Matrix Measures in the Qualitative Analysis of Parametric Uncertain Systems

Abstract

The paper considers parametric uncertain systems of the form 𝑥̇(𝑡)=𝑀𝑥(𝑡),𝑀∈ℳ,ℳ⊂ℝ^𝑛×𝑛, where ℳ is either a convex hull, or a positive cone of matrices, generated by the set of vertices 𝒱={𝑀₁,𝑀₂,…,𝑀_𝐾}⊂ℝ^𝑛×𝑛. Denote by 𝜇_‖‖ the matrix measure corresponding to a vector norm ‖‖. When ℳ is a convex hull, the condition 𝜇_‖‖(𝑀_𝑘)≤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_‖‖(𝑀_𝑘)≤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.

1. Introduction

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

For a square matrix 𝑀∈ℝ^𝑛×n, the matrix norm induced by a generic vector norm ‖‖ is defined by ‖𝑀‖=sup_{𝑦∈ℝ𝑛,y≠0}‖𝑀𝑦‖/‖𝑦‖=max_𝑦∈ℝ^𝑛_{,‖𝑦‖=1}‖𝑀𝑦‖, and the corresponding matrix measure (also known as logarithmic norm) is given by 𝜇_‖‖(𝑀)=lim_𝜃↓0(‖𝐼+𝜃𝑀‖−1)/𝜃 ([1, page 41]). The spectrum of 𝑀 is denoted by 𝜎(𝑀)={z∈ℂ∣det(𝑠𝐼−𝑀)=0} and 𝜆_𝑖(𝑀)∈𝜎(𝑀), 𝑖=1,…,𝑛, represent its eigenvalues. If 𝑀∈ℝ^𝑛×𝑛 is a symmetrical matrix, 𝑀≺0 (𝑀≻0) means that matrix 𝑀 is negative (positive) definite. If 𝑋∈ℝ^𝑛×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∈ℝ^𝑛×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 considers parametric uncertain systems of the form

𝑥̇(𝑡)=𝑀𝑥(𝑡),𝑀∈ℳ,ℳ⊂ℝ^𝑛×𝑛, (1.1) where ℳ is either a convex hull of matrices,

ℳ_ℎ={𝑀∈ℝ^𝑛×𝑛∣𝑀=^𝐾_𝑘=1𝛾_𝑘𝑀_𝑘,𝛾_𝑘≥0,^𝐾_𝑘=1𝛾_𝑘=1}, (1.2) or a positive cone of matrices,

ℳ_𝑐={𝑀∈ℝ^𝑛×𝑛∣𝑀=^𝐾_𝑘=1𝛾_𝑘𝑀_𝑘,𝛾_𝑘≥0,𝑀≠0}, (1.3) generated by the set of vertex matrices 𝒱={𝑀₁,𝑀₂,…,𝑀_𝐾}⊂ℝ^𝑛×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)=𝑒^{𝑀(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 ℳ_ℎ 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, ∀𝜀>0,∀𝑡₀∈ℝ₊,∃𝛿=𝛿(𝜀)>0∶‖𝑥₀‖≤𝛿⇒‖𝑥(𝑡;𝑡₀,𝑥₀)‖≤𝜀,∀𝑡≥𝑡₀ (1.4) 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, ∃𝑟<0∶∀𝜀>0,∀𝑡₀∈ℝ₊,∃𝛿=𝛿(𝜀)>0∶‖𝑥₀‖≤𝛿⇒‖𝑥(𝑡;𝑡₀,𝑥₀)‖≤𝜀𝑒^{𝑟(𝑡−𝑡0)},𝑡≥𝑡₀ (1.5) 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 ∀𝑀∈ℳ∶𝜎(𝑀)⊂{𝑠∈ℂ∣Re𝑠≤0}andifRe𝜆_𝑖0(𝑀)=0then𝜆_𝑖0(𝑀)issimple. (1.6) 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 ∃𝑟<0∶∀𝑀∈ℳ∶𝜎(𝑀)⊂{𝑠∈ℂ∣Re𝑠≤𝑟}. (1.7) In this case, the matrix set ℳ is said to be Hurwitz stable.

