The Order of Minimal Realization of Jordan Canonical Form Systems

Kameleh Nassiri Pirbazari and Mehdi Azari abstract: This paper presents a new method based on controllability and observability of Jordan canonical form systems useful in determining the order of minimal realization. Since any standard system is equivalent to a Jordan canonical form system, this method is applicable to any standard system.


Introduction
Consider the standard time invariant linear system: where A ∈ R n×n , B ∈ R n×m , C ∈ R p×n and D ∈ R p×m are the coefficient matrices of the system and x(t) ∈ R n , u(t) ∈ R m and y(t) ∈ R p are the state, input and output vectors respectively.The dimention of x(t) is called the order of the system.The order of system (1. is minimal if it has the smallest possible order [4,7].The realization of [A, B, C, D] is minimal if and only if it is controllable and observable [1,2,3,5,8].
Two systems [A, B, C, D] and [ Ā, B, C, D] are equivalent if they have the same order and the number of inputs and outputs is equal and there exists nonsingular matrices P and Q such that: Equivalent systems have the same minimal order, since their transfer matrices are the same [2,3].
Selecting matrices P and Q properly, each standard system [A, B, C, D] will be equivalent to a system [J, B, C, D] in which matrix J is in Jordan canonical form [6].
2. Recognizing the controllability and observability of a system using Jordan canonical form Consider the system [J, B, C, D] in which J is in the Jordan canonical form and suppose: . . .Proof: See [2] and [3].
Consider system (2.1), the linearly dependent number of uncontrollability (unobservability), for two Jordan blocks with the same eigenvalues in which the last rows (the first columns) of submatrices in B ( C) corresponding to the two Jordan blocks are linearly dependent is defined as follows: Definition 2.1.The number of linearly dependent consecutive rows with the last rows of submatrices B, corresponding to Jordan blocks having the same eigenvalues is called linearly dependent number of uncontrollability in that block.

Determining the order of minimal realization of standard systems
In this section, the system is considered in the form of [J, B, C, D] in which J is in Jordan canonical form.The following two states are considered: I) The Jordan matrix J includes Jordan blocks with distinct eigenvalues If the system [J, B, C, D] satisfies the conditions of theorems 2.1 and 2.2, the dimension of matrix J is equal to the order of minimal realization.Otherwise steps 1 and 2 are applied in order to determine the order of minimal realization: Step 1 (elimination of uncontrollable factors in system [J, B, C, D]) Considering the last rows of submatrices in B corresponding to Jordan block for each eigenvalue.If they are equal to zero, then the rows in submatrices B are eliminated together with their corresponding rows and columns in matrix J and their corresponding columns in submatrices C. The process is continued as long as the last rows of submatrices in B corresponding to Jordan block of that eigenvalue are nonzero.This process is repeated for all the eigenvalues.Finally the resultant matrices from the elimination of the rows in B and the rows and columns in J and the columns in C are called B1 , J 1 , C1 respectively.
Step 2 (elimination of unobservability factors in system [J 1 , B1 , C1 , D]) Considering the first columns of submatrices C1 which correspond to the jordan block for each eigenvalue.If the columns are zero then the columns in submatrices C1 , together with their corresponding rows and columns in matrix J 1 and their corresponding rows in submatrices B1 are eliminated.The process is continued as long as the first columns of submatrices in C1 corresponding to Jordan block of that eigenvalue are nonzero.The process is repeated for all the eigenvalues.Finally, the resultant matrix from the elimination of the rows and columns in J 1 is called J 2 .The dimension of matrix J 2 is termed the order of minimal realization of [J, B, C, D].

II) The Jordan matrix J includes Jordan blocks with repetitive eigenvalues
Consider the case in which at most two blocks of submatrices in B ( C) have the last rows (the first columns) which are linearly dependent.
Step 1 (elimination of uncontrollable factors in [J, B, C, D]) If one Jordan block with the size of n 1 has linearly dependent number of uncontrollability and linearly dependent number of unobservability n 1 , then the block is eliminated.If one other block with the size of n 2 possesses the same property, then the smaller sized block is eliminated and Stage (I) adopted, otherwise the following step is adopted: First two blocks with the same eigenvalues containing the last rows of submatrices B which correspond to the blocks which are linearly dependent, are considered.If at least one of the rows is a zero vector then (1-1) is applied, otherwise (2-1) is applied.
(1-1): The zero row (two rows) in submatrix B, its corresponding row and column in J and its corresponding column in C are eliminated.The process continues as long as the last rows of submatrices B corresponding to two Jordan blocks are not zero vectors.(The names of the matrices B, J, C in which elimination operation is performed are not changed).Now, if with the elimination of zero vectors in the submatrices of B corresponding to the two Jordan blocks, the last rows of submatrices B corresponding to the two Jordan blocks are linearly dependent, then (2-1) is applied, otherwise step 2 is applied.
(2-1): Between two Jordan blocks, the block which has the less linearly dependent number of uncontrollability (If the numbers are equal, the choice is arbitrary) is selected.Being equal to the linearly dependent number of uncontrollability, elimination is made from the last rows of submatrices B corresponding to that Jordan block, together with its corresponding rows and columns in the block of Jordan matrix J and the corresponding columns in submatrices C. The aforementioned process is performed for both the Jordan blocks with the same eigenvalues in which the last rows of submatrices B corresponding to the two Jordan blocks are linearly dependent.Stage (I) is performed for Jordan blocks with distinct eigenvalues.Finally, the resultant matrices from the eliminations are called B1 , J 1 , C1 .
The Order of Minimal Realization of Jordan Canonical Form Systems 85 Step 2 (elimination of unobservable factors [J 1 , B1 , C1 , D]) Considering the two Jordan blocks with the same eigenvalues containing linearly dependent first columns of submatrices in C1 which correspond to the two Jordan blocks, if at least one of the columns is zero then (1-2) is applied, otherwise (2-2) is applied. (1-2): The column (two columns) in submatrix C1 , together with its corresponding row and column in J 1 and its corresponding column in B1 is eliminated.The process is continued as long as the first columns of submatices in C1 corresponding to two Jordan block are not zero vectors.(The names of the matrices B1 , J 1 , C1 in which elimination operation is performed, are not changed).Now, if with the elimination of zero vectors in the submatrices of C1 corresponding to the two Jordan blocks, the first columns of submatrices C1 corresponding to the two Jordan blocks are linearly dependent, then step (2-2) is applied, otherwise the process is finished.
(2-2): Between two Jordan blocks, the block containing the less linearly dependent number of unobservability is selected (If the numbers are equal, the choice is arbitrary).Being equal to the linearly dependent number of unobservability, elimination is made from the first column of submatrix C1 corresponding to that of Jordan block, together with its corresponding rows and columns in J 1 and the corresponding rows in submatrices B1 .The aforementioned process is performed for both Jordan blocks containing the same eigenvalues, in which the first columns of submatrices C1 corresponding to the two Jordan blocks are linearly dependent.For Jordan blocks with distinct eigenvalues, step 2 of (I) is applied.Finally, the resultant matrix from the elimination of rows and columns in J 1 is called J 2 .The dimension of matrix J 2 is the order of minimal realization of the system [J, B, C, D].
Example 3.1.We consider system [J, B, C, D] in which

