Controllability of Continuous Bimodal Linear Systems

Abstract

We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions.

1. Introduction

A switched linear system (see, e.g., [1]) is a hybrid system (see, e.g., [2]) which consists of several linear subsystems and a rule that orchestrates the switching among them. The classical piecewise linear systems (see [3]) occur when the switching law depends only on the state. In recent years, piecewise linear dynamical/control systems have again attracted the attention of the researchers, indeed, because they are the most natural extension to linear systems in order to capture nonlinear phenomena as, for instance, limit cycles, heteroclinic and homoclinic orbits, and strange attractors. Also, their application has been especially successful in many engineering problems, such as the analysis and design of electronic oscillators (see [4]) or control systems.

They consist of two or more linear subsystems, each one acting in a different region—separated from the others by hypersurfaces—of the entire state space. In the basic case of bimodal systems, a couple of linear subsystems act at each side of a hyperplane. One says that the piecewise linear system is continuous if both adjacent subsystems coincide on the separating hypersurface.

For elementary circuits (see [4]), the number of state variables is typically two or three. Moreover, most of the nonlinear behavior appears already in low dimensions, even in the bimodal case. For example, the study in [5] gives a complete characterization of the focus-center limit cycle bifurcation for planar bimodal linear systems. Later works (see [6–8]) extended this analysis to the 3D case. Contrarily to what happens in 2D, in 3D, a remarkable phenomenon occurs, involving bimodal linear systems. Thus, whereas in 2D, a point placed on the separating hyperplane is asymptotically stable when both subsystems are stable; in 3D, unstable global dynamics might arise under the same hypotheses.

Concerning structural stability (i.e., with regard to small perturbations), we point the reader to [9] for an account of generic properties and to [10] for a full characterization (both studies on the 2D case). About control systems see, for example, [11] (again for the planar case).

To investigate continuous bimodal systems, several authors have used reduced forms of the matrices involved, both for dynamical systems (e.g., in [4]) and control systems (e.g., in [12, 13]). Here, we will use the reduced forms in [14, 15]. For a MAPLE program useful to compute them in 2D and 3D, see http://www.ma1.upc.edu/joanr/html/cfbpwls.html.

The goal of this paper is to find explicit and effective criteria for controllability of continuous bimodal linear control systems (for a study of the discrete-time planar case, see [16], where explicit necessary and sufficient conditions for controllability and reachability are given). Other problems concerning discrete-time systems are considered in [17], where a controller is designed. In other cases as in [18] or in [19], the system is continuous and the controller is digital. A state feedback controller for piecewise linear systems is designed in [20]. Problems as stability and stabilization of switched systems are considered in [18, 21–23] or in [24]. In [25], the problem of controllability is addressed for linear time-invariant dynamical multiagent systems. Further contributions include switched systems with stochastic perturbations (see [26] and the references therein). In our approach, the key point in the study of the controllability relies on applying the implicit general conditions in [27] to the matrices in the reduced form given in [15]. We thoroughly achieve this objective for the 2D and 3D cases (Propositions 18 and 19) and obtain partial criteria for higher dimensions (Lemma 12 and Proposition 14).

We stress that we have included the unobservable case (which appears in a natural way, e.g., when considering parametrized families of systems), and we have proved that the study of the controllability for observable systems can be reduced to the unobservable ones (Proposition 17).

The structure of this paper is as follows. First, in Section 2, we recall the definition of continuous bimodal linear control system (CBLCS), the concept of controllability, and the implicit criterion in [27]. In Section 3, we provide the reduced forms for the matrices defining the subsystems of the CBLCS (see [15]). Later on, in Section 4 we obtain some partial criteria for controllability and uncontrollability of observable systems in higher dimensions and show that observable systems can, generically, become unobservable by adding the appropriate feedback (previously, Definitions 10 and 15 suit the necessary notions of observability and feedback system to the present context). As feedbacks preserve controllability (see Remark 16), there is no loss of generality in assuming that the system is unobservable. Subsequently, Section 5 tackles the controllability of unobservable CBLCS in dimensions 2 and 3. Last section illustrates the above results with an example of a 3D system (an occasional unobservable circuit). To complete the paper, conclusions appear at the end.

To summarize, the main contribution of this paper is twofold. First, we prove that one can reduce the study of the controllability of continuous bimodal linear control systems to the study of the controllability of the unobservable ones. Secondly, we provide effective and explicit criteria for controllability. Our main motivation is to give effective tools for the study of the controllability of systems of this kind (such as circuits; see Section 6). In comparison with the previous criteria in [27], the characterization of controllability given in this paper is simple and explicit, especially for dimensions 2 and 3. Up to our knowledge, there are no criteria for the study of the controllability of the systems at hand focused on unobservable systems. In addition, we discuss the practical use of the theoretic results presented by means of an example.

Throughout the paper, ℝ will denote the set of real numbers; 𝑀_𝑛×𝑚(ℝ) the set of matrices having 𝑛 rows, 𝑚 columns, and entries in ℝ; 𝑀^⊤∈𝑀_𝑚×𝑛(ℝ) the transpose matrix of 𝑀∈𝑀_𝑛×𝑚(ℝ). We will simply write 𝑀_𝑛(𝑅) when 𝑚=𝑛; 𝐺𝑙_𝑛(ℝ) will be the group of nonsingular matrices in 𝑀_𝑛(ℝ) and 𝐼_𝑛 will be its identity element. Finally, we will denote by 𝑒₁,…,𝑒_𝑛 the natural basis of the Euclidean space ℝ^𝑛.

2. Preliminaries

We recall that a (single) linear control system is a system of type 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), (1)𝐴∈𝑀_𝑛(ℝ),𝐵∈𝑀_𝑛×1(ℝ), and 𝐶∈𝑀_1×𝑛(ℝ), where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^⊤, 𝑥_𝑖=𝑥_𝑖(𝑡), 𝑖=1,…,𝑛 are the state variables, and 𝑢(𝑡) is the control function. In 1960, Kalman (see [28]) introduced the concept of controllability (see Definition 3) and showed that it is equivalent to the controllability matrix, defined as 𝒞(𝐴,𝐵)∶=𝐵𝐴𝐵𝐴²𝐵⋯𝐴^𝑛−1𝐵, (2) having maximal rank. Therefore, the above system is controllable if and only if rank𝒞(𝐴,𝐵)=𝑛. (3) The notion of controllability (Definition 1) is valid as well for bimodal systems and Theorem 4 below (see [27]) generalizes the characterization (3) in an implicit form.

In general, a bimodal linear control system consists of two subsystems 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡),𝑖=1,2 (4) acting on disjoint semispaces. It is called continuous if both subsystems coincide in the common separatrix boundary of these semispaces, which, in particular, implies that 𝐵₁=𝐵₂. Hence, one defines the following.

Definition 1. A continuous bimodal linear control system (CBLCS) is a system of the form𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐵₁𝑢(𝑡)(𝑡)𝐴,if𝑦≤0₂𝑥(𝑡)+𝐵₂𝑢(𝑡)(𝑡)𝑦(𝑡)(𝑡),,if𝑦>0=𝐶𝑥 (5a)(5b) where 𝐴₁,𝐴₂∈𝑀_𝑛(ℝ);𝐵₁,𝐵₂∈𝑀_𝑛×1(ℝ);𝐶∈𝑀_1×𝑛(ℝ), satisfying that both subsystems in (5a) coincide for 𝑦(𝑡)=0, that is to say𝐴₁𝑥=𝐴₂𝑥,whenever𝐶𝑥=0. (6)

Remark 2. In several real applications, the systems often include state delay (for recent works on switched systems with state delay, see [23, 29–33]). However, this aspect is beyond the scope of this paper and possible extensions in this sense are contemplated as future work.
As we have pointed, the notion of controllability in [28] is also valid for these systems, and so they are controllable if they can be driven from a given initial state to a given final state by means of an appropriate control. More precisely, we have the following.

