Revista de Matema´tica: Teor´ıa y Aplicaciones 2002 9(1) : 11–23
cimpa – ucr – ccss issn: 1409-2433
on the stability of a class of polytopes of
third order square matrices and stability
radius
Efre´n Va´zquez Silva∗
Recibido/Received: 27 Julio/July 2000
Abstract
In this work we investigate the stability properties of a convex symmetric time-
invariant third order matrix’s polytope depending on a real positive parameter r. We
apply the obtained results to the calculation of the real stability radius of a third order
matrix under a certain class of affine perturbations.
Keywords: Polytope, stability radius, matrix analysis, Hurwitz stability.
Resumen
En este trabajo estudiamos las propiedads de estabilidad de un politopo sime´trico
y convexo de matrices de tercer orden invariantes en el tiempo, este politopo depende
de un para´metro real positivo r. Los resultados que se obtienen son aplicados al ca´lculo
de radio real de estabilidad, no dependiente del tiempo, de una matriz cuadrada de
tercer orden sometida a cierta multiperturbacio´n afin.
Palabras clave: Politopo, radio de estabilidad, ana´lisis matricial, estabilidad segu´n Hur-
witz.
Mathematics Subject Classification: 34K20, 34K25.
1 Introduction
In Control Theory robustness plays a very important role. This property is a determin-
ing factor in the behaviour of plants and their aggregates. Because robustness takes into
∗Faculty of Informatics and Mathematics, University of Holgu´ın, Cuba. Personal address: GP 57,
Holgu´ın, CP 80100, Cuba. E-Mail: evazquez@uho.hlg.edu.cu.
11
12 e. va´zquez
account the uncertainities that appear in the process of mathematical modelling phenom-
ena of any type, it ensures efficiency in the construction and introduction of the flexible
systems of production. Besides, a basic problem of robustness analysis is to determine
to which extent the stability of a given stable system is preserved under various types of
parameter perturbations.
The work [Nic98] analyzes the absolute stability problem of an automatic control system
of the form:
Σu : x˙ = A0x+ bu(t),
where A0 = (a0ij) ∈ IR3×3 is a constant matrix, b ∈ IR3×1, u represents the control and has
the form:
u = cTx, cT = (c1, c2, c3). (I.1)
Such systems may be encountered in Engineering of Control, where they serve as nominal
models in plants which are characterized by flexibility of their elements, such as cranes
and robots.
If information about the vector c ∈ IR3×1 is not complete, only the coordinates of this
vector are known:
ci = c0i + vi, i = 1, 3. (I.2)
c0 is a constant vector, while the vector v belongs to the polyhedron
V = {v ∈ IR3/ |vi| ≤ v0i , i = 1, 3},
v represents the control error, the values v0i are given or they are experimental data.
Afterward substitute (I.2) in (I.1) and put the result of this operation in the system Σu,
we obtain:
Σv : x˙ = (A+ bvT )x,
where A = (aij), aij = a0ij + bic
0
j ; i, j = 1, 3.
It is supposed that the system Σv is controllable, in other words, the pair (A, b) satisfies
that det[b,Ab,A2b] 6= 0, and two possibilities are considered:
1. That the perturbations v be time-invariant. In this case the stability of the system
Σv is equivalent to the stability of interval matrices of the form:
A = [aij + bivj ]i,j=1,3, vj ∈ [−v0j , v0j ].
The elements of the j-th column of A are related by the parameter vj which varies
in a symmetric interval.
2. That the perturbations v be time-varying, then the stability of the system Σv is
equivalent to the stability of the differential inclusion (see [But85]):
x˙ ∈ F (x) = {(A+ bvT )x, v ∈ V }.
