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
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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
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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 MATLAB®. Cambridge University Press, 2002. ISBN 0-521-00255-9.
Ingle, Vinay K. and John G. Proakis. Digital Signal Processing using Matlab®. Brooks/Cole 2000. ISBN 0-534-37174-4.
Smith, Steven W. The Scientist and Engineer’s 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)
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ObjectiveControl System• Manages and regulates a set
of variables in a system– SISO – single-input-single-
output– MIMO – multiple-input-
multiple-output• A quantity is measured then
controlled• Requirements
– Bandwidth– Rise time– Overshoot– Steady state error– …LIGO-G1100863
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4
ObjectiveDigital Signal Processing• Measure and filter an analog signal• Digital signal
– Created by sampling an analog signal– Can be stored
• Analog filters– Cheap, fast and have a large dynamic
range in both amplitude and frequency• Digital filters
– Can be designed and implemented “on-the-fly”
– Superior level of performance.• 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 !
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Control Theory 1
• Given a physical system– Objective: sense and control a variable in the
system• Examples– As basic as• a car’s cruise control (SISO) or
– Not so basic as• Locking the full LIGO interferometer (MIMO)
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Example: Cruise Control
• Objective– Car needs to maintain
a given speed• Physical system
includes– car’s inertia– 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
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Physical model• Physical system described
by the following equation of motion
• Simplifying and assuming friction force is proportional to speed
Direction of motion
Force from engine (f)
Normal force (n)
Weight (mg)
Friction force (ffr)
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First-order differential equation• 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
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MATLAB implementationThe linear differential equation describing the dynamics of the system
Using MATLAB’s 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)
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Results:
𝑣𝑟=𝑓𝑏=10𝑚𝑠
𝑣 (τ )=63% ∙𝑣𝑟
τ=𝑚𝑏 =20 𝑠
cruise_timedom
ain.m
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Block diagram: representing the physical system
• To illustrate a cause-and-effect relationship• A single block represents a physical system• Blocks are connected by lines – Lines represent how signals flow in the system
• In general, a physical system G has signal x(t) as input and signal y(t) as output
• G is the transfer function of the system
Gx(t) y(t)
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Car’s body
Transfer function G represents the car’s body– G converts the force from the engine (input
signal, ) to the car’s actual speed (output signal, )
with – Units:
Gf(t) v(t)
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Setting the desired speed• Second transfer function H (the controller)– Converts the desired speed (or reference) to a
required force – Sets the throttle– For simplicity, H is set to a constant
– must be dimensionless
Gf vvr H
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Plotting results• With the actual speed is the
reference: • 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
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Introducing a disturbance – a hill
• In the presence of a hill the equation of motion needs to be re-visited
• Assuming a small angle θ
Weight (mg)
Direction of motion
Force from engine (f)
Friction force (ffr)
θAdded term
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Introducing a disturbance – a hill
Weight (mg)
Direction of motion
Force from engine (f)
Friction force (ffr)
ϑ
Added term
Assuming and are constants
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Modifying the block diagram
Gvfvr H
K
θ
+ -
Summation junction
𝐾=𝑚𝑔
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Modifying the block diagram
Gvfvr H
K
θ
+ -
Summation junction
𝐾=𝑚𝑔
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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
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Negative Feedback1. Let’s measure the car’s speed and2. Correct for it by feeding back into the system
