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Continuous-time S ystems
(AKA analog systems)
Recall course objectives
Main Course Objective:Fundamentals of systems/signals interaction
(we’d like to understand how systems transform or affect signals)
Specific Course Topics:-Basic test signals and their properties-Systems and their properties-Signals and systems interaction
Time Domain: convolutionFrequency Domain: frequency response
-Signals & systems applications:audio effects, filtering, AM/FM radio
-Signal sampling and signal reconstruction
II. CT systems and their properties
Goals
I. A first classification of systems and their models:
A. Operator Systems: maps that act on signals B. Physical Systems: examples and ODE models
II. Classification of systems according to their properties:
Homogeneity, time invariance, superposition, linearity, memory, BIBO stability, controllability, invertibility, …
Systems
Systems accept excitations or input signals andproduce responses or output signals
Systems are often represented by block diagrams
SISO system
MIMO system
SISO = single input, single outputMIMO = multiple input, multiple output
Operator systems acting on signalsSystems can be “operators” or “maps” that combine signals
This is a more usual situation in the “digital world.” Howeversome analog systems can also be described through maps
Examples:1-Algebraic operators
-noise removal by averaging-motion detection through image subtraction
2-Geometric/point operators (interpolation)-image rectification-visual spatial effects: morphing, image transformation-cartography: maps of ellipsoidal/spherical bodies
3-Signal multiplication:- AM radio signal modulation before transmission
However, simple operations like these are not enough to capture allfiltering effects on signals (more on this later)
Noise* removal by averaging Original signal
Noisy versions of the signal (noise is zero mean)
Period of underlying signal Averaged signalmust be known or estimated
Matlab script in webpage: averaging.m
(*) noise = infinite-energy signal that takes random values
!
+
!
x(t) + n1(t)
!
x(t) + n2(t)
!
x(t) + n3(t)
!
x(t) +
ni(t)i"3
Motion detection by subtraction
(this is a digital system example: here image signals are a function ofdiscrete-space variables or pixels)
!
"
!
A0(x,y)
!
A1(x,y)
!
A1(x,y)
!
A0(x,y)
!
"
Geometric/point operators
E.g., fish-eye lenses (used in robotics) distort reality
input-outputtest using a knownimage for cameracalibration
(we’d like to find :
)
correctionof a real imageusing the inferred
!
A(x,y)
!
B(x,y)
!
f
!
f
!
f
!
"
!
"
!
f
!
f "1
Signal multiplication
AM signal modulation for signal transmission(used in AM radio)
!
x(t)
!
x(t)cos("ct)
!
cos("ct)!
"
II. CT systems and their properties
Goals
I. A first classification of systems and their models:
A. Operator Systems: maps that act on signals B. Physical Systems: ODE models and examples
II. Classification of systems according to their properties:
Homogeneity, time invariance, superposition, linearity, memory, BIBO stability, invertibility, controllability
ODEs and state-space system models
!
d n y( t )
dt n+ a1
d n"1y(t )
dt n"1+ ... + an"1
dy( t )dt + an y(t ) = b0
d mU ( t )
dtm+ ... + bm"1
dU (t )dt + bmU ( t )
dimensional ODEs(*) model many electro-mechanical systems
The coefficients can be time-varying or constant.When independent of y(t), ODE is linear, otherwise ODE is nonlinear
!
ai,bi
We will typically consider for To solve the ODE we need to fix some initial conditions
(*) the unknowns are the y(t) and its derivatives
!
bi = 0
!
n
!
y(t0),dydt(t0),
d2ydt 2
(t0),...,dn"1ydtn"1
(t0)!
i = 0,...,m "1
ODEs and state-space system models
Given:
Inputs: linear combination of U(t) and derivatives(represent known variables, e.g., forces in a Mech. System)
Outputs: linear combination of y(t) and/or its derivatives(represent unknown variables we would like to determine)
When coefficients are independent of y(t), the output equationsare linear; otherwise, they are nonlinear
!
d n y( t )
dt n+ a1
d n"1y(t )
dt n"1+ ... + an"1
dy( t )dt + an y(t ) = b0
d mU ( t )
dtm+ ... + bm"1
dU (t )dt + bmU ( t )
!
zl ( t ) = cl ,0d n y( t )dt n
+ ... + cl ,n"1dy( t )dt + cl ,n y( t )
!
l = 1, ..., n
Physical Systems (mechanical)
Newtonian motion
Input U(t) Output: y(t) Initial conditions y(t0), y'(t0)
Second-order linear system
Force U(t)
Distance y(t)
Resistance kfy'(t)
!
