Abstract and Applied Analysis

Volume 2013 (2013), Article ID 681523, 11 pages

http://dx.doi.org/10.1155/2013/681523

Research Article

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

Sendren Sheng-Dong Xu¹ and Chih-Chiang Chen²

¹Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, No. 43, Section 4, Keelung Road, Da’an District, Taipei 10607, Taiwan

²Institute of Electrical Control Engineering, National Chiao Tung University, No. 1001, Ta Hsueh Road, Hsinchu 30010, Taiwan

Received 18 June 2013; Revised 23 September 2013; Accepted 23 September 2013

Academic Editor: Hamid Reza Karimi

Copyright © 2013 Sendren Sheng-Dong Xu and Chih-Chiang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 unstable linear systems and 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 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 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 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 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

where we assume that neither of the two systems is stable; that is, neither nor 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:

Clearly, under this condition and by defining , one can find a proper switching law between the two systems such that for all the time. It then results in as . However, condition (2) does not provide a means to obtain such a matrix . 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:

They also proposed an algorithm to determine the existence of 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:

However, the literature search indicates that the implementation of stabilizing 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 unstable linear systems. Finally, we employ the results to design control laws and quadratically stabilizing switching laws among unstabilizable linear control systems.

3. The Proof of Equivalence

To achieve the objectives as described in Section 2, we introduce the following condition:

where

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

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. is Hurwitz

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:

The two sets have the following properties.

Lemma 3. (a) Both sets and are symmetric about the origin. That is, for , where .

(b) Each unit eigenvector associated with unstable eigenvalues of belongs to (by unstable eigenvalues one means those eigenvalues with nonnegative real parts).

(c) Either is a connected set or it consists of two disjoint connected sets with .

(d) If condition (2) holds, then and , where 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 is a diagonal matrix having the following form:

where for and for . Clearly, the set

and , 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, means that

This implies that

or

where . It follows that the set

Therefore, for any point in , there exists a smooth path lying entirely inside with an end point and the other end point in . This means that is also a connected set or a union of two connected sets as that of .

Next, if the symmetric matrix is not a diagonal matrix, then there exists an orthogonal matrix such that is a diagonal matrix; say that , and the set is the image of , under the continuous transformation , 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 and for all , this implies that

is a decreasing function of and . Since is a compact set, it follows that and for some and . Thus, for all . By the Intermediate Value Theorem, there exists a real number such that . Similarly, and for all yield that

is an increasing function of and . If , then there exists a real number such that . On the other hand, if (this corresponds to the case that all of the unstable eigenvalues of are zero), then we define . From the definitions of and , we have that

Next, we show that . Suppose, on the contrary, that . Then, there exists a nonnegative satisfying . From (18), there exist an and an such that and . It follows that

where , . Moreover, since for all and , we have that for all . This means that

The relations (19)-(20) then yield

Since, by (d) of Lemma 3, , then (21) reveals that is the union of two disjoint sets which are not symmetric about the origin. But this is impossible since, by Lemma 3, 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 .

Finally, from (18), any with will result in for all . In addition, since, for all , both and , it follows that for any and for all . Thus, the matrix for any . 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 and under these conditions. Let and be the unstable eigenvalues of and , respectively, let be a unit eigenvector of associated with , and let be a unit eigenvector of associated with . Define

Clearly, if condition (2) holds, then

for all and . The following result is a direct consequence of Theorem 5.

Corollary 6. Suppose that condition (2) holds. Then, for all and .

Proof. If condition (2) holds, then by Theorem 5 there exists a such that . It follows that

for all and . This then implies that

or for all and . This completes the proof.

Remark 7. From Corollary 6 and (23), we have that

If we define

then . Moreover, yields

This means that is negative at all of the eigenvectors associated with unstable eigenvalues of and . However, 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 .

