Modeling with uncertainty in continuous dynamical

systems: the probability and possibility approach

Gianluca Bontempi

IRIDIA, Université libre de Bruxelles,

Brussels, Belgium

e-mail: gbonte@ulb.ac.be

Modeling with uncertainty in continuous dynamical systems:

the probability and possibility approach

Gianluca Bontempi, Student Member, IEEE

IRIDIA, Université libre de Bruxelles,

Brussels, Belgium

e-mail: gbonte@ulb.ac.be

AbstractThis paper presents probability and possibility as two measures to model different types of

uncertainty in continuous dynamical systems. We analyze and compare how probability and

possibility distribution evolve in a dynamical system modeled by differential equations, where

initial conditions and/or parameter may be uncertain.

Moreover, we propose a new algorithm for the numerical resolution of a fuzzy differential system

and we relate it to existing computational methods to solve regular stochastic differential equations.

Finally, we compare results obtained by simulating a dynamical system in which uncertainty is first

represented in a possibilistic and then in a probabilistic form.

1. IntroductionThis paper compares possibility and probability as two formalisms to represent

different kinds of uncertainty in a continuous dynamical system. The notion of

continuous dynamical system is a basic concept in system theory: it is the

formalization of a natural phenomenon in a set of real variables, defined as the state,

and a set of deterministic differential equations, defined as the model. The resolution

of the system of differential equations provides the time evolution of the state

variables, defined as the behavior of the system. The differential model of a

dynamical system contains elements and operators that constitute a formal

description of features of the phenomenon under examination. However, due to

errors in the measurements, lack of precise knowledge or inherent randomness in the

process, these elements may not be uniquely defined. The idea that there can be a

lack of determinism in a deterministic set of equations appears at first to be a

contradiction in terms. What enters here is the important distinction between

mathematical precision and imprecise observations. The determinism we speak about

in dynamical models is the property of the mathematical uniqueness of the solution,

namely, given a precise initial condition of the system, there is precisely one

evolution of the system satisfying both the equations of the model and the initial

conditions. However, this mathematical determinism does not guarantee a

1

completely certain evolution, because it is not always possible to define initial

condition and parameters of the model with an infinite accuracy.

Uncertainty modeling within system theory and artificial intelligence has long been

recognized as a topic worthy of investigation. In system theory uncertainty is

classically treated in probabilistic form by the theory of stochastic processes.

Research in artificial intelligence has enriched the spectrum of available techniques

to deal with uncertainty by proposing new non probabilistic formalisms, as the

theory of possibility (based on the theory of fuzzy sets) and the theory of evidence

(Shafer, 1976).

There has been a long debate on the relationships between probability and possibility

as ways to model uncertainty (Bezdek, 1994). We suppose that in the context of

dynamical systems, uncertainty stems mainly from two sources: low accuracy of

measurement process and imprecise human knowledge (Smets, 1993).

In the first case uncertainty originates from variability of data and weakness of

sensors. Numerical quantitative measurements about initial condition and/or

parameters may be available but they cannot define with certainty the model of the

system. For example, consider a parameter representing the frequency of an event, as

the number of faults per month in an industrial machine: this variable may be not

known precisely but, if we have done a series of measurements of its value in the

past, we may model its uncertainty with a probabilistic distribution.

In the second case, uncertainty does not originate from unpredictable numerical

measurements but from imprecise human knowledge about the dynamical system: as

a consequence, only imprecise estimates of values and relations between variables

may be available. Let us consider, for example, a variable representing the amount of

heat dissipated in a steam generator in normal working conditions: it may be not

convenient, for safety and/or cost reasons, to measure this quantity but a technical

expert may qualitatively suggest its possible value (e.g. "about 90 KJoule"). In this

situation, we consider a possibilistic distribution as a proper representation of the

uncertain knowledge.

This paper focuses on dynamical systems where initial conditions and/or parameters

are described by uncertainty distributions. The main contribution of this paper is to

present and compare the laws describing the time evolution of the probability and

possibility distributions in a dynamical system. Moreover, we propose a new method

to compute how possibilistic uncertainty, represented by fuzzy numbers, evolves in

differential equations. This method is then compared with the Monte Carlo method, a

well known computational method for stochastic systems.

2

The organization of this paper is as follows. We first review some fundamentals of

dynamical systems and introduce probability measures and possibility measures as

special types of fuzzy measures (Sugeno, 1977). Next, we recall how a probabilistic

distribution evolves in a regular stochastic differential equation. Then, we present

possibilistic differential equations in the context of the theory of fuzzy sets and

numbers (Zadeh, 1965). The proposed method of numerical resolution is an extension

of the method proposed in (Bonarini, Bontempi, 1994): we show how the problem of

propagating a fuzzy distribution may be reduced to a problem of optimization, and

we present our fuzzy algorithm of integration. Also, we make an analytical and

computational comparison between probabilistic and possibilistic methods in

dynamical systems. Finally, we compare results obtained by simulating a dynamical

system in which uncertainty is first represented in a possibilistic and then in a

probabilistic form.

2. Dynamical systems and ordinary differential equations (ODE)Consider a dynamical system whose state is composed of n scalar variables yi. Let its

model be a system of n first order ordinary differential equations where the i-th

component is the following

dy

dtF ( , t; ) i (1, ,n) (c , ,c )i

i 1 k= = =y c cL L (1)

Let the initial condition be y(0)=y0 with y0∈ Rn, c a vector of control parameters

and t the variable denoting time. c will be understood to be independent of t.

