Mathematical Fuzzy Control. A Survey of Some ... - Semantic Scholar
Mathematical Fuzzy Control. A Survey of Some Recent Results SIEGFRIED GOTTWALD, Leipzig University, Institute for Logic and Philosophy of Science, Beethovenstr. 15, 04107 Leipzig, Germany. E-mail: [email protected] Abstract The core point of fuzzy control approaches are finite lists of linguistic control rules. For computerbased automatic control these lists have to be transformed into control algorithms which can be realized on a computer. The main general idea of this fuzzy control approach is that such an algorithm should yield a fuzzy subset of the output space of the control problem if confronted with a fuzzy subset of the input space. This paper surveys mathematical problems which are connected with, and arose out of these basic ideas. The main formal tools used in these mathematical considerations are fuzzy sets and fuzzy relations together with some generalized, viz. many-valued logic which underlies these considerations. And the essential way of understanding the mathematical context of fuzzy control is to look at it as an interpolation problem: one has to determine a fuzzy control function out of a finite list of interpolation nodes. Keywords: fuzzy relation equations, solvability criteria, solvability index, pseudo-solutions, optimality of pseudo-solutions, approximation and interpolation, compositional rule of inference, fuzzy control
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Introduction
In engineering science, fuzzy control methods have become a standard tool which allows to apply computerized control approaches to a wider class of problems as those which can reasonably, and effectively, treated with the more traditional mathematical methods like PD or PID control. For the industrial engineer, usually success in control applications is the main criterion. Then he even accepts methods which have, to a larger extent, only a heuristic basis. And this has been the situation with fuzzy control approaches for a considerable amount of time. Particularly with respect to the linguistic control rules which are constitutive for a lot of fuzzy control approaches. Of course, success in applications then calls for mathematical reflections about, and mathematical foundations for the methods under consideration. For linguistic control rules their transformation into fuzzy relation equations has been the core idea in a lot of such theoretical reflections. Here we discuss this type of mathematical treatment of rule based fuzzy control with particular emphasis on some more recent viewpoints which tend toward a more general view at these mathematizations.
L. J. of the IGPL, Vol. 0 No. 0, pp. 1–17 0000
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c Oxford University Press °
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Preliminaries
We use in this paper a set theoretic notation for fuzzy sets which refers a logic with truth degree set [0, 1] based upon a left continuous t-norm t, or – more general – based upon a class of (complete) prelinear residuated lattices with semigroup operation ∗. This means that we consider the logic MTL of left continuous t-norms as the formal background, cf. [6]. This logic has as standard connectives two conjunctions and a disjunction: & = ∗,
∧ = min ,
∨ = max .
In the lattice case we mean, by a slight abuse of language, by min, max the lattice meet, and the lattice join, respectively. It also has an implication → characterized by the adjointness condition u ∗ v≤w
iff
u ≤ (v → w) ,
as well as a negation − given by −H = H → 0 . The quantifiers ∀ and ∃ mean the infimum and supremum, respectively, of the truth degrees of all instances. And the truth degree 1 is the only designated one. Therefore logical validity |= H means that H has always truth degree one. The shorthand notation [[H]] denotes the truth degree of formula H, assuming that the corresponding evaluation of the (free) variables, as well as the model under consideration (in the first-order case) is clear from the context. The class term notation {x k H(x)} denotes the fuzzy set A with µA (a) = A(a) = [[H(a)]] for each a ∈ X . Occasionally we use graded identity relations ≡ and ≡∗ for fuzzy sets, based upon A j B = ∀x(A(x) → B(x)) , and defined as A≡B A ≡∗ B
= =
A j B & B j A, A j B ∧ B j A.
Obviously one has the relationships
3
|=
Bi ≡∗ Bj ↔ ∀y(Bi (y) ↔ Bj (y)) ,
|=
A ≡ B → A ≡∗ B .
Fuzzy Control and Relation Equations
In the standard approach, a fuzzy controller is determined by a list of linguistic control rules if α is Ai , then β is Bi , i = 1, . . . , n , (3.1) describing some control procedure with input variable α and output variable β.
Mathematical Fuzzy Control. A Survey of Some Recent Results 3 Mainly in engineering papers one often consider also the case of different input variables α1 , . . . , αn , in this case the linguistic control rules become the form if α1 is A1i , and . . . and αn is Ani , then β is Bi ,
i = 1, . . . , n .
But from a mathematical point of view such rules are equivalent to the former ones: one simply has to allow as the input universe for α the cartesian product of the input universes of α1 , . . . , αn .
3.1 The Compositional Rule of Inference Following the basic ideas of Zadeh [17], such a fuzzy controller is formally realized by a fuzzy relation R which connects fuzzy input information A with fuzzy output information B via the compositional rule of inference (CRI) B = A ◦ R = R00 A = {y k ∃x(A(x) & R(x, y))} .
(3.2)
Therefore, applying this idea to the linguistic control rules themselves, these rules in a natural way become transformed into fuzzy relation equations Ai ◦ R = Bi ,
for i = 1, . . . , n
(3.3)
i.e. form a system of such relation equations. The problem, to determine a fuzzy relation R which realizes via (3.2) such a list (3.1) of linguistic control rules, therefore becomes the problem to determine a solution of the corresponding system (3.3) of relation equations. This problem proves to be a rather difficult one: it often happens that a given system (3.3) of relation equations is unsolvable. This is already the case in the more specific situation that the membership degrees belong to a Boolean algebra, as discussed (as a problem for Boolean matrices) e.g. in [10]. Nice solvability criteria are still largely unknown. Thus the investigation of the structure of the solution space for (3.3) was one of the problems discussed rather early. One essentially has that this space is an upper semilattice under the simple set union determined by the maximum of the membership degrees; cf. e.g. [1]. And this semilattice has, if it is nonempty, a universal upper bound. To state the main result, one has to consider the particular fuzzy relation b= R
n \
{(x, y) k Ai (x) → Bi (y)} .
(3.4)
i=1
Theorem 3.1 b is a solution The system (3.3) of relation equations is solvable iff the fuzzy relation R of it. b is always the largest solution of the system (3.3) And in the case of solvability, R of relation equations. This result was first stated by Sanchez [15] for the particular case of the minbased G¨odel implication → in (3.4), and generalized to the case of the residuated implications based upon arbitrary left continuous t-norms—and hence to the present situation—by the author in [3]; cf. also his [4].
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Mathematical Fuzzy Control. A Survey of Some Recent Results
3.2 Modelling Strategies Besides the reference to the CRI in this type of approach toward fuzzy control, the crucial point is to determine a fuzzy relation out of a list of linguistic control rules. b can be seen as a formalization of the idea that the list (3.1) The fuzzy relation R of control rules has to be read as: if input is A1 then output is B1 and ... and if input is An then output is Bn . Having in mind such a formalization of the list (3.1) of control rules, there is immediately also another way how to read this list: input is A1 and output is B1 or ... or input is An and output is Bn . It is this understanding of the list of linguistic control rules as a (rough) description of a fuzzy function which characterizes the approach of Mamdani and Assilian [11]. b the fuzzy relation Therefore they consider instead of R RMA =
n [
(Ai × Bi ) ,
i=1
again combined with the compositional rule of inference.
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Toward a Solvability Criterion for RMA
Having in mind Theorem 3.1, one immediately is confronted with the following Problem: Under which conditions is the fuzzy relation RMA a solution of the corresponding system of relation equations. This problem is discussed in [8]. And one of the main results is the next theorem. Theorem 4.1 Let all the input sets Ai be normal. Then the fuzzy relation RMA is a solution of the corresponding system of fuzzy relation equations iff for all i, j = 1, . . . , n one has |=
∃x(Ai (x) & Aj (x)) → Bi ≡∗ Bj .