C1 C2
Eliminating zero vector in submatrix B1 , its corresponding row and column in J and its corresponding column in C 1 leads to:

C1 C2
Linearly dependent number of uncontrollability of B2 is less than B1 , consequently the last row of submatrix B2 is eliminated, also its corresponding row and column in J and its corresponding column in C which leads to:

C1 C2
The linear dependent number of unobservability of C2 is equal to 1 and the linear dependent number of unobservability of C1 is equal to 2, consequently the first column of C2 and its corresponding row and column in J 1 and its corresponding row in B1 are eliminated: The aforementioned system is of the 3 rd order and, the minimal realization of the first system is also of the 3 rd order.
The Order of Minimal Realization of Jordan Canonical Form Systems 87 Step 1: Converting the system [A, B, C, D] to the system[J, B, C, D] in which J is in Jordan canonical form.
Step 2: Elimination of zero vector from the last rows in submatrices B and also elimination of the corresponding columns in submatrices C and the corresponding rows and columns in J.
Step 3: Elimination of zero vector from first columns in submatrices C and also elimination of the corresponding rows in submatrices B and the corresponding rows and columns in J.
Step 4: If the resultant matrix J from previous step lacks Jordan blocks with the same eigenvalues, then the dimension of J is the order of minimal realization, otherwise step 5 is applied.
Step 5: If one Jordan block with the dimension of n 1 has linearly dependent number of uncontrollability and linearly dependent number of unobservability n 1 , the block is eliminated.(If there is another block with the dimension of n 2 which has similar property, the block with fewer dimension is eliminated).
Step 6: If the last rows of submatrices B corresponding to Jordan blocks with the same eigenvalues are linearly independent then step 8 is applied, otherwise step 7 is applied.
Step 7: Between two Jordan blocks containing the same eigenvalues, the block which has the less linear dependent number of uncontrollability is selected and equal to the number, the rows of B from the last row in B corresponding to that Jordan block and the corresponding columns in C and the corresponding rows and columns in J are eliminated.
Step 8: If the first column of submatrix C corresponding to Jordan blocks with the same eigenvalues are linearly dependent, then the dimension of J is the order of minimal realization, otherwise step 9 is applied.
Step 9: Between two Jordan blocks containing the same eigenvalues, the block which has the less linear dependent number of unobservability is selected and equal to the number, the columns of C from the first column in C corresponding to that Jordan block and the corresponding rows in B and the corresponding rows and columns in J are eliminated.
The dimension of matrix J which resulted from these steps is the order of minimal realization of the initial system.

Conclusion
This paper, by applying a new method and defining two new concepts, has demonstrated that with the elimination of unobservable and uncontrollable factors of a system in which the state matrix is in Jordan form, the system order can be decreased and this decrease can continue in order to remove all unobservable and uncontrollable components and the minimal order of the system can be determined.Since each system is equivalent to a system in which the state matrix is in Jordan form, then this method is efficient to achieve the minimal order of each standard system.

1 )
is equal to n.The matrix G(s) = C(sI − A) −1 B + D is called the transfer matrix of system (1.1).The system (1.1) is controllable if and only if rank [B AB ... A n−1 B] = n, and is observable if and only if rank For the sake of brevity, we 2010 Mathematics Subject Classification: 93B10, 93B20, 93C05.Submitted February 16, 2014.Published September 17, 2016 show system (1.1) as [A, B, C, D].The system [A, B, C, D] is called a realization for transfer matrix G(s) if G(s) = C(sI − A) −1 B + D. The realization [A, B, C, D] Determining the order of minimal realization of standard systems using Jordan canonical form Input: A standard system [A, B, C, D].Output: The order of minimal realization of [A, B, C, D].