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

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, 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), 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.

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 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡) (1) with 𝐴∈𝑀_𝑛×𝑛(ℂ) and 𝐵∈𝑀_𝑛×𝑝(ℂ) to the set of uncontrollable ones, denoted by 𝜇_ℂ(𝐴,𝐵), is equivalent to computing the minimum of the smallest singular value of (𝐴−𝜆𝐼_𝑛∣𝐵) over 𝜆∈ℂ: 𝜇_ℂ(𝐴,𝐵)=min_𝜆∈ℂ𝜎_𝑛𝐴−𝜆𝐼_𝑛∣𝐵, (2) this means that there aren't uncontrollable systems in a ball centered in (𝐴,𝐵) with radius 𝜇_ℂ(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 Section 4, 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 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡) (3) with 𝐴∈𝑀_𝑛×𝑛(ℝ) and 𝐵∈𝑀_𝑛×𝑝(ℝ), the distance from this system to the set of uncontrollable (real) systems and denote it by 𝜇_ℝ(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]) 𝜇_ℝ(𝐴,𝐵)≤min_𝜆∈ℝ𝜎_𝑛𝐴−𝜆𝐼_𝑛∣𝐵 (4) and the equality is not true in general (see the example in [2, page 557]). Obviously 𝜇_ℂ(𝐴,𝐵)≤𝜇_ℝ(𝐴,𝐵). In particular, any upper bound for 𝜇_ℝ(𝐴,𝐵) is also an upper bound for 𝜇_ℂ(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 weighted ℋ_∞ 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 Section 2, we summarize the definitions and some properties of norm matrices. In Section 3, we lay the foundations of the problem to be solved. In Section 4, 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 Section 5, we consider different examples.

2. Preliminaries

Let us consider the vector space of 𝑚×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 matrix 𝑀, ‖𝑀‖≥0 and ‖𝑀‖=0 if, and only if, 𝑀=0.(2)For all matrix 𝑀 and 𝜆∈𝑘, ‖𝜆𝑀‖=|𝜆|‖𝑀‖.(3)For all matrices 𝑀₁ and 𝑀₂, ‖𝑀₁+𝑀₂‖≤‖𝑀₁‖+‖𝑀₂‖.

The most frequently used matrix norms are the Frobenius norm and the 𝑝-norms.

The Frobenius norm of a matrix 𝑀=(𝑚^𝑖_𝑗)_{1≤𝑖≤𝑚1≤𝑗≤𝑛} is defined as ‖𝑀‖_𝐹=_{1≤𝑖≤𝑚1≤𝑗≤𝑛}𝑚^𝑖_𝑗². (5)

The matrix 𝑝-norm is defined for a real number 1≤𝑝≤∞ as ‖𝑀‖_𝑝=sup_𝑥≠0‖𝑀𝑥‖_𝑝‖𝑥‖_𝑝=max_{‖𝑥‖𝑝=1}‖𝑀𝑥‖_𝑝, (6) where ‖𝑥‖_𝑝=𝑥₁^𝑝+⋯+𝑥_𝑛^𝑝1/𝑝,1≤𝑝<∞ (7) and ‖𝑥‖_∞=max_{1≤𝑖≤𝑛}|𝑥_𝑖|.

In particular, ‖𝑀‖₁=max_{1≤𝑗≤𝑛}∑^𝑚_𝑖=1|𝑚^𝑖_𝑗|.The spectral norm ‖‖₂ is the square root of the maximum eigenvalue of 𝑀^𝑡𝑀 in the real case and 𝑀^𝐻𝑀 in the complex case, where 𝑀^𝑡, 𝑀^𝐻 denote the transpose and the conjugate transpose of the matrix 𝑀, respectively. And ‖𝑀‖_∞=max_{1≤𝑖≤𝑚}∑_{1≤𝑗≤𝑛}|𝑚^𝑖_𝑗|.

