ku leuven - the behavioral approach to ...sistawww/smc/jwillems/...jan c. willems (k.u. leuven) –...

European Embedded Control Institute

HYCON-EECI Graduate School on Control

Spring 2009 - Module M4

THE BEHAVIORAL APPROACH

TO

MODELING AND CONTROL

LecturersPaolo Rapisarda (Un. of Southampton)

andJan C. Willems (K.U. Leuven)

– p. 1/82

Lecture 1

Monday 02-02-2009 14.00-17.30

Models and Behaviors

Lecturer: Jan C. Willems

– p. 2/82

Outline

1. Part I:

◮ Mathematical models◮ The universum◮ The behavior◮ A variety of examples

2. Part II:

◮ Dynamical systems◮ Systems described by ODEs◮ Linearity, time-invariance◮ Properties of dynamical systems◮ Other sets of independent variables

– p. 3/82

Mathematical models

A bit of mathematics & a bit of philosophy...

– p. 4/82

Mathematical models

Assume that we have a ‘real’ phenomenon.

(like a circuit with wires, or a moving object, or a chemicalsubstance with its attributes – we do not formalize the term‘phenomenon’ further, but we will give many examples).

The phenomenon produces‘events’, synonymously called‘outcomes’.

– p. 5/82

Mathematical models

Assume that we have a ‘real’ phenomenon. The phenomenonproduces‘events’, synonymously called‘outcomes’.

Phenomenon

event, outcome

– p. 5/82

Mathematical models


Phenomenon

event, outcome

We view a deterministic model for a phenomenon as aprescription of which eventscanoccur, and whichcannot.

– p. 5/82

Mathematical models


Phenomenon

event, outcome

We view a deterministic model for a phenomenon as aprescription of which eventscanoccur, and whichcannot.

‘Can’ means: ‘in principle possible’. Which event willactually occur in a particular ‘experiment’ depends on theenvironment to which the system is exposed.

– p. 5/82

Aim of this lecture

◮ In the first part of this lecture, we develop this point ofview into a mathematical formalism.

◮ In the second part, we apply this formalism to dynamicalsystems.

– p. 6/82

The universum

– p. 7/82

Mathematization

The events can be described in the language of mathematics,in terms of mathematical concepts, by answering:

To which universum do the (unmodelled) events belong?

– p. 8/82

Mathematization



◮ Do the events belong to a discrete set?; discrete event phenomena.

◮ Are the events real numbers, or vectors of real numbers?; continuous phenomena.

◮ Are the events functions of time?; dynamical phenomena.

◮ Are the events functions of space, or time & space?; distributed phenomena.

– p. 8/82

Mathematization



◮ Do the events belong to a discrete set?; discrete event phenomena.

◮ Are the events real numbers, or vectors of real numbers?; continuous phenomena.

◮ Are the events functions of time?; dynamical phenomena.

◮ Are the events functions of space, or time & space?; distributed phenomena.

The set where the events belong to is called theuniversumdenoted byU .

– p. 8/82

Discrete event phenomena

Examples:

◮ Words in a natural language(a word is thought of as an ‘event’)U = A

∗

= all (finite) strings of letters fromthe alphabetA = {a,b,c, . . . ,x,y,z}

– p. 9/82


Examples:

◮ Words in a natural languageU = A

∗


Sentences in a natural language

◮ DNA sequences

◮ Fortran codeLATEX code

◮ Error detecting and correcting codesISBN numbers

– p. 9/82

Continuous phenomena

Examples:

◮ The pressure, volume, quantity, and temperature of a gasin a vessel

Gas

(pressure, volume, quantity, temperature)

; U = (0,∞)× (0,∞)× (0,∞)× (0,∞)

– p. 10/82


◮ A spring

F1 F2

L

Event = (forceF1, force F2, length L)

; U = R×R× (0,∞)

– p. 11/82


◮ The gravitational attraction of two bodies

+

force

position1

position2

mass1

mass2

Event = mass1, mass2, position1, position2, force

; U = (0,∞)× (0,∞)×R3×R

3×R3

– p. 12/82


◮ The voltage across and the current through a resistor

V

I

R+

−

Event = (voltage, current) ; U = R2

– p. 13/82

Dynamical phenomena

Examples:

