On state space models and their application to hydromechanic systems

ON STATE SPACE MODELS

AND THEIR APPLICATION TO

HYDROMECHANIC SYSTEMS

Michael

Schmiechen

ABS T RACT

State space models that may be identified from _time-in

dependent functional black box models, linear _in partic-ular, and their applications to hydromechanic _systems, e.g. ships, are discussed in a formal fashion, in _an attempt to establish an adequate systems _engineering theory of such models and their applications.

HYDROMECHANIC SYSTEMS

CONTENT S

Introduction 3

1.1 Black box model 3

1.2 Problem, goal, plan 5

General theory 6

2.1 State space models 6

2.2 Models of a system 8 2.3 Structures of models lo 2.4 Parameters of models 12 2.5 _{Identification of parameters 14} Linear models 17 3.1 Continuous models 17

3.2 _{Discrete, finite models 20}

3.3 Models of a system 23

3.4 _{Identification of parameters 26}

3.5 _{Transformation of models 28}

Applications 30

4.1 Models of rigid bodies 30

4.2 _{Coupling to ideal fluids 33}

4.3 Models of real fluids ₃₅

4.4 Inertia and damping 38

4.5 Models of manoeuvring ships ₄₀

Conclusions 41 5.1 Review ₄₁ 5.2 Outlook ₄₃ Acknowledgements ₄₅ References ₄₆ Notation ₄₈

1, INTRODUCTION

1.1 Black box model

It is assumed that (the dynamics of) _{a system may for input} functions

¿

F

(,/, ,d(z))

be described by the time-independent functional black box model, i.e. by the input-output equation

for the past time

where

and

R

are the input and output values of the system, _more

precisely the cause and effect values, _{real and of} dimen-sions

at the past time and the current or present time

respectively, and

¿

(x

1(_,é)1

d())

is the input-output functional describing the system (dynamics).

In a narrow sense this functional alone will be called the black box model of the system (dynamics) for the set of input functions. In the following both will be assumed

given, i.e. determined either from specifications, theories, or measurements or any combinations of these.

In a previous paper (Schmiechen, 1969) it has been shown that e.g. the set of black box models for systems in noisy feed-back loops may be defined as the set of mappings of the conditional probability densities of the inputs to the

conditional probability densities of the outputs, the

conditions being processes, which are statistically

in-dependent of the noise processes fed into the loops and

the measuring systems, i.e. under working conditions the

guiding processes. The sets of input functions are in

terms of this model defined as the unconditional probability densitIes of the inputs.

1.2 Problem, goal, plan

In order to implement or to simulate the system behaviour as described by the given black box model for the given set of input functions the first problem to be solved is to construct, from the given black box model, models of the system including internal states and structures.

The goal of this paper is to show how this problem may be solved formally and, by the way, some implications of the

solution.

The plan is to establish a general theory as far as neces-sary, to solve the problem for linear black box models _as far as possible, and apply the results to rigid bodies moving in incompressible fluids, e.g. ships.

The problem is of course basic in many fields and solu-tions are well established, e.g. in network synthesis. No attempt will be made to trace such analogies and refer to them, in order not to obscur the essentially simple and straight forward-procedures and results.

2. GENERAL THEORY

2.1 State space models

For the description of internal states state space models, i.e. state transition equations

t5()

and state-output equations

f,5L))

are adopted, where the values

5(&)

e;&5

of the state are internal values as opposed to the exter-nal or termiexter-nal values of input and output,

are the state transition functions, which are, as indicat-ed, themselves functions of the input function, the

initial time, and the present time, and

are the output functions.

While the latter may be any mappings of the state space to the output space, the set of state transition functions may be defined by three basic sentences, i.e. _axioms

of state.

The first axiom,

-

u,

implies that states are unique; the second axiom,

fl(1,

¿, ¿)

1i(4

¿

¿)

/,, ,)

implies that states do riot depend on intermediate states, i.e. not on the way, on which they are reached; and the third axiom,

if

(é»

z (é')

for

t'e!

£(é6, EL),

implies that changes of states in _{time intervals do not} depend on input outside these intervals.

In a narrow sense pairs of functions

(*fl1i)

_{alone, the} first one in accordance with the axioms, will be called state space models, and where ever possible, i.e. where no ambiguity arises, arguments will be dropped as in other cases as well.

2,2 Models of a system

Among the state space models as defined by the axioms only those models

are of interest here, that describe a system with a given black box model, the set of all such possible models being defined by the condition of output equivalence.