Definition 1.3. Consider the function 𝑉∶ℝ^𝑛→ℝ₊,𝑉(𝑥)=‖𝑥‖, (1.8) and its right Dini derivative, calculated along a solution 𝑥(𝑡) of (1.1): D⁺𝑉(𝑥(𝑡))=lim_𝜃↓0𝑉(𝑥(𝑡+𝜃))−𝑉(𝑥(𝑡))𝜃,𝑡∈ℝ₊. (1.9)(a)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∈ℳ, we have ∀𝑡∈ℝ₊∶D⁺𝑉(𝑥(𝑡))≤𝑟𝑉(𝑥(𝑡)). (1.10)(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 ∀𝑡∈ℝ₊∶D⁺𝑉(𝑥(𝑡))≤0. (1.11)

Definition 1.4. The time-dependent set 𝑋^𝜀_𝑟(𝑡;𝑡₀)={𝑥∈ℝ^𝑛∣‖𝑥‖≤𝜀𝑒^{𝑟(𝑡−𝑡0)}},𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀,𝜀>0,𝑟≤0 (1.12) is called invariant with respect to the uncertain system (1.1) if for any solution 𝑥(𝑡) of (1.1) corresponding to an 𝑀∈ℳ, we have ∀𝑡₀∈ℝ₊,∀𝑥₀∈ℝ^𝑛,‖𝑥₀‖≤𝜀⇒‖𝑥(𝑡;𝑡₀,𝑥₀)‖≤𝜀𝑒^{𝑟(𝑡−𝑡0)},∀𝑡≥𝑡₀, (1.13) meaning that any trajectory initiated inside the set 𝑋^𝜀_𝑟(𝑡₀;𝑡₀) will never leave 𝑋^𝜀_𝑟(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 be constant.

