UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME233 Advanced Control Systems IISpring 2016

Homework #5

Assigned: Apr. 6
Due:
Apr. 12(W)(Tu)

1. Figure 1 shows the feedback interconnection for a system with a disturbance observer.
When implementing the disturbance observer, we only implement the portion that
D(z)
¹U(z)

P (z)

U(z)

G(z)

+¡
¡

+

V (z)

¹U(z)

Y (z)

U(z)

Y (z)+¡¡

G¡1n (z)G¡1n (z)

Q(z)

Q(z)

^D(z)

^D(z)

Figure 1:
Structure+Figure 2: Disturbance Observer—
Controller Only

Disturbance Observer

generates U (z) from Ū (z) and Y (z), as shown in Fig. 2.
(a) Find the transfer function from Y (z) and Ū (z) to U (z) in Fig. 2.
(b) Suppose that G−1
n is proper. In this case, it is valid to choose Q(z) = α ∈ R.
Based on your answer from the previous part, note that the block diagram in Fig. 2
is not well-posed when α = 1. Does there exist α ∈ R such that the closed-looptransfer function from D(z) to P (z) is zero? If not, is there a limit to how small
we can make the transfer function from D(z) to P (z)?

12

2. Consider the feedback system in Fig. 3 where u(k) and d(k) are respectively thed(k)yd (k)r(k)T (q¡1 ; q)+¡

1R(q ¡1 )u(k)q¡d B(q¡1 )A(q ¡1 )y(k)G(q¡1)
S(q¡1 )

Figure 3: Feedback Systemcontrol and disturbance plant inputs, yd (k) is the reference model’s output, and r(k) isthe reference input to the feedback block.
The control objective is to reject the persistent deterministic disturbance d(k), place
the feedback closed-loop poles, and track the desired output yd (k).
(a) The plant transfer function G(z) is derived from a continuous time transfer function G(s) that is preceded by a zero-order hold and followed by a sampler, and isgiven by



G(s)
B̄(z)
−1−1= (1 − z ) Z L,
G(z) =sĀ(z)where
G(s) =1s(s + 1)

and the sampling time is T = 0.5 seconds.
Calculate the plant polynomials A(q −1 ) = 1 + a1 q −1 + a2 q −2 , B(q −1 ) = bo + b1 q −1
and pure delay time d.
(b) The tracking control objective is to follow the reference signal yd (k), which is theoutput of the reference model
Am (q −1 )yd (k) = q −d Bm (q −1 )ud (k) .

(1)

Select the coefficients of the second order polynomial Am (q −1 ) = 1 + am1 q −1 +am2 q −2 , so that the reference model has a natural frequency of 1 rad/sec and adamping ratio of 0.707.
Hint: Remember that, since z = esT , we can calculate the discrete time poles by
pd = epc T , where pd is the discrete time pole, pc is the continuous time pole and
T is the sampling time.
(c) Letting Bm (q −1 ) = bmo , select bmo so that the reference model has unity static
gain 1 .
1i.e. if limk→∞ ud (k) = uss then limk→∞ yd (k) = uss .

2(d) Choose the coefficients of the closed-loop system characteristic polynomial (afterpole-zero cancelation)000

Ac (q −1 ) = 1 + ac1 q −1 + ac2 q −2
so that the closed-loop feedback dynamics from r(k) to y(k) behaves as a secondorder system with a natural frequency of 2 rad/sec and a damping ratio of 0.5.
(e) Design the control system under the following specifications and assumptions:i. The closed-loop system characteristic polynomial (before pole-zero cancelation) is given by0

Ac (q −1 ) = Ac (q −1 )B s (q −1 )where
B s (q −1 ) =1
B(q −1 ),boB u (q −1 ) = bo ,and bo is the leading coefficient of B(q −1 ). This means that all of the plantzeros will be canceled by the feedback system.
ii. Assume that d(k) = 0. This means that the disturbance annihilating polynomial is selected to be Ad (q −1 ) = 1.
iii. The feedforward compensator T (q −1 , q) must be selected so that perfect tracking is achieved under a zero initial state for both the plant and the reference
model.
(f) Do a computer simulation of the control system designed in problem 2e whenyd (−1) = yd (0) = y(−1) = y(0) = 0 and
ud (k) = [us (k) − 2 us (k − 25)] + [2 us (k − 50) − 2 us (k − 75)]d(k) = 0.5us (k − 40)
where us (j) is the unit step function, i.e.
0us (j) =1(2)(3)

j<0
j≥0

Plot ud (k), yd (k), y(k) and u(k).
(g) Design the control system under the same specifications in problem 2e, exceptthat assume now that d(k) = d(k − 1).
(h) Do a computer simulation of the control system designed in problem 2g under theconditions described in problem 2f. Plot ud (k), yd (k), y(k) and u(k).
(i) Design the control system under the following specifications and assumptions:i. The closed loop system characteristic polynomial satisfies
0Ac (q −1 ) = Ac (q −1 )B s (q −1 )3whereB s (q −1 ) = 1,B u (q −1 ) = B(q −1 ) ,This means that none of the plant zeros will be canceled by the feedback
system.
ii. Assume that d(k) = d(k − 1).
iii. The feedforward compensator T (q −1 , q) is designed using the zero-phase error
tracking control approach.
(j) Do a computer simulation of the control system designed in problem 2i under theconditions described in problem 2f. Plot ud (k), yd (k), y(k) and u(k).
(k) Discuss the outcome of the simulation results. In particular• Comment on the effectiveness of the zero-phase feedforward control technique.
• Compare the control effort u(k) when the zeros are canceled vs when the zeros
are not canceled.

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