(1) may be written in the corresponding abbreviated form

y y c y c•

= ∈ ∈F( , ; ) , t R Rn k(2)

We say that such a dynamical system has n as its dynamical dimension. We denote

by y(t,y0) ≡ ( y1(t,y0), ...,yn(t,y0)) the solution of the system of differential

equations (1)1. It is also understood that no yi(t)≡t .

It is very useful to view (1) as describing the motion of a point y(t) in a n-

dimensional space, which will be called the phase space of the system. In fig. 1 we

show the phase space of a 3-th order dynamical system, where a system trajectory

starts in point y(0) at time t=0 and reaches point y(t*) at time t*.

1We remark that the solution y(t,y0) is a function of time and the initial condition: if we assume that

the inital condition is fixed, we denote the solution with y(t).

3

y1

y2

y3y(t*)

y(0)

Trajectory

fig. 1 A trajectory in the phase space of a 3rd order dynamical system

We may easily reformulate the system (1) by eliminating the control parameters and

adding for each of them a state variable whose evolution is governed by the

following differential equation

y c 0 j (1,...,k)n j j

.•

+

•= = =

(3)

The system (1) now has the form

y y y•

= ∈F( , ) t Rn(4)

If the functions Fi do not depend explicitly on time, the system is said to be

autonomous, otherwise it is non autonomous. In a non autonomous system, more

than one solution, say y(1)(t), y(2)(t) can pass through the same point of the phase

space (i.e. y(1)(t*)= y(2)(t*) for some t*) (Jackson, 1991). To ensure that only one

solution passes through each point, we will consider only autonomous systems

y y y•

= ∈F( ) Rn(5)

The general solution of this equation is of the form

y(t)=y(t,y0) (6)

where y(0)=y(0, y0)= y0 is the prescribed initial condition.

Equation (6) can be viewed as a continuous mapping in the phase space Yt:Rn→Rn,

which carries the initial point y0 to the point y(t). If the dynamical solution exists

and is unique, then to any given y(t) and t there must correspond a unique initial

point y0, that is one can write y0=K(y(t),t). This is the inverse mapping of (6) and

since the mapping is one-to-one and continuous, it is referred to as a

homeomorphism (and if it is differentiable in t and in y0, a diffeomorphism) (Arnol'd,

1992).

4

2.1. Numerical solution of ordinary differential equations

The problem of finding a solution to (5), satisfying a known initial condition y0, is

called initial value problem. The solution y(t,y0) may be expressed in analytical or

numerical form: in the first case y(t,y0) has an analytical expression, while in the

second case numerical methods are employed to compute the values y(t*,y0) at

desired instants t*.

To obtain an exact analytical solution is possible only in the simplest cases. Indeed,

several numerical methods have been developed to solve ordinary differential

equations numerically. Runge-Kutta algorithms are the most popular, with predictor

corrector and Burlish-Stoer algorithms being used where high accuracy is required,

or where the equations are 'stiff' and not susceptible to relatively simple Runge-Kutta

solution (Press et al., 1986).

3. Fuzzy measures of uncertaintySo far, we have considered continuous dynamical system where all parameters and

initial conditions are known in a precise way. We now introduce some measures that

allow us to represent uncertainty in a deterministic dynamical system.

Sugeno (1977) provided a general framework to the uncertainty measures by

introducing the notion of fuzzy measure.

A fuzzy measure g on a continuous set Y is a function

g: 2Y→ [ 0 , 1 ]

which assigns to each subset of Y a number in the unit interval [ 0 , 1 ].

In order to qualify g as a fuzzy measure, the function g must satisfy certain

properties (axioms of fuzzy measures (Klir, Folger, 1988)).

Two special types of fuzzy measures are the well-known probability measures and

possibility measures.

Let P : 2Y→ [ 0 , 1 ] be a probability measure P. By definition

P ( A) = p y dyA

( )∫ (7)

is the probability measure of A for all A ∈ 2Y. The function p : Y → R+ is theprobability density function and it satisfies the property p(y)dy

Y∫ =1.

Let us now denote a possibility measure by Π : 2Y→ [ 0 , 1 ]. We define

Π(A)= max (y)y A∈

π (8)

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as the possibility measure of A for all A ∈ 2Y where π : Y → [ 0 , 1 ] is the

possibility distribution function.

Definitions (7) and (8) reveal a substantial difference between the notion of

probability and possibility: while the probability satisfies the property of additivity,

this is not true for the possibility measure. Then, given two sets A, B ∈ 2Y with

A ∩ B = ∅- P(A∪ B)= P(A)+ P(B) while

- Π(A∪ B)= max(Π(A), Π(B)).

As we show in the rest of the paper, the above properties characterize the behavior of

probability and possibility in a dynamical system, .

4. Stochastic differential equations ( SDE )The introduction of random elements into an ODE leads to the definition of

stochastic differential equations. However, we should make precise what we intend

for a stochastic differential equation since it depends on what we mean by

derivatives of a stochastic process.

A crucial point is the regularity (Sobczyk, 1990) of random functions occurring in

the differential equation. If a very irregular random process occurs (as white noise

processes) we have the so-called Ito stochastic differential equations. In these

equations the uncertainty of variables and/or parameters is represented by adding to

their deterministic value the effect of a random process of the white noise type.

