Applied Mathematical Sciences, Vol. 7, 2013, no. 67, 3309 - 3319 HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2013.34226
A Method for the Calculation the Real Stability
Radius of Bidimensional Time-Invariant Systems
Rosa Isabel Urquiza Salgado
University of Holguin, Holguin, Cuba [email protected]
Efren Vazquez Silva
University of Informatics Sciences, Havana City, Cuba [email protected]
Copyright c 2013 Rosa 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 : ˙x = [A + BΔC] x, when (A, B, C) ∈
L2 , l , q(R), l, q ∈ Z+; A is a Hurwitz-stable matrix; B = 0, C = 0 are given matrices specifying the structure of the perturbation, and ∈ Rl×qrepresents 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
A basic problem of robustness analysis is to determine to which extent the stability of a given system is preserved under parametric perturbations. Con-sequently, is posed the problem of the determination of the greatest bound
r > 0 such 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) ∈ Ln , l , q(R) := Rn×n × Rn×l × Rq×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 ∈ Rl×q the system:
ΣΔ : ˙x = [A + BΔC]x,
and consider, for each positive number r, the differential inclusion:
Σr: ˙x∈ Fr(x) := { [A + BΔC]x : Δ ∈ Rl × q,Δ ≤ r},
(1.1)
where · is some norm on the matrix space Rl×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
r−
Ê, t
(A, B, C) = inf{r > 0 : Σr is not asymptotically stable}.
(1.2)
An upper bound for the time-varying stability radius (1.2) is the number:
r−
Ê
(A, B, C) = inf{Δ : Δ ∈ Rl× q, σ(A + BΔC)∩ C+= ∅}
(1.3)
where C+ ={λ ∈ C : e (λ) ≥ 0} and σ(M) denotes the spectrum of the ma-trix M .
Some results have been obtained in order to calculate the stability radii
r−
Ã
(A, B, C) and r−
Ã, t
(A, B, C)
of the matrix A, whenK =C or K =R, 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−
Ê
(A, B, C). The number r−
Ê
(A, B, C) is calculated by using the formula:
where E = BT
a22 −a21 −a12 a11
CT, μ = detBBTdetCTC; obtained in [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 asymp-totic 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 Fr(x), defined in (1.1), has the following properties (see [6]):
1. Fr(x) is a convex, closed and bounded subset of the plane for all x∈ R2, 2. Fr(x) depends linearly on x,
3. Fr(0) = 0, 0 /∈ Fr(x) if x= 0, 4. Fr(λx) = λFr(x), if λ∈ R, x ∈ R2,
5. λx /∈ Fr(x), for all x∈ R2, x= 0 and λ ≥ 0.
If we now define for each x ∈ R2 the sets of points in R2:
Fr+(x) ={f = (f1, f2)T ∈ Fr(x) : x2f1− x1f2 < 0},
Fr−(x) ={f = (f1, f2)T ∈ Fr(x) : x2f1− x1f2 > 0},
and consider for each x∈ R2 the optimization problems: ⎧ ⎪ ⎨ ⎪ ⎩ M aximize f, xf = f√1 x1+f2x2 f 2 1+ f 22 , subject to : f ∈ Fr+(x), (2.1) and ⎧ ⎪ ⎨ ⎪ ⎩ M aximize f, xf = f√1 x1+f2x2 f 2 1+ f 22 , subject to : f ∈ Fr−(x). (2.2)
Besides, if in the following we denote by fr+(x) and fr−(x) the solutions of the problems (2.1) and (2.2) respectively, and consider the systems of differ-ential equations:
˙x = fr−(x), (2.4)
which are defined respectively on the regions:
D+r ={x ∈ R2 : Fr+ = ∅},
D−r ={x ∈ R2 : Fr− = ∅};
then, in the work [6] was proved, taking into account the definition of the number r−
Ê, t
(A, B, C), and the fact that r−
Ê, t (A, B, C)≤ r− Ê (A, B, C), that: r− Ê,t (A, B, C) = inf{r ∈ (0, r− Ê
(A, B, C)] : at least one of the
systems (2.3) or (2.4) is not a.s.}.
(2.5)
In order to find the expressions for the vector functions fr+(x) and fr−(x), the following theorems were proved:
Theorem 1. (proof in [6]) Let (A, B, C) ∈ L2, l, q(R), where A is a stable matrix, B = 0, C = 0, r ∈ (0, r−
Ê
(A, B, C)). For each x ∈ Dr+ such that
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ γi(x) := b2 iA1x − b1iA2 x , i = 1, l, γ(x) = (γ1(x), . . . , γl(x))T := A1x B2 T − A2 x B1T, B = bij:= (det[Bi, Bj]) , i, j = 1, l.
In the previous notations we have used Mi , M i, mi j 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 fr−(x).
