Module-1:Fundamental Concepts of Modelling,System Poles and Zeros
Akshya Swain
Department of Electrical, Computer & Software Engineering, The University of Auckland,
Auckland, New Zealand.
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Summary of Topics Taught
1. Concept of Open and Closed Loop Control Systems.
2. Classification of Signals; Power & Energy Signals
3. Standard Input Signals used in Control Engineering; impulse, step, ramp etc. 4. Classification of Systems, Linear, Nonlinear,Time Invariant/Time Variant.
5. Concept of Impulse Response of a System.
6. Types of Models ( Transfer Function & Differential Equation) Models.
7. Transient Response Specifications such as Time Constant,Rise Time etc.
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Learning Outcome of this Module
I Concept of Modelling, System Order
I Classification of Systems; Static, Dynamic
I Computation of Transfer Function Models from Differential Equation Models. I Physical Significance of Poles and Zeros
I Concept of System Modes, Engineering Infinity
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Concept of Modelling
Common man observes the physical phenomena around him; the scientist encodes them into models.
Courtesy of: https://www.nationaltrust.org.uk
Figure: The iconic apple tree at Woolsthorpe Manor
Apple falls → Newton encoded: Law of Gravitation
Nothing shifts unless it is pushed → Law of inertia
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Concept of Modeling: Few Quotes
Nothing exists! → E = mc2
Courtesy of: https://apod.nasa.gov
Reality is merely an illusion, although a very persistent one.
Everything in Life is Vibration.The law of nature that states everything has a vibration. Everything is made up of atoms which are in a constant state of motion, and depending on the speed of these atoms, things appear as solid, liquid, or gas.
Well I never said E = mc2
“If a body gives off the energy L in the form of radiation, its mass
diminishes by L , the mass of a body c2
is a measure of its energy-content.”
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What is a Model and Why do we need it?
What is a Mathematical Model?
– Loosely put, a model of a system is a tool we use to answer questions about the system without having to do an experiment.
– A collection of mathematical relationships between system/process variables which purports to describe the behavior of a physical system.
– This is a convenient surrogate of the physical system.
What are its main uses?
1. For Analysis : To investigate system response under various input conditions both rapidly, and inexpensively, without tampering with the actual physical entity.
2. For Synthesis: Analytically design controllers
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DON’T CONFUSE MODELS WITH PHENOMENA!
I Model is an imaginary universe. They are not photographs of reality; they capture those aspects of the system which the designer decides to be important.
MAP IS NOT THE TERITORY; IT IS THE MODEL OF THE TERITORY
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Philosophy of Model Building
Physical Phenomena can be categorized essentially into two types:
Trinity of Electrical Engineering: the resistor, capacitor, and inductor.
I Inductance stores energy in its magnetic field (Kinetic energy) and Capacitance stores energy in its electric field ( Potential energy).
1. Energy dissipation
2. Energy absorption or storage
1. Resistance : Models the phenomenon of energy dissipation.
2. Capacitance and Inductance model the phenomenon of energy
absorption or storage.
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Concept of Static/Memory-less Systems
What is a static system?
Example: A Purely Resistive Circuit
1. A static system does not have any energy storing element.
2. It is also referred to as a system without memory.
3. The response at a particular time is dependent only on the input at
that time.
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Concept of Dynamic System or Systems with Memory
What is a Dynamic System?
1. A dynamic system is one which has at-least one energy storing element.
2. The dynamics system is also known as system with memory.
3. The response at a particular time depends both on the input as well
as the energy stored in the energy storing elements (initial conditions) (past values ).
Example: R-L, R-L-C circuits
Figure: R-L Circuit Figure: R-L-C Circuit
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Concept of System Order
What do you understand by order of a system?
The order of a system equals to the number of independent energy storing elements of the system.
I Commonly known storing elements: (Inductance and Capacitance) and dissipating element Resistance.
I Inductance stores energy in its magnetic field and capacitance stores energy in its electric field.
Example-1:Purely Resistive Circuit: The number of energy storing elements=0. Hence its order is 0 (zero).
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Examples: Concept of System Order
Exampl-2: RL Circuit: The number of energy storing elements equals to 1. Hence its order 1 (one).
Exampl-3: RC Circuit: The number of energy storing elements equals to 1. Hence its order 1 (one).
Exampl-4: RLC Circuit: The number of energy storing elements equals to 2. Hence its order equals to 2(two).
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Impulse Response Model of a Linear System
What is impulse response function of a system?
