MXEN3004/ETEN6000 - Design Assignment

Assignment Task

Control problem

You have been asked to design a controller to control the temperature of a small industrial furnace. The input is the electrical current applied through a resistor that heats up the furnace, and the output is the internal temperature. You cannot obtain a model from first principles analysis of the furnace because the furnace has been purchased off-the-shelf with no documentation and you do not know its physical characteristics. You must resort to deriving a transfer function experimentally. A unit step input has already been applied to the system and the output has been measured. A first-order plus dead time model has been deemed suitable to estimate the behaviour of the furnace making a trade-off between accurately describing the response and model simplicity.  

From the step response, identify a first-order plus dead time model

G(s) = µ /τ s+1 e −L

Then, using the identified model G(s), design a feedback compensator that satisfies the following criteria

1. gain crossover frequency ωc ≈ 0.5 τ ;

2. phase margin φm ≥ 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 < 4> 1000.5 τ .

Physical interpretation of the criteria. This is for your own understanding to give relevance to the above criteria. You do not need to discuss the five bullet points below in your report.

1. We want a closed-loop system to have a certain bandwidth, i.e. to set how rapidly the system responds to input variations and rejects load/output disturbances.

2. We accept a phase margin less than 90, i.e. we accept a bit of overshoot in the closed-loop response in order to assign a larger closed-loop bandwidth.

3. We want the furnace to reach a desired set-point temperature (step input) precisely.

4. When the set-point temperature increases linearly (a ramp) we want the furnace temperature to track with limited error.

5. We want to limit how much any noise in the measurement of the temperature, e.g. a noisy thermocouple, affects the temperature of the furnace.

Design Task

a) Identification

Use Matlab and the file systems generator.p to obtain the unit step response of the furnace by using the code: [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 µ, L and τ.

Verify your model by plotting on the same figure the step response obtained from systems generator and the unit step response of G(s).

b) Integral controller with one zero

Using G(s) with the parameters identified in the first step, design a controller that satisfies conditions 1 and 4.

C(s) = k(τ1s+1)/ s

Verify that conditions 2 and 3 are satisfied. Show that condition 5 is not satisfied.

c) Optimal integral controller with one zero

Using G(s) with the parameters identified in the first step, design a controller that minimizes the ramp taking error subject to conditions 1, 2 and 3.

C(s) = k(τ1s+1) /s

Show that condition 3 is satisfied, and plot on the same figure the ramp tacking error obtained in steps b) and c). Show that condition 5 is not satisfied.

d) Integral controller with one zero and one pole

Using G(s) with the parameters identified in the first step, design a controller that satisfies all conditions.

C(s) = k(τ1s+1)/ (τ2s+1)s

verify that the magnitude of the frequency response of the closed-loop transfer function at 1000.5 τ is less than 0.001.

If any of the design criteria cannot be achieved, then get as close as you can and explain where compromises were required.

This MXEN3004/ETEN6000 - Engineering has been solved by our PhD Experts at My Uni Paper.