤r.(R2)
Inequality (1.10) is equivalent to
(2.3)∀t0∈ℝ+,∀x0∈ℝn,∥x(t;t0,x0)∥≤er(t-t0)∥x0∥,∀t≥t0.(R3)
The matrix measureμ∥∥ fulfills the equality(2.4)∀M∈ℝn×n:μ∥∥(M)=limθ↓0∥eMθ∥-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 Mk∈ℳh, for k=1,…,K. Conversely, if (2.1) is true, then, from the convexity of the matrix measure, we get
(2.5)∀M∈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=t≥0 as x(s0)=x0, we have
(2.6)D+V(x(s0))=limθ↓0∥x(s0+θ;s0,x0)∥-∥x0∥θ≤(limθ↓0erθ-1θ)∥x0∥=rV(x(s0)).
Conversely, let t0≥0 and x0∈ℝn. 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 t≥t0.
(R3) ForM∈ℳh and θ>0, we have eMθ=I+θM+θO(θ), with limθ↓0O(θ)=0. The triangle inequality ∥I+θM∥-θ∥O(θ)∥≤∥I+θM+θO(θ)∥≤∥I+θM∥+θ∥O(θ)∥ leads to (∥I+θM∥-1)/θ-∥O(θ)∥≤(∥eMθ∥-1)/θ≤(∥I+θM∥-1)/θ+∥O(θ)∥. By taking limθ↓0, 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 M∈ℳh, we get
(2.7)∀t∈ℝ+:D+V(x(t))=D+∥x(t)∥=limθ↓0∥x(t+θ)∥-∥x(t)∥θ=limθ↓0∥eMθx(t)∥-∥x(t)∥θ≤[limθ↓0∥eMθ∥-1θ]∥x(t)∥=(R3)μ∥∥(M)∥x(t)∥≤(R1)r∥x(t)∥.
(ii)⇒(i) For all t0,θ∈ℝ+,∥eMθ∥=supx0≠0∥eMθx0∥/∥x0∥=supx0≠0∥x(t0+θ;t0,x0)∥/x0≤(R2)erθ. Hence, we have μ∥∥(M)=(R3)limθ↓0(∥eMθ∥-1)/θ≤limθ↓0(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**∈ℝ+, 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 t≥t0, 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 Theorem2.1 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 [7] shows that condition (i) in Theorem 2.1 is sufficientfor 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.
Theorem2.1 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 ∥x∥2H=∥Hx∥2=(xTHTHx)1/2, x∈ℝn, (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), ∃𝒬≻0:for allMk∈𝒱:𝒬Mk+MkT𝒬≺0 (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+MkT𝒬≺0, with 𝒬=HTH, for all Mk∈𝒱. 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 ℳ=ℳc, 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.(i)
The vertices of the positive coneℳc  fulfill the inequalities
(2.8)∀Mk∈𝒱:μ∥∥(Mk)≤0.(ii)
The functionV defined by (1.8) is a common weak Lyapunov function for the uncertain system (1.1) and (1.3).(iii)
For anyε>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 Theorem2.1 where we take r=0.Remark 2.6.
The equivalent conditions (i)–(iii) of Theorem2.5 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 [7] claims that μ∥∥(Mk)<0 for k=1,…,K (i.e., condition (i) in Theorem 2.5 with strict inequalities) is sufficientfor the Hurwitz stability of the positive cone of matrices ℳc (1.3). However this is not true. Inequalities μ∥∥(Mk)<0, k=1,…,K, imply supM∈ℳcμ(M)=0, which, together with for allM∈ℳc:maxi=1,…,n{Re{λi(M)}≤μ∥∥(M), for example, [1], yield supM∈ℳcmaxi=1,…,n{Reλi(M)}≤0. Thus, condition (1.7) for the Hurwitz stability of the positive cone of matrices ℳc 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 ℳh defined by (1.2), [7] does not discuss the necessity parts of the results.


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

Theorem 2.1.
Consider the uncertain system (1.1) with ℳ=ℳh, 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.(i)
The vertices of the convex hullℳh fulfill the inequalities
(2.1)∀Mk∈𝒱:μ∥∥(Mk)≤r.(ii)
The functionV defined by (1.8) is a common strong Lyapunov function for the uncertain system (1.1) and (1.2) with the decreasing rate r.(iii)
For anyε>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.(R1)
Inequalities (2.1) are equivalent to
(2.2)∀M∈ℳ:μ∥∥(M)≤r.(R2)
Inequality (1.10) is equivalent to
(2.3)∀t0∈ℝ+,∀x0∈ℝn,∥x(t;t0,x0)∥≤er(t-t0)∥x0∥,∀t≥t0.(R3)
The matrix measureμ∥∥ fulfills the equality(2.4)∀M∈ℝn×n:μ∥∥(M)=limθ↓0∥eMθ∥-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 Mk∈ℳh, for k=1,…,K. Conversely, if (2.1) is true, then, from the convexity of the matrix measure, we get
(2.5)∀M∈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=t≥0 as x(s0)=x0, we have
(2.6)D+V(x(s0))=limθ↓0∥x(s0+θ;s0,x0)∥-∥x0∥θ≤(limθ↓0erθ-1θ)∥x0∥=rV(x(s0)).
