Section 2–4 / Modeling in State Space 31
The outputs y 1 (t), y 2 (t),p,ym(t)of the system may be given by
(2–9)
If we define
ym(t)=gmAx 1 , x 2 ,p, xn ; u 1 , u 2 ,p, ur ; tB
y 2 (t)=g 2 Ax 1 , x 2 ,p, xn ; u 1 , u 2 ,p, ur ; tB
y 1 (t)=g 1 Ax 1 , x 2 ,p, xn ; u 1 , u 2 ,p, ur ; tB
u(t)= F
u 1 (t)
u 2 (t)
ur(t)
g(x, u,t)= F V
g 1 Ax 1 ,x 2 ,p,xn ;u 1 ,u 2 ,p,ur ;tB
g 2 Ax 1 ,x 2 ,p,xn ;u 1 ,u 2 ,p,ur ;tB
gmAx 1 ,x 2 ,p,xn ;u 1 ,u 2 ,p,ur ;tB
y(t)= F V,
y 1 (t)
y 2 (t)
ym(t)
V,
f(x, u,t)= F
f 1 Ax 1 ,x 2 ,p,xn ;u 1 ,u 2 ,p,ur ;tB
f 2 Ax 1 ,x 2 ,p,xn ;u 1 ,u 2 ,p,ur ;tB
fnAx 1 ,x 2 ,p,xn ;u 1 ,u 2 ,p,ur ;tB
x(t)= F V,
x 1 (t)
x 2 (t)
xn(t)
V,
then Equations (2–8) and (2–9) become
(2–10)
(2–11)
where Equation (2–10) is the state equation and Equation (2–11) is the output equation.
If vector functions fand/orginvolve time texplicitly, then the system is called a time-
varying system.
If Equations (2–10) and (2–11) are linearized about the operating state, then we
have the following linearized state equation and output equation:
(2–12)
(2–13)
whereA(t)is called the state matrix,B(t)the input matrix,C(t)the output matrix, and
D(t)the direct transmission matrix. (Details of linearization of nonlinear systems about
y(t)=C(t)x(t)+D(t)u(t)
x
(t)=A(t)x(t)+B(t)u(t)
y(t)=g(x, u, t)
x
(t)=f(x, u, t)
