an overview of control theory and digital signal processing
DESCRIPTION
An Overview of Control Theory and Digital Signal Processing. Luca Matone Columbia Experimental Gravity group ( GECo ) LHO Jul 18-22, 2011 LLO Aug 8-12, 2011 LIGO-G1100863. Syllabus (tentative). Objective. Control System Manages and regulates a set of variables in a system - PowerPoint PPT PresentationTRANSCRIPT
An Overview of Control Theory and Digital Signal Processing
Luca MatoneColumbia Experimental Gravity group (GECo)
LHO Jul 18-22, 2011LLO Aug 8-12, 2011
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 2
Day Topic Textbooks
1Control theory: Physical systems, models, linear systems, block diagrams, differential equations, feedback loops, cruise control example, MATLAB implementation.
Chau, Pao C. Process Control: A First Course with MATLABยฎ. Cambridge University Press, 2002. ISBN 0-521-00255-9.
Ingle, Vinay K. and John G. Proakis. Digital Signal Processing using Matlabยฎ. Brooks/Cole 2000. ISBN 0-534-37174-4.
Smith, Steven W. The Scientist and Engineerโs Guide to Digital Signal Processing. California Technical Publishing 1999. http://www.dspguide.com/
2Control theory: Laplace transform and its inverse , transfer functions, partial fraction expansion, first-order and second-order systems, dynamic response, bode plots, stability criteria, MATLAB implementation.
3Control theory: robustness, typical compensators, noise suppression, one arm cavity lock example, MATLAB implementation and time-domain simulations with SIMULINK.
4
DSP: Discrete-time signals and systems, impulse response, system stability, convolution and correlation, differential to difference equations, the Z transform, the Discrete-time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT), MATLAB implementation.
5DSP: The Fast-Fourier Transform (FFT), power spectral density, sampling theorem, aliasing, analog-to-digital transformations, digital filtering, FIR filters, IIR filters, moving average filter, filter design, ADC and DACs, MATLAB implementation.
Syllabus (tentative)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 3
ObjectiveControl Systemโข Manages and regulates a set
of variables in a systemโ SISO โ single-input-single-
outputโ MIMO โ multiple-input-
multiple-outputโข A quantity is measured then
controlledโข Requirements
โ Bandwidthโ Rise timeโ Overshootโ Steady state errorโ โฆLIGO-G1100863
4
ObjectiveDigital Signal Processingโข Measure and filter an analog signalโข Digital signal
โ Created by sampling an analog signalโ Can be stored
โข Analog filtersโ Cheap, fast and have a large dynamic
range in both amplitude and frequencyโข Digital filters
โ Can be designed and implemented โon-the-flyโ
โ Superior level of performance.โข Example: a low pass digital filter can have a
gain of , a frequency cutoff at , and a gain of less than for frequencies above . A transition of !