In [Nic98] the stability properties of the system Σv are given by the next
on the stability of a class of polytopes of third order... 13
Theorem 1.1 The set of matrices A is asymptotically stable if and only if:
i) |b1|v01 + |b2|v02 + |b3|v03 < −trA;
ii) |det[b a2 a3]|v01 + |det[a1 b a3]|v02 + |det[a1 a2 b]|v03 < −detA;
iii) dT v + 12v
TCv > (trA)
3∑
p=1
(A)pp − detA,
for the following finite set of values:
For i ∈ {1, 2, 3} and ϑi ∈ {−1, 1}, vi = ϑiv0i , {i, j, k} = {1, 2, 3}
a) If cjj ≥ 0, ckk ≥ 0, cjjckk − ckjcjk > 0
−v01 ≤ v˜1 = −
∣∣∣∣ cilvi + dl cmlcimvi + dm cmm
∣∣∣∣∣∣∣∣ cjj cjkckj ckk
∣∣∣∣ ≤ v
0
1 , {m, l} = {j, k}, m 6= l,
then vj = v˜j and vk = v˜k.
b) In other cases the third inequality must be satisfied for the following values:
For r ∈ {j, k}, ϑr ∈ {−1, 1}, vr = ϑrv0r , {r, s} = {j, k}
b’) If css > 0 and −v0s ≤ ˜˜vs = −ds + cisvi + crsvrcss ≤ v0s , then vs = ˜˜vs.
b”) Otherwise vs = ±v0s .
ai, i = 1, 2, 3, represents the i-th column of the matrix A. Here and in the sequel by
(A)ij , i, j = 1, 2, 3 will be denoted the i, j-minor of the matrix A. In iii) the elements of
the matrix C and the vector d are combinations of the elements of the matrix A and the
coordinates of the vector b.
The previous theorem is a necessary condition too for the stability of the differential
inclusion x˙ = F (x), with the set F (x) as above.
In the current note our aim is to derive the stability properties of a class of families
of third order time-invariant matrices that take their values in a convex and symmetric
polytope which in turn depends on a real positive parameter whose variation represents
an expansion or a contraction of the polytope.
The stability properties of these families are given in terms of the mentioned parameter,
and the analyzed polytope is an extension of the third order matrix intervals considered
in [Nic98] (stationary case).
The results obtained are applied to the calculation of the real stability radius of a Hurwitz-
stable matrix A ∈ IR3×3 under affine time-invariant multiple perturbations of the form
A→ A+
4∑
i=1
δiBi, A,Bi ∈ IR3×3, δi ∈ IR, i = 1, 4. Moreover, we compare our new method
with the existing possibilities for the calculation of the real stability radius of the matrix
A ∈ IR3×3 under structured perturbation. As quantitative indicator of robustness stability
14 e. va´zquez
radius has been introduced in [VanL85], [HinPri86], [HinPri86a] and it is a measure of how
large may be the perturbations that conserve the stability of the system.
The paper is organized as follows: In the next section we formulate the problem and
define the number r∗(A, (Bi)i=1,4) that plays an important role to reach our purpose. In
the Section 3 we calculate the above mentioned number and lastly (Section 4) we compare
the obtained results with other tool of robustness analysis.
2 Formulation of the problem
Let A ∈ IR3×3 be a Hurwitz-estable matrix, i.e. the spectrum σ(A) of the matrix A belongs
to the set IC− = {λ ∈ IC : <(λ) < 0}. Let Bi ∈ IR3×3, i = 1, 4, be not all null-matrices
and have the form Bi = b(v(i))T , i = 1, 4, where b ∈ IR3×1 is a constant vector, and the
vectors v(i) ∈ IR3×1, i = 1, 4, have the coordinates:
v(1) = (v01 , v
0
2 , v
0
3)
v(2) = (−v01 , v02 , v03) (1)
v(3) = (v01 ,−v02 , v03)
v(4) = (v01 , v
0
2 ,−v03).
v0q , q = 1, 2, 3, are given.
Obviously rk(Bi) = 1, i = 1, 4 ( by rk(M) we denote the rank of the matrix M).
Let us to consider for each number r > 0 the convex and symmetric polytope depending
on parameter r and formed by time-invariant matrices:
ℵ(A, (Bi)i=1,4, r) = conv{A± rBi , i = 1, 4}.
For ℵ(A, (Bi)i=1,4, r) we formulate the following problem: Find the values of r > 0 such
that the convex and symmetric polytope ℵ(A, (Bi)i=1,4, r) be estable, that is to say, that
each matrix M ∈ ℵ(A, (Bi)i=1,4, r) be estable.
We denote by:
r∗(A, (Bi)i=1,4) = inf{r > 0 : ℵ(A, (Bi)i=1,4, r) contents at least one unstable matrix}.
If we determine the number r∗(A, (Bi)i=1,4), the stated problem will be solved because
the family ℵ(A, (Bi)i=1,4, r) is asymptotically stable iff r < r∗(A, (Bi)i=1,4).