Consider a time varying parameter U(t) subject to some random effects: according to

the Ito approach its uncertainty is modeled by U(t)=D(t)+"noise" where the

probability distribution of the noise term is known and D(t) is assumed to be non

random. When the random processes occurring in a differential equation are

sufficiently regular the equations are called regular stochastic differential equations.

In this case an uncertain constant parameter U is represented by a random variable

with a known probability distribution. The theory of the Ito stochastic differential

equations differs significantly from the theory of regular stochastic equations

(Gardiner, 1985).

In this paper we only examine the regular case. In fact, we introduce uncertainty in

an autonomous dynamical system to model the incomplete knowledge about constant

parameters and/or initial conditions: otherwise, the introduction of generalized

random processes of the white noise type into the Ito SDE is adopted mainly to

model dynamical systems subjected to rapidly time-varying random excitation.

Let us now consider the regular differential equation

y y y y y0

•= ∈ =F( ) ( )Rn 0 (9)

6

where y0 is a known random variable.

The solution y(t,y0) of the stochastic initial value problem is still a regular stochastic

process, which is completely characterized by its probability density p(y(t,y0)).

4.1. Regular SDE and the Liouville equation

A well-known way to solve a regular stochastic initial-value problem is associated

with the Liouville equation (Sobczyk, 1990 and Gardiner, 1985).

Let us first introduce the concept of integral invariant.

Let Ω(t) represent an arbitrary volume at time t in the phase space of (9), and let

Ω(t+dt) represent the same volume at time t+dt (fig. 2); so Ω(t+dt) is the set of all

points y(t,y0) that propagate according to (9) and such that y(t)=y0 is contained in

Ω(t).

y1

y2

y3

Ω Ω (t)

Ω Ω (t+dt)

y 0

fig. 2 Evolution of a volume Ω(t) in the phase space of a 3rd order dynamical system

Let us consider the integral of some function f(y,t): Rn× R → R over this moving

domain Ω(t),

I(t) f( , t) dy , ,dy1(t)

n≡ ∫∫L LyΩ

(10)

f(y,t) is called an integral invariant of (9) if I(t) is constant in time.

One basic result of stochastic theory is the Liouville theorem, which states that each

function f(y,t) whose integral respect to the spatial variable is invariant during the

evolution of the system, satisfies the following equation

7

[ ]∂∂

∂∂

f( ,t)

t yii 1

nf( , t)Fi ( ) 0

yy y+

=∑ = (11)

which is commonly known as the Liouville equation.

It has been demonstrated that the probability density p (y(t),t) is an integral invariant

(Sobczyk, 1990), that is P( (t)) p( , t)d(t)

ΩΩ

= ∫ y y, is constant in time. Moreover, the

property of integral invariance implies that the Liouville equation (11) holds for the

probability density:

[ ]∂∂

∂∂

p( , t)

t yii 1

np( ,t)Fi ( ) 0

yy y+

=∑ = (12)

The Liouville equation is a first order partial differential equation and can be solved

in general by the Lagrange method.

It may be also transformed in the following ordinary differential equation holding

along the trajectories of the system (Nicolis, Prigogine, 1977):

dp( , t)

dtp( , t) * divF( )

yy y= − (13)

where div is the divergence and is defined as

div F( )F

yi

ii 1

n

y ==∑ ∂

∂(14)

The probability density of a point in the phase space may be intuitively interpreted as

the density of trajectories of the dynamical system at that point. When the divergence

is null in the whole phase space, the dynamical system is conservative: this means

that the volume of a region Ω(t), as the density of trajectories, remains constant

during the evolution of the dynamical system (fig. 3.a). When the divergence is

greater or smaller than zero, the system is said to be dissipative: if the divergence is

greater than zero the system is divergent, the volume of a region Ω(t) increases and

the density decreases (fig. 3.b). The opposite happens if the system is convergent.

8

y1y2

y3

y2

y3

y1conservative dissipative divergent

fig. 3a fig. 3b

Then, the evolution of the probability measure in a dynamical system satisfies these

two properties.

Property 1. The probability density p(y(t,y0),t) along a trajectory y(t,y0) of a

dynamical system is not constant (except for div F=0).

Property 2. The probability of finding trajectories inside the time-variant volume

Ω(t) is constant during the evolution of the dynamical system.

It is worth remembering that the Liouville equation is a special case (the diffusion

term is null) of the Fokker-Planck-Kolmogorov equation (Gardiner, 1985) that

represents the evolution of probability distribution for the Ito stochastic differential

equation. Therefore the regular SDE may be seen as a particular case of the more

general Ito SDE.

4.2. Numerical solution of regular SDE

By solution of a regular SDE (9) we mean a stochastic process satisfying the

equation together with its probabilistic properties. Generally, as in the non random

case, analytical solutions are not available. Most often, we can only obtain

approximate solutions by numerical schemes.

The Liouville equation can be used for the numerical computation of time evolution

of the probability density. However, its major problem is that the numerical

resolution of a partial differential equation implies a huge computational cost..

Another common resolution method is the MonteCarlo method (Korn, Korn, 1968).

The Monte Carlo method is a statistical method for simulation that utilizes sequences

of random numbers. It may be applied when the system is modeled by probability

density functions. Thus, it is also used to simulate the evolution of a probability

density in ordinary differential equations.

Monte Carlo simulation of ODE is composed of the repetition of these three steps:

9

- a random sampling of the probability density of the initial condition distribution

- a series of deterministic model simulations using as initial conditions the sampled

values ("trials")

- computation of the result by taking an average over the number of simulations.