This paper proves that matrix-measure-based inequalities applied to the vertices 𝑀_𝑘∈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 𝑀_𝑘∈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=ℳ_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.(i) The vertices of the convex hull ℳ_ℎ fulfill the inequalities ∀𝑀_𝑘∈𝒱∶𝜇_‖‖(𝑀_𝑘)≤𝑟. (2.1)(ii) 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.(iii) For any 𝜀>0, the exponentially contractive set 𝑋^𝜀_𝑟(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 ∀𝑀∈ℳ∶𝜇_‖‖(𝑀)≤𝑟. (2.2)(R2) Inequality (1.10) is equivalent to ∀𝑡₀∈ℝ₊,∀𝑥₀∈ℝ^𝑛,‖𝑥(𝑡;𝑡₀,𝑥₀)‖≤𝑒^{𝑟(𝑡−𝑡0)}‖𝑥₀‖,∀𝑡≥𝑡₀. (2.3)(R3)The matrix measure 𝜇_‖‖ fulfills the equality∀𝑀∈ℝ^𝑛×𝑛∶𝜇_‖‖(𝑀)=lim_𝜃↓0‖𝑒^𝑀𝜃‖−1𝜃. (2.4) 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 𝑀_𝑘∈ℳ_ℎ, for 𝑘=1,…,𝐾. Conversely, if (2.1) is true, then, from the convexity of the matrix measure, we get ∀𝑀∈ℳ_ℎ,𝑀=^𝐾_𝑘=1𝛾_𝑘𝑀_𝑘∶𝜇_‖‖(𝑀)≤^𝐾_𝑘=1𝛾_𝑘𝜇_‖‖(𝑀_𝑘)≤^𝐾_𝑘=1(𝛾_𝑘𝑟)=(^𝐾_𝑘=1𝛾_𝑘)𝑟=𝑟. (2.5)
(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 D⁺𝑉(𝑥(𝑠₀))=lim_𝜃↓0‖𝑥(𝑠₀+𝜃;𝑠₀,𝑥₀)‖−‖𝑥₀‖𝜃≤(lim_𝜃↓0𝑒^𝑟𝜃−1𝜃)‖𝑥₀‖=𝑟𝑉(𝑥(𝑠₀)). (2.6) Conversely, let 𝑡₀≥0 and 𝑥₀∈ℝ^𝑛. If inequality (1.10) holds for 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀), consider the differential equation 𝑦(̇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)=𝑒^{𝑟(𝑡−𝑡0)}𝑦(𝑡₀)=𝑒^{𝑟(𝑡−𝑡0)}𝑉(𝑥₀), for all 𝑡≥𝑡₀.
(R3) For 𝑀∈ℳ_ℎ and 𝜃>0, we have 𝑒^𝑀𝜃=𝐼+𝜃𝑀+𝜃𝑂(𝜃), with lim_𝜃↓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)‖≤(‖𝑒^𝑀𝜃‖−1)/𝜃≤(‖𝐼+𝜃𝑀‖−1)/𝜃+‖𝑂(𝜃)‖. By taking lim_𝜃↓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∈ℳ_ℎ, we get ∀𝑡∈ℝ₊∶D⁺𝑉(𝑥(𝑡))=D⁺‖𝑥(𝑡)‖=lim_𝜃↓0‖𝑥(𝑡+𝜃)‖−‖𝑥(𝑡)‖𝜃=lim_𝜃↓0‖𝑒^𝑀𝜃𝑥(𝑡)‖−‖𝑥(𝑡)‖𝜃≤[lim_𝜃↓0‖𝑒^𝑀𝜃‖−1𝜃]‖𝑥(𝑡)‖=^(R3)𝜇_‖‖(𝑀)‖𝑥(𝑡)‖≤^(R1)𝑟‖𝑥(𝑡)‖. (2.7)
(ii)⇒(i) For all 𝑡₀,𝜃∈ℝ₊,‖𝑒^𝑀𝜃‖=sup_𝑥0≠0‖𝑒^𝑀𝜃𝑥₀‖/‖𝑥₀‖=sup_𝑥0≠0‖𝑥(𝑡₀+𝜃;𝑡₀,𝑥₀)‖/𝑥₀⁽≤^R2)𝑒^𝑟𝜃. Hence, we have 𝜇_‖‖(𝑀)=^(R3)lim_𝜃↓0(‖𝑒^𝑀𝜃‖−1)/𝜃≤lim_𝜃↓0(𝑒^𝑟𝜃−1)/𝜃=𝑟.