(4.1)
This MA-solvability criterion (4.1) is a kind of functionality of the list of linguistic control rules, at least in the case of the presence of an involutive negation: because in such a case one has |=
∃x(Ai (x) & Aj (x)) ↔ Ai ∩t Aj 6≡∗ ∅ ,
Mathematical Fuzzy Control. A Survey of Some Recent Results 5 and thus condition (4.1) becomes |=
Ai ∩t Aj 6≡∗ ∅ → Bi ≡∗ Bj .
(4.2)
And this can be understood as a fuzzification of the idea “if Ai and Aj coincide to some degree, than also Bi and Bj should coincide to a certain degree”. Of course, this fuzzification is neither obvious nor completely natural, because it translates “degree of coincidence” in two different ways. Corollary 4.2 b as a solution. If condition (4.2) is satisfied, then the system of relation equations has R This leads back to the well known result, explained e.g. in [4], that the system of relation equations is solvable in the case that all the input fuzzy sets Ai are pairwise t-disjoint: Ai ∩t Aj = ∅ for all i 6= j. It is furthermore known, cf. again [4], that functionality holds true for the relational composition at least in the form |=
A ≡ B → A◦R ≡ B ◦R,
because one has the (generalized) monotonicity |=
A j B → A◦R j B ◦R.
This, by the way, gives Corollary 4.3 A necessary condition for the solvability of a system of relation equations is that one always has |= Ai ≡ Aj → Bi ≡ Bj . This condition is symmetric in i, j. Therefore one gets as a slight generalization also Corollary 4.4 Let all the input sets Ai be normal. Then the fuzzy relation RMA is a solution of our system of fuzzy relation equations iff for all i, j = 1, . . . , n one has |= Ai ∩t Aj 6≡∗ ∅ → Bi j Bj .
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Relating RMA with the Largest Solution
b However, it may happen that the system of relation equations is solvable, i.e. has R as a solution, without having the fuzzy relation RMA as a solution. An example is given in [7]. Therefore Klawonn’s condition (4.1) is only a sufficient one for the solvability of the system (3.3) of relation equations. Hence one has as a new problem to give additional assumptions, besides the solvability of the system (3.3) of relation equations, which are sufficient to guarantee that RMA is a solution of (3.3). As in [4] and already in [2], we subdivide the problem whether a fuzzy relation R is a solution of the system of relation equations into two cases.
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Definition 5.1 A fuzzy relation R has the subset property w.r.t. a system (3.3) of relation equations iff one has Ai ◦ R ⊆ Bi , for i = 1, . . . , n , (5.1) and it has the superset property w.r.t. (3.3) iff one has Ai ◦ R ⊇ Bi ,
for i = 1, . . . , n .
(5.2)
Particularly for RMA quite natural sufficient conditions for the superset property have been given, but only rather strong ones for the subset property. Proposition 5.2 If all input sets Ai are normal then RMA has the superset property. So we know with the fuzzy relation RMA , assuming that all input sets Ai are normal, at least one upper approximation of the (possible) solution for the system of relation equations. Proposition 5.3 If all input sets are pairwise disjoint (under ∩t ), then RMA has the subset property. b satisfies these It is also of interest to ask for conditions under which the relation R properties. Fortunately, for the subset property there is a nice answer. Proposition 5.4 b has the subset property. R Together with Proposition 5.2 this immediately gives Corollary 5.5 If all input sets Ai are normal then one has for all indices i the inclusions b ⊆ Bi ⊆ Ai ◦ RMA . Ai ◦ R
(5.3)
b at least one lower approximation of the Thus we know with the fuzzy relation R (possible) solution for the system of relation equations. However, the single inclusion relations (5.3) can already be proved from slightly weaker assumptions. Proposition 5.6 b ⊆ Bk ⊆ Ak ◦ RMA . If the input set Ak is normal then Ak ◦ R b always are subsets So we know that with normal input sets the fuzzy outputs Ai ◦ R of Ai ◦ RMA . Furthermore we immediately have the following global result. Proposition 5.7 If all the input sets Ai of the system of relation equations are normal and if one b then the system of relation equations is solvable, and RMA is a also has RMA ⊆ R, solution. Now we ask for conditions under which the relation RMA maps the input fuzzy b And that means again to ask for some conditions which sets Ai to subsets of Ai ◦ R. give the subset property of RMA , and thus the solvability of the system of relation equations.
Mathematical Fuzzy Control. A Survey of Some Recent Results 7 Proposition 5.8 Assume the normality of all the input sets Ai . Then to have for some index 1 ≤ k ≤ n b Ak ◦ RMA ⊆ Ak ◦ R is equivalent to the equality b = Bk . Ak ◦ RMA = Ak ◦ R Corollary 5.9 Assume the normality of all the input sets Ai . Then the condition to have for all indices 1 ≤ i ≤ n b Ai ◦ RMA ⊆ Ai ◦ R is equivalent to the fact that RMA is a solution of the system of relation equations, and hence equivalent to the second criterion of Klawonn.
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b Toward the Superset Property of R
The solvability of the system of relation equations is equivalent to the fact that the b is a solution. Therefore the solvability of our system of relation equations relation R b has the subset as well as the superset properties. is also equivalent to the fact that R Now, as seen in Proposition 5.4, the subset property is generally satisfied for the b This means we immediately have: fuzzy relation R. Corollary 6.1 b has the superset property. A system of relation equations is solvable iff its relation R Hence, to get sufficient solvability conditions for the system of relation equations b means to look for sufficient conditions for this superset property of R. And this seems to be an astonishingly hard problem. What one immediately has in general are the equivalences: b |= Bk j Ak ◦ R iff for all y |= Bk (y) → ∃x(Ak (x) &
^ (Ai (x) → Bi (y))) i
iff for all y and all i |= Bk (y) → ∃x(Ak (x) & (Ai (x) → Bi (y))) .
(6.1)
And just this last condition offers the main open problem: to find suitable conditions which are equivalent to (6.1). Particularly for i = k and continuous t-norms this is equivalent to |= Bk (y) → ∃x(Ak (x) ∧ Bk (y)) . Corollary 6.2 b is that For continuous t-norms t a necessary condition for the superset property of R hgt (Bk ) ≤ hgt (Ak ) holds for all input-output pairs (Ak , Bk ).
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Part of the present problem is to look for sufficient conditions which imply (6.1). Here a nice candidate seems to be to have for given i, k and y the existence of some x with |= Bk (y) → Ak (x) & (Ai (x) → Bi (y)) . Routine calculations show that this means that it is sufficient for (6.1) to have, for a given y, either the existence of some x with Bk (y) ≤ Ak (x) and
Ai (x) ≤ Bi (y)
or the existence of some x with Ak (x) = 1
and
Ai (x) ≤ [[Bk (y) → Bi (y)]] .
However, both these sufficient conditions look not very promising.
