Time Response & State Transition Matrix

(Uses Maple V): linalg & laplace packages, matrix, add, inverse, invlaplace, multiply

REF: Modern Control Systems, by R. Dorf & R. Bishop; Addison-Wesley, Chap 3, pp140-142, 1998.

The transient time response of the state variables of a control system is given by,

It is expressed in terms of the State Transition Matrix, initial state variables, input signal and input matrix. Example 3.5 show how such calculation is done manually for an RLC circuit with no input but has some residual initial conditions. Given below is a Maple V version of the solution.

> # ex3_5.txt
> with(linalg):
> readlib(laplace):
> A:=matrix([[0,-2],[1,-3]]);
> B:=matrix([ [1,0],[0,1]]);
> C:=add(B,A,s,-1);
> F:=inverse(C);
> phi:=matrix([ [invlaplace(F[1,1],s,t), invlaplace(F[1,2],s,t)], [invlaplace(F[2,1],s,t),invlaplace(F[2,2],s,t)]]);
> x0:=matrix([ [1],[1] ]);
> x(t):=multiply(phi,x0);
>

The result,

NOTE: In the code above, matrix C can also be calculated by,

> C:=evalm(s-A);