◮ The voltage across and the current through a capacitoror an inductor

V

I

+

−C

I

V+

−l

Event = (voltage, current): time→ R2

– p. 14/82

Dynamical phenomena

Examples:


V

I

+

−C

I

V+

−l

Event = (voltage, current): time→ R2 U =

(R

2)R

Recall the standard notation:AB := the set of maps from B to A = { f : B → A}

Dynamical phenomena; this course.– p. 14/82

Dynamical phenomena

◮ The motion of a point mass

+

mass

positionforce

Event = (force, position): time→ R3×R

3 =(R

3)R

– p. 15/82

Dynamical phenomena

◮ Planetary motion

SUN

PLANET

The events are maps fromR to R3

; U = {w : R → R3} =

(R

3)R

– p. 16/82

Dynamical phenomena

◮ The voltage across and the current into an electrical portwith ‘dynamics’

V

I

−

+��

RL

C

C

LR ��

��

The events are maps fromR to R2

; U = {(V, I) : R → R2} =

(R

3)R

– p. 17/82

Dynamical phenomena

◮ The input and the output of a signal processor

SYSTEM OUTPUTINPUT

Events: maps from Z to Rm×R

p

; U = {(u,y) : Z → Rm×R

p} = (Rm×Rp)Z

– p. 18/82

Dynamical phenomena


SYSTEM OUTPUTINPUT

Events: maps from Z to Rm×R

p

; U = {(u,y) : Z → Rm×R

p} = (Rm×Rp)Z

◮ Trajectories of variables associated with mechanicaldevices, electrical circuits, chemical reactions,multi-domain transducers, economic processes, ...phenomena with ‘memory’.

– p. 18/82

Distributed phenomena

◮ Temperature profile of, and heat absorbed by, a rod

��

��

q(x,t)

T(x,t)x

Events: maps from R×R to [0,∞)×R

; U = {(T,q) : R2 → [0,∞)×R} = ([0,∞)×R)R2

– p. 19/82


◮ EM fields.

In each point of space & at each time, there is an

electric field ~E(t,x,y,z)magnetic field ~B(t,x,y,z)current density ~j(t,x,y,z)charge density ρ(t,x,y,z)

Events: maps from R×R3 to R3×R3×R3×R

; U = {(~E,~B,~j,ρ) : R4 → R

10} =(R

10)R4

– p. 20/82


◮ EM fields.

In each point of space & at each time, there is an

electric field ~E(t,x,y,z)magnetic field ~B(t,x,y,z)current density ~j(t,x,y,z)charge density ρ(t,x,y,z)

Events: maps from R×R3 to R3×R3×R3×R

; U = {(~E,~B,~j,ρ) : R4 → R

10} =(R

10)R4

◮ Images

◮ Phenomena in which related things happen at differentpoints in space

– p. 20/82

A model is a subset: the ‘behavior’

– p. 21/82

The behavior

Given is a phenomenon with event universumU .Without further scrutiny, any event in U is possible.

After studying the situation, the conclusion is reached that theevents are constrained, that some laws are in force.

– p. 22/82

The behavior



Modeling means that certain events are declared to beimpossible, that they cannot occur.

The possibilities that remain constitute what we call the‘behavior’ of the model.

– p. 22/82

The behavior



A mathematical model:⇔ a pair (U ,B), with

U the universum of events

B a subset of U , called the behavior of the model

Notation: Mathematical model (U ,B)

– p. 22/82

The behavior

B

U

possible

allowed

forbidden

– p. 23/82

The behavior

Every “good” scientific theory is prohibition:it forbids certain things to happen...The more a theory forbids, the better it is.

Karl PopperConjectures and Refutations:The Growth of Scientific KnowledgeRouthledge, 1963

Karl Popper(1902-1994)

– p. 24/82

Examples

– p. 25/82


Examples:


∗


B = all words recognized by the spelling checker.For example, SPQR/∈ B.

B is basically defined by enumeration, by listing itselements.

– p. 26/82


Examples:


∗




Sentences in a natural language.B = all ‘legal’ sentences.SpecifyingB is a complicated issue.Usually determined using grammars.

