Part I - Design Task

You are a control engineer who has been assigned the task of developing a digital controller for the following system, where G(s) = 1/s(s+1)

(a) Design a lag compensator D(s)=K(Ts+1)/( αTs+1), α<1, such that the static velocity error constant Kv is 25 sec^-1, phase margin is 45^o, and gain margin not less than 8 dB.

If you don't know how to design the continuous compensator, set K=25 and assume some values for T and α. The phase margin can then be determined using a Matlab command, margin. Adjusting T and α by trial and error for the desired phase margin of over 45^o the lag compensator can be determined. You can also use Matlab program sisotool for the design.

Using emulation method to re-design a digital compensator which maintain the phase margin of the original compensator.

(b) Find the step response, system margin and bandwidth.

(c) Choose the sample frequency as 20*bandwidth.

(d) Analyze the effects of ZOH and AAF to the system response, redesign D(s) to maintain the phase margin of continuous time design.

(e) Approximate D(s) using bilinear method, find D(z).

(f) Compare the discrete system performance with original one with step response, and bode diagram.

(g) Compare the performance of different approximate methods for discretizing continuous compensator.

Part II - Design Task

Provide both manual solution and MATLAB code for ALL questions.

Satellites often require attitude control for proper orientation of antennas and sensors with respect to the earth. A simplified model which allows rotation only about one axis is given by θ^·· = u (double integer) where u is reference and θ is the output.

Follow the example in lectures 5 and 6 to complete the following tasks

1. Continuous time state space model.

2. Discrete time state space model (ZOH) with sample time h=0.2.

3. Discrete time controllable canonical form.

4. Discrete time transfer function.

5. Check the controllability and Observability of the system.

6. Using pole placement method to design a controller. Pick the z-plane roots of the closed loop characteristic equation so that the equivalent s-place roots have a damping ratio of ζ = 0.6 and real part of s = -2.4 rad/sec. (Hint: Refer to second order system pole position in s domain to find the s position, then z position, then controller L).

7. Design the same control using Ackermann's formula.

8. Draw the block diagram of the discrete system with the controller.

9. Compare the performance of the system with the deadbeat controller presented in the lecture. (Step response, magnitude of control signal u, bode diagram).

10. Design a full-state estimator, the poles of the estimator satisfies A_e = z²-1.2z+1.2

11. Find the transfer function of the compensator. Draw the block diagram of complete system.

12. Design a reduced order estimator with one pole at z = 0.3 for the plant.