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Getting New Pseudo-Solutions
Suppose again that all the input sets Ai are normal. The standard strategy to “solve” such a system of relation equations is to refer to its Mamdani-Assilian relation RMA and to apply, for a given fuzzy input A, the compositional rule of inference (CRI), i.e. to treat the fuzzy set A ◦ RMA as the corresponding, “right” output. Similarly one can “solve” the system of relation equations with reference to its b and to the CRI, which means to treat for any fuzzy input possible largest solution R b as its “right” output. A the fuzzy set A ◦ R But both these “solution” strategies have the (at least theoretical) disadvantage that they may give insufficient results, at least for the predetermined input sets. b may be considered as pseudo-solutions. Call R b the maximal and Thus RMA and R RMA the MA-pseudo-solution. b are upper and As was mentioned previously, these pseudo-solutions RMA and R lower approximations for the realizations of the linguistic control rules. Now one may equally well look for new pseudo-solutions, e.g. by some iteration of these pseudo-solutions in the way, that for the next iteration step in such an iteration process the system of relation equations is changed such that its (new) output sets become the real output of the former iteration step. This has been done in [7]. b from the input To formulate the dependence of the pseudo-solutions RMA and R and output data, we denote the “original” pseudo-solutions with the input-output data (Ai , Bi ) in another way and write RMA [Bk ] for RMA ,
b k ] for R b. R[B
Using the fact that for a given solvable system of relation equations its maximal b is really a solution one immediately gets pseudo-solution R Proposition 7.1 For any fuzzy relation S one has for all i: b k ◦ S] = Ai ◦ S . Ai ◦ R[A
Mathematical Fuzzy Control. A Survey of Some Recent Results 9 Hence it does not give a new pseudo-solution if one iterates the solution strategy of the maximal, i.e. Sanchez pseudo-solution after some (other) pseudo-solution. The situation changes if one uses the Mamdani-Assilian solution strategy after an other pseudo-solution strategy. Because RMA has the superset property, one should use it for an iteration step which follows a pseudo-solution step w.r.t. a fuzzy relation b This gives, cf. again which has the subset property, e.g. after the strategy using R. [7]: Theorem 7.2 One has always b k ] ⊆ Ai ◦ RMA [Ak ◦ R[B b k ]] ⊆ Ai ◦ RMA [Bk ] . Ai ◦ R[B b is a better pseudo-solution as each one of Thus the iterated relation RMA [Ak ◦ R] b RMA and R.
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Approximation and Interpolation
The standard mathematical understanding of approximation is that by an approximation process some mathematical object A, e.g. some function, is approximated, i.e. determined within some (usually previously unspecified) error bounds. Additionally one assumes that the approximating object B for A is of some predetermined, usually “simpler” kind, e.g. a polynomial function. So one may approximate some transcendental function, e.g. the trajectory of some non-linear process, by a piecewise linear function, or by a polynomial function of some bounded degree. Similarly one approximates e.g. in the Runge-Kutta methods the solution of a differential equation by a piecewise linear function, or one uses splines to approximate a difficult surface in 3-space by planar pieces. The standard mathematical understanding of interpolation is that a function f is only partially given by its values at some points of the domain of the function, the interpolation nodes. The problem then is to determine “the” values of f for all the other points of the domain (usually) between the interpolation nodes – sometimes also outside these interpolation nodes (extrapolation). And this is usually done in such a way that one considers groups of neighboring interpolation nodes which uniquely determine an interpolating function of some predetermined type within their convex hull (or something like): a function which has the interpolation nodes of the actual group as argument-value pairs – and which in this sense locally approximates the function f . In the standard fuzzy control approach the input-output data pairs of the linguistic control rules just provide interpolation nodes. And, as long as the problem is understood as a fuzzy approximation of a crisp control function, and if one assumes that the fuzzy input data are unimodal, one also has from this crisp background a notion of neighboring interpolation nodes. However, what is lacking – at least up to now – that is the idea of a local approximation of the intended crisp control function by some fuzzy function. Instead, in the standard contexts one always asks for something like a global interpolation.
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Mathematical Fuzzy Control. A Survey of Some Recent Results
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CRI as Approximation and Interpolation
In the context of fuzzy control the object which has to be determined, some control function Φ, is described only roughly, i.e. given only by its behavior in some (fuzzy) points of the state space. The standard way to roughly describe the control function is to give a list (3.1) of linguistic control rules connecting fuzzy subsets Ai of the input space X with fuzzy subsets Bi of the output space Y indicating that one likes to have Φ∗ (Ai ) = Bi ,
i = 1, . . . , n
(9.1)
for a suitable “fuzzified” version Φ∗ : IF (X ) → IF (Y) of the control function Φ : X → Y. The additional approximation idea of the CRI is to approximate Φ∗ by a fuzzy function Ψ∗ : IF (X ) → IF (Y) determined for all A ∈ IF (X ) by Ψ∗ (A) = A ◦ R
(9.2)
which refers to some suitable fuzzy relation R ∈ IF (X × Y), and understands ◦ as sup-t-composition. Formally thus the equations (9.1) become transformed into some well known system (3.3) of relation equations Ai ◦ R = Bi ,
i = 1, . . . , n
to be solved for the unknown fuzzy relation R. This approximation idea fits well with the fact that one often is satisfied with pseudo-solutions of (3.3), and particularly with the MA-pseudo-solution RMA of b of Sanchez. Both of them deMamdani/Assilian, or the S-pseudo-solution R ∗ termine approximations Ψ to the (fuzzified) control function Φ∗ .
10 Approximate Solutions of Fuzzy Relation Equations The author used in previous papers the notion of approximate solution only naively in the sense of a fuzzy relation which roughly describes the intended control behavior given via some list of linguistic control rules.
10.1 A Formal Definition A precise definition of a notion of approximate solution was given by Wu [16]. In e of a system (3.3) of FRE’s is defined as a that approach an approximate solution R fuzzy relation satisfying: 1. There are fuzzy sets Ai 0 , Bi 0 such that for all i = 1, . . . , n one has e = Bi 0 . Ai ⊆ Ai 0 and Bi 0 ⊆ Bi as well as Ai 0 ◦ R
Mathematical Fuzzy Control. A Survey of Some Recent Results 11 2. If there exist fuzzy sets Ai ∗ , Bi ∗ for i = 1, . . . , n and a fuzzy relation R∗ such that for all i = 1, . . . , n Ai ∗ ◦ R∗ = Bi ∗ and Ai ⊆ Ai ∗ ⊆ Ai 0 , Bi 0 ⊆ Bi ∗ ⊆ Bi , then one has Ai ∗ = Ai 0 and Bi ∗ = Bi 0 for all i = 1, . . . , n.
10.2 Generalizations of Wu’s Approach It is obvious that the two conditions (1), (2) of Wu are independent. What is, however, not obvious at all – and even rather arbitrary – is that condition (1) also says that the approximating input-output data (Ai 0 , Bi 0 ) should approximate the original input data from above and the original output data from below. Before we give a generalized definition we coin the name of an approximating system for (3.3) and understand by it any system Ci ◦ R = Di ,
i = 1, . . . , n
(10.1)
of relation equations with the same number of equations. Definition 10.1 A ul-approximate solution of a system (3.3) of relation equations is a solution of a ul-approximating system for (3.3) which satisfies Ai ⊆ Ci
and
Bi ⊇ Di ,
for i = 1, . . . , n .
(10.2)
An lu-approximate solution of a system (3.3) of relation equations is a solution of an lu-approximating system for (3.3) which satisfies Ai ⊇ Ci
and
Bi ⊆ Di ,
for i = 1, . . . , n .
(10.3)
An l*-approximate solution of a system (3.3) of relation equations is a solution of an l*-approximating system for (3.3) which satisfies Ai ⊇ Ci
and
Bi = Di ,
for i = 1, . . . , n .
(10.4)
In a similar way one defines the notions of ll-approximate solution, of uu-approximate solution, of u*-approximate solution, of *l-approximate solution, and of *u-approximate solution. Corollary 10.2 (i) Each *l-approximate solution of (3.3) is also an ul-approximate solution and an ll-approximate solution of (3.3). (ii) Each u*-approximate solution of (3.3) is also an ul-approximate solution and an uu-approximate solution of (3.3). Proposition 10.3 b is an *l-approximate For each system (3.3) of relation equations its S-pseudo-solution R solution. This generalizes a result of Klir/Yuan [9].