3 Calculation of the number r∗(A, (Bi)i=1,4)
In this section we study the stability of the family ℵ(A, (Bi)i=1,4, r), for this purpose we
use the well known Routh-Hurwitz criterion (see [Lan69]).
First we introduce the following definitions.
Definition 3.1 The polytope ℵ(A, (Bi)i=1,4, r) is stable if and only if all matrices
M ∈ ℵ(A, (Bi)i=1,4, r) are Hurwitz-stables.
on the stability of a class of polytopes of third order... 15
Definition 3.2 The polytope ℵ(A, (Bi)i=1,4, r) is unstable if there exists at least one ma-
trix M ∈ ℵ(A, (Bi)i=1,4, r) unstable.
With the matrices A and Bi, i = 1, 4 we form new matrices depending on real
parameter r:
M2i(r) = A− rBi, M2i−1(r) = A+ rBi, i = 1, 4,
and with their, in turns, we form a new family
M(r) = conv{Mj(r), j = 1, 8}.
Evidently ℵ(A, (Bi)i=1,4, r) =M(r) then, by the Routh-Hurwitz criterion, the considered
polytope is stable iff the following inequalities hold:
trM(r) < 0
det M(r) < 0 (2)
det M(r)− (trM(r))
∑
p
(M(r))pp > 0.
Now we introduce the following notations:
ξT = (b1v01 , b2v
0
2, b3v
0
3),
ηT = ([b1(a22 + a33)− b2a12 − b3a13]v01 , [−b1a21 + b2(a11 + a33)− b3a23]v02 ,
[−b1a31 − b2a32 + b3(a11 + a22)]v03),
ρT =
∣∣∣∣∣∣
b1 a12 a13
b2 a22 a23
b3 a32 a33
∣∣∣∣∣∣ v01,
∣∣∣∣∣∣
a11 b1 a13
a21 b2 a23
a31 b3 a33
∣∣∣∣∣∣ v02,
∣∣∣∣∣∣
a11 a12 b1
a21 a22 b2
a31 a32 b3
∣∣∣∣∣∣ v03
 , (3)
χT = ρT − (trA)ηT −
3∑
p=1
(A)ppξT ,
and the transformation
β = Gγ, G =
 1 −1 −1 1 1 −1 1 −11 −1 1 −1 −1 1 1 −1
1 −1 1 −1 1 −1 −1 1
 ,
where β ∈ IR3×1, G ∈ IR3×8 and γ ∈ IR8×1.
At this point, considering the inequalities (2), and using the notations (3), we can formulate
the next theorem for the stability of the polytope M(r).
Theorem 3.3 Let A ∈ IR3×3 be a Hurwitz-stable matrix, let be Bi ∈ IR3×3, i = 1, 4, the
matrices defined in Section 2. Then the time-invariant polytope M(r) is stable if and only
if
r < min{pi1(A, (Bi)i=1,4), pi2(A, (Bi)i=1,4), pi3(A, (Bi)i=1,4)} = r∗(A, (Bi)i=1,4), (4)
16 e. va´zquez
where the numbers pi1(A, (Bi)i=1,4), pi2(A, (Bi)i=1,4), pi3(A, (Bi)i=1,4) are the solutions of
the following optimization problems
(i)

8∑
j=1
γj −→ min
trA+ ξTβ = 0
γj ≥ 0, j = 1, 8
(ii)

8∑
j=1
γj −→ min
det A+ ρTβ = 0
γj ≥ 0, j = 1, 8
(iii)

8∑
j=1
γj −→ min
g + 〈χ, β〉 − βT ηξTβ = 0
γj ≥ 0, j = 1, 8,
respectively, g = det A− trA
3∑
p=1
(A)pp.
Proof: If we apply inequalities (2) to the considerd polytope, it is obtained that M(r)
is stable for all r > 0 such that the inequalities trA + ξTβ < 0, det A + ρTβ < 0 and
g + 〈χ, β〉 − βT ηξTβ > 0 hold. So the theorem’s statement is a direct consecuence of the
Definition 3.2 and the continuity of the LHS in the inequalities (2) like functions of the
parameter r. Finally, the equality in the expression (4) is due to Definitions 3.1, 3.2 and
definition of the number r∗(A, (Bi)i=1,4). ♦
Remark 3.4 The proved Theorem 3.3 is a necessary and sufficient condition for the sta-
bility of politope M(r). The theorem is a necessary condition too for the stability of the
corresponding time-varying polytope (M(r))(t). The polytope M(r) contains the set A
when r = 1, then if r∗(A, (Bi)i=1,4) > 1 the corresponding system Σv will be asymptoti-
cally stable (in the case when the perturbation v is time-invariant).
