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Spring 2010Advanced Topics (EENG 4010-003)
Control Systems Design (EENG 5310-001)
What is a Control System?
System- a combination of components that act together and perform a certain objective
Control System- a system in which the objective is to control a process or a device or environment
Process- a progressively continuing operations/development marked by a series of gradual changes that succeed one another in a relatively fixed way and lead towards
a particular result or end.
Control Theory
Branch of systems theory (study of interactions and behavior of a complex assemblage)
Control SystemManipulated Variable(s)
Control Variable(s)
Open Loop Control System
Control System
Manipulated Variable(s)
Control Variable(s)
Closed Loop Control System
Feedback function
Classification of Systems
Classes of Systems
Lumped ParameterDistributed Parameter (Partial Differential Equations, Transmission line example)
Deterministic
Discrete TimeContinuous Time
NonlinearLinear
Time Varying
Stochastic
Constant Coefficient
Non-homogeneous Homogeneous (No External Input; system behavior depends on initial conditions)
Example Control Systems
Mechanical and Electo-mechanical (e.g. Turntable) Control Systems Thermal (e.g. Temperature) Control System Pneumatic Control System Fluid (Hydraulic) Control Systems Complex Control Systems Industrial Controllers
– On-off Controllers– Proportional Controllers– Integral Controllers– Proportional-plus-Integral Controllers– Proportional-plus-Derivative Controllers– Proportional-plus-Integral-plus-Derivative Controllers
Mathematical Background
Why needed? (A system with differentials, integrals etc.)
Complex variables (Cauchy-Reimann Conditions, Euler Theorem)
Laplace Transformation– Definition– Standard Transforms – Inverse Laplace Transforms
Z-Transforms Matrix algebra
Laplace Transform
Definition Condition for Existence Laplace Transforms of exponential, step, ramp,
sinusoidal, pulse, and impulse functions Translation of and multiplication by Effect of Change of time scale Real and complex differentiations, initial and final value
theorems, real integration, product theorem Inverse Laplace Transform
dtetfsFtf st 0
)()()]([L
0|)(|0
tfe t
t
suchthat Limit
te)(tf
Inverse Laplace Transform
Definition Formula is seldom or never used; instead,
Heaviside partial fraction expansion is used. Illustration with a problem:
Initial conditions: y(0) = 1, y’(0) = 0, and
r(t) = 1, t >= 0. Find the steady state response Multiple pole case with Use the ideas to find and
dsesFj
tfsFjc
ic
st
)(2
1)()]([1
L
)(2342
2
trydt
dy
dt
yd
22
1
)( as
L
22
1
)( as
asL
3
2
)1(
32)(
s
sssF
Applications
Spring-mass-damper- Coulomb and viscous damper cases
RLC circuit, and concept of analogous variables Solution of spring-mass-damper (viscous case)DC motor- Field current and armature current
controlled cases Block diagrams of the above DC-motor problems Feedback System Transfer
functions and Signal flow graphs
Block Diagram Reduction
Combining blocks in a cascadeMoving a summing point ahead of a blockMoving summing point behind a blockMoving splitting point ahead of a blockMoving splitting point behind a block Elimination of a feedback loop
G1 G2 G3 G4
H2
H1
H3
R(s)Y(s)
+ +
+ -
-+
Signal Flow Graphs
Mason’s Gain Formula
Solve these two equations and generalize to
get Mason’s Gain Formula
r1
r2
x1
x2
a21 a12
a22
a11
11212111 xrxaxa 22222121 xrxaxa
k
ijkijk
ij
FG
G1 G2 G3 G4
G5G6 G7 G8
H2 H3
H7H8
R(s) Y(s)
Find Y(s)/R(s) using the formula
Another Signal Flow Graph Problem
R(s) C(s)
1 G1 G2 G3 G4 G5 G6
G8G7
-H4-H1
-H2
-H3
Homework Problem
G1(s) G2(s)
H1(s)
G4(s)
G5(s) G6(s)
H2(s)
G3(s)
+- +
+
++
++
R1(s)
R2(s)
X1(s)
X2(s)
Control System Stability: Routh-Hurwitz Criterion
Why poles need to be in Left Hand PlaneNecessary condition involving Characteristic
Equation (Polynomial) Coefficients Proof that the above condition is not sufficient
Ex: s3+s2+2s+8.Routh-Hurwitz Criterion- Necessary & Sufficient
Routh-Hurwitz Criterion: Some Typical Problems
2nd and 3rd order systems q(s)=s5+2s4+2s3+4s2+11s+10 (first element of a row 0;
other elements are not) q(s)=s4+s3+s2+s+K (Similar to above case) q(s)=s3+2s2+4s+K (for k = 8, first element of a row 0; so
are other elements of the row) q(s)= s5+4s4+8s3+8s2+7s+4 (Use auxiliary eqn.) q(s)= s5+s4+2s3+2s2+s+1: Repeated roots on imaginary
axis; Marginally stable case
Root-Locus Method: What and Why?