The set of simulated trajectories represents then a probabilistic sample of the solution

of the stochastic differential equation.

We will now consider the properties of possibility measures, as an alternative

formalism to represent uncertainty in a dynamical system. However, since the theory

of possibility is based on the theory of Fuzzy Sets (Zadeh, 1965) (Dubois, Prade,

1980), we first review some fundamentals of this theory.

5. Fuzzy sets and fuzzy numbersLet Y be the universe set whose generic elements are denoted y. The membership of

an element y to a subset A of Y can be represented by a characteristic function, or

membership function (m.f.)

µA: Y→0,1 such that

µA(y)=1 if y ∈ Α (15)µA(y)=0 if y ∉ Α

When the valuation set 0,1 is extended to the real interval [0,1], A is called a fuzzy

set.

Def. A fuzzy set A is normalized if ∃ y | µA(y)=1 (16)

Def. An α-cut of level h (0 ≤ h ≤ 1) of a fuzzy set A is the interval

Ah=y ∈ Α, µ A(y) ≥ h (17)

Def. A fuzzy set A is convex iff its α-cuts are convex.

Def. A fuzzy number is a convex, normalized fuzzy subset of the domain YThe concept of fuzzy number is an extension of the notion of real number: it encodes

approximate but non probabilistic quantitative knowledge (Kaufmann, Gupta, 1985).

For instance, let us consider a system (e.g. a steam generator) where it is not easy or

convenient to measure a certain variable (e.g. its internal pressure); furthermore

suppose that we have a qualitative, imprecise knowledge that in certain operating

conditions the variable is around 5 bar. This does not imply a probabilistic

distribution of the variable's value but rather a possibilistic distribution, which may

be represented by a fuzzy number (fig.4).

10

5 pressure

m.f.

1.0

fig. 4 A fuzzy number

Zadeh has introduced both the concept of fuzzy set (Zadeh, 1965) and the concept of

possibility measure (Zadeh, 1978). Zadeh's possibilistic principle (Zadeh, 1978)

postulates

µA(y)=πA(y) for all y∈ Y (18)

where µA(y) is the membership function of a fuzzy set A and πA(y) is the

possibilistic measure of A. We rest on this principle to justify our use of fuzzy

membership functions to represent possibilistic information.

5.1. Fuzzy functions

A fuzzy function can be understood in several ways according to where fuzziness

occurs (Dubois, Prade, 1980). We are considering deterministic models, then we will

consider only ordinary functions that just carry the fuzziness of their arguments

without generating extra fuzziness themselves. In such cases, we can apply the so-

called fuzzy extension of non fuzzy functions by the extension principle introduced

by Zadeh (1965). This principle provides a general method for extending standard

mathematical concepts in order to deal with fuzzy quantities (Dubois, Prade, 1980).Let Φ:Y1× Y2×...×Yn→Z be a deterministic mapping from Y1× Y2×...×Yn to Z

such that z=Φ(y1,y2,...,yn). The extension principle allows us to induce from n input

fuzzy sets yi

− on Yi an output fuzzy set z

− on Z through Φ given by

µ µ µ µ

µ

z t (s ,s ,...,s ) y1

y2

yn

z

1

(t) sup min (s ), (s ), , (s )

or (t) 0 if (t)1 2 n 1 2 n1

− − − −

−

=

= = ∅=

−

Φ

Φ

L(19)

where Φ− =1(t) 0 is the inverse image of t and µyi

− is the m.f. of yi

− (i=1,...n).

In the next section, we will use the extension principle to define fuzzy differential

equations.

11

6. Fuzzy differential equations (FDE)Let us consider a fuzzy differential equation

y y y y Y0

•= ∈ =F( ) R (0)n (20)

with Y0 fuzzy initial condition defined on the n-dimensional domain Y.

We interpret this notation as a fuzzy extension of a non fuzzy differential equation.We consider a fuzzy differential equation as a deterministic differential equation

where some coefficients or initial condition are uncertain and not representable in a

probabilistic form: its solution is then the time evolution of a fuzzy region of

uncertainty which corresponds to the possibility distribution in the phase space. The

following theorems characterize some properties of this solution.

Theorem 1. Let Y*=Y(t*,Y0) be the state of the fuzzy differential equation (20) at

time t*. Then its value is given by

µ µY Y 0 0*

0y* y y* y y( ) ( ) (t , )*= = . (21)

where y=y(t,y0) is the solution of the non fuzzy system y y y y y0

•= ∈ =F( ) R (0)n

.

Proof. The fuzzy value Y* of the state at time t* may be computed by the extension

principle in this wayµ µ

Yy Y y* y y

Y 0*

0 0 00

y* y( ) ( ): ( , )

=∈ =

supt*

According to the solution of the initial value problem there is only one trajectoryy(t,y0) passing through the point y* at time t*: it follows that µ µ

Y Y 0*0

y* y( ) ( )= in

which y*==y(t*,y0). +

Theorem 2. In a fuzzy differential equation the value of the probability distribution

is constant along the trajectories of the system.

Proof. According to the theorem 1, µ(y(t,y0))=µY0(y0) along a trajectory y(t,y0).

Then, according to the Zadeh's possibilistic principle (18)

π(y(t,y0))=µ(y(t,y0))=µY0(y0)= constant+

Theorem 3 Let Ω(t) represent an arbitrary volume at time t in phase space of the

fuzzy differential system (20) and Π(Ω(t)) be the possibility measure of this set. Then,

Π(Ω(t)) is time invariant.