In the two lemmas that we prove bellow, we calculate the numbers pi1(A, (Bi)i=1,4),
pi2(A, (Bi)i=1,4) and pi3(A, (Bi)i=1,4), which let us to obtain the number r
∗(A, (Bi)i=1,4).
Lemma 3.5 Let A ∈ IR3×3 and Bi ∈ IR3×3, i = 1, 4, be matrices that satisfice the
conditions of the Theorem 3.3, then:
pi1(A, (Bi)i=1,4) =
−trA
|b1|v01 + |b2|v02 + |b3|v03
,
pi2(A, (Bi)i=1,4) =
−det A
|ω1|v01 + |ω2|v02 + |ω3|v03
,
where w1 =
∣∣∣∣∣∣
b1 a12 a13
b2 a22 a23
b3 a32 a33
∣∣∣∣∣∣ , w2 =
∣∣∣∣∣∣
a11 b1 a13
a21 b2 a23
a31 b3 a33
∣∣∣∣∣∣ , w3 =
∣∣∣∣∣∣
a11 a12 b1
a21 a22 b2
a31 a32 b3
∣∣∣∣∣∣.
Proof: (i), (ii) are linear programming problems, so their solutions are reached in the
vertexs A ± rBi, i = 1, 4 of the polytope M(r). That is equivalent to looking for the
smallest positive root of the algebraic equations:
trA± r〈b, v(i)〉 = 0, det A± r〈w, v(i)〉 = 0, i = 1, 4,
where ω = (ω1, ω2, ω3), that is to say
pi1(A, (Bi)i=1,4) = inf{r > 0 / trA+ r | 〈b, v(i)〉 |= 0 for some i ∈ {1, 2, 3, 4}},
on the stability of a class of polytopes of third order... 17
pi2(A, (Bi)i=1,4) = inf{r > 0 / det A+ r | 〈w, v(i)〉 |= 0 for some i ∈ {1, 2, 3, 4}}
and the infimum of these sets is reached for the greatest values of | 〈b, v(i)〉 | and | 〈w, v(i)〉 |
respectively, this imply directly the statement of the lemma. ♦
Lemma 3.6 Let A ∈ IR3×3 and Bi ∈ IR3×3, i = 1, 4, be matrices that satisfice the
conditions of the Theorem 3.3, then pi3(A, (Bi)i=1,4) = k
∗U , where U is the solution of the
following linear programming problem:
8∑
j=1
uj −→ min
u1 +
(ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0 u2 −
(ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0 u3 +
(−ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0
u4 − (−ξ1 + ξ2 + ξ3)16‖ξ‖2 ≥ 0 u5 +
(ξ1 − ξ2 + ξ3)
16‖ξ‖2 ≥ 0 u6 −
(ξ1 − ξ2 + ξ3)
16‖ξ‖2 ≥ 0
u7 +
(ξ1 + ξ2 − ξ3)
16‖ξ‖2 ≥ 0 u8 −
(ξ1 + ξ2 − ξ3)
16‖ξ‖2 ≥ 0
(ξ1 + ξ2 + ξ3)(u1 − u2) + (−ξ1 + ξ2 + ξ3)(u3 − u4) + (ξ1 − ξ2 + ξ3)(u5 − u6)+
+(ξ1 + ξ2 − ξ3)(u7 − u8) = 0
and k∗ is the root of smallest absolute value of the equation Φ(k) = 0,
Φ(k) := − ξ
T η
4‖ξ‖2 k
2 + χ
T ξ
2‖ξ‖2 k + g − [(η1 + η2 + η3)(u1 − u2) + (−η1 + η2 + η3)(u3 − u4)
+(η1 − η2 + η3)(u5 − u6) + (η1 + η2 − η3)(u7 − u8)] |k|k2 + [(χ1 + χ2 + χ3)(u1 − u2)+
+(−χ1 + χ2 + χ3)(u3 − u4) + (χ1 − χ2 + χ3)(u5 − u6) + (χ1 + χ2 − χ3)(u7 − u8)] |k|,
ξi, ηi, χi, i = 1, 2, 3, are the coordinates of the vectors ξT , ηT , χT respectively.
