On Existence of Stabilizing Switching Laws within a Class of Unstable Linear Systems

Abstract

The equivalence of two conditions, condition (3) and condition (4) stated in Problem Statement section, regarding the existence of stabilizing switching laws between two unstable linear systems first appeared in (Feron 1996). Although Feron never published this result, it has been referenced in almost every survey on switched systems; see, for example, (Liberzon and Morse 1999). This paper proposes another way to prove the equivalence of two conditions regarding the existence of stabilizing switching laws between two unstable linear systems. One is effective for theoretical derivation, while the other is implementable, and a class of stabilizing switching laws have been explicitly constructed by Wicks et al. (1994). With the help of the equivalent relation, a condition for the existence of controllers and stabilizing switching laws between two unstabilizable linear control systems is then proposed. Then, the study is further extended to the issue concerning the construction of quadratically stabilizing switching laws among 𝑁 unstable linear systems and 𝑁 unstabilizable linear control systems. The obtained results are employed to study the existence of control laws and quadratically stabilizing switching laws within a class of unstabilizable linear control systems. The numerical examples are illustrated and simulated to show the feasibility and effectiveness of the proposed methods.

1. Introduction

The study of switched systems and switching strategies has recently attracted considerable attention (see, e.g., [1–40]). A switched system is known as a hybrid dynamical system comprising a family of continuous-time subsystems and a rule that conducts the switching among them. This kind of system arises from many practical applications such as adaptive control, intelligent control, and the control of many mechanical systems (see, e.g., [4, 5]). The key motivations for having “multiple modalities” or “variable structure” during the control period include the improvement of the transient response of an adaptive system [2] and the existence of systems (such as nonholonomic systems) that cannot be asymptotically stabilizable by a single continuous feedback control law [8, 41], which makes switch control technique especially suitable. Recently, the fault-tolerant control (FTC) issues with fault detection and diagnosis (FDD) schemes have been largely discussed [38, 42–46]. Herein, the switching strategies will also be applied in the active, passive, or data-driven framework in the current and future works. Another example in the real control application of an air-breathing hypersonic aircraft with high complexity of the motion equations can be achieved by applying a switched linear-parameter-varying systems approach [32]. In mathematical modeling of physical systems, Wu and Zheng explored the topics concerning weighted 𝐻^∞ model reduction for linear switched systems with time-varying delay [18] and dissipativity-based sliding mode control of switched stochastic systems [39]. Lian et al. studied the dwell time method and exponential stability for uncertain switched stochastic time-delay systems [37].

In general, the issue regarding stability and design of switched systems is characterized into three categories [9]. The first is to find conditions that guarantee that the switched system is asymptotically stable for any switching signal. An intuitive approach for this category is to investigate the existence of common Lyapunov function for all subsystems [2, 6, 12]. Clearly, all subsystems in this case are necessarily required to be stable. The second is to identify those classes of switching signals for which the switched system, with all subsystems being stable, is asymptotically stable. A strategy to carry out the analysis is through the use of multiple Lyapunov function technique [7]. On the other hand, instead of assuming that all subsystems are stable, the final category allowed them to be unstable. The objectives of the final category are then to explore the existence conditions for stabilizing switching signals and to explicitly construct the signals, which makes the switched system asymptotically stable [1]. In this paper, we only consider the issue of the final category.

It is known that the switching among stable systems might result in the system state being unbounded [9]. On the other hand, with suitably selecting the switching signal, the switching among unstable systems might have the property that the system state converges to zero [1]. One of the sufficient conditions regarding the existence of stabilizing switching laws between two unstable linear systems is given by (2) of Section 2 below. Though this condition is effective for theoretical derivation, it does not provide any guidelines for the design of switching laws. In 1994, Wicks et al. [1] showed that a matrix pencil condition, as given by (3) of Section 2, is also a sufficient condition for the existence of stabilizing switching laws. Although they pointed out that the latter condition is more restricted than the former one, they had presented an algorithm to facilitate the checking of the latter condition and a guideline to explicitly construct the stabilizing switching laws [1]. Recently, there are so many new research results about the switched systems that have been proposed (e.g., see [11–14, 16]). However, as far as the construction of quadratically stabilizing switching laws among 𝑁 systems is concerned, there are so many detailed and extendable parts which still need further discussion and exploration.

The issue, concerning the existence of stabilizing switching laws between two unstable linear systems, has been studied by Feron [3] and Liberzon and Morse [9]. The equivalence of two conditions (condition (3) and condition (4) stated in Section 2 below) regarding the existence of stabilizing switching laws between two unstable linear systems first appeared in Feron (1996) [3]. Although Feron never published this result, it has been referenced in almost every survey on switched systems; see, for example, [9]. In this paper, we will propose another way to show that the two conditions stated above are indeed equivalent and then apply the equivalent relation to explore the existence of controllers and stabilizing switching laws between two unstabilizable linear control systems in an implementable way. In addition, in order to further understand the geometrical interpretations of the two sufficient conditions, we also specialize the two existence conditions to a planar case. Conditions involving eigenvalues and eigenvectors for a planar system are then presented.

This paper is organized as follows. Section 2 introduces the two sufficient conditions regarding the existence of stabilizing switching laws between two unstable linear systems, stabilizing switching laws among 𝑁 unstable linear systems, and the objectives of the paper. Section 3 discusses another way to prove the equivalence of the two sufficient conditions for two linear systems. Section 4 specializes the existence condition to planar systems and gives a numerical example. Section 5 extends the study to the application of the equivalent relation to the design of controllers and stabilizing switching laws among 𝑁 unstabilizable linear systems and linear control systems and gives two numerical examples. Finally, Section 6 summarizes the main results.