((-)))

¿('()).

C

(z,é-,-'i)

In practice this condition may be relaxed and approximate state space models may be admitted, the sets of all admis-sible models being defined by some conditions replacing that of output equivalence.

According to the problem statement only _{(approximate) state} space models of the system will be considered _{further on,} which may be constructed or identified solely from the black box model given, the set of all _{identifiable state}

space models of the system

being accordingly defined by all images of the black box model in the space of all possible state space models of

the system.

If only one such image is known, i.e. if one state space model of the system has been constructed or identified

from the given black box model, the whole set may be de-fined by the transfomation rules

r» ('se )

and

nl ()

i-i (r'1 )

with where

r

denoting the group of all transformations applying to

the state space.

Consequently the only problem _{left is to construct one}

(approximate) state _{space model of the system from its} given black box model. _{In order to solve this}

problem the basic concept of

2.3 Structures of models

For the description of internal structures parametric state space models

3 ¿6)

_(i,

_,

é)

₍₉

())

/(/1)

are introduced, where the formats of the models

F;(xs, 5)

and

/2

°(xS,

})

are operational interpretations of the basic concept of model structure, and the parameters

and

7X

are model images in the spaces _2. and (the latter without exponent denoting the space of ¡Ì _{rather than the real}

space, s.8. Notation), defined by the structure of the model.

While the state-output format may be any format, the set of state transition formats is again defined by three axioms, which may be directly derived from the former,

C2 ¿-i,)

-=

(2

&f)

(z,

,

¿o)

and ¿-, ¿)

_-

_g';,

¿-,

ì

if

ì1(t')

Only pairs of formats (,k), the first one in accordance

with the axioms, will be called state space model

struc-tures.

For convenience correspondìng black box model structures may be defined by the condition

i())

(91 (-)))

resembling that of output equivalence, where

is the black box model structure and the parameter

is the black box model image in the parameter space defined by the structure, which is again subject to the axioms.

2.4 Parameters of models

Among all model images as defined by some structure (,/v)

only images of models of the system described by the given black box model are of interest, where the sets of all possible images

X0R

are defined by the condition of output equivalence

and the condition of format correspondence.

According to the problem statement only those images of models of the system will be considered further on, which may be constructed or identified solely from the given black box model, the sets of identifiable images

being defined by all images of the black box model in the model parameter spaces.

If only one such identifiable state space model image has been constructed or identified from the given black box model the whole set may be defined by the same

transformation rules as before,

i(9u,

)

Tts)

and

li (r

3 )

1)

t ¿ L

In a material mode of speech the parameters of a model of a system are called the properties of the system. Their values are not only dependent on the model structure, but also on the system of units chosen by the observer.

Consequently they have to obey another set of transformation rules to be derived from the principle of objectivity,

requiring that the structures be universal functions, in-variant under changes of units.

2.5 Identification of parameters

In terms of the concepts developed the remaining problem of the identification of one state space model for a sys-tem with a given black box model for a given set of input functions may now be broken down into the three basic operations:

Selection of a structure ( tt), identification of the

image 1b, transformation into the image (, 7').

Usually the first operation in this process of image or parameter identification is considered as the most criti-cal, in the light of the present exposition not always for the true reasons. _{As the choice of the model structure is} in principle only limited by the axioms and the principle of objectivity many more conditions may be imposed, e.g. the requirement, that the second and third operations may be performed smoothly, using well establised techniques. As a forth operation the subsequent transformation into a final state space model, _{e.g. for implementation may be} included in this consideration.

While the choice of the model to be identified is not re-stricted in principle, the final model _{may be completely} determined apriori, if e.g. from that model for the given set of input functions some other output

the present context apriori, practically e.g. determined in exactly the same way as described here. Even with linear state transition functions these other state-output f unc-tions may be non-linear, they may e.g. be cost funcunc-tions. Figure 1 shows the schematic structure of the solving process for the above mentioned problem.

generate -transf. to y implement select determine observe

Figure 1 Schematic structure for the determination of output c-(é) for given input functions

T,

black box model ¿ , and state-output function

and state space 5L

determine

3. LINEAR MODELS

3.1

Continuous models

In order to be specific and to derive some useful results in the following linear models will be considered. From the foregoing it should be well understood, that there are only (abstract) linear models that may describe (real)

systems under specified conditions, among which the restric-tion of input funcrestric-tions to a given set is only one, but

that there are no linear systems in reality.

