On the Delay Interval in Which the Control Delay Systems are Stabilizable

Abstract

In this paper, we firstly derive some criteria for considered control delay systems to be stabilizable and then get the interval of the control delay, in which the stabilizability of the systems can be kept.

1. Introduction

Recently, the phenomenon of time delay has aroused many people’s attention [1–22]. In some practical systems, such as economic systems, biological systems, and space-light industry systems, due to the transmission of the signal or the mechanical transmission, the control variables must have time delay. From reference [1, 2], we can see that time delay usually destroys the stability. So, in the study of stabilizability of the control system, we must consider control delay seriously.

References [1–4] give the basic theory of functional differential equations that give us a lot of useful method and tools to discuss time delay systems. In [7], Jean-Pierre Richard gives an overview of some recent advances and open problems on time delay systems.

As we know, we can use the controllability to determine the stabilizability. Some scholars [2–4] and [8–11, 15, 22] have given many results on the controllability of the systems with delay.

System stability theory is the foundation of stabilizability of the control system. In papers [5, 6, 15–20, 22], the authors have given some excellent results for the stability of delay systems.

In this paper, we will point out that we can use the controllability of the systems with control delay to determine the stabilizability of such systems and give the method to do so. Therefore, we can use the results of references [2, 4] for the study of stabilizability of systems with control delay.

In reference [5, 6, 15], John Chiasson, Jie Chen, and Jiang Wei give some results for computing the interval of delay values for stability of delay systems. These results will be useful in the study of the range of the control delay in which stabilizability of the systems can be kept.

The systems we consider in this paper will be the control systems with control delay:where is a state vector; is a control vector; an admissible control, that is, it is contained in the square integrable functions on every finite interval; A and B are constant matrices; is time delay; and is the initial control function.

In this paper, we will consider the stabilizability for the control systems with control delay (1). Firstly, we derive some criterions to criticize the stabilization for the control delay systems and then get the interval of the control delay, in which the stabilizability of the systems can be kept.

2. Preliminaries

For the control systems with control delay (1), we give some preliminaries involved in this paper.

Definition 1. System (1) is said to be controllable at if one can reach any state at from any admissible initial state and initial control .

Lemma 1. System (1) is controllable if and only ifThe proof of this Lemma can be easily gotten from Theorem 1 in [4], but we must assume and there.
If there exist a matrix , such thatsystem (1) will become

Definition 2. The differential systems with delay (4) are called asymptotic stable, if there exist scalars such that, for , satisfy

Definition 3. The control systems with control delay (1) are said to be stabilizable, if there is a matrix K such that system (4) is asymptotically stable. The matrix K is said to be a feedback stabilization matrix.
For an entire function , we define setwhere C is the complex plane.
For system (4), we define the entire function:From [1], we have the following.

Lemma 2. System (4) is asymptotically stable if and only ifwhere is the left complex plane.

If system (1) is controllable, that is, (2) is true, matrices A and B are equivalent to their controllable normalized forms. In the following, if (1) are controllable, we might as well let be normalized forms:where are the normalized forms as

Letwhere

Then,

From (13) and Lemma 2, we have the following.

Lemma 3. If the control systems with control delay (1) are controllable, they are stabilizable if and only iffor every is true.

Let andwhere are the polynomials of d.

Definition 4. Let be a two-variable polynomial of form (15) and , and we call the two-variable polynomial,the auxiliary polynomial of .
Suppose be the common zeros of for which and . For each such pair , let . Since and , such an exists. DefineFrom [5], we have the following.

Lemma 4. if and only ifFurthermore, there exists an for which .

3. Main Results

In this section, we will give the main results of this paper. Firstly, we give a general theorem, then for , discuss in two cases, and give some results.

Theorem 1. For system (1), if they are controllable, there exists a real number such that they are stabilizable with delay .

Proof. From Lemma 1, system (1) is controllable if and only if (2) is true. It is obvious that if (2) is true, there exists a matrix K such thatLet and take as that in (17); then, from Lemma 4, we have that Theorem 1 is true.
From Lemma 3, to discuss the stabilizability of system (1) in detail, if they are controllable, we might as well letTake , where are undetermined real numbers, and we haveIf there exist some eigenvalues of A with the negative real part, let be all the eigenvalues of A with , then there exist real numbers such thatHere, and there exists only nonnegative real part roots.
Obviously, for any leading coefficient which is 1 and order polynomial , we can setthat is,Letthere are arbitrary constants. Then, we haveLet , thenFrom (28), we can determine real numbers .
Take (28) into (22), we haveLetWe can see that, if we wantit is enough to makeSo, in the following discussion, we might as well assume that A have only nonnegative real part eigenvalues, that is,there exists only nonnegative real part roots.
Now we discuss the stabilizability of system (1) with . We discuss that in two cases.