– p. 26/82


Examples:


∗




Sentences in a natural language.B = all ‘legal’ sentences.

◮ DNA sequences.B =???

◮ LATEX code.B = all LATEX files that ‘run’.

– p. 26/82


◮ 32-bit binary strings with a parity check.

U = {0,1}32

B =

{

a1a2 · · ·a31a32 | ak ∈ {0,1} and a32(mod 2)

=31

∑k=1

ak

}

– p. 27/82



U = {0,1}32

B =

{

a1a2 · · ·a31a32 | ak ∈ {0,1} and a32(mod 2)

=31

∑k=1

ak

}

B can be expressed in other ways. For example,

B = {a1a2 · · ·a31a32 | ak ∈{0,1} and32

∑k=1

ak(mod 2)

= 0}

B =

a1a2...

a31a32

| ∃

b1b2...

b30b31

s.t.

a1a2...

a31a32

(mod 2)=

1 0 0 ··· 01 1 0 ··· 0... ... ... ...... ... ... ...0 0 ··· 1 10 0 ··· 0 1

b1b2...

b30b31

– p. 27/82



U = {0,1}32

B =

{

a1a2 · · ·a31a32 | ak ∈ {0,1} and a32(mod 2)

=31

∑k=1

ak

}

B can be expressed in other ways. For example,

B = {a1a2 · · ·a31a32 | ak ∈{0,1} and32

∑k=1

ak(mod 2)

= 0}

B =

a1a2...

a31a32

| ∃

b1b2...

b30b31

s.t.

a1a2...

a31a32

(mod 2)=

1 0 0 ··· 01 1 0 ··· 0... ... ... ...... ... ... ...0 0 ··· 1 10 0 ··· 0 1

b1b2...

b30b31

input/output representation

kernel representation

image representation– p. 27/82


Examples:

◮ The pressure, volume, quantity, and temperature of a gasin a vessel

Gas

(pressure, volume, quantity, temperature)

U = (0,∞)× (0,∞)× (0,∞)× (0,∞)

Gas law: B = {(P,V,N,T ) ∈ U | PV = NT }

– p. 28/82


◮ A massless spring

F1 F2

L

Event = (forceF1, force F2, length L)

U = R×R× (0,∞)

B ={

F1 = F2 L = ρ(F1)}

– p. 29/82


◮ The gravitational attraction of two bodies

1

+

position q mass M 1 2force F

position q 2

mass M

; U = (0,∞)× (0,∞)×R3×R

3×R3

B =

{

~F =M1M2~1M2→M1||~q1−~q2||2

}

‘inverse square law’Isaac Newton, 1642-1727

– p. 30/82


◮ The voltage across and the current through a resistor

V

I

R+

−

Event = (voltage, current) ; U = R2

‘Ohm’s law’ B = {(V, I) | V = RI }

Georg Ohm, 1789 – 1854

– p. 31/82

Dynamical phenomena

◮ The voltage across and the current through a capacitor

V

I

+

−C

Event = (voltage, current) as a function of time

; U =(R2

)R

B = {(V, I) : R → R2 | C d

dtV = I }

– p. 32/82

Dynamical phenomena


V

I

+

−C

I

V+

−l

Event = (voltage, current) as a function of time

; U =(R2

)R

B = {(V, I) : R → R2 | C d

dtV = I }

B = {(V, I) : R → R2 | V = L d

dt I }– p. 32/82

Dynamical phenomena

◮ Planetary motion U =(R3

)R

PLANET

SUN

DC

B

A 1 year

34 months

Kepler’s laws ; B

– p. 33/82

Dynamical phenomena

◮ Planetary motion U =(R3

)R

PLANET

SUN

DC

B

A 1 year

34 months

Kepler’s laws ; B = the orbits R → R3 with:

K.1 periodic, ellipses, with the sun in one of the foci;K.2 the vector from sun to planet sweeps out equal areas

in equal times;K.3 the square of the period

divided by the third powerof the major axis is thesame for all the planets

– p. 33/82

Dynamical phenomena

◮ Newton’s second law

Isaac Newtonby William Blake

+

massM

position~qforce ~F

U =(R

3×R3)R

B =

{

(~F,~q) : R → R3×R

3 | ~F = M d2

dt2~q

}

– p. 34/82

Static versus dynamic

For the pointmass, we could have chosen

event = (force, acceleration)

; U = R3×R

3B = { ~F = M~a }.

