An Alternate Method for Computation of Transfer Function Matrix

Abstract

A direct and simple numerical method is presented for calculating the transfer function matrix of a linear time invariant multivariable system (A, B, C). The method is based on the matrix-determinant identity, and it involves operations with an auxiliary vector on the matrices. The method is computationally faster compared to Liverrier and Danilevsky methods.

1. Introduction

In the design of multivariable control systems by frequency domain methods, the computation of the transfer function matrix from the state space description of the system plays an important role. The computation based on Liverrier method [1] is regarded as unsatisfactory especially when the system order is high or its eigenvalues are widely distributed. Patel [2] has presented a method whereby each numerator element of the transfer function matrix is calculated as the difference between the characteristic polynomials of two matrices which are related to the system state matrix, and the polynomials are computed by the method of Danilevsky [1]. Danilevsky method is used in many numerically intensive calculations where extreme eigenvalues have to be estimated [3–5]. Daly [6] has given a method for the transfer function matrix evaluation which involves a sequence of similarity transformations on the Rosenbrock system matrix [7] and resembles Danilevsky technique.

This paper, instead of using the Danilevsky method, presents a direct procedure to compute the characteristic polynomials associated with each numerator element of the transfer function matrix. The method makes use of an auxiliary output vector and the corresponding single-output system is transformed to its row companion form to give the coefficients of the characteristic polynomial. The above transformation is carried out by the algorithm given by Hickin and Sinha [8] which is based on the transformation of a matrix to its Hermite normal form.

2. Transfer Function Matrix

Consider the 𝑛th-order, 𝑚 inputs, ℓ outputs, linear-time invariant system described by 𝑥̇=𝐴𝑥+𝐵𝑢,𝑦=𝐶𝑥, (1) where 𝐴, 𝐵, and 𝐶 are matrices of appropriate dimensions. The ℓ×𝑚 transfer function matrix of the system is given by 𝐺(𝑠)=𝐶(𝑠𝐼−𝐴)⁻¹𝑊(𝑠)𝐵=𝐹(𝑠), (2) where 𝑊(𝑠)=𝐶adj(𝑠𝐼−𝐴)𝐵 is the ℓ×𝑚 numerator transfer function matrix and 𝐹(𝑠)=det(𝑠𝐼−𝐴)=𝑠^𝑛+𝑓₁𝑠^𝑛−1⋯+𝑓_𝑛 is the characteristic polynomial. Any typical element 𝑊_𝑖𝑗(𝑠) of 𝑊(𝑠) is 𝑊_𝑖𝑗(𝑠)=𝑐_𝑖adj(𝑠𝐼−𝐴)𝑏_𝑗,𝑖=1,…,ℓ,𝑗=1,…,𝑚. (3) Using the matrix-determinant identity [9] Det(𝑅+𝑒𝑑)=det(𝑅)+𝑑adj(𝑅)𝑒, (4) where 𝑅 is an 𝑛×𝑛 matrix and 𝑒 and 𝑑 are 𝑛×1 and 1×𝑛 vectors, respectively. 𝑊_𝑖𝑗(𝑠) can be expressed as: 𝑊_𝑖𝑗(𝑠)=det𝑠𝐼−𝐴+𝑏_𝑗𝑐_𝑖−det(𝑠𝐼−𝐴). (5) Patel [2] has suggested the method of Danilevsky for computing the polynomials in (5). A simple and direct method to compute the coefficients of the polynomials is given below.

3. Characteristic Polynomial of A

For a given matrix A, let 𝑟 represent 1×𝑛 auxiliary output vector such that (𝐴,𝑟) is observable. Form the matrix: 𝑟𝑅=𝑟𝐴𝑟𝐴²⋮𝑟𝐴^𝑛=𝑟₁𝑟₂⋮𝑟_𝑛𝑟_𝑛+1. (6)

For the above single-output system, the transformation of 𝑅 to its row companion form will result in a canonical form where the first 𝑛 rows will be unit vectors appearing in their natural order and the elements of the last row will be the coefficients of the characteristic polynomial of 𝐴. Thus, det(𝑠𝐼−𝐴) can be obtained directly from the row companion form representation of 𝑅.

