The Transform
Consider
the starred transform of
.
is not a rational function of s and this makes
Let
1
quite difficult.
, then,
Note:
is a rational function of
is a complex variable.
The z transform of
Sample and Hold Page 1
.
Table of Laplace and
Tr
Dc Gain
DC gain or steadystate gain is defined as:
Notes:
We want the output to follow the input. For example, if we excite the system
with a unit input, we want the output to be unity (one) at steadystate.
If it dosen't happen, that means the system
ContinuousTime Systems
Recall that a continuous time (CT) LDS is given by
Remarks:
 The first equation is called the state equation.
 The second equation is called the output equation.

is system, or dynamic, matrix.
is input matrix.
is output matrix.
Sample and hold
A sampler is a device which converts an analog signal into a train of
amplitude modulated pulses.
A hold device maintains the value of a pulse signal.
Sample and Hold Operation:
Simple Circuit:
is the sampling period
Sampling duration
Nyquist Stability Criterion
Frequency Stability Tests
We are looking for tests on the loop transfer function
establish stability of the closedloop system
that can be performed to
Easy to determine using a root locus.
How do this in the frequency domai
The Nyquist Criterion
Basic theorem:
Let
be the ratio of two polynomials in
Let the closed curve in the
through the mapping
.
If
.
plane be mapped into the complex plane
is analytic within and on , except at a finite number of poles, and if
has neithe
The Pulse Transfer Function
Consider the linear openloop system with continuous time input
:
Assume a sampled signal is applied to the input terminals of a linear system.
Where
Objective:
Express the digital system in terms of ztransforms
Method:
Given
Modified Ztransform:
The modified ztransform is a tool for:
Studying a systems output between samples
Considering the effect of computer processing delays
Examining multirate or nonsynchronous sampling effects
The Delayed zTransform
In general
T
Example
Let
(selected)
Design
so that the following set of performance specifications are met :
Ramp error constant
PM
Resonance peak
Solution:
1) First design
Choose
, then using
, we have,
Uncompensated system is only marginally stable and
Digital Co
Continuous Time Systems  Review
A short review over the transient response of first order and second
order continuous linear dynamical systems.
We want the response of the system to be:
Fast
Smooth
Accurate
Robust
Response Analysis Page 1
First Order Sys
DiscreteTime Signals and Systems
General System:
Or simply:
Special Class:
Linear & Shift Invariant (LSI)
DT Systems Page 1
DT Systems Page 2
Linearity
Properties:
If
and
Then:
In general:
DT Systems Page 3
ShiftInvariance
(unit sample response)
Put
EE 543aL Digital Control Systems
Side Note:
Example
Consider the system shown below.
Where
() =
(2 + 1)
( + 1)(0.2 + 1)
() = ()
Design a digital controller using a Bode diagram in the plane such that the phase margin is 50 and the
gain margin is at least
Introduction
Consider the Fourier transform of the sequence
The above Fourier transform exists if
Therefore
converges if
Equivalently for LTI systems:
The LTI system is stable iff
converges.
zTransform Page 1
converges.
Example
Does the Fourier transf
Digital PID Controller
P.I.D : ProportionalIntegralDerivative
There is a number of ways to implement continuous PID digitally. (See your
textbook)
Digital ControllerDesign Page 1
Digital ControllerDesign Page 2
Digital ControllerDesign Page 3
Digital Control Systems
SampledData Transformation of Analog filters

Need to transform analog control designs to digital ones for digital implementation.
Sampleddata transformations are techniques used to obtain numerical solutions to
integral and dif
Properties of
1)
is periodic in s with period
2) If
has a pole at
, then
Sample and Hold Page 1
.
has poles at
,
Sampling Theorem
Example:
Sampling theorem: If
satisfies
, where
is the highest
frequency component present in the continuoustime signal
, th
Stability
Continuous Time Systems
If a system is not stable, it may burn out, disintegrate, or saturate
when a signal no matter how small is applied.
In this chapter, we will discuss two types of stability:
1. External Stability
2. Internal stability
St
Root Locus
Consider the following closedloop system:
The closedloop pulse transfer function is:
Characteristic equation:
Root locus
Plot of the locus of roots of
as a function of .
Note: Basic rules for constructing root locus are the same as in the con
DiscreteTime State Space Equations
DiscreteTime Dynamical System
Consider the discretetime equation:
Let
i.e.,
Now consider the system:
Let
be the delay or shift operator, i.e.,
DT State Spate Page 1
Then
DT State Spate Page 2
DiscreteTime State Sp
Discrete Fourier Transform
For LSI (LTI) systems, the frequency response of the system is the Fourier
transform of the unitsample response.
[Side Note: Based on the properties of the frequency response of LTI system, it
has a Fourier series.]
Therefore,
Closedloop Transfer Functions
Consider the following continuous time feedback system:
The plant model can be written as:
Therefore,
And hence the closedloop response is
Where,
Loop transfer function:
Sensitivity function:
Complementary Sensitivity funct
PoleAssignment Design and State Estimation
Let us consider a digital control design using a state representation model for the servo
system. Let
sec.
A continuous state variable model is
Note:
is the position of the shaft motor.
is the shaft velocity whi
Closedloop Pulse Transfer Function
Example:
Consider the following closedloop sampleddata system:
Closedloop Pulse Transfer Function (PTF)
To find the closedloop PTF, similar to continuous time systems, write the
block diagram equations.
Remarks:
I
Continuous Time Frequency Response
We know that a sinusoidal signal can be represented by
where M is the magnitude and
is the phase.
In the steady state, sinusoidal inputs to a linear system generates sinusoidal
responses of the same frequency.
Example