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Proposition 10.4 For each system (3.3) of relation equations with normal input data its MA-pseudosolution RMA is an *u-approximate solution. Together with Corollary 10.2 these two propositions say that each system of relation equations has approximate solutions of any one of the types introduced in this section. However, it should be mentioned that these types of approximate solutions belong to a rather restricted class: caused by the fact that we considered, following Wu, only lower and upper approximations w.r.t. the inclusion relation, i.e. they are inclusion based. Other, and more general approximations of the given input-output data systems are obviously possible. But we will not discuss further versions here.
10.3 Optimality of Approximate Solutions All the previous results do not give any information about some kind of “quality” of the approximate solutions or the approximating systems. This is to some extent related to the fact that up to now we disregarded in our modified terminology Wu’s condition (2) which is a kind of optimality condition. Definition 10.5 e of a system (3.3) is called optimal iff An inclusion based approximate solution R 0 00 there does not exist a solvable system R Ci = Di 0 of relation equations whose inputoutput data (Ci 0 , Di 0 ) approximate the original input-output data of (3.3) strongly better than the input-output data (Ci , Di ) of the system which determines the fuzzy e relation R. Proposition 10.6 e is optimal, then it is also optimal as If an inclusion based *l-approximate solution R a ul-approximate solution and as an ll-approximate solution. Similar results hold true also for (l*-, u*- and) *u-approximate solutions. In those considerations we look for solutions of “approximating systems” of FRE’s: Of course, these solutions form some space of functions – and within this space one is interested to find “optimal members” for the solution problem under consideration. An obvious modification is to fix in some other way such a space R of functions, i.e. independent of the idea of approximating systems of FRE’s. In that moment one has also to specify some ranking for the members of that space R of functions. In the following we go on to discuss optimality results from both these points of view.
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Some Optimality Results for Approximate Solutions
b and RMA are optimal approxiThe problem now is whether the pseudo-solutions R mate solutions.
11.1 Optimality of the S-Pseudo-Solution b as a ul-approximate solution this optimality was shown For the S-pseudo-solution R by Klir/Yuan [9].
Mathematical Fuzzy Control. A Survey of Some Recent Results 13 Proposition 11.1 b is always an ⊆-optimal *l-approximate solution of (3.3). The fuzzy relation R From the second point of view we have, slightly reformulating and extending results presented in [13], the following result, given also in [12]. Theorem 11.2 b is the best Consider an unsolvable system of FRE’s. Then the S-pseudo-solution R approximate solution in the space Rl : Rl = {R ∈ R | Ai ◦ R ⊆ Bi for all 1 ≤ i ≤ n} . under the ranking ≤l : R0 ≤l R00
iff Ai ◦ R00 ⊆ Ai ◦ R0 for all 1 ≤ i ≤ n .
b is the best approximate solution in the Remark: Similarly one can prove that R space Rl under the ranking ≤δ : R0 ≤δ R00 for δ ∗ (R) =
n ³ ^
Bi ≡∗ Ai ◦ R
i=1
iff
´ =
δ(R00 ) ≤ δ(R0 ) n ^ ^
(Bi (y) ↔ (Ai ◦ R)(y)) .
(11.1)
i=1 y∈Y
This index δ ∗ (R) is quite similar to the solvability degree δ(R) to be introduced later on in (12.1).
11.2 Optimality of the MA-Pseudo-Solution For the MA-pseudo-solution the situation is different, as was indicated in [7]. Proposition 11.3 There exist systems (3.3) of relation equations for which their MA-pseudo-solution RMA is an *u-approximate solution which is not optimal, i.e. is an approximate solution in the approximation space Ru : Ru = {R ∈ R | Ai ◦ R ⊇ Bi for all 1 ≤ i ≤ n} , but is not optimal in this set under the preorder ≤u : R0 ≤u R00
iff
Ai ◦ R0 ≤ Ai ◦ R00 for all 1 ≤ i ≤ n.
b to the situation for RMA is that The crucial difference of the optimality result for R b in the former case the solvable approximating system has its own (largest) solution S. But a solvable approximating system may fail to have his MA-pseudo-solution RMA as a solution. The last remark leads us to a partial optimality result w.r.t. the MA-pseudosolution. The proofs of the results which shall be mentioned now can be found in [13], or easily derived from the results given there.
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Definition 11.4 Let us call a system (3.3) of relation equations MA-solvable iff its MA-pseudo-solution RMA is a solution of this system. Proposition 11.5 If a system of FRE’s has an MA-solvable *u-approximating system Ai ◦ R = Bi∗ ,
i = 1, . . . , n
(11.2)
such that for the MA-pseudo-solution RMA of the original system of FRE’s one has
then one has
Bi ⊆ Bi∗ ⊆ Ai ◦ RMA ,
i = 1, . . . , n ,
Bi∗ = Ai ◦ RMA
i = 1, . . . , n .
for all
Corollary 11.6 If all input sets of (3.3) are normal then the system Ai ◦ R = Ai ◦ RMA ,
i = 1, . . . , n ,
(11.3)
is the smallest MA-solvable *u-supersystem for (3.3). This leads back to the iterated pseudo-solution strategies. Corollary 11.7 b be the S-pseudo-solution of (3.3), let be B ci = Ai ◦ R b for i = 1, . . . , n, and Let R suppose that the modified system ci , Ai ◦ R = B
i = 1, . . . , n ,
(11.4)
b is an optimal *lis MA-solvable. Then this iterated pseudo-solution RMA [Ak ◦ R] approximate solution of (3.3). Furthermore it is a best approximate solution of the original system in the space Rl under the ranking ≤l . Let us mention also the following result (cf. [12]). Theorem: Consider an unsolvable system of FRE’s such that all input fuzzy sets Ai , 1 ≤ i ≤ n, are normal and form a semi-partition of X. Then RMA (x, y) =
n _
(Ai (x) ∗ Bi (y))
i=1
is a best possible approximate solution in the space Ru = {R ∈ R | Ai ◦ R ⊇ Bi for all 1 ≤ i ≤ n} , under the preorder ≤u : R0 ≤u R00 iff Ai ◦ R0 ≤ Ai ◦ R00 for all 1 ≤ i ≤ n. These considerations can be further generalized. Consider some pseudo-solution strategy S, i.e. some mapping from the class of families (Ai , Bi )1≤i≤n of input-output data pairs into the class of fuzzy relations, which yields for any given system (3.3) of relation equations an S-pseudo-solution RS . Then the system (3.3) will be called S-solvable iff RS is a solution of this system.
Mathematical Fuzzy Control. A Survey of Some Recent Results 15 Definition 11.8 We shall say that the S-pseudo-solution RS depends isotonically (w.r.t. inclusion) on the output data of the system (3.3) of relation equations iff the condition if
Bi ⊆ Bi0 for all i = 1, . . . , n
then
RS ⊆ RS0
holds true for the S-pseudo-solutions RS of the system (3.3) and RS0 of an “outputmodified” system Ai ◦ R = Bi0 , i = 1, . . . , n. Definition 11.9 We understand by an S-optimal *u-approximate solution of the system (3.3) the S-pseudo-solution of an S-solvable *u-approximating system of (3.3) which has the additional property that no strongly better *u-approximating system of (3.3) is Ssolvable. Proposition 11.10 Suppose that the S-pseudo-solution depends isotonically on the output data of the systems of relation equations. Assume furthermore that for the S-pseudo-solution RS of (3.3) one always has Bi ⊆ Ai ◦RS (or always has Ai ◦RS ⊆ Bi ) for i = 1, . . . , n. Then the S-pseudo-solution RS of (3.3) is an S-optimal *u-approximate (or: *l-approximate) solution of system (3.3). It is clear that Corollary 11.6 is the particular case of the MA-pseudo-solution strategy. But also Proposition 11.1 is a particular case of this Proposition 11.10: the case of the S-pseudo-solution strategy (having in mind that S-solvability and solvability are equivalent notions).