2. Problem Statement

Consider the two linear systems 𝐱̇=𝐴_𝑖𝐱,𝑖=1,2, (1) where we assume that neither of the two systems is stable; that is, neither 𝐴₁ nor 𝐴₂ is a Hurwitz matrix. It is known that a sufficient condition for the existence of stabilizing switching laws between the two unstable systems can be stated as in condition (2) as follows: ∃𝑃>0suchthat²_𝑖=1𝐱∣𝐱^𝑇𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖𝐱<0=ℝ^𝑛⧵{0}. (2) Clearly, under this condition and by defining 𝑉(𝐱)=𝐱^𝑇𝑃𝐱, one can find a proper switching law between the two systems such that 𝑉̇<0 for all the time. It then results in 𝐱(𝑡)→0 as 𝑡→∞. However, condition (2) does not provide a means to obtain such a matrix 𝑃. In 1994, Wicks et al. [1] verified that the matrix pencil condition as given by (3) is also a sufficient condition for the existence of stabilizing switching laws: ∃𝛽>0suchthat𝐴₁+𝛽𝐴₂isaHurwitzmatrix. (3) They also proposed an algorithm to determine the existence of 𝛽 and to explicitly construct a class of stabilizing switching laws if condition (3) holds. It was pointed out by [1] that condition (2) is more general than condition (3).

In 1996, Feron [3] proved that the sufficient condition (3) is also a necessary one for the existence of stabilizing switching laws. At the same time, the study concerning quadratic stabilizability was also extended to dynamic output feedback with a robust detectability condition. In 2000, Decarlo et al. [11] extended the sufficient condition (3) for two systems [1] to the sufficient condition (4) for more than two systems as follows: ∃𝛼_𝑖>0and^𝑁_𝑖=1𝛼_𝑖=1suchthat𝐴_𝑒𝑞∶=^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖isaHurwitzmatrix. (4) However, the literature search indicates that the implementation of stabilizing 𝑁 unstable linear systems needs to be further explored.

Therefore, the first goal of this paper is to propose another way to show the equivalence of conditions (2) and (3) and then to apply the equivalent relation to study the existence of controllers and stabilizing switching laws between two unstabilizable linear control systems in an implementable way. Then, we propose a way to implement the abovementioned problem for 𝑁 unstable linear systems. Finally, we employ the results to design control laws and quadratically stabilizing switching laws among 𝑁 unstabilizable linear control systems.

3. The Proof of Equivalence

To achieve the objectives as described in Section 2, we introduce the following condition: ∃𝑃>0and𝛽>0suchthat𝐿₁+𝛽𝐿₂<0, (5) where 𝐿_𝑖∶=𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖,𝑖=1,2. (6) In the following, we will show the equivalence of the three conditions (2)–(5). Firstly, we have the relation between conditions (3) and (5) as follows.

Lemma 1. Conditions (3) and (5) are equivalent.

Proof. Condition (3) holds 𝐴⟺∃𝛽>0and𝑃>0suchthat₁+𝛽𝐴₂^𝑇𝑃+𝑃𝐴₁+𝛽𝐴₂𝐴<0⟺∃𝛽>0and𝑃>0suchthat^𝑇₁𝑃+𝑃𝐴₁+𝛽𝐴^𝑇₂𝑃+𝑃𝐴₂<0. (7) This proves the result.

Next, we prove that “condition (3) ⇒condition (2),” which can also be found in [1].

Lemma 2. Condition (3) ⇒condition (2).

Proof. 𝐴₁+𝛽𝐴₂ is Hurwitz 𝐴⟹∀𝑄>0,∃𝑃>0suchthat₁+𝛽𝐴₂^𝑇𝑃+𝑃𝐴₁+𝛽𝐴₂𝐱=−𝑄<0,⟹∀𝐱≠0,_𝑇𝐴^𝑇₁𝑃+𝑃𝐴₁+𝛽𝐴^𝑇₂𝑃+𝑃𝐴₂𝐱=−𝐱^𝑇𝑄𝐱<0,⟹∀𝐱≠0.Atleastoneof𝐱^𝑇𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖𝐱,𝑖=1,2,islessthanzero. (8) The result then follows.

Finally, we will show that “condition (2) ⇒condition (5).” Before the proof, we define two subsets of the unit sphere as follows: Ω_𝑖∶=𝐱∣𝐱^𝑇𝐿_𝑖𝐱≥0,‖𝐱‖=1,𝑖=1,2. (9) The two sets have the following properties.

Lemma 3. (a) Both sets Ω₁ and Ω₂ are symmetric about the origin. That is, Ω_𝑖=−Ω_𝑖 for 𝑖=1,2, where −Ω_𝑖∶={−𝐱∣𝐱∈Ω_𝑖}.
(b) Each unit eigenvector associated with unstable eigenvalues of 𝐿_𝑖 belongs to Ω_𝑖 (by unstable eigenvalues one means those eigenvalues with nonnegative real parts).
(c) Either Ω_𝑖 is a connected set or it consists of two disjoint connected sets Ω_𝑖=Ω⁺_𝑖∪Ω⁻_𝑖 with Ω⁺_𝑖=−Ω⁻_𝑖.
(d) If condition (2) holds, then Ω₁∩Ω₂=∅ and 𝑆⧵{Ω₁∪Ω₂}≠∅, where 𝑆={𝐱∣‖𝐱‖=1} denotes the unit sphere.

