# Upper Bounds for the Distance between a Controllable Switched Linear System and the Set of Uncontrollable Ones

Authors: Josep Clotet; M. Dolors Magret
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The set of controllable switched linear systems is an open and dense set in the space of all switched linear systems. Therefore it makes sense to compute the distance from a controllable system to the nearest uncontrollable one. In the case of a standard system,x˙t=Axt+But, R. Eising, D. Boley, and W. S. Lu obtain some results for this distance, both in the complex and real cases. In this work we explore this distance, for switched linear systems in the real case, obtaining upper bounds for it. The main contribution of the paper is to prove that a natural generalization of the upper bound obtained by D. Boley and W. S. Lu is true in the case of switched linear systems.


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## Body

## 1. Introduction

A natural question in system theory (and in many other areas of applied mathematics) is whether a certain qualitative property of a system is, or not, preserved under small changes in its parameters (external perturbations, uncertainty on their measure, etc.). If the answer is affirmative, a second natural question concerns the “smallness" of these admissible changes. See, for example, [1, 2].In particular, the controllability of a system is a generic property; that is to say, the set of controllable systems is open and dense in the corresponding space of matrices. Hence, it makes sense to ask how far a controllable system is from the nearest uncontrollable ones (see, e.g., [2–7]) and to find bounds for the distance between a controllable system and the set of uncontrollable ones (see, e.g., [2, 3, 8]).It is known (Eising [4]) that the problem of computing the distance from a given controllable standard system
(1)x˙(t)=Ax(t)+Bu(t)
with A∈Mn×n(ℂ) and B∈Mn×p(ℂ) to the set of uncontrollable ones, denoted by μℂ(A,B), is equivalent to computing the minimum of the smallest singular value of (A-λIn∣B) over λ∈ℂ:
(2)μℂ(A,B)=minλ∈ℂσn(A-λIn∣B),
this means that there aren't uncontrollable systems in a ball centered in (A,B) with radius μℂ(A,B), with respect to a given matrix norm.In this paper we tackle a generalization of this result. As we will see in an example in Section4, in the case where switched linear systems are considered, the natural generalization does not provide the distance from a controllable switched linear system to the set of uncontrollable ones.We can also consider, given a controllable standard system(3)x˙(t)=Ax(t)+Bu(t)
with A∈Mn×n(ℝ) and B∈Mn×p(ℝ), the distance from this system to the set of uncontrollable (real) systems and denote it by μℝ(A,B). Actually, complex perturbations of real systems do not always correspond to a system with physical meaning. Note that in this case (Boley and Lu [3])
(4)μℝ(A,B)≤minλ∈ℝσn(A-λIn∣B)
and the equality is not true in general (see the example in [2, page 557]). Obviously μℂ(A,B)≤μℝ(A,B). In particular, any upper bound for μℝ(A,B) is also an upper bound for μℂ(A,B).Many practical systems can be modelled as switched linear systems. Due to their theoretical and practical importance switched linear systems have attracted a lot of interest during the last years. For continuous-time linear switched systems, with time-varying delay, the weightedℋ∞ model reduction problem has been investigated in [9] and for continuous-time switched stochastic systems, the problems of dissipativity analysis and dissipativity-based sliding mode control have been studied in [10]. In [11] problems of stability and stabilization for discrete-time switched nonlinear systems with time-varying delay are studied (see also [12] for the study of stability and stabilization problems for a class of discrete-time Takagi-Sugeno fuzzy systems with time-varying state delay). In [13] the position control for a two-wheeled mobile robot is studied and in [14] an application to a dynamic output feedback controller design is described.Several authors have recently studied controllability and observability of switched linear systems. Algebraic characterization of controllability in [15] shows that controllability is also in this case a generic property. Therefore, the analogous questions arise and it makes sense to study the bounds for the distances from a controllable switched linear system to the nearest uncontrollable one, as it was studied for standard systems. Up to now, no bound for switched linear systems had been obtained, except that in [16] (by the same authors). In this work, we concretely obtain an upper bound for this distance which improves substantially that obtained in [16].The structure of the paper is as follows.In Section2, we summarize the definitions and some properties of norm matrices.In Section3, we lay the foundations of the problem to be solved.In Section4, we obtain (Theorem 9) an explicit upper bound for the distance between a controllable switched linear system and the set of uncontrollable ones.Finally, in Section5, we consider different examples.


