Page 196 - Modern Control Systems
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Chapter 3 State Variable Models
where we use the relationship for p given in Equation (3.28) and the relationship
for q given in Equation (3.29). But p = x 3 and q = x 4, so Equation (3.32) can be
written as
*i *i h bi 1
3 3 (3.34)
Mi Mi M t Mi Mi
and Equation (3.33) as
. _ k\ k\ + k 2 b\ bi + b 2
X4 XI X2 + X3 XA (3.35)
~ ~M~ 2 ~ ~mr w 2 ~ ~MT -
In matrix form, Equations (3.30), (3.31), (3.34), and (3.35) can be written as
x = Ax + BH
where
0 0
0 0
A = _A and B = j _
Mi
Mi
0
W 2
and u is the external force acting on the system (see Figure 3.6). If we choose/? as the
output, then
y = [1 0 0 0]x = Cx.
Suppose that the two rolling carts have the following parameter values: ki = 150 N/m;
= = 15 N s/m; b 2 = 30 N s/m; Mi = 5 kg; and = 20 kg. The
k 2 700 N/m; b x M 2
+ <7 •+P
k\(Q-p) k x(p-q) «-
M 2 Mi •+>«
b x{q-p) b-iip-q) •*•
(a) (b)
FIGURE 3.6 Free-body diagrams of the two rolling carts, (a) Cart 2; (b) Cart 1.