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Proof. Given that the value of the possibility distribution π(y(t)) is time invariantalong a trajectory y(t) for the theorem 2, and that Π Ω

Ω( (t)) sup ( (t))

(t) (t)=

∈yyπ

we may

deduce that also Π(Ω(t)) is time invariant. +

To interpret intuitively these results let us consider a solution (trajectory) of the

system passing through the point y1=y(t1,y0) at time t1 and through the point

y2=y(t2,y0) at time t2. What the theorem 2 tells us is that the possibility that a

trajectory passes through y1 at t1 is equal to the possibility that a trajectory passes

through y2 at time t2. Moreover, if we consider a region Ω(t) that evolves according

to the model of the system, for the theorem 3 the possibility of finding a trajectory in

Ω(t1) at time t1 equals the possibility of finding a trajectory in Ω(t2) at time t2.

6.1. Numerical solution of a fuzzy differential equation

The Extension Principle provides a method to compute the fuzzy value of a fuzzy

mapping but, in practice, its application is not feasible because of the infinite number

of computations it would require.

Our approach considers instead fuzzy calculus as an extension of interval

mathematics by the α-cut notion. A fuzzy computation may be decomposed into

several interval computations, each one having as arguments the α-cut intervals of

the fuzzy arguments at the same level. The results are the α-cuts of the fuzzy results

(Nguyen, 1978). Then, the following relation holds for the mapping (19)

zh

y1

y2

yn h

y1h

y2h

ynh

= =ϕ ϕ( , ,..., ) ( , ,..., )

with zh

and yih

α-cuts of h-level of the fuzzy numbers z and yi, respectively.

Every fuzzy computation, even if differential, may be then decomposed into the

repetition of more interval computations. From now on we will treat only interval

computation.

6.1.1 Interval differential equations (IDE)

Let us consider a system of interval differential equations (IDE) where the initial

condition Y0αα is the α-cut of the fuzzy initial condition Y0 in the FDE (20).

y y y y Y0

•= ∈ =F( ) R (0)n

αα (22)

This is the differential equation α-cut of the corresponding fuzzy differential

equation (20).

Let us define as solution Yαα (t) of (22) the set of solutions of the ordinary differential

equation whose initial condition belongs to Y0αα .

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6.1.2 Numerical solution of IDE

To solve numerically the system (22) it might seem intuitive to apply the interval

mathematics (Moore, 1966) directly to the numerical algorithms introduced in

Section 2.1. However, in (Bonarini, Bontempi, 1994) we have shown that a

straightforward use of interval mathematics for the resolution of fuzzy (or interval)

differential equation can produce incorrect results. This is due to the fact that the

fuzzy (and interval) formalism is unable to represent the interaction that the

differential equation establishes between variables: as a consequence, spurious values

are introduced into the solution and the evolution of the system may reach region

where no numerical solution exists. Figure 5 illustrates the problem. The figure plots

the evolution of the initial condition (i.e. the rectangle ABCD) in a oscillatory 2-nd

order system: if we represent the uncertain state of the dynamical system at time t' in

the interval formalism, we introduce in the solution spurious points (black regions)

which are not the evolution of points belonging to ABCD.

A B

CA'

B'

D'

D

C'

fig. 5 Introduction of spurious trajectories in the non interacting simulation of an oscillating system

Our approach is instead the following: the initial condition of the interval differential

equation defines an hyper cube. We call this n-cube the region of uncertainty of the

system at time t=0. To solve the IDE means to compute the evolution in time of the

region of uncertainty.

We proved that under general conditions of continuity and differentiability it is

sufficient to compute the evolution of the external surface of the uncertainty region

(Bonarini, Bontempi, 1994). We demonstrated that

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Theorem 4. An ordinary autonomous differential equation maps the external surface

of its region of uncertainty at time t into the external surface of its region of

uncertainty at time t+dt.

The validity of the theorem leads to the conclusion that it is sufficient to calculate the

trajectories of the points belonging to the external surface of the region of

uncertainty to know the evolution in time of the region itself. However, the quantity

of trajectories to be computed is still infinite, also if it is of a lower order. The

problem remains how to sample, in the most convenient way, the external surface of

the region of uncertainty during its time evolution.

We extend the approach, presented in (Bonarini, Bontempi, 1994) by reducing this

sampling problem to an optimization problem. To find the numerical solution of an

IDE means to know at each time step t* the maximum and the minimum value of

each state variable yi. These values are obtained by trying to maximize and minimize

at time t* the functions yi(t*,y0). We may then use a constrained multivariable

optimization algorithm to find the extremes of the functions yi(t*,y0) i=1...n where

t* is fixed and the argument y0 is constrained to range over the external surface of

the initial region of uncertainty.

Let us remind that the performance of a multivariable optimization algorithm may be

greatly increased by providing to it the partial derivatives (gradient) of the function

value respect to the arguments: in our case we should provide the derivatives at time

t * of yi respect to y0∈ Y0αα . To obtain these derivatives we use the connection

matrix (Moore, 1966), as follows.

We denote by y(t,y0) the n-dimensional, real vector solution to the numeric initial-

value problem. We define the connection matrix for the solution y(t,y0) as the matrix

C(t,y0) with elements

cy t

yij

i

j

=−

−

=−

∂

∂

( , )y

y y0

(23)

We denote by J(t,y0) the Jacobian matrix of the vector function evaluated at y(t,y0).