– p. 35/82




; U = R3×R

3B = { ~F = M~a }.

For the capacitor and for the inductor, we could have chosen

event = (voltage, charge); U = R×R B = { Q = CV }

event = (current, flux) ; U = R×R B = { NΦ = LI }with N = number of turns

Michael Faraday1791 – 1867

– p. 35/82




; U = R3×R

3B = { ~F = M~a }.

For the capacitor and for the inductor, we could have chosen

event = (voltage, charge); U = R×R B = { Q = CV }

event = (current, flux) ; U = R×R B = { NΦ = LI }with N = number of turns

Michael Faraday1791 – 1867

Combining with

~a = d2

dt2~q, I = ddt Q, V = N d

dt Φ,

recovers the dynamical equations.

– p. 35/82


Which of these descriptions is most appropriate depends onthe application.

But if we think in terms of ‘observable’ variables, or, better,in terms of the variables by which a device interacts with itsenvironment, it is most logical to choose

(~F,~q)

and(V, I)

as the phenomenological variables.

– p. 36/82

Dynamical phenomena


System outputs inputs

Events: maps fromZ toRm×R

p; U = {(u,y) : Z → R

m+p}

– p. 37/82

Dynamical phenomena


System outputs inputs

Events: maps fromZ toRm×R

p; U = {(u,y) : Z → R

m+p}

For an MA system

B ={

(u,y) : Z → R2 | y(t) = 12T+1 ∑t+T

t′=t−Tu(t ′)

}

many variations.– p. 37/82


◮ The temperature profile of, and heat absorbed by, a rod

��

��

q(x,t)

T(x,t)x

Events: maps fromR×R to [0,∞)×R

U = {(T,q) : R2 → [0,∞)×R}

B ={

(T,q) : R2 → [0,∞)×R | ∂

∂ t T = ∂ 2

∂ x2 T +q}

‘the diffusion equation’– p. 38/82


◮ Maxwell’s equations for EM fields in free space

∇ ·~E =1ε0

ρ ,

∇×~E = −∂∂ t

~B,

∇ ·~B = 0 ,

c2∇×~B =1ε0

~j +∂∂ t

~E.

B = maps fromthe independent variables:(t,x,y,z) time and spaceto the dependent variables:(~E,~B,~j,ρ)

electric & magnetic field, current & charge densitythat satisfy Maxwell’s equations.

James Clerk Maxwell1831 – 1879

– p. 39/82

Behavioral models

Behavioral models fit the tradition of modeling, but have notbeen approached or formalized as such in a deterministicsetting.The behavior captures the essence of what a model is.

– p. 40/82

Behavioral models


The behavior is all there is.Equivalence of models, properties of models,

symmetries, system identification, etc.,must all refer to the behavior.

– p. 40/82

Behavioral models


The behavior is all there is.Equivalence of models, properties of models,

symmetries, system identification, etc.,must all refer to the behavior.

Every ‘good’ scientific theory is prohibition: it forbidscertain things to happen...The more a theory forbids, the better it is.

Replace ‘scientific theory’ by ‘mathematical model’ !– p. 40/82

Recapitulation

◮ A phenomenon produces ‘events’, ‘outcomes’.; the event universumU

◮ A mathematical model specifies a subsetB of U .B is the behavior of the model.

◮ A mathematical model is a pair(U ,B).

– p. 41/82

End of Part I

– p. 42/82

Part II

– p. 43/82

Outline

1. Part I:

◮ Mathematical models◮ The universum◮ The behavior◮ A variety of examples

2. Part II:

◮ Dynamical systems

◮ Systems described by ODEs◮ Linearity, time-invariance◮ Properties of dynamical systems◮ Other sets of independent variables

– p. 44/82

Dynamical systems

– p. 45/82

The dynamic behavior

In dynamical systems, the ‘events’ are maps, with thetime-axis as domain, hence events are functions of time.