## 2. Preliminaries

Let us consider the vector space ofm×n matrices with coefficients in a field k (k=ℝ or ℂ).A matrix norm∥∥ is a mapping associating to each matrix M a nonnegative number ∥M∥ having the following properties. (1)
For all matrixM, ∥M∥≥0 and ∥M∥=0 if, and only if, M=0.(2)
For all matrixM and λ∈k, ∥λM∥=|λ|∥M∥.(3)
For all matricesM1 and M2, ∥M1+M2∥≤∥M1∥+∥M2∥.The most frequently used matrix norms are the Frobenius norm and thep-norms.The Frobenius norm of a matrixM=(mji)1≤i≤m1≤j≤n is defined as
(5)∥M∥F=∑1≤i≤m‍∑1≤j≤n‍|mji|2.The matrixp-norm is defined for a real number 1≤p≤∞ as
(6)∥M∥p=supx≠0∥Mx∥p∥x∥p=max∥x∥p=1∥Mx∥p,
where
(7)∥x∥p=(|x1|p+⋯+|xn|p)1/p,1≤p<∞
and ∥x∥∞=max1≤i≤n|xi|.In particular,∥M∥1=max1≤j≤n∑i=1m|mji|.The spectral norm ∥∥2 is the square root of the maximum eigenvalue of MtM in the real case and MHM in the complex case, where Mt, MH denote the transpose and the conjugate transpose of the matrix M, respectively. And ∥M∥∞=max1≤i≤m∑1≤j≤n|mji|.The Frobenius and thep-norms satisfy the following inequality (submultiplicative property): for all m×n matrix M1 and n×p matrix M2, ∥M1M2∥≤∥M1∥∥M2∥, and some further inequalities, relating them and which are commonly used in matrix analysis,(i)
∥M∥2≤∥M∥F≤n∥M∥2,(ii)
maxi,j|mji|≤∥M∥2≤mnmaxi,j|mji|,(iii)
(1/m)∥M∥1≤∥M∥2≤n∥M∥1,(iv)
(1/n)∥M∥∞≤∥M∥2≤m∥M∥∞.


## 3. Set-Up

Firstly, we will make precise the systems to which the results in Section4 can be applied. Let us consider a switched linear system Σ defined by
(8)x˙(t)=Aθ(t)x(t)+Bθ(t)u(t),
where Aθ(t)∈Mn×n(ℝ),Bθ(t)∈Mn×p(ℝ),x(t)∈ℝn is the state; u(t)∈ℝp is the piecewise continuous input function; and θ is the piecewise constant signal having range an index set Λ={1,…,ℓ}.A switching signal (see [15]) is said to be a switching path if it is a function of time. Given an initial time t0, a switching path is a function of time.θ
:
[
t
0
,
T
)
→
Λ, t0<T≤∞. A switching path θ is said to be well defined on [t0,T) if it is defined in [t0,T) and for all t∈[t0,T), both lims→t+θ(s) and lims→t-θ(s) exist and the set
(9){t∈[t0,T)∣lims→t+θ(s)≠lims→t-θ(s)}
is finite for any finite sub interval of [t0,T) (in the case where t=t0, we will consider lims→t0-θ(s)=θ(t0)).Sun-Ge proved in [15] that the system above is controllable if, and only if, ℝn is equal to the vector space
(10)𝒱=∑i0,…,in-1∈{1,…,ℓ}j1,…,jn-1∈{0,…,n-1}‍Im(Ain-1jn-1⋯Ai1j1Bi0),
where Im(M) stands for image of matrix M. Hence controllability depends only on the set of matrices {(Ai,Bi)i∈{1,…,ℓ}} defining the subsystems of Σ.Therefore, the set of matrices{(Ai,Bi)i∈{1,…,ℓ}} defining subsystems of a controllable system is an open dense subset of Mn×n(ℝ)ℓ×Mn×p(ℝ)ℓ. In particular, for each set of matrices {(Ai,Bi)i∈{1,…,ℓ}} there exists an open neighborhood of this set in Mn×n(ℝ)ℓ×Mn×p(ℝ)ℓ with the whole set of matrices in it defining subsystems of a controllable system.The norm (Frobenius norm, 2-norm, etc.) of the set of matrices{(Ai,Bi)i∈{1,…,ℓ}} is taken as
(11)∥{(Ai,Bi)i∈{1,…,ℓ}}∥=∥(A1|B1|⋯|Aℓ|Bℓ)∥
and thus the distance between two sets of matrices is
(12)d({(Ai,Bi)i∈{1,…,ℓ}},{(Ai′,Bi′)i∈{1,…,ℓ}})=∥{(Ai′-Ai,Bi′-Bi)i∈{1,…,ℓ}}∥.Given a controllable systemΣ with subsystems defined by the set of matrices {(Ai,Bi)i∈{1,…,ℓ}}, the distance (with regard to a given matrix norm) from this system to the nearest uncontrollable one is
(13)μℝ({(Ai,Bi)i∈{1,…,ℓ}})=minδAi∈Mn×n(ℝ),δBi∈Mn×p(ℝ)∥{(δAi,δBi)i∈{1,…,ℓ}}∥
with {(Ai+δAi,Bi+δBi)i∈{1,…,ℓ}} being the set of matrices defining the subsystems of an uncontrollable system.


