INTERCONNECTED SYSTEMS

INTERCONNECTED SYSTEMS

Jan C. WillemsK.U. Leuven, Flanders, Belgium

Mini-symposium, TU Delft October 15, 2008

– p. 1/106

Outline

◮ Open, connected, and modular

◮ [Classical dynamical systems]

◮ [Input/output systems]

◮ Modeling by tearing, zooming, and linking

◮ [On canceling poles and zeros]

◮ [DAEs]

◮ Signal flow graphs

◮ Bond graphs

◮ Circuit diagrams

◮ [Control as interconnection]

– p. 2/106

Systems

– p. 3/106

– p. 4/106

Features

◮ open

◮ interconnected

◮ modular

◮ dynamic

– p. 5/106

Features

◮ open

◮ interconnected

◮ modular

◮ dynamic

Aim:

develop a suitable mathematical language

aimed at computer-assisted modeling.

Modeling ⇔ Describing reality accurately

– p. 5/106

Open, connected, modular

– p. 6/106

Open

SYSTEM

ENVIRONMENT

Boundary

Systems interact with their environment

– p. 7/106

Connected

Systems consist of an architecture of interconnected subsystems

– p. 8/106

Modular

Systems are modular: composed of‘building blocks’

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SYSTEM

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– p. 9/106

The development of the notion

of a dynamical system

– p. 10/106

Theme

– p. 11/106

First things first

1. Get the physics right

2. The rest is mathematics

R.E. KalmanOpening lecture

IFAC World CongressPrague, July 4, 2005

– p. 12/106

First things first

1. Get the physics right

2. The rest is mathematics

R.E. KalmanOpening lecture

IFAC World CongressPrague, July 4, 2005

Prima la fisica, poi la matematica

– p. 12/106

The missing link

◮ Get the physics right

◮ Translate the physics into mathematics

◮ The rest is mathematics

What are the ‘right’ concepts?

What is the ‘natural’ generalization?

What are the ‘relevant’ questions?

– p. 13/106

Closed dynamical systems

– p. 14/106


K.1, K.2, & K.3

;d2

dt2w(t)+1w(t)

| ddt w(t)|2

= 0

; with x = (w,

ddt w) ;

ddt x = f (x)

– p. 15/106


K.1, K.2, & K.3

;d2

dt2w(t)+1w(t)

| ddt w(t)|2

= 0

; with x = (w,

ddt w) ;

ddt x = f (x)

ddt x = f (x) ; ‘dynamical systems’, flows

; flows as paradigm of dynamics ; closed systems

Motion determined by initial conditions: a (inadequate)paradigm for modeling dynamics. Very frequently inmathematics and physics (chaos theory, synchronization,flows on manifolds, classical Hamiltonian mechanics, ... )

– p. 15/106

Inputs and outputs

– p. 16/106

Input/output systems

SYSTEMstimulus response

causeinput

effectoutput

Transfer functions, impedances, convolutions,Volterra series, ...

– p. 17/106

Input/output systems


causeinput

effectoutput

Oliver Heaviside (1850-1925)

Norbert Wiener (1894-1964)

and the many electrical circuit theorists

– p. 17/106

Mathematical description


causeinput

effectoutput

y(t) =∫ t

0 or −∞ H(t − t ′)u(t ′) dt ′

– p. 18/106

Mathematical description


causeinput

effectoutput

y(t) =∫ t

0 or −∞ H(t − t ′)u(t ′) dt ′

y(t) = H0(t)+∫ t

−∞H1(t − t ′)u(t ′) dt ′+

∫ t

−∞

∫ t ′

−∞H2(t − t ′,t ′− t ′′)u(t ′)u(t ′′) dt ′dt ′′ + · · ·

Awkward nonlinear — far from the physicsFail to deal with ‘initial conditions’.

– p. 18/106

Input/state/output systems

Around 1960: aparadigm shift to

ddt x = f (x,u), y = g(x,u)

Rudolf Kalman (1930- )

◮ open

◮ ready to be interconnectedoutputs of one system7→ inputs of another

◮ deals with initial conditions

◮ incorporates nonlinearities, time-variation

◮ models many physical phenomena

◮ · · ·

This framework turned out to be very effective and useful!– p. 19/106

Theme of this lecture

We are accustomed to view an open dynamical system as aninput/output structure (with or without the state)

outputsI/O SYSTEMinputs

– p. 20/106


We are accustomed to view an open dynamical system as aninput/output structure (with or without the state)

outputsI/O SYSTEMinputs

Is this an appropriate abstractionof models of physical systems?

– p. 20/106


and interconnection as output-to-input assignment

– p. 21/106



– p. 21/106



Feedback

Series

Parallel

– p. 21/106



Feedback

Series

Parallel

Is this an appropriate abstraction ofinterconnection of physical systems?

