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.

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.