Conversely, let t0≥0 and x0∈ℝn. 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 t≥t0.
(R3) ForM∈ℳh and θ>0, we have eMθ=I+θM+θO(θ), with limθ↓0O(θ)=0. The triangle inequality ∥I+θM∥-θ∥O(θ)∥≤∥I+θM+θO(θ)∥≤∥I+θM∥+θ∥O(θ)∥ leads to (∥I+θM∥-1)/θ-∥O(θ)∥≤(∥eMθ∥-1)/θ≤(∥I+θM∥-1)/θ+∥O(θ)∥. By taking limθ↓0, 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 M∈ℳh, we get
(2.7)∀t∈ℝ+:D+V(x(t))=D+∥x(t)∥=limθ↓0∥x(t+θ)∥-∥x(t)∥θ=limθ↓0∥eMθx(t)∥-∥x(t)∥θ≤[limθ↓0∥eMθ∥-1θ]∥x(t)∥=(R3)μ∥∥(M)∥x(t)∥≤(R1)r∥x(t)∥.
(ii)⇒(i) For all t0,θ∈ℝ+,∥eMθ∥=supx0≠0∥eMθx0∥/∥x0∥=supx0≠0∥x(t0+θ;t0,x0)∥/x0≤(R2)erθ. Hence, we have μ∥∥(M)=(R3)limθ↓0(∥eMθ∥-1)/θ≤limθ↓0(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**∈ℝ+, 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 t≥t0, 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 Theorem2.1 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 [7] shows that condition (i) in Theorem 2.1 is sufficientfor 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.
Theorem2.1 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 ∥x∥2H=∥Hx∥2=(xTHTHx)1/2, x∈ℝn, (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), ∃𝒬≻0:for allMk∈𝒱:𝒬Mk+MkT𝒬≺0 (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+MkT𝒬≺0, with 𝒬=HTH, for all Mk∈𝒱. 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 ℳ=ℳc, 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.(i)
The vertices of the positive coneℳc  fulfill the inequalities
(2.8)∀Mk∈𝒱:μ∥∥(Mk)≤0.(ii)
The functionV defined by (1.8) is a common weak Lyapunov function for the uncertain system (1.1) and (1.3).(iii)
For anyε>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 Theorem2.1 where we take r=0.Remark 2.6.
The equivalent conditions (i)–(iii) of Theorem2.5 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 [7] claims that μ∥∥(Mk)<0 for k=1,…,K (i.e., condition (i) in Theorem 2.5 with strict inequalities) is sufficientfor the Hurwitz stability of the positive cone of matrices ℳc (1.3). However this is not true. Inequalities μ∥∥(Mk)<0, k=1,…,K, imply supM∈ℳcμ(M)=0, which, together with for allM∈ℳc:maxi=1,…,n{Re{λi(M)}≤μ∥∥(M), for example, [1], yield supM∈ℳcmaxi=1,…,n{Reλi(M)}≤0. Thus, condition (1.7) for the Hurwitz stability of the positive cone of matrices ℳc 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 ℳh 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 Theorems2.1 and 2.5

Condition (i) of both Theorems2.1 and 2.5 represents inequalities of the form(3.1)μ∥∥(Mk)≤r,k=1,…,K,
which involve all the vertices 𝒱={M1,M2,…,MK}⊂ℝn×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*∈ℝn×n such that the satisfaction of inequality(3.2)μ∥∥(M*)≤r
guarantees the fulfillment of (3.1).Given a real matrixA=(aij)∈ℝn×n, let us define its comparison matrixA̅=(a̅ij)∈ℝn×n by(3.3)a̅ii=aii,i=1,…,n;a̅ij=|aij|,i,j=1,…,n,i≠j.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 μ∥∥ is the corresponding matrix measure.(H2)
MatrixM*∈ℝn×n satisfies the componentwise inequalities
(3.4)PkTMk¯Pk≤M*,k=1,…,K,
for some permutation matrices Pk∈ℝn×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.
(a)
We organize the proof in two parts. Part I proves the following results.
