T H I R T E E N
Digital Control Systems
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: Transient Design via Gain
a. From the answer to the antenna control challenge in Chapter 5, the equivalent forward transfer function found by neglecting the dyna
Root Locus : Design
Design Example :
The differential equation of a DC motor is given by
Motor speed
Applied voltage
Determining the parameters and inserting their actual
values we have
The transfer function representing the system is then
Which correspon
Nyquist Plots / Nyquist Stability Criterion
Given
Nyquist plot is a polar plot for
vs
using the Nyquist contour (K=1 is assumed)
Applying the Nyquist criterion to the Nyquist plot we can
determine the stability of the closedloop system.
Nyquist Criterion
Lead Compensator
Design Example :
For the system with the following block diagram
representation
Find
so that the dominant closed loop poles are at
Solution : Start with
Compensator design with
is not sufficient
However if the compensator is in the form
w
Time Response
After the engineer obtains a mathematical representation
of a subsystem, the subsystem is analyzed for its transient
and steadystate responses to see if these characteristics
yield the desired behavior.
This section is devoted to the analysi
Modeling
This lecture we will consentrate on how to do system
modeling based on two commonly used techniques
In frequency domain using Transfer Function (TF)
representation
In time domain via using State Space representation
Transition between the TF to S
Frequency Response Analysis
Consider
let the input be in the form
Assume that the system is stable and the steady state
response of the system to a sinusoidal inputdoes not
depend on the initial conditions
We have
PFE
yields
Complex
conjugate of
Inverse L
Stability
This lecture we will concentrate on
How to determine the stability of a system represented as
a transfer function
How to determine the stability of a system represented in
statespace
How to determine system parameters to yield stability
Definit
Block Diagram Representation
This lecture we will concentrate on
Representing system components with block diagrams
Analyze and design transient response for systems
consisting of multiple subsystems
Reduce a block diagram of multiple systems to a single
Root Locus
This lecture we will learn
What is root locus
How to sketch rootlocus
How to determine the closed loop poles via root locus
How to use root locus to describe the transient
response, and stability of a system as a system
parameter is varied
Roo
T H R E E
Modeling in the Time Domain
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: StateSpace Representation
. Ea(s) 150 For the power amplifier, V (s) = s+150 . Taking the inverse Laplace transform, ea +150ea = p 150vp. Thus, the state equation
F O U R
Time Response
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: OpenLoop Response
The forward transfer function for angular velocity is, 0(s) 24 G(s) = V (s) = (s+150)(s+1.32) P a. 0(t) = A + Be150t + Ce1.32t 24 b. G(s) = 2 . Therefore, 2n
F I V E
Reduction of Multiple Subsystems
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: Designing a ClosedLoop Response
a. Drawing the block diagram of the system:
Pots
Pre amp
Power amp
Motor, load and gears
ui +
10
K
150 s+150

0.16 s (s+1.32)
T W O
Modeling in the Frequency Domain
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: Transfer Functions
Finding each transfer function: Vi(s) 10 = ; i(s) Vp(s) PreAmp: V (s) = K; i Ea(s) 150 Power Amp: V (s) = s+150 p Pot: 50 Motor: Jm = 0.05 + 5
T W E L V E
Design via State Space
SOLUTION TO CASE STUDY CHALLENGE
Antenna Control: Design of Controller and Observer
a. We first draw the signalflow diagram of the plant using the physical variables of the system as state variables.
Writing the state e
O N E
Introduction
ANSWERS TO REVIEW QUESTIONS
1. Guided missiles, automatic gain control in radio receivers, satellite tracking antenna 2. Yes  power gain, remote control, parameter conversion; No  Expense, complexity 3. Motor, low pass filter, inertia
E L E V E N
Design via Frequency Response
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: Gain Design
a. The required phase margin for 25% overshoot ( = 0.404), found from Eq. (10.73), is 43.49o. 50.88K From the solution to the Case Study Challenge
T E N
Frequency Response Techniques
SOLUTION TO CASE STUDY CHALLENGE
Antenna Control: Stability Design and Transient Performance
First find the forward transfer function, G(s). Pot: K1 = Preamp: K Power amp: 100 G1(s) = s(s+100) Motor and load: Kt 1 1 1 J
E I G H T
Root Locus Techniques
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: Transient Design via Gain
a. From the Chapter 5 Case Study Challenge: 76.39K G(s) = s(s+150)(s+1.32) 1 Since Ts = 8 seconds, we search along  2 , the real part of poles
N I N E
Design via Root Locus
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: LagLead Compensation
76.39K a. Uncompensated: From the Chapter 8 Case Study Challenge, G(s) = s(s+150)(s+1.32) = 7194.23 1 6.9 s(s+150)(s+1.32) with the dominant poles a
S E V E N
SteadyState Errors
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: SteadyState Error Design via Gain
76.39K a. G(s) = s(s+150)(s+1.32) . System is Type 1. Step input: e() = 0; Ramp input: 1 2.59 = 76.39K = K ; Parabolic input: e() = . 15
S I X
Stability
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: Stability Design via Gain
From the antenna control challenge of Chapter 5, 76.39K T(s) = 3 s +151.32s2+198s+76.39K Make a Routh table: s3 s2 s1 s0 1 151.32 29961.3676.39K 151.32 76.39K
ELM 322, Control Systems
Control Systems
Spring 2015
Dr. Erkan Zergerolu
202 Computer Engineering Department
Email: [email protected]
CLASS TIME: See the Schedule
OFFICE HOURS: To be announced.
Introduction to Control Systems
A control system is