ECE 416 Spring 2015
Homework # 2
Due Thursday, January 29, 2015
1. For each of the following cases,
(i) compute by hand and table look-up the discrete transfer function Gd (z) when the
given continuous transfer function G(s) is preceded by ZOH;
(ii) Verif

Sampling Theory
Spectrum of a Sampled signal: Let f (t) be a continuous
signal and F () be its Fourier Transform
Z
F () = F [f (t)] =
f (t)ejt dt
f (t) =
f (t)
T
F () = F [f (t)] =
Z
P
X
k= f (kT )(t
kT )
f (kT )(t kT )ejt dt
k=
p. 1/
F () =
F () =
Z

> % Example of Calculation of Closed-loop Poles
> a = 1; Ti = 0.1; T = 0.01; K = 1;
> G = tf(a,[1 a]); Gd = c2d(G,T,'zoh')
Gd =
0.00995
-z - 0.99
Sample time: 0.01 seconds
Discrete-time transfer function.
> C = K*(1+tf(T/Ti,[1 -1],T)
C=
z - 0.9
-z-1
Sampl

ECE 416 Spring 2014
Homework # 1
Due Thursday, January 16, 2014
1. Consider the electric circuit in Figure 1 with input v and output i2 .
(a) Let vC1 , vC2 , vL1 , and vL2 be the voltages across the capacitors and inductors
(with positive polarity on the

Lab 12: Digital Control 2 Design by Emulation
1. Laboratory Objectives
The objectives of this lab are to
a) build a LabView data acquisition system to implement a phase-lead digital
position controller of a DC motor, designed by emulation of an analog con

Lab 10 Inverted Pendulum
1. Laboratory Objectives
The objectives of this lab are to
(1) estimate the pendulum moment of inertia experimentally;
(2) design and implement an LQR controller for balancing the inverted pendulum at
the upward equilibrium positi

> % Calculation of the transfer function of a state model
> A = [1 1 0;0 0 1;0 -1 -1], B = [1;0;2], C = [1 0 1]
A=
1 1 0
0 0 1
0 -1 -1
B=
1
0
2
C=
1 0 1
> % First, using the symbolic math toolbox
> s = sym('s'); G = C*inv(s*eye(3)-A)*B
G=
1/(s - 1) + 2/(s

ECE 416 Spring 2014
Homework # 7
Due Thursday, March 13, 2014
1. Consider a digital control system with continuous plant
G(s) =
1
s+1
discrete controller
C(z) = K 0
and sampling period T .
(a) Using the Routh-Hurwitz criterion, determine the range of K, a

ECE 416 Spring 2014
Homework # 6
Due Thursday, February 20, 2014
1. For each of the following cases,
(i) compute by hand and table look-up the discrete transfer function Gd (z) when the
given continuous transfer function G(s) is preceded by ZOH;
(ii) Veri

ECE 416 Spring 2014
Homework # 5
Due Thursday, February 13, 2014
1. A discrete-time system with input u and output y is represented by the dierence
equation
y(k + 3) = 43 y(k + 2) 18 y(k + 1) + 2u(k + 3) 18 u(k + 2) 58 u(k + 1) + 81 u(k)
(a) Find the tran

ECE 416 Spring 2014
Homework # 4
Due Thursday, February 6, 2014
1. Consider the system
x 1 = x2 ,
x 2 = 2x1 + 3x2 + 2 sat(3x2 )
(a) Find all equilibrium points.
(b) Using linearization, determine the type of each equilibrium point.
(c) Construct the phase

ECE 416 Spring 2014
Homework # 3
Due Thursday, January 30, 2014
1. Consider the position control of a DC motor. The system is represented by the state
model
[
]
[ ]
[
]
0
0 1
x =
x + k u, y = 1 0 x
1
0
where x1 = , x2 = , and u is the voltage input. Let

ECE 416 Spring 2014
Homework # 2
Due Thursday, January 23, 2014
1. Consider the system
[
]
[ ]
1 1
1
x =
x+
u,
4 1
1
[
]
y = 1 2 x+u
(a) Find the transition matrix
(b) Find the transfer function.
(c) Is the system stable?
[ ]
1
(d) Find y(t) when x(0) =
a

Nonlinear Systems
x 1 = f1 (x1 , x2 , u)
x 2 = f2 (x1 , x2 , u)
Unforced (Autonomous) System:
x 1 = f1 (x1 , x2 )
x 2 = f2 (x1 , x2 )
Equilibrium Points: A point x = (x1 , x2 ) is said to be an
equilibrium point if
x(t0 ) = x x(t) x , t t0
Equilibrium poi

ECE 416 Laboratory
Experiment # 7 VTOL I
1
Preliminaries
1.1
Laboratory Objectives
1. Study the dynamics of the Vertical Take-Off and Landing (VTOL) trainer
2. Introduce the idea of minor-loop feedback control to separate the dynamics of the plant
from th

Lab 9 Gantry Control
(Rotary Pendulum Gantry Trainer)
1. Laboratory Objectives
In industry, the crane is often used to transport items from one place to another. The
gantry experiment involves developing a control system for a crane traveling on a
moving

Lab 11: Digital Control 1 State Feedback
1. Laboratory Objectives
The objectives of this lab are to
a) build a Labview data-acquisition system to implement state-feedback digital
position control of a DC motor;
b) investigate the effect of the controller

Lab 13: Digital Control 3 Direct Digital Design
1. Laboratory Objectives
The objectives of this lab are to
a) build a LabView data acquisition system to implement a phase-lead digital
position controller of a DC motor, designed by direct design in the z -

ECE 416 Laboratory
Experiment # 1 DC Motor I
1
Preliminaries
1.1
Laboratory Objectives
1. Identify the parameters of a DC motor.
2. Design a PI controller to regulate the speed of the motor. The controller parameters are
chosen by (a) online tuning; (b) c

Before you start
1. ELVIS
National Instruments ELVIS workstation is the hardware for building and testing
customized circuit boards. It provides interfaces to plug in a prototype board, as well as
other instruments, as shown in Figure 1.
Figure 1: Front

ECE 416 Laboratory
Experiment # 3 HVAC I
1
Preliminaries
1.1
Laboratory Objectives
1. Learn the principles of a simple heating relay control to regulate the chamber temperature
2. Develop a simple model of the heating process
1.2
Hardware Components
Gene

Lab 6: LABVIEW Programming for Data Acquisition
1. Laboratory Objectives
In previous labs you used front panels to run the experiments and perform input and output data
acquisition. The objective of this experiment is to learn how to use LabView tools to

ECE 416 Laboratory
Experiment # 4 HVAC II
1
Preliminaries
1.1
Laboratory Objectives
1. Study the behavior of the proportional-integral (PI) controller for the heating plant
2. Vary PI gains heuristically to control temperature
3. See the effect of anti-wi

ECE 416 Laboratory
Experiment # 2 DC Motor II
1
Preliminaries
1.1
Laboratory Objectives
Study
1. The effect of Set-Point Weight
2. Filtering of computed speed
3. Integrator Anti-Windup
4. Speed tracking of triangular waveforms
5. Position control in the p

ECE 416 Laboratory
Experiment # 8 VTOL II
1
Preliminaries
1.1
Laboratory Objectives
1. Study feedback control of the VTOL trainer using a PID controller
2. Study the steady-state error with and without integral action
3. Design the PID controller gains
4.

Lab 5: LABVIEW Simulation for Control System
1. Laboratory Objectives
The objective of this experiment is to simulate a control system using LABVIEW
simulation module in schematic programming language.
2. Background Material
LabVIEW Simulation Module is a