. Quantified Set Inversion with Applications to Control

2004 IEEE International Symposium on Computer Aided Control Systems Design Taipei, Taiwan, September 2-4.2004

Quantified Set Inversion with Applications to Control

Pau Herrero,

Miguel A. Sainz,

Josep

Vehi and

Luc

Jaulin

Abstract-This paper decribes a new reliable method, based on Modal Interval Analysis (MZd) and Set Inver- sion (SI) techniques, for the characterization of solution sets defined by Quantified Constraints Satisfaction Problems ( Q C S P ) over continuous domains. The presented method- ology, called Quantified Set lnvenion (QSI), can be used over a wide range of engineering problems involving uncer- tain nonlinear models. Finally, an application on parameter identification is pmented.

I. INTRODUCTION

Many engineering problems, like in control engineering, can he formulated in a logical form by means of some kind of first order predicate formulas: formulas with the logical quantifiers, universal and existential, a set of real continuous functions, equalities or inequalities and variables ranging over real interval domains. More recently, this for- mulation has been referenced by different authors under the names: Generalized Constraints Satisfaction Problems [27]

or Quantified Constraints Satisfaction Problems ( Q C S P ) A. Stare-of-the-Art

Up to now, Cylindrical Algebraic Decomposition [29], [ 7 ] , [12], for which a practical implementation exists [4], has been the most extended method to solve this type of problems. However, this technique is only well suited for small or middle-size problems because of its computational complexity. Moreover, it often generates huge output consisting on highly complicated algebraic expressions which are not useful for many applications and it does not provide partial information before computing the total result.

~21. ~241.

Methods that appear lately [ I l l , [2] try to avoid some of these problems restricting oneself to approximate instead of exact solutions, using solvers based on numerical methods. However, these algorithms are also restricted to very special cases (e.g. quantified variables only occur once, only one quantifier,etc.). Recently, some of these deficiencies have been partially removed by Ratschan [24]

but, a lot of work remains to he done before obtaining an efficient and general method.

Many practical examples exist on the resolution of Q C S P using the different existing approaches, for

P. Hemro, M. A. SvinZ and J. Vrhi m with l n r t i ~ r dlnformatlca i Aplicacians. Universital de Gi", Blds. P4, Campus dc MonWivi. E- 17071 Gimm, CaWloniu, Spain. {pherreroseia, vehiseia.

sainz@ima}.udg.es

L. Iaulin is wilh Laboramire d'Ing6nicrie des SysPmes Automatis&.

Univcrrit6 d h g e r s . 62 avenue Nom Dame du Lac 49 OOO Angers, F m c e . jaulinsuniv-angers.fr

example in control engineering [SI, [181, (91, [251, [201, electrical engineering [28], mechanical engineering [ 141, [13], biology [6] and various others [3].

11. PROBLEM STATEMENT

A Quantified Constraint (QC) is an algebraic expression over the reah which contains quantifiers (3,V), predicate symbols (e.g.,=,

<,

5). function symbols (e.g.+,

-,

x , sin, e x p ) , rational constants and variables x = { z l , .

. . ,

zn} ranging over reds domains D = { D I > .

. . >Dn}.

An example of a QC is the following one,

Vx

E psx4 +p z 2 + q z + r

2

0, (1) where x is a universally (V) quantified variable and p and r are free variable.

As defined in [27], a numerical constraint satisfaction problem, is a triple C S P = (z, D , C ( z ) ) defined by

(i) a set of numeric variables z = {ZI,

. . .

,zn}, (ii) a set of domains D = {Dl,

. . . ,

D,} where

Di,

a set

of numeric values, is the domain associated with the variable x i .

(iii) a set of constraints C(z) = { C ~ ( Z ) ,

...

,&(E)}

where a constraint Ci(z) is determined by any nu- meric relation (equation, inequality, inclusion, etc.) linking a set of variables under consideration.