Matone: An Overview of Control Theory and Digital Signal Processing (1)LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 5
Control Theory 1
โข Given a physical systemโ Objective: sense and control a variable in the
systemโข Examplesโ As basic asโข a carโs cruise control (SISO) or
โ Not so basic asโข Locking the full LIGO interferometer (MIMO)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 6
Example: Cruise Control
โข Objectiveโ Car needs to maintain
a given speedโข Physical system
includesโ carโs inertiaโ friction
Direction of motion
Normal force (n)
Force from engine (f)
Weight (mg)
Friction force (ffr)
Force or free body diagram: allows to analyze the forces at play
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 7
Physical modelโข Physical system described
by the following equation of motion
โข Simplifying and assuming friction force is proportional to speed
Direction of motion
Force from engine (f)
Normal force (n)
Weight (mg)
Friction force (ffr)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 8
First-order differential equationโข Solving for first-order
differential equation (assuming is a constant)
yields the solution
Direction of motion
Engine force (f)
Friction force (ffr)
Speed at regime
Time constant
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 9
MATLAB implementationThe linear differential equation describing the dynamics of the system
Using MATLABโs Symbolic Math Toolbox
>> dsolve('m*Dy=f-b*y','y(0)=0')ans =(f - f/exp((b*t)/m))/b
Direction of motion
Engine force (f)
Friction force (ffr)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 10
Results:
๐ฃ๐=๐๐=10๐๐
๐ฃ (ฯ )=63% โ๐ฃ๐
ฯ=๐๐ =20 ๐
cruise_timedom
ain.m
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 11
Block diagram: representing the physical system
โข To illustrate a cause-and-effect relationshipโข A single block represents a physical systemโข Blocks are connected by lines โ Lines represent how signals flow in the system
โข In general, a physical system G has signal x(t) as input and signal y(t) as output
โข G is the transfer function of the system
Gx(t) y(t)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 12
Carโs body
Transfer function G represents the carโs bodyโ G converts the force from the engine (input
signal, ) to the carโs actual speed (output signal, )
with โ Units:
Gf(t) v(t)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 13
Setting the desired speedโข Second transfer function H (the controller)โ Converts the desired speed (or reference) to a
required force โ Sets the throttleโ For simplicity, H is set to a constant
โ must be dimensionless
Gf vvr H
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 14
Plotting resultsโข With the actual speed is the
reference: โข Simulate: setting desired
speed to 25 m/s (55 mph)
Generated force by controller H
Resulting speed v
Desired speed vr
cruisefeedback_timedom
ain.m
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 15
Introducing a disturbance โ a hill
โข In the presence of a hill the equation of motion needs to be re-visited
โข Assuming a small angle ฮธ
Weight (mg)
Direction of motion
Force from engine (f)
Friction force (ffr)
ฮธAdded term
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 16
Introducing a disturbance โ a hill
Weight (mg)
Direction of motion
Force from engine (f)
Friction force (ffr)
ฯ
Added term
Assuming and are constants
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 17
Modifying the block diagram
Gvfvr H
K
ฮธ
+ -
Summation junction
๐พ=๐๐
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 18
Modifying the block diagram
Gvfvr H
K
ฮธ
+ -
Summation junction
๐พ=๐๐
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 19
Plotting resultsSetting desired speed to 25 m/s and slope of
Generated force by controller H
Resulting speed v
Desired speed
Problem!
cruisefeedback_timedom
ain.m
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 20
Negative Feedback1. Letโs measure the carโs speed and2. Correct for it by feeding back into the system
a measure of the actual speed
Gvfvr H
K
ฮธ
+ -
c
e
Correction signal