The matrix J(t,y0) has elements

J (t, )F ( )

ijiyy

0

y y y0

==

∂∂y j t( , )

(24)

Moore (1966) demonstrated that

∂∂

C(t, )

tJ(t, ) C(t, )

yy y0

0 0= ⋅ (25)

15

with C(0,y0) = I where I is the identity matrix.

The elements of the matrix give the sensitivity of the solution components yi(t,y0)

with respect to small changes of the initial values (y0)i.

By coupling the IDE system with the set of n2 equations coming from the connection

matrix differential system we obtain an augmented differential system 2 which gives

at time t* the values of the variables yi and the values of the derivatives of the

variables yi respect to the initial condition y0. We may then use these combined

values as the value and the gradient of the function to be maximized (or minimized),

respectively.

The central part of the algorithm is structured in the following steps:

(i) the optimization routine takes a starting point y0 in the external surface of the

initial uncertainty region;

(ii) this point is passed to the routine of ODE numerical integration which uses y0 as

initial condition and returns the value y(t*,y0) and the gradient values;

(iii) if the optimization routine considers y(t*,y0) as a maximum (minimum) the

algorithm stops; otherwise, the value y(t*,y0) and the gradient are used to sample a

new point y0 and the execution returns to point (ii).

2Let us consider for example the Van Der Pol system

z.

(y y 1) z y

y.

z

= − ⋅ − ⋅ −

=

The augmented system, including the terms of connection matrix, with initial conditions is

z y y z y

y z

d

dt

z

z

d

dt

z

y

d

dt

y

z

d

dt

y

y

y y y z

z

z

z

y

y

z

y

y

z z

y y

z

z

z

y

y

z

y

y

.( )

.

( ) ( )

( )

( )

( )

( )

( )

( )

= − ⋅ − ⋅ −

=

=− ⋅ − − ⋅ ⋅ −

⋅

=

=

=

=

=

=

1

0 0

0 0

1 2 1 0 0

0 0

00

00

00 1

00 0

00 0

00 1

1 0

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂∂

∂∂

∂∂

∂∂

16

Let us consider as example a second order IDE (fig. 6): the external surface at time

t=0 is the square ABCD: it is transformed by the differential equation into the region

A'B'C'D' at time t*. For each of the 4 parts (e.g., the segment AB) of the external

surface of ABCD, the optimization algorithm searches the maximum and the

minimum of the correspondent transformed part (e.g., the curve A'B'). The greatest

of the maxima and the smallest of the minima are the extrema values of yi.

y1

y2

A B

C D

A' B'

C'

D'

t=0

t=t*

y2_max(t*)

y2_min(t*)

y1_max(t*)y1_min(t*)

fig. 6 Evolution of the external surface of the region of uncertainty for a 2-nd order system

Let us remind, anyway, that we want a method of numerical resolution for a fuzzy

differential equation. This procedure must be then repeated for each α-cut that

decomposes the fuzzy initial value. The number of the α-cuts to be considered

depends on the desired degree of accuracy. At this regard, it is important to remark

that the region of uncertainty corresponding to an α-cut of level h1 is always

contained into the region of uncertainty corresponding to an α-cut of level h2, if

h1>h2 (Bonarini, Bontempi, 1994).

Here it is the complete fuzzy algorithm in a pseudo programming language

17

for each α-cut

for t*=initial_time to final_time

for each variable yi

yi_max(t*)=-Infinity

yi_min(t*)=Infinity

for each component Y0j of the external surface of the initial region of uncertainty

j=1...2n

yij_max(t*)= max(yi(t*,y0)) (y0 ∈ Y0j & yi(t*,y0) is a solution of (20))

yij_min(t*)= min(yi(t*,y0)) (y0 ∈ Y0j & yi(t*,y0) is a solution of (20))

yi_max(t*)=max(yi_max,yij_max)

yi_min(t*)=min(yi_min,yij_min)

end for

end for

end for

end for

table 1

This algorithm has been implemented in Matlab language (The Mathworks, 1992), by

combining together the algorithms of optimization, resolution of differential

equations and symbolic computation provided by this mathematical tool.

The Symbolic Toolbox is used to automatically create the augmented differential

system by computing the Jacobian of the original differential system.

The Optimization Toolbox provides the algorithm of constrained optimization constr.

The Matlab routine ode23 performs numerical integration.

Moreover we used the graphical facilities of Matlab to present results of fuzzy

simulation.

6.1.3 Remarks

The proposed method goes one step further with respect to the non interacting

algorithm proposed in (Bonarini, Bontempi, 1994): the heuristic criterion to choose

the trajectories is now replaced by an optimization algorithm whose task is to choose

the points to be simulated in order to reconstruct the region of uncertainty at a given

t*. This does not guarantee to obtain the absolute maximum and minimum but the

degree of performance is analogous to that of a gradient-based algorithm of

optimization in front of a multivariable function.

In terms of computational complexity it is necessary to remember two drawbacks of

the present approach: (i) the complexity is an exponential function of the order n of

18

the dynamical system (2n is the number of components of the external surface of the

initial region of uncertainty) (ii) the fuzzy solution at time t* is not computed by

starting from the fuzzy solution at the previous instant (t*-dt): in fact, there is no

guarantee that the trajectories considered to compute the extremes of the region of

uncertainty at time (t*-dt) are the same trajectories to be considered at time t*. Then,

at each time t* the trajectories must be computed starting from the time t=0. This

implies an heavy computational charge when t* becomes high.

