Exponential Stability of Time-Switched Two-Subsystem Nonlinear Systems with Application to Intermittent Control

Abstract

This paper studies the exponential stability of a class of periodically time-switched nonlinear systems. Three cases of such systems which are composed, respectively, of a pair of unstable subsystems, of both stable and unstable subsystems, and of a pair of stable systems, are considered. For the first case, the proposed result shows that there exists periodically switching rule guaranteeing the exponential stability of the whole system with (sufficient) small switching period if there is a Hurwitz linear convex combination of two uncertain linear systems derived from two subsystems by certain linearization. For the second case, we present two general switching criteria by means of multiple and single Lyapunov function, respectively. We also investigate the stability issue of the third case, and the switching criteria of exponential stability are proposed. The present results for the second case are further applied to the periodically intermittent control. Several numerical examples are also given to show the effectiveness of theoretical results.

1. Introduction

In the recent years, motivated by the fact that many practical systems are inherently multimodal in the sense of that several dynamical systems are required to describe their behavior which may depend on various environmental factors [1, 2], and by the fact that the methods of intelligent control design are based on switching between different controllers [2, 3], the study of switched systems has been received an increasing attention in control theory and applications [2, 4–13]. It is also worth noting that the switching rule is naturally identified as two classes. One is known as state-switched system, and another is time-switched system. So far, most contributions to switched systems were made for the state-switched system. For example, the stability properties for state-dependent switched systems have been characterized in [4–7], and the switched control synthesis designs have been presented in [8, 9]. For the time-switched systems, we refer the readers to [14–16]. One can see that the study of time-switched systems is almost limited to the linear subsystems, at most with the nonlinear perturbations. In [14], the switched system consists of only Hurwitz stable subsystems. The papers [15, 16] deal with the switched systems with both stable and unstable subsystems by means of average dwell time approach.

In the present paper, we study the exponential stability of time-periodically switched systems composed of a pair of nonlinear subsystems with neural-type Lipstchiz nonlinearities (i.e., the nonlinear vector-valued function is of the form ). Three cases of such systems will be dealt with.

Case 1. Periodically switched system with a pair of unstable nonlinear subsystems.

Case 2. Periodically switched system with both stable and unstable nonlinear subsystems.

Case 3. Periodically switched system with a pair of stable nonlinear subsystems.

For the first case, a linearization transformation is introduced to shift the exponential stability issue of the original system into the robustly exponential stability of the transformed system with a pair of linear time-varying subsystems, and we will use the average-system approach to analyze the robustly exponential stability of the transformed systems. For the second and third cases, the general theoretical frameworks based on the multiple Lyapunov functions will be established. We also suggest that if there exists a common Lyapunov function such that (i) for the second case, for the first subsystem and for the second subsystem, then the system as a whole will be globally exponentially stable for any switching period and any switching rate satisfying ; (ii) for the third case, , , for the th subsystem, then the system as a whole will be globally exponentially stable for any switching period and any switching rate satisfying . Note also that for the first case because both subsystems are unstable there is no such common Lyapunov function that is negative definite for any subsystem. Based on the stability analysis of the second case, we address the periodically intermittent output feedback control problem. Several numerical examples will be presented to show the validity of the theoretical results.

The rest of the paper is organized as follows. In the next section, the problem to be dealt with is formulated and the necessary preliminaries are presented. Then, the theoretical results for three cases of time-switched systems are established in Sections 3–5, respectively. Section 6 deals with the stabilization problem by means of periodically intermittent control. In Section 7, several examples are given to verify the effectiveness of the theoretical results. Finally, conclusions are drawn in Section 8.

2. Problem Formulation and Preliminaries

Consider a class of periodically time-switched systems with s ubsystems

(2.1)

where denotes the state vector, and , are all constant matrices, is called the switching period, is called the time duration of the th subsystem with . It is clear that the system (2.1) is a class of periodically time-switched systems with m nonlinear subsystems. Throughout this paper, we also assume that with are continuous functions satisfying the following condition.

(H1) There exist constant scalar numbers such that

(2.2)

In order to linearize system (2.1), we define functions as follows:

(2.3)

Let . Then, system (2.1) can be rewritten as

(2.4)

Notice that assumption (H1) implies that . Therefore, we have, for any and ,

(2.5)

Furthermore, let, for , be

(2.6)

Then, for any , we have

(2.7)

In order to formulate the transformation, we still need the following quantities:

(2.8)

Because the entries of matrix are nonnegative, we further define

(2.9)

Simple computation yields, for ,

(2.10)

We then have the following lemma which is of key importance for our results.