12
Introducing the Solvability Degree
Following [4, 5] one may consider for a system of relation equations the (global) solvability degree n Y ξ = [[∃X (Ai ◦ X ≡ Bi )]] , (12.1) i=1
and for any fuzzy relation R their solution degree δ(R) = [[
n Y
(Ai ◦ R ≡ Bi )]] .
(12.2)
i=1
Q Here means the finite iteration of the strong conjunction connective &, and is defined in the standard way. The following result was first proved in [3], and has been further discussed in [4, 5]. Theorem 12.1 b ≤ ξ. ξ n ≤ δ(R) Of course, the n-th power here is again the iteration of the strong Q conjunction operation ∗, i.e. the semantical counterpart of the syntactic operation . Obviously this result can be rewritten in a slightly modified form which makes it more transparent that Theorem 12.1 really gives an estimation for the solvability degree ξ in terms of a particular solution degree.
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Corollary 12.2 b n ≤ ξ n ≤ δ(R). b δ(R) One has for continuous t-norms that they are ordinal sums of isomorphic copies of two basic t-norms, of the L Ã ukasiewicz t-norm tL given by tL (u, v) = max{u + v − 1, 0} and of the arithmetic product tP . (Sometimes also G¨odel’s t-norm min is allowed for these summands. However, this is unimportant because of the definition of an ordinal sum of t-norms.) Corollary 12.3 b and ξ always belong to In the case that ∗ is a continuous t-norm t, the values δ(R) the same ordinal t-summand. A further property is of interest for the case of t-norm based structures L. Proposition 12.4 For each continuous t-norm t and each 1 ≤ n ∈ N there exists n-th roots. Having this in mind, one can immediately rewrite Theorem 12.1 for this particular case in an even nicer form as we did it in Corollary 12.2. Proposition 12.5 For t-norms which have n-th roots one has the inequalities q b ≤ ξ ≤ n δ(R) b . δ(R) Using as in [14] the notation z(u) for the largest t-idempotent below u, this last result allows for the following slight modification. Corollary 12.6 For t-norms which have n-th roots one has the inequalities q n b b . z(δ(R)) ≤ ξ ≤ δ(R) b of the S-pseudoBesides these core results which involve the solution degree δ(R) solution of the system (3.3), the problem appears to determine the solution degree of the relation RMA . Proposition 12.7 If all input sets Ai are normal then Y^ δ ∗ (RMA ) = [[ (Ai ∩t Aj 6≡ ∅ → Bi j Bj )]] . i
j
This is a generalization of the former Klawonn criterion. We also find, as explained in [7], a second result which indicates that RMA [Ak ◦ b k ]] is at least sometimes as good a pseudo-solution as RMA . R[B Proposition 12.8 If all input sets Ai are normal and if one has b j Aj ◦ R b, |= Bi j Bj → Ai ◦ R then
b k ]]) . δ ∗ (RMA ) ≤ δ ∗ (RMA [Ak ◦ R[B
Mathematical Fuzzy Control. A Survey of Some Recent Results 17
References [1] A. DiNola, S. Sessa, W. Pedrycz and E. Sanchez, Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Theory and Decision Libr., ser. D, Kluwer Acad. Publ.: Dordrecht, 1989. [2] S. Gottwald, Criteria for non-interactivity of fuzzy logic controller rules. In: Large Scale Systems: Theory and Applications, Proc. 3rd IFAC/IFORS Symp. Warsaw 1983 (A. Straszak, ed.), Pergamon Press: Oxford, 1984, 229–233. [3] S. Gottwald, Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarb. Kybernet. 22 (1986) 67–91. [4] S. Gottwald, Fuzzy Sets and Fuzzy Logic. The Foundations of Application – From a Mathematical Point of View, Vieweg: Braunschweig/Wiesbaden and Teknea: Toulouse, 1993. [5] S. Gottwald, Generalised solvability behaviour for systems of fuzzy equations, in: V. Nov´ ak, I. Perfilieva (Eds.), Discovering the World with Fuzzy Logic, Advances in Soft Computing, Physica-Verlag: Heidelberg, 2000, 401–430. [6] S. Gottwald, A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9, Research Studies Press: Baldock, 2001. [7] S. Gottwald, V. Nov´ ak, I. Perfilieva, Fuzzy control and t-norm-based fuzzy logic. Some recent results, in: Proc. 9th Internat. Conf. IPMU’2002, ESIA – Universit´ e de Savoie, Annecy, 2002, 1087–1094. [8] Klawonn, F. (2001): Fuzzy points, fuzzy relations and fuzzy functions. In: Discovering the World with Fuzzy Logic (V. Nov´ ak, I. Perfilieva eds.) Advances in Soft Computing, PhysicaVerlag: Heidelberg 2000, 431–453. [9] G. Klir, B. Yuan, Approximate solutions of systems of fuzzy relation equations, in: FUZZ-IEEE ’94. Proc. 3rd Internat. Conf. Fuzzy Systems, Orlando FL, 1994, 1452–1457. [10] R.D. Luce, A note on Boolean matrix theory, Proc. Amer. Math. Soc. 3 (1952) 382–388. [11] A. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man-Machine Studies 7 (1975) 1–13. [12] I. Perfilieva, Fuzzy function as an approximate solution to a system of fuzzy relation equations, Fuzzy Sets Systems, 147 (2004), 363–383. [13] I. Perfilieva, S. Gottwald, Fuzzy function as a solution to a system of fuzzy relation equations, Internat. J. General Systems, 32 (2003) 361–372. [14] I. Perfilieva, A. Tonis, Compatibility of systems of fuzzy relation equations, Internat. J. General Systems, 29 (2000) 511–528. [15] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control, 30 (1976) 38–48. [16] W. Wu, Fuzzy reasoning and fuzzy relation equations, Fuzzy Sets Systems, 20 (1986) 67–78. [17] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Systems, Man and Cybernet. SMC-3 (1973) 28–44.
Received 13 October 2004.
1
Introduction
In engineering science, fuzzy control methods have become a standard tool which allows to apply computerized control approaches to a wider class of problems as those which can reasonably, and effectively, treated with the more traditional mathematical methods like PD or PID control. For the industrial engineer, usually success in control applications is the main criterion. Then he even accepts methods which have, to a larger extent, only a heuristic basis. And this has been the situation with fuzzy control approaches for a considerable amount of time. Particularly with respect to the linguistic control rules which are constitutive for a lot of fuzzy control approaches. Of course, success in applications then calls for mathematical reflections about, and mathematical foundations for the methods under consideration. For linguistic control rules their transformation into fuzzy relation equations has been the core idea in a lot of such theoretical reflections. Here we discuss this type of mathematical treatment of rule based fuzzy control with particular emphasis on some more recent viewpoints which tend toward a more general view at these mathematizations.
L. J. of the IGPL, Vol. 0 No. 0, pp. 1–17 0000
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c Oxford University Press °
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Preliminaries
We use in this paper a set theoretic notation for fuzzy sets which refers a logic with truth degree set [0, 1] based upon a left continuous t-norm t, or – more general – based upon a class of (complete) prelinear residuated lattices with semigroup operation ∗. This means that we consider the logic MTL of left continuous t-norms as the formal background, cf. [6]. This logic has as standard connectives two conjunctions and a disjunction: & = ∗,
∧ = min ,
∨ = max .
