Section 9–4 / Solving the Time-Invariant State Equation 661
By substituting this assumed solution into Equation (9–28), we obtain
(9–30)
If the assumed solution is to be the true solution, Equation (9–30) must hold for all t.Thus,
by equating the coefficients of like powers of ton both sides of Equation (9–30), we obtain
By substituting t=0into Equation (9–29), we obtain
x(0)=b 0
Thus, the solution x(t)can be written as
The expression in the parentheses on the right-hand side of this last equation is an n*n
matrix. Because of its similarity to the infinite power series for a scalar exponential, we
call it the matrix exponential and write
In terms of the matrix exponential, the solution of Equation (9–28) can be written as
(9–31)
Since the matrix exponential is very important in the state-space analysis of linear
systems, we shall next examine its properties.
Matrix Exponential. It can be proved that the matrix exponential of an n*n
matrixA,
converges absolutely for all finite t.(Hence, computer calculations for evaluating the
elements of eAtby using the series expansion can be easily carried out.)
eAt= a
qk= 0