Definition 3. One says that system (5a) and (5b) is (completely) controllable if for any pair of states (𝑥₀,𝑥_𝑓) there exists a locally integrable input 𝑢(𝑡) such that the solution of (5a) and (5b) transfers the state 𝑥(0)=𝑥₀ to the state 𝑥_𝑓, that is,𝑥(𝑇)=𝑥_𝑓 for some 𝑇>0.

Obviously, a necessary condition is 𝐵≠0. The next theorem extends to CBLCS the characterization of controllability given in [28] for single control systems.

Theorem 4 (Çamhbel et al., 2004; see [27]). Consider a CBLCS defined by (5a) and (5b) and the assumption (6). Let one denote by 𝑒∈𝑀_𝑛×1(ℝ) the matrix such that 𝑒𝐶=𝐴₂−𝐴₁. Then, this system is controllable if and only if the following conditions hold.(C1) The pair (𝐴₁,[𝐵∣𝑒]) is controllable. (C2) Whenever there exist 𝜆∈ℝ,𝑣∈ℝ^𝑛⧵{0}, and 𝜇₁,𝜇₂∈ℝ² such that 𝑣^⊤𝜇_𝑖𝜆𝐼_𝑛−𝐴_𝑖𝐵𝐶0=0,𝑖=1,2, (7) then, necessarily, 𝜇₁𝜇₂>0.

Remark 5. Concerning the above theorem, two points are worth mentioning.(1)Because of (6), 𝐴₂−𝐴₁ is at most of rank 1 and the kernel of 𝐶 is contained in the kernel of 𝐴₂−𝐴₁, and so the columns of 𝐴^⊤₂−𝐴^⊤₁ belong to the span of 𝐶^⊤∈ℝ^𝑛. Therefore, there exists a matrix 𝑒∈𝑀_𝑛×1(ℝ) such that 𝑒𝐶=𝐴₂−𝐴₁ (see [27]).(2)In condition (C1), 𝐵∶=[𝐵∣𝑒] denotes the 𝑛×2 matrix formed by the column matrices 𝐵 and 𝑒. Thus, this condition means that the controllability matrix 𝒞(𝐴,𝐵)—see (2)—has full rank.

Our aim is to derive, from Theorem 4, explicit criteria for controllability by means of simplifying the involved matrices. Firstly, note that, as an immediate consequence of Definition 3, linear transformations of the state variables do not change the controllable character of the system. Hence, one can assume 𝐶=10⋯0, (8) so that the separating subspace is Ker𝐶=𝑥∈ℝ^𝑛∶𝑥₁=0 (9) and (6) in Definition 1 becomes 𝐴₂𝑒_𝑖=𝐴₁𝑒_𝑖,2≤𝑖≤𝑛; (10) namely, 𝐴₁ and 𝐴₂ differ only in their first column, 𝐴₁𝑒₁ and 𝐴₂𝑒₁, respectively, and 𝑒=𝐴₂𝑒₁−𝐴₁𝑒₁. (11) From now on, these simplifications are assumed. Then we will refer to a CBLCS simply as the triple (𝐴₁,𝐴₂,𝐵) verifying (10).

3. Admissible Basis Changes

Here we introduce, in Definition 6, the admissible basis changes as those changes in the state variables space preserving the hyperplanes 𝑥₁=𝑘 and hence the matrix 𝐶=(10⋯0). Next, in Proposition 8, we see that Conditions (C1) and (C2) of Theorem 4 are invariant under these changes, so one can check the controllability on any set of transformed matrices. Finally, Theorem 11 introduces the specific reduced forms which we will use in the sequel.

Definition 6. Admissible basis changes are basis changes in the state space variables preserving the hyperplane 𝑥₁=𝑘. One will write 𝒮 for the subgroup of the corresponding matrices in 𝐺𝑙_𝑛(ℝ). Therefore, 𝑆∈𝒮⟺𝑆=10𝑈𝑇,𝑇∈𝐺𝑙_𝑛−1(ℝ),𝑈∈𝑀_𝑛−1×1(ℝ). (12)

Remark 7. We emphasize two points.(1)The basis changes in Definition 6 transform the triple of matrices (𝐴₁,𝐴₂,𝐵) into (𝐴^′₁,𝐴^′₂,𝐵^′)=(𝑆⁻¹𝐴₁𝑆,𝑆⁻¹𝐴₂𝑆,𝑆⁻¹𝐵), whereas 𝐶^′=𝐶𝑆=𝐶.(2)Moreover, these transformations are just the same as the ones in [15] for dynamical systems, so the reduced forms are also valid for CBLCS here.

Proposition 8. For a given CBLCS, admissible basis changes preserve Condition (C1) and the values 𝜇₁, 𝜇₂ in Condition (C2).

Proof. Consider a CBLCS defined by (𝐴₁,𝐴₂,𝐵). Let us see that the vector 𝑒^′ verifying 𝑒^′𝐶=𝐴^′₂−𝐴^′₁ is just 𝑒^′=𝑆⁻¹𝑒 as follows: 𝑆⁻¹𝑒𝐶=𝑆⁻¹𝑒(𝐶𝑆)=𝑆⁻¹𝐴₂−𝐴₁𝑆=𝑆⁻¹𝐴₂𝑆−𝑆⁻¹𝐴₁𝑆=𝐴^′₂−𝐴^′₁. (13) Then, the pair (𝐴₁,[𝐵∣𝑒]) is controllable if and only if (𝑆⁻¹𝐴₁𝑆,[𝑆⁻¹𝐵∣𝑆⁻¹𝑒]) is controllable, sincerank𝑆⁻¹𝐵∣𝑆⁻¹𝑒𝑆⁻¹𝐴₁𝑆𝑆⁻¹𝐵∣𝑆⁻¹𝑒⋯𝑆⁻¹𝐴₁𝑆^𝑛−1𝑆⁻¹𝐵∣𝑆⁻¹𝑒=rank𝑆⁻¹[𝐵∣𝑒]𝐴₁[𝐵∣𝑒]⋯𝐴₁^𝑛−1[𝐵∣𝑒]=rank[𝐵∣𝑒]𝐴₁[𝐵∣𝑒]⋯𝐴₁^𝑛−1[𝐵∣𝑒]. (14)This proves that Condition (C1) is invariant under admissible basis changes. Besides, from 𝜆𝐼_𝑛−𝑆⁻¹𝐴_𝑖𝑆𝑆⁻¹𝐵=𝑆𝐶𝑆0⁻¹𝐼_𝑛𝜆𝐼_𝑛−𝐴_𝑖𝐵𝑆𝐶0𝐼_𝑛, (15) it is clear that 𝑣^𝑡𝜇_𝑖𝜆𝐼_𝑛−𝑆⁻¹𝐴_𝑖𝑆𝑆⁻¹𝐵𝐶𝑆0=0 (16) is equivalent to 𝑣^𝑡𝑆⁻¹𝜇_𝑖𝜆𝐼_𝑛−𝐴_𝑖𝐵𝐶0=0, (17) and, hence, admissible basis changes also preserve the values of 𝜇₁,𝜇₂ in Condition (C2).

