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
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.
References