Error signal
-+
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 21
Negative Feedback1. Letโs measure the carโs speed and2. Correct for it by feeding back into the system
a measure of the actual speed
Gvfvr H
K
ฮธ
+ -
c
e
Correction signal c: in this case it is just a measure of the actual speed
Error signal e: the difference between the desired speed and the measured speed. If null, then
-+
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 22
Negative feedbackโข Plot of force vs. time
and speed vs. time with negative feedback
โข Setting โข Result:โ Faster response with
feedback (compare blue against red curves)
โ Speed at regime: (error of )
cruisefeedback_timedom
ain2.m
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 23
Negative feedbackโข Increasing the
controllerโs gain (H)โ decreases the rise time โ while decreasing the
steady state errorโข Setting โข Result:โ Even faster responseโ Speed at regime: (error
of )
cruisefeedback_timedom
ain3.m
Yes butโฆ how does it work?LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 24
๐=๐ฃ๐ โ๐๐ฃ=๐บ โ( ๐ โ๐พ โ ๐)
Gvfvr H
K
ฮธ
+ -e
-
+
c
๐=๐ฃ
๐ =๐ป โ๐
Signal flow and block diagrams
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 25
๐=๐ฃ๐ โ๐
Gvfvr H
K
ฮธ
+ -e
-
+
c
ยฟ๐ฃ๐โ๐ฃยฟ๐ฃ๐โ๐บ โ ( ๐ โ๐พ โ๐ )ยฟ๐ฃ๐โ๐บ โ (๐ป โ๐โ๐พ โ๐ )
๐= 11+๐บ โ๐ป โ๐ฃ๐+
๐พ โ๐บ1+๐บ โ๐ป โ ๐
Systemโs open loop gain (dimensionless)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1)
Gvfvr H
K
ฮธ
+ -e
-
+
c
๐= 11+๐บ โ๐ป โ๐ฃ๐+
๐พ โ๐บ1+๐บ โ๐ป โ ๐
(high gain, closed loop, with feedback)
26
(low gain, open loop, or no feedback)
Error signal in closed loop: close to zero, proportional to angle
The higher the controllerโs gain, the lower e
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 27
๐ฃ=๐บ โ๐ป โ๐โ๐พ โ๐บ โ๐
Gvfvr H
K
ฮธ
+ -e
-
+
c
ยฟ๐บ โ๐ป โ(๐ฃยฟยฟ๐ โ๐ฃ)โ๐พ โG โ๐ ยฟยฟ๐บ โ๐ป โ๐ฃ๐ โ๐บ โ๐ป โ๐ฃโ๐พ โ๐บ โ๐
๐ฃ=๐บ โ๐ป1+๐บ โ๐ป โ๐ฃ๐โ(
๐พ โ๐บ1+๐บ โ๐ป ) โ๐
With no feedback๐ฃ=๐บ โ๐ป โ๐ฃ๐โ๐พ โ๐บ โ ๐
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 28
Gvfvr H
K
ฮธ
+ -e
-
+
c
๐ฃ=๐บ โ๐ป1+๐บ โ๐ป ๐ฃ๐โ
๐พ โ๐บ1+๐บ โ๐ป ๐
(high gain, closed loop, with feedback)
(low gain, open loop or no feedback)
Actual speed : close to with an error proportional to when in closed loop. The higher the controllerโs gain, the lower the speed error.
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 29
๐ =๐ป โ๐
Gvfvr H
K
ฮธ
+ -e
-
+
c
ยฟ๐ป โ( 11+๐บ โ๐ป ๐ฃ๐+
๐พ โ๐บ1+๐บ โ๐ป ๐)
๐ = ๐ป1+๐บ โ๐ป โ๐ฃ๐+
๐พ โ๐บ โ๐ป1+๐บ โ๐ป โ ๐
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 30
Gvfvr H
K
ฮธ
+ -e
-
+
c
๐ = ๐ป1+๐บ โ๐ป โ๐ฃ๐+
๐พ โ๐บ โ๐ป1+๐บ โ๐ป โ ๐
(high gain, closed loop, with feedback) (low gain, open loop, or no
feedback)
Force : at regime, it does not depend on the gain in while proportional to angle
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 31
Plotting and โข Open loop
โข Closed loop
โข Setting โข Plotting the open loop
transfer function vs. time and the closed loop transfer function vs. time
โข Notice the rapid rise time for the closed loop case
cruisefeedback_timedom
ain2.m
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 32
The error signal eโข Plot of error signal e vs
timeโข Error signal decreases
to 3 m/s.โข Notice a steady state
error
cruisefeedback_timedom
ain2.m
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 33
Open and closed loop TF with
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 34
Error signal e with
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 35
Cruise control example
โข First-order differential equation
โข Simplest controller: simply a gain with no time constants involved
โข How to handle more complicated problems?
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 36
P1x yP2P1P2 yx
P1x y
P2
ยฑ+ยฟ
P1 ยฑP2 yx
P1x y
P2
โ+ยฟ
๐11ยฑ ๐1 ๐2
yx
Block diagram reduction
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 37
Practice
Determine the output C in terms of inputs U and R.
LIGO-G1100863
U
CR ๐บ1+ +
-
+ ๐บ2
Matone: An Overview of Control Theory and Digital Signal Processing (1) 38
PracticeDetermine the output in terms of inputs and .
