A Method for the Calculation the Real Stability Radius of Bidimensional Time-Invariant Systems

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1 Applied Mathematical Sciences, Vol. 7, 2013, no. 67, HIKAI Ltd, A Method for the Calculation the eal Stability adius of Bidimensional Time-Invariant Systems osa Isabel Urquiza Salgado University of Holguin, Holguin, Cuba rurquiza@facinf.uho.edu.cu Efren Vazquez Silva University of Informatics Sciences, Havana City, Cuba vazquezsilva@uci.cu Copyright c 2013 osa Isabel Urquiza Salgado and Efren Vazquez Silva. This 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. Abstract. In this paper we obtain a method for the calculation the stability radius of the perturbed family Δ : ẋ =[A + BΔC] x, when (A, B, C) L 2,l,q (), l, q Z + ; A is a Hurwitz-stable matrix; B 0,C 0 are given matrices specifying the structure of the perturbation, and l q represents the uncertainty of the perturbation. Mathematics Subject Classification: 34D20, 37C20, 34D10, 34D05 Keywords: Perturbed systems of linear differential equations, Asymptotic stability of the solutions, Stability radius 1. Introduction In the solutions of many problems, dynamical linear systems are used as models. Hence, we must take some system as nominal and consider its perturbations. In the applications it is very important to guarantee the asymptotic stability of all the systems: the nominal system and the corresponding perturbated system. We recall that a system is asymptotically stable (a.s.) if all its solutions x(t) converge to the origin of coordinates as t +.

2 3310 osa Isabel Urquiza Salgado and Efren Vazquez Silva A basic problem of robustness analysis is to determine to which extent the stability of a given system is preserved under parametric perturbations. Consequently, is posed the problem of the determination of the greatest bound r>0such that the stability will be preserved under perturbations of norm strictly least of r in the given space of perturbations. Such upper bound is named stability radius (see, for example, [3]). In this paper we consider a family of systems of differential equations, namely, let (A, B, C) L n,l,q () := n n n l q n, where A is a Hurwitz-stable matrix, i.e. the spectrum of A is contained in the left open complex semiplane. B 0, C 0 are given matrices specifying the structure of the perturbation, l, q Z +. We consider for each matrix Δ l q the system: Σ Δ :ẋ =[A + BΔC]x, and consider, for each positive number r, the differential inclusion: (1.1) Σ r :ẋ F r (x) :={ [A + BΔC]x :Δ l q, Δ r}, where is some norm on the matrix space l q. Following [3], [4], [2], we define the real stability radius of the matrix A, for linear time-varying perturbations of structure (B,C), as (1.2) r,t (A, B, C) = inf{r >0:Σ r is not asymptotically stable}. An upper bound for the time-varying stability radius (1.2) is the number: (1.3) r (A, B, C) = inf{ Δ :Δ l q,σ(a + BΔC) C + } where C + = {λ C : e (λ) 0} and σ(m) denotes the spectrum of the matrix M. Some results have been obtained in order to calculate the stability radii r (A, B, C) and r K K,t (A, B, C) of the matrix A, when K =C or K =, and the matrix A is under different type of perturbations (see, for example, [8], [7], [2], [9], [4], [10], [13], [12], [5]). In this paper we obtain a method for the calculation the stability radius (1.2) for the perturbed family (differential inclusion) (1.1), when r ( 0,r (A, B, C)). The number r (A, B, C) is calculated by using the formula: r tr A 2 det A (A, B, C) = min, CB F E 2F + E 4 F 4μ det2 A,

3 A method for the calculation the real stability radius 3311 [ ] where E = B T a22 a 21 C a 12 a T, μ = det ( BB ) T det ( C T C ) ; obtained in 11 [12]. Here the size of the perturbation is measured by the Frobenius norm. The present note is organized as follows. In Section 2 we study the asymptotic stability of the so called auxiliary Barabanov s systems, which determine the stability of the inclusion (1.1). In Section 3 the main results are presented; in Section 4 we calculate the stability radius for some triples A, B, C, with A stable. 2. Asymptotic stability of the auxiliary Barabanov s systems In this section we apply a resultad obtained by the authors in the paper [6]. So, the set F r (x), defined in (1.1), has the following properties (see [6]): 1. F r (x) is a convex, closed and bounded subset of the plane for all x 2, 2. F r (x) depends linearly on x, 3. F r (0) = 0, 0 / F r (x) ifx 0, 4. F r (λx) =λf r (x), if λ, x 2, 5. λx / F r (x), for all x 2, x 0 and λ 0. If we now define for each x 2 the sets of points in 2 : F + r (x) ={f =(f 1,f 2 ) T F r (x) :x 2 f 1 x 1 f 2 < 0}, F r (x) ={f =(f 1,f 2 ) T F r (x) :x 2 f 1 x 1 f 2 > 0}, and consider for each x 2 the optimization problems: (2.1) and (2.2) Maximize subject to : Maximize subject to : f, x = f 1 x 1 +f 2 x 2 f f f 2 2 f F + r (x), f, x = f 1 x 1 +f 2 x 2 f f f 2 2 f F r (x). Besides, if in the following we denote by f r + (x) and f r (x) the solutions of the problems (2.1) and (2.2) respectively, and consider the systems of differential equations:,, (2.3) ẋ = f + r (x),