The linear time-independent functional black box model is the well known Duhamel integral

(e)

JI/e_e')

t(C'J d,

where the memory (or weighting or pulse response) function

¿ 1r

₍

R

¿) & (

describing the dynamics of a system for a given set of input functions is assumed given in the following and in a

narrow sense will alone be called the black box model of the system now.ÇNote the convenient change of interpretation as compared with that of Chapter 2.).

From the input-output equation in the form

(í)

JL(6_2

z/)'

the output is not a state, i.e.

due to the fact that, in general, the input-output equation is no state transition equation, i.e. it does not conform to the axioms, the second one in particular.

Due to the memory of the system instead of the present output alone the whole future output as a result of the whole past input may be considered as a state. The cor-responding state space model

(é)

_57D,é)

with the future time

and the state

a)

E$

A

does infact conform to the axioms, _{to the first and the} third trivially, and to the second as follows:

4

i(y1 )+J/)z//'

2') 7'(')a(

The state so defined ìs the most general in the present context, depending only on the assumption of linearity. The fact that, in general, the state space has one dimen-sion more than the output space will of course not be changed by the following transformations.

3.2 Discrete, finite models

For the purposes of implementation and simulation models with discrete, finite states rather than continuous states are of interest. _{The simplest way to arrive at one such} state is based on the assumption that the continuous state may be expanded (approximately) into the finite _Taylor series

(o)

t')é,

/u!,

where

f('9, O(c)),

i.e. under the assumption that the _{output derivatives in} question exist.

Consequently a discrete state may be defined by the trans-formation (o)

i0()

(3(,/)/

/

eflo

where

4 (j

D(er)

with the well-known state space model

OC ,

=

This model is so far completely universal _{under the} pre-vailing assumption. _{The individual features of the}

-- 1

indicating that the additional state component is a linear function of the state _{components necessary for the} descrip-tion of the system and of the past input.

If it is further on assumed that the input, as the state,

may expanded into a finite Taylor _series

(-'j/!,

I,

1'(û,

D(,iLf)

the last transition equation may be put into the form where

V

f(,

(t)),

system under consideration are completely incorporated in the missing interpretation of the formally introduced state component

Due to the assumption that the _{state is finite and the} model is linear this interpretation may be rendered in the

form t

a,

(é)

f) (i"

s-'

o u, with I

¿

-a,u3. Xei

C)

y

where

The black box model corresponding to this state space model is of course the well-known differential equation

¿'o')

_(y)

_(y)

¿O)

---2e

ü(4)x= ¿

uF _y

As both models are invariant underl linear transformations of the output except for the parameter components _which transform as the output, they are particularly convenient starting points for the following discussion.

3.3 Models of a system

The invariant state space models developed, conforming _to the axioms, as may be concluded from their integral form

Ì)

₌

()

+

f

I /

are possible state space models of a system with a given black box model, if they conform to the condition of output equivalence.

In order to facilitate the discussion Fourier transforms are introduced, e.g.

L

//

(1

₎

b,

where

is the imaginary variable, in deviation from the interpre-tation in Chapter 2.,

L-

T('//21,

dt't;1

is the transfer

function,

and

ì

is the imaginary unit.

Consequently the condition of output equivalence takes the form

-

.-Under the condition

¿7(z',ì

'r)

the original model may for convenience in a first stage be transformed into a closely related model, which may most easily be derived from the black box model in the form

where now

with

('z7

?(- i,

directly corresponding to the state space model

-

_-

be,'

U/)_ç,

D

¿r.!rr

,'

_*

i

i

(i

(Along the same lìnes another model may be derived, if the black box model is further tranformed to Its partial

frac-tians representation}.

As in the general theory only state space models will f ur-ther on be considered, which may be identified from the

given black box model, i.e. here the tranfer function. Accordingly the problem to be solved is to identify the parameter components from the transfer function, i.e. map

the latter into the parameter spaces.

Any such model, and infact all models derived from it by subsequent transformations, may according to the exposition most conveniently be classified by the code

)

(a),

'(i'), cLPz7),

7('r) )

-Of course any of the equivalent representations may be chosen, in particular those referring to the state

dimen-s ion

d(i)

((fJ().

In the present context and in general they do not offer any advantages.

3,4 Identification of parameters

The condition of output equivalence in the form

/o°L(p)0

-

6tLp)

is linear in the parameter components and will _{be used for} their identification, conveniently starting with the in-variants, after the high frequency asymtotes have been identified separately.