In the lattice case we mean, by a slight abuse of language, by min, max the lattice meet, and the lattice join, respectively. It also has an implication → characterized by the adjointness condition u ∗ v≤w
iff
u ≤ (v → w) ,
as well as a negation − given by −H = H → 0 . The quantifiers ∀ and ∃ mean the infimum and supremum, respectively, of the truth degrees of all instances. And the truth degree 1 is the only designated one. Therefore logical validity |= H means that H has always truth degree one. The shorthand notation [[H]] denotes the truth degree of formula H, assuming that the corresponding evaluation of the (free) variables, as well as the model under consideration (in the first-order case) is clear from the context. The class term notation {x k H(x)} denotes the fuzzy set A with µA (a) = A(a) = [[H(a)]] for each a ∈ X . Occasionally we use graded identity relations ≡ and ≡∗ for fuzzy sets, based upon A j B = ∀x(A(x) → B(x)) , and defined as A≡B A ≡∗ B
= =
A j B & B j A, A j B ∧ B j A.
Obviously one has the relationships
3
|=
Bi ≡∗ Bj ↔ ∀y(Bi (y) ↔ Bj (y)) ,
|=
A ≡ B → A ≡∗ B .
Fuzzy Control and Relation Equations
In the standard approach, a fuzzy controller is determined by a list of linguistic control rules if α is Ai , then β is Bi , i = 1, . . . , n , (3.1) describing some control procedure with input variable α and output variable β.
Mathematical Fuzzy Control. A Survey of Some Recent Results 3 Mainly in engineering papers one often consider also the case of different input variables α1 , . . . , αn , in this case the linguistic control rules become the form if α1 is A1i , and . . . and αn is Ani , then β is Bi ,
i = 1, . . . , n .
But from a mathematical point of view such rules are equivalent to the former ones: one simply has to allow as the input universe for α the cartesian product of the input universes of α1 , . . . , αn .
3.1 The Compositional Rule of Inference Following the basic ideas of Zadeh [17], such a fuzzy controller is formally realized by a fuzzy relation R which connects fuzzy input information A with fuzzy output information B via the compositional rule of inference (CRI) B = A ◦ R = R00 A = {y k ∃x(A(x) & R(x, y))} .
(3.2)
Therefore, applying this idea to the linguistic control rules themselves, these rules in a natural way become transformed into fuzzy relation equations Ai ◦ R = Bi ,
for i = 1, . . . , n
(3.3)
i.e. form a system of such relation equations. The problem, to determine a fuzzy relation R which realizes via (3.2) such a list (3.1) of linguistic control rules, therefore becomes the problem to determine a solution of the corresponding system (3.3) of relation equations. This problem proves to be a rather difficult one: it often happens that a given system (3.3) of relation equations is unsolvable. This is already the case in the more specific situation that the membership degrees belong to a Boolean algebra, as discussed (as a problem for Boolean matrices) e.g. in [10]. Nice solvability criteria are still largely unknown. Thus the investigation of the structure of the solution space for (3.3) was one of the problems discussed rather early. One essentially has that this space is an upper semilattice under the simple set union determined by the maximum of the membership degrees; cf. e.g. [1]. And this semilattice has, if it is nonempty, a universal upper bound. To state the main result, one has to consider the particular fuzzy relation b= R
n \
{(x, y) k Ai (x) → Bi (y)} .
(3.4)
i=1
Theorem 3.1 b is a solution The system (3.3) of relation equations is solvable iff the fuzzy relation R of it. b is always the largest solution of the system (3.3) And in the case of solvability, R of relation equations. This result was first stated by Sanchez [15] for the particular case of the minbased G¨odel implication → in (3.4), and generalized to the case of the residuated implications based upon arbitrary left continuous t-norms—and hence to the present situation—by the author in [3]; cf. also his [4].
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Mathematical Fuzzy Control. A Survey of Some Recent Results
3.2 Modelling Strategies Besides the reference to the CRI in this type of approach toward fuzzy control, the crucial point is to determine a fuzzy relation out of a list of linguistic control rules. b can be seen as a formalization of the idea that the list (3.1) The fuzzy relation R of control rules has to be read as: if input is A1 then output is B1 and ... and if input is An then output is Bn . Having in mind such a formalization of the list (3.1) of control rules, there is immediately also another way how to read this list: input is A1 and output is B1 or ... or input is An and output is Bn . It is this understanding of the list of linguistic control rules as a (rough) description of a fuzzy function which characterizes the approach of Mamdani and Assilian [11]. b the fuzzy relation Therefore they consider instead of R RMA =
n [
(Ai × Bi ) ,
i=1
again combined with the compositional rule of inference.
4
Toward a Solvability Criterion for RMA
Having in mind Theorem 3.1, one immediately is confronted with the following Problem: Under which conditions is the fuzzy relation RMA a solution of the corresponding system of relation equations. This problem is discussed in [8]. And one of the main results is the next theorem. Theorem 4.1 Let all the input sets Ai be normal. Then the fuzzy relation RMA is a solution of the corresponding system of fuzzy relation equations iff for all i, j = 1, . . . , n one has |=
∃x(Ai (x) & Aj (x)) → Bi ≡∗ Bj .
(4.1)
This MA-solvability criterion (4.1) is a kind of functionality of the list of linguistic control rules, at least in the case of the presence of an involutive negation: because in such a case one has |=
∃x(Ai (x) & Aj (x)) ↔ Ai ∩t Aj 6≡∗ ∅ ,
Mathematical Fuzzy Control. A Survey of Some Recent Results 5 and thus condition (4.1) becomes |=
Ai ∩t Aj 6≡∗ ∅ → Bi ≡∗ Bj .
(4.2)
And this can be understood as a fuzzification of the idea “if Ai and Aj coincide to some degree, than also Bi and Bj should coincide to a certain degree”. Of course, this fuzzification is neither obvious nor completely natural, because it translates “degree of coincidence” in two different ways. Corollary 4.2 b as a solution. If condition (4.2) is satisfied, then the system of relation equations has R This leads back to the well known result, explained e.g. in [4], that the system of relation equations is solvable in the case that all the input fuzzy sets Ai are pairwise t-disjoint: Ai ∩t Aj = ∅ for all i 6= j. It is furthermore known, cf. again [4], that functionality holds true for the relational composition at least in the form |=
A ≡ B → A◦R ≡ B ◦R,
because one has the (generalized) monotonicity |=
A j B → A◦R j B ◦R.
This, by the way, gives Corollary 4.3 A necessary condition for the solvability of a system of relation equations is that one always has |= Ai ≡ Aj → Bi ≡ Bj . This condition is symmetric in i, j. Therefore one gets as a slight generalization also Corollary 4.4 Let all the input sets Ai be normal. Then the fuzzy relation RMA is a solution of our system of fuzzy relation equations iff for all i, j = 1, . . . , n one has |= Ai ∩t Aj 6≡∗ ∅ → Bi j Bj .
5
Relating RMA with the Largest Solution
b However, it may happen that the system of relation equations is solvable, i.e. has R as a solution, without having the fuzzy relation RMA as a solution. An example is given in [7]. Therefore Klawonn’s condition (4.1) is only a sufficient one for the solvability of the system (3.3) of relation equations. Hence one has as a new problem to give additional assumptions, besides the solvability of the system (3.3) of relation equations, which are sufficient to guarantee that RMA is a solution of (3.3). As in [4] and already in [2], we subdivide the problem whether a fuzzy relation R is a solution of the system of relation equations into two cases.
