Minicourse on behavioral systems theory Lecture 1: General introduction

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The Behavioral Approach

to

Systems Theory

Paolo Rapisarda, Un. of Southampton, U.K. &

Jan C. Willems, K.U. Leuven, Belgium

MTNS 2006

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Lecture 1: General Introduction

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Questions

• What is a mathematical model , really?

• How is this specialized to dynamics ?

• How are models arrived at?

• From basic laws:‘first principles’ modeling

• Combined with interconnection:

tearing, zooming, & linking

• From measured data:SYSID(system identification)

• What is the role of (differential) equations ?

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Questions

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Questions

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Questions

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Questions

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The seminal idea

Consider a ‘phenomenon’; produces ‘outcomes’, ‘events’ .

Mathematization: events belong to a set,U.

Modeling question: Which events can really occur ?

The model specifies: Only those in the subsetB⊆U!

⇒⇒ a mathematical model, withbehaviorB ⇐⇐

Before modeling: events inU are possible

After modeling: only events inBare possible

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The seminal idea

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The seminal idea

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Examples

Gas law

Phenomenon: A balloon filled with a gas

Gas

¡¡ Model the relation between

volume, quantity, pressure, & temperature !!

Event: (vol. V, quant. N, press. P, temp. T) ;U=R4+

Charles Boyle and Avogadro

; model

PV

NT = a universal constant=:R

⇒⇒ B= n

(T,P,V,N)∈_R4 + |

PV

NT =R

o

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Examples

Gas law

Gas

; model

PV

⇒⇒ B= n

(T,P,V,N)∈_R4 + |

PV

NT =R

o

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Examples

Gas law

Gas

; model

PV

⇒⇒ B= n

(T,P,V,N)∈_R4 + |

PV

NT =R

o

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Examples

An economy Phenomenon: trading of a product

price, sales & production !!

Event: price P , demand D, supply S ;U= R3+

U =R2+

B=intersection of two graphs : ;usually point(s) The price P becomes a ‘hidden’

variable. Modeling using ‘hidden’, ‘auxiliary’, ‘latent’ intermediate variables is very common.

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Examples

Event: price P , demand D _;U=R2+

Typical model: B= graph of a curve

Demand

Price

U =R2+

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Examples

Event: price P, supply S _;U= R2+

Typical model: B= graph of a curve

Supply

Price

U =R2+

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Examples

U =R2+

B=intersection of two graphs : ;usually point(s)

Price DemandSupply

The price P becomes a ‘hidden’ variable. Modeling using ‘hidden’, ‘auxiliary’, ‘latent’ intermediate variables is very common.

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Examples

sales & production !! Price only to explain mechanism

Event: demand D, supply S _; U=R2+

B=intersection of two graphs : ;usually point(s)

DemandSupply The price P becomes a ‘hidden’

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Examples

Newton’s 2-nd law

Phenomenon: A moving mass FORCE

MASS

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Examples

MASS

force, mass, & acceleration !!

Event: (force F, mass m, acceleration a)

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Examples

MASS

force, mass, & acceleration !!

Event: (force F, mass m, acceleration a)

;U= R3×R+×R3 Model due to Newton:

F =ma

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Examples

MASS

But, the aim of Newton’s law is really:

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Examples

MASS

But, the aim of Newton’s law is really:

force, mass, & position !!

Event: (force F, mass m, position q)

F =ma, a= d

2

dt2q

not ‘instantaneous’ relation between F,m,q _; dynamics

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Dynamical systems

Phenomenon produces ‘events’ that are functions of time

Mathematization: It is convenient to distinguish

domain (‘independent’variables) T⊆R ‘time-axis’

co-domain(‘dependent’variables) W ‘signal space’

A dynamical system :=

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Dynamical systems

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Dynamical systems

Σ = (T,W,B) B⊆(W)T the behavior

T=R,R+,or interval inR: continuous-timesystems

T=Z,N, etc.: discrete-timesystems

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Dynamical systems

W=Rw,etc. lumpedsystems

W=finite: finitary systems

T=ZorN,Wfinite: DES(discrete event systems)

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Dynamical systems

Wvector space,B ⊂(W)Tlinear subspace: linearsystems controllability, observability, stabilizability, dissipativity, stability, symmetry, reversibility, (equivalent) representa-tions, etc.: to be defined in terms of the behaviorB

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Examples

FORCE

MASS

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Examples

FORCE

MASS

force & position of a pointmass !!

Event: (force F (a f’n of time), position q (a f’n of time))

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Examples

FORCE

MASS

force & position of a pointmass !!

