Stability radius of second order linear structured differential inclusions

For arbitrary second order square matrices A, B, C; A Hurwitz stable, the minimum positive value R for which the differential inclusion ẋ ∈ FR(x) := {(A + B∆C)x, ∆ ∈ R2×2, ‖∆‖ ≤ R} fails to be asymptotically stable is calculated, where ‖ · ‖ denotes the operator norm of a matrix.


Introduction
Let A be a second order stable matrix (it means that all eigenvalues of A have negative real part), and let B, C be second order arbitrary matrices and R a positive real number.For each vector x of the plane we consider the set of vectors where • denotes the operator norm of a matrix.The object of investigation in this work is the global asymptotic stability (g.a.s.) of the parameter-dependent differential inclusion ẋ ∈ F R (x), (1.2) and the main problem considered is the computation of the number R i (A, B, C) = inf{R > 0 : ẋ ∈ F R (x) is not g.a.s.}. (1.3) The number R i (A, B, C) is closely related to the robustness of stability of the linear system ẋ = Ax under real perturbations of different classes.As in [5] we consider perturbed systems H. González of the type Σ ∆ : ẋ(t) = Ax(t) + B∆Cx(t) Σ N : ẋ(t) = Ax(t) + BN(Cx(t)) Σ ∆(t) : ẋ(t) = Ax(t) + B∆(t)Cx(t) Σ N(t) : ẋ(t) = Ax(t) + BN(Cx(t), t) (1.4) where • ∆ belongs to the class of matrices R 2×2 provided with the operator norm; • N belongs to the class P n (R) of functions N : R 2 → R 2 , N(0) = 0, N is differentiable at 0, is locally Lipschitz and there exists γ ≥ 0 such that N(x) ≤ γ x for all x ∈ R 2 provided with the norm • ∆(•) belongs to the class P t (R) of functions of the space L ∞ (R + , R 2×2 ) provided with the norm ) is locally Lipschitz in x continuous in t and there exists γ ≥ 0 such that N(x, t) ≤ γ x for all x ∈ R 2 , t ∈ R + provided with the norm Following [5] (see [3,4] also), we define the stability radii of A with respect to the considered perturbations classes For the defined stability radii in [5] it has been shown that for arbitrary triple Effective methods for the calculation of the complex stability radius are exposed in [5], and for the real time invariant linear structured stability radius a method is given in [7].For the real time-varying and nonlinear structured stability radii we do not have general methods, but for the class of positive systems the problem has been solved in [6].The problem of the calculation of the real linear structured time-varying stability radius of second order systems taken Frobenius norm as the perturbation norm is considered in works [8] and [9].The main result of this work is a characterization of the number R i (A, B, C) in terms of the radius R(A, B, C) and a pair of extremal elliptic integrals associated with the differential inclusion (1.1)-(1.2) in case that the differential inclusion has orbits that are spirals in the plane of faces turning around the origin in positive or negative sense.We also prove that: for all triple (A, B, C) of second order matrices, where A is Hurwitz stable.The organization of the paper is as follows: in Section 2 we give a formula for the computation of R(A, B, C).In Section 3 we enunciate a Filippov's Theorem [1] about the asymptotic stability of differential inclusions, which will help us in the fundamentation of the results.In Section 4 we apply this theorem and obtain conditions for the stability of our differential inclusion (1.1)-(1.2) in terms of the number R(A, B, C) and two elliptic integrals.In Section 5 we prove the relations (1.6) and in Section 7 we give some examples for the applications of the main results of this work.The results of this work are a continuation of the paper [2], where the problem of the calculation of the number R i (A) is solved when the perturbations of the linear inclusion are unstructured, i.e., the matrices B, C are the second order identity matrix.

Computation of R(A, B, C)
Lemma 2.1.Let A, B, C be arbitrary second order matrices and A Hurwitz stable.Then where tr(M) denotes the trace of the matrix M, s 1 , s 2 are the singular values of the matrix BC, and σ 1 (CA −1 B) denotes the greatest singular value of the matrix CA −1 B. (The singular values of a square matrix M are the square roots of the eigenvalues of the symmetric matrix M * M.) Proof.λ 2 − tr(A + B∆C)λ + det(A + B∆C) is the characteristic polynomial of the matrix A + B∆C.The roots of this polynomial have negative real parts if and only if tr(A + B∆C) > 0 and det(A + B∆C) > 0. From this it follows that Let BC = U * diag(s 1 , s 2 )V the singular value decomposition of the matrix BC, where U and V are orthogonal matrices and denote ∆ = V∆U * , then we have

.3)
A is a Hurwitz matrix, so det(A + B∆C) = 0 ⇔ det(I + A −1 B∆C) = 0.This equality is equivalent to the existence of 0 = v ∈ R 2 such that (I + A −1 B∆C)v = 0, and this last assertion is equivalent to the existence of 0 = w ∈ R 2 such that (I + CA −1 B∆)w = 0. Then from H. González this fact and making use of the equality inf ∆ : det(I which is a direct consequence of Lemma 1 of [7] we obtain The assertion of the lemma follows from (2.2), (2.3) and (2.4).