Why it is so important?
The convolution relation is given as:
∞ 0
∞ 0
The response (output) of a system when the input is an impulse.
If we know the impulse response function (model) of a system i.e the response of a system to an impulse input, the response of the system to any arbitrary input u(t) can be found by convolving the impulse response function g(t) with the input u(t).
y(t) = =
g(τ )u(t − τ )dτ u(τ)g(t − τ)dτ
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Transfer Function Model
What is the transfer function of a system?
The transfer function of a linear system is the Fourier transform of its impulse response.
If g(τ) is the impulse response then the transfer function is given by, ∞
Alternative Definition:
G(jω) =
g(τ )e−jωτ dτ
0
It is the ratio of the Laplace transform of the output to the Laplace transform of the input with initial conditions zero. Thus
G(jω) = Y (jω) U(jω)
I This formula is used if the differential equation models of the system is known.
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Computation of Transfer Function Model of a System from its
Differential Equation Models
Problem: Given the differential equation model of a linear time invariant system, how would we compute the transfer function.
Step-1: Apply Laplacian operator to the differential equation.
Step-2: Assume all the initial conditions equal to zero.
Step-3: Divide the Laplace transform of output say Y (s) to the Laplace Transform of input; say U(s) to get the transfer function G(s)
Example-1: Consider a linear time invariant system which is modelled ( represented) by the differential equation
y ̈(t) + a1y ̇(t) + a2y(t) = b0u(t)
Compute the transfer function model assuming y as the output and u as the input.
Solution:
Taking the Laplace transform of the above equation with intitial conditions zero gives
s2Y (s) + a1sY (s) + a2Y (s) = b0U(s)
or, G(s)=Y(s)= b0
U(s) s2 +a1s+a2
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Example: Computation of Transfer Function Model of a System from its Differential Equation Models
Example-2: The differential equation model of a linear time invariant system is given by
…
y (t) + 9y ̈(t) + 26y ̇(t) + 24y(t) = u ̇ + 5u(t)
Compute the transfer function model assuming y as the output and u as the input.
Solution:
Taking the Laplace transform of the above equation with intitial conditions zero gives
s3Y(s)+9s2Y(s)+26sY(s)+24Y(s) = sU(s)+5U(s) or, G(s)=Y(s)= s+5 = s+5
(s+2)(s+3)(s+4)
The system has 3-poles;located at s = −2,s = −3 and s = −4 and 1-zero; located at s = −5
U(s) s3 +9s2 +26s+24
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Example: Computation of Transfer Function Model of a System from its
Differential Equation Models
Example-3: The differential equation model of a linear time invariant system is given by
…
y (t) + 5y ̈(t) + 4y ̇(t) + 20y(t) = u ̈ + 16u(t)
Compute the transfer function model assuming y as the output and u as the input.
Solution:
Taking the Laplace transform of the above equation with initial conditions zero gives
s3Y(s)+5s2Y(s)+4sY(s)+20Y(s) = s2U(s)+16U(s)
or, G(s)= Y(s) = s2 +16 = s2 +16
U(s) s3 +5s2 +4s+20 s3 +5s2 +4s+20
= (s+j4)(s−j4) (s+j2)(s−j2)(s+5)
The system has 3-poles;located at s = +j2,s = −j2 and s = −5 and two-zeros; located at s = +j4 and s = −j4.
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Physical Significance of Poles & Zeros
I We know that everything in life is vibration i.e.everything vibrates.
I Poles are the natural frequencies of a system.
I If the system is excited by an input whose frequency equals to its natural frequency, the output of the system will be unbounded ( infinity).
I Zeros are the frequencies which are blocked by the system.
I Thus, if the system is excited by an input whose frequency corresponds to a zero, the output of the system will be zero i.e. the system will block this frequency component .
Before we further elaborate about their significance, let us first understand the concept of complex frequency.
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Concept of Complex Frequency jω
What do you understand by a signal of frequency ‘jω’?
I In electrical engineering we often come across the term complex frequency, jω or
s = σ ± jω.
I Let us understand first what does the complex frequency imply? I Complex frequency always comes in pairs. ± is implicit
I Hint: De Moivre’s Formula:ej x = cosx + j sinx
I Example: if a signal has a complex frequency j314 rad/s, then this corresponds to a pure sinusoid of frequency 314 rad/s (i.e. 50 Hz).
Complex frequency jω represents a pure sinusoidal signal of frequency ω rad/s.