Proof: Let us to apply the Lagrange’s method (see, for example [GalTij91]) in order to
solve the problem iii).
The corresponding Lagrange’s function has the form:
L(γ, λ) = λ0f0(γ1, . . . , γ8)−
8∑
j=1
λjfj(γ1, . . . , γ8) + λ9f9(γ1, . . . , γ8), (5)
where
f0(γ1, . . . , γ8) =
8∑
j=1
γj, fj(γ1, . . . , γ8) = γj , j = 1, 8,
f9(γ1, . . . , γ8) = g + 〈GTχ, γ〉 − γTGT ηξTGγ.
Apllying the necessary extreme condition to the function L(γ, λ) given in (5), we obtain
the following algebraic equations system:
λ0
 1...
1
−
λ1...
λ8
+ λ9[GTχ− 2GT ηξTGγ] = 0. (6)
18 e. va´zquez
In the system (6) we consider λ9 6= 0 (in other case the problem lacks of interest) and we
write it in the form:
2GT ηξTGγ = GTχ+Λ, (7)
where
Λ =
1
λ9
[
λ0
 1...
1
−
λ1...
λ8
] ∈ IR8.
The system (7) has solution when Λ ∈ Im(GT ), i.e. Λ = GTh, h ∈ IR3. We can rewrite
the system (7) as follows:
2GT ηξTGγ = GT (χ+ h). (8)
¿From the last equation we have that:
GT (2ηξTGγ − (χ+ h)) = 0,
and we obtain directly
2ηξTGγ = (χ+ h), (9)
by virtue of injectivity of the linear application GT : IR3 → IR8. So, every solution of the
equation (8) is solution of the equation (9) too and vice versa.
Now consider the equation
16ηξTβ = (χ+ h). (10)
If β is solution of (10), then γ = GTβ is solution of (9) by virtue of the relationship
GGT = 8I3, where I3 is the unit matrix of third order.
Let us to look for the solution of (10). Such solution will exist if the vector χ+h is multiple
of the vector η: χ+ h = kη, k-constant. Then, in place of (10) we have:
16ηξTβ = kη. (11)
We look for a particular solution of the equation (11) in the form: β = sξ. By (11) it is
obtained that
16ηξT sξ = kη ⇒ 16sξT ξη = kη ⇒ s = k
16‖ξ‖2 ,
from where β = kξ
16‖ξ‖2 .
But we must find the general solution of the system (8). Taking into account that, by
virtue of equivalence beetwen the systems (8) and (9), GTβ is a particular solution of (8)
and the general solution that we want has the form:
γ˜ = γ +W,
on the stability of a class of polytopes of third order... 19
where
γ = GTβ =
k
16‖ξ‖2

ξ1 + ξ2 + ξ3
−(ξ1 + ξ2 + ξ3)
−ξ1 + ξ2 + ξ3
−(−ξ1 + ξ2 + ξ3)
ξ1 − ξ2 + ξ3
−(ξ1 − ξ2 + ξ3)
ξ1 + ξ2 − ξ3
−(ξ1 + ξ2 − ξ3)

,
W ∈ ker(ηξTG), besides
ker(ηξTG) = {W ∈ IR8/ (ξ1 + ξ2 + ξ3)(W1 −W2) +
+(−ξ1 + ξ2 + ξ3)(W3 −W4) +
+(ξ1 − ξ2 + ξ3)(W5 −W6) +
+(ξ1 + ξ2 − ξ3)(W7 −W8) = 0}.