(R1)
IfP∈ℝn×n is a permutation matrix, then
(3.5)∀A∈ℝn×n:μ∥∥(PTAP)=μ∥∥(A).(R2)
GivenA∈ℝn×n, if the componentwise inequality
(3.6)PTA¯P≤M*
is fulfilled for a permutation matrix P∈ℝn×n, then
(3.7)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*∈ℝn×n, ∥x*∥=1, such that ∥A∥=∥Ax*∥. Since the considered vector norm ∥∥ is permutation invariant, we have ∥x*∥=∥PTx*∥=1 for a permutation matrix P∈ℝn×n. This leads to ∥A∥=∥Ax*∥=∥(AP)(PTx*)∥≤∥AP∥∥PTx*∥=∥AP∥. Let us prove that the strict inequality ∥A∥<∥AP∥ does not hold. Assume, by contradiction, that ∥A∥<∥AP∥. Then, there exists y*∈ℝn×n, ∥y*∥=1, such that ∥APy*∥=∥AP∥>∥A∥. Hence, x**=Py*∈ℝn×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 limθ↓0 we obtain equality (3.5).
(R2) First, we exploit the componentwise matrix inequality (3.6). For small θ>0, we get 0≤I+θPTA̅P≤I+θM* that leads to the following componentwise vector inequality |(I+θPTA̅P)y|≤|(I+θPTA̅P)|y∥≤|(I+θM*)|y∥, with y∈ℝn.
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)y∥≤∥I+θM*∥∥|y|∥=∥I+θM*∥∥y∥. Thus, we get ∥(I+θPTA̅P)∥=max∥y∥=1∥(I+θPTA̅P)y∥≤∥I+θM*∥ and (∥(I+θPTA̅P)∥-1)/θ≤(∥I+θM*∥-1)/θ. By taking limθ↓0 we obtain the inequality μ∥∥(PTA̅P)≤μ∥∥(M*).
Similarly, the componentwise matrix inequalityA≤A¯ 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 equalityμ∥∥(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).(H1)
The vector norm∥∥ is an absolute vector norm  and μ∥∥ denotes the corresponding matrix measure.(H2)
MatrixM*∈ℝn×n satisfies the componentwise inequalities
(3.8)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 Proposition3.1 to show that, for a given A∈ℝn×n satisfying the componentwise inequality A≤A¯≤M*, the monotonicity of ∥∥ implies μ∥∥(A)≤μ∥∥(A̅)≤μ∥∥(M*).
(b) The proof of necessity is identical to Theorem2.1.Remark 3.3.
Proposition3.2 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 ℳI={M∈ℝn×n∣M-≤M≤M+}. This system is said to be CWEAS if there exist di>0,i=1,…,n, and r<0 such that for allt,t0∈ℝ+,t≥t0:-di≤xi(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 [12] the interval system is CWEAS if and only if M̃d≤rd, where d=[d1⋯dn]T∈ℝn 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+|}, i≠j,i,j=1,…,n. On the other hand, Theorem 2.1 characterizes CWEAS if applied for the vector norm ∥x∥∞D=∥Dx∥∞=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 ℳI such that mii**=m̃ii, i=1,…,n, |mij**|=m̃ij, i≠j, i,j=1,…,n, which implies μ∥∥∞D(M̃)=maxi=1,…,n{m̃ii+∑j=1,j≠inm̃ij(dj/di)}=maxi=1,…,n{mii**+∑j=1,j≠in|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,j≠inm̃ij(dj/di)≤r, i=1,…,n, showing that the CWEAS characterization M̃d≤rd derived in [12] for interval systems is incorporated into the current approach to parametric uncertain systems.Remark 3.4.
Propositions3.1 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 Mℓ*,ℓ=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 𝒱ℓ⊆𝒱, such that ⋃ℓ=1L𝒱ℓ=𝒱.


## 4. Illustrative Examples

This section illustrates the applicability of our results to three examples. Examples4.1 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]:
(4.1)𝒱={M1,M2,M3}⊂ℝ3×3,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 [18] shows that matrices M1,M2, and M3 have the following common quadratic Lyapunov function:
(4.2)VC𝒬LF(x)=xT𝒬x,𝒬=[12.6-5.705.70-5.707.50-2.405.70-2.403.12],𝒬≻0,
since 𝒬Mk+MkT𝒬≺0, k=1,3¯.
We define the convex hull of matricesℳh having the set of vertices 𝒱 (4.1), that is,
(4.3)ℳh={M∈ℝ3×3∣M=∑k=13γkMk,γk≥0,∑k=13γk=1},Mk∈𝒱,
and the positive cone of matrices ℳc having the set of vertices 𝒱 (4.1), that is,
(4.4)ℳc={M∈ℝ3×3∣M=∑k=13γkMk,γk≥0,M≠0},Mk∈𝒱.