Phenomenon

eventsignal space

time

– p. 46/82



Phenomenon

eventsignal space

time

It is convenient to distinguish in the very notation

the domain of the maps, thetime setand the codomain, thesignal space

the set where the functions take on their values.

– p. 46/82



It is convenient to distinguish in the very notation

the domain of the maps, thetime setand the codomain, thesignal space

the set where the functions take on their values.

The behavior of a dynamical system, hence a family oftime-trajectories, is often described by a system of ordinarydifferential equations (ODEs) or difference equations.

– p. 46/82


Formal definition: A dynamical system:⇔ (T,W,B)

T ⊆ R time setW signal space

B ⊆ WT the behavior

a family of trajectories T → W

w : T → Rw ∈ B ⇔ w is compatible with the modelw : T → Rw /∈ B ⇔ the model forbids w

– p. 47/82


Formal definition: A dynamical system:⇔ (T,W,B)

T ⊆ R time setW signal space

B ⊆ WT the behavior

a family of trajectories T → W

w : T → Rw ∈ B ⇔ w is compatible with the modelw : T → Rw /∈ B ⇔ the model forbids w

mostly, T = R,R+,Z, or N (= {0,1,2, . . .}),and, in this course,W = Rw, for somew ∈ N

B is a family of time trajectoriestaking values in a (finite-dimensional) vector space.

T = R or R+ ; ‘continuous-time’ systems and ODEsT = Z or N ; ‘discrete-time’ systems and difference eqn’sWe mainly deal with the caseT = R.

– p. 47/82

ODEs as dynamical systems

– p. 48/82

ODEs as models for dynamical systems

Consider the ODE

f

(

w,ddt

w,d2

dt2w, . . . ,dn

dtnw

)

= 0

withf : W×R

w×·· ·×Rw

︸︷︷︸

n times

→ R•

W ⊆ Rw

Notational purists may prefer to write

f ◦

(

w,ddt

w,d2

dt2w, . . . ,dn

dtnw

)

= 0

But we leave this to puritans.

– p. 49/82

ODEs as models for dynamical systems

Consider the ODE

f

(

w,ddt

w,d2

dt2w, . . . ,dn

dtnw

)

= 0

withf : W×R

w×·· ·×Rw

︸︷︷︸

n times

→ R•

W ⊆ Rw

Defines the dynamical system(R,W,B) with

B = {w : R → W, sufficiently smooth|

f

(

w(t),dwdt

(t),d2wdt2 (t), . . . ,

dnwdtn

(t)

)

= 0 ∀ t ∈ R}

other solution concepts may be appropriate as well ...

– p. 49/82

Examples

◮ Newton’s second law

Isaac Newtonby William Blake

+

massM

position~qforce ~F

~F = M d2

dt2~q w =

[

~F~q

]

W = R3×R

3

f : (~F,ddt

~F,d2

dt2~F ,~q,

ddt

~q,d2

dt2~q) 7→ ~F −Md2

dt2~q

– p. 50/82

Examples

◮ A pointmass in a gravitational field

~q

SUN

PLANET

inverse square law

d2~qdt2 + k

~1~q

||~q||2= 0 (∗)

f : (~q,ddt

~q,d2

dt2~q) 7→d2

dt2~q+ k~1~q

||~q||2

– p. 51/82

Examples

◮ A pointmass in a gravitational field

~q

SUN

PLANET

inverse square law

d2~qdt2 + k

~1~q

||~q||2= 0 (∗)

All orbits satisfying K1, K2, and K3 are solutions.

PLANET

SUN

DC

B

A 1 year

34 months

Is (∗) a ‘representation’ of K1, K2, and K3?Do K1, K2, and K3 give all the solutions of(∗)?

– p. 51/82

Examples

The convolution

y(t) =

∫ t

0e(t−t ′)u(t ′)dt ′ w(t) =

[

u(t)y(t)

]

can be represented by

ddt

y = y+u

f : (u,ddt

u,y,ddt

y) 7→ddt

y− y−u

Note advantages/disdavantages of each of theserepresentations.

