ontrol problem

From the step response of an industrial plant, identify a first-order plus dead time model

G(s) =

m

t s+1

eLs:

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.

From the step response of an industrial plant, identify a first-order plus dead time model

G(s) =

m

t s+1

eLs:

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.

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