## 4. Upper Bound for the Distance

Note that if we generalize, in a natural way, (2) to the case of switched linear systems and define
(14)μℂ({(Ai,Bi)i∈{1,…,ℓ}})=minδAi∈Mn×n(ℂ),δBi∈Mn×p(ℂ)∥{(δAi,δBi)i∈{1,…,ℓ}}∥
with {(Ai+δAi,Bi+δBi)i∈{1,…,ℓ}} being the set of matrices defining the subsystems of an uncontrollable system, then this distance μℂ({(Ai,Bi)i∈{1,…,ℓ}}) is not equal to
(15)minλ1,…,λℓ∈ℂσn(A1-λ1In|B1|⋯|Aℓ-λℓIn|Bℓ)
as the following example shows.Example 1.
Consider a switched linear system with two subsystems defined by matrices(16)A1=(0000200.112),A2=(011001001),B1=B2=(100).
This system is controllable, as it is straightforward to be checked (dim𝒱=3) and
(17)minλ1,λ2∈ℂσ3(A1-λ1I3|B1|⋯|A2-λ2I3|B2)=minλ1,λ2∈ℂσ3(λ1001-λ211102-λ1000-λ2100.112-λ10001-λ20)=0.348160,
but the system consisting of two subsystems defined by
(18)A1=(000020012),A2=(011001001),B1=B2=(100),
is uncontrollable and the distance (both considering the Frobenius norm and the 2-norm) between this system and the previous one is 0.1<0.348160.
The main result in this section is the generalization of inequality (4) to the case of switched linear systems. The natural generalization seems to be
(19)μℝ({(Ai,Bi)i∈{1,…,ℓ}})≤minλ1,…,λℓ∈ℝσn(A1-λ1In|B1|⋯|Aℓ-λℓIn|Bℓ).
This generalization is proved in Theorem 9.
Consider the switched linear system (8) and let us assume that this system is controllable. For each i∈{1,…,ℓ}, let us consider the vector subspace
(20)∑k1,…,kℓ∈{0,1}k1+⋯+kℓ=1j1,…,jn-2∈{0,1,…,n-2}j1+⋯+jn-2≤n-2h1,…,hn-2∈{1,…,ℓ}Ai≠Ah1‍[∑0≤j≤n-1‍Im(Aij(Ah1j1⋯Ahn-2jn-2B1k1⋯Bℓkℓ))]
and let us denote by Gi(Σ) the matrix having as columns the generators of the vector subspace above and by G(Σ) the matrix
(21)G(Σ)=(G1(Σ)∣⋯∣Gℓ(Σ)).
Throughout the paper, the notationrkM will denote the rank of matrix M.Lemma 2.
The rank of matrixG(Σ) (written rkG(Σ)) is equal to the dimension of the vector space
(22)∑k1,…,kℓ∈{0,1}k1+⋯+kℓ=1j1,…,jn-1∈{0,1,…,n-1}i1,…,in-1∈{1,…,ℓ}Im(Ai1j1⋯Ain-1jn-1B1k1⋯Bℓkℓ).Proof.
It is a straightforward consequence of Cayley-Hamilton Theorem and the fact that this vector subspace and𝒱 defined above are the same subspace.As a consequence of Lemma2, another characterization of controllable switched linear systems is that system Σ is controllable if, and only if, rkG(Σ)=n.A first upper bound for the distance, from our system to the set of uncontrollable ones, may be deduced from the singular value decomposition of matrixG(Σ), in a similar way to the one in [3], in the case where rkGi(Σ)=n, for all i∈{1,…,ℓ}. Namely, let P, Q be orthogonal matrices such that