A solution to a numeric constraint satisfaction problem C S P = (z, D , C ( z ) ) is an instantiation of the variables of x for which both inclusion in the associated domains and all the constraints of C ( z ) are satisfied. All the solutions of a constraint satisfaction problem thus constitute the set

(2) Now suppose that the constraints C ( x ) depend on some parameters p 1 , p 2 , .

. .

, P I about which we only know that they belong to some intervals P I , Pz,

. . . ,

Pl. Moreover, these parameters have an associated quantifier Q E {V, 3).

Taking into account the dual character of interval uncer- tainty, the most general definition of the set of solutions to such Quantified Constraint Satisfaction problem

QCSP

should have the form

C = { z € D I Q i ( p , , , P , , ) . . . Q i ( p , , , P , , ) C ( z ) } , (3) C = {z E D

I

C ( x ) i s satisfied}.

where

. ^{Q i}

are logical quantifiers V or 3 (in this paper, only the case of universal quantifiers preceding the existential ones will he dealt).

.

( ~ 1 . ~ 2 , . . . , P I } is the set of parameters of the con- straints system considered,

{ P I , P2,.

. . ,

Pl} is a set of intervals containing the possible values of these parameters,

.

^{c< E C I} is a permutation of the numbers 1,.

. .

,1.

The sets of the form (3) will he referred to as quanti- fied solutions sets to the numerical quantified constraints satisfaction problem QCSP = ( x , D , C ( z ) ) .

111. METHODOLOGY A. Set Inversion

One way of solving a C S P is through the characterization of its solution set by means of the Set lnversion

(SI)

approach.

Let CSP he a constraint satisfaction problem CSP ⁼ (x:

D ,

C(x)). Set inversion aims at characterizing the set C of all x such that C is satisfied.

Remark: All constraints are considered under the form C ( x ) := f ( x ) = 1. where f a continuous function from Rn to R m .

Given a box X (ca?esian product of intervals), an algorithm which does set inversion is based on a branch- and-hound technique and the 3 followings set of rules:

Rule 1 : V(a, X)C(x) q X

i

C

This logic formula, used to prove that a box

X

is contained in the solution set, is equivalent to the following interval computation and interval inclusions

O u t ( f ( X ) ) G y ,

where f ( X ) are the ranges of the function components over the interval vector

X

and O u t ( f ( X ) ) are outer approximations o f f

(X).

Rule 2 :

V(z,

X)-C(z) @

X LE.

This logic formula, used to prove that a box

X

does not belongs to the solution set, is easily proved by means of the following interval computation and interval inclusions

O u t ( f ( X ) ) G T .

Finally, if Rule 1 and Rule 2 are not accomplished the position of the box

X

is undefined.

Rule 3 : Otherwise, X i s undefined.

Fig. 1 shows a two dimensional example of the three Then the algorithm which does set inversion is as follows where

.

^{c: SI stops} the bisecting procedure over X when this possible situations corresponding to the 3 rules.

precision is reached,

TABLE 1 SI ALGORITHM

~ i g ~ l i ~ l m

sr(Ln:

c,

xo,

e , out: E-, AX) - I ) Initialization: Stack=Xo;C- := 0 A Z := 0

2) Repeat 1) llnrtack ~~~~~~~~~~ X : ~

4) if with(xj 5 e , then AX :=

ac

U

x,

5) 6 )

else if Rule 1 is satisfied, then C- := C- U X, else If Rule 2 is satisfied, then X is non solution, 7)

8) Until Stack=0

else Bisect X and stack resulting boxes;

I

.

^E-: Subpaving (list of nonoverlapping boxes) repre- senting an inner approximation of the solution set,

. ^AX:

Subpaving representing all the boxes for which nothing could he proved.

These suhpavings provide the following bracketing of the solution set:

e- c c c e- u n c

B. Quanrified Set Inversion via Modal Interval Ana/ysix Classical

SI

is well suited characterizing solution sets of the form (2). The problem arises when the sets are of the form (3). Several authors have proposed solutions to this problem using classical interval analysis and constraint propagation approaches [16], 121, [24]. In this section, a new algorithm for the characterization of quantified solution sets based on Modal Interval Analysis ( M A )

[IO]

is presented. This algorithm will he referred to as Quantified Set Inversion

(esp.