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Concept of Complex Frequency s = σ + jω
I What do you understand by a signal of frequency ‘s = σ + jω’?
Hint: De Moivre’s Formula ejx = cosx + jsinx
Example: The signal e−10tsin40πt would look as:
Complex frequency s = σ + jω represents an exponentially damped sinusoidal signal of frequency ω rad/s and this decays/increases at a rate decided by σ.
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Interpretation of Zeros
I Consider the RC-Circuit shown in the Figure
The transfer function is given by
VOUT(s) = s VIN(s) s+ 1
I The system has a zero located at s = 0.
I This implies that if we apply a signal of frequency 0 rad/s (DC), the output
would be zero.
I This is obvious from the physics of the circuit, as the capacitor at the input acts as a blocking capacitor.Because, at f = 0, the impedance offered by capacitor is infinite.
R1 C1
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Interpretation of Zeros
I Consider the RL-Circuit shown in the Figure
The transfer function of the system is given by
VOUT(s) = s VIN(s) s+ R2
L2
I Thesystemhaszeroats=0.
I This implies if we apply a signal of 0 rad/s (DC), the output would be zero.
I This is obvious from the physics of the circuit. The impedance offered by inductor at f = 0 is zero.
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Example: Interpretation of Zeros
I Consider a Mass Spring Damper Mechanical System:
Forc1 =1,c2 =2,d=1,m1 =1andm2 =4,thetransferfunctionbetweenthe
position xa and force xe is given as:
G(s)= Xa(s) = s2 +1
Xe(s) s4 + 0.5s3 + 1.75s2 + 0.5s + 0.5
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Interpretation of Zeros
I The zeros of this system are at s = ±j1.
I As discussed before, this corresponds to a signal sint.
I Let us see what will be the response when the system is excited by the signal
sint.
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Concept of Engineering Infinity
I Consider a signal e−at.
I The time constant associated with this signal: T = 1 .
a
I Theoretically: This signal will decay to zero as time approaches to infinity. I However, in practice, its value will be very very small after 5-time constants.
The engineering infinity is equal to five times the time constant i.e t∞ = 5T
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Example-1: Concept of System Modes
I Consider the systems with transfer function
G1(s)= 10 , G2(s) = 10 , G3(s)= 10 , G4(s) = 10
s+1 s+5 s+10 s+20 I The poles of these systems are located at −1,−5,−10 and −20.
I The impulse response of these systems are shown in the Figure.
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Example-1:Concept of System Modes
Figure: The impulse response of these systems: which system is the fastest ?
1. The system whose poles are located farthest from the jω axis is the fastest ( fastest mode) (System G4.)
2. The system whose poles are located nearest to the jω axis is the slowest ( slowest mode)(System G1)
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Example-2:Concept of System Modes
I Consider a system, whose transfer function model is resolved into partial fraction and is expressed as
G(s)=Y(s)= k1 + k2 + k3 − k4 U(s) s+1 s+2 s+5 s+10
I The poles of this system are located at −1,−2,−5 and −10.
I The characteristic modes are e−t, e−2t,e−5t and e−10t
I The time constants associated with these modes: T = 1 for e−t, T = 0.5 for e−2t,T = 0.2 for e−5t,and T = 0.1 for e−10t.
I Thus, the fastest mode (i.e. e−10t) corresponds to the pole which is farthest from the jω axis and the slowest mode (i.e. e−2t) corresponds to the pole which is nearest to the jω axis.
I The impulse response of this system is given by
y(t) = k1e−t + k2e−2t + k3e−5t + k4e−10t
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Example-2:Concept of System Modes
I The contribution of various modes to the impulse response function for k1 = 10, k2 =−5,k3 =8andk4 =−2isshownintheFigure.
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Example-2:Concept of System Modes
I The impulse response of the system, which is equal to sum of the responses due to all modes, is shown in Figure.
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Learning Summary of Module-1
1. A signal described by a complex frequency represents either a pure sinusoid or exponentially damped sinusoid.
2. For all practical purposes, the infinity time can be considered equal to 5 times the time constant of the system.
3. Model is an imaginary universe without having any existence; it is abstract. However, it is used for solving analysis and synthesis problems of physical systems.
4. A dynamic system has memory due to presence of at least one energy storing elements. It has, therefore, memory.
5. The order of a system is the number of independent energy storing elements of the system.
6. The impulse response function of a linear time invariant (LTI) system completely characterizes it.
7. Poles are the natural frequencies of the system.
8. Zeros of a system are the frequencies which are blocked by the system.
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