Numerical computation method for the evaluation of the transition matrix A. K. Mandal, M.Tech., D. Roy Choudhury, M.Tech., A. K. Choudhury, M.Sc, D.Phil., and A. K. Bhattacharyya, M.Tech. Abstract A technique for the evaluation of the transition matrix of a linear time-invariant system by a numerical computation method is described. The method is simple and accurate, and is applicable to higher-order systems. The error propagated in the transient response of the system at different intervals, due to the truncation error committed in the numerical computation of the transition matrix, has also been estimated. 1 Introduction In control systems or practical circuits, the dynamical behaviour of a homogeneous /nth-order, linear, time-invariant system can be represented by an mth-order, linear, differential equation having the following form: x° n \t) + a m _,*<"»-»(/) + . . . + a,x(O + a o x(t) = 0 . . . . (1) where x(t) is the system variable, a 0 , a u . . ., a m _, are the constant parameters of the system, and the primes (w), (w — 1) etc. denote the order of the derivatives. This equation can be written in the matrix form X(t) = AX(t) (2) where X(t) is a column matrix of the state variable X x (t), X 2 (t) . . ., X m (t), and A is a (m x m) square matrix given by A = 0 0 0 0 -a 2 0 1 tn — L fti—I (3) The solution of the matrix differential eqn. 2 is given by X(t) = c A 'X(0) = <f>(t)X(O) (4) where X(0) is A^O at t = 0, and e At = <f>(t) is called the transition matrix, which is a (m x m) square matrix given by <f> m2 (t) • (5) It is evident from eqn. 4 that the equation defining the dynamics of a fixed linear system is determined by the initial conditions and the transition matrix of the system only. If we let t = nT, where T is a small interval of time and n is any positive integer, eqn. 4 gives the response at the wth instant, X ( n T ) = e A " ' X ( 0 ) = {<(>(T)}"X(0) . . . . ( 6 ) Using eqn. 6, the response at (n + l)th instant is given by X{(n + 1)T} = <f>(T)X(nT) (7) It is evident from eqns. 6 and 7 that, once the transition matrix is computed for a small interval of time T, the response at any future interval can be found, either by computing the matrix power as in eqn. 6 or by using the recursive relation Paper 5764 E, first received 13th February and in revised form 5th August 1968 The authors are with the Institute of Radio Physics & Electronics, University of Calcutta, Calcutta 9, India 500 of eqn. 7. This shows the importance of the transition matrix in linear-system response, and the problem of obtaining the response of a linear system reduces to that of the computation of the transition matrix. There are several methods for computing the transition matrix, but they can all be classified into two groups: (a) methods that require the complete knowledge of the eigenvalues of the characteristic equation (b) those that do not The difficulty encountered in the determination of the eigenvalues of a higher-order system, and the associated manipulation of matrix polynomials, make the former method of computation of the transition matrix a formidable task. The latter methods prove suitable when it is desired to compute the transition matrix numerically. A method of numerical computation, by which the transi- tion matrix can be computed within a prescribed amount of accuracy, is presented here, along with the details of error analysis and its control. 2 Numerical computation method The time response of the z'th state (eqn. 4) can be written as = .2 / = 1, 2, . . ., m (8) Now if we assume that Xj(0) = 1, and all other states are zero initially, eqn. 8 reduces to *,(') = <f>ij(O (9) Thus, placing a unit initial condition on the yth state, and zero initial conditions on the remaining states, the yth column of the transition matrix can be found by Taylor- series expansion of x t (t). This process of determining the transition matrix is valid, because, for a linear system, it is independent of the initial conditions of the system, and is determined by the system parameters alone. 2.1 Derivation The Taylor-series expansion of x t {f) over a small interval of time T is given by 3 (10) where jc^ r) (O) is the rth derivative of x t (t) with respect to time, evaluated at t — 0. The different elements of the transition matrix of a system described by eqn. 1 can be found with the help of eqn. 10, as shown below. 2.1.1 First column The elements of the first column of the transition matrix are found by putting and PROC IEE, Vol. 116, No. 4, APRIL 1969