Let us consider the case when the constraints are under the form C(x) := f ( x ) 0, with f a continuous function from

Rn

to

E.

The main difference between the classical SI Algorithm and the quantified one lies on the used set of rules. For the proposed algorithm the following rules will be used:

<

Fig. 1. Solution s a

Rule 1 : v ( z , X ) V ( P U ,

P u ) ~ ( P E ,

P E ) C ( z )

*

x ⁱ

E.

This logic formula, used to prove that a box

X

belongs to the solution set. can not he easily proved by means of classical interval computations. For this reason, M I A is proposed. M I A is a powerful mathematical tool which allows the evaluation of quantified interval formulas by means of interval computations. Concretely, to evaluate the set of logic formulas, the *-semantic theorem given by M I A is used to reduce equivalently the logical formula to the interval inclusion

O U t ( r ’ ( X . P U , P E ) )

i Z.

where

X , Po

are proper intervals, PE improper one, O u t ( J * ( X , P o , P E ) ) is an outer approximation of the the *-semantic extension of the continuous function f and

Z

= [O,O],

Z

= [-w,O] or

2

= [O,ca] depending on if the constraints are under the form C ( x ) := f ( x ) = 0.

C ( x ) := f ( z )

<

0 or C ( x ) :=

f(z) >

0, respectively.

R e m a r k A modal interval X is defined as a couple

X

= (X’,V) or X = ( X ’ , 3 ) where X’ is its classic interval domain, X’ E I @ ) , and the quantifiers ‘d and 3 are a selection modality The modal intervals of the type X = ( X ’ , E ) are called proper intervals or exisrenriul intervals, the intervals of the type X = (X’,V) are called impmper intervals or universal inrelvals. A modal interval can be represented using their canonical coordinates in the form

{

([a,b]’,3) i f a < b ([b,a]‘,V) if a

2

b

X

= [a, b] =

For example, the interval [2,5] is equal to ([Z, 5 ] , 3 ) and the interval [8,4] is equal to ([418],V).

In order to obtain the second rule, used to prove that a box X does not belongs to the solution set, the following implication is used:

Rule 2 :

This logical formula is, analogously, equivalent to the - ( ~ ( P u , P u ) ~ ( P E , P E ) ~ ( x , X ) C ( x ) )

*

x c E.

following interval exclusion:

IMfYx,

P u , P E ) )

P

^Z,

where P O is a proper interval, X , P E improper ones, I n n ( f ’ ( X , P u , P E ) ) is an inner approximation of the the *-semantic extension of the continuous function f , and Z = [O,O],

2

= [-w,O] or Z = [0, 001 depending on if the constraints are under the form C ( x ) := f ( r ) = 0, C ( r ) ^:= f ( z )

<

0 or C ( z ) := f ( z )

>

0, respectively.

Finally, if none of these rules are accomplished, the box

X

is undefined.

Rule 3 : otherwise,

X

is undefined

Computing the semantic extension of a continuous function f by means of any of their interpretable rational extensions provokes an overestimation of the interval evaluation, due to the multi-occurrences of variable, when the rational computations is not optimal. An algorithm, based on results of M I A and branch-and-hound techniques which allows to efficiently compute an inner and an outer approximation o f f ‘ has been recently built.

When the constraints are under the form C ( x ) :=

f (x)

>=

0, with f a continuous function from

B”

to Wm and each variable existentially quantified appears in only a component function, the problem is reduced to m different problems, one for each component function.

<

IV. APPLICATION

Interval model based techniques, like robust control [26], (171, [301, [151 or robust fault detection [I], requires from a well knowledge of the process model to he treated. This section describes an application of the Q S I algorithm which consists on identifying on a guaranteed way 1311 the parameter of a nonlinear model.