As the highest derivative of the output necessary is not known in advance the basic equation is transformed twice and solved recursively. At first all parameter components are divided by one of the invariants of lower order taken

negative, e.g. -, if the system is stable, _{resulting in}

A_

_/

_LYE)

"-'C., with

L(fl)

_¡oL(p)

L'p)

and

r[(1,

)1).

This equation is now further transformed to

L'(fr)

and the (triangular) transformation _{matrix Vare built up} recursively and the process ìs terminated, _{as soon as no} further such functions may be generated.

Consequently the orthonormal invariants are obtained accord-ing to the rule

// 1/ _/

J

L() L

or, if the orthonormal functions are defined sum-orthonormal as in practical applications, according to a corresponding summing rule.

Further on the final invariants have to be derived by two inverse transformations, i.e. in detail

and

with

f.

A similar procedure may be employed to identify the remain-ing parameter components from the residual equation

_'p

=

This procedure is terminated as soon as the right hand side of the equation may be described by the terms identified so far.

35 Transformation of models

The linear state space model, which has been introduced so far and the identification of which has been discussed in some detail, is not necessarily the most convenient in all cases, but as mentioned before other models may be derived by transformations.

Of greatest interest for practical applications are all the transforms of the above model without input derivatives as

far as possible, i.e. the canonical state space models

(w;

-

x

_'40V

':

V

_,

-9.--

s__1 (-J where now

& Jz

d(o(,-)

,d(cçi*a

and

r ¿? f(t

Z?(1).

For convenience in the following only standard canonical state space models defined by the conditions

J

-will be considered. This condition results in

/r5

-o

)

while all the other matrices vanish, i.e. _{it results in the} standard canonical state space model

-=

2. ¿, z

)

- cjl

-SO3

s

5Z1(e9I

o

i _i c:---í'_7 7L ,5L5

The transformation of the image of the intermediate model

b0,

into that of the standard canonical model

9

(4g;,

,

does thus conveniently not require any involved extra com-putations.

respectively, will be considered, as its output the linear velocity of the reference point and the angular velocity, i.e. the motion

all quantities referred to the observation space and in terms of body fixed Cartesian coordinates.

From classical mechanics the black box model of this system is known to be

bi'6

&bCblGba=.

T"2

where

4. APPLICATIONS

4.1 Models of rigid bodies

The first example to be considered is that of a rigid body moving in an observation space

R.

As input of the body the forces and moments, about some body fixed reference

point, acting on its surface and in its volume, i.e. momentum fluxes and sources

If this basic model is linearized about some mean state the resulting state space model is

4'O

1b,

1' _{'6 )}

where all variables denote now deviations from the mean values and

denotes the reactive "damping".

If "small" deviations from mean positions of tracks are of interest, e.g. for buoyant systems as referred to the free surface, the state space model

e62, --

'tbj

7L

e' v1

'p

ho

tPba,

a generalized &-operator, providing the "crossproduct" f

the motion and the momentum (Schmiechen, 1962, 1964, 1965)

According to the foregoing theory this black box model corresponds to the nonlinear state space model

7L _J,60

is employed, where all the components are now in terms of space fixed Cartesian coordinates

In view of the following the state space models are not transformed to the canonical forms.

The corresponding black box models are for both linear models of the type

4.2 Coupling to ideal fluids

In ship dynamics, in the theory of hydrornechanic systems in general, the coupling of rigid and fluid _{systems is of} interest, in the present context the coupling of rigid _bodies to surrounding (infinite or semiinfinite) incompressible

fluids, which will be considered at rest in the inertial observation space, if the body was not moving.

The input of a fluid system surrounding _{a body is the motion} of the body, the deviation from some mean motion of the

body, or the deviation from some position, i.e.

while the output of the fluid system is equal to the forces and moments acting on the surface of _{the body, i.e. the} momentum flux across the surface or

1g=

The two equations are the coupling _{conditions for the two} systems, describing a feed-hack coupling of _{the two systems.}

In the case of an (infinite) fluid _{the blac)ç box model of} the fluid system

£=-

-

-;e 21e1

2L

corresponding to the state space model

i1 ..-

-

2

-is known from classical hydromechan.ics.

The complete state space model of the total system, the rigid body and the surrounding ideal fluid, _{may under due} consideration of the coupling conditions be reduced to the model

b °° /

with the total inertia

of the system, if the state of the fluid _{system is eliminated.}

In this special case the output of the fluid _{ìs a state,} due to the fact that in ideal fluids only reversible changes of momentum occur, the state defined being the rate of these changes.