Proof. Parts (a), (b), and (d) are trivial, so we omit their proofs.
We now prove part (c). Firstly, we assume that 𝐿_𝑖 is a diagonal matrix having the following form: 𝐿_𝑖=diag𝜆⁺₁,…,𝜆⁺_𝑚𝑖,𝜆⁻_𝑚𝑖+1,…,𝜆⁻_𝑛, (10) where 𝜆⁺_𝑗≥0 for 𝑗=1,…,𝑚_𝑖 and 𝜆⁻_𝑗<0 for 𝑗=𝑚_𝑖+1,…,𝑛. Clearly, the set Γ_𝑖∶=𝐱=𝑥₁,…,𝑥_𝑚𝑖,0,…,0^𝑇∣‖𝐱‖=1⊂Ω_𝑖, (11) and Γ_𝑖, a subset of the unit sphere, is either a connected set or a union of two connected sets which are symmetric about the origin. Now, 𝐱∈Ω_𝑖 means that 𝐱^𝑇𝐿_𝑖𝐱=^𝑚𝑖_𝑗=1𝜆⁺_𝑗𝑥²_𝑗+^𝑛_{𝑗=𝑚𝑖+1}𝜆⁻_𝑗𝑥²_𝑗≥0. (12) This implies that ^𝑚𝑖_𝑗=1𝜆⁺_𝑗𝑥²_𝑗+^𝑛_{𝑗=𝑚𝑖+1}𝜆⁻_𝑗𝑡𝑥_𝑗²≥0,∀0≤𝑡≤1, (13) or 𝐱^𝑇_𝑡𝐿_𝑖𝐱_𝑡≥0,∀0≤𝑡≤1, (14) where 𝐱_𝑡∶=(𝑥₁,…,𝑥_𝑚𝑖,𝑡𝑥_𝑚1+1,…,𝑡𝑥_𝑛)^𝑇. It follows that the set 𝐱_𝑡𝐱_𝑡∣0≤𝑡≤1⊆Ω_𝑖. (15) Therefore, for any point 𝐱 in Ω_𝑖, there exists a smooth path lying entirely inside Ω_𝑖 with an end point 𝐱 and the other end point in Γ_𝑖. This means that Ω_𝑖 is also a connected set or a union of two connected sets as that of Γ_𝑖.
Next, if the symmetric matrix 𝐿_𝑖 is not a diagonal matrix, then there exists an orthogonal matrix 𝑈 such that 𝑈^𝑇𝐿_𝑖𝑈 is a diagonal matrix; say that 𝐷_𝑖=𝑈^𝑇𝐿_𝑖𝑈, and the set Ω_𝑖 is the image of Ω^new_𝑖∶={𝐲∣𝐲^𝑇𝐷_𝑖𝐲≥0, ‖𝐲‖=1} under the continuous transformation 𝐱=𝑈𝐲, since the image of a connected set under a continuous transformation is also a connected set. The result then follows.

Now, we are in the position to show that “condition (2) ⇒ condition (5).”

Lemma 4. Condition (2) ⇒condition (5).

Proof. Since 𝐱^𝑇𝐿₁𝐱≥0 and 𝐱^𝑇𝐿₂𝐱<0 for all 𝐱∈Ω₁, this implies that 𝑔₁𝛽∶=sup_𝐱∈Ω1𝐱^𝑇𝐿₁+𝛽𝐿₂𝐱 (16) is a decreasing function of 𝛽 and 𝑔₁(0)≥0. Since Ω₁ is a compact set, it follows that sup_𝐱∈Ω1𝐱^𝑇𝐿₁𝐱=𝑙₁ and sup_𝐱∈Ω1𝐱^𝑇𝐿₂𝐱=−𝑙₂ for some 𝑙₁>0 and 𝑙₂>0. Thus, 𝑔₁(𝛽)<0 for all 𝛽>𝑙₁/𝑙₂. By the Intermediate Value Theorem, there exists a real number 𝛽^∗₁≥0 such that 𝑔₁(𝛽^∗₁)=0. Similarly, 𝐱^𝑇𝐿₁𝐱<0 and 𝐱^𝑇𝐿₂𝐱≥0 for all 𝐱∈Ω₂ yield that 𝑔₂𝛽∶=sup_𝐱∈Ω2𝐱^𝑇𝐿₁+𝛽𝐿₂𝐱 (17) is an increasing function of 𝛽 and 𝑔₂(0)<0. If sup_𝐱∈Ω2𝐱^𝑇𝐿₂𝐱>0, then there exists a real number 𝛽^∗₂>0 such that 𝑔₂(𝛽^∗₂)=0. On the other hand, if sup_𝐱∈Ω2𝐱^𝑇𝐿₂𝐱=0 (this corresponds to the case that all of the unstable eigenvalues of 𝐿₂ are zero), then we define 𝛽^∗₂=∞. From the definitions of 𝛽^∗₁ and 𝛽^∗₂, we have that 𝛽>𝛽^∗₁⟺𝐱^𝑇𝐿₁+𝛽𝐿₂𝐱<0,∀𝐱∈Ω₁,𝛽<𝛽^∗₂⟺𝐱^𝑇𝐿₁+𝛽𝐿₂𝐱<0,∀𝐱∈Ω₂. (18)
Next, we show that 𝛽^∗₁<𝛽^∗₂. Suppose, on the contrary, that 𝛽^∗₂≤𝛽^∗₁. Then, there exists a nonnegative 𝛽^∗ satisfying 𝛽^∗₂≤𝛽^∗≤𝛽^∗₁. From (18), there exist an 𝐱₁∈Ω₁ and an 𝐱₂∈Ω₂ such that 𝐱^𝑇₁(𝐿₁+𝛽^∗𝐿₂)𝐱₁≥0 and 𝐱^𝑇₂(𝐿₁+𝛽^∗𝐿₂)𝐱₂≥0. It follows that Ω^∗∩Ω₁≠∅,Ω^∗∩Ω₂≠∅, (19) where Ω^∗={𝐱∣𝐱^𝑇(𝐿₁+𝛽^∗𝐿₂)𝐱≥0, ‖𝐱‖=1}. Moreover, since 𝐱^𝑇𝐿_𝑖𝐱<0 for all 𝑖=1,2 and 𝐱∈𝑆⧵(Ω₁∪Ω₂), we have that 𝐱^𝑇(𝐿₁+𝛽^∗𝐿₂)𝐱<0 for all 𝐱∈𝑆⧵(Ω₁∪Ω₂). This means that Ω^∗⊆Ω₁∪Ω₂. (20) The relations (19)-(20) then yield Ω^∗=Ω^∗∩Ω₁∪Ω^∗∩Ω₂. (21) Since, by (d) of Lemma 3, Ω₁∩Ω₂=∅, then (21) reveals that Ω^∗ is the union of two disjoint sets which are not symmetric about the origin. But this is impossible since, by Lemma 3, Ω^∗ should also have the property that it is a connected set or it consists of two disjoint connected sets which are symmetric about the origin. This contradiction then implies that 𝛽^∗₁<𝛽^∗₂.
Finally, from (18), any 𝛽 with 𝛽^∗₁<𝛽<𝛽^∗₂ will result in 𝐱^𝑇(𝐿₁+𝛽𝐿₂)𝐱<0 for all 𝐱∈Ω₁∪Ω₂. In addition, since, for all 𝐱∈𝑆⧵(Ω₁∪Ω₂), both 𝐱^𝑇𝐿₁𝑥<0 and 𝐱^𝑇𝐿₂𝐱<0, it follows that 𝐱^𝑇(𝐿₁+𝛽𝐿₂)𝐱<0 for any 𝛽∈(𝛽^∗₁,𝛽^∗₂) and for all 𝐱∈𝑆. Thus, the matrix 𝐿₁+𝛽𝐿₂<0 for any 𝛽∈(𝛽^∗₁,𝛽^∗₂). This completes the proof.