A. Parumerer Identification

The problem treated in this section is a well known problem of the literature. It has been taken from [16], which at the same time has been inspired from 1231.

B. Pmblem Srutement

by two main characteristics:

The present problem of parameter identification is defined i. The process model A nonlinear process which de- pends on the variable t and two parameters x1 _and x 2 is used. The theoretical model of the process is:

y ( x . t) = 20exp(-z1t)

-

8 e x p ( - x z t )

ii. The eonctraints to he satisfied The constraints im- posed by the system are:

y ( x : t i ) E K , t i E T,,

vz

E 1, I

. .

^{, l o ,}

where Y, corresponds to the uncertainty associated to the measure yi and Ti represents the uncertainty associated to the measurement time t i .

Table ll shows the uncertainty associated to y and t.

Fig. 2 is a graphic representation of the unceaainty rectangles associated to the vectors t and y of the table

n.

The accepted parameter set is defined by

TABLE II

UNCERTAINTY ASSOCIATED TO t ^{A N D} y

I i II T. ". I

12

4 2 0

Fip. 3. Pavings generated by QSZ algoriuUn Grouping the existential quantifiers and expressing it under

a

vectorial form

which is implied by the following exclusion test E = {Z E X13t E T,3y E Y , y ( z , t j - y = 0 ) .

I . . ( f : ( x l , XZ, Ti, Y,

j j

g

^[O,O],

For one sample i ( i = ( 1 , .

. .

,lo}), the logic formula

which fulfils the points belonging to the solution set Ci is with XI,

X,,

Ti and Y , improper intervals.

~~

~(xl,x:)~((zZ,~~)3(~l,T!)3(2/2, "&,ti) - vi = 0

which is equivalent to the following inclusion test O u t ( f : ( X i , X z .

Ti, Y,)) C

[0,01, with XI and

XZ

proper intervals and T; and ones.

improper

The logic formula which fulfils the points not belonging to the solution set Ci is

.4 -2

I

I

0

5

10 15 20 25

't

Fig. 2 . Graphic representation of the uncnWinry associated 10 the veclors t m d y

Then.

c

=

xi n . . . n cl0.

C. Test Case

For

X

= [0,1.2] x [0,0.5] and a precision of e = 0.01.

QSZ generates in 20 seconds on a Peutium llI lGHz, the paving of fig. 3.

where the darker region corresponds to the solution set C, the grey region corresponds to the non solution set

E

and the white region is undefined.

D.

Analysis of the Results

Comparing the obtained results with the ones obtained by other existing algorithms [161, [19], for which an efficient implementation [22] exists for the second one, it can be said that any relevant difference can be observed in terms of the solution and computational performances.

However, the method proposed in 1191 should be better in terms of computational complexity for a higher order problem (e.g. more parameter to identify) due to the use of constraint propagation techniques [SI, 1211.

The main difference between the presented algorithm and the mentioned ones does not lie on the computational complexity but on the conceptual complexity. While in the Q S l algorithm the set tules used to prove if a box

X

is inside or not from the solution set are achieved by means of simple interval computations provided by

M A

the other algorithms needs from more complex strategies to carry on the same task.

V. CONCLUSIONS AND FUTURE WORKS A. Conclusions

The contribution of this paper has been to introduce a new algorithm, based on M I A and S I techniques, for the characterization of solution sets defined by numerical QCSP. The applicability of the method to engineering problems has been shown by means of a well known problem of the literature on parameter identification. A comparison with other existing techniques has also been carried out concluding that the presented algorithm intro- duces more simplicity to the problem of characterizing the set defined by a QCSP.

B. Future Works

I ) Reducing the complexity via Constraint Propagation:

In order to reduce the non polynomial complexity of the SI algorithm due to the branching, a narrowing operator (a contractor) for quantified constraints will be provided.

This contractor, based on constraint propagation techniques and M I A , allows the contraction of an initial box X containing the solution set C to another one

X'

such that

X' still contains E.

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