Applied and Computational Mathematics
Volume 5, Issue 3, June 2016, Pages: 165-168

The Application of Matrix in Control Theory

Yuanyuan Zhang

College of Science, China Three Gorges University, Yichang, China

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To cite this article:

Yuanyuan Zhang. The Application of Matrix in Control Theory. Applied and Computational Mathematics. Vol. 5, No. 3, 2016, pp. 165-168. doi: 10.11648/j.acm.20160503.21

Received: June 21, 2016; Accepted: July 21, 2016; Published: July 25, 2016

Abstract: This paper presents a new way to justify the controllability of linear ordinary systems. This way is based on the maximum geometric multiplicity of eigenvalues for the coefficient matrix of the linear ordinary equation. This method is equivalent to other discrimination laws for controllability.

Keywords: Controllability, Ordinary Differential Equation, Geometric Multiplicity, Eigenvalues

Contents

1. Introduction

The controllability problems of ordinary differential equations have been a great interest for decades due to their practical applications [1-5]. There are a lot of literatures on this issue ( see for instance [6-15]). The current paper presents a new way to justify the controllability of linear ordinary systems. This way is based on the maximum geometric multiplicity of eigenvalues for the coefficient matrix of the linear ordinary equation.

The rest of the paper is organized as follows: Section 2 presents a necessary and sufficient condition for justifying the controllability of the linear ordinary differential system. Section 3 gives several examples to show the efficiency of the new method. Section 4 is the conclusion of this paper.

2. Main Results

Let  and  be two matrix in  and  respectively, where . Consider the following controlled ordinary differential equation:

(1)

Here  and  is a control function in the following admissible set:

(2)

where  is a positive constant. Throughout this paper, we use  to denote the linear differential system given by equation (1). The main result of the paper is as follows:

Theorem The linear differential systemis controllable if and only if the number of columns of the matrix is no less than the maximum geometric multiplicity of eigenvalues for the matrix.

Proof. Suppose the system  is controllable, and  are the eigenvalues of the matrix with the geometric multiplicity and algebraic multiplicity. Then there exists an invertible matrix such that

(3)

where

(4)

In equation (4)  represents the Jordan’s block corresponding to, .

Suppose the order of the Jordan block  is , and

(5)

Let  We use  to represent the system  is controllable and  to represent the systemis controllable, then the problems  and  are equivalent.

Suppose  is controllable, then for any eigenvalues

, take  for example, we have

(6)

where

(7)

It is obvious that the number of nonzero rows for equation (6) is , where . Then from equation (6) we can see that the number of linearly independent rows of the matrix  must be no less than . So the number of columns for equation  must be no less than . Because  is any eigenvalue for the matrix , so the necessity of the theorem has been proved.

On the other hand, if the number of columns for equation  is no less than , , then for any eigenvalues , we have

(8)

So

(9)

From equation (9) we can obtain that all the rows of the matrix  are linearly independent, which completes the proof of the sufficiency of the theorem.

3. Applications

In this section we provide several examples to show the efficiency of the above-mentioned method.

Example 1. Let

(10)

We aim to proof that system (10) is controllable.

Proof. Let

(11)

then the system (10) can be simplified as

(12)

It is easy to obtain

(13)

(14)

So the Jordan’s normal form of matrix  is as follows

(15)

From equation (15), it is obvious that the maximum geometric multiplicity of the eigenvalues of the matrix  is 2, and the number of the non-zero columns of the matrix  is 3. According to the given method in section 2, system (10) is controllable.

Example 2. Let

(16)

Our aim is to proof that system (16) is controllable.

Proof. For

(17)

then the minimum polynomials of the matrix  is

(18)

Let

(19)

then

(20)

So

(21)

From equation (21), it is easy to obtain

(22)

So the elementary divisors of the matrix  is

(23)

Then the Jordan’s normal form of the matrix  is as follows

(24)

From equation (24), it is easy to obtain the maximum geometric multiplicity of the eigenvalues of the matrix  is 1, and the number of the non-zero columns of the matrix  is 2. So according to the given method in section 2, system (16) is controllable.

Example 3. Let

(25)

Our aim is to proof that system (25) is non-controllable.

Proof. For

(26)

and the minimum polynomials of the matrix  is

(27)

then the invariant factors of the matrix  is

(28)

So the Jordan’s normal form of matrix  is as follows

(29)

From equation (29), it is obvious that the maximum geometric multiplicity of the eigenvalue of the matrix  is 2, and the number of the non-zero columns of the matrix  is 1. According to the given method in section 2, system (25) is non-controllable.

Example 4. Let

(30)

Proof. For system (30) there are different cases:

Case 1. When , the maximum geometric multiplicity of the eigenvalue of the matrix  is 3, and the number of the non-zero columns of the matrix  is 1. According to the given method in section 2, system (30) is non-controllable.

Case 2. When , the maximum geometric multiplicity of the eigenvalue of the matrix  is 2 and the number of the non-zero columns of the matrix  is 1. According to the given method in section 2, system (30) is non-controllable.

Case 3. When, the maximum geometric multiplicity of the eigenvalue of the matrix  is 1 and the number of the non-zero columns of the matrix is 1. According to the given method in section 2, system (30) is controllable.

In (30), if the number of the non-zero columns of the matrix  is 2, for example,

(31)

then in case 1 the system  is non-controllable; in case 2 and case 3 the system  is controllable.

Still in (30), if the number of the non-zero columns of  the matrix  is 3, for example,

(32)

then the number of non-zero columns of the matrix  is 3. According to the given method in section 2, whether  are equal or not, the system  is controllable.

In fact, as long as the number of non-zero columns of the matrix  is no less than 3, the system  is always controllable.

4. Conclusion

This paper presents a new necessary and sufficient condition for justifying the controllability of the linear ordinary differential system. This way is based on the maximum geometric multiplicity of eigenvalues for the coefficient matrix of the linear ordinary equation. This method is equivalent to other discrimination laws for controllability. Section 2 presents a necessary and sufficient condition for justifying the controllability of the linear ordinary differential system. Section 3 gives several examples to show the efficiency of the new method. Section 4 is the conclusion of this paper. This method can also be used in other linear ordinary differential systems.

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