– p. 21/106

An example

– p. 22/106

Tearing

(pressure, flow) (pressure, flow)

– p. 23/106

Tearing


(p’,f’)(p,f)

– p. 23/106

Tearing


(p’,f’)(p,f)

21 3

– p. 23/106

Zooming

Subsystems 1 and 3 (tanks ):


– p. 24/106

Zooming



– p. 24/106

Zooming



p’, f’p, fh

– p. 24/106

Zooming



p’, f’p, fh

A ddt h = f + f ′,

B f =

√

|p− p0−ρh| if p− p0 ≥ ρh,

−√

|p− p0−ρh| if p− p0 ≤ ρh,

C f ′ =

√

|p′− p0−ρh| if p′− p0 ≥ ρh,

−√

|p′− p0−ρh| if p′− p0 ≤ ρh,

– p. 24/106

Zooming

Subsystem 2 (pipe ):

p’, f’p, f

– p. 25/106

Zooming

Subsystem 2 (pipe ):

p’, f’p, f

f = − f ′, p− p′ = α f

– p. 25/106

Linking

Interconnection laws:

p, f p’, f’

– p. 26/106

Linking


p, f p’, f’

p = p′, f + f ′ = 0

– p. 26/106

Linking


p, f p’, f’

p = p′, f + f ′ = 0

Leads to the complete model:

– p. 26/106

A1ddt h1 = f1 + f ′1,

B1 f1 =

√

|p1− p0−ρh1| if p1− p0 ≥ ρh1,

−√

|p1− p0−ρh1| if p1− p0 ≤ ρh1,(blackbox 1)

C1 f ′1 =

√

|p′1− p0−ρh1| if p′1− p0 ≥ ρh1,

−√

|p′1− p0−ρh1| if p′1− p0 ≤ ρh1,

f2 = − f ′2, p2− p′2 = α f2, (blackbox 2)

A3ddt h3 = f3 + f ′3,

C f3 =

√

|p3− p0−ρh3| if p3− p0 ≥ ρh3,

−√

|p3− p0−ρh3| if p3− p0 ≤ ρh3,(blackbox 3)

C3 f ′3 =

√

|p′3− p0−ρh3| if p′3− p0 ≥ ρh3,

−√

|p′3− p0−ρh3| if p′3− p0 ≤ ρh3,

p′1 = p2, f ′1 + f2 = 0, p′2 = p3, f ′2 + f3 = 0. (interconnection)

pleft = p1, fleft = f1, pright = p′3, fright = f ′3. (manifest variable assignment)

– p. 27/106

NB

This tableau of equations is the endpoint of a straightforwardfirst-principles-modeling procedure.

◮ Unclear (and, in fact, irrelevant) input/output structurefor the terminal variables,both in the overall system and in the subsystems

◮ Many variables, indivisibly, at the same terminal

◮ Interconnection = variable sharing

◮ No signal flows, no output-to-input assignment

– p. 28/106

NB

This tableau of equations is the endpoint of a straightforwardfirst-principles-modeling procedure.

◮ Unclear (and, in fact, irrelevant) input/output structurefor the terminal variables,both in the overall system and in the subsystems

◮ Many variables, indivisibly, at the same terminal

◮ Interconnection = variable sharing

◮ No signal flows, no output-to-input assignment

These remarks pertain to every physical interconnection.And, ultimately, every interconnection is physical.

– p. 28/106

Behavioral systems

– p. 29/106

Behavioral approach

A dynamical system

:⇔ a family of time trajectories, ‘the behavior’

Interconnection ⇔ ‘variable sharing’

Control ⇔ interconnection

Modeling of interconnected physical systems is the strongestcase for ‘behaviors’.

– p. 30/106

Terminals

We consider systems that interact with their environmentthrough terminals

– p. 31/106

Terminals


There are manyelectrical, mechanical, hydraulic, thermal,civil engineering, pneumatic, ... connectionsthat can be viewed this way,exactly, literally.

For mechanical systems, think of interconnections asscrewing, gluing, welding, ...Hinges, hooks, etc. ought to be thought of as devices(modules).

– p. 31/106

Terminals


There are manyelectrical, mechanical, hydraulic, thermal,civil engineering, pneumatic, ... connectionsthat can be viewed this way,exactly, literally.

For mechanical systems, think of interconnections asscrewing, gluing, welding, ...Hinges, hooks, etc. ought to be thought of as devices(modules).

The clearest example is anelectrical connection.A terminal = a single wire.

– p. 31/106

Interconnection architecture

– p. 32/106

Objective

Formalize mathematically interconnection of systems.

– p. 33/106

Graph with leaves

Architecture:

graph with leaves leaf

vertex

edge

vertices ; systems with terminalsedges ; connected terminalsleaves ; interaction with environment

terminals ; system variables– p. 34/106

Behavioral equations

1. Module equations for each vertex.Relation among the variables on the terminals.

2. Interconnection equations for each edge.Equating the variables on the terminals associatedwith the same edge.

3. Manifest variable assignmentSpecifies the variables of interest.

– p. 35/106

Behavioral equations

1. Module equations for each vertex.Relation among the variables on the terminals.

Behavioral equations stored as (parametrized) modulesin a data-base.