LIGO-G1100863
U1
CR ๐บ1 ++
+
+ ๐บ2
U2
๐ป1+
+๐ป2
Matone: An Overview of Control Theory and Digital Signal Processing (1) 39
More practice
Determine C/R for the following systems.
(a)
(b)(c)
LIGO-G1100863
R ๐บ1 ++
++
๐บ2
C
๐ป1
R ๐บ1 ++
++
๐บ2
C
๐ป1
R๐บ1 +
+++
๐บ2
C
๐ป1
Matone: An Overview of Control Theory and Digital Signal Processing (1) 40
How do we MEASURE the OL TF of a system when the loop is closed?
1. Add an injection point in a closed loop system
2. Inject signal and read signal (just before the injection) and (right after the injection)
3. Solve for the ratio ๐ฅ
๐ฆ 1 ๐ฆ 2
++
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 41
So farโฆโข Control theory builds on differential equationsโข Block diagrams help visualize the signal flow in a
physical systemโข The cause-and-effect relationship between
variables is referred to as a transfer function (TF)โข The systemโs open-loop TF is the product of
transfer functionsโ cruise control example: โ Two cases: and
โข MATLAB implementationโ Functions used: dsolve
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 42
Laplace Transforms
โข The technique of Laplace transform (and its inverse) facilitates the solution of ordinary differential equations (ODE).
โข Transformation from the time-domain to the frequency-domain.
โข Functions are complex, often described in terms of magnitude and phase
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 43
Linear systemsโข To map a model to frequency spaceโ System must be linearโ Output proportional to input
โข Given system Pโ Input signals: and โ Output signals (response): and
โข System P is linearโ If input signal: โ Then output signal: โ Superposition principle
Px y
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 44
Example
โข Is a linear system?Knowing that and If input is , output is
System is linear
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 45
Example
โข Is a linear system?Knowing that andIf input is , output is
System is not linear
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 46
In generalOutput
Input
For a stable system
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 47
Laplace Transform Lโข Transforms a linear differential equation into an
algebraic equationโข Tool in solving differential equationsโข Laplace transform of function f
โข Laplace inverse transform of function F
where is the transform variableImaginary unit 2๐ ๐
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 48
Time domain โ Laplace domain
๐ =๐๐ฆ๐๐กy (๐ก ) f (๐ก )
๐บ (๐ )๐ (๐ ) F (๐ )
Inpu
t
Out
put
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 49
Laplace Transform L
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 50
(Some) Laplace transform pairs
Unit stepUnit ramp
Exponential
Sinusoid
SHO
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 51
Laplace transform propertiesโข Linearity
โข Derivativesโ First-order: โ Second-order:
โข Integral
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 52
Solution to ODEs
1. Laplace transform the systemโs ODE2. Solve the algebraic equation in s3. Inverse transform back to the time domain
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1)
Transfer Function
โ๐=0
๐
๐๐๐ท๐ ๐ฆ (๐ก )=โ
๐=0
๐
๐๐๐ท๐ ๐ฅ (๐ก )
โ๐=0
๐
๐๐ ๐ ๐๐ (๐ )=โ
๐= 0
๐
๐๐๐ ๐ ๐ (๐ )
๐บ (๐ )=๐ (๐ )๐ (๐ )=
โ๐=0
๐
๐๐๐ ๐
โ๐=0
๐
๐๐ ๐ ๐
53
Using derivative property
Algebraic equation in s, the ratio of 2 polynomials in s
Transfer function relates input to output .
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1)
Transfer Function
๐บ (๐ )=๐ (๐ )๐ (๐ )=
โ๐=0
๐
๐๐๐ ๐
โ๐=0
๐
๐๐ ๐ ๐
54
The roots of the numerator are referred to as zeros.
The roots of the denominator are referred to as poles.