Theorem 2. (see [6]) Let (A, B, C) ∈ L2, l, q(R), where A is a stable matrix,
B = 0, C = 0, r ∈ (0, r−
Ê
(A, B, C)). For each x∈ Dr− such that γ(x) = 0, the vector fr−(x) , in the case when f = (f1, f2)T ∈ Fr−(x), has the expression:
fr−(x) = Λ
2
Λ2+ μ1ϕ2(x)[Ax + α(x, r)Bγ(x)] . (2.7)
Now we define the vector functions:
g+r(x) = Ax− α(x, r)Bγ(x), (2.8)
gr−(x) = Ax + α(x, r)Bγ(x), (2.9)
and consider the systems:
˙x = gr+(x), (2.10)
˙x = gr−(x). (2.11)
The systems (2.10) and (2.11) are the so called auxiliary Barabanov’s sys-tems. Its asymptotic stability imply the stability of the family (1.1). And so, in the paper [6] was proved that:
r−
Ê, t
(A, B, C) = infr∈ (0, r−
Ê
(A, B, C)] : at least one of the
systems (2.10) or (2.11) is not a.s.} .
(2.12)
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:
˙x1 = g1(x1, x2) ˙x2 = g2(x1, x2), (2.13)
where the functions gi(x), i = 1, 2; x = (x1, x2)T, are defined and continuous in all the phase-plane R2, and gi(λx) = λgi(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 {c x : c ≥ 0};
b): if for ‘almost all’ k ∈ R is g2(k, 1)k− g1(k, 1) > 0; or for ‘almost all’
k∈ R is g2(k, 1)k− g1(k, 1) < 0, then +∞ −∞ g1(k + 1) k + g2(k + 1) |g2(k + 1) k +−g1(k + 1)| 1 1 + k2dk < 0. (2.14)
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−
Ê
(A, B, C)). We introduce the functions of the variable k:
h+r(k) = gr,2+(k, 1)k− gr,1+(k, 1), (2.15)
h−r(k) = gr,2−(k, 1)k− gr,1−(k, 1); (2.16)
the numbers:
r0+(A, B, C) = inf{r > 0 : h+r(k) > 0 f or almost all k ∈ R }, (2.17)
r0−(A, B, C) = inf{r > 0 : h−r(k) < 0 f or almost all k ∈ R }, (2.18)
and the functions of the variable r:
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)| , (2.21) G−r(k) := g − r,1(k, 1)k + gr,2−(k, 1) |g− r,2(k, 1)k− gr,1−(k, 1)| . (2.22)
We have that gr+(k, 1) = (A + BΔC)(k, 1)T for some Δ∈ Rl× q, ΔF ≤ r, and as we take r in (0, r−
Ê
(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) ∈ L2, l, q(R), where A is a stable matrix, B = 0,
C = 0. Then for r ∈ (0, r−
Ê
(A, B, C)) the system (2.10) is asymptotically stable if and only if r ≤ r0+(A, B, C) or I+(r) < 0; while the system (2.11) is asymptotically stable if and only if r ≤ r−0(A, B, C) or I−(r) < 0.
Proof: In the following, we will write the proof of each result only for the
system ˙x = gr+(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 condi-tion 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 ˙x = gr+(x) is a.s. if and only if I+(r) < 0.♦
Lemma 5. Let (A, B, C) ∈ L2, l, q(R), where A is a stable matrix, B = 0,
C = 0 such that r0+(A, B, C) < r−
Ê
(A, B, C). Then there exists a num-ber ˘r in the interval (r+0(A, B, C), r−
Ê
(A, B, C)), such that I+(r) < 0 for
r ∈ (r+0(A, B, C), ˘r). Analogously, if r−0(A, B, C) < r−
Ê
(A, B, C) then there exists a number
¯
r∈ (r0−(A, B, C), r−
Ê
(A, B, C)), such that I−(r) < 0 for r∈ (r0−(A, B, C), ¯r).
Proof: Note that by the continuity of the function gr+(k, 1) with respect to
k and r, with r ∈ (0, r−
Ê
(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 ∈ (r0+(A, B, C), r−
Ê
(A, B, C)). Then, from definition (2.17) it holds that:
i): h+r(k) > 0 for almost all k∈ R. On the other side,
ii): there exists k0 ∈ R such that h+
r+
0(A,B,C)(k0) = 0.
It is not difficult to see that ii) holds. Let us suppose that for some ¯k ∈ R is h+
r+
0(A,B,C)(¯k) < 0. By continuity of h +
r+
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) on r, h+r(k) < 0 for each k ∈ [˜α, ˜β]; however it is not possible, since
r > r0+(A, B, C). Therefore, h+ r+
0(A,B,C)(k0)≥ 0 for each k ∈ R. Let us assume now that h+
r+
0(A,B,C)(k) > 0 for all k ∈ R. Hence h +
r(k) > 0 in each real k also for r in a left ε-neighbourhood of r0+(A, B, C), but it contradicts the definition (2.17) of r0+(A, B, C).