Lemma 2.1 (See [17, 18]).

Let

(2.11)

Then, .

Proof. A proof of this lemma is presented in the appendix.

Now, we consider the following uncertain linear system with ,

(2.12)

It follows from Lemma 2.1 that the parameter uncertainties in (2.4) and (2.12) are identical, which implies that (2.12) is equivalent to system (2.4). It is also observed that the robust stability property of (2.4) implies the stability property of system (2.1). Therefore, in order to derive the sufficient conditions for stability of system (2.1), we consider the robust stability of system (2.12).

Remark 2.2.

If the nonlinear function satisfies the assumption

(H2) for and , .

Then, and . This implies that , where denotes a matrix with as the th line and th column entry.

For briefness, we mainly focus on system (2.1) with two subsystems in the sequel, that is, . In this case, we rewrite, respectively, systems (2.1) and (2.12) as

(2.13)

(2.14)

where is called the switching rate. The results for (2.13) and/or (2.14) are easily extended to the general systems (2.1) and/or (2.12) with subsystems.

The following two lemmas are useful in the sequel.

Lemma 2.3 (Sanchez and Perez [19]).

Given any real matrices of appropriate dimensions and a scalar such that . Then, the following inequality holds:

(2.15)

where the superscript means the transpose of a matrix.

Lemma 2.4 (Schur complement, Boyd et al. [20]).

The following LMI:

(2.16)

where , , and depend affinely on , is equivalent to

(2.17)

Throughout this paper, we denote by the transpose of matrix ; and the minimal and maximal eigenvalues of a real symmetric matrix , respectively; the symmetrical and positive (semipositive, negative, seminegative) definite matrix , and the Euclidian norm of the square matrix.

3. Stability Analysis for the First Case

In this section, we consider system (2.13) composed of a pair of unstable subsystems with neural-type nonlinearities. Since the robust stability property of system (2.14) implies the stability property of the original system (2.13), we will derive a sufficient condition of globally exponential stability. The theoretical result shows that similar switching criterion guaranteeing the exponential stability of the origin of time-switched LTI systems still holds for time-switched nonlinear systems with neural-type nonlinearities.

The main result in this section is as follows.

Theorem 3.1.

Suppose that there exist symmetric and positive definite matrix , positive constants , and such that

(3.1)

where are defined, respectively, in (2.8). Then, there exists (small) switching period T such that the origin of time-switched system (2.14) is globally robustly exponentially stable, and therefore system (2.13) is globally exponentially stable.

Proof.

We only need to show that inequality (3.1) implies the robustly exponential stability of system (2.14) with small switching period. Without loss of generality, we assume that the initial value is with the starting time . Then, by piecewise integration, we have

(3.2)

Along this idea and omitting the terms when tends to zero, one observes that if we let tend to zero, on any fixed time interval the solution of (2.14) will tend to the solution (with the same initial condition) of the following "averaged" system (see [21] for more details), for any ,

(3.3)

In particular, the stability properties of the switched system (2.14) will for sufficiently small T be determined by the robust stability properties of the averaged system (3.3). Note that system (3.3) is globally exponentially stable if and only if there exists such that is Hurwitz for any , which is equivalent to the fact that there exists symmetric and positive definite matrix such that . A sufficient condition for this inequality is just , that is, inequality (3.1). This is because, for ,

(3.4)

This concludes the proof.

Remark 3.2.

This result is seen as the natural extension of that of time-switched linear systems [16].

4. Stability Analysis for the Second Case

In this section, we consider system (2.13) composed of both stable and unstable subsystems with neural-type nonlinearities. By means of, respectively, multiple and single Lyapunov function, we propose two general criteria, together with simple but effective sufficient conditions guaranteeing the globally exponential stability.

Reconsider the time-switched system (2.13). Here we assume that the first subsystem is globally exponentially stable at the origin, while the second one unstable. It is natural that the system as a whole will be globally exponentially stable for any given when the switching period T approaches to infinite and . Our interest here is to determine a region of the binary composed of the switching period and switching rate such that the system as a whole is globally exponentially stable therein.

Theorem 4.1.

Suppose that there exist two scalar functions , a continuous and monotonously increasing function with and constants , and such that the following conditions hold:

(i);

(ii)for any , when , , and when ;

(iii), for any and ;

(iv).