4 3312 osa Isabel Urquiza Salgado and Efren Vazquez Silva (2.4) ẋ = f r (x), which are defined respectively on the regions: D + r = {x 2 : F + r }, D r = {x 2 : F r }; then, in the work [6] was proved, taking into account the definition of the number r,t (A, B, C), and the fact that r,t (A, B, C) r (A, B, C), that: r,t (A, B, C) = inf{r (0,r (A, B, C)] : at least one of the (2.5) systems (2.3) or (2.4) is not a.s.}. In order to find the expressions for the vector functions f + r (x) and f r (x), the following theorems were proved: Theorem 1. (proof in [6]) Let (A, B, C) L 2,l,q (), where A is a stable matrix, B 0, C 0, r (0,r (A, B, C)). For each x D+ r such that γ(x) 0, the vector f + r (x), in the case when f =(f 1,f 2 ) T F + r (x), has the expression: (2.6) where f + r (x) = Λ 2 [Ax α(x, r)bγ(x)], Λ 2 + μ 1 ϕ 2 (x) α(x, r) = ϕ(x) Λ = r Cx γ(x) 2 r 2 μ 1 Cx 2, Λ= Cx γ(x) 2 r rc 1 x 2 μ 1 Cx 2,ϕ(x) = Cx 2 C 1 x, and μ 1 = i =1,...,l i<j l b2 ij = det ( BB ) T, j=1

5 A method for the calculation the real stability radius 3313 γ i (x) := b 2 i A 1 x b 1i A 2 x, i= 1,l, γ(x) = (γ 1 (x),...,γ l (x)) T := A 1 xb2 T A 2 xb1 T, B ( ) = bij := (det[b i,b j ]), i,j = 1,l. In the previous notations we have used M i, M i, m ij respectively for the i-th row, the i-th column and the elements of the matrix M. Analogously was obtained the expression for the function f r (x). Theorem 2. (see [6]) Let (A, B, C) L 2,l,q (), where A is a stable matrix, B 0, C 0, r (0,r (A, B, C)). For each x D r such that γ(x) 0, the vector fr (x), in the case when f =(f 1,f 2 ) T Fr (x), has the expression: fr (x) = Λ 2 (2.7) [Ax + α(x, r)bγ(x)]. Λ 2 + μ 1 ϕ 2 (x) Now we define the vector functions: (2.8) g + r (x) =Ax α(x, r)bγ(x), (2.9) gr (x) =Ax + α(x, r)bγ(x), and consider the systems: (2.10) ẋ = g + r (x), (2.11) ẋ = g r (x). The systems (2.10) and (2.11) are the so called auxiliary Barabanov s systems. Its asymptotic stability imply the stability of the family (1.1). And so, in the paper [6] was proved that: r,t (A, B, C) = inf { r (0,r (A, B, C)] : at least one of the (2.12) systems (2.10) or (2.11) is not a.s.}. Thus, we now will investigate the asymptotic stability of the auxiliary Barabanov s systems (2.10) and (2.11). To achieve this goal, we present a theorem, that represents a particular case of a known result of Filippov (see [1]), which states the necessary and sufficient conditions for the stability of homogeneous

6 3314 osa Isabel Urquiza Salgado and Efren Vazquez Silva second order systems of differential equations. This result will be used in order to investigate the stability of the systems (2.10) and (2.11). Consider the system: (2.13) { ẋ1 = g 1 (x 1,x 2 ) ẋ 2 = g 2 (x 1,x 2 ), where the functions g i (x), i =1, 2; x =(x 1,x 2 ) T, are defined and continuous in all the phase-plane 2, and g i (λx) =λg i (x) for each λ 0, i =1, 2. Theorem 3. (proof in [1]) The system (2.13) is asymptotically stable if and only if the following conditions hold: a): for each x 0 the vector g(x) is not on {cx: c 0}; b): if for almost all k is g 2 (k, 1)k g 1 (k, 1) > 0; or for almost all k is g 2 (k, 1)k g 1 (k, 1) < 0, then (2.14) + g 1 (k +1)k + g 2 (k +1) 1 dk < 0. g 2 (k +1)k + g 1 (k +1) 1+k2 In the next we investigate the conditions of Theorem 3 for the systems (2.10) and (2.11) when the parameter r varies in the interval (0,r (A, B, C)). We introduce the functions of the variable k: (2.15) h + r (k) =g + r,2(k, 1)k g + r,1(k, 1), (2.16) the numbers: h r (k) =g r,2 (k, 1)k g r,1 (k, 1); (2.17) r 0 + (A, B, C) = inf{r >0:h+ r (k) > 0 for almost all k }, (2.18) r0 (A, B, C) = inf{r >0:h r (k) < 0 for almost all k }, and the functions of the variable r: (2.19) (2.20) where I + (r) = I (r) = + + G + r (k) G r (k) dk 1+k 2, dk 1+k 2,

