ME 578
Quiz 1
Spring 2011
Review of Control Systems Fundamentals+Delta Functions
January 21 2011
Problem 1.
What is the value of the following integral? What is the range of values of a and for which
it exists?
I a,
d
at
dt t e cos tdt
Solution
t h t
ME 578 Digital Control
Quiz 4 Spring 2015
Sample and Hold
16 February 2015
a. Show how the zero order hold and first order hold are implementable as
FIR filters.
b. Can a second order hold be implemented via an FIR filter? Why and why
not?
1
ME 578 Digital Control
Quiz 6 Spring 2015
Stability
4 March 2015
State if the differential equation
&
x = x 2 , x ( 0 ) =x0
has equilibria. Are the equilibria stable? If so, under what conditions? If not, does
it even have solutions?
1
Linear Quadratic (LQ) Optimal Control
Continuous-time LQ Regulation (LQR) and Riccati Equation
Infinite Horizon Solution (ARE)
Discrete-time LQ control
Principle of Optimality
Discrete-time RE and DARE
Properties of LQR
Robustness
Pole locations (root l
Design of Discrete Time Controller State Space Approaches
(Full) State Feedback
Pole (eigenvalue) placement, when can it be done?
Ackermanns Formula
Deadbeat (Finite settling time) control
Other issues (input scaling, observability)
Comparison with ou
Chapter 10 Linear Quadratic Gaussian Control
Discrete-time random process
Linear system with stochastic input
Discrete-time Kalman filter
Kartik B. Ariyur 2011
1
Discrete-time Random Process
A random process can be considered as a function x(, k) of
two
ME578 Digital Control
Spring 2011
MWF 10:30 11:20
Kartik B. Ariyur
Office: ME187
E-mail: kariyur@purdue.edu
Phone: (765) 494-8613
Office hours: Friday 14pm
Introduction and Motivation
About this course
Course objectives
Major course content
Project and
ME578 Digital Control
Spring 2009
Linear Quadratic Gaussian (LQG) Problem
System
Linear time invariant system subjected to random input disturbance
w(k), random measurement noise v(k), and uncertain random initial
condition (state) x(0):
x(k 1) A x(k ) B
Design of Discrete Time Controller Polynomial Approach
Introduction
Assumptions
Problem statement
TDOF design (not just feedback)
Polynomial design approach The Diophantine equation
Pole placement (only works on poles)
Model matching (achieve desired p
The z-Transform and Difference Equations
The z-Transform
Definition
Properties
Inverse z-Transform
Solving Linear Difference Equations Using z-Transform
Pulse Transfer Function
Impulse Response Sequence
Frequency Response of Discrete-Time Systems
Kar
Sampled Data Analysis
Issues with sampling
Sampling theory
Impulse sampling
Frequency domain interpretation
Shannons sampling theorem
Reconstruction
Ideal (non-causal) hold
Zero-order-hold
Aliasing
Kartik B. Ariyur 2011
1
What is Sampling?
A continuou
Spring 2011
Quiz 2
ME 578
Quantization, NyquistShannon Theorem, SampEe and Hoid
E? MW
3
x
:
(gix ~94" 25: EMA it.
a
January 28 2011
mmgi -~ EEQ)
Write the frequency respunse 0f the rst mder hem
fiE : g?)
Preblem 1.
def
cfw_La
mi
Luau,
hiv
F617: Spring 201
ME 578 Quiz 5 Spring 20:
State space representations
February 18 2011
Problem 1. Write the transfer function
2+1
G(z) =
22+z+1
In the controllable and observable canonical forms.
gait/than i. z: 715%): a: nm%:jm M %1M1%<%4&
U gang, 2;an
_ / l
w \ [0 l [
ME 578 Quiz 4 Spring 2011
Pulse Transfer Functions
February 4 2011
Probiem 1. Derive the ZDH equivaient transfer functiam of the continueus time system
4e2.1S
GP (S) =
S +1
~ (:3 '1
Sampling is performed at 100Hz. my 7% *- @
T3. way :3 T$+Tg '3? TLZZ
Quiz 6
Stabihty: Part 1
February 25 2011
Probiem 1. Find the equilibria (3'? the differentiai equation
Spring 2011
X I X X2
and check for their stability.
'2 _ Cf)
on Ex, , W m.
guhhm : quugkh W (x "m
\C a ) *zg C21 cfw_it 9;
Wg/akk
ME 578 Digital Control
Quiz 5 Spring 2015
Representations
24 February 2015
Consider the state-space system
&
x =Ax +Bu
y =Cx +Du
y
u
1
1
y2
u
u =2 y =
,
M
M
y
u
p
m
With n states, m inputs, and y outputs or measurements. Suppose the sampl