2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 59
we obtain a relationship between system state and inputF(s),andchoosing
Y(s)
Q(s)=s+0. 001we obtain a relationship between system state and outputY(s).Here,weare
assigning an arbitrary stateQ(s). From these two relationships, a state-variable
model can be developed.
From the state and input relationship, we obtain
(s^2 +1. 05 s+0.001)Q(s)=F(s)The time-domain differential equation, with zero initial conditions is then
q^00 (t)=− 1. 05 q^0 (t)− 0. 001 q(t)+f(t)Similarly, from the state and output relationship, we have
Y(s)=(s+0.001)Q(s)The time-domain equation is then
y(t)=q^0 (t)+0. 001 q(t)If we choosex 1 (t)=q(t),then
x^01 (t)=q^0 (t)=x 2 (t)and
x^02 (t)=q^00 (t)=− 1. 05 x 2 (t)− 0. 001 x 1 (t)+f(t)
represents the set of state-variable equations. The corresponding output equa-
tioncanthenbewrittenasy(t)=x 2 (t)+0. 001 x 1 (t). These equations can be
written in standard vector-matrix form as
∑
x^01 (t)
x^02 (t)
∏
=
∑
01
− 0. 001 − 1. 05
∏∑
x 1 (t)
x 2 (t)∏
+
∑
0
1
∏
f(t)[y(t)] =£
0 .001 1
§
∑
x 1 (t)
x 2 (t)∏
A solution of the set of state and output equations given above can be ob-
tained using aMatlabsimulation package called Simulink. In the simulation
diagram in Figure 2.23, we have set up a simulation to obtain the step response
of the modified system described earlier. It can be seen in Figure 2.24 that while
the rise time is well within the required specification, the steady-state error is
large and has to be minimized using integral control action.
Choosing valuesKP = 800,KI =40,andKD=0,wegettheforward
transfer function as
G(s)=Gc(s)Gp(s)=μ
KDs^2 +KPs+KI
s∂μ
1
ms+b∂
=
μ
800 s+40
s∂μ
1
1000 s+50∂
=
0. 8
s