7. Probability vs. possibility in differential equationsThere exists a certain amount of literature debating the problem of the relation

between possibility and probability (Bezdek, 94). In the sections 4 and 6 we have

shown how probability density and possibility distribution evolve differently along

the trajectories of a dynamic system: while probability changes according to the

Liouville equation, possibility remains constant. We us now discuss the relation

between these two formalisms in analytical and computational terms.

7.1 Probability vs. possibility in analytical terms

Let y(t) be a point in the phase space of the dynamical system. The probability that

the state belongs to a infinitesimal volume Ω(t) around y(t) is p(y(t),t)⋅Ω(t). Given

the property of integral invariance p(y(t),t)⋅Ω(t) = p(y(t+dt),t+dt)⋅Ω(t+dt). Then

along a trajectory, the quantity p(y(t),t)⋅Ω(t) remains constant.

That is the following relation holds

d

dt( p( (t),t) (t) ) 0y ⋅ =Ω (26)

This implies

p( (t),t) (t) p( (t), t) (t) 0• •

⋅ + ⋅ =y yΩ ΩFrom the Liouville equation (13) we have

− ⋅ ⋅ + ⋅ =•

p( (t), t) div F( (t)) (t) p( (t),t) (t) 0y y yΩ ΩThen along the trajectories where p(y(t),t) is different from zero we have

Ω Ω(t) (t) div F( (t))•

= ⋅ y (27)

We may use the formula (27) to obtain

d

dt( ( (t), t) (t))

d ( (t), t)

dt(t) ( (t), t) (t)

( (t),t) (t) div F( (t))

ππ

π

π

yy

y

y y

⋅ = ⋅ + ⋅

= ⋅ ⋅

•Ω Ω Ω

Ω (28)

19

Consequently, the evolutions in time of probability and possibility are summarized

by the following equations

dp( (t),t)

dtp( (t), t) div F( (t)) (29a)

d(p( (t), t) (t))

dt0 (29b)

d( ( (t), t))

dt0 (29c)

d( ( (t), t) (t))

dt( (t),t) div F( (t)) (29d)

yy y

y

y

yy y

= − ⋅

⋅=

=

⋅= ⋅

Ω

Ω

π

ππ

(1)

The first consideration we may deduce concerns what Zadeh called

possibility/probability consistency principle.

Zadeh (1978) defined a degree of consistency γ of the probability density p with the

possibility distribution π. We extend Zadeh's definition to a continuous domain Y by

γ π(t) ( (t), t) p( (t), t) dY

= ⋅ ⋅∫ y y y (30)

Let us notice that when γ is closer to 1, the consistency is higher.

We may demonstrate the following theorem:

Theorem 5. Consider a dynamic system where the uncertainty of the initial condition

is represented both in probabilistic and in possibilistic form. If the initial probability

density is consistent with the initial possibility distribution (γ=1) then this consistency

is conserved during the system's evolution.

Proof. γ =1 means that p(y)>0 implies π(y)=1. Then, from the differential laws of

evolution (29a) and (29c), we can see that the consistency is conserved during thesystem evolution.

+

Our second consideration refers to the definition of a parameter relating possibility

and probability density evolution along a system trajectory.

Theorem 6. Let us define η(y(t),t) so that

η(y(t),t)=p t t

t t

( ( ), )

( ( ), )

y

yπ(31)

The time evolution of η is governed by the following differential law

η•

(y(t),t)= -η(y(t),t)*div F (y(t),t) with η(0)=p( ( ), )

( ( ), )

y

y

0 0

0 0π(32)

20

Proof. For the time invariance of π(y(t),t) (formula (29c)) and the formula (29a) we

have η•

(y(t),t)=p( (t),t)

( (t), t)

( (t), t) divF( (t))

( (t), t)( (t),t) divF( (t))

•

=− ⋅

= − ⋅yy

y yy

y yπ π

ηp

. +This differential relation permits to obtain a probability density value p(y(t),t) from

the related possibility distribution π(y(t),t) and vice versa, once we know the initial

distribution p( ( ), )

( ( ), )

y

y

0 0

0 0π. We may then conclude that not only the consistency property

is satisfied but also that it is possible to define a parameter η relating probability and

possibility, whose evolution depends on the model of the dynamical system.

7.2 Monte Carlo and FDE

The different analytical properties between probability and possibility, studied in the

previous section, suggest the adoption of specific methods to solve numerically

SDEs and FDEs.

However, Fishwick (1990) proposed the stochastic Monte Carlo method also for the

numerical resolution of a fuzzy differential equation when the initial distribution is a

normalized fuzzy distribution. We will now demonstrate that the Monte Carlo

approach cannot be correctly used when we are dealing with possibility, even if its

distribution is normalized.

Let us consider the differential laws of evolution for probability (29b) and possibility

and (29d). In the stochastic case, the homeomorphism between Ω(t) and Ω(t+dt) and

(29b) implies that the probability of being at time t in Ω(t) equals the probability to

be at time t+dt in Ω(t+dt). Then if we make a random sampling in Ω(t), the evolution

of these points according to (9) determines a random sampling in Ω(t+dt). As a

consequence, a random sampling of the initial condition produces a consistent

random sampling of the stochastic solution. This property justifies the adoption of a

Monte Carlo method in the probabilistic setting.