We have transformed the problem iii) with cuadratic restriction in a new linear program-
ming problem. Concretely:
8∑
j=1
Wj −→ min
W1 +
k(ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0 W2 −
k(ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0 W3 +
k(−ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0
W4 − k(−ξ1 + ξ2 + ξ3)16‖ξ‖2 ≥ 0 W5 +
k(ξ1 − ξ2 + ξ3)
16‖ξ‖2 ≥ 0 W6 −
k(ξ1 − ξ2 + ξ3)
16‖ξ‖2 ≥ 0
W7 +
k(ξ1 + ξ2 − ξ3)
16‖ξ‖2 ≥ 0 W8 −
k(ξ1 + ξ2 − ξ3)
16‖ξ‖2 ≥ 0
(ξ1 + ξ2 + ξ3)(W1 −W2) + (−ξ1 + ξ2 + ξ3)(W3 −W4) + (ξ1 − ξ2 + ξ3)(W5 −W6)+
+(ξ1 + ξ2 − ξ3)(W7 −W8) = 0
By change of variables Wj|k| = uj, j = 1, 8, the above problem is transformed in a new
problem:
8∑
j=1
uj −→ min
u1 + sign(k)
(ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0 u2 − sign(k)
(ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0
u3 + sign(k)
(−ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0 u4 − sign(k)
(−ξ1 + ξ2 + ξ3)
16‖ξ‖2 ≥ 0
u5 + sign(k)
(ξ1 − ξ2 + ξ3)
16‖ξ‖2 ≥ 0 u6 − sign(k)
(ξ1 − ξ2 + ξ3)
16‖ξ‖2 ≥ 0
u7 + sign(k)
(ξ1 + ξ2 − ξ3)
16‖ξ‖2 ≥ 0 u8 − sign(k)
(ξ1 + ξ2 − ξ3)
16‖ξ‖2 ≥ 0
20 e. va´zquez
(ξ1 + ξ2 + ξ3)(u1 − u2) + (−ξ1 + ξ2 + ξ3)(u3 − u4) + (ξ1 − ξ2 + ξ3)(u5 − u6)+
+(ξ1 + ξ2 − ξ3)(u7 − u8) = 0
Here we need to consider the cases k > 0 and k < 0, however it is sufficient to solve
one of it because the solution for k coincides with the solution for −k. At this moment we
can calculate Wj = |k|uj , j = 1, 8, and thus the coordenates of the vector γ˜. The value
of the parametr k is obtained from the fact that γ˜ must satisfy the restriction f9(γ˜) = 0
corresponding to the problem iii), that is to say, we obtain that k is the root of samllest
absolute value of the polynomial Φ(k), denoted by k∗. ♦
Obviously, in accordance with definition of real time-invariant stability radius and defini-
tions of the numbers pi1(A, (Bi)i=1,4), pi2(A, (Bi)i=1,4) and pi3(A, (Bi)i=1,4), we conclude
that rIR(A, (Bi)i=1,4, IC+) = r
∗(A, (Bi)i=1,4).
In the next we expose in an example how to apply the results obtained.
Example 3.7
A =
−1 −1 13 −1 3
−2 1 −4
 , b =
 1−1
1
 , v0 =
−12
1
 .
pi1(A, (Bi)i=1,4) = 3, pi2(A, (Bi)i=1,4) =
1
2
.
By the other hand ξT = (−1,−2, 1), ηT = (7,−2, 1), χT = (67,−2, 1), g = 60
γ˜T = (−k48 +W1,
k
48 +W2,W3,W4,
k
48 +W5,
−k
48 +W6,
−k
48 +W7,
k
48 +W8)
T
ker(ηξTG) = {W ∈ IR8/ −W1 +W2 +W5 −W6 + 2(W8 −W7) = 0}.
In the new variables we need to solve the following linear programming problem:
8∑
j=1
uj −→ min
u1 − 148 ≥ 0 u2 + 148 ≥ 0 u3 ≥ 0 u4 ≥ 0
u5 + 148 ≥ 0 u6 − 148 ≥ 0 u7 − 124 ≥ 0 u8 + 124 ≥ 0
−u1 + u2 + u5 − u6 − 2(u7 − u8) = 0
The solution is U = 18 , the equations for the parametr k are:
− 1
12
k2 − 90
12
k + 60 = 0,
1
4
k2 − 5
6
k + 60 = 0,
so k∗ = 7.39273 and thus pi3(A, (Bi)i=1,4) = 0.924094.