2. Interconnection equations for each edge.Equating the variables on the terminals associatedwith the same edge.

Interconnection laws stored in a data-base.Laws depend on terminal type:electrical / mechanical / hydraulic / etc.

3. Manifest variable assignmentSpecifies the variables of interest.

– p. 35/106

An example

– p. 36/106

RLC circuit

Model the port behavior of

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by tearing, zooming, and linking.

– p. 37/106

RLC circuit

Model the port behavior of

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f

b

c

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4

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by tearing, zooming, and linking.

In each vertex there is a module; module equationseach terminal involves 2 variables (potential, current)in each edge there is an electrical interconnection;

interconnection equations– p. 37/106

Modules

1

3

2

22

11 1

11

2

322

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connector1

capacitor connector2

inductorresistor1

resistor2– p. 38/106

Modules

1

3

2

22

11 1

11

2

322

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connector1 n = 3

capacitor C connector2 n = 3

inductor Lresistor1 RC

resistor2 RL

– p. 38/106

Module equations

vertex 1 : Vconnector1,1 = Vconnector1,2 = Vconnector1,3

Iconnector1,1 + Iconnector1,2 + Iconnector1,3 = 0

vertex 2 : VRC,1−VRC,2 = RCIRC,1, IRC,1 + IRC,2 = 0

vertex 3 : L ddt IL,1 = VL,1−VL,2, IL,1+ IL,2 = 0

vertex 4 : C ddt

(

VC,1−VC,2)

= IC,1, IC,1 + IC,2 = 0

vertex 5 : VRL,1−VRL,2 = RLIRL,1

IRL,1 + IRL,2 = 0



– p. 39/106

Module equations

Vconnector1,1 = Vconnector1,2 = Vconnector1,3


VRC,1−VRC,2 = RCIRC,1, IRC,1 + IRC,2 = 0

L ddt IL,1 = VL,1−VL,2, IL,1 + IL,2 = 0

vertex 4: C ddt

(

VC,1 − VC,2

)

= IC,1 , IC,1 + IC,2 = 0

VRL,1−VRL,2 = RLIRL,1

IRL,1+ IRL,2 = 0

Vconnector2,1 = Vconnector2,2 = Vconnector2,3


IC,1

IC,2

IV,1

VC,2

C

– p. 39/106

Interconnection

All interconnections are of electrical type

Ileft

Ileft

Iright

Iright

Vleft

Vleft

Vright

Vright

Interconnection equations:

potential left = potential right ; Vleft = Vright

current left + current right = 0 ; Ileft + Iright = 0

– p. 40/106

Interconnection equations

edge c: VRC,1 = Vconnector1,2 IRC,1 + Iconnector1,2 = 0

edge d: VL,1 = Vconnector1,3 IL,1 + Iconnector1,3 = 0

edge e: VRC,2 = VC,1 IRC,2 + IC,1 = 0

edge f: VL,2 = VRC,1 IL,2 + IRL,1 = 0

edge g: VC,2 = Vconnector2,1 IC,2 + Iconnector2,1 = 0

edge h: VRL,2 = Vconnector2,2 IRL,2 + Iconnector2,2 = 0

– p. 41/106

Interconnection equations

VRC,1 = Vconnector1,2 IRC,1 + Iconnector1,2 = 0

edge d: VL,1 = Vconnector1,3

IL,1 + Iconnector1,3 = 0

VRC,2 = VC,1 IRC,2 + IC,1 = 0

VL,2 = VRC,1 IL,2 + IRL,1 = 0

VC,2 = Vconnector2,1 IC,2 + Iconnector2,1 = 0

VRL,2 = Vconnector2,2 IRL,2 + Iconnector2,2 = 0

L

��connector1

Iconnector1,3

IL,1Vconnector1,3

VL,1

– p. 42/106

Manifest variable assignment

Vexternalport = Vconnector1,1−Vconnector2,3

Iexternalport = Iconnector1,1

−

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+

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connector2

Iconnector1,1Vconnector1,1

Vconnector2,3

Iexternalport

Vexternalport

– p. 43/106

Complete model



vertex 2 : VRC ,1−VRC ,2 = RCIRC ,1, IRC ,1 + IRC ,2 = 0

vertex 3 : L ddt IL,1 = VL,1−VL,2, IL,1 + IL,2 = 0

vertex 4 : C ddt

(

VC,1−VC,2)