If instead we want to interpret the initial normalized distribution as a possibility

distribution, the law (29d) states that a random sampling on the initial condition does

no more imply a random sampling on the possibilistic solution. In a dynamical

system, generally, the quantity π(t,y(t))dΩ(t) is not time invariant.

Therefore, the use of the Monte Carlo approach to evaluate the evolution of a

normalized fuzzy distribution in a deterministic system is not justified.

7.3 Probabilistic vs. possibilistic computational methods

The type of uncertainty affecting the model determines what method of simulation

use. While the Monte Carlo method appears the most effective in the simulation of a

21

regular SDE, our algorithm (table 1) seems well suited to simulate a fuzzy

differential equation.

Even if they cover different situations, it may be interesting to compare the two

methods in terms of computational complexity and reliability.

Monte Carlo is simply the repetition of previously fixed number of numerical

simulations. This means better performance in terms of complexity (see par. 6.1.3)

with respect to the fuzzy method, but worse reliability. In fact, for a general

differential equation there is no indication about the number of random trajectories to

be considered. A random sampling on the initial condition transforms itself into a

consistent random sampling but we cannot be sure that the number of samples which

characterizes with a certain approximation the initial probability distribution are still

significant with the same tolerance during the system evolution.

Let us consider for example a divergent system whose initial probability distribution

is normal with mean µ. Let us take n random samples from this distribution and

assume that m and S represent the sample mean and the variance, respectively. The

confidence interval for µ is3 m±tS

nα

2( ) (Weiss, 1993): this means that the

probability P tm

Sn

t( )− ≤ − ≤α αµ

2 2 that the real value of µ is inside the interval

m±tS

nα

2( ) is 1-α. If we consider a successive time instant, the variance S of the

sample increases and the confidence in the value of m, as estimate of µ, decreases

(i.e. we have a greater interval of confidence). Then, in this case a fixed number of

samples determines a continuous deterioration of the estimate.

In our fuzzy approach, on the contrary, we use a gradient based optimization

algorithm at each time step to determine how many trajectories are necessary to

describe the possibility distribution. The number of trajectories does not only depend

on the initial distribution of possibility but also on the model's structure.

8. ExperimentsWe present two simulations of a dynamical system, model of a real mechanism, in

which uncertainty is first represented in a possibilistic and then in a probabilistic

form.

Let us consider the physical system in fig. 7 (Borrie, 1992): a rod of mass M and

moment of inertia I, which is rotable at one end round a horizontal axis and driven

3 t is the Student t-distribution. tα denotes the t-value with area α to its right under a t-curve (Weiss,

93)

22

by a motor gearbox that supplies a torque u. The whole arrangement is mounted on a

table subject to vertical displacement whose unknown acceleration is w.

θ

motor

rod

w table

fig. 7 Experimental non linear device

The angular behavior of the rod is modeled by the following (heuristic) non linear

equation, derived using Lagrangian mechanics:

I f Mgl u Ml wθ θ θ θ.. .

cos cos + + = −where θ is the angular position of the rod, f is the frictional torque, l is the distance

from the axis to the center of gravity .

Let us model the uncertain acceleration in the possibilistic case by the uniform fuzzy

set described in fig. 8, and in the probabilistic case by an uniform probability

distribution between 0.0 and 1.0.

Let us simulate the system by our fuzzy resolution method and by Monte Carlo.

In fig. 9 we have the evolution of the possibility distribution of θ in time: we have

plotted the α-cut corresponding to the base (in this case a single α-cut describes the

whole fuzzy number). In fig. 10 we have the evolution of the 100 random trajectories

of θ obtained by sampling its initial probability density .m.f.

1.0

w 0.0 1.0 0.5

fig. 8 Fuzzy uniform distribution of the vertical acceleration

23

24

0 5 10 15 20 25 30 35 40 45−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7

fig. 9 Possibilistic system evolution according our fuzzy algorithm

25

0 5 10 15 20 25 30 35−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7

fig. 10 Probabilistic system evolution according Monte Carlo method

Analyzing the random Monte Carlo evolution it is possible to have an indication of the

evolution of the probability density but as stressed before we must suppose that the

number of samples is sufficient for that.

9. Conclusion

In this paper we have presented and compared probability and possibility as two

formalisms to represent and propagate uncertainty in a continuous dynamical system.

Probability and possibility can be used as fuzzy measures on the dynamical phase

space to describe two different kinds of uncertainty: the uncertainty that originates from

random measurements, and the uncertainty that stems from imprecise human

knowledge. Starting from the basic properties of system theory and fuzzy measures,

we have showed that different, but related, analytical laws describe the behavior of

probability and possibility in dynamical systems. We have made this relation precise

26

by the introduction of a parameter, whose evolution is characterized by a differential

equation derived by the system model.

The different analytical properties suggest the adoption of specific computational

methods to simulate the evolution of possibilistic and probabilistic dynamical systems.

While Monte Carlo methods are traditionally used when the uncertainty is represented

in probabilistic form, we have proposed a simulation algorithm for systems where the

initial conditions and/or parameters are represented by possibilistic distributions.

Unlike the Monte Carlo method, our method may adapt the number of trajectories

considered to the changing complexity of system dynamics.

Some fields of engineering are interested in a non stochastic representation of

uncertainty in dynamical systems, like Fuzzy Control and Identification. We believe

that a comparative analysis with related stochastic methods may be useful to better

understand the differences and the relative merits of both these approaches. We hope

that the study proposed in this paper, may help in this analysis.

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