## 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?

1

(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.

3

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.

4