Event: (force F (a f’n of time), position q (a f’n of time))

;T=R,W =R3×R3 Model:

F =ma, a = d

2

dt2q

; Σ = (R,R3×R3,B) with

⇒⇒ B =n (F,q) :R→R3×R3 | F =m d2 dt2q

o

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Examples

RLC circuit

Phenomenon: the port voltage and current, f’ns of time

I + − V R_L C R C L system environment I + − V R_L C R C L

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Examples

RLC circuit

; Σ = (R,R2,B)

behaviorBspecified by:

Case 1: CRC 6= L RL RC RL +

1+RC RL

CRC d dt +CRC

L RL d2 dt2 V =

1+CRC d

dt 1+ L RL

d dt

RCI

Case 2: CRC = L RL

RC RL

+CRC d dt

V = (1+CRC) d dtRCI

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Examples

input/output models u1 u2 u 1 y u m y 2 p

input SYSTEM output

y(t) =f(y(t−1),· · ·,y(t−n),u(t),u(t−1),u(t−n)), w =

u y

Differential equation analogue

P(d

dt)y =P( d

dt)u,w =

u y

, P,Q: polynomial matrices

or matrices of rational functions as in y =G(s)u

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Examples

input/output models

State models

R.E. Kalman

d

dtx =Ax+Bu, y =Cx+Du; d

dtx =f◦(x,u),y =h◦(x,u)

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Examples

input/output models

State models

d

dtx =Ax+Bu, y =Cx+Du; d

dtx =f◦(x,u),y =h◦(x,u)

¿¿ What is the behavior of this system ??

In applications, we care foremost about i/o pairs u,y

; Σ = (R,U×Y,B)

B={(u,y) :R→U×Y|

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Latent variables

Auxiliary variables. We call them ‘latent’ . They are ubiqui-tous:

• states in dynamical systems

• prices in economics

• the wave function in QM

• the basic probability spaceΩ

• potentials in mechanics, in EM

• interconnection variables

• driving variables in linear system theory

• etc., etc.

Their importance in applications merits formalization.

Latent variable model:= (U,L,Bfull) withBfull ⊆(U×L)

U: space of manifest variables L: space of latent variables

Bfull: ‘full behavior’

B={u ∈U|∃`∈L: (u, `)∈B_full}: ‘manifest behavior’.

This is readily generalized to dynamical systems.

Alatent variable dynamical system :=

(T,W,L,Bfull) withBfull ⊆(W×L)T

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Latent variables

Latent variable model:= (U,L,B_full) withB_full ⊆(U×L)

B_full: ‘full behavior’

B={u ∈U|∃`∈L: (u, `)∈Bfull}: ‘manifest behavior’.

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Latent variables

Latent variable model:= (U,L,B_full) withB_full ⊆(U×L)

B_full: ‘full behavior’

B={u ∈U|∃`∈L: (u, `)∈Bfull}: ‘manifest behavior’.

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Example

The price in our economic example

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Example

RLC circuit

a

V Vb

Ia

Ib

R_L C

C

L R

environment system

Model voltage/current histories as a f’n of time !

How do we actually go about this modeling ?

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Example

RLC circuit

TEARING

11

R 7 _L

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Example

RLC circuit

ZOOMING

Thelist of the modules& theassociated terminals:

Module Type Terminals Parameter

RC resistor (1, 2) in ohms

RL resistor (3, 4) in ohms

C capacitor (5, 6) in farad

L inductor (7, 8) in henry

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Example

RLC circuit

TEARING

Theinterconnection architecture:

11

R 7 L

8 C 6 5 R_L 4 3 10 9 12 13 14 2 1 C connector1 connector2 Pairing

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Example

RLC circuit

Manifest variable assignment:

the variables

V9,I9,V12,I12

on the external terminals{9, 12}, i.e,

Va = V9,Ia =I9,Vb =V12,Ib =I12.

The internal terminals are

{1,2,3,4,5,6,7,8,10,11,13,14}

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Example

RLC circuit

Faraday Ohm

Henry Coulomb

Equations for the full behavior:

Modules Constitutive equations

RC I1+I2=0 V1−V2=RCI1

RL I7+I8=0 V7−V8=RLI7

C I5+I6=0 C_dtd(V5−V6) =I5

L I7+I8=0 V7−V8=LdtdI7

connector1 I9+I10+I11=0 V9=V10=V11

Kirchhoff

Interconnection pair Interconnection equations

{10,1} V10=V1 I10+I1=0 {11,7} V11=V7 I11+I7=0

{2,5} V2=V5 I2+I5=0

{8,3} V8=V3 I8+I3=0

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Example

RLC circuit

All these eq’ns combined define a latent variable system in themanifest ‘external’ variables

w = (Va,Ia,Vb,Ib)

with‘internal’ latent variables

`= (V1,I1,V2,I2,V3,I3,V4,I4,V5,I5,V6,I6,V7,I7,

V8,I8,V10,I10,V11,I11,V13,I13,V14,I14).