Theorem 5 below summarizes the conclusions.

Theorem 5. Condition (2) ⇔condition (3) ⇔condition (5).

We have shown that conditions (2)–(5) are equivalent. We now investigate the relation of the unstable eigenpairs of 𝐿₁ and 𝐿₂ under these conditions. Let {𝜆⁺₁,…,𝜆⁺_𝑚1} and {𝜇⁺₁,…,𝜇⁺_𝑚2} be the unstable eigenvalues of 𝐿₁ and 𝐿₂, respectively, let 𝐱⁺_1𝑖 be a unit eigenvector of 𝐿₁ associated with 𝜆⁺_𝑖, and let 𝐱⁺_2𝑗 be a unit eigenvector of 𝐿₂ associated with 𝜇⁺_𝑗. Define 𝛿_1𝑗∶=𝐱^+𝑇_2𝑗𝐿₁𝐱⁺_2𝑗,𝑗=1,…,𝑚₂,𝛿_2𝑖∶=𝐱^+𝑇_1𝑖𝐿₂𝐱⁺_1𝑖,𝑖=1,…,𝑚₁. (22) Clearly, if condition (2) holds, then 𝛿_1𝑗<0,𝛿_2𝑖<0 (23) for all 1≤𝑖≤𝑚₁ and 1≤𝑗≤𝑚₂. The following result is a direct consequence of Theorem 5.

Corollary 6. Suppose that condition (2) holds. Then, 𝜆⁺_𝑖𝜇⁺_𝑗<𝛿_1𝑗𝛿_2𝑖 for all 1≤𝑖≤𝑚₁ and 1≤𝑗≤𝑚₂.

Proof. If condition (2) holds, then by Theorem 5 there exists a 𝛽>0 such that 𝐿₁+𝛽𝐿₂<0. It follows that 𝐱^+𝑇_1𝑖𝐿₁+𝛽𝐿₂𝐱⁺_1𝑖=𝜆⁺_𝑖+𝛽𝛿_2𝑖𝐱<0,_+𝑇2𝑗𝐿₁+𝛽𝐿₂𝐱⁺_2𝑗=𝛿_1𝑗+𝛽𝜇⁺_𝑗<0 (24) for all 1≤𝑖≤𝑚₁ and 1≤𝑗≤𝑚₂. This then implies that 𝜆⁺_𝑖<−𝛽𝛿_2𝑖,𝜇⁺_𝑗𝛿<−^1𝑗𝛽, (25) or 𝜆⁺_𝑖𝜇⁺_𝑗<𝛿_1𝑗𝛿_2𝑖 for all 1≤𝑖≤𝑚₁ and 1≤𝑗≤𝑚₂. This completes the proof.

Remark 7. From Corollary 6 and (23), we have that 0≤max_{1≤𝑖≤𝑚1}−𝜆⁺_𝑖𝛿_2𝑖<min_{1≤𝑗≤𝑚2}−𝛿_1𝑗𝜇⁺_𝑗. (26) If we define 𝛽₁∶=max_{1≤𝑖≤𝑚1}−𝜆⁺_𝑖𝛿_2𝑖,𝛽₂∶=min_{1≤𝑗≤𝑚2}−𝛿_1𝑗𝜇⁺_𝑗, (27) then 𝛽₁<𝛽₂. Moreover, 𝛽∈(𝛽₁,𝛽₂) yields 𝐱^+𝑇_1𝑖𝐿₁+𝛽𝐿₂𝐱⁺_1𝑖=𝜆⁺_𝑖+𝛽𝛿_2𝑖<𝜆⁺_𝑖+𝛽₁𝛿_2𝑖𝐱<0,_+𝑇2𝑗𝐿₁+𝛽𝐿₂𝐱⁺_2𝑗=𝛿_1𝑗+𝛽𝜇⁺_𝑗<𝛿_1𝑗+𝛽₂𝜇⁺_𝑗<0. (28) This means that 𝐱^𝑇(𝐿₁+𝛽𝐿₂)𝐱 is negative at all of the eigenvectors associated with unstable eigenvalues of 𝐿₁ and 𝐿₂. However, 𝐿₁+𝛽𝐿₂ might not be a negative definite matrix, which will be discussed in more details in the following section. From this observation, we then have the relation 𝛽₁≤𝛽^∗₁<𝛽^∗₂≤𝛽₂.

4. Specialization to Planar Systems

In order to explore the geometrical interpretations of conditions (2)-(3), (5), in this section, we focus the study on planar systems.

