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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 490903, 11 pages
Controllability Analysis of Linear Discrete Time Systems with Time Delay in State
1Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing 102617, China
2State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
3School of Electrical and Electronics Engineering, East China Jiaotong University, Nanchang 330013, China
4School of Electric and Information Engineering, Guangxi University of Science and Technology, Liuzhou 545006, China
5Guangxi Key Laboratory of Automobile Components and Vehicle Technology, Liuzhou 545006, China
Received 27 August 2012; Accepted 10 October 2012
Academic Editor: Valery Y. Glizer
Copyright © 2012 Hong Shi et al. 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.
The controllability issues for linear discrete-time systems with delay in state are addressed. By introducing a new concept, the minimum controllability realization index (MinCRI), the characteristic of controllability is revealed. It is proved that the MinCRI of a system with state delay exists and is finite. Based on this result, a necessary and sufficient condition for the controllability of discrete-time linear systems with state delay is established.
The concept of controllability of dynamical systems was first proposed by Kalman in 1960s . Since then, controllability of dynamical systems has been studied by many authors in various contexts [2–11], because controllability turns out to be a fundamental concept in modern control theory and has dose connections with pole assignment, structure decomposition, quadratic control, and so forth [12, 13]. On the other hand, time delay phenomena are very common in practical systems, for instance, in economic, biological, and physiological systems. Studying the time delay phenomena in control systems has become an important topic in control theory. Chyung studied the controllability for linear time-invariant systems with constant time delay in the control functions. Simple algebraic-type necessary and sufficient criteria are established [3, 4]. However, once the time delay appears in state, the problem becomes much more complex. There are some preliminary results in [5, 6]. However, all these results are not suitable for verification and application.
In this paper, we consider the discrete-time case, the system model described as follows: where is the state, is the input, , , are constant matrices, and positive integer is the length of the steps of time delay. The initial states are given arbitrarily.
The controllability discussed here refers to the unconstrained controllability, or say completely controllability.
Definition 1.1 (controllability). System (1.1) is said to be (completely) controllable, if for any initial state and any final state , there exist a positive integer and input such that .
Definition 1.2 (controllability realization index, (CRI)). For system (1.1), if there exists a positive integer such that for any initial state and any terminal state , there exist input such that ; then one calls the controllability realization index (CRI) of system (1.1). Obviously, if exists, such is not unique, so one calls the smallest among them the minimum controllability realization index (MinCRI).
In this paper, our main concern is the following. Can the controllability of system (1.1) be realized completely in finite steps, or say can the MinCRI of the system be finite?
Here we demonstrate that the answer to this question is yes. And in the case of planar systems, the exact value of the controllability realization index is obtained, and we will prove it is independent of the choices of and .
This paper is organized as follows. In Section 2 some concepts are introduced, which will be used in the later discussion. Section 3 contains the main results. Finally, the conclusion is provided in Section 4.
Denote by the nonnegative integer set, the real number set, respectively. The matrices are said to be linearly dependent in , if there exist scalars , not all zero, such that . In what follows, will be used to denote the linear subspace constructed by the linear combinations of matrices .
Definition 2.1 (symmetrical subspace). Given matrices , one defines a subset of as follows: Then a one-to-one mapping is constructed as follows:
give any , supposing , one defines Then we get a subspace of by We call the symmetrical subspace of .
By Definition 2.1, it is easy to verify that , where is the identity matrix in .
Now, we introduce a matrix sequence as follows:
Then the solution of system (1.1) can be expressed as where is the part of the solution with zero input.
Lemma 2.2. Given the matrix sequence in (2.3), the following statements hold:(a) there are no similar terms between and ;(b) for any , it can be expressed as where is a subset of defined as follows: (c) for all .
Proof. Statement (a) is nearly self-evident because any term from is ended by , whereas any term from is ended by .
To prove the result of statement (b), mathematical induction is invoked.(I)The first terms of are , and it is obvious that , for . Thus, for , (2.5) holds.(II)Assume that, for , is expressed in form (2.5). We will prove that can be expressed in form (2.5) as well.
First, we prove that, for any , we have where the sets , are defined as By the assumption, we have Thus, we have .
On the contrary, given any , suppose that where , .