We consider two differential equations systems: ˙x = g+
r+0(A,B,C)(x), (2.23)
˙x = gr+(x). (2.24)
Let l be the line with slope k0 across the origin of coordinates and let x0 ∈ l such that x0 = 1. From i) we have that the trajectory of the system (2.23) through x0 coincides with the ray that passes for x0 and begins at the origin. As gr+(x) = (A + BΔC)x for some Δ∈ Rl× q with ΔF ≤ r < r−
Ê
(A, B, C), the perturbed matrix A is stable and so the movement on the ray is produced toward the origin of coordinates.
Let xr(t) be the solution of (2.24) that satisfies xr(0) = x0. Thus, from condition ii) it is easy to see that there exists a number ζ > 0 such that
xr(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 > r0+(A, B, C) it is clear that xr(ζ)→ 0 if r → r0+(A, B, C), and so |xr(ζ)| < 1 if r is sufficiently small, however the solution xr(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∈ (r0+(A, B, C), r−
Ê
(A, B, C)) ((r0−(A, B, C), r−
Ê
(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)
∂r ≥ 0.
This is a direct consequence from the fact that:
and from the inequalities: ∂ α(x, r) ∂ r = Cxγ(x)2 (γ(x)2− r2μ1Cx2)3|2 ≥ 0, ∂Gr+(k) ∂α = ∂ ∂α A1(k, 1)T − αB1γ(k, 1)k + A2 (k, 1)T − αB2 γ(k, 1) [A2 (k, 1)T − αB2 γ(k, 1)]k− A1(k, 1)T + αB1γ(k, 1) = A1(k, 1)TB2 − A2 (k, 1)TB1γ(k, 1)(1 + k2) [A2 (k, 1)T − αB2 γ(k, 1)]k− A1(k, 1)T + αB1γ(k, 1) 2 = γ(k, 1) 2(1 + k2) [A2 (k, 1)T − αB2 γ(k, 1)]k− A1(k, 1)T + αB1γ(k, 1) 2 > 0 for all k ∈ R.♦ 3. Main results Let us define the numbers:
r+(A, B, C) = ⎧ ⎪ ⎨ ⎪ ⎩ r− Ê (A, B, C) if r0+(A, B, C) /∈ (0, r− Ê (A, B, C)) or lim r↑r−Ê(A,B,C) I+(r)≤ 0,
root of I+(r) = 0 in other cases, (3.1) r−(A, B, C) = ⎧ ⎪ ⎨ ⎪ ⎩ r− Ê (A, B, C) if r0−(A, B, C) /∈ (0, r− Ê (A, B, C)) or lim r↑r−Ê(A,B,C) I−(r)≤ 0,
root of I−(r) = 0 in other cases. (3.2)
Theorem 7. Let (A, B, C)∈ L2, l, q(R), such that A is a stable matrix, B = 0,
C = 0. Then r− Ê, t (A, B, C) = min r+(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
4. Examples
Now we apply Theorem 7 to the calculation of the stability radius r−
Ê, t
(A, B, C) for some triples (A, B, C) ∈ L2, l, q(R).
Example 1: A = −5 −2 2 0 ; B = 0 −1 2 1 0 0 ; C = ⎡ ⎣21 10 0 −1 ⎤ ⎦. Applying the definition of r−
Ê
(A, B, C), in Section 1, (2.17) and (2.18) was obtained that: r− Ê (A, B, C) = √ 4 35+√745 ≈ 0.506797; r + 0 (A, B, C) = r−0(A, B, C) = +∞.
Then, by Theorem 7 we have that r−
Ê, t (A, B, C) = 4 35 +√745 . Example 2: A = −2 0 −3 −2 ; B = 1 0 0 3 ; C = ⎡ ⎣−1 −11 1 1 0 ⎤ ⎦. Was obtained that:
r− Ê (A, B, C) = 4 39 + 3√137 ≈ 0.46463, r0+(A, B, C) = +∞ ⇒ r+(A, B, C) = r− Ê (A, B, C), r0−(A, B, C) = 0. The equation I−(r) = 0 has its root between the numbers 0.15065 and 0.15066. Then, r− Ê, t (A, B, C) = r−(A, B, C)≈ 0.15065. Example 3: A = −1 0 1 −1 ; B = 1 −1 0 0 1 −1 ; C = 1 0 0 2 . Was obtained that:
r−
Ê
(A, B, C) = 1
5 +√13 ≈ 0.340887,
r0+(A, B, C) = 0, r0−(A, B, C) = +∞ ⇒ r−(A, B, C) = r−
Ê
(A, B, C). The function I+(r) changes its sign on the interval (r0+(A, B, C), r−
Ê
(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; 0.340887) and we conclude that the equation
I+(r) = 0 has its root between the numbers 0.13721 and 0.13722. Then,
r−
Ê, t
(A, B, C) = r+(A, B, C)≈ 0.13721.
Conclusion
Acknowledgements. The present paper has been supported by
Mathe-matics Research Support Fund, Atlanta, USA. The authors are indebted to Professor Dr. Theodore P. Hill, School of Mathematics, State Institute of Technology, Atlanta, Georgia, USA for his support.
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