Then, the origin of the time-switched system (2.13) is globally exponentially stable.

Proof.

When , it follows from condition (ii) that

(4.1)

Similarly, when , the differential inequality implies

(4.2)

From (4.1)-(4.2) and condition (iii), we have the following

(a)When , .

(b)When ,

(4.3)

(c)When ,

(4.4)

(d)When ,

(4.5)

By induction, we have the following.

(e) When which implies ,

(4.6)

(f)When which implies ,

(4.7)

Therefore, we can conclude the proof from (e)-(f) and conditions (i) and (iv).

Based on Theorem 4.1, if we choose the quadratic Lyapunov function for the th subsystem, the following result is immediate.

Corollary 4.2.

Suppose that there exist symmetric and positive definite matrices and , the positive constants such that

(i),

(ii),

(iii),

where . Then, by the switching period T with switching rate , the origin of time-switched system (2.13) with both stable and unstable subsystems is globally exponentially stable. Moreover, the norm of state vector satisfies the following inequality:

(4.8)

Proof.

Consider the Lyapunov function . When , from condition (i) and Lemma 2.3 the derivative of along the trajectories of the first subsystem is calculated and estimated as follows:

(4.9)

Similarly, based on condition (ii) and Lemma 2.3, for , we have

(4.10)

Therefore, conditions (i)-(ii) in Theorem 4.1 hold. Obviously, the definition of implies which leads to condition (iii) in Theorem 4.1. Thus, we complete the proof.

Remark 4.3.

Condition (iv) in Theorem 4.1 and condition (iii) in Corollary 4.2 hold for large enough switching period T if . This is completely consistent with the extreme case that only the stable system is activated, that is, .

Remark 4.4.

Condition (iii) in this theorem can help to derive an estimated region of period T and switching rate , where each binary can guarantee the exponential stability of system (2.13). For computational consideration, we suggest the following steps.

(a)Find the maximum and the corresponding from condition (i) which is equivalent to the following linear matrix inequality with respect to and (this follows from Lemma 2.4):

(4.11)

(b)Therefore, this step is changed into solving the optimization problem

(4.12)

(c)Find the minimum and the corresponding from condition (ii) by solving the optimization problem

(4.13)

(d)Estimate the region of

(4.14)

Then, for any , system (2.13) is globally exponentially stable.

Similarly, following the idea of the proof of Theorem 4.1, we have the simpler result when a common Lyapunov function is chosen.

Theorem 4.5.

Suppose that there exist a Lyapunov function , a continuous and monotonously increasing function with , and constants , , and such that the following conditions hold:

(i);

(ii)for any , when , , and when , ;

(iii).

Then, the origin of the time-switched system (2.13) is globally exponentially stable for any switching period .

Proof.

The proof is similar with that of Theorem 4.1, and omitted here.

From this theorem, a standard Lyapunov function will yield the following corollary.

Corollary 4.6.

Suppose that there exist symmetric and a positive definite matrix , four positive constants such that

(i),

(ii),

(iii).

Then, for any switching period , the origin of time-switched system (2.13) is globally exponentially stable. Moreover, the norm of state vector satisfies the following inequality:

(4.15)

We now study this problem by using the linearization system (2.14) and Theorem 4.5. A simpler sufficient condition for exponential stability of system (2.13) with both stable and unstable subsystems is established.

Corollary 4.7.

Suppose that there exist symmetric and positive definite matrix , positive constants such that

(i)

(ii),

then, for arbitrary , if , the origin of time-switched system (2.13) with both stable and unstable subsystems is globally exponentially stable.

Proof.

Consider the common Lyapunov function for both subsystems. The rest of the proof is similar to that of Theorem 4.1, and hence omitted.

Remark 4.8.

In Corollaries 4.6 and 4.7, condition (i) is to guarantee the exponential stability of the first subsystem, while the instability of the second subsystem follows condition (ii).

Remark 4.9.

For computational consideration, we suggest the following algorithm.

(a)Find the maximum and the corresponding P from condition (i) in Corollary 4.6 (or Corollary 4.7) by using the convex optimization algorithm.

(b)Find the minimum from condition (ii) in Corollary 4.6 (or Corollary 4.7) by using the convex optimization algorithm. Note that the matrix is solved in (a).

(c)Calculate the low bound of , . Then, for any , if , system (2.13) is globally exponentially stable.