= IC,1, IC,1 + IC,2 = 0

vertex 5 : VRL ,1−VRL ,2 = RLIRL,1

IRL,1 + IRL ,2 = 0



edge c: VRC,1 = Vconnector1,2

IRC,1 + Iconnector1,2 = 0

edge d: VL1 = Vconnector1,3

IL1 + Iconnector1,3 = 0

edge e: VRC,2 = VC1

IRC,2 + IC1 = 0

edge f: VL2 = VRC,1

IL2 + IRL,1 = 0

edge g: VC2 = Vconnector2,1

IC2 + Iconnector2,1 = 0

edge h: VRL,2 = Vconnector2,2

IRL,2 + Iconnector2,2 = 0

Vexternalport = Vconnector,1,1−Vconnector2,3


– p. 44/106

Port behavior

B = { (Vexternalport, Iexternalport) : R → R2 |

∃ latent variables trajectories

(Vconnector1,1, Iconnector1,1, . . . , . . .) : R → R28

such that

Vconnector1,1 = Vconnector1,2 = Vconnector1,3, . . . ,

all 24 equations are satisfied}

– p. 45/106

Port behavior

B = { (Vexternalport, Iexternalport) : R → R2 |

∃ latent variables trajectories

(Vconnector1,1, Iconnector1,1, . . . , . . .) : R → R28

such that

Vconnector1,1 = Vconnector1,2 = Vconnector1,3, . . . ,

all 24 equations are satisfied}

Can we simplify this expression forB?

– p. 45/106

Port behavior

; the dynamical system with behaviorB specified by:

Case 1: CRC 6=L

RL

(

RCRL

+(

1+ RCRL

)

CRCddt +CRC

LRL

d2

dt2

)

V =(

1+ LRL

ddt

)

(

1+CRCddt

)

RCI

; B = all solutions (V, I) : R → R2

– p. 46/106

Port behavior


Case 1: CRC 6=L

RL

(

RCRL

+(

1+ RCRL

)

CRCddt +CRC

LRL

d2

dt2

)

V =(

1+ LRL

ddt

)

(

1+CRCddt

)

RCI

Case 2: CRC =L

RL

(

RCRL

+CRCddt

)

V =(

1+CRCddt

)

RCI


– p. 46/106

Port behavior

Thm: In LTIDSs latent variables can be eliminated !


Case 1: CRC 6=L

RL

(

RCRL

+(

1+ RCRL

)

CRCddt +CRC

LRL

d2

dt2

)

V =(

1+ LRL

ddt

)

(

1+CRCddt

)

RCI

Case 2: CRC =L

RL

(

RCRL

+CRCddt

)

V =(

1+CRCddt

)

RCI


– p. 46/106

The complete model is a linear constant coefficient DAE




vertex 3 : L ddt IL,1 = VL,1−VL,2 IL,1 + IL,2 = 0

vertex 4 : C ddt

(

VC,1−VC,2)

= IC,1 IC,1 + IC,2 = 0


IRL ,1 + IRL,2 = 0



edge c: VRC,1 = Vconnector1,2


edge d: VL1 = Vconnector1,3

IL1 + Iconnector1,3 = 0

edge e: VRC,2 = VC1

IRC,2 + IC1 = 0

edge f: VL2 = VRC,1

IL2 + IRL,1 = 0

edge g: VC2 = Vconnector2,1

IC2 + Iconnector2,1 = 0

edge h: VRL,2 = Vconnector2,2

IRL,2 + Iconnector2,2 = 0

Vexternalport = Vconnector,1,1−Vconnector2,3


– p. 47/106

Canceling poles and zeros

– p. 48/106

Can poles and zeros be cancelled?

Case 1: CRC →L

RL

(

1+ LRL

ddt

)(

RCRL

+CRCddt

)

V =(

1+ LRL

ddt

)

(

1+CRCddt

)

RCI

Case 2: CRC =L

RL

(

RCRL

+CRCddt

)

V =(

1+CRCddt

)

RCI

– p. 49/106


Case 1: CRC →L

RL

(

1+ LRL

ddt

)(

RCRL

+CRCddt

)

V =(

1+ LRL

ddt

)

(

1+CRCddt

)

RCI

Case 2: CRC =L

RL

(

RCRL

+CRCddt

)

V =(

1+CRCddt

)

RCI

Cancellation appears allowed.

– p. 49/106


Case 1: CRC →L

RLand RC = RL

(

1+ LRL

ddt

)

(

1+CRCddt

)

V =(

1+ LRL

ddt

)

(

1+CRCddt

)

RCI

Case 2: CRC =L

RLand RC = RL

(

1+CRCddt

)

V =(

1+CRCddt

)

RCI

– p. 50/106


Case 1: CRC →L

RLand RC = RL

(

1+ LRL

ddt

)

(

1+CRCddt

)

V =(

1+ LRL

ddt

)

(

1+CRCddt

)

RCI

Case 2: CRC =L

RLand RC = RL

(

1+CRCddt

)

V =(

1+CRCddt

)

RCI

One common factor can be cancelled, the other not.