4.3 Models of real fluids

In real fluids irreversible changes of momentum occur and consequently the classical model is no longer useful. If

further on only linear models for the description of devia-tions from some mean states are considered general state space models of the type developed in Chapter 3 have to be employed, e.g. the standard canonical model

_ -

A

t,0

₎

The modified coupling condition is

-as the term 24 h-as been introduced to represent, or rather

remind of, inputs to the fluid other than the output of _the body, e.g. due to settings of control surfaces, etc.

Under the assumption that the range of fr is the same for the rigid body and the fluid, the total standard canonical state space model for the case of a rigid body moving in a real fluid, where only deviations from a mean motion are of interest, may consequently be stated as follows:

b

b0,

-

-

o

11t_I

V(eo

if

,L Z

i

a--

- A)

O,:I!

Ç-1

-and this may be reduced to the st-andard canonical model of the total system

and the total input

r

12

For quasisteady inputs the complete model may be replaced by the asymptotic model

+qÓh=

a,

eft

with the low frequency asymptotic damping

e /

kv(O,,

z,g1

--

-

aD(a,

.

- .t6o ,I

and inertia

A corresponding high frequency asymptotic model is meaningful only, if the inputs are really limited to high frequencies and are not broadband processes as e.g. in collisions.

In exactly the same way the case may be treated, where devia-tions from posidevia-tions are of interest, there is only one

more state component to be taken into consideration. The resulting standard canonical model is

-Clearly the interpretation of equal symbols in both models is different.

4.4 Inertia and damping

With direct reference to the classical model and the asymp-totic models naval architects usually represent the dynamics of fluid systems in terms of the black box model

L

_{£2? I}

_-

;t

2 A -

)

calling and _{the frequency dependent inertia and} damping, respectively, or by a corresponding model for the second case.

From the black box models standard canonical state _space models may be derived exactly according to the procedure described.

From the foregoing it should be clearly understood that

the concepts of inertia and damping in their usual application are completely inadequate for the modelling of fluid systems and, consequently, of hydromechanic systems, but that the additional aggregate properties introduced may adquately solve the problem.

In practical identification problem very often one dimen-sional cases are considered arid consequently the formal apparatus reduces to its bare minimum. In many of these cases models of the type

-

(1, 1, f,

/ )

Çi(éi

Figure 2 : Analogue network for the simulation

of

one-dimensional hydromechanic systems,

= (f, f, t

f f)

o

Figure 2 shows the analogue network for this type of model, assuming ail parameter components to be positive.

4.5 Models of manoeuviing ships

in order to develop state space models of manoeuvring marine systems, e.g. ships, the way outlined in the fore-going may be continued to appropriately include _{the effect-C} ors, or the effector settings may be directly considered _a

system inputs and the whole procedure described may be applied to the resulting black box model without reference to the concept of force. In practice both approaches are combined, as e.g. in Nomoto's second order _model.

While under service conditions of ships the low frequency asymptotic equation including rather nonlinear terms is

usually sufficient in special "manoeuvres", _{e.g. collisions,} or for high performance crafts, e.g. hydrofoil boats, the complete models described are necessary.

5. CONCLUSIONS

5.1 Review

The formal problem which has been formally dealt with in these notes is the construction _{of state space models from} black box models describing real systems for given sets of input functions. _{In order to fix the ideas and in view of} the applications time-independent functional black box model have been treated. _{At first the general theory has}

been estab1ised as far as necessary _{in order to provide} a sound basis for the subsequent construction of linear state space models and their applications to rigid bodies

moving in incompressible _fluids.

Concerning the geiea _{cheory the exposition is heavily} leaning on Wymor&s fundamental _{theory of systems}

engi-neering (Wymore, 1967), but certainly not living up to the standards proposed. _{Concerning the linear models only} well established basic theorems _{have been put together,} while in the applicaticn- basic problems in marine systems dynamics have been touched upon.

In all the whole exercise is hoped to provide a coherent context and an adeq' a _e f )I _{n i apparatus for the solution}

of the problem stated, fo _{ìeeral discussions as well}

as

detailed analysis.

This paper brings prior efforts of the author in the same direction (Schmiechen, 1962-1972) _{extending over more} than a decade now, finally to a certain end. _{At least a}

general linear model for (fluid) _{systems has been} estab-lished, that may serve as the basis for future work.