(23)G(Σ)=Pt(diag(σ1,…,σn)∣0)Q,
where σ1≥⋯≥σn are the singular values of G(Σ). Then the following statement holds.Theorem 3 (see [16]).
Given a controllable switched linear systemΣ consisting of subsystems defined by a set of matrices {(Ai,Bi)i∈{1,…,ℓ}}, such that rkGi(Σ)=n, for all i∈{1,…,ℓ}, with regard to the 2-norm,
(24)μℝ({(Ai,Bi)i∈{1,…,ℓ}})≤∑i∈{1,…,ℓ}‍σn(1+∥AiC∥2σn,i),
where AiC is the companion matrix for Ai and σ1,i≥⋯≥σn,i are the singular values of Gi(Σ).Remember that, given a square matrixA with characteristic polynomial Q(t)=tn+a1tn-1+⋯+an-1t+an, the matrix
(25)(00⋯0-an10⋯0-an-101⋯0-an-2⋯⋯⋯⋯00⋯1-a1)
is called the companion matrix of matrix A.Proof.
First of all, note that{(Ai,Bi)i∈{1,…,ℓ}} are associated to a controllable system if, and only if, {(PAiPt,PBi)i∈{1,…,ℓ}} are also associated to a controllable system.
Denoting by𝒫(26)diag(ℓ(PtIp),…,(PtIp))
and since
(27)P(A1-X1|B1-Y1|⋯|Aℓ-Xℓ|Bℓ-Yℓ)𝒫=(P(A1-X1)Pt|P(B1-Y1)|⋯|P(Aℓ-Xℓ)Pt|P(Bℓ-Yℓ)P(A1-X1)Pt)
and P and 𝒫 are orthogonal matrices, we deduce that
(28)∥(A1-X1|B1-Y1|⋯|Aℓ-Xℓ|Bℓ-Yℓ)∥2=∥(P(A1-X1)Pt|P(B1-Y1)|⋯|P(Aℓ-Xℓ)Pt|P(Bℓ-Yℓ)P(A1-X1)Pt)∥2.
Therefore
(29)μℝ({(Ai,Bi)i∈{1,…,ℓ}})=μℝ({(PAiPt,PBi)i∈{1,…,ℓ}}).
Let us considerPBi=(Bi1Bi2), with Bi1∈M(n-1)×p(ℝ), for i∈{1,…,ℓ}.
We can write(30)Bi2=ent(PBi)=ent(PG(Σ)(0⋯0Ip(ji)0⋯0)t)=ent((diag(σ1,…,σn)∣0)Q(0⋯0Ip(ji)0⋯0)t),
where ji indicates the column ji of matrix G(Σ) where matrix Bi first appears, with e1,…,en being the natural basis of the Euclidean space ℝn. Then
(31)∥Bi2∥2≤σn∥Q(0⋯0Ip(ji)0⋯0)t∥2=σn
since Q is an orthogonal matrix.
It is straightforward to check thatAiGi(Σ)=Gi(Σ)(I⊗AiC), where I is the identity matrix of suitable order. Then if we denote by Gi(Σ)† the Moore-Penrose inverse of matrix Gi(Σ), Ai=Gi(Σ)(I⊗AiC)Gi(Σ)† and
(32)PAiPt=PGi(Σ)(I⊗AiC)Gi(Σ)†Pt=P(G(Σ)(0⋯0Ici(i)0⋯0)t)(I⊗AiC)Gi(Σ)†Pt=(diag(σ1,…,σn)∣0)Q(0⋯0Ici(i)0⋯0)t×(I⊗AiC)Gi(Σ)†Pt,
where ci is the number of columns of matrix Gi(Σ).
In an analogous way, let us consider(33)PAiPt=(Ai1Ai2Ai3Ai4)
with Ai1∈M(n-1)×(n-1)(ℝ),
(34)Ai3=ent(diag(σ1,…,σn)∣0)Q×(0⋯0Ici(i)0⋯0)t(I⊗AiC)Qit×(diag(1σ1,i,…,1σn,i)0)PiPt(In-10),
where σ1,i≥⋯≥σn,i, are the singular values of Gi(Σ) (Gi(Σ)=Pit(diag(σ1,i,…,σn,i)|0)Qi). Therefore