Plotting the trajectories of the poles of a closed loop control system with free parameter variations
Useful in the design for stability with out sacrificing much on performance
Closed Loop Transfer Function
Let Open Loop Gain Roots of the closed loop characteristic
equation depend on K.
G(s)
H(s)
Y(s)R(s)+ -
)()(1
)(
)(
)(
sHsG
sG
sR
sY
n
jj
m
ii
n
m
ps
zsK
pspsps
zszszsKsHsG
1
1
21
21
)(
)(.
))...()((
))...()(()()(
0)()(1 sHsG
Relationship between closed loop poles and open loop gain
When K=0, closed loop poles match open loop poles
When closed loop poles match open loop zeros.
Hence we can say, the closed loop poles start at open loop poles and approach closed loop zeros as K increases and thus form trajectories.
,K
Mathematical Preliminaries of Root Locus Method
Complex numbers can be
expressed as (absolute value,
angle) pairs.Now,
The loci of closed loop poles can
be determined using the above
constraints (particularly, the
angle constraint) on G(s)H(s).
s=+j
|s|
s=|s|.ej
s
s=+j
-s1=-j
|s+s1|
s+s1=|s+s1|ej
1)()(0)()(1 sHsGsHsG
)12(180)()(&1|)()(| 0 ksHsGsHsG
Root Locus Method- Step1 thru 3 of a 7-Step Procedure
Step-1: Locate poles and zeros of G(s)H(s).
Step 2: Determine Root Locus on the real-axis using angle constraint. Value of K at any particular test point s can be calculated using the magnitude constraint.
Step 3: Find asymptotes by using angle constraint in
. Find asymptote centroid
. This formula may be
obtained by setting
Illustrative Problem:
1)(;)2)(1(
)(
sHsss
KsG
)()(0
SHsGLimits
mn
zpn
j
m
i ij
A
1 1
)()(
0-1-2
3j
3j
A
n
jj
m
ii
mnA ps
zsKsHsG
s
K
1
1
)(
)(1)()(1
)(1
Root Locus Method- Step 4
Step 4: Determine breakaway points (points where two or more loci coincide giving multiple roots and then deviate). Now, from the characteristic equation
where , we get, at a multiple pole s1, , because at s1,
. Thus we get at s1,
Since at s = s1. Thus, we get break points by setting dK/ds=0. In the
example, we get s = -0.4226 or -1.5774 (invalid).
0)(
)(.1)()(1
sP
sZKsHsG0)(.)()( sZKsPsf
0| 1ssds
df
))...(()()( 21 nr sssssssf
0)()('
)(')()(;
)('
)(' sZ
sZ
sPsPsf
sZ
sPK 0)().(')(').( sZsPsZsP
0)(
)().(')(')(,)(
)(2
sZ
sZsPsZsP
ds
dK
sZ
sPK
Root Locus Method- Step 5
Step 5: Determine the points (if any) where the root loci cross the imaginary axis using Roth-Hurwitz Stability Criterion.
Illustration with the Example Problem
Characteristic equation for the problem:s3+3s2+2s+KFrom the array, we know that the system
is marginally stable at K=6. Now, we can
get the value of (imaginary axis
crossing) either by solving the second row
3s2+6 =0 or the original equation with s=j.
