Page 143 - Mechatronics with Experiments

CLOSED LOOP CONTROL 129
2. What is the steady-state error when a PD-type controller is used?
3. What is the steady-state error when a PI-type controller is used?
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Confirm your results with a Simulink or MATLAB simulation.
5. Consider the dynamics of a DC motor speed control system and its current mode amplifier.
Consider the the motor torque–speed transfer function is a first-order filter and the current mode
amplifier input–output dynamics is also a first-order filter.

w(s) 100
= (2.232)
T(s) 0.02s + 1
T(s) = 10 ⋅ i(s) (2.233)
i(s) 10
= (2.234)
i cmd (s) 0.005s + 1

Consider that the current command is generated by an analog controller (PID type) using motor speed
as feedback signal and commanded velocity signals.
(a) Ignore the filtering effect of the current amplifier, and determine the locus of closed loop poles
(root locus) of the closed loop control system under three different controllers: P, PD, PI. In each
case vary proportional gain as the root locus parameter and select different values of derivative
and integral gains.

i (s) = K ⋅ (w (s) − w(s); P control (2.235)
cmd p cmd
= (K + K s) ⋅ (w (s) − w(s); PD control (2.236)
p d cmd
( )
K
= K + i ⋅ (w cmd (s) − w(s); PI control (2.237)
p
s
(b) Repeat the same root locus analysis by including the amplifier dynamics in the analysis. Discuss
your results in terms of the effect of selecting different controller types (P, PD, PI), their gains,
and the effect of additional filter type dynamics in the control loop.
(c) Consider the same DC motor control problem, except this time consider the closed loop position
control problem. All other components are same, except that the commanded signal is the desired
position signal and the feedback signal is the position signal. As a result, the only difference
in the model is the torque–position transfer function, which has one additional integrator term
compared to the speed control problem,
(s) 100
= (2.238)
T(s) s(0.02s + 1)
Repeat the analysis for part (a) and (b) for the closed loop position control of a DC motor problem.

6. Consider the dynamics of a DC motor speed control system and its current mode amplifier.
Consider the the motor torque–speed transfer function is a first-order filter and the current mode
amplifier input–output dynamics is also a first-order filter.

w(s) K m
= (2.239)
T(s) s + 1
m
T(s) = K ⋅ i(s) (2.240)
T
i(s) K a
= (2.241)
i cmd (s) s + 1
a
Assume that there is a load torque acting as a disturbance on the motor and that it is in the form of
(i) step function, (ii) a ramp function. If we want to make sure that the steady-state speed error due
to the load torque is zero, what kind of controller is required in order to deal with each type of
disturbance (load) torque? Why can’t a PD type controller do the job?