The transformation of the matrix 𝑅 to its row companion form is done in n steps on a row basis starting from the first row. For each row select the element 𝑟_𝑖𝑖 as pivot and calculate 𝑟_𝑖=𝑟_𝑖−𝑒_𝑖,𝑅_𝑖=𝑅_𝑖−1−𝑟_𝑖𝑖⁻¹𝑅_𝑖−1𝑒^𝑇_𝑖𝑟_𝑖,𝑖=1,2,…,𝑛,𝑅₀=𝑅, (7) where 𝑒_𝑖 is a unit row vector. Each step of transformation will result in the elements of the row 𝑟_𝑖 being expressed as the unit vector 𝑒_𝑖.

After the 𝑛 step transformation, the matrix 𝑅_𝑛 will be in the Hermite normal form where the unit row vectors appear in their natural order in the first 𝑛 rows and the elements of the last row will be (−𝑓_𝑛,−𝑓_𝑛−1,…,−𝑓₁). Thus, the coefficients of the characteristic polynomial 𝐹(𝑠) of 𝐴 are obtained.

The polynomial of degree (𝑛−1) corresponding to the element 𝑤_𝑖𝑗(𝑠) is obtained as the difference between the characteristic polynomials of (𝐴−𝑏_𝑗𝑐_𝑖) and 𝐴. Thus in all (ℓ𝑚+1), characteristic polynomials have to be computed to evaluate the complete transfer function matrix 𝐺(𝑠).

4. Numerical Example One

𝐴=0−21100000,𝐵=100001,𝐶=2000−22. (8)

4.1. Computation of det(𝑠𝐼−𝐴)

Let 𝑟=(110) as per (6): 𝑅=1101−21−2−21−24−2. (9) The transformation with (7) gives 𝑅₁11=,𝑅=𝑅−−2−20101001−31−201−26−2₂=𝑅₁−002=,𝑅−11−41100010−2010−20₃=𝑅₂−0010−200=.1000100010−20 (10) Hence det(𝑠𝐼−𝐴)=𝑠³+2𝑠.

4.2. To Find the Element 𝑤₁₁(𝑠) of 𝑊(𝑠)

The characteristic polynomial of (𝐴−𝑏₁𝑐₁) is computed using (6) and (7) and is (𝑠³+2𝑠²+2𝑠). Hence the elements 𝑤₁₁(𝑠) as per (5) are 2𝑠². Similarly all the other elements of 𝑊(𝑠) can be computed.

5. Numerical Example Two

The jet engine model which used NEC International form [10] is considered to compare the numerical properties of the present method with other known methods. The fifth-order model of the jet engine is considered for computing the coefficients of the characteristic polynomial. The state matrix is: 𝐴=−0.3245𝐸+01−0.2158𝐸+01−0.9155𝐸+030.5731𝐸+000.1342𝐸+030.1642𝐸+01−0.5941𝐸+01−0.2816𝐸+010.1897𝐸+010.5705𝐸+000.1685𝐸−01−0.2854𝐸−01−0.1003𝐸+010.7994𝐸−020.5887𝐸+010.0000.0000.0001.000𝐸+020.000−0.2613𝐸+010.6862𝐸+010.7405𝐸+030.1195𝐸+01−0.1715𝐸+01. (11)The computation was done on FORTRAN program, and results are shown in Table 1. In the above matrix 𝐴, the ratio between the largest and the smallest element is of the order 10⁵. The present method gives identical results as the Danilevsky method, whereas the Liverrier method shows a small difference in the coefficients values in the last two rows. The execution time for Liverrier, Danilevsky, and present methods was found to be 0.43, 0.36, and 0.14 seconds, respectively. The eigenvalues are −174.653, −10, −6.671 +  𝑗 1.293, −6.671 −  𝑗 1.293, and −2.629. The eigenvalues computed by MATLAB are in agreement with the above results.

Table 1 

Comparison of computation of coefficients of characteristic polynomial.

6. Conclusions

A simple and direct method of computing the transfer function matrix of the system (𝐴,𝐵,𝐶) has been presented. All the (ℓ𝑚+1) polynomials of the transfer function matrix are computed from the characteristic polynomials of (ℓ𝑚+1) matrices related to the system state matrix. The coefficients of the above polynomials have been evaluated by a simple transformation of an auxiliary single-output system to its row companion form. The transformation is easily programmed on the computer.