a measure of the actual speed
Gvfvr H
K
θ
+ -
c
e
Correction signal
Error signal
-+
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Negative Feedback1. Let’s measure the car’s speed and2. Correct for it by feeding back into the system
a measure of the actual speed
Gvfvr H
K
θ
+ -
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
-+
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Negative feedback• Plot of force vs. time
and speed vs. time with negative feedback
• Setting • Result:– Faster response with
feedback (compare blue against red curves)
– Speed at regime: (error of )
cruisefeedback_timedom
ain2.m
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Negative feedback• Increasing the
controller’s gain (H)– decreases the rise time – while decreasing the
steady state error• Setting • Result:– Even faster response– Speed at regime: (error
of )
cruisefeedback_timedom
ain3.m
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𝑒=𝑣𝑟 −𝑐𝑣=𝐺 ∙( 𝑓 −𝐾 ∙ 𝜃)
Gvfvr H
K
θ
+ -e
-
+
c
𝑐=𝑣
𝑓 =𝐻 ∙𝑒
Signal flow and block diagrams
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𝑒=𝑣𝑟 −𝑐
Gvfvr H
K
θ
+ -e
-
+
c
¿𝑣𝑟−𝑣¿𝑣𝑟−𝐺 ∙ ( 𝑓 −𝐾 ∙𝜃 )¿𝑣𝑟−𝐺 ∙ (𝐻 ∙𝑒−𝐾 ∙𝜃 )
𝑒= 11+𝐺 ∙𝐻 ∙𝑣𝑟+
𝐾 ∙𝐺1+𝐺 ∙𝐻 ∙ 𝜃
System’s open loop gain (dimensionless)
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Gvfvr H
K
θ
+ -e
-
+
c
𝑒= 11+𝐺 ∙𝐻 ∙𝑣𝑟+
𝐾 ∙𝐺1+𝐺 ∙𝐻 ∙ 𝜃
(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 controller’s gain, the lower e
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𝑣=𝐺 ∙𝐻 ∙𝑒−𝐾 ∙𝐺 ∙𝜃
Gvfvr H
K
θ
+ -e
-
+
c
¿𝐺 ∙𝐻 ∙(𝑣¿¿𝑟 −𝑣)−𝐾 ∙G ∙𝜃 ¿¿𝐺 ∙𝐻 ∙𝑣𝑟 −𝐺 ∙𝐻 ∙𝑣−𝐾 ∙𝐺 ∙𝜃
𝑣=𝐺 ∙𝐻1+𝐺 ∙𝐻 ∙𝑣𝑟−(
𝐾 ∙𝐺1+𝐺 ∙𝐻 ) ∙𝜃
With no feedback𝑣=𝐺 ∙𝐻 ∙𝑣𝑟−𝐾 ∙𝐺 ∙ 𝜃
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Gvfvr H
K
θ
+ -e
-
+
c
𝑣=𝐺 ∙𝐻1+𝐺 ∙𝐻 𝑣𝑟−
𝐾 ∙𝐺1+𝐺 ∙𝐻 𝜃
(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 controller’s gain, the lower the speed error.
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𝑓 =𝐻 ∙𝑒
Gvfvr H
K
θ
+ -e
-
+
c
¿𝐻 ∙( 11+𝐺 ∙𝐻 𝑣𝑟+
𝐾 ∙𝐺1+𝐺 ∙𝐻 𝜃)
𝑓 = 𝐻1+𝐺 ∙𝐻 ∙𝑣𝑟+
𝐾 ∙𝐺 ∙𝐻1+𝐺 ∙𝐻 ∙ 𝜃
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Gvfvr H
K
θ
+ -e
-
+
c
𝑓 = 𝐻1+𝐺 ∙𝐻 ∙𝑣𝑟+
𝐾 ∙𝐺 ∙𝐻1+𝐺 ∙𝐻 ∙ 𝜃
(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
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Plotting and • Open loop
• Closed loop
• Setting • Plotting the open loop
transfer function vs. time and the closed loop transfer function vs. time
• Notice the rapid rise time for the closed loop case
cruisefeedback_timedom
ain2.m
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The error signal e• Plot of error signal e vs
time• Error signal decreases
to 3 m/s.• Notice a steady state
error
cruisefeedback_timedom
ain2.m
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Open and closed loop TF with
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Error signal e with
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Cruise control example
• First-order differential equation
• Simplest controller: simply a gain with no time constants involved
• How to handle more complicated problems?
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P1x yP2P1P2 yx
P1x y
P2
±+¿
P1 ±P2 yx
P1x y
P2
∓+¿
𝑃11± 𝑃1 𝑃2
yx
Block diagram reduction
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Practice
Determine the output C in terms of inputs U and R.
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U
CR 𝐺1+ +
-
+ 𝐺2
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PracticeDetermine the output in terms of inputs and .
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U1
CR 𝐺1 ++
+
+ 𝐺2
U2
𝐻1+
+𝐻2
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More practice
Determine C/R for the following systems.
(a)
(b)(c)
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R 𝐺1 ++
++
𝐺2
C
𝐻1
R 𝐺1 ++
++
𝐺2
C
𝐻1
R𝐺1 +
+++
𝐺2
C
𝐻1
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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 𝑥
𝑦 1 𝑦 2
++
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So far…• Control theory builds on differential equations• Block diagrams help visualize the signal flow in a
physical system• The cause-and-effect relationship between
variables is referred to as a transfer function (TF)• The system’s open-loop TF is the product of
transfer functions– cruise control example: – Two cases: and
• MATLAB implementation– Functions used: dsolve
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Laplace Transforms
• The technique of Laplace transform (and its inverse) facilitates the solution of ordinary differential equations (ODE).
• Transformation from the time-domain to the frequency-domain.