5. Stability Analysis for the Third Case

In this section, we consider the time-switched system (2.13) with a pair of stable subsystems. The contribution is twofold. Firstly, for the linear case, we characterize the stability properties in four aspects. Secondly, we extend the results for the linear case to the nonlinear system with neural-type nonlinearities.

5.1. Linear Case

Consider the following time-switched system with a pair of stable linear subsystems:

(5.1)

where denotes the state vector, , are Hurwitz.

We have the following results.

Theorem 5.1.

System (5.1) with a pair of stable subsystems is globally exponentially stable for any switching rule if there exists a symmetric and positive definite matrix P such that both the following inequalities hold:

(i),

(ii).

Proof.

Consider the common Lyapunov function . Integrating by part V(x) with respect to time t along the trajectories of the system (5.1), we have, for any ,

(5.2)

where , and is positive constant and satisfies . Note that for any , and therefore, (5.2) concludes the proof.

When the condition in Theorem 5.1 does not hold, we suggest the following claim.

Theorem 5.2.

Let

(5.3)

(5.4)

Then, system (5.1) is globally exponentially stable if the switching period T satisfies

(5.5)

where .

Proof.

Consider the multiple Lyapunov function

(5.6)

Note that . We calculate the derivatives of with respect to time t along the trajectories of the system (5.1) as follows:

(5.7)

Therefore, we have

(a)for any ,

(5.8)

(b)for any ,

(5.9)

Hence, if , that is, , system (5.1) is globally exponentially stable. The proof is thus completed.

5.2. Nonlinear Case

Consider again the nonlinear time-switched system (2.13) and the "linearization" system (2.14), but in this section we assume that both subsystems in (2.14) are robust Hurwitz-stable.

Arguing similarly with the previous subsection, we have the analogs of the results described by Theorems 5.1 and 5.2.

Theorem 5.3.

Assume that, for ,

(5.10)

exist. Then, the origin of system (2.13) is globally exponentially stable if any one of the following conditions holds.

(i)There exists a symmetric and positive definite matrix P such that

(5.11)

is satisfied for any .

(ii)The switching period T satisfies

(5.12)

where .

Notice that, for any ,

(5.13)

Then, the following corollary is immediate.

Corollary 5.4.

System (2.13) is globally exponentially stable if any one of the following conditions holds.

(i)There exist a symmetric and positive definite matrix P and positive constants such that

(5.14)

(ii)If, for , there exist symmetric and positive definite matrices and positive constants satisfying

(5.15)

and further the switching period and switching rate satisfy , where .

6. Intermittent Control with Time Duration

Intermittent control is a straightforward engineering approach to process control of any type. As a special form of switching control, intermittent control is also divided into two classes: state-dependent switching rule and time-switching rule. The former implies that the control operation is activated only when the states enter the certain region which is often pregiven; while the later activates the control only in some finite time intervals; the system evolves freely when the time goes out of those intervals. Therefore, these intermittent control systems are open-loop. Intermittent control has been used for a variety of purposes in such engineering fields as manufacturing, transportation, air-quality control, and communication, and so on. In the chaos control context, Carr and Schwartz [22–24] demonstrated a method to control unstable steady states in high-dimensional flows using the duration of time for which feedback control was applied as an addition parameter. Starrett [25] proposed the so-called "occasional bang-bang" method to stabilize a periodic saddle point in a strange attractor. The main idea of occasional bang-bang is to apply control only at regular intervals and for fixed durations. An application of intermittent control in air-quality control was reported in [26] where the authors reported a kind of intermittent control system which was designed to achieve ambient air quality standards constantly by varying emission rates in response to changing atmospheric dispersion conditions. Recently, the authors in [27] proposed a scheme to decouple neighboring qubits in quantum computers through bang-bang pulse control and demonstrated that two similar sequence of pulses with different time intervals not only suppress decoherence but entirely or selectively decouple two neighboring qubits. The authors in [27–31] suggested to controlling the evolution of a system by using strong, short pulses as a new means for quantum error prevention. Also, the authors in [32] discussed chaotic synchronization by using intermittent control.

An extreme case of intermittent control is impulsive control which has been gained increasing interest and intensively researched [33, 34]. The prominent characteristic of impulsive control is that the states of controlled system will "jump" at certain discrete time moments, namely, the control is with zero duration of time. Because the states of controlled systems are changed directly, impulsive control is an effective approach when the states are observable, but it seems to be invalid when the states of controlled systems are unobservable.

