Question

The parameters are as follows

k=10 a=0.50 b=0.3 c=0.6 d=9 w_1=12 w_2=15 Kv=30

A feedback control system (illustrated in Figure 1) needs to be designed such that the closed-loop system is asymptotically stable and such that the following design criteria are met:

1. the gain crossover frequency w_c should be between w₁ and w₂.

2. the steady-state error should be zero in response to a unit step reference.

3. the velocity constant should be greater than K_v (in other words, the steady-state unit ramp error should be less than 1=K_v).

4. the phase margin should be at least 55^o.

If the four performance criteria are met, further iteration of the controller may be undertaken (if you wish) to minimise the settling time of the step response from r(t) to y(t). If you cannot meet any of the design criteria, get as close as you can while ensuring closed-loop stability, and explain where and why compromises were needed.

This task will be approached incrementally, beginning with a proportional controller and finishing with a lead-lag controller.

All graphs should be clearly labelled and legible, and all design steps
should include some justification. MATLAB or a similar computational package may be used for any of the
calculations or graphs requiqired

θ(r) r(t) ut) elo) Figure 1: Feedback control system A pulley and belt transmission has a linearized relationship between the driven pulley angle e() in degrees and the input torque u(t) in Newton meters given by the following differential equation du(t) A feedback control system (illustrated in Figure 1) needs to be designed such that the closed-loop system is asymptotically stable and such that the following design criteria are met: Consider a phase-lead compensator ts +1 Show that this controller cannot satisfy all of the design requirements (hint: select Kp to satisfy the velocity constant requirement and show that the conditions to apply the inversion formulae do not hold). Find parameters for a phase-lead compensator that satisfies the constraints on the gain crossover frequency and on the phase margin (crieria 1 and 2), while keeping the velocity error as small as possible. Draw the bode plot of the resulting system, showing the required phase margin, and plot the closed-loop response to a unit step input. Plot the tracking error in response to a unit ramp, and show that the velocity error requirement rion 3) is not met.

Answer 1

Since s dt2 dt lt3 elt It dt2 or funétion ic deined ta Ihe above Trarsfer Zero . sleady stat eu -

zeros -0.6 H(s) poles -0.3 more 0.32.98 more 2.98 Plot! 20log(IH(S)|) Real plot Asymptotic plot 50 -100 -150 10-1 100 101 phase(H(s)) Real plot 50 Asymptotic plot 100 150 200 250 300 350 -400 10-1 100