• Functions are complex, often described in terms of magnitude and phase
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Linear systems• To map a model to frequency space– System must be linear– Output proportional to input
• Given system P– Input signals: and – Output signals (response): and
• System P is linear– If input signal: – Then output signal: – Superposition principle
Px y
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Example
• Is a linear system?Knowing that and If input is , output is
System is linear
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Example
• Is a linear system?Knowing that andIf input is , output is
System is not linear
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In generalOutput
Input
For a stable system
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Laplace Transform L• Transforms a linear differential equation into an
algebraic equation• Tool in solving differential equations• Laplace transform of function f
• Laplace inverse transform of function F
where is the transform variableImaginary unit 2𝜋 𝑓
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Time domain ↔ Laplace domain
𝑓 =𝑑𝑦𝑑𝑡y (𝑡 ) f (𝑡 )
𝐺 (𝑠 )𝑌 (𝑠) F (𝑠)
Inpu
t
Out
put
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Laplace Transform L
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(Some) Laplace transform pairs
Unit stepUnit ramp
Exponential
Sinusoid
SHO
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Laplace transform properties• Linearity
• Derivatives– First-order: – Second-order:
• Integral
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Solution to ODEs
1. Laplace transform the system’s ODE2. Solve the algebraic equation in s3. Inverse transform back to the time domain
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Transfer Function
∑𝑖=0
𝑛
𝑎𝑖𝐷𝑖 𝑦 (𝑡 )=∑
𝑖=0
𝑚
𝑏𝑖𝐷𝑖 𝑥 (𝑡 )
∑𝑖=0
𝑛
𝑎𝑖 𝑠𝑖𝑌 (𝑠 )=∑
𝑖= 0
𝑚
𝑏𝑖𝑠𝑖 𝑋 (𝑠)
𝐺 (𝑠)=𝑌 (𝑠)𝑋 (𝑠)=
∑𝑖=0
𝑚
𝑏𝑖𝑠𝑖
∑𝑖=0
𝑛
𝑎𝑖 𝑠𝑖
53
Using derivative property
Algebraic equation in s, the ratio of 2 polynomials in s
Transfer function relates input to output .
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Matone: An Overview of Control Theory and Digital Signal Processing (1)
Transfer Function
𝐺 (𝑠)=𝑌 (𝑠)𝑋 (𝑠)=
∑𝑖=0
𝑚
𝑏𝑖𝑠𝑖
∑𝑖=0
𝑛
𝑎𝑖 𝑠𝑖
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 by• The coefficients of s or• Its poles and zeros
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Matone: An Overview of Control Theory and Digital Signal Processing (1) 55
Let’s apply it to the cruise control example: transfer function (the car’s 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
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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
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Dynamic response: using lookup tables to inverse transform
Laplace inverse transform using lookup tables
Pole LIGO-G1100863
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First-order system step response
𝐺 (𝑠 )= 𝑝𝑠+𝑝
63 %
Pole p
first
orde
r.m
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MATLAB implementation
The step response of transfer function
>> G=tf(5, [1 5]);>> step(G);
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Partial fraction expansion1. Reduce a complex function to a collection of
simpler ones2. Then use lookup table Order m
Order n
𝑚≤𝑛
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Comments
1. Poles of F(s) determine the time evolution of f(t)
2. Zeros of F(s) affect coefficients3. Poles closer to origin → larger time constants
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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 = []
𝐹 (𝑠 )= 3𝑠+2+
1𝑠−2 +
2𝑠+1
𝑓 (𝑡 )=3𝑒−2 𝑡+𝑒2 𝑡+2𝑒− 𝑡
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Example: LRC circuit
𝑣 𝑖𝑛(𝑡) 𝑣𝑜𝑢𝑡 (𝑡)
𝐿 𝑅
𝐶
𝑣 𝑖𝑛 (𝑡 )=𝑣 𝐿 (𝑡 )+𝑣𝑅 (𝑡 )+𝑣𝐶 (𝑡 )
¿𝐿 𝑑𝑑𝑡 𝑖 (𝑡)+𝑅𝑖(𝑡)+1𝐶∫
0
𝑡
𝑖 (𝜏 )𝑑𝜏
¿𝐿𝐷𝑖 (𝑡 )+𝑅𝑖 (𝑡 )+ 1𝐶1𝐷 𝑖(𝑡)
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Example: LRC circuit
𝑣 𝑖𝑛(𝑡) 𝑣𝑜𝑢𝑡 (𝑡)
𝐿 𝑅
𝐶
𝑣 𝑖𝑛(𝑡)=𝐿𝐷𝑖 (𝑡 )+𝑅𝑖 (𝑡 )+ 1𝐶1𝐷 𝑖(𝑡 )
𝑖 (𝑡 )=𝐶 𝑑𝑑𝑡 𝑣𝑜𝑢𝑡 (𝑡 )=𝐶 𝐷𝑣𝑜𝑢𝑡 (𝑡 )
𝑣 𝑖𝑛 (𝑡 )=(𝐿𝐶𝐷2+𝑅𝐶 𝐷+1 )𝑣𝑜𝑢𝑡 (𝑡)
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Matone: An Overview of Control Theory and Digital Signal Processing (1) 65
Example: LRC circuit
𝑣 𝑖𝑛(𝑡) 𝑣𝑜𝑢𝑡 (𝑡)
𝐿 𝑅
𝐶𝑣 𝑖𝑛 (𝑡 )=(𝐿𝐶𝐷2+𝑅𝐶 𝐷+1 )𝑣𝑜𝑢𝑡 (𝑡)
𝑉 𝑖𝑛 (𝑠 )= (𝐿𝐶𝑠2+𝑅𝐶𝑠+1 )𝑉 𝑜𝑢𝑡(𝑠)
L