(ii)⇒(iii) By contradiction, assume that there exists 𝜀^∗>0 such that the exponentially contractive set 𝑋^𝜀∗_𝑟(𝑡;𝑡₀) is not invariant with respect to the uncertain system (1.1) and (1.2). Then there exists a trajectory 𝑥(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 ‖𝑥(𝑡^∗)‖≤𝜀^∗𝑒^{𝑟(𝑡∗−𝑡0)} and ‖𝑥(𝑡^∗∗)‖>𝜀^∗𝑒^{𝑟(𝑡∗∗−𝑡0)}. This leads to 𝑒^{𝑟(𝑡∗∗−𝑡∗)}‖𝑥(𝑡^∗)‖<‖𝑥(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 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 [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 ‖𝑥‖^𝐻₂=‖𝐻𝑥‖₂=(𝑥^𝑇𝐻^𝑇𝐻𝑥)^1/2, 𝑥∈ℝⁿ, (where 𝐻 is a nonsingular matrix) and the corresponding matrix measure 𝜇_‖‖^𝐻₂, we get the following well-known characterization of the quadratic stability of uncertain system (1.1) and (1.2), ∃𝒬≻0∶forall𝑀_𝑘∈𝒱∶𝒬𝑀_𝑘+𝑀^𝑇_𝑘𝒬≺0 (e.g., [16, page 213]). Indeed, according to ([1, page 41]) inequality (2.1) in Theorem 2.1 means 𝜆_max((1/2)𝐻𝑀_𝑘𝐻⁻¹+(1/2)(𝐻𝑀_𝑘𝐻⁻¹)^𝑇)=𝜇_‖‖^𝐻₂(𝑀_𝑘)≤𝑟<0, which is equivalent to the condition 𝒬𝑀_𝑘+𝑀^𝑇_𝑘𝒬≺0, with 𝒬=𝐻^𝑇𝐻, for all 𝑀_𝑘∈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=ℳ_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.(i)The vertices of the positive cone ℳ_𝑐 fulfill the inequalities ∀𝑀_𝑘∈𝒱∶𝜇_‖‖(𝑀_𝑘)≤0. (2.8)(ii) The function V defined by (1.8) is a common weak Lyapunov function for the uncertain system (1.1) and (1.3).(iii) For any 𝜀>0, the constant set 𝑋^𝜀₀(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 [7] claims that 𝜇_‖‖(𝑀_𝑘)<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 ℳ_𝑐 (1.3). However this is not true. Inequalities 𝜇_‖‖(𝑀_𝑘)<0, 𝑘=1,…,𝐾, imply sup_{𝑀∈ℳ𝑐}𝜇(𝑀)=0, which, together with forall𝑀∈ℳ_𝑐∶max_{𝑖=1,…,𝑛}{Re{𝜆_𝑖(𝑀)}≤𝜇_‖‖(𝑀), for example, [1], yield sup_{𝑀∈ℳ𝑐}max_{𝑖=1,…,𝑛}{Re𝜆_𝑖(𝑀)}≤0. Thus, condition (1.7) for the Hurwitz stability of the positive cone of matrices ℳ_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 ℳ_ℎ 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