The Frobenius and the 𝑝-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)max_𝑖,𝑗|𝑚^𝑖_𝑗|≤‖𝑀‖₂≤√𝑚𝑛max_𝑖,𝑗|𝑚^𝑖_𝑗|, (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 Section 4 can be applied. Let us consider a switched linear system Σ defined by 𝑥̇(𝑡)=𝐴_𝜃(𝑡)𝑥(𝑡)+𝐵_𝜃(𝑡)𝑢(𝑡), (8) where 𝐴_𝜃(𝑡)∈𝑀_𝑛×𝑛(ℝ),𝐵_𝜃(𝑡)∈𝑀_𝑛×𝑝(ℝ),𝑥(𝑡)∈ℝ^𝑛 is the state; 𝑢(𝑡)∈ℝ^𝑝 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∶[𝑡₀,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 lim_𝑠→𝑡+𝜃(𝑠) and lim_{𝑠→𝑡−}𝜃(𝑠) exist and the set 𝑡∈𝑡₀,𝑇∣lim_𝑠→𝑡+𝜃(𝑠)≠lim_{𝑠→𝑡−}𝜃(𝑠) (9) is finite for any finite sub interval of [𝑡₀,𝑇) (in the case where 𝑡=𝑡₀, we will consider lim_𝑠→𝑡⁻₀𝜃(𝑠)=𝜃(𝑡₀)).

Sun-Ge proved in [15] that the system above is controllable if, and only if, ℝ^𝑛 is equal to the vector space 𝒱=^{𝑗1,…,𝑗𝑛−1∈{0,…,𝑛−1}}_{𝑖0,…,𝑖𝑛−1∈{}1,…,ℓ}Im𝐴^𝑗𝑛−1_𝑖𝑛−1⋯𝐴^𝑗1_𝑖1𝐵_𝑖0, (10) where Im(𝑀) stands for image of matrix 𝑀. Hence controllability depends only on the set of matrices {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}} defining the subsystems of Σ.

Therefore, the set of matrices {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}} defining subsystems of a controllable system is an open dense subset of 𝑀_𝑛×𝑛(ℝ)^ℓ×𝑀_𝑛×𝑝(ℝ)^ℓ. In particular, for each set of matrices {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}} there exists an open neighborhood of this set in 𝑀_𝑛×𝑛(ℝ)^ℓ×𝑀_𝑛×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 {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}} is taken as 𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}}=𝐴₁𝐵₁⋯𝐴_ℓ𝐵_ℓ (11) and thus the distance between two sets of matrices is 𝑑𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}},𝐴^′_𝑖,𝐵^′_{𝑖𝑖∈{1,…,ℓ}}=𝐴^′_𝑖−𝐴_𝑖,𝐵^′_𝑖−𝐵_{𝑖𝑖∈{1,…,ℓ}}. (12)

Given a controllable system Σ with subsystems defined by the set of matrices {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}, the distance (with regard to a given matrix norm) from this system to the nearest uncontrollable one is 𝜇_ℝ𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}}=min_{𝛿𝐴𝑖∈𝑀𝑛×𝑛(ℝ),𝛿𝐵𝑖∈𝑀𝑛×𝑝(ℝ)}𝛿𝐴_𝑖,𝛿𝐵_{𝑖𝑖∈{1,…,ℓ}} (13) with {(𝐴_𝑖+𝛿𝐴_𝑖,𝐵_𝑖+𝛿𝐵_𝑖)_{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 𝜇_ℂ𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}}=min_{𝛿𝐴𝑖∈𝑀𝑛×𝑛(ℂ),𝛿𝐵𝑖∈𝑀𝑛×𝑝(ℂ)}𝛿𝐴_𝑖,𝛿𝐵_{𝑖𝑖∈{1,…,ℓ}} (14) with {(𝐴_𝑖+𝛿𝐴_𝑖,𝐵_𝑖+𝛿𝐵_𝑖)_{𝑖∈{1,…,ℓ}}} being the set of matrices defining the subsystems of an uncontrollable system, then this distance 𝜇_ℂ({(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}) is not equal to min_{𝜆1,…,𝜆ℓ∈ℂ}𝜎_𝑛𝐴₁−𝜆₁𝐼_𝑛𝐵₁⋯𝐴_ℓ−𝜆_ℓ𝐼_𝑛𝐵_ℓ (15) as the following example shows.