M d2y(t)dt 2
= "k fdy(t)dt
+U(t)
d2y(t)dt2 +
k fM
dy(t)dt
=1M
U (t) , y(t0 ), ʹ′ y (t0 )
!
F" = ma
!
"# = Ia
Physical Systems (mechanical)
Mass-spring-damper
Input U(t)Outputs y(t), y’(t), or combinationInitial conditions y(t0), y'(t0)
Second-order linear system
M
D K
U(t)
y(t)
!
M d2y(t)dt 2
+ D dy(t)dt
+ Ky(t) =U(t)
d2y(t)dt2 +
DM
dy(t)dt
+KM
y(t) =1M
U(t), y(t0 ), ʹ′ y (t0 )
Physical Systems (mechanical)
Simple pendulum
I=ML2, moment of inertiaInput (force at the ball) U(t)Output θ(t)Initial conditions θ(t0), θ'(t0)
Second-order nonlinear system (why?)
Simulink script in webpage: PendulumWithoutDamping.mdl
L
U(t)
θ(t)
M Mg sin θ(t)!
I d2"(t)dt 2
= LU(t) #MgLsin(")
!
d2"(t)dt 2
+gLsin(") =
1ML
U(t)
!
"(t0), # " (t0)
Physical Systems (electrical)
Kirchhoff’s lawsCurrent law:
sum of (signed) currents at a “node” is zero(node=electrical juncture of two or more devices)
Voltage law:sum of (signed) voltages around a “loop” is zero(loop=closed path passing through ordered sequence of nodes)
Circuit element laws:Resistor:
Capacitor:
Inductor:!
iR =VR
!
i = C dVdt
!
V = L didt
Physical Systems (electrical)
Objective: Find a model relatingthe input and output
2 nodes and 2 loopsEquation of upper node:Equation of left loop:Equation of right loop:
Circuit element equations:
Putting it all together:
!
i
!
VC
!
i1
!
i2
!
i " i1 " i2 = 0
!
V +V1 = 0
!
"V1 +VC = 0
!
V1 = i1R
!
C dVCdt
= i2
!
i = i2 + i1
!
i = C dVCdt
+1RV1
!
C dVCdt
+1RVC = i
!
i
!
VC
ODEs and state-space models
A state-space representation of an nth order ODE describing aphysical system is obtained as follows:
State
The state-space representation of the system is a systemof first-order differential equations in the new variables
If the original nth order ODE is linear, then the state-spacerepresentation can be expressed in matrix form:
A linear output equation is expressed as ,This can be generalized for several variables
!
x = x1, x2, x3, ..., xn( )T = y, dydt, d2y
dt 2, ..., dn"1y
dtn"1#
$ %
&
' (
T
!
A =
0 1 0 ... 00 0 1 ... 0... ... ... ... ...0 0 0 ... 1
"an1 "an"1 "an"2 ... "a1
#
$
% % % % % %
&
'
( ( ( ( ( (
!
B =
00...0bm
"
#
$ $ $ $ $ $
%
&
' ' ' ' ' '
!
dxdt
= Ax + Bu,
!
x(t0)
!
z = Cx
!
x1, x2, x3, ..., xn
!
C = (cl ,k )
Why state-space models are used
State-space formulation allowsto lump multiple variables ina single state vector
Distillation column:Hundreds of state variables
Concentration and temp at eachtray positionLots of structure
Output of one tray is theinput to the next
Several inputsBoiler power, reflux ratio,
feed rateMany outputs
Some tray temperatures,final concentration
!
x
Why state-space models are used
Well suited for MIMO systems
MIMO and SISO systemshave same form instate-space formulation
This allows for uniformtreatment• Analysis of system
properties• Linearization• Simulation (matlab,
simulink)
f1(t)f2(t)
v1(t)v2(t)
d(t)
!
