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

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 ) = b

0

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,b

i

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),dy

dt(t0),d2y

dt2(t0),...,

dn"1y

dtn"1(t0)!

i = 0,...,m "1

ODEs and state-space system models

Given:

Possible inputs: linear combination of U(t) and derivatives(represent known variables, e.g., forces in a Mech. System)

Possible outputs: linear combination of y(t) and/or itsderivatives (represent unknown variables we would like todetermine)

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 ) = b

0

d mU ( t )

dtm+ ... + bm"1

dU (t )

dt+ bmU ( t )

!

zl( t ) = cl ,0

dny( t )

dtn + ... + cl ,n"1

dy( 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)

!

Md2y(t)

dt2

= "k fdy(t)

dt+U(t)

d2y(t)dt2 +

k fM

dy(t)dt

=1M

U (t) , y(t0 ), ′ 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)

!

Md2y(t)

dt2

+ Ddy(t)

dt+ Ky(t) =U(t)

d2y(t)dt2 +

DM

dy(t)dt

+KM

y(t) =1M

U(t), y(t0 ), ′ 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?)

L

U(t)

θ(t)

M Mg sin θ(t)!

Id2"(t)

dt2

= LU(t) #MgLsin(")

!

" " # (t) + gL$1#(t) = (ML)

$1U(t)

!

d2"(t)

dt2

+g

Lsin(") =

1

MLU(t)

!

"(t0), # " (t

0)

Linearization for small θ is

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 = CdV

dt

!

V = Ldi

dt

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" i

2= 0

!

V +V1

= 0

!

"V1

+VC

= 0

!

V1

= i1R

!

CdV

C

dt= i

2

!

i = i2

+ i1

!

i = CdV

C

dt+1

RV1

!

i = CdV

C

dt+1

RVC

!

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, x

2, x

3, ..., xn( )

T= y,

dy

dt,

d2y

dt2, ...,

dn"1y

dtn"1

#

$ %

&

' (

T

!

A =

0 1 0 ... 0

0 0 1 ... 0

... ... ... ... ...

0 0 0 ... 1

"an1

"an"1 "a

n"2 ... "a1

#

$

% % % % % %

&

'

( ( ( ( ( (

!

B =

0

0

...

0

bm

"

#

$ $ $ $ $ $

%

&

' ' ' ' ' '

!

dx

dt= Ax + Bu,

!

x(t0)

!

z = Cx

!

x1, x

2, x

3, ..., x

n

!

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)

f1(t)f2(t)

v1(t)v2(t)

d(t)

!

˙ x 1(t)

˙ x 2(t)

˙ x 3(t)

"

#

$ $ $

%

&

' ' '

=

(k f 1

M1

0 0

0(k f 2

M2

0

(1 1 0

"

#

$ $ $ $ $ $ $

%

&

' ' ' ' ' ' '

x1(t)

x2(t)

x3(t)

"

#

$ $ $

%

&

' ' '

+

M1(1

0

0

0

M2(1

0

"

#

$ $ $

%

&

' ' '

f1(t)

f2(t)

"

# $

%

& '

v1(t)

v2(t)

d(t)

"

#

$ $ $

%

&

' ' '

=

1 0 0

0 1 0

0 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

!

dx

dt= 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" R

m# $ # R

n

!

dx

dt="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"x

1

..."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),

!

dx

dt= f (x,u),

!

x(t0),

!

h :Rn" # " R

p

!

dx

dt="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 single input, single output system

The system is excited by a voltage and responds with a voltage

We assume the circuit has no initial energy at the capacitor!

vin(t)

!

vout(t)

Response of a RC Low-pass filter

If the RC low-pass filter is excited by a step voltage

Its response is

vint( ) = Au t( )

vou t

t( ) = A 1! e! t /RC( )u t( )

That is, if the excitation is doubled, the response doubles

Homogeneity

In a homogeneous system, multiplying the excitation byany constant (including complex constants), multiplies theresponse by the same constant.

Homogeneity Test

Homogeneity Test: 1) apply an arbitrary as input and obtain as output 2) then apply and obtain its output, h(t)

If h(t) = K then the system is homogeneous!

y1(t)

!

g(t)

!

Kg(t)

!

y1(t)

If g(t)H

! "! y1 t( )and K g(t)H

! "! K y1 t( )#H is Homogeneous

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 if, for any arbitrary excitations, the sum ofthe two excitations causes a response which is the sum of the tworesponses, the system is said to 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 non-linear can beaccurately approximated for analytical purposes bya linear system for small excitations (recall thediscussion on 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 produceunique responses

In other words, in an invertible system, knowledge ofthe response is sufficient to determine theexcitation

Any system with input and output just described by alinear ODE of the form

is invertible. A system with input and output describedby the operator map is non-invertible becausesin(y) does not have an inverse.

!

dny( t )

dtn

+ a1

dn"1

y(t )

dtn"1

+ ... + an"1

dy( t )

dt+ an y(t ) = x( t )!

x(t)

!

y(t)

!

z(t) = sin(y(t))

!

x(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

!

dx

dt= Ax + Bg,

!

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+1

RCy(t) =

1

RCg(t)

!

dy( t)

dt= "

1

RCy( t)+

1

RCg( t)

!

A = "1

RC

!

A

!

det(" #1+1

RC) = 0

!

" = #1

RC

!

"

System controllability

A physical system described by a linear state-space model

is controllable if and only if for any initial condition thereexists a control so that we can reach any final stateafter finite time. That is, if the system is controllable we cando anything with it!

Controllability Theorem: A linear system is controllable if andonly if

is the dimension of the state of the system

Control how? The Controllability Grammian is used to find theright g(t) (out of the scope of this course)

!

x(t0)

!

dx

dt= Ax + Bg,

!

x(t0)

!

g(t)

!

x( tf )

!

rank[B,AB,A2B,A

3B,...,A

n"1B] = n

!

n

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.