𝜇_‖‖(𝑀_𝑘)≤𝑟,𝑘=1,…,𝐾, (3.1) which involve all the vertices 𝒱={𝑀₁,𝑀₂,…,𝑀_𝐾}⊂ℝ^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^∗∈ℝ^𝑛×𝑛 such that the satisfaction of inequality

𝜇_‖‖(𝑀^∗)≤𝑟 (3.2) guarantees the fulfillment of (3.1).

Given a real matrix 𝐴=(𝑎_𝑖𝑗)∈ℝ^𝑛×𝑛, let us define its comparison matrix 𝐴=(𝑎_𝑖𝑗)∈ℝ^𝑛×𝑛 by

𝑎_𝑖𝑖=𝑎_𝑖𝑖,𝑖=1,…,𝑛;𝑎_𝑖𝑗=|𝑎_𝑖𝑗|,𝑖,𝑗=1,…,𝑛,𝑖≠𝑗. (3.3)

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)Matrix 𝑀^∗∈ℝ^𝑛×𝑛 satisfies the componentwise inequalities 𝑃_𝑘^𝑇𝑀_𝑘𝑃_𝑘≤𝑀^∗,𝑘=1,…,𝐾, (3.4) for some permutation matrices 𝑃_𝑘∈ℝ^𝑛×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) If 𝑃∈ℝ^𝑛×𝑛 is a permutation matrix, then ∀𝐴∈ℝ^𝑛×𝑛∶𝜇_‖‖(𝑃^𝑇𝐴𝑃)=𝜇_‖‖(𝐴). (3.5)(R2)Given 𝐴∈ℝ^𝑛×𝑛, if the componentwise inequality 𝑃^𝑇𝐴𝑃≤𝑀^∗ (3.6) is fulfilled for a permutation matrix 𝑃∈ℝ^𝑛×𝑛, then 𝜇_‖‖(𝐴)≤𝜇_‖‖(𝑀^∗). (3.7) 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^∗∈ℝ^𝑛×𝑛, ‖𝑥^∗‖=1, such that ‖𝐴‖=‖𝐴𝑥^∗‖. Since the considered vector norm ‖‖ is permutation invariant, we have ‖𝑥^∗‖=‖𝑃^𝑇𝑥^∗‖=1 for a permutation matrix 𝑃∈ℝ^𝑛×𝑛. This leads to ‖𝐴‖=‖𝐴𝑥^∗‖=‖(𝐴𝑃)(𝑃^𝑇𝑥^∗)‖≤‖𝐴𝑃‖‖𝑃^𝑇𝑥^∗‖=‖𝐴𝑃‖. Let us prove that the strict inequality ‖𝐴‖<‖𝐴𝑃‖ does not hold. Assume, by contradiction, that ‖𝐴‖<‖𝐴𝑃‖. Then, there exists 𝑦^∗∈ℝ^𝑛×𝑛, ‖𝑦^∗‖=1, such that ‖𝐴𝑃𝑦^∗‖=‖𝐴𝑃‖>‖A‖. Hence, 𝑥^∗∗=𝑃𝑦^∗∈ℝ^𝑛×𝑛 with ‖𝑥^∗∗‖=‖𝑃𝑦^∗‖=𝑦^∗=1 satisfies ‖𝐴‖<‖𝐴𝑥^∗∗‖that contradicts the definition of ‖𝐴‖. Consequently, ‖𝐴‖=‖𝐴𝑃‖.
Similarly we prove that ‖𝐴𝑃‖=‖𝑃^𝑇𝐴𝑃‖, yielding ‖𝐴‖=‖𝑃^𝑇𝐴𝑃‖. Thus, we get ‖𝐼+𝜃𝐴‖=‖𝑃^𝑇(𝐼+𝜃𝐴)𝑃‖=‖𝐼+𝜃𝑃^𝑇𝐴𝑃‖ and, consequently, (‖𝐼+𝜃𝐴‖−1)/𝜃 = (‖𝐼+𝜃𝑃^𝑇𝐴𝑃‖−1)/𝜃. By taking lim_𝜃↓0 we obtain equality (3.5).
(R2) First, we exploit the componentwise matrix inequality (3.6). For small 𝜃>0, we get 0≤𝐼+𝜃𝑃^𝑇𝐴𝑃≤𝐼+𝜃𝑀^∗ that leads to the following componentwise vector inequality |(𝐼+𝜃𝑃^𝑇𝐴𝑃)𝑦|≤|(𝐼+𝜃𝑃^𝑇𝐴𝑃)|𝑦‖≤|(𝐼+𝜃𝑀^∗)|𝑦‖, with 𝑦∈ℝ^𝑛.
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+𝜃𝑃^𝑇𝐴𝑃)𝑦‖≤‖(𝐼+𝜃𝑃^𝑇𝐴𝑃)|𝑦|‖≤‖(𝐼+𝜃𝑀^∗)|𝑦|‖, that implies ‖(𝐼+𝜃𝑃^𝑇𝐴𝑃)𝑦‖≤‖𝐼+𝜃𝑀^∗‖‖|𝑦|‖=‖𝐼+𝜃𝑀^∗‖‖𝑦‖. Thus, we get ‖(𝐼+𝜃𝑃^𝑇𝐴𝑃)‖=max_‖𝑦‖=1‖(𝐼+𝜃𝑃^𝑇𝐴𝑃)𝑦‖≤‖𝐼+𝜃𝑀^∗‖ and (‖(𝐼+𝜃𝑃^𝑇𝐴𝑃)‖−1)/𝜃≤(‖𝐼+𝜃𝑀^∗‖−1)/𝜃. By taking lim_𝜃↓0 we obtain the inequality 𝜇_‖‖(𝑃^𝑇𝐴𝑃)≤𝜇_‖‖(𝑀^∗).