Example 1. Consider a switched linear system with two subsystems defined by matrices 𝐴₁=0000200.112,𝐴₂=,𝐵011001001₁=𝐵₂=100. (16) This system is controllable, as it is straightforward to be checked (dim𝒱=3) and min_{𝜆1,𝜆2∈ℂ}𝜎₃𝐴₁−𝜆₁𝐼₃𝐵₁⋯𝐴₂−𝜆₂𝐼₃𝐵₂=min_{𝜆1,𝜆2∈ℂ}𝜎₃𝜆₁001−𝜆₂11102−𝜆₁000−𝜆₂100.112−𝜆₁0001−𝜆₂0=0.348160, (17) but the system consisting of two subsystems defined by 𝐴₁=000020012,𝐴₂=,𝐵011001001₁=𝐵₂=100, (18) 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 𝜇_ℝ𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}}≤min_{𝜆1,…,𝜆ℓ∈ℝ}𝜎_𝑛𝐴₁−𝜆₁𝐼_𝑛𝐵₁⋯𝐴_ℓ−𝜆_ℓ𝐼_𝑛𝐵_ℓ. (19) 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 Im𝐴_{𝑘1,…,𝑘ℓ∈{}𝑘0,11+⋯+𝑘ℓ𝑗=11,…,𝑗𝑛−2∈{}𝑗0,1,…,𝑛−21+⋯+𝑗𝑛−2ℎ≤𝑛−21,…,ℎ𝑛−2∈{}𝐴1,…,ℓ𝑖≠𝐴ℎ10≤𝑗≤𝑛−1}^𝑗_𝑖𝐴^𝑗1_ℎ1⋯𝐴^𝑗𝑛−2_ℎ𝑛−2𝐵^𝑘1₁⋯𝐵^𝑘ℓ_ℓ (20) and let us denote by 𝐺_𝑖(Σ) the matrix having as columns the generators of the vector subspace above and by 𝐺(Σ) the matrix 𝐺(Σ)=𝐺₁(Σ)∣⋯∣𝐺_ℓ(Σ). (21)
Throughout the paper, the notation rk𝑀 will denote the rank of matrix 𝑀.

Lemma 2. The rank of matrix 𝐺(Σ) (written rk𝐺(Σ)) is equal to the dimension of the vector space Im𝐴_{𝑘1,…,𝑘ℓ∈{}𝑘0,11+⋯+𝑘ℓ𝑗=11,…,𝑗𝑛−1∈{}𝑖0,1,…,𝑛−11,…,𝑖𝑛−1∈{}1,…,ℓ}^𝑗1_𝑖1⋯𝐴^𝑗𝑛−1_𝑖𝑛−1𝐵^𝑘1₁⋯𝐵^𝑘ℓ_ℓ. (22)

Proof. It is a straightforward consequence of Cayley-Hamilton Theorem and the fact that this vector subspace and 𝒱 defined above are the same subspace.

As a consequence of Lemma 2, 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 matrix 𝐺(Σ), in a similar way to the one in [3], in the case where rk𝐺_𝑖(Σ)=𝑛, for all 𝑖∈{1,…,ℓ}. Namely, let 𝑃, 𝑄 be orthogonal matrices such that 𝐺(Σ)=𝑃^𝑡diag𝜎₁,…,𝜎_𝑛∣0𝑄, (23) where 𝜎₁≥⋯≥𝜎_𝑛 are the singular values of 𝐺(Σ). Then the following statement holds.

Theorem 3 (see [16]). Given a controllable switched linear system Σ consisting of subsystems defined by a set of matrices {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}, such that rk𝐺_𝑖(Σ)=𝑛, for all 𝑖∈{1,…,ℓ}, with regard to the 2-norm, 𝜇_ℝ(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}≤_{{}𝑖∈1,…,ℓ}𝜎_𝑛𝐴1+^𝐶𝑖2𝜎_𝑛,𝑖, (24) where 𝐴^𝐶_𝑖 is the companion matrix for 𝐴_𝑖 and 𝜎_1,𝑖≥⋯≥𝜎_𝑛,𝑖 are the singular values of 𝐺_𝑖(Σ).