Consider the two linear systems as given by (44) with 𝑛=2. Since neither 𝐴₁ nor 𝐴₂ is Hurwitz, we have that, for any 𝑃>0, none of 𝐿_𝑖=𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖, 𝑖=1,2, is a Hurwitz matrix. This means that both 𝐿₁ and 𝐿₂ contain exactly one unstable eigenvalue if condition (2) holds. Denote by (𝜆⁺₁,𝐱⁺₁₁) and (𝜇⁺₁,𝐱⁺₂₁) the unstable eigenpairs of 𝐿₁ and 𝐿₂, respectively, with ‖𝐱⁺₁₁‖=‖𝐱⁺₂₁‖=1 and 𝐱^+𝑇₁₁𝐱⁺₂₁≥0. (29) Then, by Corollary 6, a necessary condition for condition (2) is 𝜆⁺₁𝜇⁺₁<𝛿₁₁𝛿₂₁, (30) where 𝛿₁₁ and 𝛿₂₁ are given by (22). Clearly, condition (30) implies that 𝜆⁺₁𝜇⁺₁<𝜆⁻₁𝜇⁻₁, (31) where 𝜆⁻₁ and 𝜇⁻₁ denote stable eigenvalues of 𝐿₁ and 𝐿₂, respectively. Condition (30) or (31) is only a necessary one for condition (2). However, if we add an extra condition as described in (32), we have a necessary and sufficient one for condition (2).

Theorem 8. Consider the two systems as given by (44) with 𝑛=2. Then, condition (2) holds if and only if condition (30) or (31) holds and cos²𝜆𝜃<⁺¹𝜇⁺₁+𝜆⁻₁𝜇⁻₁−2𝜆⁺₁𝜇⁺₁𝜆⁻₁𝜇⁻₁𝜆⁺₁−𝜆⁻₁𝜇⁺₁−𝜇⁻₁𝑜𝑟𝜃>cos⁻¹𝜆⁻₁𝜇⁻₁−𝜆⁺₁𝜇⁺₁𝜆⁺₁−𝜆⁻₁𝜇⁺₁−𝜇⁻₁, (32) where 𝜃 is the angle between 𝐱⁺₁₁ and 𝐱⁺₂₁.

Proof. It is not difficult to see that the angles of the two-sided cones {𝐱∣𝐱^𝑇diag(𝜆⁺₁,𝜆⁻₁)𝐱≥0} and {𝐱∣𝐱^𝑇diag(𝜇⁺₁,𝜇⁻₁)𝐱≥0} are 2tan⁻¹−𝜆⁺₁/𝜆⁻₁ and 2tan⁻¹−𝜇⁺₁/𝜇⁻₁, respectively. These angles are invariant under orthogonal transformation. Moreover, since 𝐿₁ and 𝐿₂ are symmetric matrices, they possess complete orthonormal set of eigenvectors. Thus, condition(2)holds⟺𝜃>tan⁻¹−𝜆⁺₁𝜆⁻₁+tan⁻¹−𝜇⁺₁𝜇⁻₁⟺cos𝜃<costan⁻¹−𝜆⁺₁𝜆⁻₁+tan⁻¹−𝜇⁺₁𝜇⁻₁⟺cos𝜃<𝜆⁻₁𝜇⁻₁−𝜆⁺₁𝜇⁺₁𝜆⁺₁−𝜆⁻₁𝜇⁺₁−𝜇⁻₁. (33) The result then follows.

From the previous Theorem 8, it requires not only the relation of eigenvalues having the relation (31) but also the angle of eigenvectors satisfying (32) for two unstable planar systems to match condition (2). However, if both 𝐿₁ and 𝐿₂ are diagonal matrices or have the same set of eigenvectors with 𝐱⁺₁₁=±𝐱⁻₂₁ and 𝐱⁺₂₁=±𝐱⁻₁₁, where 𝐱⁻₁₁ and 𝐱⁻₂₁ denote unit eigenvectors of 𝐿₁ and 𝐿₂ associated with stable eigenvalues, then condition (32) is automatically true and condition (30) or (31) is then a necessary and sufficient one for condition (2) as described by the following result.

Corollary 9. Suppose that 𝐿₁ and 𝐿₂ have the same set of eigenvectors with 𝐱⁺₁₁=±𝐱⁻₂₁ and 𝐱⁺₂₁=±𝐱⁻₁₁. Then, condition (2) holds if and only if condition (30) or (31) holds.

4.1. Application to Unstabilizable Linear Control Systems

Consider the two linear control systems 𝐱̇=𝐴𝐱+𝐵_𝑖𝐮_𝑖,𝑖=1,2. (34) Here, 𝐴∈ℝ^𝑛×𝑛, 𝐵₁∈ℝ^𝑛×𝑚1, and 𝐵₂∈ℝ^𝑛×𝑚2, and we assume that neither of the two control systems is stabilizable. Although neither of the two control systems is stabilizable by any linear state feedback, it is still possible to have a set of controllers 𝐮₁=𝐾₁𝐱 and 𝐮₂=𝐾₂𝐱 and stabilizing switching laws between the two systems, which makes 𝐱(𝑡)→0 as 𝑡→∞. In this section, we will study the existence condition and implement it through the equivalent relation as described in Section 3.

According to the results of Section 3, there exist 𝐾₁ and 𝐾₂ and a stabilizing switching law between the two control systems given in (34) if there exists 𝛽>0 such that 𝐴+𝐵₁𝐾₁+𝛽𝐴+𝐵₂𝐾₂isaHurwitzmatrix, (35) or, equivalently, if there exist 𝑃>0 and 𝛽>0 such that 𝐴+𝐵₁𝐾₁^𝑇𝑃+𝑃𝐴+𝐵₁𝐾₁+𝛽𝐴+𝐵₂𝐾₂^𝑇𝑃+𝑃𝐴+𝐵₂𝐾₂<0. (36) If we choose 𝐾₁=−𝐵^𝑇₁𝑃,𝐾₂=−𝐵^𝑇₂𝑃, (37) then inequality (36) becomes 𝐴^𝑇𝑃+𝑃𝐴−2𝑃𝐵₁𝐵^𝑇₁𝑃+𝛽𝐴^𝑇𝑃+𝑃𝐴−2𝑃𝐵₂𝐵^𝑇₂𝑃<0 (38) or 𝐴^𝑇_𝛽𝑃+𝑃𝐴_𝛽−2𝑃𝐵_𝛽𝐵^𝑇_𝛽𝑃<0, (39) where 𝐴_𝛽∶=(1+𝛽)𝐴 and 𝐵_𝛽∶=(𝐵₁⋮𝛽𝐵₂). Note the existence of 𝑃>0 and 𝛽>0 such that inequality (39) holds if and only if (𝐴_𝛽,𝐵_𝛽) is stabilizable. We, thus, have the following result.