From Proposition 8, it follows that one can check the controllability conditions of Theorem 4 on any triple of matrices (𝐴^′₁,𝐴^′₂,𝐵^′) obtained, via an admissible basis change, from the triple (𝐴₁,𝐴₂,𝐵) defining the CBLCS. Hence, there is no loss of generality in assuming that the matrices in (𝐴₁,𝐴₂,𝐵) are in the reduced form derived in [15] (see Remark 7 above). We recall this result for the reader’s convenience.

Lemma 9 (Ferrer et al. 2010, see [15]). Given a CBLCS (𝐴₁,𝐴₂,𝐵), then Kerℒ₁=Kerℒ₂⊂Ker𝐶, (18) where ℒ₁, ℒ₂ are the observability matrices, that is, ℒ_𝑖𝐶∶=𝐶𝐴_𝑖⋮𝐶𝐴_𝑖^𝑛−1,𝑖=1,2. (19)

Definition 10. Under the above conditions, one writes 𝐾∶=Kerℒ₁=Kerℒ₂⊆Ker𝐶, (20) and one defines 𝑟∶=rankℒ₁=rankℒ₂=𝑛−dim𝐾 (21) as the observability rank of the system. In particular, the system is observable if 𝑟=𝑛.

Theorem 11 (Ferrer et al., 2010; see [15]). With the notation of Lemma 9 and Definition 10 above, one has the following.(a) The subspace 𝐾 is invariant under the endomorphisms 𝒜₁ and 𝒜₂ corresponding to the matrices 𝐴₁ and 𝐴₂, and their restrictions coincide, that is, 𝒜_1𝐾=𝒜_2𝐾, (22) and one will write 𝐽 for the Jordan normal form of these restricted endomorphisms. (b)There exists an admissible basis change such that the following matrices are obtained: 𝐴^′₁=𝑆⁻¹𝐴₁𝐾𝑆=¹0𝐴₁𝐽,𝐴^′₂=𝑆⁻¹𝐴₂𝐾𝑆=²0𝐴₂𝐽, (23) where the matrices 𝐾₁, 𝐾₂, 𝐴₁, and 𝐴₂ are given, respectively, by 𝐾₁=𝑎₁𝑎10⋯0₂𝑎01⋯0⋮⋮⋮⋱⋮_𝑟−1𝑎00⋯1_𝑟,𝐾00⋯0₂=𝑎₁10⋯0𝑎₂01⋯0⋮⋮⋮⋱⋮𝑎_𝑟−100⋯1𝑎_𝑟,𝐴00⋯0₁=𝑎^′_𝑟+1𝑎00⋯0⋮⋮⋮⋱⋮_′𝑛,𝐴00⋯0₂=𝑎^′_𝑟+100⋯0⋮⋮⋮⋱⋮𝑎^′_𝑛,00⋯0 (24) and hence the following hold.(b1) The coefficients 𝑎₁,…,𝑎_𝑟, 𝑎₁,…,𝑎_𝑟 are univocally determined by imposing the following relations: 𝑄_𝐴1(𝑠)=𝑄_𝐾1(𝑠)𝑄_𝐽(𝑠),𝑄_𝐴2(𝑠)=𝑄_𝐾2(𝑠)𝑄_𝐽(𝑠), (25) on the characteristic polynomials of 𝐴₁, 𝐾₁, 𝐴₂, 𝐾₂, and 𝐽. (b2) The coefficients 𝑎^′_𝑟+1,…,𝑎^′_𝑛, 𝑎^′_𝑟+1,…,𝑎^′_𝑛 depend on the admissible basis change applied. Moreover, 𝑆⁻¹𝐵^⊤=𝑏₁𝑏^′₂⋯𝑏^′_𝑛, (26) where 𝑏₁=𝐶𝐵 and 𝑏^′₂,…,𝑏^′_𝑛 also depend on the admissible basis change. In particular, 𝑏₁=𝑏^′₂=⋯=𝑏^′_𝑛=0 if and only if 𝐵=0.

4. General Explicit Criteria

Firstly, from the results in Section 3, we derive in Lemma 12 and Proposition 13 general sufficient conditions for uncontrollability of CBLCS. Next, from Theorems 4 and 11, Proposition 14 provides a sufficient condition for controllability of observable systems. Finally, Proposition 17 states that, if an observable CBLCS does not satisfy the hypothesis of Proposition 14, one can bring it into an unobservable feedback system (see Definition 15) in order to study its controllability.

Lemma 12. Let (𝐴₁,𝐴₂,𝐵) be a CBLCS and 𝐾 as in Theorem 11. If 𝐵∈𝐾, then the system is uncontrollable. In particular, it is uncontrollable whenever 𝐵=0.

Proof. If 𝐵∈𝐾, a trajectory having initial conditions in 𝐾 remains in 𝐾 whatever the applied control, and so the system is uncontrollable.

The next proposition states a more specific condition for uncontrollability. We recall that 𝑟 is the observability rank (see (21)).

Proposition 13. Let (𝐴₁,𝐴₂,𝐵) be a CBLCS in the form of Theorem 11; then, if 𝑏₁=⋯=𝑏_𝑟=0, the system is uncontrollable.

Proof. The assertion is obvious for 𝑟=𝑛. If 𝑟<𝑛, it is clear that the last 𝑛−𝑟 vectors of the admissible basis considered in Theorem 11 span the subspace 𝐾, and so the result follows from Lemma 12.

Now, we focus on observable systems.

Proposition 14. Let (𝐴₁,𝐴₂,𝐵) be an observable CBLCS in the form of Theorem 11. Then, if 𝑏₁=⋯=𝑏_𝑛−1=0,𝑏_𝑛≠0, the system is controllable.

Proof. For observable CBLCS, 𝑟=𝑛; thus, the reduced matrices in Theorem 11 take the following form: 𝐴₁=𝑎₁𝑎10⋯0₂𝑎01⋯0⋮⋮⋮⋱⋮_𝑛−1𝑎00⋯1_𝑛,𝐴00⋯0₁=𝑎₁10⋯0𝑎₂01⋯0⋮⋮⋮⋱⋮𝑎_𝑛−100⋯1𝑎_𝑛𝑏00⋯0,𝐵=¹𝑏₂⋮𝑏_𝑛−1𝑏_𝑛, (27) from which it follows at once that det(𝐵𝐴₁𝐵⋯𝐴₁^𝑛−1𝐵)≠0, so Condition (C1) of Theorem 4 holds. Condition (C2) is also trivial, because no 𝑣≠0 makes the last 𝑛 columns of 𝑣^⊤𝜇_𝑖𝜆𝐼_𝑛−𝐴_𝑖𝐵𝐶0∶𝜆𝑣₂−𝑣₁,…,𝜆𝑣_𝑛−𝑣_𝑛−1,𝑣_𝑛𝑏_𝑛 (28) equal 0.

If the hypothesis in Proposition 14 fails, we can study the controllability of a given observable CBLCS on the corresponding unobservable one obtained by the feedback described in Proposition 17.

As in the classical case of (single) control systems, feedback transformations result from applying additional controls of the form 𝐹𝑥(𝑡).

Definition 15. If (𝐴₁,𝐴₂,𝐵) is a CBLCS, one says that the system (𝐴₁+𝐵𝐹,𝐴₂+𝐵𝐹,𝐵) is the feedback system obtained from (𝐴₁,𝐴₂,𝐵) by means of the feedback 𝐹∈𝑀_1×𝑛(ℝ).