Filippov's theorem
In this section we enunciate Filippov's Theorem [1], which will be the fundamental tool in the analysis of the stability of the differential inclusion (1.1)-(1.2).Let: be a differential inclusion which satisfies the following properties (i) for all x the set F(x) is nonempty, bounded, closed and convex; (ii) F(x) is upper semi-continuous with respect to the set's inclusion as function of x; (iii) F(cx) = cF(x) for all x and c ≥ 0.
We will use the notations Theorem 3.1 (Filippov's Theorem).The differential inclusion (3.1) satisfying the conditions (i)-(iii) is asymptotically stable if and only if for all x = 0 the set F(x) does not have common points with the ray cx, 0 ≤ c < +∞ and when the set F + (ϕ) (resp.F − (ϕ)) for almost all ϕ is nonempty, the inequality

Application of Filippov's theorem
From the definition (1.1) we have that for all R > 0 the set F R (x) for all x ∈ R 2 is non empty, bounded, closed and convex set of the plane.So the differential inclusion (1.1)-(1.2) satisfies properties (i)-(iii) and Filippov's Theorem is applicable.
The following lemma allows us to write the set F R (x) in the form we will use in the for suitable 0 ≤ r(t) ≤ R 0 , 0 ≤ θ(t) < 2π, and so all solutions of the perturbed system ẋ = Ax + BN(Cx, t) is a solution of the differential inclusion (4.1) with R = R 0 , from what follows the inequality (4.2) So from this lemma and (1.5) we can restrict the analysis of the asymptotic stability of the differential inclusion (1.1)-(1.2) for R < R(A, B, C).
Proof.a) The set F R (x) := {(A + B∆C)x, ∆ ∈ R 2×2 , ∆ ≤ R}, with R < R(A, B, C) does not have common points with the ray cx, 0 ≤ c < +∞ for all x = 0 because the matrix 3 + p 2 4 > 0, which, according to definitions (4.6), is equivalent to the assertion b) of this lemma.c) 3 + p 2 4 < 0, which, according to definitions (4.6), is equivalent to the assertion c) of this lemma.
In what follows with the aim of shorten the expressions for the functions f i , i = 1, 2, g and p i , i = 1, 2, 3, 4 we omit the ϕ argument.
We denote: then for R ∈ (R + (A, B, C), R(A, B, C)) the function K + (ϕ) that appears in the Filippov's theorem can be written as Similarly for R ∈ (R − (A, B, C), R(A, B, C)) the function K − (ϕ) can be written as
Theorem 4.5.The differential inclusion (4.1) depending on the parameter R is asymptotically stable if and only if the following conditions hold: where the functions K + R (ϕ) and K − R (ϕ) are defined by the expressions (4.10) and (4.11).
Proof.The assertion of the theorem follows directly from Filippov's Theorem and Lemmas 4.3 and 4.4.
. The condition about the sign of integrals I + (R) and I − (R) must be checked only when R < R + (A, B, C), respectively R < R − (A, B, C).Note that from expressions (4.8), (4.9) we have that the integrals I + (R), I − (R) are monotone increasing functions of R, and so the value of R for which the integrals are equal to zero can be obtained using bisection method.Note also that R Remark 5.2.The differential inclusion (4.3) can be written as where K(θ, ϕ, r) is given by (4.7).Then for R ∈ (R + (A, B, C), R(A, B, C)) the equation where K R + (ϕ) is given by (4.10), is the equation in polar coordinates corresponding to the differential system obtained from inclusion (4.1), after the substitution of cos(θ) and sin(θ) by (4.12) and (4.13), respectively.This last system has the form Put Then the function N + R (x) belongs to the class P n (R) defined in the introduction of this work and has norm which is equal to R. Furthermore, the differential system (5.1) can be written as ẋ = Ax + R Cx BN + R (x).
We can conclude that for R ∈ (R + (A, B, C), R(A, B, C)) the system (5.1), which we will name the positive extremal system of the differential inclusion (1.1)-(1.2) is the perturbation of the nominal linear system ẋ = Ax with the nonlinear perturbation N + R (x) of the class P n (R) which has norm equal to R. Furthermore, the trajectories of this system are spirals which turn around the origin in positive sense and the value of the integral I + (R) is the Lyapunov

5 Remarks to the Theorem 4.5 Remark 5.1.
Theorem 4.5gives a method for the calculation of the number R i (A, B, C) for arbitrary triples of matrices (A, B, C) ∈ R 2×2 , where A is Hurwitz stable.We have formulas (2.1), (4.14), (4.10) and (4.11) for the computation of the numbers R(A, B, C), R + (A, B, C), R − (A, B, C) and the functions K