Theorem 10. Consider the two linear control systems as given by (34). Suppose that neither of the two control systems is stabilizable, but (𝐴,𝐵) is stabilizable, where 𝐵=(𝐵₁⋮𝐵₂). Then, there exist 𝐾₁∈ℝ^𝑚1×𝑛, 𝐾₂∈ℝ^𝑚2×𝑛, and a stabilizing switching law between the two systems 𝐴+𝐵₁𝐾₁ and 𝐴+𝐵₂𝐾₂. Moreover, 𝐾₁ and 𝐾₂ can be chosen in the form of (37), and 𝑃 is solved by the following Riccati equation: 𝐴^𝑇𝑃+𝑃𝐴−2𝑃𝐵₁𝐵^𝑇₁𝑃+𝛽𝐴^𝑇𝑃+𝑃𝐴−2𝑃𝐵₂𝐵^𝑇₂𝑃=−𝐻, (40) where 𝐻>0 and 𝛽>0.

Proof. The theorem can be easily verified by noting the following fact that (𝐴_𝛽,𝐵_𝛽) is stabilizable if and only if (𝐴,𝐵) is stabilizable and the discussions before this theorem.

After deriving the control laws for the two control systems given in (34), the stabilizing switching laws between the two control systems can then be implemented through an existing algorithm proposed by [1]. To demonstrate the use of the result, we give an example as follows.

4.2. A Numerical Example

Example 1. Consider the two linear control systems given by (34) with 𝐴=−32−1414−70−2729−140−5860,𝐵₁=2510,𝐵₂=0−1−1. (41) By direct checking, neither (𝐴,𝐵₁) nor (𝐴,𝐵₂) is stabilizable, but (𝐴,𝐵) is stabilizable, where 𝐵=(𝐵₁⋮𝐵₂). The solution of the Riccati equation given by (40) with 𝛽=2 and 𝐻 being the identity matrix is found to be 𝑃=36.418.1−18.218.121.1−15.1−18.2−15.112.1. (42) Choose 𝐾₁ and 𝐾₂ in the form of (37); then we have two linear systems with system matrices 𝐴_𝑖=𝐴+𝐵_𝑖𝐾_𝑖, 𝑖=1,2, where 𝐴₁=,𝐴5.74.1−5.324.218.3−19.348.532.6−36.7₂=.−32.0−14.014.0−69.9−33.031.9−139.9−64.062.9 (43) Clearly, neither of the two systems is stable, while, by Theorem 10, they will satisfy condition (3). A stabilizing switching law can then be constructed through the algorithm proposed by [1]. Figure 1 demonstrates the timing response of the three states and phase trajectory with initial (2,−1,1)^𝑇. The three states are observed to converge to zero, which agrees with the theoretical results.

(a)

(b)

Figure 1 

(a) Timing response of the three states and (b) state space trajectory.

5. Switching among 𝑁 Linear Systems

In this section, we extend the derivation stated above to the equivalence of two sufficient conditions for the existence of quadratically stabilizing switching laws among 𝑁 linear systems. Consider the 𝑁  𝑛-dimensional unstable linear systems 𝐱̇=𝐴_𝑖𝐱,𝑖=1,2,…,𝑁, (44) where 𝐱∈ℝ^𝑛 and 𝐴_𝑖∈ℝ^𝑛×𝑛, and we assume that all 𝐴_𝑖, 𝑖=1,…,𝑁, are unstable matrices; that is, none of the 𝑁 matrices is a Hurwitz matrix. It is known that a sufficient condition for the existence of quadratically stabilizing switching laws among the 𝑁 unstable systems can be stated as in condition (45) as follows: ∃𝑃>0suchthat^𝑁_𝑖=1𝐱∣𝐱^𝑇𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖𝐱<0=ℝ^𝑛⧵{0}. (45) However, condition (45) does not provide a means to obtain such a matrix 𝑃. On the other hand, Decarlo et al. [11] verified that the matrix pencil condition as given by (4) is also a sufficient condition for the existence of stabilizing switching laws. They also proposed an algorithm to determine the existence of 𝛼_𝑖 and to explicitly construct a class of quadratically stabilizing switching laws if condition (4) holds. However, it is not difficult to prove that condition (4) is equivalent to condition (45).

5.1. Implementation for Switching among 𝑁 Systems

To achieve the second objective as described in Section 2, an LMI condition is proposed in this section to implement the existence of 𝛼_𝑖 given in (4). To this end, if we can find 𝛼_𝑖>0 and ∑^𝑁_𝑖=1𝛼_𝑖=1 such that 𝐴_𝑒𝑞∶=∑^𝑁_𝑖=1𝛼_𝑖𝐴^𝑠_𝑖 is a Hurwitz matrix, where 𝐴^𝑠_𝑖∶=(𝐴_𝑖+𝐴^𝑇_𝑖)/2, then it can be inferred by Lemma 2 below that the quadratically stabilizing switching for systems (44) is feasible.

Lemma 2 (see [47]). For any 𝐴_𝑖∈ℝ^𝑛×𝑛, 𝑖=1,2,…,𝑁, there exists 𝐴_𝑖=(𝐴_𝑖+𝐴^𝐻_𝑖)/2+(𝐴_𝑖−𝐴^𝐻_𝑖)/2, where (𝐴_𝑖+𝐴^𝐻_𝑖)/2≡𝐴^𝑠_𝑖 means the symmetric part of 𝐴_𝑖. By the Rayleigh quotient of 𝐴^𝑠_𝑖, one can derive that Re𝜆𝐴_𝑖⊆𝜆_min𝐴^𝑠_𝑖,𝜆_max𝐴^𝑠_𝑖. (46)

Proof. It is trivial; thus, it is omitted here.