=⇒ r∗(A, (Bi)i=1,4) =
1
2
.
the matrix of the corresponding system Σv is stable, while the real time-invariant stability
radius of the matrix system x˙ = Ωx, Ω ∈M(r), is rIR(A, (Bi)i=1,4, IC+) = 0.5
on the stability of a class of polytopes of third order... 21
4 Comparison with a previous result
In the previous section we have obtained a new method in order to calculate the real
time-invariant stability radius of a matrix A ∈ IR3×3 under certain affine perturbation
class (the rank of the perturbation is one). By this reason we refer an earlier method by
Hinrichsen and Pritchard, obtained from results of Qiu et alter ([QBRDYD95]). (Other
results closely related to results obtained by Hinrichsen and Pritchard can be found in
[Doy82],[PacDoy93]).
Let us to consider the system
x˙ = Ax, (12)
where the matrix A ∈ IR3×3 is Hurwitz stable. Joining with the system (12) we consider
the perturbed system
x˙ = (A+ bvT )x, (13)
b ∈ IR3×1 is a constant vector, while v ∈ IR3×1, |vi| ≤ v(0)i , i = 1, 3, represents the
uncertainity of the perturbation.
Let us to put C = diag(v(0)1 , v
(0)
2 , v
(0)
3 ) and ∆ = [δ1, δ2, δ3], hence we can rewrite the
perturbed system (13) in the form:
x˙ = (A+ b∆C)x. (14)
In the system (14) the uncertainity in the perturbation is represented in the form
vi = δiv
(0)
i , |δi| ≤ 1 (recall that we consider the values v(0)i known). As perturbation
norm we use infinity norm such that ‖∆‖∞ ≤ 1.
Definition 4.1 The real stability radius of the matrix A with respect to time-invariant
perturbations of structure (b, C), is defined by:
rIR(A, b, C, IC+) = inf{‖∆‖ : σ(A+ b∆C) ∩ IC+ 6= ∅},
where ‖ · ‖ is an arbitrary norm.
Thus, when the class of perturbations is ∆ = IRl×q and l = 1 or q = 1, it holds the
following:
Proposition 4.2 ([HinPri2000])
rIR(A, b, C, IC+) = [max
ω∈IR
dist(X(iω), IRY (iω))]−1,
where X(s), Y (s) are the real and the imaginary parts of the transfer matrix
G(s) = C(sI −A)−1b associated with the perturbed system (14).
22 e. va´zquez
4.1 Calculation of rIR(A, b, C, IC+)
Let us apply the Proposition 4.2 to the Example 3.7.
A =
−1 −1 13 −1 3
−2 1 −4
 , b =
 1−1
1
 , v0 =
−12
1
 , C =
−1 o 00 2 0
0 0 1
 .
(ω2I +A2)−1 =
1
ϕ(ω)
ω4 + 18ω2 − 19 −3ω2 − 27 8ω2 − 2812ω2 + 48 ω4 + 13ω2 + 36 12ω2 + 48
−13ω2 + 23 3ω2 + 27 ω4 − 3ω2 + 32
 ,
ϕ(ω) := det(ω2I +A2) = (ω2 + 1)(ω2 + 4)(ω2 + 9), A.b =
 17
−7
 .
It is easy to see that:
X(iω) = −C(ω2I +A2)−1Ab, Y (iω) = −ωC(ω2I +A2)−1b,
then
X(iω) =
1
ϕ(ω)
 ω4 − 95ω2 − 12−14ω4 − 38ω2 + 712
7ω4 − 29ω2 + 12
 , Y (iω) = ω
ϕ(ω)
 ω4 − 7ω2 − 202ω4 − 22ω2 − 128
−ω4 + 19ω2 − 28
 .
dist(X(iω), IRY (iω)) = ‖ · ‖1 = min
α∈IR
3∑
k=1
|Xk(iω)− αYk(iω)| =
=
1
ϕ(ω)
min
α∈IR
[|ω4 − 95ω2 − 12− α(ω5 − 7ω3 − 20ω)| + | − 14ω4 − 38ω2 + 72−
−α(2ω5 − 22ω3 − 128ω)| + |7ω4 − 29ω2 + 12− α(ω5 − 19ω3 + 28ω)|] .
By Proposition 4.2 an algorithm may be obtained in order to calculate the stability radius,
but in order to solve the above extremal problem the great difficulty is related with the
determination of the sign, respect to α, of each of the polynomials on variable ω. The
new method that we propose is more simple than the above way, and it would help in the
problem of computing the real time-invariant stability radius for the studied perturbation
class with bigger rank, i.e. rank 2 and 3.
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on the stability of a class of polytopes of third order... 23
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