Cancellation appearssometimes, but only sometimes allowed

– p. 50/106


Case 1: CRC →L

RLand RC = RL

(

1+ LRL

ddt

)

(

1+CRCddt

)

V =(

1+ LRL

ddt

)

(

1+CRCddt

)

RCI

Case 2: CRC =L

RLand RC = RL

(

1+CRCddt

)

V =(

1+CRCddt

)

RCI

One common factor can be cancelled, the other not.

Cancellation appearssometimes, but only sometimes allowed

Important? Look at short circuit behavior V = 0.Incorrect cancellation gives spurious exponential responses!

– p. 50/106

Seshu graphs, trees, & admittance formula

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– p. 51/106


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– p. 51/106


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Circuit admittance =

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=Σ

graph1Π

branchesbranch admittances

Σgraph2

Πbranches

branch admittances


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– p. 51/106

Seshu admittance formula

Seshu gives:

I(s)

V (s)=

RCRL

+(

1+ RCRL

)

CRCs+CRCL

RLs2

(

1+ LRL

s)

(1+CRCs)RC

Gives the correct admittance/impedance/transfer function.

– p. 52/106


Seshu gives:

I(s)

V (s)=

RCRL

+(

1+ RCRL

)

CRCs+CRCL

RLs2

(

1+ LRL

s)

(1+CRCs)RC


But, it does not give the correct behavior.It does not cope with common factors.

Same remarks for Mason’s formula, for series connection, ...

– p. 52/106


Seshu gives:

I(s)

V (s)=

RCRL

+(

1+ RCRL

)

CRCs+CRCL

RLs2

(

1+ LRL

s)

(1+CRCs)RC


But, it does not give the correct behavior.It does not cope with common factors.

Same remarks for Mason’s formula, for series connection, ...

Respect common factors

Respect the uncontrollable

Beware of transfer function thinking– p. 52/106

Other methodologies

– p. 53/106

Differential-algebraic equations (DAEs)

– p. 54/106

DAEs

Differential-algebraic equations:

ddt Ex = Ax+Bu, y = Cx+Du w =

[

uy

]

– p. 55/106

DAEs



[

uy

]

Is this a sensible first-principles-model alternative to theoverly structured

ddt

x = Ax+Bu, y = Cx+Du w =

[

uy

]

?

Where does the input/output partition come from?

– p. 55/106

DAEs



[

uy

]

Where does the input/output partition come from?

Often the additional assumption is made that the pencildeterminant(Eµ −Aξ ) is regular

; determinant(Eµ −Aξ ) 6= 0.

Where does this assumption come from?

Where does the fact thatE and A are regular come from?Why should they even be square?

– p. 55/106




vertex 3 : L ddt IL,1 = VL,1−VL,2, IL,1 + IL,2 = 0

vertex 4 : C ddt

(

VC,1−VC,2)

= IC,1, IC,1 + IC,2 = 0


IRL,1 + IRL ,2 = 0



edge c: VRC,1 = Vconnector12


edge d: VL1 = Vconnector13

IL1 + Iconnector13 = 0

edge e: VRC,2 = VC1

IRC,2 + IC1 = 0

edge f: VL2 = VRC,1

IL2 + IRL,1 = 0

edge g: VC2 = Vconnector21

IC2 + Iconnector21 = 0

edge h: VRL,2 = Vconnector22

IRL,2 + Iconnector22 = 0Vexternalport = Vconnector1,1−Vconnector2,3 Iexternalport = Iconnector11

Eddt

x = Ax+B Iexternalport Vexternalport = Cx+D Iexternalport

with x = the terminal variables (14 term’s, 28 variables),1 input (say,Iexternalport), 1 output (say,Vexternalport) and 24 eq’ns.

E,A ∈ R23×28

,B ∈ R23×1

,C ∈ R1×23

,D ∈ R

– p. 56/106

Signal flow graphs

– p. 57/106

input/output thinking

There are many many examples where output-to-inputconnection is eminently natural:

– p. 58/106


There are many many examples where output-to-inputconnection is eminently natural:

fin

outf

– p. 58/106

input/output partition

terminal with 2 physical variables

Assume that one of these variables acts as input, the other asoutput.

– p. 59/106

input/output partition

input

output

Assume that one of these variables acts as input, the other asoutput.

– p. 59/106

Block diagram

variable 2

variable 1

◮ shows terminal variables separate

◮ suggests that inputs and outputs occur at differentphysical points

Pedagogically awkward, confusing.

– p. 60/106


variable 2

variable 1

◮ allows impossible input-output connections

Does not respect the physics.– p. 61/106


– p. 62/106


variablesshared

– p. 62/106


variablesshared

partitionsi/o

– p. 62/106


variablesshared

partitionsi/o

?

?

– p. 62/106


variablesshared

partitionsi/o

For physical systemsinput-to-input & output-to-output

assignment very prevalent:force to force; pressure to pressure; heat flow to heat flow;temperature to temperature; mass flow to mass flow; ...

Physical systems are not signal processors

– p. 62/106

The input/output approach as the primary and universalview of open systems is a historical misconception.