In ℝ3 we define the vector norm
(4.5)∥x∥2H=∥Hx∥2,withH=[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.(i)
Theorem2.1 applied to the qualitative analysis of uncertain system (1.1) and (4.3) reveals the following properties.(a)
The function(4.6)V:ℝ3→ℝ+,V(x)=∥x∥2H
is a common strong Lyapunov function for the uncertain system (1.1) and (4.3) with the decreasing rate r=-0.1999.(b)
Any exponentially contractive setX-0.1999ε(t;t0) of the form
(4.7)X-0.1999ε(t;t0)={x∈ℝ3∣∥x∥2H≤εe-0.1999(t-t0)},t,t0∈ℝ+,t≥t0,ε>0
is invariant with respect to the uncertain system (1.1) and (4.3).
(ii)
Theorem2.5 applied to the qualitative analysis of uncertain system (1.1) and (4.4) reveals the following properties.
(a)
The functionV 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(4.8)X0ε(t;t0)={x∈ℝ3∣∥x∥2H≤ε},t,t0∈ℝ+,t≥t0,ε>0
is invariant with respect to the uncertain system (1.1) and (4.4).Example 4.2.
Let us consider the interval matrix [19]:
(4.9)ℳI={M∈ℝ2×2∣M=M0+∆,|∆|≤R},M0=[-3.81.60.6-4.2],R=0.17[1111].
Obviously, the set ℳI can be regarded as a convex hull with K=24 vertices
(4.10)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 Proposition3.1 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
(4.11)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.(a)
The function(4.12)V:ℝ2→ℝ+,V(x)=∥x∥p,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,∞}.(b)
Any exponentially contractive set of the form(4.13)Xrpε(t;t0)={x∈ℝ2∣∥x∥p≤εerp(t-t0)},t,t0∈ℝ+,t≥t0,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 vertexM* is an essentially positive matrix (all off-diagonal entries are positive) and we can use the Perron Theorem, in accordance with [20]. 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 ℝ2 by ∥x∥pDp=∥Dpx∥p 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
(4.14)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.(a)
The function(4.15)V:ℝ2→ℝ+,V(x)=∥Dpx∥p,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.(b)
Any exponentially contractive set of the form(4.16)X-2.6456ε(t;t0)={x∈ℝ2∣∥Dpx∥p≤εe-2.6456(t-t0)},t,t0∈ℝ+,t≥t0,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
(4.17)ℳI′={M∈ℝ2×2∣M-≤M≤M+},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 ℳI (4.9).Example 4.3.
Let us consider the translation of the mechanical system in Figure1. 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
(4.18)[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 1.5kg≤m≤2kg, 3N/mm≤k≤4N/mm. Therefore we introduce the notation M(m;k)=[-2kk-1/m-2/m] that allows describing the set of system matrices ℳ={M(m;k)∣1.5≤m≤2,3≤k≤4} as the convex hull of form (1.2) defined by the vertices
(4.19)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 conditionsF(t0)=F0,v(t0)=v0, we analyze the free response of the system. We want to see if there exists r<0 such that
(4.20)∀F0,|F0|≤3N,∀v0,|v0|≤4mm/s⇒|F(t)|≤3er(t-t0),|v(t)|≤4er(t-t0),∀t≥t0.
The problem can be approached in terms of Theorem2.1, by considering in ℝ2 the vector norm ∥x∥∞D=∥Dx∥∞, with D=diag{1/3,1/4}, and the exponentially contractive set
(4.21)Xr1(t;t0)={x∈ℝ2∣∥x∥∞D≤er(t-t0)},t,t0∈ℝ+,t≥t0.
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=-0.625s-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/mm≤k≤4N/mm.
The graphical plots in Figures2 and 3 present the simulation results for a system belonging to the considered family, that corresponds to the concrete values m=1.75kg, k=3.5N/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
(4.22)∀F0,|F0|≤F*,∀v0,|v0|≤v*⇒|F(t)|≤F*er*(t-t0),|v(t)|≤v*er*(t-t0),∀t≥t0,
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
(4.23)r*=max[3(v*F*-2)k1,0.5(F*v*-2)].
The requestγ2<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.Figure 1
The mechanical system used in Example4.3.Time-evolution of the state-space trajectories corresponding to four initial conditions.
(a)(b)(a) 3D representation of the exponentially contractive setXr1(t;0). (b) State-space portrait. The same four trajectories as in Figure 2.
(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.



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