660 Chapter 9 / Control Systems Analysis in State Space9–4 Solving the Time-Invariant State Equation
In this section, we shall obtain the general solution of the linear time-invariant state equa-
tion. We shall first consider the homogeneous case and then the nonhomogeneous case.
Solution of Homogeneous State Equations. Before we solve vector-matrix
differential equations, let us review the solution of the scalar differential equation
(9–25)
In solving this equation, we may assume a solution x(t)of the form
x(t)=b 0 +b 1 t+b 2 t^2 +p+bktk+p (9–26)
By substituting this assumed solution into Equation (9–25), we obtain
(9–27)
If the assumed solution is to be the true solution, Equation (9–27) must hold for any t.
Hence, equating the coefficients of the equal powers of t,we obtain
The value of b 0 is determined by substituting t=0into Equation (9–26), or
x(0)=b 0
Hence, the solution x(t)can be written as
We shall now solve the vector-matrix differential equation
(9–28)
where
By analogy with the scalar case, we assume that the solution is in the form of a vector
power series in t,or
x(t)=b 0 +b 1 t+b 2 t^2 +p+bktk+p (9–29)
A=n*n constant matrix
x=n-vector
x
=Ax
=eatx(0)
x(t)= a 1 +at+
1
2!
a^2 t^2 +p+
1
k!
aktk+pbx(0)
bk=
1
k!
akb 0
b 3 =
1
3
ab 2 =
1
3 * 2
a^3 b 0
b 2 =
1
2
ab 1 =
1
2
a^2 b 0
b 1 =ab 0
=aAb 0 +b 1 t+b 2 t^2 +p+bk tk+pB
b 1 +2b 2 t+3b 3 t^2 +p+kbk tk-^1 +p
x# =ax
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