Remark 16. Obviously, feedback transformations preserve controllability.

Proposition 17. Let (𝐴₁,𝐴₂,𝐵) be an observable CBLCS which, again, one assumes to be in the form of Theorem 11. If 𝑏₁=⋯=𝑏_𝑗−1=0,𝑏_𝑗≠0,1≤𝑗<𝑛; the feedback system determined by the triple 𝐴₁+𝐵𝐹,𝐴₂+𝐵𝐹,𝐵 (29) with 𝐹=0⋯0𝑓_𝑗+10⋯0,𝑓_𝑗+11=−𝑏_𝑗 (30) is unobservable, having rank 𝑟=𝑗 (recall Definition 15).

Proof. From the reduced forms of Theorem 11, it is clear that 𝐴₁𝑎+𝐵𝐹=¹10⋯0𝑎000⋯0₂01⋯0000⋯0⋮⋮⋮⋱⋮𝑎⋮⋮⋮⋱⋮_𝑗−100⋯1𝑎000⋯0_𝑗00⋯0000⋯0𝑎_𝑗+100⋯0𝑎⋆10⋯0_𝑗+200⋯0⋆01⋯0⋮⋮⋮⋱⋮𝑎⋮⋮⋮⋱⋮_𝑛−100⋯0𝑎⋆00⋯1_𝑛00⋯0⋆00⋯0, (31) where the upper and lower left blocks are of sizes 𝑗×𝑗 and (𝑛−𝑗)×𝑗, respectively, and, hence, 𝐶𝐶𝐴𝑟=rank¹⋮𝐶𝐴+𝐵𝐹₁+𝐵𝐹^𝑛−1=rank10⋯00⋯0⋆1⋯00⋯0⋮⋮⋱⋮⋮⋱⋮⋆⋆⋯10⋯0⋆⋆⋯⋆0⋯0⋆⋆⋯⋆0⋯0⋮⋮⋱⋮⋮⋱⋮⋆⋆⋯⋆0⋯0=𝑗, (32) which closes the proof.

Taking into account this proposition, one concludes that, beyond the cases covered by Propositions 13 and 14, the study of the controllability can be restricted to the study of unobservable systems.

5. Explicit Criteria for 𝑛=2 and 𝑛=3

With the above tools, we will obtain explicit simple criteria when 𝑛=2 (in Proposition 18) and 𝑛=3 (in Proposition 19).

Firstly, let us see that the criterion in [21] for observable planar systems is also valid for the unobservable ones.

Proposition 18. Let one consider a planar CBLCS defined by (𝐴₁,𝐴₂,𝐵), with 𝐴₁=𝑎₁𝑎₃𝑎₂𝑎₄,𝐴₂=𝑎₁𝑎₃𝑎₂𝑎₄𝑏,𝐵=¹𝑏₂. (33) One writes 𝒞₁, 𝒞₂ for the controllability matrices of the corresponding subsystems, that is, 𝒞₁=𝐵𝐴₁𝐵,𝒞₂=𝐵𝐴₂𝐵. (34) It turns out that(i) the system is unobservable if and only if 𝑎₃=0; (ii) then, it is controllable if and only if det𝒞₁det𝒞₂>0 (35) or, equivalently, 𝑏₁≠0,𝑏₁𝑎₂−𝑎₁𝑏₂+𝑎₄𝑏₂𝑏₁𝑎₂−𝑎₁𝑏₂+𝑎₄𝑏₂>0. (36)

Proof. (i) One checks immediately that 𝑟=2 if and only if 𝑎₃≠0 (see Definition 10). (ii) Take 𝑎₃=0. If 𝑏₁=0, the system is uncontrollable from Proposition 13. Now, assume 𝑏₁≠0. One can rewrite Condition (C2) of Theorem 4 in the following form: 𝑏₁𝑣₁+𝑏₂𝑣₂=0,𝑣₂𝜆−𝑎₄𝑣=0,₁𝜆−𝑎₁−𝑣₂𝑎₂+𝜇₁𝑏=0for𝑖=1,₁𝑣₁+𝑏₂𝑣₂=0,𝑣₂𝜆−𝑎₄𝑣=0,₁𝜆−𝑎₁−𝑣₂𝑎₂+𝜇₂=0for𝑖=2, (37)𝜆∈ℝ, and (𝑣₁,𝑣₂)≠(0,0) implies 𝜇₁𝜇₂>0. Then, 𝑣₁=𝑏₂,𝑣₂=−𝑏₁,𝜆=𝑎₄,𝜇₁=𝑏₂𝑎₄−𝑎₁+𝑏₁𝑎₂,𝜇₂=𝑏₂𝑎₄−𝑎₁+𝑏₁𝑎₂; (38) therefore, Condition (C2) of Theorem 4 is equivalent to (36). In addition, (36) implies Condition (C1) in the same theorem, because one can rewrite this condition as 𝑏rank¹𝑎₁𝑏₁𝑎₁−𝑎₁𝑎₁𝑎₁−𝑎₁𝑏₂𝑎₂𝑏₁+𝑎₄𝑏₂𝑎₂−𝑎₂𝑎₂𝑎₁−𝑎₁+𝑎₄𝑎₂−𝑎₂=2, (39) which completes the proof of Proposition 18.

Finally, the proposition below details the case 𝑛=3.

Proposition 19. For 𝑛=3, let one consider an unobservable CBLCS defined by (𝐴₁,𝐴₂,𝐵), which one assumes to have the reduced form in Theorem 11. Let one denote by 𝒞₁, 𝒞₂ the controllability matrices of both subsystems, that is, 𝒞₁=𝐵𝐴₁𝐵𝐴²₁𝐵,𝒞₂=𝐵𝐴₂𝐵𝐴²₂𝐵 (40) and by det^𝑗𝑘₁₂𝒞_𝑖, 𝑖=1,2, the determinant of the submatrix corresponding to the rows 𝑗,𝑘 and the two first columns.(1) If 𝑟=2, the reduced forms are 𝐴₁=𝑎₁𝑎10₂𝑎00₃0𝜆₀,𝐴₂=𝑎₁10𝑎₂00𝑎₃0𝜆₀,𝑏𝐵=𝑏¹₂𝑏₃, (41) and then one has the following.(1.1) If 𝑏₁=𝑏₂=0, the system is uncontrollable. (1.2) If 𝑏₁=0 and 𝑏₂≠0, the system is controllable if and only if det𝒞₁det𝒞₂>0. (42)(1.3) If 𝑏₁≠0, the system is controllable if and only if det𝒞₁det𝒞₂>0,det¹²₁₂𝒞₁det¹²₁₂𝒞₂>0. (43)(2) If 𝑟=1, one has the following possibilities.(2.1) If 𝑏₁=0, the system is uncontrollable. (2.2) If 𝑏₁≠0, one distinguishes between the following corresponding four different reduced forms.

(a) If 𝐴₁=𝑎₁𝑎00₂𝜆₁0𝑎₃0𝜆₁,𝐴₂=𝑎₁00𝑎₂𝜆₁0𝑎₃0𝜆₁,𝑏𝐵=𝑏¹₂𝑏₃, (44) then the system is uncontrollable.

(b) If 𝐴₁=𝑎₁𝑎00₂𝜆₁0𝑎₃1𝜆₁,𝐴₂=𝑎₁00𝑎₂𝜆₁0𝑎₃1𝜆₁,𝑏𝐵=𝑏¹₂𝑏₃, (45) then the system is controllable if and only if det¹²₁₂𝒞₁det¹²₁₂𝒞₂>0; (46) moreover, det𝒞_𝑖=(1/𝑏₁)(det¹²₁₂𝒞_𝑖)², 𝑖=1,2, so that det𝒞₁det𝒞₂>0.