7 A method for the calculation the real stability radius 3315 (2.21) G + r (k) := g+ r,1(k, 1)k + g r,2(k, + 1) g r,2 + (k, 1)k g+ r,1 (k, 1), G r (k) := g r,1(k, 1)k + gr,2(k, (2.22) 1) gr,2 (k, 1)k g r,1 (k, 1). We have that g r + (k, 1) = (A + BΔC)(k, 1)T for some Δ l q, Δ F r, and as we take r in (0,r (A, B, C)), the matrix A + BΔC is stable. Thus, the auxiliary system (2.10) satisfies the condition a) of Theorem 3. A similar analysis tells us that the mentioned condition is satisfied also for the second auxiliary system (2.11). Theorem 4. Let (A, B, C) L 2,l,q (), where A is a stable matrix, B 0, C 0. Then for r (0,r (A, B, C)) the system (2.10) is asymptotically stable if and only if r r 0 + (A, B, C) ori+ (r) < 0; while the system (2.11) is asymptotically stable if and only if r r0 (A, B, C) ori (r) < 0. Proof: In the following, we will write the proof of each result only for the system ẋ = g r + (x). For the other extremal auxiliary system the proof is similar. Let r<r 0 + (A, B, C). Then, from definition (2.17) of r 0 + (A, B, C) the condition b) of Theorem 3 holds. Let r r 0 + (A, B, C). Then by definition of the number r 0 + (A, B, C), the condition b) of Theorem 3 and the expression (2.19), we conclude that the system ẋ = g r + (x) is a.s. if and only if I+ (r) < 0. Lemma 5. Let (A, B, C) L 2,l,q (), where A is a stable matrix, B 0, C 0 such that r 0 + (A, B, C) < r (A, B, C). Then there exists a number r in the interval (r 0 + (A, B, C),r (A, B, C)), such that I+ (r) < 0 for r (r 0 + (A, B, C), r). Analogously, if r0 (A, B, C) <r (A, B, C) then there exists a number r (r0 (A, B, C),r (A, B, C)), such that I (r) < 0 for r (r0 (A, B, C), r). Proof: Note that by the continuity of the function g r + (k, 1) with respect to k and r, with r (0,r (A, B, C)), and from definition (2.15); the function h + r (k) is continuous with respect to k and r also for the parameter values into consideration. Let r (r 0 + (A, B, C),r (A, B, C)). Then, from definition (2.17) it holds that: i): h + r (k) > 0 for almost all k. On the other side, ii): there exists k 0 such that h + r + (A,B,C)(k 0)=0. 0 It is not difficult to see that ii) holds. Let us suppose that for some k is h + (A,B,C)( k) < 0. By continuity of h + r + 0 r + 0 (A,B,C)(k) with respect to k, we have

8 3316 osa Isabel Urquiza Salgado and Efren Vazquez Silva that h + r + 0 (A,B,C)(k) < 0 for each k in some [α, β]. Then, for parameter values sufficiently close to r 0 + (A, B, C) but larger than r+ 0 (A, B, C), by continuity of h + r (k) onr, h+ r (k) < 0 for each k [ α, β]; however it is not possible, since r>r 0 + (A, B, C). Therefore, h+ r + (A,B,C)(k 0) 0 for each k. 0 Let us assume now that h + r + (A,B,C)(k) > 0 for all k. Hence h+ r (k) > 0in 0 each real k also for r in a left ε-neighbourhood of r 0 + (A, B, C), but it contradicts the definition (2.17) of r 0 + (A, B, C). We consider two differential equations systems: (2.23) ẋ = g + r + 0 (A,B,C)(x), (2.24) ẋ = g r + (x). Let l be the line with slope k 0 across the origin of coordinates and let x 0 l such that x 0 = 1. From i) we have that the trajectory of the system (2.23) through x 0 coincides with the ray that passes for x 0 and begins at the origin. As g r + (x) =(A + BΔC)x for some Δ l q with Δ F r<r (A, B, C), the perturbed matrix A is stable and so the movement on the ray is produced toward the origin of coordinates. Let x r (t) be the solution of (2.24) that satisfies x r (0) = x 0. Thus, from condition ii) it is easy to see that there exists a number ζ > 0 such that x r (ζ) l. We take the minimal ζ for which the inclusion holds. By continuity of the solutions of the system (2.24) respect to the parameter r>r 0 + (A, B, C) it is clear that x r (ζ) 0ifr r 0 + (A, B, C), and so x r (ζ) < 1ifr is sufficiently small, however the solution x r (t) 0 when t + and by homogeneity of the system, the same happens with all the solutions of the system (2.24), i.e. this system is a.s. Lemma 6. The function r I + (r)(r I (r)) for r (r 0 + (A, B, C),r (A, B, C)) ((r0 (A, B, C),r (A, B, C))) is monotone increasing. Proof: The monotone character of the function I + (x) follows at once from the monotone character with respect to r of the function G + r (k) given by (2.21). This is an immediate consequence of the definition (2.19) of I + (r). We have to prove that I + (r) is an increasing function of r, i.e. that G + r (k) 0. r This is a direct consequence from the fact that: G + r (k) r = G+ r (k) α(k, r) α(k, r), r