Remember that, given a square matrix 𝐴 with characteristic polynomial 𝑄(𝑡)=𝑡^𝑛+𝑎₁𝑡^𝑛−1+⋯+𝑎_𝑛−1𝑡+𝑎_𝑛, the matrix 00⋯0−𝑎_𝑛10⋯0−𝑎_𝑛−101⋯0−𝑎_𝑛−2⋯⋯⋯⋯00⋯1−𝑎₁ (25) is called the companion matrix of matrix 𝐴.

Proof. First of all, note that {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}} are associated to a controllable system if, and only if, {(𝑃𝐴_𝑖𝑃^𝑡,𝑃𝐵_𝑖)_{𝑖∈{1,…,ℓ}}} are also associated to a controllable system.
Denoting by 𝒫ℓdiag𝑃^𝑡𝐼_𝑝𝑃,…,^𝑡𝐼_𝑝 (26) and since 𝑃𝐴₁−𝑋₁𝐵₁−𝑌₁⋯𝐴_ℓ−𝑋_ℓ𝐵_ℓ−𝑌_ℓ𝒫=𝑃𝐴₁−𝑋₁𝑃^𝑡𝑃𝐵₁−𝑌₁⋯𝑃𝐴_ℓ−𝑋_ℓ𝑃^𝑡𝑃𝐵_ℓ−𝑌_ℓ (27) and 𝑃 and 𝒫 are orthogonal matrices, we deduce that 𝐴₁−𝑋₁𝐵₁−𝑌₁⋯𝐴_ℓ−𝑋_ℓ𝐵_ℓ−𝑌_ℓ2=𝑃𝐴₁−𝑋₁𝑃^𝑡𝑃𝐵₁−𝑌₁⋯𝑃𝐴_ℓ−𝑋_ℓ𝑃^𝑡𝑃𝐵_ℓ−𝑌_ℓ2. (28) Therefore 𝜇_ℝ𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}}=𝜇_ℝ𝑃𝐴_𝑖𝑃^𝑡,𝑃𝐵_{𝑖𝑖∈{1,…,ℓ}}. (29)
Let us consider 𝑃𝐵_𝑖=^𝐵1𝑖_𝐵2𝑖, with 𝐵¹_𝑖∈𝑀_{(𝑛−1)×𝑝}(ℝ), for 𝑖∈{1,…,ℓ}.
We can write 𝐵²_𝑖=𝑒^𝑡_𝑛𝑃𝐵_𝑖=𝑒^𝑡_𝑛𝑃𝐺(Σ)0⋯0𝐼^𝑗𝑖_𝑝0⋯0^𝑡=𝑒^𝑡_𝑛diag𝜎₁,…,𝜎_𝑛∣0𝑄0⋯0𝐼^𝑗𝑖_𝑝0⋯0^𝑡, (30) where 𝑗_𝑖 indicates the column 𝑗_𝑖 of matrix 𝐺(Σ) where matrix 𝐵_𝑖 first appears, with 𝑒₁,…,𝑒_𝑛 being the natural basis of the Euclidean space ℝ^𝑛. Then 𝐵²_𝑖2≤𝜎_𝑛𝑄0⋯0𝐼^𝑗𝑖_𝑝0⋯0^𝑡₂=𝜎_𝑛 (31) since 𝑄 is an orthogonal matrix.
It is straightforward to check that 𝐴_𝑖𝐺_𝑖(Σ)=𝐺_𝑖(Σ)(𝐼⊗𝐴^𝐶_𝑖), where 𝐼 is the identity matrix of suitable order. Then if we denote by 𝐺_𝑖(Σ)^† the Moore-Penrose inverse of matrix 𝐺_𝑖(Σ), 𝐴_𝑖=𝐺_𝑖(Σ)(𝐼⊗𝐴^𝐶_𝑖)𝐺_𝑖(Σ)^† and 𝑃𝐴_𝑖𝑃^𝑡=𝑃𝐺_𝑖(Σ)𝐼⊗𝐴^𝐶_𝑖𝐺_𝑖(Σ)^†𝑃^𝑡=𝑃𝐺(Σ)0⋯0𝐼^(𝑖)_𝑐𝑖0⋯0^𝑡𝐼⊗𝐴^𝐶_𝑖𝐺_𝑖(Σ)^†𝑃^𝑡=diag𝜎₁,…,𝜎_𝑛∣0𝑄0⋯0𝐼^(𝑖)_𝑐𝑖0⋯0^𝑡×𝐼⊗𝐴^𝐶_𝑖𝐺_𝑖(Σ)^†𝑃^𝑡, (32) where 𝑐_𝑖 is the number of columns of matrix 𝐺_𝑖(Σ).
In an analogous way, let us consider 𝑃𝐴_𝑖𝑃^𝑡=𝐴¹_𝑖𝐴²_𝑖𝐴³_𝑖𝐴⁴_𝑖 (33) with 𝐴¹_𝑖∈𝑀_{(𝑛−1)×(𝑛−1)}(ℝ), 𝐴³_𝑖=𝑒^𝑡_𝑛diag𝜎₁,…,𝜎_𝑛𝑄×∣00⋯0𝐼^(𝑖)_𝑐𝑖0⋯0^𝑡𝐼⊗𝐴^𝐶_𝑖𝑄^𝑡_𝑖×1diag𝜎_1,𝑖1,…,𝜎_𝑛,𝑖0𝑃_𝑖𝑃^𝑡𝐼_𝑛−10, (34) where 𝜎_1,𝑖≥⋯≥𝜎_𝑛,𝑖, are the singular values of 𝐺_𝑖(Σ) (𝐺_𝑖(Σ)=𝑃^𝑡_𝑖(diag(𝜎_1,𝑖,…,𝜎_𝑛,𝑖)|0)𝑄_𝑖). Therefore 𝐴³_𝑖2≤𝜎_𝑛𝐼⊗𝐴^𝐶_𝑖2𝐺_𝑖(Σ)^†₂≤𝜎_𝑛𝜎_𝑛,𝑖𝐴^𝐶_𝑖2. (35)

