S OME ANALYSIS PROBLEMS OF THE LINEAR SYSTEMS

 Abstract: New approaches to the transformations of the uncontrollable and unobservable matrices of linear systems to their canonical forms are proposed. It is shown that the uncontrollable pair ( A , B ) and unobservable pair ( A , C ) of linear systems can be transform to their controllable (𝐴̅,𝐵̅) and observable (𝐴̅,𝐶̅) canonical forms by suitable choice of nonsingular matrix M satisfying the condition 𝑀[𝐴 𝐵] = [𝐴̅ 𝐵̅] and [𝐴 𝐵]𝑀 = [𝐴̂ 𝐵̂] , respectively. It


INTRODUCTION
The concepts of the controllability and observability introduced by Kalman [8,9] have been the basic notions of the modern control theory. It well-known that if the linear system is controllable then by the use of state feedbacks it is possible to modify the dynamical properties of the closedloop systems [1,2,[5][6][7][8][9][10][11][12][13][14]. If the linear system is observable then it is possible to design an observer which reconstruct the state vector of the system [1,2,[5][6][7][8][9][10][11][12][13][14]. Descriptor systems of integer and fractional order has been analyzed in [6,13]. The stabilization of positive descriptor fractional linear systems with two different fractional order by decentralized controller have been investigated in [13]. The eigenvalues assignment in uncontrollable linear continuoustime systems has been analyzed in [4].
In this paper new approaches to the transformations of the uncontrollable and unobservable linear systems will be proposed. In Section 2 some basic theorems concerning matrix equations with non-square matrices and their solutions are given. Transformations of the uncontrollable pairs to their canonical forms are presented in Section 3 and of the unobservable pairs in Section 4. Transformation of the controllable pairs in one canonical forms to other one is analyzed in Section 5. Elimination of the singularity in descriptor linear systems is considered in Section 6. Reduction of the descriptor linear systems to their standard forms by the use of the feedbacks is analyzed in Section 7. Concluding remarks are given in Section 8.
The following notation will be used:  -the set of real numbers, m n  -the set of m n real matrices, n I -the n n identity matrix.

THEIR SOLUTIONS
Consider the matrix equation Proof follows immediately from the Kronecker-Cappelly Theorem [3]. Theorem 2. If the condition (2) is satisfied then the solution X of the equation (1) is given by where n m r P    is the right inverse of the matrix P given by Proof is similar (dual) to the proof of Theorem 1. Theorem 4. If the condition (8) is satisfied then the solution of the equation (7) is given by where the left inverse of the matrix P is given by Proof is similar (dual) to the proof of Theorem 2.

THEIR CANONICAL FORMS
Consider the continuous-time linear system [ rank (13) to the canonical forms (12) if and only if Proof. From Theorem1 it follows that there exists a nonsingular matrix if and only if the condition (14) is satisfied. □ If the condition (14) is satisfied then for the given matrices A,B and A , B the matrix M can be computed by the use of the following procedure. Procedure 1.
Check the condition (14).The problem has a solution if and only if the condition (14) is satisfied.
Using the equality

B MB
 find the corresponding column of the matrix M.
Using the equality A MA  find the remaining columns of the matrix M.
The theorem will be illustrated by the following simple example. (16b) Using Procedure 1 we obtain the following.
Step 1. The condition (14) is satisfied for the matrix Step 2. From the equality Step 3. Taking into account (17) and we obtain: Therefore, the desired nonsingular matrix M has the form The approach based on the equation can be also used to transform the controllable pair (A,B) to the desired standard controllable form The procedure will be shown on the following simple example.
Example 2. For the controllable pair (20) such that the pair

III. TRANSFORMATIONS OF THE UNOBSERVABLE PAIRS TO THEIR CANONICAL FORMS
To simplify the notation we assume p = 1(single output systems).
Proof is similar (dual) to the proof of Theorem 5. If the condition (28) is satisfied then for the given matrices A, C and ̂, ̂ the matrix ̂ can be computed by the use of the following procedure. Procedure 2.

Check the condition (28). The problem has a solution if and only if the condition (28) is satisfied.
Step 2.
Using the equality C M C find the corresponding column of the matrix M .
Using the equality Using Procedure 2 we obtain the following.
Example 3. Consider the controllable pairs in their canonical forms In this case the condition (36) takes the form 3 0 3 1 Therefore, does not exists the nonsingular matrix Proof is similar(dual) to the proof of Theorem 8.

Consider the continuous-time linear system
Bu Ax Therefore we have the following conclusion. By suitable choice of the matrix M it is not possible to transform the descriptor system (45) to the standard one of the form  Bv Ax which satisfies the condition (56) since

VII. CONCLUSIONS
Two approaches to the transformations of the uncontrollable and unobservable linear systems to their canonical forms has been proposed (Theorems 5 and 6) and procedures for calculation of transformation matrices have been given (Procedures 1 and 2).The procedures have been illustrated by simple numerical examples. It has been shown that the pair (12a) Carnot been transformed to the pair (12b) by the nonsingular matrix M satisfying (35) (Theorem 7). Necessary and sufficient conditions have been established for the reduction of the descriptor linear systems to their standard forms(Theorem 8). The considerations can be extended to the discrete-time linear systems and to the fractional orders linear systems. An open problem is an extension of these approaches to the different orders linear systems.