Description
1 Introduction
In this homework, you will be evaluating several Pad´e approximations, followed by model-based
design methods. Distributed parameter systems as well as feedforward/cascade controllers will
also be examined.
2 Questions
1. Pad´e Approximation
In mathematics, a Pad´e approximant is the ”best” approximation of a function by a
rational function of given order. This definition is obtained from Wikipedia article titled
Pad´e approximant. Finding the approximation requires you to first find the Taylor series
expansion of any given function around a point x0 and then equate the coefficients of Pad´e
polynomial to that of the Taylor series. Now, as an example, let’s prove the equation given
in lectures. Let m > 0 denote the order of the numerator and n ≥ 1 denote the order of
the denominator of the approximant R[m/n] (that is, R[m/n] =
Pm
j=0 aj s
j
1+Pn
j=1 bj s
j ). For a 1 by 1
approximant R[1/1], using the first m + n + 1 = 3 coefficients from Taylor series expansion
around s = 0,
e
−θs = 1 − θs +
θ
2
s
2
2! − . . . ∼=
a0 + a1s
1 + b1s
a0 + a1s = (1 + b1s)(1 − θs +
θ
2
2
s
2
) + higher order terms of order greater than s
2
a0 = 1, a1 = b1 − θ, 0 =
θ
2
2
− b1θ
=⇒ b1 = θ/2 and a1 = −θ/2
(1)
(a) Calculate the R[2/2] Pad´e approximation of e
−θs around s = 0 using m+n+1 coefficients
of the Taylor series expansion around s = 0.
(b) Calculate the R[0/1] and R[0/2] Pad´e approximations of e
−θs as well.
(c) Plot the magnitude and phase responses of R[0/1], R[0/2], R[1/1] and R[2/2] for a fixed
θ = 1(sec). Compare them to those of the e
−s
. Justify which one is expected to
represent the original system better?
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(d) Plot the step responses of those five systems. Does your justification hold from the
previous step? How do the systems with non-minimum phase zeros behave arount
t = 0?
2. Model-Based Design Methods
You have learnt two model based controller design methods in your classes. Both methods
are closely related and may lead to the same controller parameters if design parameters
are specified consistently. In the first part of this problem, you are required to find the
IMC controller parameters for two systems as a verification of DS PID parameters. In the
second part, you are required to design a PID controller using the DS method and observe
the behavior of the plant under uncertainties.
(a) Derive the PI controller parameters using Internal Model Control design method for
the following plant. Assume that r = 1.
G˜
p(s) = Kp
1
τps + 1
(2)
(b) Derive the PID controller parameters using Internal Model Control design method for
the following plant. Assume that r = 1.
G˜
p(s) = Kp
1
(τ1s + 1)(τ2s + 1) (3)
(c) Derive the PID controller parameters using Internal Model Control design method for
the following plant. Assume that r = 1.
G˜
p(s) = Kp
(−βs + 1)
(τ1s + 1)(τ2s + 1), β > 0
Now, consider the following process.
G˜
p(s) = K
(10s + 1)(5s + 1) (4)
where K = 1. Find the PID controller parameters using Direct Synthesis mehod with
τc = 5 (min).
(d) Simulate the system for the perfect model.
(e) Suppose that K changes unexpectedly from 1 to 1 + α. Find the range of α for which
the closed loop system is stable.
(f) What is the limiting value of τc if you are given that |α| < 0.2?
3. Distributed Parameter Systems
Consider the following normalized partial differential equation
∂T
∂x +
∂T
∂t = 0 (5)
2
subject to the following conditions.
T(x, 0) = Te for 0 < x ≤ 1 (initial condition)
T(0, t) = V (t) + Te for t > 0 (boundary condition)
Let η(x, t) = T(x, t) − Te. Rewrite the differential equation & boundary conditions. Then
use them to find T(x,t) using Laplace transform. Note that as the DE is normalized
algebraic equations between x and t are allowed (e.g. x/t).
4. Feedforward Control
For the feedforward control structure given in Fig. 1, let Gp =
1
(2s+1)(3s+1) , Gd =
1
s+1 .
Figure 1
(a) Find the ideal Gf f in order to have the transfer function D(s)/Y (s) = 0.
(b) Discuss why the transfer function you found above is not physically realizable (an
improper transfer function is unrealizable, but why?).
(c) Approximate the numerator by a first order transfer function & find Gf f . Use the
same approach as what you do when you approximate pure time delay as e
−θps =
1 − θps +
(θps)
2
2! − . . . ∼= 1 − θps.
(d) Simulate the system for a unit step disturbance with and without the feedforward
controller. Comment on the effect of the feedforward controller.
5. Cascade Control
Consider the cascade control structure given in Fig. 2. Let Gv =
5
s+1 , Gp =
4
(4s+1)(2s+1) ,
Gc2 = Kc2 = 4, Gm1 = 0.05 and Gm2 = 0.2, where all time constants have the units of
minutes.
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Figure 2
(a) Compare the open and closed loop time constants of the inner loop.
(b) Find the proportional only gain using Ziegler-Nichols continuous cycling method.
(c) Find the proportional only gain with the same method, but this time without the
inner controller (that is, set Gm2 = 0 and Kc2 = 1).
(d) Find the steady state error values (E1) for a unit step change in the disturbance D for
both systems (with and without the inner controller).
(e) Simulate the systems to verify your results. You may assume that R = 0.
(f) Compare the two systems in terms of stability, disturbance rejection performance in
the inner loop and speed.
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