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Definition 5.1 A fuzzy relation R has the subset property w.r.t. a system (3.3) of relation equations iff one has Ai ◦ R ⊆ Bi , for i = 1, . . . , n , (5.1) and it has the superset property w.r.t. (3.3) iff one has Ai ◦ R ⊇ Bi ,
for i = 1, . . . , n .
(5.2)
Particularly for RMA quite natural sufficient conditions for the superset property have been given, but only rather strong ones for the subset property. Proposition 5.2 If all input sets Ai are normal then RMA has the superset property. So we know with the fuzzy relation RMA , assuming that all input sets Ai are normal, at least one upper approximation of the (possible) solution for the system of relation equations. Proposition 5.3 If all input sets are pairwise disjoint (under ∩t ), then RMA has the subset property. b satisfies these It is also of interest to ask for conditions under which the relation R properties. Fortunately, for the subset property there is a nice answer. Proposition 5.4 b has the subset property. R Together with Proposition 5.2 this immediately gives Corollary 5.5 If all input sets Ai are normal then one has for all indices i the inclusions b ⊆ Bi ⊆ Ai ◦ RMA . Ai ◦ R
(5.3)
b at least one lower approximation of the Thus we know with the fuzzy relation R (possible) solution for the system of relation equations. However, the single inclusion relations (5.3) can already be proved from slightly weaker assumptions. Proposition 5.6 b ⊆ Bk ⊆ Ak ◦ RMA . If the input set Ak is normal then Ak ◦ R b always are subsets So we know that with normal input sets the fuzzy outputs Ai ◦ R of Ai ◦ RMA . Furthermore we immediately have the following global result. Proposition 5.7 If all the input sets Ai of the system of relation equations are normal and if one b then the system of relation equations is solvable, and RMA is a also has RMA ⊆ R, solution. Now we ask for conditions under which the relation RMA maps the input fuzzy b And that means again to ask for some conditions which sets Ai to subsets of Ai ◦ R. give the subset property of RMA , and thus the solvability of the system of relation equations.
Mathematical Fuzzy Control. A Survey of Some Recent Results 7 Proposition 5.8 Assume the normality of all the input sets Ai . Then to have for some index 1 ≤ k ≤ n b Ak ◦ RMA ⊆ Ak ◦ R is equivalent to the equality b = Bk . Ak ◦ RMA = Ak ◦ R Corollary 5.9 Assume the normality of all the input sets Ai . Then the condition to have for all indices 1 ≤ i ≤ n b Ai ◦ RMA ⊆ Ai ◦ R is equivalent to the fact that RMA is a solution of the system of relation equations, and hence equivalent to the second criterion of Klawonn.
6
b Toward the Superset Property of R
The solvability of the system of relation equations is equivalent to the fact that the b is a solution. Therefore the solvability of our system of relation equations relation R b has the subset as well as the superset properties. is also equivalent to the fact that R Now, as seen in Proposition 5.4, the subset property is generally satisfied for the b This means we immediately have: fuzzy relation R. Corollary 6.1 b has the superset property. A system of relation equations is solvable iff its relation R Hence, to get sufficient solvability conditions for the system of relation equations b means to look for sufficient conditions for this superset property of R. And this seems to be an astonishingly hard problem. What one immediately has in general are the equivalences: b |= Bk j Ak ◦ R iff for all y |= Bk (y) → ∃x(Ak (x) &
^ (Ai (x) → Bi (y))) i
iff for all y and all i |= Bk (y) → ∃x(Ak (x) & (Ai (x) → Bi (y))) .
(6.1)
And just this last condition offers the main open problem: to find suitable conditions which are equivalent to (6.1). Particularly for i = k and continuous t-norms this is equivalent to |= Bk (y) → ∃x(Ak (x) ∧ Bk (y)) . Corollary 6.2 b is that For continuous t-norms t a necessary condition for the superset property of R hgt (Bk ) ≤ hgt (Ak ) holds for all input-output pairs (Ak , Bk ).
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Part of the present problem is to look for sufficient conditions which imply (6.1). Here a nice candidate seems to be to have for given i, k and y the existence of some x with |= Bk (y) → Ak (x) & (Ai (x) → Bi (y)) . Routine calculations show that this means that it is sufficient for (6.1) to have, for a given y, either the existence of some x with Bk (y) ≤ Ak (x) and
Ai (x) ≤ Bi (y)
or the existence of some x with Ak (x) = 1
and
Ai (x) ≤ [[Bk (y) → Bi (y)]] .
However, both these sufficient conditions look not very promising.
7
Getting New Pseudo-Solutions
Suppose again that all the input sets Ai are normal. The standard strategy to “solve” such a system of relation equations is to refer to its Mamdani-Assilian relation RMA and to apply, for a given fuzzy input A, the compositional rule of inference (CRI), i.e. to treat the fuzzy set A ◦ RMA as the corresponding, “right” output. Similarly one can “solve” the system of relation equations with reference to its b and to the CRI, which means to treat for any fuzzy input possible largest solution R b as its “right” output. A the fuzzy set A ◦ R But both these “solution” strategies have the (at least theoretical) disadvantage that they may give insufficient results, at least for the predetermined input sets. b may be considered as pseudo-solutions. Call R b the maximal and Thus RMA and R RMA the MA-pseudo-solution. b are upper and As was mentioned previously, these pseudo-solutions RMA and R lower approximations for the realizations of the linguistic control rules. Now one may equally well look for new pseudo-solutions, e.g. by some iteration of these pseudo-solutions in the way, that for the next iteration step in such an iteration process the system of relation equations is changed such that its (new) output sets become the real output of the former iteration step. This has been done in [7]. b from the input To formulate the dependence of the pseudo-solutions RMA and R and output data, we denote the “original” pseudo-solutions with the input-output data (Ai , Bi ) in another way and write RMA [Bk ] for RMA ,
b k ] for R b. R[B
Using the fact that for a given solvable system of relation equations its maximal b is really a solution one immediately gets pseudo-solution R Proposition 7.1 For any fuzzy relation S one has for all i: b k ◦ S] = Ai ◦ S . Ai ◦ R[A
Mathematical Fuzzy Control. A Survey of Some Recent Results 9 Hence it does not give a new pseudo-solution if one iterates the solution strategy of the maximal, i.e. Sanchez pseudo-solution after some (other) pseudo-solution. The situation changes if one uses the Mamdani-Assilian solution strategy after an other pseudo-solution strategy. Because RMA has the superset property, one should use it for an iteration step which follows a pseudo-solution step w.r.t. a fuzzy relation b This gives, cf. again which has the subset property, e.g. after the strategy using R. [7]: Theorem 7.2 One has always b k ] ⊆ Ai ◦ RMA [Ak ◦ R[B b k ]] ⊆ Ai ◦ RMA [Bk ] . Ai ◦ R[B b is a better pseudo-solution as each one of Thus the iterated relation RMA [Ak ◦ R] b RMA and R.
