ECE509 – Analysis of Linear Systems

Final Computer Project Due: Tuesday, 27-July-2021

1. Consider Problem 1 from Project 1, where the plant was given by

x˙ =

1 0 −1−3 4 3

2 5 4

x+

1−1

1

u

y =

[

3 0 0

]

x

As was verified then, this system is controllable and observable. Here you no longer have access to the

state vector x, only to the output y. using your state feedback control design from Project 1, together with

a state observer, design a system that is able to track the same reference r used in Project 1, a train of

steps alternating between −2 and 2 with a frequency of your choice – but again, make sure the frequency

is chosen so that steady-state tracking can take place. Do the following:

(a) Describe your overall control design solution.

(b) Simulate the system with your control solution. Demonstrate your design by plotting, in one figure, the

real state and its estimate (you may also plot each pair separately in a single figure), and in another,

the real and estimated output together with the reference to be tracked. Show two full periods of your

reference.

Consider using the Simulink observer example I have provided (see Isidore) to help you do this problem.

Make sure the plant and observer have different initial conditions (of your choice), so that the

effect of the observer can be clearly seen.

Bonus (5 pts.): Why does the output always first go in the opposite direction before tracking the alternating

step signal?

2. Mechanical Press

Figure 1: A Minster Machine 45 ton mechanical press.

A simple mechanical press, such as the one shown in Figure 1 (a Minster Machine 45 ton press) can be

modeled as two rotating masses joined by a torsional spring, as depiced in Figure 2. θ1 and ω1 = θ˙1 are,

θ1, ω1 θ2, ω2k1, c1

J1 J2

Figure 2: Schematic diagram of a mechanical press.

respectively, the angle and angular speed of the press motor spindle. Similarly, ω2 = θ˙2. In a typical press

application, it is of interest to control the motor angle, θ1. The parameters of the system are as follows:

• J1 = 0.1722kg ·m2: inertia of rotating mass 1;

• J2 = 0.1392kg ·m2: inertia of rotating mass 2;

• K1 = 20491N ·m/rad: torsional spring constant; and

• c1 = 4.7521N · s/m: damping coefficient.

By choosing the states x = [θ1, ω1, θ2, ω2]

>, the press can be modeled by the state equation

x˙ =

0 1 0 0

−K1/J1 −c1/J1 K1/J1 c1/J1

0 0 0 1

K1/J2 c1/J2 −K1/J2 −c1/J2

x+

0

1/J1

0

0

u

y =

[

1 0 0 0

]

x,

and the output y is in radians. It is desired to track a reference r(t) that can be produced by using the

following code:

% Set time vector

t = [0:0.001:3];

% Set reference signal

r = (t<=0.6).*t*12/0.6 + (t>0.6 & t<=1)*12 +...

(t>1 & t<=2).*(18-6*t) +...

(t>2 & t<=2.5)*6 + (t>2.5 & t<=3)*12;

figure(1), plot(t, r , ’LineWidth’, 2), grid

xlabel(’Time (s)’), ylabel(’Reference (rad}’)

title(’Desired reference position’)

(a) Set the initial conditions as follows: x(0) = [10, 0, 0, 0]>, and xˆ(0) = 0. Let the tracking error be

defined as e(t) = r(t) − y(t). Design a state feedback controller together with a state observer that

can track the given reference while satisfying the following design specifications:

i. |e(t)| ≤ 10rad for 0 ≤ t ≤ 3s.

ii. |e(t)| ≤ 0.3rad for t = 0.6s.

iii. The settling time (98% criterion) for the step at t = 2.5s should be less than 0.05s.

A good way to solve such a problem is to first design the state feedback, temporarily assuming that

the state vector x is available, and meet the requirements. Then, use xˆ from the observer instead of

x and design the observer to still meet the requirements.

Provide enough support via plots to show that your design meets all the given specifications.

(b) Design a PID controller that meets the same design specifications as those in part (a).

(c) Discuss and compare the state feedback design and the PID design. Provide insight into their respective

performance limitations, advantages and disadvantages.

General notes and hints:

• Your report should be well presented and typed. Your report should be in PDF format. No other format

will be accepted, and will be returned ungraded.

• In your report, only include your result plots and any pertinent analysis, but no code. Do upload all your

code to Isidore.

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• A short and concise report is best. All you need to do is show me your design steps, and prove that your

controller works by including the requested plots.

• All figures must be carefully labeled. The x and y axes should be labeled, a title used, and a legend should

be included when more than one signal is plotted in the same figure.

• Presentation, English, and professional appearance count 10% of the total grade. A poorly presented report

will automatically be deducted 10%.

• The honor code will be strictly enforced. You may discuss the project with your classmates, but youmaynot-

share a design or provide a technical explanation that are similar (beyond pure coincidence) to those of

another student. When in doubt, talk to me!

• I will conduct short interviews with each of you on July 29 and will ask you specificquestions about your

designs and your report. The result of this interview will be a significant factor in determining your grade.

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