– p. 52/82

Linear time-invariant differential systems

LTIDSs

– p. 53/82

LTIDSs

The dynamical system(R,Rw,B) ; B is said to be

[[ linear ]] :⇔ [[ [[w1,w2 ∈ B,α ∈ R]] ⇒ [[αw1 +w2 ∈ B]] ]]

– p. 54/82

LTIDSs



[[ time-invariant ]] :⇔ [[ [[w ∈ B, σ t the t-shift]] ⇒ [[σ t(w) ∈ B]] ]]

σ t( f )(t ′) := f (t ′ + t)

σ

t−shift

f

map

t f

t

– p. 54/82

LTIDSs



[[ time-invariant ]] :⇔ [[ [[w ∈ B, σ t the t-shift]] ⇒ [[σ t(w) ∈ B]] ]]

σ t( f )(t ′) := f (t ′ + t)

σ

t−shift

f

map

t f

t

[[ differential ]] :⇔ [[B can be ‘described’ by an ODE]].

– p. 54/82

Linearity

In many examples,B is the graph of a map

u 7→ y = F(u) w =

[

uy

]

with F a map: u ∈ a space of inputs7→ y ∈ a space of outputs.

Then B = {w =

[

uy

]

| y = F(u)} = the ‘graph’ of F

When F is a linear map, the resulting system is linear.Def. of linear system has the classical one as a special case.

– p. 55/82

Linearity

In many examples,B is the graph of a map

u 7→ y = F(u) w =

[

uy

]

with F a map: u ∈ a space of inputs7→ y ∈ a space of outputs.

Then B = {w =

[

uy

]

| y = F(u)} = the ‘graph’ of F

When F is a linear map, the resulting system is linear.Def. of linear system has the classical one as a special case.

But a dynamical system, even in input/output form(say p( d

dt )y = q( ddt )u) is seldom a map !

y depends onu and on the initial conditions.– p. 55/82

LTIDSs

The dynamical system(R,Rw,B) is

a linear time-invariant differential system (LTIDS) :⇔the behavior consists of the set of solutions of a system oflinear constant coefficient ODEs

R0w+R1ddt

w+ · · ·+Rn

dn

dtnw = 0.

R0,R1, · · · ,Rn ∈ R•×w real matrices that parametrize thesystem, andw : R → Rw.

– p. 56/82

LTIDSs

The dynamical system(R,Rw,B) is

a linear time-invariant differential system (LTIDS) :⇔the behavior consists of the set of solutions of a system oflinear constant coefficient ODEs

R0w+R1ddt

w+ · · ·+Rn

dn

dtnw = 0.

R0,R1, · · · ,Rn ∈ R•×w real matrices that parametrize thesystem, andw : R → Rw. In polynomial matrix notation

; R(

ddt

)w = 0

with R(ξ ) = R0 +R1ξ + · · ·+Rnξ n ∈ R [ξ ]•×w

a polynomial matrix , usually ‘wide’ or square.

– p. 56/82

Examples

Examples:

◮d2

dt2~q = ~F ,w =

[

~F~q

]

; R(ξ ) =[

I3×3... −I3×3ξ 2

]

– p. 57/82

Examples

Examples:

◮d2

dt2~q = ~F ,w =

[

~F~q

]

; R(ξ ) =[

I3×3... −I3×3ξ 2

]

◮ddt x = Ax+Bu, y = Cx+Du, w =

xuy

;

R(ξ ) =

[

−ξ In×n +A B 0C D −Ip×p

]

◮ p0w+ p1ddt w+ · · ·+ pn dn

dtn w = 0; R(ξ ) = p(ξ ) with p(ξ ) = p0 + p1ξ + · · ·+ pnξ n.

– p. 57/82

LTIDSs

We should define what we mean by a solution of

R(

ddt

)w = 0

For ease of exposition, we takeC ∞ (R,Rw) solutions.Hence the behavior defined is

B =

{

w ∈ C∞ (R,Rw) | R

(ddt

)

w = 0

}

– p. 58/82

LTIDSs

We should define what we mean by a solution of

R(

ddt

)w = 0

For ease of exposition, we takeC ∞ (R,Rw) solutions.Hence the behavior defined is

B =

{

w ∈ C∞ (R,Rw) | R

(ddt

)

w = 0

}

B = kernel(R

(ddt

))‘kernel representation’ of this B.