Similarly, the componentwise matrix inequality 𝐴≤𝐴 leads to 𝜇_‖‖(𝑃^𝑇𝐴𝑃)≤𝜇_‖‖(𝑃^𝑇𝐴𝑃). Finally, we have 𝜇_‖‖(A)=𝜇_‖‖(𝑃^𝑇𝐴𝑃)≤𝜇_‖‖(𝑃^𝑇𝐴𝑃)≤𝜇_‖‖(𝑀^∗).
Proof of Part II. From (3.4), according to (R2) we get 𝜇_‖‖(𝑀_𝑘)≤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).(H1)The vector norm ‖‖ is an absolute vector norm and 𝜇_‖‖ denotes the corresponding matrix measure.(H2)Matrix 𝑀^∗∈ℝ^𝑛×𝑛 satisfies the componentwise inequalities 𝑀_𝑘≤𝑀^∗,𝑘=1,…,𝐾. (3.8)(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 𝐴∈ℝ^𝑛×𝑛 satisfying the componentwise inequality 𝐴≤𝐴≤𝑀^∗, the monotonicity of ‖‖ implies 𝜇_‖‖(𝐴)≤𝜇_‖‖(𝐴)≤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 [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 ℳ_𝐼={𝑀∈ℝ^𝑛×𝑛∣𝑀⁻≤𝑀≤𝑀⁺}. This system is said to be CWEAS if there exist 𝑑_𝑖>0, 𝑖=1,…,𝑛, and 𝑟<0 such that forall𝑡,𝑡₀∈ℝ₊, 𝑡≥𝑡₀: −𝑑_𝑖≤𝑥_𝑖(𝑡₀)≤𝑑_𝑖⇒−𝑑_𝑖𝑒^{𝑟(𝑡−𝑡0)}≤𝑥_𝑖(𝑡)≤𝑑_𝑖𝑒^{𝑟(𝑡−𝑡0)}, 𝑖=1,…,𝑛, where 𝑥_𝑖(𝑡₀), 𝑥_𝑖(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 𝑀𝑑≤𝑟𝑑, where 𝑑=[𝑑₁⋯𝑑_𝑛]^𝑇∈ℝ^𝑛 and the matrix 𝑀=(𝑚_𝑖𝑗) is built from the entries of the matrices 𝑀⁻=(𝑚⁻_𝑖𝑗) and 𝑀⁺=(𝑚⁺_𝑖𝑗) by 𝑚_𝑖𝑖=𝑚⁺_𝑖𝑖,𝑖=1,…,𝑛, and 𝑚_𝑖𝑗=max{|𝑚⁻_𝑖𝑗|,|𝑚⁺_𝑖𝑗|}, 𝑖≠𝑗, 𝑖,𝑗=1,…,𝑛. On the other hand, Theorem 2.1 characterizes CWEAS if applied for the vector norm ‖𝑥‖^𝐷_∞=‖𝐷𝑥‖_∞=max_{𝑖=1,…,𝑛}{𝑥_𝑖/𝑑_𝑖}, with 𝐷=diag{1/𝑑₁,…,1/𝑑_𝑛}. At the same time, we can use Proposition 3.2(b), since (3.8) is satisfied with 𝑀^∗=𝑀, ‖‖^𝐷_∞ is an absolute vector norm, and there exist 𝑀^∗∗=(𝑚^∗∗_𝑖𝑗) belonging to the set of vertices of ℳ_𝐼 such that 𝑚^∗∗_𝑖𝑖=𝑚_𝑖𝑖, 𝑖=1,…,𝑛, |𝑚^∗∗_𝑖𝑗|=𝑚_𝑖𝑗, 𝑖≠𝑗, 𝑖,𝑗=1,…,𝑛, which implies 𝜇_‖‖^𝐷_∞(𝑀)=max_{𝑖=1,…,𝑛}{𝑚_𝑖𝑖+∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑚_𝑖𝑗(𝑑_𝑗/𝑑_𝑖)}=max_{𝑖=1,…,𝑛}{𝑚^∗∗_𝑖𝑖+∑^𝑛_{𝑗=1,𝑗≠𝑖}|𝑚^∗∗_𝑖𝑗|(𝑑_𝑗/𝑑_𝑖)}=𝜇_‖‖^𝐷_∞(𝑀^∗∗). Thus 𝜇_‖‖^𝐷_∞(𝑀)≤𝑟 is a necessary and sufficient condition for the CWEAS of the interval system. Finally we notice that 𝜇_‖‖^𝐷_∞(𝑀)≤𝑟 is equivalent to 