Let us set the blocks 𝐵²_𝑖, 𝐴³_𝑖 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 𝜎_𝑛+(‖𝐴^𝐶_𝑖‖₂𝜎_𝑛/𝜎_(𝑛,𝑖)), for all 𝑖∈{1,…,ℓ}, and the statement follows.

Observe that, in the case where ℓ=1 (there is only one subsystem) we obtain the same result as that in [3] because the product 𝑃_𝑖𝑃^𝑡 in (34) is equal to 𝐼_𝑛, and therefore ‖𝐴³_𝑖‖₂≤(𝜎_𝑛/𝜎_𝑛−1)‖𝐴^𝐶_𝑖‖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 {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}. Given 𝜆₁,…,𝜆_ℓ∈ℝ, we will denote by 𝑀(Σ,𝜆₁,…,𝜆_ℓ) the following matrix: 𝑀Σ,𝜆₁,…,𝜆_ℓ=𝐴₁−𝜆₁𝐼_𝑛𝐵₁⋯𝐴_ℓ−𝜆_ℓ𝐼_𝑛𝐵_ℓ. (36)

Lemma 4. Given a switched linear system Σ with subsystems defined by a set of matrices {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}, rk𝐺(Σ)≤rk𝑀Σ,𝜆₁,…,𝜆_ℓ,∀𝜆₁,…,𝜆_ℓ∈ℝ^ℓ. (37)

