Recent Question/Assignment

ontrol problem
From the step response of an industrial plant, identify a first-order plus dead time model
G(s) =
m
t s+1
e􀀀Ls:
Then, using the model G(s), design a feedback compensator that satisfies the following criteria
1. gain crossover frequency wc  0:5
t ;
2. phase margin fm  60o;
3. given a step reference signal, the tracking error is zero;
4. given a ramp reference signal with slope R, the tracking error satisfies er 3t R;
5. the closed-loop system attenuates of at least 1
1000 a measurement noise at frequencies wd 1000:5
t .
1 Identification
Use Matlab and the file systems generator.p obtain the unit step response of the plant as
[y,t]=systems generator(N)
where N is your student number. Please ensure that you use the correct student number. From the
step response identify the parameters m, L and t. (hint: find m imposing the dc-gain, then find L as the
time when the response of the real systems reaches 20% of the steady-state and finally find t imposing
that the response of G(s) matches the one of the plant at 63% of the steady state; see also Example 4 of
“delay and higher order.pdf” from Lecture 4). Verify your model by plotting on the same figure the step
response obtained from systems generator and the unit step response of G(s).
2 Integral controller
Using G(s) with the parameters identified in the previous step, design a controller
C(s) =
k
s
1
that satisfies condition 1). (Hint: find k imposing that wc = 0:5
t ). Show that condition 2) is not satisfied.
Show that the conditions 3) and 4) are satisfied.
3 PI controller
Using G(s) with the parameters identified in the first step, design a controller
C(s) =
k(t1s+1)
s
that satisfies conditions 1) and 2). (Hint: find t1 such that condition 2 is satisfied, and then find k imposing
that wc = 0:5
t . Always impose conditions with some margin, for example, impose a phase margin larger than
the bare minimum of 60.) Verify that conditions 3) and 4) are satisfied. Show that condition 5 is not satisfied.
4 Filtered PI controller
Using G(s) with the parameters identified in the first step, design a controller
C(s) =
k(t1s+1)
(t2s+1)s
that satisfies all conditions. (Hint: use t1 from the previous step, obtained with a sufficiently large phase
margin, find t2 such that condition 5 is satisfied, and then find k imposing that wc = 0:5
t . Show that all
conditions are satisfied.
To verify your final design, using the transfer function G(s) obtained identified in step 1
• plot the closed-loop unit step response;
• plot the error in response to a unit ramp input;
• plot the closed-loop response to a sinusoid with angular frequency wc;
• show the Bode (or margin) plot of the loop-gain transfer function L(s) to show that the required
crossover frequency and phase margin have been achieved;
• verify that the magnitude of the frequency response of the closed-loop transfer function at 1000:5
t is
less than 0:001.
2
If any of the design criteria cannot be achieved, then get as close as you can and explain where compromises
were required.