A New Algorithm for On-Line Relational Identification of Nonlinear Dynamic Systems P. J. Costa Branco J.A. Dente Instituto Superior Tknico (I.S.T.) Departamento de Eng. Electr&nica e de Computadms Sec@ de MAquinasElktricas e Electrdnica de Pot&ncia E-mail :
[email protected]
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Abstract Fuzzy relations can model nonlinear dynamic systems since they a r e universal approximators and can perform nonlinear mappings. A model which takes into account input-output relations is more adequate for identification of complex non-linear processes. Fuzzy relation R describes existing relation between input-output variables treated as fuzzy sets. Although, this relation must be modified when a time-variant systems on-line identification is done. For each convenient set of sampling periods, a time-invariant fuzzy relation R can be found. A new on-line identification algorithm for fuzzy model systems, based on simplified max-min relational equation i s proposed. This algorithm presents a gravitycenter adjustment method based on error value between measured and predicted system sates and a time-variant universes of discourse concept. This last concept intends to avoid predicted signal saturation. The fuzzy algorithm is tested in two numerical examples: the traditional gas-furnace experimental data-set and a bilinear system. Its performance for different situations are discussed and compared with other published results.
investigated for the gas furnace data of Box and Jenkins [5] and a bilinear system.
lI.FUZZY SYSTEMS Systems under study are modeled as fuzzy systems described bY
- -
-
where is mux-min composition operator, U1, Us, ..., U,, denote input fuzzy sets, y stands for an output fuzzy set and R is a fuzzy relation expressing system input-output - -
relationship 7 and U1, U2, ..., Un are fuzzy reference sets defined on space universes of discourse Y and U respectively,
YY+[O,l],
u:u+ [O, 13. They satisfy the condition of completeness:
I. INTRODUCI'ION
Many system-dynamic identification approaches have been proposed during the last two decades. However, for real Every fuzzy set y and defined in Y and U, can be expressed complex systems with nonlinear time-variant structure many difficulties still exist. For this class of so called "ill-defined by fuzzy referential sets and represented by a possibility systems", fuzzy-set theory is applied in order to perform vector [PI, pz, ...,pn] where n is the number of the operations on fuzzy imprecise information. Zadeh [l] defines system identification as "the determination on the basis of referential fuzzy sets. This referential sets are linguistic input and output of a system, within a specified class of representations of input-output variables. Each possibility systems, to which the system under test is equivalent". Fuzzy measure value is calculated in same way for y and E. In the identification defines a linguistic model describing temporal dynamic system behaviour either by linguistic approach [2,3] same manner than to Y , possibility measure Pi for is or using fuzzy-relational equations [4]. Based on fuzzy model, calculatedaccordingto a controller can be constructed. Its relation is determined by input-output data being predicates in the strategy-control rules [89]. Therefore, a fuzzy model predictive controller can be pi = POS@lUj = sup,&Ji(u) t i?(u)], (9 implemented and modified according to the new process conditions. In this paper, a new algorithm that builds fuzzy model systems is presented where on-line gravity-centers where Pi E [0311 and t is a triangular norm (in that case min adjustment and variable universe of discourse concept are ) 151. Equation (6) can be reduced to applied.The proposed algorithm characteristics are 1173
c
m i n . Next, collecting the pair
(uk,
y$ and doing its
Fk,
fuzzification Ut),the imposed structure parameters are calculated resulting in (k)relational matrix,
6 is a degenerated fuzzy set (fuzzy singleton) such that ~ ( u o=) 1 .This approach takes fuzzy and nonfuzzy data being treated uniformly and reduce memory load for the algorithm Computer impk"tati0n [2]. TO tranSlatt? linguistic variables into numeric ones, gravity-center method can be used, i.e. if
U
=
Pi.Wi/
W~RR wi
is
,
(14)
where x denotes the Cartesian product. Each element of R,
R(i, j) ,is calculatedby the Cartesian product of the elements
Pi- Pj of b k , Yd by,
wi i- 1
i=l
R k = 6kx TIC
R(i, j) = min@i, Pj) . i, j = 1 ,..., n
(8)
Ui fuzzy reference gravitycenterset.