Our interest focuses on the class of intermittent control with time duration, namely, the control is activated in certain nonzero time intervals, and off in other time intervals. Specifically, the control law is of the form

(6.1)

Consider the nonlinear system described by

(6.2)

where denotes the state vector, and are constant matrices, and the nonlinear function, , is continuous with , and satisfies the Lipstchitz condition with the Liptchiz constant , namely, . presents the output of the system with the coefficient matrix , and is the external input with the form of (6.1). For analytical simplification, we assume in this paper the input is periodical switching with the fixed duration of time. Specifically, we take , in the sequel, as the form

(6.3)

with

(6.4)

Our object is to find the appropriate , and such that system (6.2) is stabilized exponentially at the origin under the periodically intermittent control (6.3) with (6.4).

Note that the controlled system (6.2) with (6.3)-(6.4) can be rewritten as

(6.5)

According to (2.8), it is not difficult to obtain , and , where and . Because , both the equalities and hold. For notational simplification, let us define , and . From Corollary 4.7, we have the following result.

Theorem 6.1.

There exist a matrix , and a symmetric, positive definite matrix P, and positive constants , and such that

(i),

(ii),

(iii)

Then, the origin of system (6.5) is exponentially stable for any , and the corresponding control gain matrix is determined by .

Proof.

We only need to show that the conditions in Corollary 4.7 are satisfied. The proof is trivial, and therefore omitted here.

To end this section, we consider a special external input of the form where

(6.6)

in which is a constant scalar. Then, from Theorem 6.1, the following corollary is immediate.

Corollary 6.2.

Let , and . The origin of controlled system (6.5) is exponentially stable for any if .

7. Illustrating Examples

In this section, we will give two examples to show the validity of the proposed results.

Example 7.1.

Consider system (2.13) with

(7.1)

Both subsystems are unstable, as shown in Figure 1(a). Note that

(7.2)

Taking (identity matrix) and , we have, when , with the eigenvalues and . Hence, it follows from Theorem 3.1 that when the system in this example is globally exponentially stable for some small enough , as shown in Figure 1(b).

Figure 1

(a) Time response curves of norm of solution vectors of two subsystems in Example 7. 1. (b) Phase diagram of the switched system in Example 7.1. The switching period , switching rate , the initial value .

Example 7.2.

Consider system (2.13) with

(7.3)

Based on Corollary 4.2, solving the linear matrix inequalities by LMI ToolBox involved in the engineering software MATLAB, we obtain

(7.4)

Therefore, and an estimated region of is

(7.5)

which covers the whole region above the curve with and .

We also obtain the switching law by using Corollary 4.6. Solving condition (i) in Corollary 4.6, we get the maximum and the corresponding P; then solving the condition (ii) yields ; finally, we calculate the feasible interval of switching rate, . It follows from Corollary 4.6 that the system described in Example 7.2 is globally exponentially stable for any switching period , if the switching rate satisfies . For the simulation, we choose and . The convergence behavior of this system is shown in Figure 2.

Figure 2

Time response curves of states of the system in Example 7. 2.

8. Conclusions

Globally exponential stability of a class of periodically time-switched systems with two nonlinear subsystems has been investigated in this paper. Based on the stability property of subsystems, we divided the considered systems into three subclasses, namely, system composed of a pair of unstable subsystems; system composed of both stable and unstable subsystems; system composed of a pair of stable subsystems. A least one sufficient condition guaranteeing global exponential stability of each of three subclasses was derived by different method including the "averaged" system approach, multiple and single Lyapunov function, and robust analysis of linear time-variant systems. The periodically intermittent control design problem was also addressed. This paper focuses on only the neural-type Lipstchitz nonlinearity, therefore, the further work may deal with the general nonlinear subsystems. Also, the effect of short and strong control strength on the reduction of the control cost in the presence of output noise seems to be an interesting topic.

Appendix A

A Proof of Lemma 2.1

For any , it is easy to see that there exist real constants , such that

(A1)

where satisfies

(A2)

Since , it can be decomposed as

(A3)

where is the th column-vector of identity matrix.

From (A.1) and (A.3), we have

(A4)

From (4.1) and (4.2), we can see that there exists a such that , that is, . Hence, . Noting that the process above is inverse, we can also derive the relation . Therefore, . Similarly, one can show . The proof is thus completed.