TABLE OF CONTENTS Let us again consider the state equations pair
where for two or more simultaneous differential equations A and C are 2 2 or higher order matrices, and b and d are column vectors with two or more rows. In this section we will introduce the state transition matrix e At, and we will prove that the solution of the matrix differential equation
with initial conditions
is obtained from the relation
Proof:
Let A be any n n matrix whose elements are constants. Then, anothe n n matrix denoted as ?(t), is said to be the state transition matrix of (5.34), if it is related to the matrix A as the matrix power series
where I is the n n identity matrix.
From (5.124), we find that
Differentiation of (5.124) with respect to t yields
and by comparison with (5.124) we obtain
To prove that (5.123) is the solution of the first equation of (5.120), we must prove that it satisfies both the initial condition and the matrix differential equation. The initial condition is satisfied from the relation
where we have used (5.125) for the initial condition. The integral is zero since the upper and lower limits of integration are the same.
To prove that the first equation of (5.120) is also satisfied, we differentiate the assumed solution
with respect to t and we use (5.127), that is,
Then,
or
We recognize the bracketed terms in (5.129) as x(t),...