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LRC circuit: transfer function
Setting L = 1 H, C = 1 F and R = 1 Ω
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LRC circuit: dynamic response to step
Setting the input to a step of amplitude 1 V
The unit step response is
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LRC circuit: dynamic response to step
Using MATLAB for the solution
>> n = [1];>> d = [1 1 1 0];>> [α, a, k] = residue(n, d);
Coefficient vector for polynomial at numerator
Coefficient vector for polynomial at denominator
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Plotting results of two methods
>> y=α.'*exp(a*t);>> plot(t, y, 'bo');
>> y2=step(1,[1 1 1],t);>> plot(t, y2, ‘r.’);
OR
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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 ±5% 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
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Matone: An Overview of Control Theory and Digital Signal Processing (1) 71
Verify the following
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Back to cruise control: system’s step response
Gvfvr H
K
θ
+ -
c
e-+
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Step response
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Back to cruise control
>> n = [k/m];>> d = [1 (b+k)/m 0];>> [α, a, k] = residue(n, d);>> y=α.'*exp(a*t);>> plot(t, y, 'bo');
ORLIGO-G1100863
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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.');
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Transfer function, poles and zeros• It is convenient to express a transfer function
G(s) in terms of its poles and zeros:
• k is the gain of the transfer function
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Summary of pole characteristics• Real distinct poles (often negative)
• Real poles, repeated m times (often negative)
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Summary of pole characteristics• Complex-conjugate poles
often re-written as a second-order term
• Poles on imaginary axis– Sinusoid– Pole at zero: step function
• Poles with a positive real part– Unstable time-domain solution
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Summary• The Laplace transform is a tool to facilitate solving for ODEs.• Systems need to be linear• No need to do the transform (integral)
– Use transform pairs, transform tables– Laplace transform properties: linearity, derivatives and integrals.
• Once in the Laplace domain, a TF is simply the ratio of two polynomials in s. Carry out algebra to solve the problem.
• No need to do the inverse transform– Use transform pairs, transform tables– For high-order TFs, use the partial-fraction expansion to reduce the problem to
simpler parts• IMPORTANT:
– Poles of a transfer function determine the time evolution of the system– Poles with a real positive part correspond to unstable and unphysical systems– The system TF needs to have poles with a negative real part
• MATLAB implementation– Functions used: step, residue, tf
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Solutions
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Practice
Determine the output C in terms of inputs U and R.Sol:
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U
CR 𝐺1+ +
-
+ 𝐺2
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PracticeDetermine the output in terms of inputs and .Sol:
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U1
CR 𝐺1 ++
+
+ 𝐺2
U2
𝐻1+
+𝐻2
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Matone: An Overview of Control Theory and Digital Signal Processing (1) 83
More practiceDetermine C/R for the following systems. Sol:
(a)
(b)(c)
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R 𝐺1 ++
++
𝐺2
C
𝐻1
R 𝐺1 ++
++
𝐺2
C
𝐻1
R𝐺1 +
+++
𝐺2
C
𝐻1
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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 𝑥
𝑦 1 𝑦 2Sol:
++
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Matone: An Overview of Control Theory and Digital Signal Processing (1) 85
Partial-fraction examples
• Denominator: has distinct, real roots– Example 2.4, 2.5, 2.6
• Denominator: complex roots– Example 2.7, 2.8
• Denominator: repeated roots– Example 2.9
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Practice: verify the following
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