Transfer function can be defined byโข The coefficients of s orโข Its poles and zeros
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 55
Letโs apply it to the cruise control example: transfer function (the carโs body)
Direction of motion
Engine force (f)
Friction force (ffr)
G(t)f(t) v(t)
G(s)F(s) V(s)
Pole at LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 56
Dynamic response: using lookup tables to inverse transform
Laplace inverse transform using lookup tables
Input: step function, amplitude
The response (in frequency space) isThe time-domain response is
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 57
Dynamic response: using lookup tables to inverse transform
Laplace inverse transform using lookup tables
Pole LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 58
First-order system step response
๐บ (๐ )= ๐๐ +๐
63 %
Pole p
first
orde
r.m
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 59
MATLAB implementation
The step response of transfer function
>> G=tf(5, [1 5]);>> step(G);
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 60
Partial fraction expansion1. Reduce a complex function to a collection of
simpler ones2. Then use lookup table Order m
Order n
๐โค๐
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 61
Comments
1. Poles of F(s) determine the time evolution of f(t)
2. Zeros of F(s) affect coefficients3. Poles closer to origin โ larger time constants
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 62
ExampleFind of the Laplace transform
Sol: Using MATLAB
>> [R,P,K]=residue([6 0 -12],[1 1 -4 -4])R = 3.0000 1.0000 2.0000P = -2.0000 2.0000 -1.0000K = []
๐น (๐ )= 3๐ +2+
1๐ โ2 +
2๐ +1
๐ (๐ก )=3๐โ2 ๐ก+๐2 ๐ก+2๐โ ๐ก
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 63
Example: LRC circuit
๐ฃ ๐๐(๐ก) ๐ฃ๐๐ข๐ก (๐ก)
๐ฟ ๐
๐ถ
๐ฃ ๐๐ (๐ก )=๐ฃ ๐ฟ (๐ก )+๐ฃ๐ (๐ก )+๐ฃ๐ถ (๐ก )
ยฟ๐ฟ ๐๐๐ก ๐ (๐ก)+๐ ๐(๐ก)+1๐ถโซ
0
๐ก
๐ (๐ )๐๐
ยฟ๐ฟ๐ท๐ (๐ก )+๐ ๐ (๐ก )+ 1๐ถ1๐ท ๐(๐ก)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 64
Example: LRC circuit
๐ฃ ๐๐(๐ก) ๐ฃ๐๐ข๐ก (๐ก)
๐ฟ ๐
๐ถ
๐ฃ ๐๐(๐ก)=๐ฟ๐ท๐ (๐ก )+๐ ๐ (๐ก )+ 1๐ถ1๐ท ๐(๐ก )
๐ (๐ก )=๐ถ ๐๐๐ก ๐ฃ๐๐ข๐ก (๐ก )=๐ถ ๐ท๐ฃ๐๐ข๐ก (๐ก )
๐ฃ ๐๐ (๐ก )=(๐ฟ๐ถ๐ท2+๐ ๐ถ ๐ท+1 )๐ฃ๐๐ข๐ก (๐ก)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 65
Example: LRC circuit
๐ฃ ๐๐(๐ก) ๐ฃ๐๐ข๐ก (๐ก)
๐ฟ ๐
๐ถ๐ฃ ๐๐ (๐ก )=(๐ฟ๐ถ๐ท2+๐ ๐ถ ๐ท+1 )๐ฃ๐๐ข๐ก (๐ก)
๐ ๐๐ (๐ )= (๐ฟ๐ถ๐ 2+๐ ๐ถ๐ +1 )๐ ๐๐ข๐ก(๐ )
L
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 66
LRC circuit: transfer function
Setting L = 1 H, C = 1 F and R = 1 ฮฉ
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 67
LRC circuit: dynamic response to step
Setting the input to a step of amplitude 1 V
The unit step response is
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 68
LRC circuit: dynamic response to step
Using MATLAB for the solution
>> n = [1];>> d = [1 1 1 0];>> [ฮฑ, a, k] = residue(n, d);
Coefficient vector for polynomial at numerator
Coefficient vector for polynomial at denominator
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 69
Plotting results of two methods
>> y=ฮฑ.'*exp(a*t);>> plot(t, y, 'bo');
>> y2=step(1,[1 1 1],t);>> plot(t, y2, โr.โ);
OR
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1)
Second-order system step response
Overshoot
Period of oscillation
Settling time: time to settle to ยฑ5% of final value (determined by the term)
seco
ndor
der.m
70
Note: often the response of a high-order system is similar to the second-order one
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 71