(35)∥Ai3∥2≤σn∥(I⊗AiC)∥2∥Gi(Σ)†∥2≤σnσn,i∥AiC∥2.Let us set the blocksBi2, Ai3 to zero, for all i; it is obvious that the system thus obtained is uncontrollable. The real perturbation we have committed has a norm which is bounded by σn+(∥AiC∥2σn/σ(n,i)), for all i∈{1,…,ℓ}, and the statement follows.Observe that, in the case whereℓ=1 (there is only one subsystem) we obtain the same result as that in [3] because the product PiPt in (34) is equal to In, and therefore ∥Ai3∥2≤(σn/σn-1)∥AiC∥2.In order to obtain an improved upper bound, we need the following matrix.Let us consider a switched linear system with subsystems defined by the set of matrices{(Ai,Bi)i∈{1,…,ℓ}}. Given λ1,…,λℓ∈ℝ, we will denote by M(Σ,λ1,…,λℓ) the following matrix:
(36)M(Σ,λ1,…,λℓ)=(A1-λ1In|B1|⋯|Aℓ-λℓIn|Bℓ).Lemma 4.
Given a switched linear systemΣ with subsystems defined by a set of matrices {(Ai,Bi)i∈{1,…,ℓ}},
(37)rkG(Σ)≤rkM(Σ,λ1,…,λℓ),∀(λ1,…,λℓ)∈ℝℓ.Proof.
Let us assume thatλ10,…,λℓ0∈ℝ are such that
(38)minλ1,…,λℓ∈ℝrkM(Σ,λ1,…,λℓ)=rkM(Σ,λ10,…,λℓ0).
Note that in the case where rkM(Σ,λ10,…,λℓ0)=n, the statement is obvious. Let us assume that the rank of M(Σ,λ10,…,λℓ0)=n-m, m>0. Then there exist linearly independent vectors v1,…,vm∈ℝn such that
(39)v1tM(Σ,λ10,…,λℓ0)=⋯=vmtM(Σ,λ10,…,λℓ0)=0.
Therefore
(40)vit(A1-λ1In)=⋯=vit(Aℓ-λℓIn)=0,i∈{1,…,m},vitB1=⋯=vitBℓ=0,i∈{1,…,m}.
It is straightforward to check thatv1,…,vm are left eigenvectors of matrices A1,…,Aℓ and therefore
(41)v1tG(Σ)=⋯=vmtG(Σ)=0.
ThusrkG(Σ)≤n-m.As a consequence of Lemma4, we can conclude that if there exist λ10,…,λℓ0∈ℝ such that rkM(Σ,λ10,…,λℓ0)<n, the system can not be controllable.Remark 5.
Note that there exist uncontrollable systems such that for allλ1,…,λℓ∈ℝ, rkM(Σ,λ1,…,λℓ)=n. For example, we consider a system Σ with subsystems defined by matrices
(42)A1=(000020012),A2=(011001001),B1=B2=(100).
It is straightforward that rkG(Σ)=1 and that the rank of M(Σ,λ1,λ2)=3, for all λ1,λ2∈ℝ.In particular, we may obtain a new upper bound for the distance between a controllable system and the set of uncontrollable ones, with no restriction onrkGi(Σ), i∈{1,…,ℓ}.Theorem 6.
Given a controllable system with subsystems defined by the set of matrices{(Ai,Bi)i∈{1,…,ℓ}},
(43)μℝ({(Ai,Bi)i∈{1,…,ℓ}})≤d({(Ai,Bi)i∈{1,…,ℓ}},𝒳),
where 𝒳={{(Ai,Bi)i∈{1,…,ℓ}}|∃λ1,…,λℓ∈ℝ such that rkM(Σ,λ1,…,λℓ)≤n-1}.Proof.