S3 1 2
S2 3 K
S1 (6-K)/3
S0 K
Root Locus Method- Step 6 and 7
Step 6: Determine angles of departure at complex poles and arrival at complex zeros using angle criterion.
Step 7: Choose a test point in the broad neighborhood of imaginary axis and origin and check whether sum of the angles is an odd multiple of +180 or -180. If it does not satisfy, select another one. Continue the process till sufficient number of test points satisfying angle condition are located. Draw the root loci using information from steps 1-5.
Root Locus approach to Control System Design
Effect of Addition of Poles to Open Loop Function: Pulls the root locus right; lowers system’s stability and slows down the settling of response.
Effect of Addition of Zeros to Open Loop Function: Pulls the Root Locus to Left; improves system stability and speeds up the settling of response
j
x
j
xx
j
x x x
j
xxxo
j
xxx o
j
xxx o
Performance Criteria Used In Design
We consider 2nd order systems here, because higher order systems with 2 dominant poles can be approximated to 2nd order systems e.g.
when
For 2nd order system
For unit step input
Where . Two types of performance criteria (Transient and Steady State) Stability is a validity criterion (Non-negotiable).
))(2(
1)( 22 assssT
nn
||10|| na
)(2
)( 22
2
sRss
sYnn
n
)1sin(1
11)(;
1)( 2
2
tety
ssR n
tn
1cos
Transient Performance Criteria
t
y(t)1.0
TR
ess
overshoot
TP
TS
)1sin(1
11)( 2
2
tety n
tn
•Rise Time TR= Time to reach Value 1.0
•Rise Time Tr1= Time from 0.1 to 0.9
Empirical Formula is
for• Settling time (Time to settle to within 98% of 1.0)=4/n
• Peak Time
Percentage Overshoot =
nrT
6.016.21
2.0
0.6
nTr1
8.03.0
21
n
PT
21/100 e
Series Compensators for Improved Design
RC OP-Amp Circuit for phase lead (or lag) compensator
Lead Compensator for Improved Transient Response; Example: Required to
reduce rise time to half keeping = 0.5. Lag Compensator for Improved steady-state
performance. Example:
)2(
4)(
sssG
)2)(1(
06.1)(
ssssG
)3386.2)(5864.03307.0)(5864.03307.0(
06.1
06.1)2)(1(
06.1
)(
)(
sjsjsssssR
sC
Frequency Response Analysis
Response to x(t) = X sin(t)G(s) = K/(Ts+1) and G(s)=(s+1/T1)/(s+1/T2) cases Frequency response graphs- Bode, and Nyquist
plots of Resonant frequency and peak valueNichols ChartNyquist Stability Criterion
1
])/()/(21[,)1(,)(, 211
nn jjTjjK
Control System Design Using Frequency Response Analysis
Lead Compensation Lag Compensation Lag-Lead Compensation
State Space Analysis
State-Space Representation of a Generic Transfer Function in Canonical Forms:– Controllable Canonical Form– Observable Canonical Form– Diagonal Canonical Form– Jordan Canonical Form
Eigenvalue Analysis
Solution of State Equations
Solution of Homogeneous Equations Interpretation of Show that the state transition matrix is
given by Properties of Solution of Nonhomogeneous EquationsCayley-Hamilton Theorem
Axx
11 )()( AIA set t L
teA
)(t
Controllability and Observability
Definitions of Controllable and Observable Systems
Controllabililty and Obervability Conditions Principle of Duality
Control System Design in State Space
Necessary and Sufficient Condition for Arbitrary Pole Placement
Determination of Feedback Gain Matrix by Ackerman’s formula
Design of Servo Systems
Introduction to Sampled Data Control Systems
Z-transform and Inverse Z-transform Properties of Z-Transform and Comparison
with the Corresponding Laplace Transform Properties
Transfer Functions of Discrete Data Systems
Analysis of Sampled Data Systems
Input and Output Response of Sampled Data Systems
Differences in the Transient Characteristics of Continuous Data Systems and Corresponding Discrete (Sampled) Data Systems
Root Locus Analysis of Sampled Data Systems