Proof. Let us assume that 𝜆⁰₁,…,𝜆⁰_ℓ∈ℝ are such that min_{𝜆1,…,𝜆ℓ∈ℝ}rk𝑀Σ,𝜆₁,…,𝜆_ℓ=rk𝑀Σ,𝜆⁰₁,…,𝜆⁰_ℓ. (38) Note that in the case where rk𝑀(Σ,𝜆⁰₁,…,𝜆⁰_ℓ)=𝑛, the statement is obvious. Let us assume that the rank of 𝑀(Σ,𝜆⁰₁,…,𝜆⁰_ℓ)=𝑛−𝑚, 𝑚>0. Then there exist linearly independent vectors 𝑣₁,…,𝑣_𝑚∈ℝ^𝑛 such that 𝑣^𝑡₁𝑀Σ,𝜆⁰₁,…,𝜆⁰_ℓ=⋯=𝑣^𝑡_𝑚𝑀Σ,𝜆⁰₁,…,𝜆⁰_ℓ=0. (39) Therefore 𝑣^𝑡_𝑖𝐴₁−𝜆₁𝐼_𝑛=⋯=𝑣^𝑡_𝑖𝐴_ℓ−𝜆_ℓ𝐼_𝑛{},𝑣=0,𝑖∈1,…,𝑚_𝑡𝑖𝐵₁=⋯=𝑣^𝑡_𝑖𝐵_ℓ=0,𝑖∈{1,…,𝑚}. (40)
It is straightforward to check that 𝑣₁,…,𝑣_𝑚 are left eigenvectors of matrices 𝐴₁,…,𝐴_ℓ and therefore 𝑣^𝑡₁𝐺(Σ)=⋯=𝑣^𝑡_𝑚𝐺(Σ)=0. (41)
Thus rk𝐺(Σ)≤𝑛−𝑚.

As a consequence of Lemma 4, we can conclude that if there exist 𝜆⁰₁,…,𝜆⁰_ℓ∈ℝ such that rk𝑀(Σ,𝜆⁰₁,…,𝜆⁰_ℓ)<𝑛, the system can not be controllable.

Remark 5. Note that there exist uncontrollable systems such that for all 𝜆₁,…,𝜆_ℓ∈ℝ, rk𝑀(Σ,𝜆₁,…,𝜆_ℓ)=𝑛. For example, we consider a system Σ with subsystems defined by matrices 𝐴₁=000020012,𝐴₂=,𝐵011001001₁=𝐵₂=100. (42) 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 on rk𝐺_𝑖(Σ), 𝑖∈{1,…,ℓ}.

Theorem 6. Given a controllable system with subsystems defined by the set of matrices {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}, 𝜇_ℝ𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}}≤𝑑𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}},𝒳, (43) where 𝒳={{(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}|∃𝜆₁,…,𝜆_ℓ∈ℝ such that rk𝑀(Σ,𝜆₁,…,𝜆_ℓ)≤𝑛−1}.

Proof. It is easily deduced from Lemma 4.

In the particular case of an uncontrollable system with rk𝐺(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 {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}} with rk𝐺(Σ)=𝑛−1, there exist 𝜆⁰₁,…,𝜆⁰_ℓ∈ℝ such that rk𝑀(Σ,𝜆⁰₁,…,𝜆⁰_ℓ)=𝑛−1.

Proof. Let us consider the controllability canonical form of {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}𝐴¹_𝑖𝐴²_𝑖0𝐴³_𝑖,𝐵¹_𝑖0_{𝑖∈{1,…,ℓ}}. (44) Since rk𝐺(Σ)=𝑛−1, 𝐴³_𝑖=𝑎³_𝑖, 1≤𝑖≤ℓ, it suffices to take 𝜆⁰₁=𝑎³₁, …, 𝜆⁰_ℓ=𝑎³_ℓ.

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 𝐴₁=000100010,𝐵₁=001,𝐴₂=000100010,𝐵₂=00.0.1 (45)
It is easy to check that this system is controllable and that the nearest uncontrollable one is a system Σ^′ with subsystems defined by 𝐴₁=000100010,𝐵₁=001,𝐴₂=000100010,𝐵₂=000, (46) which is such that rk𝐺(Σ^′)=1.
More concretely, there is no uncontrollable system with distance (with regard to the Frobenius norm) less than 0.1, from the first system.

Finally, the following upper bound is obtained.