The manifest behaviorBis given by

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Example

RLC circuit

Elimination:

Case 1: CRC 6=

L

RL

.

(RC RL

+(1+RC RL

)CRC d dt+CRC

L RL

d2

dt2)(Va−Vb)= (1+CRC d dt)(1+

L RL

d dt)RCIa. Ia+Ib =0

Case 2: CRC =

L RL

.

(RC RL

+CRC d

dt)(Va−Vb)= (1+CRC d dt)RCIa Ia+Ib =0

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Example

RLC circuit

Note: the eliminated equations are differential equations!

Does this follow from some general principle ?

Algorithms for elimination?

The modeling of this RLC circuit is an example of tearing, zooming & linking . It is the most prevalent way of modeling. See my website for formalization. Crucial role of latent variables.

Note: no input/output thinking;

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System properties

In this framework, system theoretic notions like

Controllability, observability, stabilizability,...

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System properties

In this framework, system theoretic notions like

Controllability, observability, stabilizability,...

become simpler, more general, more convincing.

For simplicity, we consider only

time-invariant, continuous-time systems withT=R

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Controllability

The time-invariant systemΣ = (T,W,B)is said to be

controllable

if for allw1,w2 ∈B ∃ w ∈Band T ≥0 such that

w(t) =

w1(t) t <0

w2(t −T) t ≥T

Controllability :⇔

legal trajectories must be‘patch-able’, ‘concatenable’.

2 0 1 w w W time 2 0 T 1 w w

!Tw W

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Controllability

2 0

1 w

w

W

time

2

0 T

1

w

!Tw W

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Controllability

2 0

1 w

w

W

time

2

0 T

1

w

!Tw W

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Examples

d

dtx =Ax +Bu; d

dtx = f ◦(x,u)

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Examples

d

dtx =Ax +Bu; d

dtx = f ◦(x,u)

withw = (x,u), controllable ⇔ ‘state point’ controllable.

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Examples

d

dtx =Ax +Bu; d

dtx = f ◦(x,u)

RLC circuit Case 2: CRC =

L RL

(RC RL

+CRC d

dt)(Va−Vb)= (1+CRC d dt)RCIa Ia+Ib =0

Assume also RC =RL . Controllable ?

Va−Vb =RCIa+constant·e

− t

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Examples

d

dtx =Ax +Bu; d

dtx = f ◦(x,u)

p(d

dt)y =q( d

dt)u

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Examples

d

dtx =Ax +Bu; d

dtx = f ◦(x,u)

w =M(d

dt)`

M a polynomial matrix, always has a controllable manifest behavior.

In fact, this characterizes the

controllable linear time-invariant differentiable systems (‘image representation’).

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Examples

w =M(d

dt)`

M a polynomial matrix, always has a controllable manifest behavior. Likewise,

w =F(d

dt)`

F matrix ofrat. f’nshas controllable manifest behavior. But we need to give this ‘differential equation’ a meaning.

Whence

y =G(d

dt)u, w =

u y

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Observability

SYSTEM

w₁

variables

to!be!deduce variables

w₂

observed

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Observability

Consider the systemΣ = (T,W1×W2,B). Each element

ofBhence consists of a pair of trajectories(w1,w2):

w1 :observed;

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Observability

w1 :observed;

w2 :to-be-deduced.

Definition:w2is said to be

observable from w1

if[[(w1,w20) ∈B, and(w1,w200)∈B]]⇒[[(w20 =w200)]],

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Observability

w1 :observed;

w2 :to-be-deduced.

observable from w1

if[[(w1,w20) ∈B, and(w1,w200)∈B]]⇒[[(w20 =w200)]],

i.e., if onB, there exists a mapw1 7→w2.

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Examples

d

dtx =Ax+Bu,y = Cx+Du; d

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Examples

a V Vb Ia Ib R_L C C L R environment system

Controllability of this system (referring to external terminal variables) is a well-defined question.

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Examples

Faraday Ohm

Henry Coulomb

Equations for the full behavior:

Modules Constitutive equations

RC I1+I2=0 V1−V2=RCI1

RL I7+I8=0 V7−V8=RLI7

C I5+I6=0 Cdtd(V5−V6) =I5

L I7+I8=0 V7−V8=L_dtdI7

Kirchhoff

Interconnection pair Interconnection equations

{10,1} V10=V1 I10+I1=0 {11,7} V11=V7 I11+I7=0

{2,5} V2=V5 I2+I5=0

{8,3} V8=V3 I8+I3=0

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Examples

All these eq’ns combined define a latent variable system in

themanifest variables

w = (Va,Ia,Vb,Ib)

withlatent variables

`= (V1,I1,V2,I2,V3,I3,V4,I4,V5,I5,V6,I6,V7,I7,

V8,I8,V10,I10,V11,I11,V13,I13,V14,I14).