The sooner it is abandoned as a starting point, the better.

– p. 63/106

The input/output approach as the primary and universalview of open systems is a historical misconception.

◮ It fails in the most elementary examples.

◮ It does not deal adequately with interconnections.

◮ It breaks symmetries.

◮ It does not respect the physics.

◮ It is pedagogically ineffective.

The sooner it is abandoned as a starting point, the better.

– p. 63/106

“Block diagrams unsuitable for serious physical modeling

- the control/physics barrier”

“Behavior based (declarative) modeling is a good alternative”

Karl Astr om

from K.J. Astr om, Present Developments in Control Applications

IFAC 50-th Anniversary CelebrationHeidelberg, September 12, 2006.

– p. 64/106

Notes & arrows

My dear young man, don’t take it too hard.Your work is ingenious. It’s quality work.But there are simply too many notes that’s all ...

– p. 65/106

Notes & arrows

Ingenious. Quality work.But there are simply too many arrows , that’s all ...

– p. 65/106

Bond graphs

– p. 66/106

Bond graphs

variablesshared

– p. 67/106

Bond graphs

effortflow

Interconnection variables consist of

an effort and a flow effort × flow = power

Interconnection ⇔[efforts equal] & [flows sum to 0]

⇒ power equal

‘Power is the universal currency of physical systems’

– p. 67/106

Bond graphs

Interconnection variables:

◮ voltage & current

◮ force & velocity

◮ pressure & mass flow

◮ temperature & heat flow

temperature &heat flow

temperature◮ ...

– p. 68/106

Remarks

◮ Mechanical interconnections equate positions, notvelocities.

◮ Not all interconnections involve equating energy transfer.

◮ Terminals are for interconnection, ports for energytransfer.

– p. 69/106

Remarks

◮ Mechanical interconnections equate positions, notvelocities.

◮ Not all interconnections involve equating energy transfer.

◮ Terminals for interconnection, ports for energy transfer

This last point is illustrated for electrical interconnections.

– p. 69/106

Terminals versus ports

(potential, current)

Electricalcircuit

Terminal variables and behavior:

(V1, I1,V2, I2, . . . ,Vn, In) ; behavior B ⊆(

R2n)R

– p. 70/106


Port 1

Port 2

Port k

Circuit

Port :⇔

sum currents = 0potentials + constant

⇒ potentials

– p. 71/106


Port 1

Port 2

Port k

Circuit

Port :⇔


⇒ potentials

(

V1, I1 . . . ,Vp, Ip ,Vp+1, . . . , In)

∈ B,α : R → R

⇓

(

V1 +α , I1, . . . ,Vp +α , Ip ,Vp+1, . . . , In)

∈ B

I1 + · · ·+ Ip = 0

– p. 71/106


Port 1

Port 2

Port k

Circuit

Port :⇔


⇒ potentials

The behavioral equations contain the variablesV1,V2 . . . ,Vp

only as the differences

Vi −Vj for i, j = 1, ...p

and contain the equation

I1 + I2 + · · ·+ Ip = 0

– p. 71/106



Electricalcircuit

All the terminals together form a port

(

V1, I1 . . . ,Vn, In)

∈ B,α : R → R

⇓(

V1 +α , I1, . . . ,Vn +α , In)

∈ B

I1 + · · ·+ In = 0

– p. 72/106



Electricalcircuit

All the terminals together form a port

(

V1, I1 . . . ,Vn, In)

∈ B,α : R → R

⇓(

V1 +α , I1, . . . ,Vn +α , In)

∈ B

I1 + · · ·+ In = 0

Viewed as universal ‘laws’ governing electrical circuits,thesemay be thought of as theKirchhoff laws, KVL & KCL

This property is closed under interconnection.– p. 72/106


Circuit 1 Circuit 2

Circuit 3

Circuit 1 Circuit 2

Circuit 3

– p. 73/106


Circuit 1 Circuit 2

Circuit 3

Circuit 1 Circuit 2

Circuit 3

Interconnection via terminals, energy transfer via ports.One cannot speak about

“the energy transferred from circuit 1 to circuit 2”

unless their interconnected terminals form a port.

– p. 73/106

Hierarchy

– p. 74/106

New modules from old ones

Tearing, zooming, linking is hierarchical :

leaf

vertex

edge

– p. 75/106



leaf

vertex

edge

Embed modules in vertices, obtain behavioral equations forthe interconnected system, eliminate the latent variables,

– p. 75/106



leaf

vertex

edge

Terminals

MODULE

Embed modules in vertices, obtain behavioral equations forthe interconnected system, eliminate the latent variables, anduse interconnected systemas a module with terminals in anew interconnection architecture.