It is easily deduced from Lemma4.In the particular case of an uncontrollable system withrkG(Σ)=n-1, the results above may be summarized as follows.Theorem 7.
Given an uncontrollable switched linear system with subsystems defined by the set of matrices{(Ai,Bi)i∈{1,…,ℓ}} with rkG(Σ)=n-1, there exist λ10,…,λℓ0∈ℝ such that rkM(Σ,λ10,…,λℓ0)=n-1.Proof.
Let us consider the controllability canonical form of{(Ai,Bi)i∈{1,…,ℓ}}(44){((Ai1Ai20Ai3),(Bi10))i∈{1,…,ℓ}}.
Since rkG(Σ)=n-1,   Ai3=(ai3), 1≤i≤ℓ, it suffices to take λ10=a13, …, λℓ0=aℓ3.Remark 8.
Note that given a controllable system (rkG(Σ)=n) it is possible that the nearest uncontrollable one is a system Σ′ such that rkG(Σ′)<n-1. For example, let us consider the system with subsystems defined by
(45)A1=(000100010),B1=(001),A2=(000100010),B2=(0.100).
It is easy to check that this system is controllable and that the nearest uncontrollable one is a systemΣ′ with subsystems defined by
(46)A1=(000100010),B1=(001),A2=(000100010),B2=(000),
which is such that rkG(Σ′)=1.
More concretely, there is no uncontrollable system with distance (with regard to the Frobenius norm) less than0.1, from the first system.Finally, the following upper bound is obtained.Theorem 9.
Given a controllable systemΣ with subsystems defined by the set of matrices {(Ai,Bi)i∈{1,…,ℓ}}, the distance (with regard to the Frobenius norm) from this system to the nearest uncontrollable one is
(47)μℝ({(Ai,Bi)i∈{1,…,ℓ}})≤minλ1,…,λℓ∈ℝσn(M(Σ,λ1,…,λℓ))
and the distance (with regard to the 2-norm) from this system to the nearest uncontrollable one is
(48)μℝ({(Ai,Bi)i∈{1,…,ℓ}})≤minλ1,…,λℓ∈ℝσn(M(Σ,λ1,…,λℓ)),
where σn(M(Σ,λ1,…,λℓ)) is the smallest singular value of the matrix M(Σ,λ1,…,λℓ).Proof.
The statement summarizes the results above. We apply theorem in the appendix (Eckart-Young-Mirsky matrix approximation theorem) to the matrixM(Σ,λ1,…,λℓ) which has full rank r=n for any values of λ1,…,λℓ∈ℝ, because system Σ is controllable. A perturbed uncontrollable system Σ′ would have associated the matrix M(Σ′,λ1,…,λℓ) with rank k=n-1. Theorem in the appendix shows that the minimal distance from M(Σ,λ1,…,λℓ) to M(Σ′,λ1,…,λℓ) (with regard to the Frobenius norm or the 2-norm) is equal to the smallest singular value of matrix M(Σ,λ1,…,λℓ). The nearest uncontrollable system is obtained minimizing the values of λ1,…,λℓ∈ℝ.