The manifest behaviorBis given by

B={(Va,Ia,Vb,Ib):R→ R4 | ∃ `:R→R24. . .}

Are the latent variables observable from the manifest ones?

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Examples

p(d

dt)y =q( d

dt)u

u is observable from y ⇔ q = non-zero constant

( no zeros ).

A controllable linear time-invariant differential system al-ways has an observable ‘image’ representation

w =M(d

dt)`.

In fact, this again characterizes the

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Kalman definitions

Special case: classical Kalman definitions for

d

dtx = f ◦(x,u), y =h◦(x,u).

R.E. Kalman

controllability: variables =(input, state)

If a system is not (state) controllable, why is it?

Insufficient influence of the control?

Orbad choice of the state?

observability: _;observed = (input, output),

to-be-deduced = state.

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Kalman definitions

d

dtx = f ◦(x,u), y =h◦(x,u).

R.E. Kalman

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Kalman definitions

d

dtx = f ◦(x,u), y =h◦(x,u).

R.E. Kalman

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Stabilizability

The systemΣ = (T,Rw,B)is said to be stabilizable if, for allw ∈B, there existsw0 ∈Bsuch that

w(t) =w0(t)for t <0 and w0(t) −→

t→∞ 0.

Stabilizability :⇔

legal trajectories can be steered to a desired point.

w’

w

0

W

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Stabilizability

The systemΣ = (T,Rw,B)is said to be stabilizable if, for allw ∈B, there existsw0 ∈Bsuch that

w(t) =w0(t)for t <0 and w0(t) −→

t→∞ 0.

Stabilizability :⇔

legal trajectories can be steered to a desired point.

w’

w

0

W

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Detectability

SYSTEM

w₁

variables

w₂

observed

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Detectability

SYSTEM

w₁

variables

w₂

observed

¿ Is it possible to deducew2asymptotically fromw1 ?

detectable from w1 if

[[(w1,w20)∈B, and(w1,w200)∈B]]

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Btw

• A model is not a map, but a relation.

• A flow is a very limited model class.

;closed dynamical systems.

• An open dynamical system is not an input/output map .

• input/state/output systems, although still limited, are the first class of suitably general models

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Btw

• A flow

d

dtx =f(x)with or without y =h(x)

is a very limited model class.

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Btw

Heaviside Wiener Nyquist Bode

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Btw

R.E. Kalman

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Btw

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Summary

• A mathematical model = a subset

• A dynamical system = a behavior

= a family of trajectories

• Latent variables are ubiquitous in models

• Important properties of dynamical systems

• Controllability : concatenability of trajectories • Observability : deducing one trajectory from another • Stabilizability : driving a trajectory to zero

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Summary

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Summary

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Summary

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Summary

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Stochastic models

We only considerdeterministicmodels. Stochastic models:

Laplace Kolmogorov

there is a map P (the’probability’)

P :A →[0,1]

withAa ‘σ-algebra’ of subsets ofU.

P(B) =‘degree of certainty’ (relative frequency, propensity, plausibility, belief) that outcomes are inB; ∼

=the degree of validity ofBas a model.

Determinism: P is a‘{0,1}-law’

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Stochastic models

We only considerdeterministicmodels. Stochastic models: there is a map P (the’probability’)

P :A →[0,1]

withAa ‘σ-algebra’ of subsets ofU.

P(B) =‘degree of certainty’ (relative frequency, propensity, plausibility, belief) that outcomes are inB; ∼

=the degree of validity ofBas a model.

Determinism: P is a‘{0,1}-law’

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Fuzzy models

L. Zadeh

Fuzzy models:there is a mapµ(‘membership f’n’)

µ:U→[0,1] µ(x) =‘the extent to which x belongs to

the model’s behavior’.

Determinism:µis‘crisp’:

image(µ) ={0,1},

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Fuzzy models

Fuzzy models:there is a mapµ(‘membership f’n’)

µ:U→[0,1]

µ(x) =‘the extent to which x belongs to

the model’s behavior’.

Determinism:µis‘crisp’:

image(µ) ={0,1},

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Every ‘good’ scientific theory is prohibition: it forbids certain things to happen...

The more a theory forbids, the better it is.

Karl Popper (1902-1994)