Verify the following
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 72
Back to cruise control: systemโs step response
Gvfvr H
K
ฮธ
+ -
c
e-+
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 73
Step response
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 74
Back to cruise control
>> n = [k/m];>> d = [1 (b+k)/m 0];>> [ฮฑ, a, k] = residue(n, d);>> y=ฮฑ.'*exp(a*t);>> plot(t, y, 'bo');
ORLIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 75
Back to cruise control
>> G = tf([1/m],[1 b/m]);>> H = k;>> CL = G * H / (1 + G * H);>> [y, t] = step(CL);>> plot(t, y, 'b.');
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 76
Transfer function, poles and zerosโข It is convenient to express a transfer function
G(s) in terms of its poles and zeros:
โข k is the gain of the transfer function
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 77
Summary of pole characteristicsโข Real distinct poles (often negative)
โข Real poles, repeated m times (often negative)
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 78
Summary of pole characteristicsโข Complex-conjugate poles
often re-written as a second-order term
โข Poles on imaginary axisโ Sinusoidโ Pole at zero: step function
โข Poles with a positive real partโ Unstable time-domain solution
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 79
Summaryโข The Laplace transform is a tool to facilitate solving for ODEs.โข Systems need to be linearโข No need to do the transform (integral)
โ Use transform pairs, transform tablesโ Laplace transform properties: linearity, derivatives and integrals.
โข Once in the Laplace domain, a TF is simply the ratio of two polynomials in s. Carry out algebra to solve the problem.
โข No need to do the inverse transformโ Use transform pairs, transform tablesโ For high-order TFs, use the partial-fraction expansion to reduce the problem to
simpler partsโข IMPORTANT:
โ Poles of a transfer function determine the time evolution of the systemโ Poles with a real positive part correspond to unstable and unphysical systemsโ The system TF needs to have poles with a negative real part
โข MATLAB implementationโ Functions used: step, residue, tf
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 80
Solutions
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 81
Practice
Determine the output C in terms of inputs U and R.Sol:
LIGO-G1100863
U
CR ๐บ1+ +
-
+ ๐บ2
Matone: An Overview of Control Theory and Digital Signal Processing (1) 82
PracticeDetermine the output in terms of inputs and .Sol:
LIGO-G1100863
U1
CR ๐บ1 ++
+
+ ๐บ2
U2
๐ป1+
+๐ป2
Matone: An Overview of Control Theory and Digital Signal Processing (1) 83
More practiceDetermine C/R for the following systems. Sol:
(a)
(b)(c)
LIGO-G1100863
R ๐บ1 ++
++
๐บ2
C
๐ป1
R ๐บ1 ++
++
๐บ2
C
๐ป1
R๐บ1 +
+++
๐บ2
C
๐ป1
Matone: An Overview of Control Theory and Digital Signal Processing (1) 84
How do we MEASURE the OL TF of a system when the loop is closed?
1. Add an injection point in a closed loop system
2. Inject signal and read signal (just before the injection) and (right after the injection)
3. Solve for the ratio ๐ฅ
๐ฆ 1 ๐ฆ 2Sol:
++
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 85
Partial-fraction examples
โข Denominator: has distinct, real rootsโ Example 2.4, 2.5, 2.6
โข Denominator: complex rootsโ Example 2.7, 2.8
โข Denominator: repeated rootsโ Example 2.9
LIGO-G1100863
Matone: An Overview of Control Theory and Digital Signal Processing (1) 86
Practice: verify the following
LIGO-G1100863