5.2 Outlook

No attempt has been made in these notes to indicate or even follow all the evident and less evident sidelines, or to explicitely state the philosophical implications of the whole procedure.

The general theory may, as Wymore has shown, be extended to far more general models than those considered here, thus permitting the application to fai: more complex

sys-tems as e.g. marine traffic control syssys-tems for congested waters, or marine transport systems, in any case including man-machine interaction.

While here only linear models have been identified, a

large variety of non-linear models may be treated, i.e. corresponding state space models may be constructed.

In practical applications to hydromechanic systems state space models according tó the present theory will have to be constructed for individual systems, they will have to be utilized, and interpreted in physical terms as far as necessary.

There is no need for physical interpretation at least not as far as the initial conditions are concerned, which have not been dealt with at all. As they are essentially a representation of the past history of the system, the simplest way to arrive at the initial conditions for a

certain situation is to simulate the past history leading to that state, starting e.g. from some steady state.

Physical interpretations may be useful for the theoretical determination or the system properties. _{As with the} hy-drodynamic inertia and damping

_C,

and , respectively,

the properties ¿2e, and & _{are aggregate properties of the} fluid systems with an infinite, but not independent number of state components.

These few remarks may indicate _{the diversity of problems,} which in continuation of this _{work, may be studied.}

6. ACKNOWLEDGEMENTS

The author is greatful to Prof. _{W. Stein, Senator tür}

Wissenschaft und Kunst, and Prof. S. _{Schuster, Direktor der} Versuchsaustalt für Wasserbau und _{Schiffbau, Berlin, and}

to Dr.-Ing. A. Witkowski, Präsident _{der Technischen}

Universität Berlin, for granting a leave of absence from the duties at the Berlin Model _{Basin and as Privatdozent,} respectively, and he is greatly indebted to Prof. S. Kaya, President of the Japan Society for the Promotion of

Science, Tokyo, for granting a Visiting Professorship, and to Prof. K. Nomoto, University of _{Osaka, and Prof. S.}

Motora, University of Tokyo for the appointment at the Department of Naval Architecture _{at the University of} Tokyo from August through November 1973, during which period this study has been completed. _{Special thanks are} due to Prof. T. Koyama for many critical discussions, and to Miss R. Shimura for the expert typing of the manuscript.

7. REFERENCES

Schmiechen, M:

Eine allgemeine Gleichung für Bewegungen starrer K6rper

in Flüssigkeiten und ihre Anwendung auf ebene Bewegungen

von Doppelkörpern.

Berlin, 1962, Dissertation.

Berlin: VWS, Mitt. Heft 48, 1964.

Fragen der Kursstabilität und Steuerfähigkeit von Schiffen.

Jb. STG: 58 (1964) 319/340.

A general equation of motion for rigid bodies _in incompressible fluids.

Berlin: VWS, 1965.

ONR Contract N 62-558-2552.

On high-frequency damping of screw propellers. Proc. 11. ITTC (1966) 311/315.

Design and evaluation of experiments for the identifi-cation of physical systems.

Cambridge, Mass.: MIT, NAME Rep. 69-1, 1969.

Über mögliche Kopplungen der Bewegungen bei Tauchfahr-zeugen.

Written discussion.

International Symposium on Directional Stability and Control of Bodies Moving in Water.

J. Mech. Engg. Science 14 (1972) suppi. _{issue, 227.}

Wymore, A.W..:

A mathematical theory of systems engineering: the elements.

tï

integer space interval including _the limiting values indicated

sets of state space models

8. NOTATION

General notation Chapter 1., 2.

Concept s

concept input output state

functions J' values

_{J' ()}

dimensions _c(Oì spaces Model

model _{black box}

state space

equations input- state

state-output _{transition output}

functions

i

) formats parameters

r

spaces

_p

a

R

Details

F (

, ) _{function spaces, i.e. sets of mappings}

of first to the second space indicated

rG

format spaces

real space interval _{including the} limiting values indicated

gk

real space of dimension k

current, present time

past time initial time t, running time

T

transformations

u

identity transformations

X

_{Cartesian product}

Special notation Chapter 3., 4. a1 model parameters

A

b

25

e

C

Do

highest derivative of 2 imaginary unit indices ¿ memory function

L

transfer function, Fourier transform of ¿

L"

p

o.,v,

ky

V

orthonormal functions imaginary variable future time operational indices orthomormalizing matrix circular frequency