LTIDS B ↔ kernel(R

(ddt

))

The theory of LTIDSs is basically the study of real polynomialmatrices, matrices over the ringR [ξ ].

– p. 58/82

LTIDSs

B = kernel(R

(ddt

))‘kernel representation’ of this B.

LTIDS B ↔ kernel(R

(ddt

))

The theory of LTIDSs is basically the study of real polynomialmatrices, matrices over the ringR [ξ ].

The theory of LTIDSs is the topic of lectures 2 and 3.

– p. 58/82

Properties of dynamical systems

– p. 59/82

Controllability

Assume thatΣ = (R,W,B) is time-invariant(to avoid irrelevant complications)

and T = R (for the sake of concreteness)

Σ is said to be controllable :⇔for all w1,w2 ∈ B, there existsT ≥ 0 and w ∈ B such that

w(t) =

{

w1(t) for t < 0w2(t −T ) for t ≥ T

– p. 60/82

A picture is worth a thousand words

1

w2

w

time

W

2w

1

w

w

time

W W

controllability ⇔ concatenability of trajectories after a delay

– p. 61/82

Stabilizability

Assume thatΣ = (R,Rw,B)T = R,W = Rw (for the sake of concreteness)

Σ is said to be stabilizable :⇔for all w ∈ B, there existsw′ ∈ B such that

w′(t) = w(t) for t < 0

andw′(t) → ∞ for t → ∞

– p. 62/82

In pictures

w’

w

0

time

Rw

stabilizability ⇔ trajectories can be steered to0

– p. 63/82

Autonomous systems

Assume thatΣ = (R,W,B)T = R (for the sake of concreteness)

Σ is said to be autonomous :⇔

w1,w2 ∈ B and w1(t) = w2(t) for t < 0 ⇒ w1 = w2.

– p. 64/82

In pictures

time

time

FUTURE

PAST

W

W

In an autonomous system, the past determines the future.

– p. 65/82

Autonomous systems

Assume thatΣ = (R,W,B)T = R (for the sake of concreteness)



Equivalently, there is a map

F : W(−∞,0) → W

[0,+∞)

such that

w ∈ B ⇒ w|[0,+∞)= F

(

w|(−∞,0)

)

– p. 66/82

Autonomous systems




Examples:

◮ Kepler’s laws.

◮dndtnw = f

(

w, ddt w, . . . , dn−1

dtn−1w)

◮ In particular, ddt x = f (x)

– p. 67/82

Autonomous systems




Examples:

◮ Kepler’s laws.

◮dndtnw = f

(

w, ddt w, . . . , dn−1

dtn−1w)

◮ In particular, ddt x = f (x)

Autonomous models aim at‘closed’ systems.Our aim is principally ‘open’ systems.

– p. 67/82

Observability

Consider Σ = (T,W1×W2,B).

w2 is said to be observable fromw1 in Σ :⇔

(w1,w′2),(w1,w

′′2) ∈ B ⇒ w′

2 = w′′2

– p. 68/82

In pictures

observed w2to−be−deduced SYSTEMw1variables variables

– p. 69/82

In pictures

observed w2to−be−deduced SYSTEMw1variables variables

observability ⇔ unobserved can be deduced from observed

Note: knowing the laws that govern the system !!

– p. 69/82

Observability



(w1,w′2),(w1,w

′′2) ∈ B ⇒ w′

2 = w′′2

Equivalently, iff there exists a mapF : WT1 → W

T2 such that

(w1,w2) ∈ B ⇒ w2 = F (w1)

– p. 70/82

Observability



(w1,w′2),(w1,w

′′2) ∈ B ⇒ w′

2 = w′′2

Equivalently, iff there exists a mapF : WT1 → W

T2 such that

(w1,w2) ∈ B ⇒ w2 = F (w1)

In applications, often‘observed’ = manifest, ‘to-be-deduced’ = latent (see lecture 2).In this case we can speak about‘an observable model’But in general observability depends on the partitionW = W1×W2.

– p. 70/82

Detectability

Consider Σ = (T,Rw1 ×Rw2,B).

w2 is said to be detectable fromw1 in Σ :⇔

(w1,w′2),(w1,w

′′2) ∈ B ⇒ w′

2(t)−w′′2(t) →t→∞ 0

w2 can be asymptotically deduced fromw1 (knowing the lawsof the system).