9 and from the inequalities: A method for the calculation the real stability radius 3317 α(x, r) r = Cx γ(x) 2 0, ( γ(x) 2 r 2 μ 1 Cx ) G + r (k) α = [ A1 (k, 1) T αb 1 γ(k, 1) ] k + A 2 (k, 1) T αb 2 γ(k, 1) α [A 2 (k, 1) T αb 2 γ(k, 1)]k A 1 (k, 1) T + αb 1 γ(k, 1) = [ ] A1 (k, 1) T B 2 A 2 (k, 1) T B 1 γ(k, 1)(1 + k 2 ) { } 2 [A 2 (k, 1) T αb 2 γ(k, 1)]k A 1 (k, 1) T + αb 1 γ(k, 1) = { γ(k, 1) 2 (1 + k 2 ) } 2 > 0 [A 2 (k, 1) T αb 2 γ(k, 1)]k A 1 (k, 1) T + αb 1 γ(k, 1) for all k. Let us define the numbers: (3.1) (3.2) 3. Main results r (A, B, C) if r+ 0 (A, B, C) / (0, r (A, B, C)) r + (A, B, C) = or lim I + (r) 0, r r (A,B,C) root of I + (r) =0 in other cases, r (A, B, C) if r 0 (A, B, C) / (0, r (A, B, C)) r (A, B, C) = or lim I (r) 0, r r (A,B,C) root of I (r) =0 in other cases. Theorem 7. Let (A, B, C) L 2,l,q (), such that A is a stable matrix, B 0, C 0. Then } r,t {r (A, B, C) = min + (A, B, C), r (A, B, C), where the numbers r + (A, B, C), r (A, B, C) were defined respectively in (3.1) and (3.2). Proof: Taking into account the conclusion (2.12), it is sufficient to prove that the systems (2.10) and (2.11) are both a.s. if and only if r<r + (A, B, C) and r<r (A, B, C) respectively, which is a direct consequence of the definitions (3.1), (3.2), Theorem 4, Lemma 5 and Lemma 6.

10 3318 osa Isabel Urquiza Salgado and Efren Vazquez Silva 4. Examples Now we apply Theorem 7 to the calculation of the stability radius r,t (A, B, C) for some triples (A, B, C) L 2,l,q (). [ ] [ ] Example 1: A = ; B = ; C = Applying the definition of r (A, B, C), in Section 1, (2.17) and (2.18) was obtained that: r (A, B, C) = ; 745 r+ 0 (A, B, C) =r0 (A, B, C) =+. Then, by Theorem 7 we have that r,t (A, B, C) = [ ] [ ] Example 2: A = ; B = ; C = Was obtained that: r (A, B, C) = , r 0 + (A, B, C) =+ r+ (A, B, C) =r (A, B, C), r 0 (A, B, C) =0. The equation I (r) = 0 has its root between the numbers and Then, r,t (A, B, C) =r (A, B, C) [ ] [ ] [ ] Example 3: A = ; B = ; C = Was obtained that: r (A, B, C) = , r 0 + (A, B, C) =0,r 0 (A, B, C) =+ r (A, B, C) =r (A, B, C). The function I + (r) changes its sign on the interval (r 0 + (A, B, C),r (A, B, C)). Hence, to obtain r + (A, B, C), we compute the integral I + (r) for different values of r in the interval (0; ) and we conclude that the equation I + (r) = 0 has its root between the numbers and Then, r,t (A, B, C) =r+ (A, B, C) Conclusion We have obtained a method (algorthm) for the calculation the real stability radius for the considered bidimensional time-invariant systems, when they are perturbed in the considered way.

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