˙ x 1(t)˙ x 2(t)˙ x 3(t)
"
#
$ $ $
%
&
' ' '
=
(k f 1M1
0 0
0(k f 2M2
0
(1 1 0
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
x1(t)x2(t)x3(t)
"
#
$ $ $
%
&
' ' '
+
M1(1
00
0M2(1
0
"
#
$ $ $
%
&
' ' '
f1(t)f2(t)
"
# $
%
& '
v1(t)v2(t)d(t)
"
#
$ $ $
%
&
' ' '
=
1 0 00 1 00 0 1
"
#
$ $ $
%
&
' ' '
x1(t)x2(t)x3(t)
"
#
$ $ $
%
&
' ' '
Why do we linearize about equilibrium points?
Unfortunately, there are no general formulas to solvenonlinear ODEs. Then we are forced to look for (1)particular solutions and (2) approximations to the solutions
How can we find particular solutions to nonlinear ODEs?Equilibrium points are always particular constant solutions
How to approximate the solutions of a nonlinear ODE?(a) We know how to solve linear ODEs
(b) The qualitative behavior of a nonlinear ODE with aninitial condition close to an equilibrium point, under inputs ofsmall magnitude, can be found by solving the linearizedequation about that equilibrium point with zero inputs
Linearization about equilibrium point
An equilibrium point is such that
Equilibrium point = constant solution to ODEThe system remains at rest at all times if initiallyplaced at the equilibrium and no inputs are applied
Pendulum example. The state is Thependulum has two equilibrium points:
(vertical bottom position, zero velocity)
(vertical top position, zero velocity)!
x = (", # " )T
!
x1 = (0,0)T
!
x2 = (" ,0)T
!
dxdt
= f (x,u),
!
f (x0,0) = 0
!
x0
!
x(t0),
Linearization is easy in state-space formulation
Suppose the map is nonlinear(as in the pendulum example)
Linearization of the system about with is:
with constant matrices
!
f :Rn " Rm # $ # Rn
!
dxdt
="f"x#
$ %
&
' ( |x= x0,u= 0
(x ) x0) +"f"u#
$ %
&
' ( |x= x0 ,u= 0
u,
!
x(t0)!
x0
!
u = 0
!
"f"x#
$ %
&
' ( |x=x0 ,u=0
=
"f1"x1
... "f1"xn
... ... ..."fn"x1
... "fn"xn
#
$
% % % % %
&
'
( ( ( ( ( |x=x0 ,u=0
!
"f"u#
$ %
&
' ( |x=x0,u=0
=
"f1"u..."fn"u
#
$
% % % %
&
'
( ( ( ( |x=x0,u=0
Linearization with additional output map
For systems with an additional nonlinear output map:
where , linearization becomes:!
z = h(x),
!
dxdt
= f (x,u),
!
x(t0),
!
h :Rn " # " Rp
!
dxdt
="f"x#
$ %
&
' ( |x= x0,u= 0
(x ) x0) +"f"u#
$ %
&
' ( |x= x0 ,u= 0
u,
!
z ="h"x#
$ %
&
' ( |x= x0,
(x ) x0)
!
h(x0) = 0,
CT systems and their properties
Goals
I System examples and their models e.g. using basic principles
A. Operator systems: maps that act on signals
B. Physical systems: ODE models and examples
II System properties
Homogeneity, time invariance, superposition, linearity, memory, invertibility, BIBO stability, controllability
Response of a RC Low-pass filter
An RC low-pass filter is a simple circuit
It can be modeled as a SISO system
The system is excited by a voltage and responds with a voltage
Circuit might have initial voltage at capacitor!
vin (t)
!
vout (t)
!
vout (0)
Response of a RC Low-pass filter
If excited by a step voltage
Resp onds with
Unless otherwise said, by ‘response’ we mean ‘zero-state response’
If the excitation is doubled, (zero-state) response doubles
!
vout (t) = vout (0)e" t /RCu(t)
zero" input1 2 4 4 3 4 4
+ A(1" e" t /RC )u(t)zero"state
1 2 4 4 3 4 4 !
vin (t) = Au(t)
Homogeneity
In a homogeneous system, multiplying the excitation by anyconstant (including complex constants), multiplies the response bythe same constant
Homogeneity Test: 1) apply arbitrary input and obtain output, 2) then apply and obtain its output,
If then the system is homogeneous
If g(t) H! "! y1 t( )and K g(t) H! "! K y1 t( )#H is Homogeneous!
g(t)
!
y1(t)
!
Kg(t)
!
h(t)
!
h(t) = Ky1(t)
Time invariance
If an excitation causes a response and delaying the excitationsimply delays the response by the same amount of time,then the system is time invariant
If g(t) H! "! y1 t( )and g(t # t0 ) H! "! y1 t # t0( )$H is Time Invariant
This test must succeed for any g and any t0 .