– p. 71/82

Controllability versus observability

In our setting, controllability is an intrinsic property of asystem.

Observability depends on the variable partition.

There is no ‘duality’ - we need more structure for that.

– p. 72/82

LTIDSs

Controllability, observability, stabilizability, auton omy, etc.will be studied in great detail for LTIDSs in lecture 2.

– p. 73/82

State controllability and observability

Consider classical Kalman definitions for

x = f (x,u) y = f (x,u)state spaceX, input spaceU, output spaceY

– p. 74/82




state controllability :⇔∀ x1,x2 ∈ X,∃ T ≥ 0 and u : [0,T ] → U

such that x(0) = x1 yields x(T ) = x2.

1

x2X2

x

This is aspecial caseof our controllability:variables: w = x = stateor w = (x,u) = (input, state)

– p. 74/82




Similar for state observabilityKalman definition: observed (u,y); to-be-deducedx.

– p. 74/82




Why should we be so concerned with the state?

If a system is not (state) controllable, what is it caused by?insufficient influence of the control?bad choice of the state?not properly editing of the equations?

Kalman’s definitions address a rather special situation.

– p. 74/82

Other sets of independent variables

– p. 75/82

Other time sets

The setting is identical for LTIDSs with time set

[0,∞),(−∞,0] or [t1, t2] .

– p. 76/82

Other time sets


[0,∞),(−∞,0] or [t1, t2] .

For discrete-time systems with time-axisN := {0,1,2, . . .},the differential operator should be replaced by the shiftσ ,σ( f )(t) := f (t +1).Only non-negative powers ofσ make sense.

– p. 76/82

Other time sets


[0,∞),(−∞,0] or [t1, t2] .

For discrete-time systems with time-axisN := {0,1,2, . . .},the differential operator should be replaced by the shiftσ ,σ( f )(t) := f (t +1).Only non-negative powers ofσ make sense.

For discrete-time systems with time-axisZ, the differentialoperator should be replaced by the shiftσσ( f )(t) := f (t +1).Both positive and negative powers ofσ make sense.σ−1( f )(t) := f (t −1).

– p. 76/82


In distributed systems, ‘events’ are maps from a set ofindependent variables to a set ofdependent variables .Usually the independent variables are time and space.

��

��

��

��

��

��

event

��

��

Phenomenon

t

spacex

w

time

– p. 77/82


In distributed systems, ‘events’ are maps from a set ofindependent variables to a set ofdependent variables .Usually the independent variables are time and space.

The behavior of a distributed system is usually described bya system of partial differential equations (PDEs).

– p. 77/82

Example: heat diffusion

��

��

q(x,t)

T(x,t)x

Events: maps fromR×R to R×R

B ={

(T,q) : R2 → [0,∞)×R | ∂∂ t T = ∂ 2

∂ x2 T +q}

set of independent variables:R2 or R+×R, time and space.set of dependent variables:R2 or R+×R,

temperature and heat flow.

– p. 78/82

Example: Maxwell’s equations

∇ ·~E =1ε0

ρ ,

∇×~E = −∂∂ t

~B,

∇ ·~B = 0 ,

c2∇×~B =1ε0

~j +∂∂ t

~E.

independent variables:(t,x,y,z) time and space ; R4

dependent variables:(~E,~B,~j,ρ)

electric & magnetic field, current & charge density ; R10.

James Clerk Maxwell1831 – 1879

– p. 79/82

Summary of Lecture 1

– p. 80/82

The main points

◮ A model is a subsetB of a universumU .B is the behavior of the model.

– p. 81/82

The main points


◮ Dynamical systems: events are time trajectories.

– p. 81/82

The main points



◮ Important system properties: controllability,stabilizability, observability, detectability, autonomy.

– p. 81/82

The main points




◮ Controllability and observability generalize theirclassical counterparts in a meaningful way.

– p. 81/82

The main points




◮ Controllability and observability generalize theirclassical counterparts in a meaningful way.

◮ Distributed phenomena fit well in this setting by taking amore general set of independent variables than just time.

– p. 81/82

End of Lecture 1

– p. 82/82

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