ECE5610 - Digital Control Systems  - Solved

Problem 1 (45points)
The block diagram of a rigid-body satellite is given in Figure 1. The control signal is the voltage !(#). The zero-order hold output %(#) is converted into a torque &(#) by an amplifier and the thrusters. The system output is the attitude angle Θ(#) of the satellite.

 

Figure 1: Block diagram for a satellite.

(a)    Find the transfer function G(z) = Θ(+)/-(+).

(b)   Use the results of part (a) to find the system’s unit-step response, that is, the response with !(#) = .(#). (You need to find Θ(z)   and Θ(kT))

(c)    Sketch the zero-order-hold output %(#) in (b).

Figure 2 shows a simplified diagram of the attitude control of a satellite. Note that A/D is an ideal sampler with sampling time T>0, and D/A is ZOH.

 

Figure 2: Simplified diagram of a satellite digital attitude control

(d)   Assume that 2(+) =   3 > 0    and the sampling time is T    >         0. Find the steady state error to a step input as a function of 3. (Hit: What is the type of system? Is it stable?)

(e)   Can you find a value of 3 such that the following performance specifications are satisfied?

a.      Zero steady state error to a step input

b.      No overshoot in the step response

(Hint: for checking stability you can use any methods you like!)

If so, give a suitable value for K, if not, fully justify you answer. A simple yes or no answer will not give you partial credit. 

,         Fall        2019,    Final      Exam,   100        minutes

Problem 2 (30 points)
Consider the model of a first order hold shown in Figure 3

 

Figure 3: Block Diagram for Problem 2.

(a)    Find an analytic expression for the impulse response of this system c(t). [Hint: The continuous-time transfer function of a first hold is given by:

 

Then you need to take inverse Laplace transform to obtain c(t)!]

(b)   Find the discrete transfer function E(z)/R(z). (Hint: You need to use the general procedure! Do you remember it? J 1)- find the original flow graph 2) assign variables to each sampler outputs 3) consider each sampler output as source. Find the sampler inputs and the system outputs in terms of each sampler output and the system input 4) take the starred transform)

(c)    Find the discrete transfer function C(z)/R(z). Does your result make sense (Hint: Recall that this block diagram is supposed to implement a first-order hold!) A simple yes or no answer will not give you partial credit.

Page left blank for calculations.

4

,  Fall        2022,    Final      Exam,   100        minutes

Problem 3 (20 points)
Find the values of K for which the following system is stable for 6(+) = 1/(+ − 2).

 


Page left blank for calculations.

6

,  Fall        2022,    Final      Exam,   100        minutes