A First Course in FUZZY and NEURAL CONTROL

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 73

Design specifications Since the DC motor is being used as an actuator, we
wish to position the rotor very precisely. Therefore the steady-state error of
the motor angular position should be zero. We will also require the steady-
state error due to a disturbance to be zero as well. The other performance
requirement is that the motor reach its desired position very quickly. In this
case, let us specify a settling time requirement of 40 milliseconds. We also want
to have an overshoot of less than 5%.

System parameters For our example, let us assume the following parameters
for the electromechanical system:

ïmoment of inertia of the rotor:J=3. 0 ∗ 10 −^6 kg m^2 /s^2

ïdamping ratio of the mechanical system:b=3. 5 ∗ 10 −^6 Nm/s

ïelectromotive force constant:K=Ke=Kt=0.03 N m/A

ïelectric resistance:R=4Ω

ïelectric inductance:L=2. 0 ∗ 10 −^6 H

Step response We first simulate the open-loop response of this system to
determine its behavior to a step input. For this, theMatlabm-filecan be set
up as follows:

J=3.0E-6; %Inertia constant
b=3.5E-6; %Damping constant
K=0.03; %Motor constant
R=4; %Armature resistance
L=2.0E-6; %Armature inductance
num=K; %Transfer function numerator
den=[(JL) ((JR)+(Lb)) ((bR)+K^2) 0];
%Transfer function denominator
sys1=tf(num,den); %Matlabfunction ìtfî establishes the transfer function
step(sys1,0:0.001:0.2); %Step response with plot
As illustrated in Figure 2.40, the rotor position continues to grow when a
1.0 volt step input is applied to the motor terminals. The open-loop response
clearly shows the need for a feedback that will stabilize the system for a step
input.
Considering a unity feedback system, the closed-loop transfer function of the
system can be developed as follows. LetGp(s)represent the open-loop transfer

function. The closed-loop transfer function then becomes1+GGp(ps()s).Thistransfer

function can be obtained by usingMatlabfunctions as illustrated below.

sys2=1 %Unity feedback transfer function

sys=feedback(sys1,sys2);

%Function ìfeedbackî gives closed-loop transfer function