the starred transform of
is not a rational function of s and this makes
is a rational function of
is a complex variable.
The z- transform of
Sample and Hold Page 1
Table of Laplace and
DC gain or steady-state gain is defined as:
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 steady-state.
If it dosen't happen, that means the system
Recall that a continuous time (CT) LDS is given by
- 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:
is the sampling period
Nyquist Stability Criterion
Frequency Stability Tests
We are looking for tests on the loop transfer function
establish stability of the closed-loop system
that can be performed to
Easy to determine using a root locus.
How do this in the frequency domai
The Nyquist Criterion
be the ratio of two polynomials in
Let the closed curve in the
through the mapping
-plane be mapped into the complex plane
is analytic within and on , except at a finite number of poles, and if
The Pulse Transfer Function
Consider the linear open-loop system with continuous time input
Assume a sampled signal is applied to the input terminals of a linear system.
Express the digital system in terms of z-transforms
The modified z-transform is a tool for:
Studying a systems output between samples
Considering the effect of computer processing delays
Examining multi-rate or non-synchronous sampling effects
The Delayed z-Transform
so that the following set of performance specifications are met :
Ramp error constant
1) First design
, then using
, we have,
Uncompensated system is only marginally stable and
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:
Response Analysis Page 1
First Order Sys
Discrete-Time Signals and Systems
Linear & Shift Invariant (LSI)
DT Systems Page 1
DT Systems Page 2
DT Systems Page 3
(unit sample response)
EE 543aL Digital Control Systems
Consider the system shown below.
(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
Consider the Fourier transform of the sequence
The above Fourier transform exists if
Equivalently for LTI systems:
The LTI system is stable iff
z-Transform Page 1
Does the Fourier transf
Digital PID Controller
P.I.D : Proportional-Integral-Derivative
There is a number of ways to implement continuous PID digitally. (See your
Digital ControllerDesign Page 1
Digital ControllerDesign Page 2
Digital ControllerDesign Page 3
Digital Control Systems
Sampled-Data Transformation of Analog filters
Need to transform analog control designs to digital ones for digital implementation.
Sampled-data transformations are techniques used to obtain numerical solutions to
integral and dif
is periodic in s with period
has a pole at
Sample and Hold Page 1
has poles at
Sampling theorem: If
is the highest
frequency component present in the continuous-time signal
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
Consider the following closed-loop system:
The closed-loop pulse transfer function is:
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
Discrete-Time State Space Equations
Discrete-Time Dynamical System
Consider the discrete-time equation:
Now consider the system:
be the delay or shift operator, i.e.,
DT State Spate Page 1
DT State Spate Page 2
Discrete-Time State Sp
Discrete Fourier Transform
For LSI (LTI) systems, the frequency response of the system is the Fourier
transform of the unit-sample response.
[Side Note: Based on the properties of the frequency response of LTI system, it
has a Fourier series.]
Closed-loop Transfer Functions
Consider the following continuous time feedback system:
The plant model can be written as:
And hence the closed-loop response is
Loop transfer function:
Complementary Sensitivity funct
Pole-Assignment Design and State Estimation
Let us consider a digital control design using a state representation model for the servo
A continuous state variable model is
is the position of the shaft motor.
is the shaft velocity whi
Closed-loop Pulse Transfer Function
Consider the following closed-loop sampled-data system:
Closed-loop Pulse Transfer Function (PTF)
To find the closed-loop PTF, similar to continuous time systems, write the
block diagram equations.
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.