(c) If 𝐴₁=𝑎₁𝑎00₂𝜆₁0𝑎₃0𝜆₂,𝐴₂=𝑎₁00𝑎₂𝜆₁0𝑎₃0𝜆₂,𝑏𝐵=𝑏¹₂𝑏₃, (47)𝜆₁, 𝜆₂∈ℝ with 𝜆₁≠𝜆₂, then the system is controllable if and only if det¹²₁₂𝒞₁det¹²₁₂𝒞₂>0,det¹³₁₂𝒞₁det¹³₁₂𝒞₂>0; (48) moreover, det𝒞_𝑖=((𝜆₂−𝜆₁)/𝑏₁)det¹²₁₂𝒞_𝑖det¹³₁₂𝒞_𝑖, 𝑖=1,2, so that det𝒞₁det𝒞₂>0.

(d) If 𝐴₁=𝑎₁𝑎00₂𝜂₁𝜂₂𝑎₃−𝜂₂𝜂₁,𝐴₂=𝑎₁00𝑎₂𝜂₁𝜂₂𝑎₃−𝜂₂𝜂₁,𝑏𝐵=𝑏¹₂𝑏₃, (49) with 𝜂₂≠0, then the system is controllable.

Proof. We will go through the following different cases.
Case  1.1. It follows straightforward from Proposition 13.
Case  1.2. In this case we have 𝒞₁=0𝑏₂𝑎₁𝑏₂𝑏₂0𝑎₂𝑏₂𝑏₃𝜆₀𝑏₃𝑎₃𝑏₂+𝜆²₀𝑏₃ (50) and analogously for 𝒞₂. On the other hand, for the triple of matrices (41), Condition (C2) of Theorem 4 can be rewritten as 𝜇₁𝜇₂=𝑎₁𝑣₁+𝑎₂𝑣₂+𝑎₃𝑣₃−𝜆𝑣₁𝑎₁𝑣₁+𝑎₂𝑣₂+𝑎₃𝑣₃−𝜆𝑣₁>0 (51) for any pair 𝜆∈ℝ,𝑣≠0 such that 𝜆𝑣₂−𝑣₁=0,𝜆𝑣₃−𝜆₀𝑣₃=0,𝑣₂𝑏₂+𝑣₃𝑏₃=0 (52) and, as 𝑏₂≠0, 𝑣₂𝑏=−³𝑏₂𝑣₃,𝑣₁=𝜆𝑣₂𝑏=−𝜆³𝑏₂𝑣₃. (53) Hence, 𝑣₃≠0. Obviously, we can assume 𝑣₃=1, so that 𝜆=𝜆₀,𝑣=−𝜆₀𝑏₃𝑏₂𝑏,−³𝑏₂,𝜇,1₁=−𝜆₀𝑏₃𝑏₂𝑎₁−𝑏₃𝑏₂𝑎₂+𝑎₃+𝜆²₀𝑏₃𝑏₂1=−𝑏³₂det𝒞₁,𝜇₂1=−𝑏³₂det𝒞₂; (54) therefore, Condition (C2) of Theorem 4 is equivalent to det𝒞₁det𝒞₂>0, (55) which obviously implies Condition (C1).
Case  1.3. First note that, as 𝑏₁≠0, we can assume 𝑏₁=1. Then, 𝒞₁=1𝑎₁+𝑏₂𝑎₁𝑎₁+𝑏₂+𝑎₂𝑏₂𝑎₂𝑎₂𝑎₁+𝑏₂𝑏₃𝑎₃+𝜆₀𝑏₃𝑎₃𝑎₁+𝑏₂+𝜆₀𝑎₃+𝜆₀𝑏₃. (56) In particular, det¹²₁₂𝒞₁=𝑎₂−𝑏₂(𝑎₁+𝑏₂), so we can write det𝒞₁ as det𝒞₁=det𝐵𝐴₁𝐵𝐴²₁𝐵−𝑎₁+𝑏₂𝐴₁𝐵=det1𝑎₁+𝑏₂det¹²₁₂𝒞₁𝑏₂𝑎₂0𝑏₃𝑎₃+𝜆₀𝑏₃𝜆₀𝑎₃+𝜆₀𝑏₃−𝜆₀𝑏₃𝑎₁+𝑏₂=𝑏₂𝑎₃−𝑏₃𝑎₂+𝜆₀𝑎₃+𝜆²₀𝑏₃−𝜆₀𝑏₃𝑎₁det¹²₁₂𝒞₁ (57) and analogously for 𝒞₂. Now the pairs 𝜆∈ℝ, 𝑣≠0 in Condition (C2) of Theorem 4 must verify 𝜆𝑣₂−𝑣₁=0,𝜆𝑣₃−𝜆₀𝑣₃𝑣=0,₁+𝑣₂𝑏₂+𝑣₃𝑏₃=0. (58) For 𝑣₃=0, necessarily 𝑣₂≠0 (otherwise, 𝑣₁=0 by the first equation). We can assume 𝑣₂=1, so that 𝑣=−𝑏₂,1,0,𝜆=−𝑏₂,𝜇₁=−𝑏₂𝑎₁+𝑎₂−𝑏²₂=det¹²₁₂𝒞₁,𝜇₂=det¹²₁₂𝒞₂. (59) Thus, a necessary condition for the controllability of the system is det¹²₁₂𝒞₁det¹²₁₂𝒞₂>0. (60) Furthermore, we must consider the pairs 𝜆,𝑣 with 𝑣₃≠0. We can assume 𝑣₃=1, so that 𝜆=𝜆₀ and 𝑣₁=𝜆₀𝑣₂=−𝑏₂𝑣₂−𝑏₃. (61) Now we distinguish the following three situations, depending on the solutions of the last equation.(i) If 𝜆₀+𝑏₂=0,𝑏₃≠0, none 𝑣₂ is a solution of (61), so that Condition (C2) of Theorem 4 is equivalent todet¹²₁₂𝒞₁det¹²₁₂𝒞₂>0, (62) but det𝒞₁=𝑏₃−𝑎₂+𝑏²₂+𝑏₂𝑎₁det¹²₁₂𝒞₁=−𝑏₃det¹²₁₂𝒞₁²,det𝒞₂=−𝑏₃det¹²₁₂𝒞₂². (63) Hence, Condition (C1) of Theorem 4 holds, and also det𝒞₁det𝒞₂>0. (ii) If 𝜆₀+𝑏₂≠0, (61) has a unique solution: 𝑣₂𝑏=−³𝜆₀+𝑏₂,𝑣₁𝜆=−⁰𝑏₃𝜆₀+𝑏₂,𝜇₁1=−𝜆₀+𝑏₂𝑎₁𝜆₀𝑏₃+𝑎₂𝑏₃−𝑎₃𝜆₀+𝑏₂−𝜆²₀𝑏₃=1𝜆₀+𝑏₂×det𝒞₁det¹²₁₂𝒞₁,𝜇₂=1𝜆₀+𝑏₂×det𝒞₂det¹²₁₂𝒞₂; (64) therefore, Condition (C2) of Theorem 4 is equivalent to det¹²₁₂𝒞₁det¹²₁₂𝒞₂>0,det𝒞₁det𝒞₂>0 (65) which obviously implies Condition (C1). (iii) Finally, if 𝜆₀+𝑏₂=𝑏₃=0, we have infinitely many solutions of (61) and then 𝜆=𝜆₀,𝑣=𝜆₀𝑣₂,𝑣₂,1,𝑣₂∈ℝ. (66) Notice that det𝒞₁=det𝒞₂=0. (67) Hence, (43) does not hold. Let us see that conditions of Theorem 4 also fail. From Condition (C2), we have 𝜇₁=𝑎₁𝜆₀𝑣₂+𝑎₂𝑣₂+𝑎₃−𝜆²₀𝑣₂=𝑣₂det¹²₁₂𝒞₁+𝑎₃,𝜇₂=𝑣₂det¹²₁₂𝒞₂+𝑎₃ (68) and it follows that 𝜇₁𝜇₂>0 holds for any 𝑣₂ if and only if 𝑎₃𝑎₃𝑎>0,³det¹²₁₂𝒞₁=𝑎₃det¹²₁₂𝒞₂, (69) but then Condition (C1) fails, since rank𝐵𝐴₁𝐵𝐴²₁𝐵𝑒𝐴₁𝑒𝐴²₁𝑒=2. (70) Indeed, the first two columns are linearly independent because (by hypothesis) det¹²₁₂𝒞₁≠0 and, as det𝒞₁=0, they span the third one. Finally, it is enough to verify that column 𝑒 is a linear combination of columns 𝐵 and 𝐴₁𝐵, as direct computation shows det𝐵𝐴₁𝐵𝑒=det1𝑎₁−𝜆₀𝑎₁−𝑎₁−𝜆₀𝑎₂𝑎₂−𝑎₂0𝑎₃𝑎₃−𝑎₃=−det1𝑎₁−𝜆₀𝑎₁−𝜆₀−𝜆₀𝑎₂𝑎₂0𝑎₃𝑎₃=𝑎₃det¹²₁₂𝒞₂−𝑎₃det¹²₁₂𝒞₁=0. (71)Next, let us assume 𝑟=1. Then we have the following.