Theorem 9. Given a controllable system Σ with subsystems defined by the set of matrices {(𝐴_𝑖,𝐵_𝑖)_{𝑖∈{1,…,ℓ}}}, the distance (with regard to the Frobenius norm) from this system to the nearest uncontrollable one is 𝜇_ℝ𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}}≤min_{𝜆1,…,𝜆ℓ∈ℝ}𝜎_𝑛𝑀Σ,𝜆₁,…,𝜆_ℓ (47) and the distance (with regard to the 2-norm) from this system to the nearest uncontrollable one is 𝜇_ℝ𝐴_𝑖,𝐵_{𝑖𝑖∈{1,…,ℓ}}≤min_{𝜆1,…,𝜆ℓ∈ℝ}𝜎_𝑛𝑀Σ,𝜆₁,…,𝜆_ℓ, (48) where 𝜎_𝑛(𝑀(Σ,𝜆₁,…,𝜆_ℓ)) is the smallest singular value of the matrix 𝑀(Σ,𝜆₁,…,𝜆_ℓ).

Proof. The statement summarizes the results above. We apply theorem in the appendix (Eckart-Young-Mirsky matrix approximation theorem) to the matrix 𝑀(Σ,𝜆₁,…,𝜆_ℓ) which has full rank 𝑟=𝑛 for any values of 𝜆₁,…,𝜆_ℓ∈ℝ, because system Σ is controllable. A perturbed uncontrollable system Σ^′ would have associated the matrix 𝑀(Σ^′,𝜆₁,…,𝜆_ℓ) with rank 𝑘=𝑛−1. Theorem in the appendix shows that the minimal distance from 𝑀(Σ,𝜆₁,…,𝜆_ℓ) to 𝑀(Σ^′,𝜆₁,…,𝜆_ℓ) (with regard to the Frobenius norm or the 2-norm) is equal to the smallest singular value of matrix 𝑀(Σ,𝜆₁,…,𝜆_ℓ). The nearest uncontrollable system is obtained minimizing the values of 𝜆₁,…,𝜆_ℓ∈ℝ.

5. Examples

Example 1. Let us consider a boost converter (see [17]) which is represented in Figure 1 and with the equations 𝑒_𝐶̇(𝑡)𝑖_𝐿̇(𝑡)=−11𝑅𝐶𝐶−1𝐿0𝑒_𝐶(𝑡)𝑖_𝐿(𝑡)+0𝑒_𝑆𝐿 (49) if 𝑠(𝑡)=0, and 𝑒_𝐶̇(𝑡)𝑖_𝐿̇(𝑡)=−10𝑒𝑅𝐶00^𝐶(𝑡)𝑖_𝐿(𝑡)+0𝑒_𝑆𝐿 (50) if 𝑠(𝑡)=1.

Figure 1 

Boost converter.

Denoting 𝑡=𝜏𝑇, with 𝑇 being the switching time period, 𝑒_𝐶̇(𝜏)𝑖_𝐿̇(𝜏)=−𝑇𝑇𝑅𝐶𝐶−𝑇𝐿0𝑒_𝐶(𝜏)𝑖_𝐿(𝜏)+0𝑇𝐿𝑒_𝑆 (51) if 𝑠(𝑡)=0, and 𝑒_𝐶̇(𝜏)𝑖_𝐿̇(𝜏)=−𝑇0𝑒𝑅𝐶00^𝐶(𝜏)𝑖_𝐿(𝜏)+0𝑇𝐿𝑒_𝑆 (52) if 𝑠(𝑡)=1.

Thus the system can be described as a switched linear system Σ(𝑅,𝐶,𝐿,𝑇,𝑒_𝑆): 𝑥̇(𝜏)=𝐴₁𝑥(𝜏)+𝐵₁𝑢(𝜏),𝑥̇(𝜏)=𝐴₂𝑥(𝜏)+𝐵₂𝑢(𝜏) (53) with 𝑢(𝜏)=1, 𝐴₁=−𝑇𝑇𝑅𝐶𝐶−𝑇𝐿0,𝐵₁=0𝑇𝐿𝑒_𝑆,𝐴₂=−𝑇0𝑅𝐶00,𝐵₂=0𝑇𝐿𝑒_𝑆. (54)

Assume that 𝑒_𝑆(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 𝐴₁=−11603−2075,𝐵₁=02,𝐴15₂=−106000,𝐵₂=02.15 (55) In this case, 𝐺₁(Σ)=𝐵₁𝐴₁𝐵₁𝐵₂𝐴₁𝐵₂=02024524502150,𝐺15₂(Σ)=𝐵₁𝐴₂𝐵₁𝐵₂𝐴₂𝐵₂=2000002150.15 (56)