According to (46), condition (4) can be turned into the following LMIs: 𝐴^𝑠_𝑒𝑞=𝛼₁𝐴^𝑠₁+𝛼₂𝐴^𝑠₂+⋯+𝛼_𝑁𝐴^𝑠_𝑁<0. (47)

From Lemma 2 and letting 𝛽_𝑘∶=𝛼_𝑘+1/𝛼₁, 𝑘=1,2,…,𝑁−1, if we can find positive constants 𝛽₁,𝛽₂,…,𝛽_𝑁−1 such that 𝐴^𝑠_𝑒𝑞=𝐴^𝑠₁+𝛽₁𝐴^𝑠₂+⋯+𝛽_𝑁−1𝐴^𝑠_𝑁<0, (48) then condition (4) automatically holds. Although condition (48) is less general than condition (4), it can be solved by LMI toolbox in MATLAB to carry out the implementation.

5.2. Switching among 𝑁 Unstabilizable Control Systems

Consider the 𝑁 linear control systems 𝐱̇=𝐴𝐱+𝐵_𝑖𝐮_𝑖,𝑖=1,2,…,𝑁, (49) where 𝐴∈ℝ^𝑛×𝑛 and 𝐵_𝑖∈ℝ^{𝑛×𝑚𝑖}, and we assume that each of (𝐴,𝐵_𝑖) is unstabilizable. Although neither of the (𝐴,𝐵_𝑖) is stabilizable by any linear state feedback, it is still possible to have a set of controllers 𝐮_𝑖(𝐱)=𝐾_𝑖𝐱 and quadratically stabilizing switching laws among the 𝑁 systems, which makes 𝐱(𝑡)→𝟎 as 𝑡→∞. In this section, we will study the existence condition and implement it through the equivalent relation of condition (4) and condition (2).

According to the abovementioned equivalence result, there exist 𝐾₁,…,𝐾_𝑁 and stabilizing switching laws among the 𝑁 control systems given in (49) if there exist positive constants 𝛽₁,…,𝛽_𝑁−1 such that 𝐴+𝐵₁𝐾₁+^𝑁−1_𝑘=1𝛽_𝑘𝐴+𝐵_𝑘𝐾_𝑘 (50) is a Hurwitz matrix or, equivalently, if there exist 𝑃>0 and positive constants 𝛽₁,…,𝛽_𝑁−1 such that 𝐴+𝐵₁𝐾₁^𝑇𝑃+𝑃𝐴+𝐵₁𝐾₁+^𝑁−1_𝑘=1𝛽_𝑘𝐴+𝐵_𝑘𝐾_𝑘^𝑇𝑃+𝑃𝐴+𝐵_𝑘𝐾_𝑘<0. (51) If we choose 𝐾_𝑖=−𝐵^𝑇_𝑖𝑃, (52) then inequality (51) becomes 𝐴^𝑇𝑃+𝑃𝐴−2𝑃𝐵₁𝐵^𝑇₁𝑃+^𝑁−1_𝑘=1𝛽_𝑘𝐴^𝑇𝑃+𝑃𝐴−2𝑃𝐵_𝑘𝐵^𝑇_𝑘𝑃<0, (53) or 𝐴^𝑇_𝛽𝑃+𝑃𝐴_𝛽−2𝑃𝐵_𝛽𝐵^𝑇_𝛽𝑃<0, (54) where 𝐴_𝛽∶=1+^𝑁−1_𝑘=1𝛽_𝑘𝐵𝐴,_𝛽=𝐵₁⋮𝛽₁𝐹₂⋮⋯⋮𝛽_𝑁−1𝐵_𝑁. (55)(56) Note the existence of 𝑃>0 and positive constants 𝛽₁,…,𝛽_𝑁−1 such that inequality (56) holds if and only if (𝐴_𝛽,𝐵_𝛽) is stabilizable. We, thus, have the following result.

Theorem 3. Consider a class of 𝑁 linear control systems as given by (49). Suppose that none of the N control systems is stabilizable, but (𝐴,𝐵_𝑖) is stabilizable, where 𝐵=(𝐵₁⋮𝐵₂⋮⋯⋮𝐵_𝑁). Then, there exist 𝐾_𝑖∈ℝ^{𝑚𝑖×𝑛} and a quadratically stabilizing switching law among systems 𝐴+𝐵_𝑘𝐾_𝑘, 𝑘=1,2,…,𝑁. Moreover, 𝐾_𝑘 can be chosen in the form of (52), and 𝑃 is solved by the following Riccati equation: 𝐴^𝑇_𝛽𝑃+𝑃𝐴_𝛽−2𝑃𝐵_𝛽𝐵^𝑇_𝛽𝑃=−𝐻, (57) where 𝐻>0 and 𝛽>0.

Proof. To prove the theorem, we just need to prove that (𝐴_𝛼,𝐵_𝛼) is stabilizable if and only if (𝐴,𝐵) is stabilizable, where 𝐴_𝛼=(∑^𝑁_𝑖=1𝛼_𝑖)𝐴=𝛼𝐴, 𝐵_𝛼=(√𝛼₁𝐵₁⋮⋯⋮√𝛼_𝑁𝐵_𝑁), and 𝐵=(𝐵₁⋮⋯⋮𝐵_𝑁). If (𝐴,𝐵) is stabilizable, then 𝐾=(𝐾₁⋮⋯⋮𝐾_𝑁)^𝑇 must exist such that 𝐴+𝐵𝐾 is Hurwitz and 𝐴+𝐵𝐾=𝐴+∑^𝑁_𝑖=1𝐵_𝑖𝐾_𝑖. We select 𝐾=((𝛼/√𝛼₁)𝐾₁⋮⋯⋮(𝛼/√𝛼_𝑁)𝐾_𝑁)^𝑇, and then we can get 𝐴_𝛼+𝐵_𝛼𝐾=𝐴_𝛼+^𝑁_𝑖=1√𝛼_𝑖𝐵_𝑖𝛼√𝛼_𝑖𝐾_𝑖=𝛼𝐴+^𝑁_𝑖=1𝛼𝐵_𝑖𝐾_𝑖=𝛼𝐴+^𝑁_𝑖=1𝐵_𝑖𝐾_𝑖. (58) Since 𝐴+∑^𝑁_𝑖=1𝐵_𝑖𝐾_𝑖 is a Hurwitz matrix and 𝛼>0, therefore, (𝐴_𝛼,𝐵_𝛼) is stabilizable, and then we can find 𝑃 by (57).