Case  2.1. This case follows from Proposition 13.
Case  2.2. (a) The proof proceeds in an analogous way to the proof of Case 1.2. We compute 𝜇₁,𝜇₂ for any pair 𝜆∈ℝ,𝑣≠0 such that 𝜆−𝜆₁𝑣₂=0,𝜆−𝜆₁𝑣₃𝑣=0,₁𝑏₁+𝑣₂𝑏₂+𝑣₃𝑏₃=0. (72) Hence, if 𝜆≠𝜆₁, then 𝑣₂=𝑣₃=0, and none 𝑣≠0 verifies the last equation. On the contrary, if 𝜆=𝜆₁, by taking 𝑣₃=−𝑎₁−𝜆₁𝑏₂𝑏₁+𝑎₂,𝑣₂=𝑎₁−𝜆₁𝑏₃𝑏₁−𝑎₃,𝑣₁𝑏=−²𝑏₁𝑣₂−𝑏₃𝑏₁𝑣₃ (73) we have 𝜇₁=0. Therefore, Condition (C2) of Theorem 4 does not hold.
(b) For the triple of matrices of this case, we have𝒞₁=𝑏₁𝑎₁𝑏₁𝑎²₁𝑏₁𝑏₂𝑎₂𝑏₁+𝜆₁𝑏₂𝑎₂𝑎₁+𝜆₁𝑎₂𝑏₁+𝜆²₁𝑏₂𝑏₃𝑎₃𝑏₁+𝑏₂+𝜆₁𝑏₃𝑎₃𝑎₁+𝑎₂+𝜆₁𝑎₃𝑏₁+2𝜆₁𝑏₂+𝜆²₁𝑏₃ (74)and analogously for 𝒞₂. Now, the pairs 𝜆∈ℝ, 𝑣≠0 in Condition (C2) of Theorem 4 must verify𝜆−𝜆₁𝑣₂−𝑣₃=0,𝜆−𝜆₁𝑣₃𝑣=0,₁𝑏₁+𝑣₂𝑏₂+𝑣₃𝑏₃=0. (75) For 𝜆≠𝜆₁, the only solution is 𝑣₃=𝑣₂=𝑣₁=0. For 𝜆=𝜆₁, there are infinitely many solutions: 𝑣₁𝑏=−²𝑏₁𝑣₂,𝑣₂≠0,𝑣₃𝜇=0,₁=𝑣₂𝑏₁𝑎₂𝑏₁−𝑎₁𝑏₂+𝜆₁𝑏₂=𝑣₂𝑏²₁det¹²₁₂𝒞₁ (76) and analogously for 𝜇₂. Therefore, Condition (C2) of Theorem 4 is equivalent to det¹²₁₂𝒞₁det¹²₁₂𝒞₂>0 (77) which implies Condition (C1) because det𝒞₁=det𝐵𝐴₁𝐵𝐴²₁𝐵=det𝐵𝐴₁𝐵𝐴²₁𝐵−𝑎₁+𝜆₁𝐴₁𝐵+𝑎₁𝜆₁𝐵𝑏=det¹𝑎₁𝑏₁0𝑏₂𝑎₂𝑏₁+𝜆₁𝑏₂0𝑏₃𝑎₃𝑏₁+𝑏₂+𝜆₁𝑏₃𝑎₂𝑏₁+𝜆₁𝑏₂−𝑎₁𝑏₂=1𝑏₁det¹²₁₂𝒞₁². (78)
(c) For the triple in this case, one sees that 𝒞₁=𝑏₁𝑎₁𝑏₁𝑎²₁𝑏₁𝑏₂𝑎₂𝑏₁+𝜆₁𝑏₂𝑎₂𝑎₁𝑏₁+𝜆₁𝑎₂𝑏₁+𝜆₁𝑏₂𝑏₃𝑎₃𝑏₁+𝜆₂𝑏₃𝑎₃𝑎₁𝑏₁+𝜆₂𝑎₃𝑏₁+𝜆₂𝑏₃ (79) and analogously for 𝒞₂. Now, the pairs 𝜆∈ℝ, 𝑣≠0 in Condition (C2) of Theorem 4 must verify 𝜆−𝜆₁𝑣₂=0,𝜆−𝜆₂𝑣₃𝑣=0,₁𝑏₁+𝑣₂𝑏₂+𝑣₃𝑏₃=0. (80) If 𝑣₂=𝑣₃=0, none 𝑣≠0 verifies the last equation. Hence, we can assume either 𝑣₂=1 or 𝑣₃=1, which yields, respectively, −𝑏𝑣=²𝑏₁,1,0,𝜆=𝜆₁,𝜇₁=1𝑏²₁det¹²₁₂𝒞₁,𝜇₂=1𝑏²₁det¹²₁₂𝒞₂,−𝑏𝑣=³𝑏₁,0,1,𝜆=𝜆₂,𝜇₁=1𝑏²₁det¹³₁₂𝒞₁,𝜇₂=1𝑏²₁det¹³₁₂𝒞₂. (81) Therefore, Condition (C2) of Theorem 4 is equivalent to det¹²₁₂𝒞₁det¹²₁₂𝒞₂>0,det¹³₁₂𝒞₁det¹³₁₂𝒞₂>0 (82) which implies Condition (C1) because det𝒞₁=det𝐵𝐴₁𝐵𝐴²₁𝐵−𝑎₁𝐴₁𝐵𝑏=det¹𝑎₁𝑏₁0𝑏₂𝑎₂𝑏₁+𝜆₁𝑏₂𝜆₁𝑏₁det¹²₁₂𝒞₁𝑏₃𝑎₃𝑏₁+𝜆₂𝑏₃𝜆₂𝑏₁det¹³₁₂𝒞₁𝜆=−¹−𝜆₂𝑏₁det¹²₁₂𝒞₁det¹³₁₂𝒞₁. (83)
(d) In this case, the pairs 𝜆∈ℝ, 𝑣≠0 in Condition (C2) of Theorem 4 must verify 𝜆−𝜂₁𝑣₂+𝜂₂𝑣₃=0,−𝜂₂𝑣₂+𝜆−𝜂₁𝑣₃𝑣=0,₁𝑏₁+𝑣₂𝑏₂+𝑣₃𝑏₃=0. (84) Hence, (𝜂²₂+(𝜆−𝜂₁)²)𝑣₃=0. As 𝜂₂≠0, the only solution is 𝑣₃=0, which implies 𝑣₂=0 and, in turn, 𝑣₁=0. Therefore, there is no pair 𝜆,𝜇 verifying Condition (C2) of Theorem 4. Condition (C1) also holds, since the matrix consisting of the three first columns there has maximal rank. Indeed, a straightforward computation leads to det𝒞₁=det𝐵𝐴₁𝐵𝐴²₁𝐵=1𝑏²₁det𝐵𝐴₁𝐵−𝑎₁𝐵𝑏₁𝐴₁𝐴₁𝐵−𝑎₁𝐵𝑏₁=1𝑏₁detdet¹²₁₂𝒞₁𝜂₁det¹²₁₂𝒞₁+𝜂₂det¹³₁₂𝒞₁det¹³₁₂𝒞₁−𝜂₂det¹²₁₂𝒞₁+𝜂₁det¹³₁₂𝒞₁≠0 (85) because the second column is just the image of the first one by 𝜂₁𝜂₂−𝜂₂𝜂₁, with 𝜂₂≠0.

