Modeling and Systems theory - lecture 1

8/14/2019 Modeling and Systems theory - lecture 1

1/7

Modeling and Nonlinear Systems Theory

SC4092

Dimitri Jeltsema, Delft Institute of Applied Mathematics

Lecture 1 Physical Modeling

Mo deli ng and N onl ine ar Systems The ory Lec ture 1 1 / 2 7

Who are we...

Lecturer:

Dr. ing. Dimitri Jeltsema, M.Sc.

Delft Institute of Applied Mathematics

Mekelweg 4, Room HB 5.280

Course assistant:

Paolo Forni (DCSC)

Questions:

Email: [email protected]

Blackboard:

Slides, reader, assignments, etc.

Enroll!

Mod eli ng a nd N onl ine ar Syst ems Theory Lec ture 1 2 / 2 7

Content and study goals

After successful completion of the course, you are able to

construct models of systems from the knowledge of

physics, with an emphasize on the internal energy of the

system.

write models of systems described by differential and

algebraic equations in a control systems form.distinguish between linear and nonlinear systems

properties.

decide when to apply linear and when to apply nonlinear

theory.

determine several stability properties for nonlinear

systems.


Content and study goals

apply dissipativity and passivity concepts for stabilization

and to analyze input-output stability.

determine controllability and observability properties fornonlinear systems.

design linearizing feedback transformations.



2/7

Mo deli ng and N onl ine ar Systems The ory Lec ture 1 3 / 2 7 Mod eli ng a nd N onl ine ar Syst ems Theory Lec ture 1 4 / 2 7

Examination

Written exam (80%)

Friday, January 31 2013, 09:00 to 12:00 hours

Please subscribe in time!

Case study (10%) + 2 home works (5% + 5%)

Case study: modeling + stability problemHW1: controllability/observability HW2: FB linearization

Simulation with Matlab / Simulink

Groups of two

Final grade = 0.8exam + 0.1case + 0.05HW1 + 0.05HW2


Models for multi-physics systems

System: Object or set of objects from which we want to

study the properties.

Model of a system: A tool we use to answer questions about

the system without having to do an experiment.

Types of models:

mental: intuition and experience, verbal: if..., then...

physical: scale models, laboratory set-ups

mathematical: equations that describe relation between

quantities that are important for behaviour of system,

e.g., laws of nature. Focus of SC4092!



In discussing a (lumped-parameter) system, we deal with:

Sets ofelements such as ideal springs, masses, dampers,

inductors, capacitors, resistors, tubes, tanks,

transformers, gyrators, etc..

Sets ofvariables (signals) such as forces, velocities,

voltages, currents, pressure, temperature, etc..

Sets ofrelationships between variables.

Interactionsbetween elements.



Two type of elements:

Dynamic(mass, spring, inductor, capacitor, etc.)

energetic (energy storage)

Static(resistor, damper, transformer, etc.)

non-energetic(dissipation, scaling)

Two type of element relationships:

Constitutiverelationships (all elements)

Dynamicalrelationships (dynamic elements)



3/7


Other types of relationships:

Interconnective relationships. Interconnection of

elements (Kirchhofflaws, DAlembert, etc.)

Combination of constitutive and dynamical relationships

yields the component relationships.These relationships define our

dynamical system model.



Restriction to continuous-time modeling based on our

knowledge of the laws of nature in a structured way.

Deterministic dynamical lumped-parameter

systems (ODEs) of the form

x(t) = fx(t), u(t), t, x(t) = dx(t)

dty(t) = h

x(t), u(t), t,

with u Rm, x Rn, y Rp, t R. Furthermore,

f : Rn Rm R Rn, h: Rn Rm R Rp.

Such set of ODEs is called a state-space modelwith state x.

M od el in g a nd N on li ne ar S yst em s T he or y L ec tu re 1 1 0 / 2 7


Important subclass:

Time-Invariant:x(t) = f

x(t), u(t)

,

y(t) = h

x(t), u(t).

Linear and Time-Invariant (LTI):

x(t) = Ax(t) +Bu(t)

y(t) = Cx(t) +Du(t).

Autonomous: x=f(x) or x(t) = Ax(t).

How to obtain such models? Consider following domains:

Electrical Hydraulical

Mechanical translational Chemical

Mechanical rotational Thermo-dynamical

M od el in g a nd N on li ne ar S ys te ms T he or y L ec tu re 1 1 1 / 2 7

Signals versus ports

In previous courses, you have learned to model and analyse

systems using transfer functions:

H(s) = Y(s)

U(s) =C(sI A)1B+ D.