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 . Since neither nor is Hurwitz, we have that, for any , none of , , is a Hurwitz matrix. This means that both and contain exactly one unstable eigenvalue if condition (2) holds. Denote by and the unstable eigenpairs of and , respectively, with and

Then, by Corollary 6, a necessary condition for condition (2) is

where and are given by (22). Clearly, condition (30) implies that

where and denote stable eigenvalues of and , 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 . Then, condition (2) holds if and only if condition (30) or (31) holds and

where is the angle between and .

Proof. It is not difficult to see that the angles of the two-sided cones and are and , respectively. These angles are invariant under orthogonal transformation. Moreover, since and are symmetric matrices, they possess complete orthonormal set of eigenvectors. Thus,

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 and are diagonal matrices or have the same set of eigenvectors with and , where and denote unit eigenvectors of and 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 and have the same set of eigenvectors with and . 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

Here, , , and , 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 and and stabilizing switching laws between the two systems, which makes as . 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 and and a stabilizing switching law between the two control systems given in (34) if there exists such that

or, equivalently, if there exist and such that

If we choose

then inequality (36) becomes

or

where and . Note the existence of and such that inequality (39) holds if and only if 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 is stabilizable, where . Then, there exist , , and a stabilizing switching law between the two systems and . Moreover, and can be chosen in the form of (37), and is solved by the following Riccati equation:

where and .

Proof. The theorem can be easily verified by noting the following fact that is stabilizable if and only if 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

By direct checking, neither nor is stabilizable, but is stabilizable, where . The solution of the Riccati equation given by (40) with and being the identity matrix is found to be

Choose and in the form of (37); then we have two linear systems with system matrices , , where

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 . 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 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 linear systems. Consider the   -dimensional unstable linear systems

where and , and we assume that all , , are unstable matrices; that is, none of the matrices is a Hurwitz matrix. It is known that a sufficient condition for the existence of quadratically stabilizing switching laws among the unstable systems can be stated as in condition (45) as follows:

However, condition (45) does not provide a means to obtain such a matrix . 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 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 Systems

To achieve the second objective as described in Section 2, an LMI condition is proposed in this section to implement the existence of given in (4). To this end, if we can find and such that is a Hurwitz matrix, where , 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 , , there exists , where means the symmetric part of . By the Rayleigh quotient of , one can derive that

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

According to (46), condition (4) can be turned into the following LMIs:

From Lemma 2 and letting , , if we can find positive constants such that

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 Unstabilizable Control Systems

Consider the linear control systems

where and , and we assume that each of is unstabilizable. Although neither of the is stabilizable by any linear state feedback, it is still possible to have a set of controllers and quadratically stabilizing switching laws among the systems, which makes as . 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 and stabilizing switching laws among the control systems given in (49) if there exist positive constants such that

is a Hurwitz matrix or, equivalently, if there exist and positive constants such that

If we choose

then inequality (51) becomes

or

where

Note the existence of and positive constants such that inequality (56) holds if and only if is stabilizable. We, thus, have the following result.

Theorem 3. Consider a class of linear control systems as given by (49). Suppose that none of the N control systems is stabilizable, but is stabilizable, where . Then, there exist and a quadratically stabilizing switching law among systems , . Moreover, can be chosen in the form of (52), and is solved by the following Riccati equation:

where and .

Proof. To prove the theorem, we just need to prove that is stabilizable if and only if is stabilizable, where , , and . If is stabilizable, then must exist such that is Hurwitz and . We select , and then we can get

Since is a Hurwitz matrix and , therefore, is stabilizable, and then we can find by (57).

After the determination of , 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

Indeed, we have that , , and . By using the results of Section 3, we can easily find that and via solving (48). It follows from Lemma 1 that is a Hurwitz matrix and

By solving the Lyapunov equation , we have that

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 . 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

By direct checking, neither of , , and is stabilizable, but is stabilizable, where . The solution of the Riccati equation given by (57) with and being the identity matrix is found to be

Choose , , and in the form of (52); then we have three linear systems with system matrices , , , and , where

Of course, all of the three matrices , , and 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 . 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 unstable linear systems and extended the result to organize the controllers and the quadratically stabilizing switching laws of a class of 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.

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