After the determination of 𝐾_𝑖, the stabilizing switching law can be implemented by the results of Section 5.1.

5.3. Numerical Examples

Example 1. Consider the three unstable matrices in the form of (44) with 𝐴₁=,𝐴0−10−10206−9−40₂=,𝐴−100−4−1−10−9−1−105₃=.7−8−21−1473−9 (59) Indeed, we have that 𝜆(𝐴₁)={10.3039,−5.1520±5.7990𝑖}, 𝜆(𝐴₂)={−9.5994,−15.0750,9.6744}, and 𝜆(𝐴₃)={−1.7049,3.8525±4.0791𝑖}. By using the results of Section 3, we can easily find that 𝛽₁=4.2349 and 𝛽₂=4.1726 via solving (48). It follows from Lemma 1 that 𝐴_𝑒𝑞∶=𝐴₁+𝛽₁𝐴₂+𝛽₂𝐴₃ is a Hurwitz matrix and 𝐴_𝑒𝑞=−13.1408−43.3808−35.28481.9377−46.5215−15.423715.9733−33.8312−16.3788. (60) By solving the Lyapunov equation 𝐴^𝑇_𝑒𝑞𝑃+𝑃𝐴_𝑒𝑞+𝐼=0, we have that 𝑃=0.06050.0073−0.01730.00730.0194−0.0253−0.0173−0.02530.0660. (61) The quadratically stabilizing switching law is then easily derived by the algorithm presented by [1]. Figure 2(a) describes the three states response with initial (2,−1,1)^𝑇. The three states are observed to converge to zero. Figure 2(b) shows the switching sequences. Figure 3 describes the phase plane trajectory of the switched systems. The trajectory is observed to converge to the origin. The simulation results agree with the theoretical results.

(a)

(b)

Figure 2 

(a) Timing response of the three states and (b) switching sequences.

Figure 3 

Phase plane trajectory of the three systems.

Example 2. Consider the three linear control systems in the form of (49) with 𝐴=33−32−328−7−816−16−15,𝐵₁=4,𝐵−5−6₂=5111,𝐵₃=.−21−1−16 (62) By direct checking, neither of (𝐴,𝐵₁), (𝐴,𝐵₂), and (𝐴,𝐵₃) is stabilizable, but (𝐴,𝐵) is stabilizable, where 𝐵=(𝐵₁⋮𝐵₂⋮𝐵₃). The solution of the Riccati equation given by (57) with 𝛽=2 and 𝐻 being the identity matrix is found to be 𝑃=125.814−153.305−178.213−153.305307.177224.444−178.213224.444294.794. (63) Choose 𝐾₁, 𝐾₂, and 𝐾₃ in the form of (52); then we have three linear systems with system matrices 𝐴_𝑖=𝐴+𝐵_𝑖𝐾_𝑖, 𝑖=1, 2, and 3, where 𝐸₁=10³∗,𝐸−0.83203.21080.06840.7000−2.6013−0.0883−1.02203.87540.1055₂=10³∗,𝐸−4.7984−0.84575.9446−2.1881−0.37692.7087−0.4232−0.09000.5283₃=10⁴∗.−0.3483−0.1674−1.2177−0.0133−0.0073−0.0494−0.0494−0.1067−0.7788 (64) Of course, all of the three matrices 𝐴₁, 𝐴₂, and 𝐴₃ are not stable, while, by Theorem 3, they satisfy condition (4). A quadratically stabilizing switching law can then be constructed through the algorithm proposed by [1]. Figure 4(a) demonstrates the three states response which are observed to converge to zero. Figure 4(b) shows the switching sequences. Figure 5 describes the phase plane trajectory with initial (2,−1,1)^𝑇. The phase plane trajectory is observed to converge to the origin. The simulation results agree with the theoretical results.

(a)

(b)

Figure 4 

(a) Timing response of three states and (b) switching sequences.

Figure 5 

Phase plane trajectory of the three control systems.

6. Conclusions

In this study, we propose a way to show the equivalence of two sufficient conditions for the existence of stabilizing switching laws between two unstable linear systems. One of the conditions is powerful for theoretical derivation, while the other is easily implemented through an existing algorithm. By the use of the equivalent relation, a simple existence condition of controllers and stabilizing switching laws between two unstabilizable linear control systems is then proposed and found to be easily implemented. An illustrative example has demonstrated the use and benefit of the equivalent relation. In addition, the existence conditions are also specialized to planar systems to facilitate the understanding of their geometrical interpretations. It requires not only the relation between the eigenvalues of the associated matrices but also the angles between the unstable eigenvectors to match the existence condition. Moreover, we have discussed a feasible way to construct the quadratically stabilizing switching laws among 𝑁 unstable linear systems and extended the result to organize the controllers and the quadratically stabilizing switching laws of a class of 𝑁 unstabilizable linear control systems. The illustrative examples and simulation results demonstrate the feasibility and effectiveness of the proposed method for switching control issues.

Acknowledgments

The authors would like to thank the Lead Editor, Professor Chang-Hua Lien, and the Editor, Professor Hamid Reza Karimi, for their assistance. The authors would also like to thank the anonymous reviewers for their constructive comments and helpful suggestions.