Often very useful, but

Only applicable to linear systems

Does not reflect physical structure

Initial conditions are assumed to be zero

Not suitable for library building

Not easy to interconnect systems...



4/7

Signals versus ports

With physical systems, signal modeling is often not

suitable

Always a bi-directional effect

To model/control OPEN systems, signal modeling is

NOT the solution

This is true also between domains: typical example DCmotor gyration

Physical system rather share variables instead of signals

Physical systems are not signal processors!

We need something more!

PORT-BASED MODELING



5/7

A motivating example

Spring subsystem (let q= z z0)

q= fk

ek= dEk

dq (q), Ek(q) =

k

2q2,

withfk the velocity of the endpoint of the spring (where it

is attached to the mass), and ekis the force at this point.Mass subsystem:

p= fm

em = dEm

dp (p), Em(p) =

p2

2m,

where fm denotes the force exerted on the mass, and

em= pm

is the velocity of the mass.


A motivating exampleFinally, we couple the subsystems through the interconnection

fk=em, fm= ek (Newtons third law),

leading to the final equations of the overall system

q

p=

0 1

1 0"

Hq

(q, p)

H

p

(q, p)

#, H(q, p) =

k

2q2 +

p2

2m.

These are the well-known Hamiltonian equations.

Observe that the energy is conserved since

H= H

q(q, p)q+

H

p(q, p)p

= H

q(q, p)

H

p(q, p) +

H

p(q, p)

H

q(q, p)

= 0.


Electrical LC circuit

Inductor 1:

1= f1 (voltage)

(current) e1= dH1

d1(1)

Inductor 2:

2= f2 (voltage)

(current) e2= dH2

d2(2)

Capacitor:

Q= f3 (current)

(voltage)e3= dH3

dQ(Q)



6/7


Kirchhoffs voltage and current laws yieldf1f2f3

I

=

0 0 1 1

0 0 1 0

1 1 0 0

1 0 0 0

e1e2e3V

, ()

which, after substituting the individual equations, yields12

Q

=

0 0 10 0 1

1 1 0

H1

(1,2, Q)

H2

(1,2, Q)

HQ

(1,2, Q)

+

10

0

V,

I = H

1(1,2, Q),

with total energy H(1,2, Q) = H1(1) +H2(2) +H3(Q).



Observe that if we consider the rate of change of energy:

H= VI. (Verify!)

This is simply stating that the change in stored energy (power)

is equal to the supplied power and motivated the choice for the

output

I = H

1(1,2, Q).

We call(V, I) a power conjugated input-output pair.

The way the energy flows through the circuit is defined by the

interconnection structure (), where we observe that

f1e1 f2e2 f3e3+VI =0.


Dirac structure

Think of a Dirac structure as the wiring in an electrical circuit:



7/7

Port-Hamiltonian systems

For a large class of lossless systems, the algebraic equations

can be eliminated, as to obtainI 0

0 I

x

y

=

J(x) g(x)

gT(x) 0

Hx

(x)

u

,

which yields the port-Hamiltonian system

x=J(x)H

x(x) +g(x)u,

y=gT(x)H

x(x),

where x Rn, H(x) is the total stored energy, J(x) is a

skew-symmetric matrix, i.e., J(x) = JT(x), and u,y Rm is

a conjugated input-output pair.


Port-Hamiltonian systems

Due to skew-symmetry ofJ(x), the power-balance reads

H=TH

x (x)x

=TH

x (x)J(x)

H

x(x)

| {z }0+

TH

x (x)g(x)u

=yTu,

or, equivalently, after integrating from 0 to T:

H

x(T) H

x(0)

| {z }stored energy

=

TZ0

yT(t)u(t)dt

| {z }supplied energy

.


Port-based modeling of physical system

Physical system can be viewed as a set of simpler

subsystems that exchange energy (dynamics is exchange

of energy!);

Energy is neither allied to a particular physical domain nor

restricted to linear elements and systems;

Energy can serve as a lingua franca to facilitatecommunication among scientists and engineers from

different fields;

Role of energy and the interconnections between

subsystems provide the basis for various control

techniques.

Modeling and Systems theory - lecture 1

Documents

Transcript of Modeling and Systems theory - lecture 1