𝑚_𝑖𝑖(𝑑_𝑖/𝑑_𝑖)+∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑚_𝑖𝑗(𝑑_𝑗/𝑑_𝑖)≤𝑟, 𝑖=1,…,𝑛, showing that the CWEAS characterization 𝑀𝑑≤𝑟𝑑 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 𝑀^∗, several test matrices 𝑀^∗₁,𝑀^∗₂,…,𝑀^∗_𝐿, with 𝐿 being significantly smaller than 𝐾. Each 𝑀^∗_ℓ,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 𝒱_ℓ⊆𝒱, such that ⋃^𝐿_ℓ=1𝒱_ℓ=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 [18]: 𝒱={𝑀₁,𝑀₂,𝑀₃}⊂ℝ^3×3,𝑀₁=,𝑀−3.4610.951−0.410−0.480−2.725−0.172252.903−2.504−1.014₂=−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 (4.1) Paper [18] shows that matrices 𝑀₁, 𝑀₂, and 𝑀₃ have the following common quadratic Lyapunov function: 𝑉_C𝒬LF(𝑥)=𝑥^𝑇𝒬𝑥,𝒬=12.6−5.705.70−5.707.50−2.405.70−2.403.12,𝒬≻0, (4.2) since 𝒬𝑀_𝑘+𝑀^𝑇_𝑘𝒬≺0, 𝑘=1,3.
We define the convex hull of matrices ℳ_ℎ having the set of vertices 𝒱 (4.1), that is,
ℳ_ℎ={𝑀∈ℝ^3×3∣𝑀=³_𝑘=1𝛾_𝑘𝑀_𝑘,𝛾_𝑘≥0,³_𝑘=1𝛾_𝑘=1},𝑀_𝑘∈𝒱, (4.3) and the positive cone of matrices ℳ_𝑐 having the set of vertices 𝒱 (4.1), that is, ℳ_𝑐={𝑀∈ℝ^3×3∣𝑀=³_𝑘=1𝛾_𝑘𝑀_𝑘,𝛾_𝑘≥0,𝑀≠0},𝑀_𝑘∈𝒱. (4.4) In ℝ³ we define the vector norm ‖𝑥‖^𝐻₂=‖𝐻𝑥‖₂,with𝐻=3.2079−0.91731.2116−0.91732.5580−0.33931.2116−0.33931.2397, (4.5) where 𝐻²=𝐻^𝑇𝐻=𝒬, in accordance with Remark 2.4, and consider the corresponding matrix measure 𝜇_‖‖^𝐻₂(𝑀)=𝜇_‖‖2(𝐻𝑀𝐻⁻¹). For the vertex-matrices in 𝒱 (4.1) simple computations give 𝜇_‖‖^𝐻₂(𝑀₁)=−1.5915, 𝜇_‖‖^𝐻₂(𝑀₂)=−0.5271, and 𝜇_‖‖^𝐻₂(𝑀₃)=−0.1999.(i)Theorem 2.1 applied to the qualitative analysis of uncertain system (1.1) and (4.3) reveals the following properties.(a)The function 𝑉∶ℝ³→ℝ₊,𝑉(𝑥)=‖𝑥‖^𝐻₂ (4.6) 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 set 𝑋^𝜀_−0.1999(𝑡;𝑡₀) of the form 𝑋^𝜀_−0.1999(𝑡;𝑡₀)={𝑥∈ℝ³∣‖𝑥‖^𝐻₂≤𝜀𝑒^{−0.1999(𝑡−𝑡0)}},𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀,𝜀>0 (4.7) 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 𝑉 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 𝑋^𝜀₀(𝑡;𝑡₀)={𝑥∈ℝ³∣‖𝑥‖^𝐻₂≤𝜀},𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀,𝜀>0 (4.8) is invariant with respect to the uncertain system (1.1) and (4.4).