(15)
On-line identification algorithm is divided in two steps: first
Fuzzy relational equation (1) describes multiple-input singleoutput fuzzy systems. From a system theory point of view, following simplified version of (1) can be considered as single-input single-output fuzzy system,
step adjusts estimated relational matrix R used in output system prediction: the second one realize fuzzy union
- -
operation between the adjusted matrix R(k) with the one
Y=UR
(9)
and
its discretized version for each time moment k .Equation (8) can be rewritten as following
-
Y(yd = supukeu[min(Gdud, R dub Yd)] ,
(1 1)
whereu , y membership functions and R are used with y k E y. F~~~ =lation R is written as a set of fuzzy rules in terms of referential sets defied on each universe of discourse. For single-input single-output system (9) defined with n referential sets for and y , R is a n x n matrix of possibility measures, R(i, j) = Pij
(12)
Each matrix element is translated as a linguistic simple rule, if Ui then Yj with possibility pij
computed from the measured (k+l) values R'k+l . This proposed identification algorithm is completed by an gravitycenters adjustment method and the variable universes of discourse concept. A. Gravity-Center Adjustment Method Membership functions used by simplicity to characterize input-output system variables are presented in Figure 1. In this figure, universe of discourse is symmetric (WnivDisc). There are n=7 referential sets separating the universe of discourse in six equal segments. Each referential set has its gravity-cenkr represented by CGi, i = NB
,..., PB
(16)
Membership functions or referential sets are: NB - Negative Big: NM - Negative Medium: NS - Negative Small; ZE Null; PS - Positive Small: PM - Positive Medium: PB Positive Big.
(13)
and for each Vi there are n simple rules that form a called compound rule. -UnivDisc +UnivDisc Fig. 1. Reference fuzzy set definition.
m.PROPOSEDON-LINE FUZZYSYSTEM IDENTIFICATION The first stage of identification process is system structure specification determined by the composition operator sup-min . For discretized universes of discourse sup-min stays m x -
A numerical value
vector: 1174
uk
will be represented by its possibility
b, are reached. Suppose that UInic and YInic are, respectively, initial values of the input and output universes of discourse. Define UnivDiscl, and Uni~Disc2~ as time instant k limit The elements of c k are linguistics values of u k in such referential sets. Each set is characterized by a gravity-center and has two limits depending on universe of discourse magnitude. Numerical values translated from its linguistic representation are calculated by the discretized gravity-center method. For a linguistic vector y
y = [NB,NM, NS, ZE, PS, PM, PB]
UnivDisc2 = YInic
1.O
C . Identipcation Algorithm Description
On-line identification of system relational structure is computed at each instant k resulting in a nonlinear system
(NB.CG,) + ( N M . C G ~+ ... + (PB.CG~) CGM+ C G m +
[ (211. . + [ (&!-q.
UnivDisclk=UInic . 1.0 +
(18)
its numerical value becomes: Y=
values for input-output universes of discourse. The expansion processed at same time in the two universes, are then described by:
... -tCGw
*
(19)
m i s value is a media of linguisticvalues, weighted by the gravity-centersofach referential set- n e proposed methodof gravity-centers adjustment is based on error between measured
ikafter linguistic-
Output Vdue y k and its predicted value numerical translation:
relational matrix R k used for the fuzzy prediction of system output. Suppose three time instants in identification process (k-l), (k) and (k+l). Each pair Uk, y k representing system
cl,
input-output, has its correspondents fuzzy values and Y k obtained from already adjusted referential sets. The identification algorithm is divided in two steps (Fig. 2): first
h
ek=Yk-yk
(20)
is done in Samedirectionof error value andis The proportional to each linguistic value "weight" (NB, NS, ..., 3
PB) of the output estimated fuzzy vector, Y k . The adjustment stays :
step adjusts relational matrix Rk, used in system output prediction, resulting in a matrix R(k) ; second step realizes fuzzy union between R (k) with that computed from measured VdUeS
R'k+l.
Relational matrix R k is obtained by: C G= ~ CG~" + (a. e . 3 i j i =NB,...,PB
(21)
Parameter a is responsible for dynamics adjust. To (a>> l), the identification algorithm Dresents some oscillations in values the algorithm performance deteriorates because error between measured and predicted values grows up. Output signal dynamics is incorporated by the gravity-center adjustment method, reducing the error t" G k and Yk
.
Rk=( vkx Y k ) uR(k- 1). that is a fuzzy union between
(23) uk, Yk
Cartesian product With
time instant (k-1) in its first step. Equation (10) is used to predict one-step forward fuzzy system output. With next input signal
ck+and using mar-min composition, results:
e Yk+l=Uk+l*Rk
(24)
B. Time-Variant Universes of Discourse
Estimated fuzzy output value is translated to numerical one Proposed identification algorithm presents saturation in predicted values near from universe of discourse limits. T~ overcome this, a percentage expansion b in the initial limit values is proposed. Therefore, if input-output system values are near from universe of discourse initial limits, they will be expanded until the new limit values, determined by parameter
h
by gravity-centa a d j m " I"d resulting in y k + I . This value is compared with real one, generating error signal ek 1175
.
.. .
+
= Yk
+
- ik +
(25)
used in gravity-center adjustment. By other side, an error
This error vector corrects matrix R k by
vector E k + is obtained with each element being the bounded difference [101 between referential set values:
Boundeddif€emce is defined as
with R (4 being the bounded sum [101between R and error
- - -
-
C = AeB
Ek+l. To each column j of matrix R k , a
factor
3
proportional to the component j of E L + 1 is added. Parameter q is a value that can be modified (o< q