– p. 75/106

Example

Model the behavior of the external terminal voltages andcurrents of the following circuit:

– p. 76/106

Example


One section:

;

– p. 76/106

Example


One section:

;

Hierarchical combination:

; ⇒

– p. 76/106

Circuit diagrams

– p. 77/106

Circuits and graphs

Classical circuit theory evolves around adigraph with2-terminal elements or external ports in the edges andconnections in the vertices.

vertex

edge

– p. 78/106

Circuits and graphs

Classical circuit theory evolves around adigraph with2-terminal elements or external ports in the edges andconnections in the vertices. For example,

��

RL

C

C

LRI

V−

+��

��

��

;

��

��

��

– p. 78/106

Circuits and graphs

Classical circuit theory evolves around adigraph with2-terminal elements or external ports in the edges andconnections in the vertices.

vertex

edge

Associate a voltage drop and a current with each edge, and

embed an element (say,R, L, or C) in each ‘internal’ edge.

– p. 78/106

KVL & KCL

Basic laws:

Kirchhoff’s current law for each vertex:

∑edges adjacent to vertex

± currents in edges= 0

Kirchhoff’s voltage law for each cycle:

∑edges in the cycle

± voltage drops over edges= 0

Equivalently, the vertices have an electric potential.

– p. 79/106

KVL & KCL

Basic laws:

Kirchhoff’s current law for each vertex:

∑edges adjacent to vertex

± currents in edges= 0

Kirchhoff’s voltage law for each cycle:

∑edges in the cycle

± voltage drops over edges= 0

Equivalently, the vertices have an electric potential.

Combined with the constitutive laws of the elements in the‘internal’ edges, this yields equations for the behavior (say, ofthe voltages and currents of the external ports).

– p. 79/106

Limitations

This methodology is very limited:

◮ It can only deal with 2-terminal elements and 2-terminalexternal ports.

◮ It is purely port oriented. It does not articulate thatterminals, not ports make the interconnections.

◮ It is not hierarchicalAn already-modeled-circuit cannot be reused as asubsystem in a larger circuit diagram.

– p. 80/106

Embedding a circuit in a graph

edge

vertex

��

��

��

��

��

��

��

– p. 81/106


edge

vertex

Perfect for 2-terminal one-ports

��

��

��

��

��

��

��

– p. 81/106


edge

vertex

��

��

��

��

R

R

R

There is no way to embed a 3-terminal circuit in a circuitgraph,

��

��

��

��

– p. 81/106


edge

vertex

��

��

��

��

R

R

R

There is no way to embed a 3-terminal circuit in a circuitgraph, unless we tear the blackbox into its components

��

��

��

��

R

RR ��

R/3

R/3R/3

– p. 81/106


edge

vertex

If we imbed a 4-terminal circuit into a circuit graph, it has t obe a 2-port.

��

��

��

��

��

��

– p. 82/106


edge

vertex

If we imbed a 4-terminal circuit into a circuit graph, it has t obe a 2-port.

��

��

��

��

��

��

R

R

R

R

not embeddable embeddable– p. 82/106

Vertices and edges

In circuit graphs,subsystems are in the edges ,connections are in the vertices

vertex

edge

– p. 83/106

Vertices and edges

In circuit graphs,subsystems are in the edges ,connections are in the vertices

vertex

edge

leaf

vertex

edge

Contrast with tearing, zooming, linking:subsystems are in the vertices , connections are in the edges

– p. 83/106

Various facets of control

– p. 84/106

Path planning

ddt

x = f (x,u)

Choose time-function u(·) : [0,T ] → U so as to achieve(optimal) state transfer.

‘open loop control’

1

x2X2

x

– p. 85/106

Decision making

��

��

CONTROLLER

controlinputs

FEEDBACK

to−be−controlled outputsexogenous inputs

measuredoutputs

Actuators PLANT

CLOSED−LOOPSYSTEM

Sensors

Choose afeedback system that processes sensor outputs andgenerates actuator inputs so as to achieve good (optimal)performance.

‘feedback control’‘closed loop control’‘intelligent control’

– p. 86/106

Embedded systems control

ControllerPlant

Choose controller so as to achieve good (optimal)performance of the interconnected system

‘control as interconnection’‘integrated system design’

– p. 87/106

Example

suspension spring

road profile

axle mass

body mass

damper

tire tire

axle mass

body mass

mechanical impedance

road profile

– p. 88/106

Control as interconnection

– p. 89/106

Interconnecting a controller

Before interconnection:to−be−controlled

terminals

Plant terminals Controllerterminals

control

exogenous

After interconnection:to−be−controlled

terminals

Plantterminals

Controller

exogenous

Contrast with:

control outputsmeasured

inputsoutputs

to−be−controlled

Controller

Actuators

exogenous

Controlled system

SensorsPlant

inputs

– p. 90/106

Many controllers are not sensor-to-actuator

Controlling turbulence:

– p. 91/106


Controlling turbulence:

– p. 91/106

Nagano 1998 Winter Olympics

– p. 92/106


controlling drag:

– p. 93/106

Many stabilizers are not sensor-to-actuator

Stabilization:

– p. 94/106

Disturbance attenuation

tire

axle mass

body mass

mechanical impedance

road profile

suspension spring

road profile

axle mass

body mass

damper

tire

active damper

actuator

road profile

sensors

tire

control algorithm

body mass

axle mass

spring

– p. 95/106

Control as Interconnection


Controlled system

Plant Controller

controlterminals

terminals

◮ Are all interconnections ‘reasonable’?