Case I. and and there exists only positive real part zeros.
If the quadratic polynomial equationthere exists only positive real part roots, and we can rewrite as

Theorem 2. For system (1), when and if they are controllable, then they are stabilizable with delay .

Proof. For we have thatand there exists only negative real part roots.
Letthen .From (38), we haveTaking it into (39), we haveThe solutions of (41) areHere, . Taking into (40), we haveForThen, we haveUsing the same method, we haveSo,Then, we have that Theorem 2 is true.

Case II. and has only pure imaginary zeros.
If the quadratic polynomial equationthere exists only pure imaginary roots, and we can rewrite as

Theorem 3. For system (1), when and if they are controllable, then they are stabilizable with delay .

Proof. Letthen .From (51), we haveTaking it into (52), we haveThe solutions of (54) areTaking them into (53), we haveForThen, we haveUsing the same method, we haveSo,Then, we have Theorem 3 is true.
Now, we discuss the general case: for any positive integer n, the polynomialand there exists only positive real part roots.
It is obvious that the polynomial equationand there exists only negative real part roots.
Letthen .
When n is an even number, let , and we haveFrom (64), we haveTaking it into (65), we obtainWe can prove that all the roots of (67) are pure imaginary roots, let them be ; then (67) will becomeForWe obtainthenSo,From (66), we havewhere .
In the same way, we can also prove that when n is an odd number, we still have .
From and , we have or .
Let . It is obvious that . Then, we haveFrom the discussion above, we obtain the following.

Theorem 4. For system (1), if they are controllable andthere exists only positive real part roots, and they are stabilizable with control delay .

Now, we give an example to illustrate the use of results.

Example 1. Consider systemswhere .
The solutions of (76) areThey are obviously unstable.
Adding a control with the delay term in (76), we have control systemswhere , , and are the initial control function.
Comparing (78) with (1), we haveForfrom Lemma 1, we have that system (78) is controllable.
Foruse Theorem 2, for , we have that system (78) is stabilizable with delay
We get an interval of the control delay, in which the stabilizability of system (78) can be kept.
We must take notice that if the control delay is outside of interval , system (78) is perhaps unstable. In fact, if we take the feedback matrix as , the control delay and system (78) is unstable. For , is outside of interval .
For satisfies equationSystem (78) has solutionsThey are obviously unstable.

4. Concluding Remarks

In practice, we often meet many unstable systems such aswhere is a state vector. We usually add control terms in them. But unfortunately, due to the transmission of the signal or the mechanical transmission takes a period of time, time delay may appear in control variables, and the control systems will become as system (1). The results of this paper give the methods to find the interval of the control delay, in which the stabilizability of the systems can be kept. From Example 1, we can see that, on the outside of this interval, the systems may be unstable. So, the method given in this paper has practical significance.

Now, we give some Remarks to finish this paper.

Remark 1. By Lemmas 1–4 and Theorem 1, system (1) is controllable andThere exists some negative real part roots, and we can also get analogous results and find the interval of control delay with which the systems are stabilizable.

Remark 2. Although Theorem 4 requires system (1) to be controllable, if it is uncontrollable but the uncontrollable subsystems are stable, we can also get analogous results and find the interval of control delay with which the systems are stabilizable.

Remark 3. In Theorem 4, interval of control delay h is obtained, with which the systems are stabilizable. But whether includes all delay values for which system (1) is stabilizable can be left as an open problem.

Remark 4. In papers [5, 6, 15–20], the authors have given some excellent results for the stability of delay systems. But all systems, they study, are not control systems, we cannot use their results directly to judge the stabilizability for the control systems with control delay. In comparison with these, we can see that the results of this paper are very useful.

Remark 5. We note that, in recent years, many scholars researched delayed systems with fast time-varying delay [20, 21] and gave many wonderful consequences. Usually the methods they used to deal with stability are linear matrix inequalities (LMIs). In this paper, we use the characteristic root methods to deal with stability (or the stabilizability) for the systems with time-invariant delay. If we want to use the characteristic root methods to deal with the stability (or the stabilizability) for the delayed systems with fast time-varying delay, or to use the results of this paper to the delayed systems with fast time-varying delay, we will meet more difficulties.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (nos. 11371027, 11471015, and 1601003), Anhui Provincial Natural Science Foundation (nos. 1608085MA12 and 1508085MA01), and Program of Natural Science of Colleges of Anhui Province (no. KJ2013A032).