There are two cases: (i) the final term of is , , , then there exists a matrix polynomial such that , then it is easy to verify that , and it follows that ; (ii) the final term of is , , , then there exists a matrix polynomial such that , then it is easy to verify that , and it follows that .
Thus, we have . Hence, we know that (2.7) holds.
Secondly, since , we have Thus, (2.5) holds for . Hence, statement (b) holds.
As for statement (c), it can be deduced naturally from statement (b). Note that if , then as well. So they must appear in . By Definition 2.1, .
3. Main Results
First, we consider the general case.
Theorem 3.1. System (1.1) is controllable if and only if .
Theorem 3.2. For system (1.1), there exists such that
Proof. It is obvious that . Consider an auxiliary scalar sequence , where . By definition, for any . Hence, there exists a constant such that, . Then for , there exists such that for all , we have . It implies that .
Remark 3.3. From Theorem 3.2, it is clear that the controllability of the system (1.1) can be realized completely within first steps during the evolving process, which means is just a CRI of system (1.1), so the existence of MinCRI of linear discrete-time delay systems is ensured and the value of MinCRI is finite. Moreover, is dependent on , , and .
Now we consider the second-order case, that is, .
Theorem 3.4. If , then is a CRI of system (1.1).
Before giving the proof of Theorem 3.4, we first give some lemmas.
Lemma 3.5. Given , for any , one has .
Proof. According to the different cases, we formulate the proof into two cases.
(a) are linearly dependent. Without loss of generality, suppose that Then we have It is obvious that .
(b) are linearly independent. If , the proof is completed. Otherwise, suppose that . We use mathematical induction.(i)For , by the assumption, .(ii)Suppose that for , . Assume For , we have It is obvious that .
Lemma 3.6. Given , one has .
Proof. First, it is obvious that
Now we prove that .
For any , assume that where ; .
It is easy to see that if we could prove that each then we would have proved that It is easy to verify that can only be rewritten in three different forms:(a), where ;(b), where ;(c), where .
By Lemma 3.5, we know that (3.8) holds. Thus, we have It follows that
Now we are in a position to prove Theorem 3.4.
Proof of Theorem 3.4. By the definition of CRI, we only need to prove that By Lemmas 2.2 and 3.6, we have Now we prove that By the definition of , we have , , , , . Note that can be linearly expressed by ; then, we have Consider the term ; it first appears in . It is easy to verify that all other terms in can be linearly expressed by . Thus, we know that (3.14) holds. This implies that (3.8) holds.
Without proof, we can get the following corollaries directly.
Corollary 3.7. Assuming , system (1.1) is controllable if and only if
Corollary 3.8. Assuming , if system (1.1) is controllable, then the controllability can be realized in steps; that is, one can select appropriate such that system (1.1) can be driven from any initial state to any terminal state.
Remark 3.9. Note that the parameter does not appear in Corollary 3.7. This implies that the specific number of the delay steps has nothing to do with the controllability of the system.
In this section, we present some numerical examples to illustrate the validity of our theoretic results.
Example 4.1. Consider the system (1.1) with , , and By simple calculation, we have . By Corollary 3.7, the system should be controllable. In fact, we have Leting , we can select suitable and such that be any state in . Thus, the system is controllable indeed.
Example 4.2. Consider the system (1.1) with , , and By simple calculation, we have . By Corollary 3.7, the system should not be controllable. In fact, it is easy to see that the second element of the state is not affected by any input . Thus, the system is not controllable indeed.
This paper discussed the controllability of linear discrete-time systems with delay in state. After introducing a new concept called MinCRI to describe the controllability feature of delay systems, we proved the existence and finiteness of MinCRI. Based on this, a necessary and sufficient condition for the controllability of linear delay systems has been derived.
The authors are grateful to the Associate Editor and the reviewers for their helpful and valuable comments and suggestions for improving this paper. This work is supported by the National Natural Science Foundation (NNSF) of China (60774089, 10972003, and 60736022). This work is supported by N2008-07 (08010702014) from Beijing Institute of Petrochemical Technology and supported by the Opening Project of Guangxi Key Laboratory of Automobile Components and Vehicle Technology(NO.2012KFZD03).
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