Example 4.2. Let us consider the interval matrix [19]: ℳ_𝐼={𝑀∈ℝ^2×2∣𝑀=𝑀₀+▵,|▵|≤𝑅},𝑀₀=−3.81.60.6−4.2,𝑅=0.171111. (4.9) Obviously, the set ℳ_𝐼 can be regarded as a convex hull with 𝐾=2⁴ vertices 𝒱={𝑀_𝑘=𝑀₀+▵_𝑘,𝑘=1,…,16},where▵_𝑘=0.17±1±1±1±1. (4.10) The comparison matrices 𝑀_𝑘, 𝑘=1,…,16, of the vertices of 𝒱 (4.10) are built in accordance with (3.3) yielding 𝑀_𝑘=𝑀_𝑘, 𝑘=1,…,16 (since all 𝑀_𝑘 are essentially nonnegative matrices). The dominant vertex 𝑀^∗=𝑀₀+𝑅=[^−3.631.77_0.77−4.03]∈𝒱 satisfies inequalities (3.8), meaning that 𝑀^∗ also satisfies inequalities (3.4) with all the permutation matrices equal to the unity matrix, 𝑃_𝑘=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 𝑟_𝑝=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(𝑀_𝑘)≤𝑟_𝑝,𝑘=1,…,16,𝑝∈{1,2,∞}. (4.11) 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 𝑉∶ℝ²→ℝ₊,𝑉(𝑥)=‖𝑥‖_𝑝,𝑝∈{1,2,∞} (4.12) is a common strong Lyapunov function for the uncertain system (1.1) and (4.9) with the decreasing rate 𝑟_𝑝, 𝑝∈{1,2,∞}.(b)Any exponentially contractive set of the form 𝑋^𝜀_𝑟𝑝(𝑡;𝑡₀)={𝑥∈ℝ²∣‖𝑥‖_𝑝≤𝜀𝑒^{𝑟𝑝(𝑡−𝑡0)}},𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀,𝜀>0,𝑝∈{1,2,∞} (4.13) 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 [20]. Denote by 𝜆_max(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‖^𝐷𝑝_𝑝=‖𝐷_𝑝𝑥‖_𝑝 we have 𝜇_‖‖^𝐷_𝑝𝑝(𝑀^∗)=𝜆_max(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=𝜆_max(𝑀^∗)=−2.6456 proving that the vertex matrices in 𝒱 (4.10) satisfy the inequalities
𝜇_‖‖^𝐷_𝑝𝑝(𝑀_𝑘)≤𝑟,𝑘=1,…,16,𝑝∈{1,2,∞}. (4.14) 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 𝑉∶ℝ²→ℝ₊,𝑉(𝑥)=‖𝐷_𝑝𝑥‖_𝑝,𝑝∈{1,2,∞} (4.15) 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 𝑋^𝜀_−2.6456(𝑡;𝑡₀)={𝑥∈ℝ²∣‖𝐷_𝑝𝑥‖_𝑝≤𝜀𝑒^{−2.6456(𝑡−𝑡0)}},𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀,𝜀>0,𝑝∈{1,2,∞} (4.16)
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 ℳ^′_𝐼={𝑀∈ℝ^2×2∣𝑀⁻≤𝑀≤𝑀⁺},𝑀⁺=−3.631.770.77−4.03,𝑀⁻=,𝑎𝑏𝑐𝑑𝑎≤−3.63,|𝑏|≤1.77,|𝑐|≤0.77,𝑑≤−4.03, (4.17) which has the same dominant vertex 𝑀^∗ as the original interval matrix ℳ_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 ̇̇=−𝑘𝐹(𝑡)𝑣(𝑡)𝛾₂𝑘−1𝑚−𝛾₁𝑚𝐹(𝑡)𝑣(𝑡), (4.18) 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
𝒱={𝑀₁,𝑀₂,𝑀₃,𝑀₄},where𝑀₁𝑀=𝑀(1.5;3),₂=𝑀(1.5;4),𝑀₃=𝑀(2;3),𝑀₄=𝑀(2;4). (4.19)
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)|≤3𝑒^{𝑟(𝑡−𝑡0)},|𝑣(𝑡)|≤4𝑒^{𝑟(𝑡−𝑡0)},∀𝑡≥𝑡₀. (4.20)
The problem can be approached in terms of Theorem 2.1, by considering in ℝ² the vector norm ‖𝑥‖^𝐷_∞=‖𝐷𝑥‖_∞, with 𝐷=diag{1/3,1/4}, and the exponentially contractive set
𝑋¹_𝑟(𝑡;𝑡₀)={𝑥∈ℝ²∣‖𝑥‖^𝐷_∞≤𝑒^{𝑟(𝑡−𝑡0)}},𝑡,𝑡₀∈ℝ₊,𝑡≥𝑡₀. (4.21)
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_∞(𝑀_𝑘)=𝜇_‖‖∞(𝐷𝑀_𝑘𝐷⁻¹) for the vertices 𝑀_𝑘, 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 Figures 2 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 ±3e^𝑟𝑡, ±4e^𝑟𝑡 as used in condition (4.20), with 𝑡₀=0. Figure 3 offers a 3D representation of the exponentially contractive set 𝑋¹_𝑟(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^∗𝑒^{𝑟∗(𝑡−𝑡0)},|𝑣(𝑡)|≤𝑣^∗𝑒^{𝑟∗(𝑡−𝑡0)},∀𝑡≥𝑡₀, (4.22) 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^∗𝑣=max[3(^∗𝐹^∗−2)𝑘₁𝐹,0.5(^∗𝑣^∗−2)]. (4.23)
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.

Figure 1 

The mechanical system used in Example 4.3.

(a)

(b)

Figure 2 

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

(a)

(b)

Figure 3 

(a) 3D representation of the exponentially contractive set 𝑋1𝑟(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.