Remark 20. We note that the use of the reduced forms in Propositions 18 and 19 leads to explicit and simple criteria (in comparison with [27]) for controllability of unobservable systems. These results, combined with the feedback induced unobservability stated in Proposition 17, provide an effective approach to the study of the controllability of the systems at hand. Of course, for piecewise linear systems, one can find in the literature a lot of examples of feedback design for several different purposes (for recent contributions, see [18–20]), but we have not found previous results on using feedbacks to discuss the controllability of a given system. In its current state, the approach presented here is valid for CBLCS. Possible extensions to general switched systems or to other interesting cases, such as systems with state delay, are subject of future research.

6. Example

In this section, we will apply the results in Section 5 to a circuit analyzed in [4] (see Figure 1) which is an example of a 3D CBLCS that becomes unobservable for some values of the parameters. As we will see, the study of the controllability of this system will be an easy computation thanks to the use of Proposition 19.

Figure 1 

An occasional unobservable circuit.

Application of Kirchhoff laws yields 𝑅₂𝐶₁𝑉̇=−𝑅₂𝑖(𝑉)−𝑅₂𝐼_𝐿−𝑉+𝑉_𝐶2𝐿𝐼̇_𝐿=𝑉−𝐼_𝐿𝑅₁𝑅₂𝐶₂𝑉̇_𝐶2=𝑉−𝑉_𝐶2, (86) where 𝑖(𝑉)=𝑉−𝑉_𝑆sgn(𝑉)𝑅|𝑉|≥𝑉,if^𝑆2−𝑉𝑅|𝑉|<𝑉,if^𝑆2 (87) with 𝑉_𝑆 being a constant saturation voltage (see [4] and the references therein for further details). Making the change of variables 𝑥=𝑥₁,𝑥₂,𝑥₃=2𝑉_𝑆𝑉,𝐼_𝐿,𝑉_𝐶2, (88) the system becomes ̇−𝑥=𝑅+𝑅₂𝑅𝑅₂𝐶₁−1𝐶₁1𝑅₂𝐶₁1𝐿−𝑅₁𝐿01𝑅₂𝐶₂10−𝑅₂𝐶₂2𝑥+𝐶₁𝑅00sat_𝑁𝑥₁ (89) when only positive values of 𝑥₁ are considered. Here, sat_𝑁(𝑥₁) denotes the normalized saturation function, defined by sat_𝑁(𝑣)=|𝑣|(𝑣)|𝑣|𝑣,if≤1sgn,if≥1. (90)

We ask if the system is controllable by means of a control parameter 𝑢 acting as follows: sat𝑥₁=𝑥;𝑢¹,if𝑥₁≤𝑢𝑢,if𝑥₁≥𝑢. (91) Now, introducing the new variable 𝑥₂=−𝑅+𝑅₂𝑅𝑅₂𝐶₁𝑥₁−1𝐶₁𝑥₂+1𝑅₂𝐶₁𝑥₃ (92) yields𝑥̇₁𝑥̇‌₂𝑥̇₃=010−𝑅𝑅²₂𝐶₂−𝑅₁𝑅₂𝐶₂𝑅+𝑅₂+𝑅𝐿𝑅𝑅²₂𝐶₁𝐶₂𝐿−𝐿𝑅+𝑅₂−𝑅𝑅₁𝑅₂𝐶₁𝑅𝑅₂𝐶₁𝐿𝑅₁𝑅₂𝐶₂−𝐿𝑅²₂𝐶₁𝐶₂𝐿1𝑅₂𝐶₂10−𝑅₂𝐶₂𝑥₁𝑥₂𝑥₃+2𝐶₁𝑅−2𝑅+𝑅₂𝑅²𝑅₂𝐶²₁0sat𝑥₁;𝑢. (93)

Notice that the separating hyperplane is 𝑥₁=𝑢; therefore, the shift 𝑥₁=𝑥₁−𝑢 places it into 𝑥₁=0. If, in addition, we write 𝑥₃=𝑥₃, then, in the new coordinates 𝑥=(𝑥₁,𝑥₂,𝑥₃), the above system takes the following form:̇‌𝐴𝑥=¹𝑥+𝐵𝑢,if𝑥₁𝐴≤0₂𝑥+𝐵𝑢,if𝑥₁≥0, (94) where 𝐴₁=2𝑅𝐶₁10−𝑅𝑅²₂𝐶₂−𝑅₁𝑅₂𝐶₂𝑅+𝑅₂+𝑅𝐿𝑅𝑅²₂𝐶₁𝐶₂𝐿−2𝑅+𝑅₂𝑅²𝑅₂𝐶²₁−𝐿𝑅+𝑅₂−𝑅𝑅₁𝑅₂𝐶₁𝑅𝑅₂𝐶₁𝐿𝑅₁𝑅₂𝐶₂−𝐿𝑅²₂𝐶₁𝐶₂𝐿1𝑅₂𝐶₂10−𝑅₂𝐶₂,𝐴₂=010−𝑅𝑅²₂𝐶₂−𝑅₁𝑅₂𝐶₂𝑅+𝑅₂+𝑅𝐿𝑅𝑅²₂𝐶₁𝐶₂𝐿−𝐿𝑅+𝑅₂−𝑅𝑅₁𝑅₂𝐶₁𝑅𝑅₂𝐶₁𝐿𝑅₁𝑅₂𝐶₂−𝐿𝑅²₂𝐶₁𝐶₂𝐿1𝑅₂𝐶₂10−𝑅₂𝐶₂,2𝐵=𝑅𝐶₁−𝑅𝑅²₂𝐶₂−𝑅₁𝑅₂𝐶₂𝑅+𝑅₂+𝑅𝐿𝑅𝑅²₂𝐶₁𝐶₂𝐿−2𝑅+𝑅₂𝑅²𝑅₂𝐶²₁1𝑅₂𝐶₂. (95)