System Σ 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 𝜇_ℝ({(𝐴_𝑖,𝐵_𝑖)_𝑖∈{1,2}}) with regard to the 2-norm can be obtained: min_{𝜆1,𝜆2∈ℝ}𝜎₂𝑀Σ,𝜆₁,𝜆₂=0.186217. (57)

Note that for any real value of 𝑅≠0, 𝐶≠0, 𝐿≠0, 𝑇≠0 and for any constant function 𝑒_𝑆≠0, system Σ(𝑅,𝐶,𝐿,𝑇,𝑒_𝑆) is controllable. In order to obtain an uncontrollable system we should apply a parameter variation to the values of the parameters, setting 𝑒_𝑆=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 system Σ 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 𝑒_𝑆(𝑡) and an appropriate switching signal.

Example 2. Let us consider the following first order time-varying plant (see [18]) with the equations 𝑦̇+2𝑦=0.1𝑢,𝑦̇+0.25𝑦=10𝑢. (58)

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 as 𝑥=(𝑢̇,𝑢,𝑦)^𝑡, the following state space representation of the closed-loop system Σ is obtained: 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐵₁𝑢_𝑐(𝑡),𝑥̇(𝑡)=𝐴₂𝑥(𝑡)+𝐵₂𝑢_𝑐(𝑡) (59) with 𝑢_𝑐(𝑡)=1, 𝐴₁=−4.41.56−43.210000.1−2,𝐵₁=00,𝐴12₂=−6.15−5.70.0225100010−0.25,𝐵₂=00.0.12 (60) In this case,𝐺₁(Σ)=𝐵₁𝐴₁𝐵₁𝐴²₁𝐵₁𝐴₂𝐵₁𝐴₁𝐴₂𝐵₁𝐴²₁𝐴₂𝐵₁𝐵₂𝐴₁𝐵₂𝐴²₁𝐵₂𝐴₂𝐵₂𝐴₁𝐴₂𝐵₂𝐴²₁𝐴₂𝐵₂=,𝐺12−52.8251.0−73.8343.4−1678.10.1−0.52.5−0.73.4−16.8012−52.812−73.8343.400.1−0.50.1−0.73.4001.201.2−9.8000.000.0−0.1₂(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 (61)(62)

Since rk𝐺(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 𝜇_ℝ({(𝐴_𝑖,𝐵_𝑖)_𝑖∈{1,2}}) with regard to the 2-norm is obtained: _𝑖∈{1,2}𝜎₃𝐴1+^𝐶𝑖2𝜎_3,𝑖=123.756921, (63) and applying Theorem 9, an upper bound for 𝜇_ℝ({(𝐴_𝑖,𝐵_𝑖)_𝑖∈{1,2}}) with regard to the 2-norm is obtained: min_{𝜆1,𝜆2∈ℝ}𝜎₃𝑀Σ,𝜆₁,𝜆₂=1.359749. (64) Thus, a substantial improvement for the upper bound has been achieved.

The ad hoc study of the closed-loop system Σ 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 (Theorem 9) 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.

Appendix

Eckart-Young-Mirsky Matrix Approximation Theorem

Let the singular value decomposition (SVD) of 𝐶∈ℝ^𝑚×𝑛 be given by 𝐶=∑^𝑟_𝑖=1𝜎_𝑖𝑢_𝑖𝑣^𝑡_𝑖 with 𝑟=rk𝐶. If 𝑘<𝑟 and 𝐶_𝑘=∑^𝑘_𝑖=1𝜎_𝑖𝑢_𝑖𝑣^𝑡_𝑖, then min_rk𝐷=𝑘‖𝐶−𝐷‖₂=𝐶−𝐶_𝑘2=𝜎_𝑘+1,min_rk𝐷=𝑘‖𝐶−𝐷‖_𝐹=𝐶−𝐶_𝑘𝐹=^𝑞_𝑖=𝑘+1𝜎²_𝑖, (A.1) where 𝑞=min{𝑚,𝑛}.

This theorem can be found in [19].

Acknowledgment

This work was supported by MTM 2011-23892.