◮ Which controlled behaviors can be achieved?

◮ Parametrize all stabilizing controllers

◮ ...

– p. 96/106

Implementability


controlterminals

terminals

w

c

B C

K

For simplicity, restrict attention to LTIDSs, L •.

Let B ∈ L w+c be the plant behavior,C ∈ L c be the controller behavior, and

K = {w : R → Rw | ∃ c ∈ C such that (w,c) ∈ B}

be thecontrolled behavior

– p. 97/106

Implementability


controlterminals

terminals

w

c

B C

K

For simplicity, restrict attention to LTIDSs, L •.

Let B ∈ L w+c be the plant behavior,C ∈ L c be the controller behavior, and

K = {w : R → Rw | ∃ c ∈ C such that (w,c) ∈ B}

be thecontrolled behavior

Elimination theorem ⇒ K ∈ L w

– p. 97/106

Implementability

(b)


controlterminals

terminals

w

c

B C

K

For a givenB ∈ L w+c, call K ∈ L w implementable if thereexistsC ∈ L c such thatK is the controlled behavior.

Which K ∈ L w are implementable ?

– p. 98/106

Every lecture must have at least one theorem


Hidden behavior

Plant

controlterminals

terminals

wc = 0

Define thehidden behavior

N := {w|(w,0) ∈ B}

– p. 99/106



Hidden behavior

Plant

controlterminals

terminals

wc = 0


N := {w|(w,0) ∈ B}


Uncontrolled plant behavior

Plant

controlterminals

terminals

wc = free

Define theuncontrolled plant behavior

P := {w|∃ c : (w,c) ∈ B}

– p. 99/106



Hidden behavior

Plant

controlterminals

terminals

wc = 0


N := {w|(w,0) ∈ B}


Uncontrolled plant behavior

Plant

controlterminals

terminals

wc = free

Define theuncontrolled plant behavior

P := {w|∃ c : (w,c) ∈ B}

Implementability theorem

K ∈ L w is implementable ⇔ N ⊆ K ⊆ P

– p. 99/106

Summary

– p. 100/106

Main points

◮ Interconnection = variable (terminal) sharing

– p. 101/106

Main points


◮ Modeling by physical systems proceeds bytearing, zooming, and linking

– p. 101/106

Main points



◮ Hierarchical procedure

– p. 101/106

Main points




◮ Importance of latent variables and theelimination theorem

– p. 101/106

Main points





◮ Limitations of input/output thinking

– p. 101/106

Main points






◮ Control is interconnection, sensor output to actuatorinput feedback important special case

– p. 101/106

Main points






◮ Control is interconnection, sensor output to actuatorinput feedback important special case

◮ Need generalization to distributed terminals, etc.

– p. 101/106

Overview

– p. 102/106

Behavioral systems

◮ Gets the physics right

– p. 103/106

Behavioral systems


◮ Starts with first principles models

– p. 103/106

Behavioral systems



◮ Latent variables with state as a special case

– p. 103/106

Behavioral systems




◮ Avoids universal use of signal flow graphs

– p. 103/106

Behavioral systems





◮ i/o and i/s/o are important special cases

– p. 103/106

Behavioral systems





◮ i/o and i/s/o are important special cases

◮ Extends seamlessly to PDEs

– p. 103/106

Behavioral systems

◮ Controllability becomes genuine system property

– p. 104/106

Behavioral systems


◮ Deals faithfully with interconnections: variable sharing

– p. 104/106

Behavioral systems



◮ Views control as interconnection

– p. 104/106

Behavioral systems




◮ Advantages in SYSID with the MPUM, etc.

– p. 104/106

Behavioral systems





◮ Natural approach to dissipative systems

– p. 104/106

Behavioral systems





◮ Natural approach to dissipative systems

◮ Far easier pedagogically

– p. 104/106

1. A dynamical system = a family of trajectories.

2. Interconnection = variable sharing

3. Control = interconnection

– p. 105/106

Want to read about it? SeeThe behavioral approach to open and interconnected systems,Control Systems Magazine, Volume 27, pages 46-99, 2007.

The lecture frames are available from/athttp://www.esat.kuleuven.be/∼jwillems

– p. 106/106

Want to read about it? SeeThe behavioral approach to open and interconnected systems,Control Systems Magazine, Volume 27, pages 46-99, 2007.

The lecture frames are available from/athttp://www.esat.kuleuven.be/∼jwillems

Thank youThank you

Thank youThank you

Thank you

Thank you– p. 106/106

Documents

INTERCONNECTED SYSTEMS