A simple computation shows that for 𝐿=𝑅₁𝑅₂𝐶₂ the system is unobservable, having the following reduced form (see [15]): 𝐴^𝑅₁=𝑎₁𝑎10₂𝑎00₃0𝜆₀,𝐴^𝑅₂=𝛼₁𝛼10₂𝛼00₃0𝜆₀,𝐵^𝑅=𝑏₁𝑏₂𝑏₃, (96) where 𝜆₀1=−𝑅₂𝐶₂,𝑎₁=2𝑅𝐶₁−𝑅𝐶₂+𝑅₂𝐶₂+𝑅𝐶₁𝑅𝑅₂𝐶₁𝐶₂,𝑎₂=𝑅₁−𝑅𝑅𝑅₁𝑅₂𝐶₁𝐶₂,𝑎₃=1𝑅₂𝐶₂,𝛼₁=−𝑅𝐶₂+𝑅₂𝐶₂+𝑅𝐶₁𝑅𝑅₂𝐶₁𝐶₂,𝛼₂=−𝑅+𝑅₁𝑅𝑅₁𝑅₂𝐶₁𝐶₂,𝛼₃=1𝑅₂𝐶₂,𝑏₁=2𝑅𝐶₁,𝑏₂=𝑅₁−𝑅𝑅𝑅₁𝑅₂𝐶₁𝐶₂,𝑏₃=1𝑅₂𝐶₂. (97) Thus, according to Case 1.3 in Proposition 19, the system will be controllable if and only if det𝒞₁det𝒞₂>0,det¹²₁₂𝒞₁det¹²₁₂𝒞₂>0. (98) As in the proof there,det𝒞₁=𝑏₂𝑎₃−𝑏₃𝑎₂+𝜆₀𝑎₃𝑏₁+𝜆²₀𝑏₃−𝜆₀𝑏₃𝑎₁det¹²₁₂𝒞₁ (99) and analogously for det𝒞₂. In our case one obtains for the product det𝒞₁det𝒞₂=𝑅+𝑅₂²𝑅²𝑅⁴₂𝐶²₁𝐶³₂det¹²₁₂𝒞₁det¹²₁₂𝒞₂. (100) Hence, we need to check only the second condition which, for the above reduced form, writes det¹²₁₂𝒞₁det¹²₁₂𝒞₂=𝑎₂𝑏²₁−𝑎₁𝑏₁𝑏₂−𝑏²₂𝛼₂𝑏²₁−𝛼₁𝑏₁𝑏₂−𝑏²₂>0; (101) but taking into account the values of the parameters in (97), it is equivalent to 𝑅₁−𝑅2𝑅𝐶₂+𝑅₂𝐶₂+𝑅𝐶₁𝑅₁−𝑅₁−𝑅𝑅𝐶₁×−4𝑅+𝑅₁𝑅₁𝑅₂𝐶₂+2𝑅₁−𝑅𝑅𝐶₂+𝑅₂𝐶₂+𝑅𝐶₁𝑅₁−𝑅₁−𝑅²𝑅𝐶₁>0. (102) Clearly, the above inequality holds if 𝑅₁<𝑅, but it does not hold if 𝑅₁=𝑅, whereas for 𝑅₁>𝑅, it holds or not depending on the other parameters. Thus, for 𝐿=𝑅₁𝑅₂𝐶₂ we have the following.(i)If 𝑅₁<𝑅, the system is controllable. (ii)If 𝑅₁=𝑅, the system is uncontrollable. (iii)If 𝑅₁>𝑅, then controllability requires additional conditions. For example, if 𝑅𝑅₁−𝑅>𝑅₂3𝑅+𝑅₁or𝑅₁−𝑅>4𝑅₁𝑅₂𝐶₂𝑅𝐶₁, (103) one sees at once that the system is controllable. On the other hand, necessary conditions for controllability are 𝑅₁𝐶₁+𝐶₂>𝑅₂𝐶₂,2𝐶₂𝑅₂−𝑅𝑅₁+𝑅<𝐶₁. (104)

For 𝐿≠𝑅₁𝑅₂𝐶₂, the system is observable, and so to study the controllability we make use of Proposition 17 with 𝑏₁≠0. Hence (see Remark 16), the system is controllable if and only if the feedback system defined by (𝐴₁+𝐵𝐹,𝐴₂+𝐵𝐹,𝐵) is controllable, being1𝐹=0−𝑏₁0=10−2𝑅𝐶₁0,𝐴₁2+𝐵𝐹=𝐶₁𝑅00−𝑅𝑅²₂𝐶₂−𝑅₁𝑅₂𝐶₂𝑅+𝑅₂+𝑅𝐿𝑅𝑅²₂𝐶₁𝐶₂𝐿−2𝑅+𝑅₂𝑅²𝑅₂𝐶²₁𝐽̃1𝑅₂𝐶₂,𝐴₂0+𝐵𝐹=00−𝑅𝑅²₂𝐶₂−𝑅₁𝑅₂𝐶₂𝑅+𝑅₂+𝑅𝐿𝑅𝑅²₂𝐶₁𝐶₂𝐿𝐽̃1𝑅₂𝐶₂,2𝐵=𝑅𝐶₁−𝑅𝑅²₂𝐶₂−𝑅₁𝑅₂𝐶₂𝑅+𝑅₂+𝑅𝐿𝑅𝑅²₂𝐶₁𝐶₂𝐿−2𝑅+𝑅₂𝑅²𝑅₂𝐶²₁1𝑅₂𝐶₂, (105)where the block 𝐽̃ in the matrices above is given by 𝐽̃=−𝑅₁𝑅²₂𝐶₂+𝑅𝑅²₂𝐶₂+𝑅𝑅₁𝑅₂𝐶₂−𝑅𝐿2𝑅²₂𝐶₂𝐿𝑅₁𝑅₂𝐶₂−𝐿𝑅²₂𝐶₁𝐶₂𝐿−𝐶₁𝑅2𝑅₂𝐶₂−1𝑅₂𝐶₂. (106) It can be readily seen that this feedback system is unobservable with 𝑟=1, so that we can apply Proposition 19 again, now in the Case 2.2. In particular, it will be controllable if 𝐽̃ has complex eigenvalues. For example,(iv) if 𝐿=𝑅₁𝑅₂𝐶₂/2, the system is controllable if 4𝑅𝑅²₂+4𝑅𝑅₁𝑅₂+𝑅𝑅²₁<8𝑅²₁𝑅₂. (107)

7. Conclusions

We obtain simple explicit criteria for the controllability of a continuous bimodal linear control system by means of the implicit ones in [27] and the reduced matrix forms in [15]. We cover fully dimensions 𝑛=2 and 𝑛=3, and partially higher ones. In particular, we prove that we can restrict ourselves to the nongeneric case of unobservable systems. Some interesting problems for further research could be the characterization of controllability for similar systems with multiple inputs and the extension of the present results to systems with delays in the state variables, or even to general switched systems.

Acknowledgments

The second author was supported by the Spanish MINECO-FEDER Grants MTM2009-06973 and MTM 2012-31714 and the Catalan CUR-DIUE Grant 2009SGR859. The other two authors were partially supported by the Project MTM 2011-23892.