## 5. Examples

Example 1.
Let us consider a boost converter (see [17]) which is represented in Figure 1 and with the equations
(49)(e˙C(t)i˙L(t))=(-1RC1C-1L0)(eC(t)iL(t))+(0eSL)
if s(t)=0, and
(50)(e˙C(t)i˙L(t))=(-1RC000)(eC(t)iL(t))+(0eSL)
if s(t)=1.Figure 1
Boost converter.Denotingt=τT, with T being the switching time period,
(51)(e˙C(τ)i˙L(τ))=(-TRCTC-TL0)(eC(τ)iL(τ))+(0TLeS)
if s(t)=0, and
(52)(e˙C(τ)i˙L(τ))=(-TRC000)(eC(τ)iL(τ))+(0TLeS)
if s(t)=1.Thus the system can be described as a switched linear systemΣ(R,C,L,T,eS):
(53)x˙(τ)=A1x(τ)+B1u(τ),x˙(τ)=A2x(τ)+B2u(τ)
with u(τ)=1,
(54)A1=(-TRCTC-TL0),B1=(0TLeS),A2=(-TRC000),B2=(0TLeS).Assume thateS(t)=5 V, L=0.75mH, the switching time period is T=20 μs, C=60μF, and R=20Ω. Then the system obtained, which we will denoted by Σ, has
(55)A1=(-16013-2750),B1=(0215),A2=(-160000),B2=(0215).
In this case,
(56)G1(Σ)=(B1A1B1B2A1B2)=(0245024521502150),G2(Σ)=(B1A2B1B2A2B2)=(000021502150).SystemΣ is controllable, since rkG(Σ)=rk(G1(Σ)∣G2(Σ))=2. Note that Theorem 3 cannot be applied because rkG2(Σ)<2, but applying Theorem 9, an upper bound for μℝ({(Ai,Bi)i∈{1,2}}) with regard to the 2-norm can be obtained:
(57)minλ1,λ2∈ℝσ2(M(Σ,λ1,λ2))=0.186217.Note that for any real value ofR≠0, C≠0, L≠0, T≠0 and for any constant function eS≠0, system Σ(R,C,L,T,eS) is controllable. In order to obtain an uncontrollable system we should apply a parameter variation to the values of the parameters, setting eS=0, or T=0, or L=∞, or C=∞. Straightforward calculations show that the distance, with regard to the 2-norm, from system Σ to a system having any of these values is at least 0.188562.This ad hoc study of this circuit leads to the conclusion that the obtained upper bound (0.186217) is useful to guarantee that, if we take any system (with physical meaning) at a distance from the systemΣ less than this upper bound, it will be controllable. That is to say, we can obtain any desired value for the voltage over the load R, with an appropriate source voltage eS(t) and an appropriate switching signal.Example 2.
Let us consider the following first order time-varying plant (see [18]) with the equations
(58)y˙+2y=0.1u,y˙+0.25y=10u.Adopting a pole placement design, being the goal to place the poles of the closed-loop system at fixed locations, if we choose the state vector asx=(u˙,u,y)t, the following state space representation of the closed-loop system Σ is obtained:
(59)x˙(t)=A1x(t)+B1uc(t),x˙(t)=A2x(t)+B2uc(t)
with uc(t)=1,
(60)A1=(-4.41.56-43.210000.1-2),B1=(1200),A2=(-6.15-5.70.0225100010-0.25),B2=(0.1200).
In this case,(61)G1(Σ)=(B1A1B1A12B1A2B1A1A2B1A12A2B1B2A1B2A12B2A2B2A1A2B2A12A2B2)=(12-52.8251.0-73.8343.4-1678.10.1-0.52.5-0.73.4-16.8012-52.812-73.8343.400.1-0.50.1-0.73.4001.201.2-9.8000.000.0-0.1),(62)G2(Σ)=(B1A2B1A22B1A1B1A2A1B1A22A1B1B2A2B2A22B2A1B2A2A1B2A22A1B2)=(12-73.8385.5-52.8256.3-1272.70.1-0.73.9-0.52.6-12.7012-73.812-52.8256.300.1-0.70.1-0.52.6001200120-558001.201.2-5.6).SincerkG(Σ)=rk(G1(Σ)∣G2(Σ))=3, system Σ is controllable. In this case, rkG1(Σ)rkG2(Σ)=3. Applying Theorem 3, an upper bound for μℝ({(Ai,Bi)i∈{1,2}}) with regard to the 2-norm is obtained:
(63)∑i∈{1,2}‍σ3(1+∥AiC∥2σ3,i)=123.756921,
and applying Theorem 9, an upper bound for μℝ({(Ai,Bi)i∈{1,2}}) with regard to the 2-norm is obtained:
(64)minλ1,λ2∈ℝσ3(M(Σ,λ1,λ2))=1.359749.
Thus, a substantial improvement for the upper bound has been achieved.The ad hoc study of the closed-loop systemΣ leads to the conclusion that the obtained upper bound (1.359749) is useful to guarantee that, if we take any system (with physical meaning) at a distance from the system Σ less than this upper bound, it will be controllable.


## 6. Conclusions

The main contribution of the paper is to obtain (Theorem9) an upper bound for the distance from a controllable switched linear system to the set of uncontrollable ones, generalizing the upper bound obtained by D. Boley and W. S. Lu for a standard system. Up to now no bound for a switched linear system has been obtained before that in [16] and the one presented here, which improves it. On the other hand, this upper bound can be effectively computed with the current resources.



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*Source: 948147-2013-05-27.xml*