Additivity property
If one excitation causes a response and another excitation causesanother response and the sum of the two excitations causes aresponse which is the sum of the two responses, the system is saidto be additive
If g(t) H! "! y1 t( )and h(t) H! "! y2 t( )and g t( ) + h t( ) H! "! y1 t( ) + y2 t( )#H is Additive
Linearity and LTI systems
If a system is both homogeneous and additive, it islinear
If a system is both linear and time-invariant, it iscalled an LTI (linear, time-invariant) system
Some systems which are nonlinear can be accuratelyapproximated for analytical purposes by a linearsystem for small excitations (recall the discussionon linearization)
We will mainly focus on LTI systems because we cancharacterize their response to any signal
System Invertibility
A system is invertible if unique excitations produce uniqueresponses
In an invertible system, knowledge of the response is sufficientto determine the excitation
Any system with input and output described by a linearODE of the form
is invertible.
A system with input and output described by theoperator map is non-invertible because sin(U) doesnot have an inverse.
!
d n y( t )
dt n+ a1
d n"1y(t )
dt n"1+ ... + an"1
dy( t )dt + an y(t ) = U ( t )!
U(t)
!
y(t)
!
z(t) = sin(U(t))
!
U(t)
!
z(t)
Memory
This concept reflects the extent to which the present behavior ofa system (its outputs) is affected by its past (initialconditions or past values of the inputs)
Physical systems modeled through ODEs have memory: this isassociated with the system inability to dissipate energy orredistribute it instantaneously
Example: Think about how a pendulum initially off thevertical winds down to the equilibrium position. The time ittakes to do it captures the pendulum memory
If a system is well understood then one can relate its memoryto specific properties of the system (e.g. “system stability”)
In fact, all filtering methods in signal processing are based onexploiting the memory properties of systems
Memory
A system is said to be memoryless if for any timethe output at depends only on the input at time
Example:If y(t) = K u(t), then the system is memorylessIf y(t) = K u(t-1), then it has memory
(Operator systems described through static maps areusually memoryless)
Any system that contains a derivative in it hasmemory; e.g., any systemdescribed through an ODE
!
t1
!
t1
!
t1
Stability
Any system for which the response is bounded for any arbitrarybounded excitation is said to be bounded-input-bounded-output (BIBO) stable system, otherwise it is unstable
Intuition: All systems for which outputs “do not explode” (i.e.outputs can only reach finite values) are BIBO stable.Intuitively, if a system has a “small memory” or “dissipatesenergy quickly,” then it will be stable.
If an ODE describing the system is available, then we can applythe following test for BIBO stabilityA system described by a differential equation is stable if theeigenvalues of the solution of the equation all have negativereal parts
Stability
Stable systems return to equilibrium despite input disturbances
How to check Stability when ODEs available
Suppose a state-space model for the physical system isavailable
Then, the system is stable if and only if the eigenvalues of thematrix (= the eigenvalues of the ODE) have all negativereal parts
The eigenvalues of are the solutions to the equation
Here, is the dimension of the state
!
dxdt
= Ax + Bu,
!
x(t0)
!
det("In # A) = 0!
A
!
A
!
"
!
n
!
x
How to check Stability when ODEs available
Simple example: Low-pass filter
ODE:
state-space representation:
Matrix is just a number:
Calculation of eigenvalues:
Thus, the system is BIBO stable
!
dy(t)dt
+1RC
y(t) =1RC
g(t)
!
dy( t)dt
= "1RC
y( t)+ 1RC
g( t)
!
A = "1RC
!
A
!
det(" #1+1RC) = 0
!
" = #1RC
!
"
Summary
Important points to remember:
1. We can model simple (mechanical/electric) systems by resorting tobasic principles and producing ODEs.
2. A special system representation is the state-space representation,useful for simulation, linearization and to check system controllability.
3. Special system properties are homogeneity, additivity, time-invariance,LTI, invertibility, memory, and stability. These properties can be checked bylooking at input-output experiments (no models required in principle.)
4. If an ODE model of the system is available, we can check systemstability by finding the eigenvalues of the ODE.
5. If an state space representation of the system is available, we can checkthe system controllability properties by applying the controllabilitytheorem.