Problem Solving Task - Time Response and Controller Design

Please provide your solutions to the below questions. Show all work and present your work logically, justifying all assumptions. The questions marked with an asterisk (*) include a MATLAB component.

Exercise 1: Reduction of subsystems

Find the transfer function T (s) = C(S)/C(R) of the block diagram in Figure given that:

G₁ = 2/(s+2), G₂ = 1/s, G₃ = 5

G₄ = ((s+5)/(s+3)(s+10)), G₅ = 2, G₆ = s, G₇ = 5/s

Also, write a MATLAB script using feedback, series and parallel commands and confirm that your handwritten answer and MATLAB agree.

Exercise 2: What's the damping?

Consider the motor/load feedback control system shown in Figure 2 below. The load damping D_L is unknown, but when R(s) is a step, the response of θ_L has an overshoot of 20%. Do the following:

a) Find the transfer function θ_L(s)/R(s) (you'll have a D_L in your equation since you don't know it yet)

b) Calculate the damping ratio that corresponds to 20% overshoot

c) Find D_L using the information from a and b).

Exercise 3: A proportional controller

For the velocity control system in Figure 3:

a) Find the closed-loop transfer function T(s) = c(s)/R(s).

b) Find a value of K_p that will yield less than 10% overshoot for the closed-loop system. (Note: ignore the zero dynamics to calculate K_p initially).

c) Is the second order component of the closed-loop system dominant?

d) What is the magnitude of step input that will give a steady-state velocity of c(∞) = 50 m/s?

e) Using your K_p from part b), write a MATLAB script that calculates the closed-loop transfer function, T(s) = c(s)/R(s).

f) Simulate the step response of T(s). Is the overshoot 10% as you designed? Discuss.

Exercise 4: Linearization of a vehicle moving through air

Consider the simple model of a vehicle moving through air

mv^· = F - kv²

where v is the velocity of the vehicle, F is the forward traction force, and k is a drag parameter that depends on the properties of the air and the geometry of the vehicle. The parameters are m = 1000kg and k = 10N·s²/m². Do the following:

a) Linearize the model about the steady-state F^- = 1000N.

b) Find the transfer function v'(s)/F'(s) for the linearized system in a)

c) Consider a vehicle that is at the steady-state you found in a). A step change in the traction force is introduced, F'(s) = 200/s. Use your transfer function model to predict time response of the velocity v(t).

Exercise 5: Stability

Consider a unity feedback (H(s) = 1) configuration Figure 4. For the following systems, determine the range of K that will maintain stability.

(a) G_c(s) = K, G_a(s) = 1/s(s²+1), and G_p(s) = 1/(s²+s+1)

(b) G_c(s) = Ks, G_a(s) = 1/(s+1), and G_p(s) =  1/((s+2)(s+5))

Exercise 6: Disc drive problem

Recall the disc drive problem introduced in tutorial. Last time we developed the transfer functions seen in Figure 5. Do the following:

a) Find the transfer function G_d(s) = P_o(s)/T_d(s) when G_c(s) = K_c.

b) For G_c(s) = K_p, find the range of K_p for which the closed-loop system is stable.

c) Write a MATLAB script using feedback, series and parallel commands to obtain P_o(s)/P_t(s) and P_o(s)/T_d(s) when K_c = 12.14.

Exercise 7: Disc drive controller

Recall the disc drive problem from Tutorials, where we demonstrated that the open-loop system can be written as

a) Consider the controller that we designed in tutorial:

Gc(s) = K_c = 12.14

Find the steady-state error to a ramp input with this controller. If we wish to reduce this error to 0.005, can we do it with a different K_c? (Hint: consider your stability limits!)

b) Well try a more complex controller of the form

G_c(s) = K_c(s + a)

which is sometimes called a proportional-derivative controller. Find the closed-loop transfer function, T(s) = P_o(s)/P_t(s).

c) Find the conditions (inequalities) on K_c and a such that the closed-loop system is stable. Use MATLAB to plot the stability boundaries (again, inequalities) on a K_c vs a plot.

d) Find values of K_c and a such that the steady-state error to a ramp is less than 0.005.

Exercise 8: Using Root Locus

Consider the feedback control system in Figure 7. In this exercise, we'll walk through designing G_c(s) with different levels of complexity.

To this ends, do the following by hand (unless otherwise stated):

a) Sketch (by hand) the root locus and find the dosed loop poles when G_c(s) = K_c = 1. Also: find the steady-state error to a step and ramp inputs, ζ and the settling time.

b) In order to improve the transient response, a PD controller of the form

G_c(s) = K_c(s + a)

is being considered. Determine the values of K and a so that the closed loop system has an overshoot of 1.6 and a settling time of 2s for a step input. (Hint: use a to ensure that the poles are on the root locus). You may ignore the effect of the zero dynamics for this part.

c) What is the steady-state error to a ramp?

d) We now require that steady-state error to a ramp is eliminated. Your boss has told you to "just add an integrator" to the controller to eliminate the steady-state error. Sketch the root locus and use it to tell your boss why this won't work.

e) A smarter way to eliminate the steady-state error is to use a RID controller of the form

G_c(s) =  (K_c(s + a)(s + b)/s)

where a is the same as in b) and b is close to the origin. Select a suitable value for b so that the desired poles from b) are on the root locus. If necessary (it may or may not be!), adjust the value of K_c to ensure that the desired closed-loop poles are achieved. You may use MATIAB to check the location of the closed-loop poles.

f) Simulate the closed-loop step response in MATLAB using the PID controller from e). Twiddle K_c a and b (if necessary) to achieve your targets (i.e. 16% overshoot and a settling time of 2s).

Exercise 9: Lead-Lag design using Root Locus

Recall the disc drive problem from Tutorials, where we demonstrated that the open-loop system can be written as

We will now try to design a lead-lag compensator with the requirements that

• Overshoot ≤ 10
• T_s ≤ 75MS
• e_ramp(∞)≤ 0.001

Do the following (you may use MATLAB at your leisure, but be sure to explain your logic for your design choices):

a) Use MATLAB to draw the root locus when G_c = K_c.

b) Use MATLAB to draw the region where the dominant closed-loop poles must be to satisfy the transient requirements. Comment on your ability to achieve these requirements with a gain-only controller.

c) Design a lead compensator(s) to meet the transient requirements (i.e. overshoot and settling time).

d) Design a lag compensator to achieve the steady-state tracking requirement.

e) Use MATLAB to compute the resulting closed-loop poles and discuss second order dominance.

Some Hints:

• You may need to place more than one lead compensator for part c)
• When assessing second order dominance of the closed-loop system, be sure to cancel poles and zeros (i.e. use minreal).