8
Approximation and Interpolation
The standard mathematical understanding of approximation is that by an approximation process some mathematical object A, e.g. some function, is approximated, i.e. determined within some (usually previously unspecified) error bounds. Additionally one assumes that the approximating object B for A is of some predetermined, usually “simpler” kind, e.g. a polynomial function. So one may approximate some transcendental function, e.g. the trajectory of some non-linear process, by a piecewise linear function, or by a polynomial function of some bounded degree. Similarly one approximates e.g. in the Runge-Kutta methods the solution of a differential equation by a piecewise linear function, or one uses splines to approximate a difficult surface in 3-space by planar pieces. The standard mathematical understanding of interpolation is that a function f is only partially given by its values at some points of the domain of the function, the interpolation nodes. The problem then is to determine “the” values of f for all the other points of the domain (usually) between the interpolation nodes – sometimes also outside these interpolation nodes (extrapolation). And this is usually done in such a way that one considers groups of neighboring interpolation nodes which uniquely determine an interpolating function of some predetermined type within their convex hull (or something like): a function which has the interpolation nodes of the actual group as argument-value pairs – and which in this sense locally approximates the function f . In the standard fuzzy control approach the input-output data pairs of the linguistic control rules just provide interpolation nodes. And, as long as the problem is understood as a fuzzy approximation of a crisp control function, and if one assumes that the fuzzy input data are unimodal, one also has from this crisp background a notion of neighboring interpolation nodes. However, what is lacking – at least up to now – that is the idea of a local approximation of the intended crisp control function by some fuzzy function. Instead, in the standard contexts one always asks for something like a global interpolation.
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Mathematical Fuzzy Control. A Survey of Some Recent Results
9
CRI as Approximation and Interpolation
In the context of fuzzy control the object which has to be determined, some control function Φ, is described only roughly, i.e. given only by its behavior in some (fuzzy) points of the state space. The standard way to roughly describe the control function is to give a list (3.1) of linguistic control rules connecting fuzzy subsets Ai of the input space X with fuzzy subsets Bi of the output space Y indicating that one likes to have Φ∗ (Ai ) = Bi ,
i = 1, . . . , n
(9.1)
for a suitable “fuzzified” version Φ∗ : IF (X ) → IF (Y) of the control function Φ : X → Y. The additional approximation idea of the CRI is to approximate Φ∗ by a fuzzy function Ψ∗ : IF (X ) → IF (Y) determined for all A ∈ IF (X ) by Ψ∗ (A) = A ◦ R
(9.2)
which refers to some suitable fuzzy relation R ∈ IF (X × Y), and understands ◦ as sup-t-composition. Formally thus the equations (9.1) become transformed into some well known system (3.3) of relation equations Ai ◦ R = Bi ,
i = 1, . . . , n
to be solved for the unknown fuzzy relation R. This approximation idea fits well with the fact that one often is satisfied with pseudo-solutions of (3.3), and particularly with the MA-pseudo-solution RMA of b of Sanchez. Both of them deMamdani/Assilian, or the S-pseudo-solution R ∗ termine approximations Ψ to the (fuzzified) control function Φ∗ .
10 Approximate Solutions of Fuzzy Relation Equations The author used in previous papers the notion of approximate solution only naively in the sense of a fuzzy relation which roughly describes the intended control behavior given via some list of linguistic control rules.
10.1 A Formal Definition A precise definition of a notion of approximate solution was given by Wu [16]. In e of a system (3.3) of FRE’s is defined as a that approach an approximate solution R fuzzy relation satisfying: 1. There are fuzzy sets Ai 0 , Bi 0 such that for all i = 1, . . . , n one has e = Bi 0 . Ai ⊆ Ai 0 and Bi 0 ⊆ Bi as well as Ai 0 ◦ R
Mathematical Fuzzy Control. A Survey of Some Recent Results 11 2. If there exist fuzzy sets Ai ∗ , Bi ∗ for i = 1, . . . , n and a fuzzy relation R∗ such that for all i = 1, . . . , n Ai ∗ ◦ R∗ = Bi ∗ and Ai ⊆ Ai ∗ ⊆ Ai 0 , Bi 0 ⊆ Bi ∗ ⊆ Bi , then one has Ai ∗ = Ai 0 and Bi ∗ = Bi 0 for all i = 1, . . . , n.
10.2 Generalizations of Wu’s Approach It is obvious that the two conditions (1), (2) of Wu are independent. What is, however, not obvious at all – and even rather arbitrary – is that condition (1) also says that the approximating input-output data (Ai 0 , Bi 0 ) should approximate the original input data from above and the original output data from below. Before we give a generalized definition we coin the name of an approximating system for (3.3) and understand by it any system Ci ◦ R = Di ,
i = 1, . . . , n
(10.1)
of relation equations with the same number of equations. Definition 10.1 A ul-approximate solution of a system (3.3) of relation equations is a solution of a ul-approximating system for (3.3) which satisfies Ai ⊆ Ci
and
Bi ⊇ Di ,
for i = 1, . . . , n .
(10.2)
An lu-approximate solution of a system (3.3) of relation equations is a solution of an lu-approximating system for (3.3) which satisfies Ai ⊇ Ci
and
Bi ⊆ Di ,
for i = 1, . . . , n .
(10.3)
An l*-approximate solution of a system (3.3) of relation equations is a solution of an l*-approximating system for (3.3) which satisfies Ai ⊇ Ci
and
Bi = Di ,
for i = 1, . . . , n .
(10.4)
In a similar way one defines the notions of ll-approximate solution, of uu-approximate solution, of u*-approximate solution, of *l-approximate solution, and of *u-approximate solution. Corollary 10.2 (i) Each *l-approximate solution of (3.3) is also an ul-approximate solution and an ll-approximate solution of (3.3). (ii) Each u*-approximate solution of (3.3) is also an ul-approximate solution and an uu-approximate solution of (3.3). Proposition 10.3 b is an *l-approximate For each system (3.3) of relation equations its S-pseudo-solution R solution. This generalizes a result of Klir/Yuan [9].
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Mathematical Fuzzy Control. A Survey of Some Recent Results
Proposition 10.4 For each system (3.3) of relation equations with normal input data its MA-pseudosolution RMA is an *u-approximate solution. Together with Corollary 10.2 these two propositions say that each system of relation equations has approximate solutions of any one of the types introduced in this section. However, it should be mentioned that these types of approximate solutions belong to a rather restricted class: caused by the fact that we considered, following Wu, only lower and upper approximations w.r.t. the inclusion relation, i.e. they are inclusion based. Other, and more general approximations of the given input-output data systems are obviously possible. But we will not discuss further versions here.
10.3 Optimality of Approximate Solutions All the previous results do not give any information about some kind of “quality” of the approximate solutions or the approximating systems. This is to some extent related to the fact that up to now we disregarded in our modified terminology Wu’s condition (2) which is a kind of optimality condition. Definition 10.5 e of a system (3.3) is called optimal iff An inclusion based approximate solution R 0 00 there does not exist a solvable system R Ci = Di 0 of relation equations whose inputoutput data (Ci 0 , Di 0 ) approximate the original input-output data of (3.3) strongly better than the input-output data (Ci , Di ) of the system which determines the fuzzy e relation R. Proposition 10.6 e is optimal, then it is also optimal as If an inclusion based *l-approximate solution R a ul-approximate solution and as an ll-approximate solution. Similar results hold true also for (l*-, u*- and) *u-approximate solutions. In those considerations we look for solutions of “approximating systems” of FRE’s: Of course, these solutions form some space of functions – and within this space one is interested to find “optimal members” for the solution problem under consideration. An obvious modification is to fix in some other way such a space R of functions, i.e. independent of the idea of approximating systems of FRE’s. In that moment one has also to specify some ranking for the members of that space R of functions. In the following we go on to discuss optimality results from both these points of view.
11
Some Optimality Results for Approximate Solutions
b and RMA are optimal approxiThe problem now is whether the pseudo-solutions R mate solutions.
11.1 Optimality of the S-Pseudo-Solution b as a ul-approximate solution this optimality was shown For the S-pseudo-solution R by Klir/Yuan [9].
Mathematical Fuzzy Control. A Survey of Some Recent Results 13 Proposition 11.1 b is always an ⊆-optimal *l-approximate solution of (3.3). The fuzzy relation R From the second point of view we have, slightly reformulating and extending results presented in [13], the following result, given also in [12]. Theorem 11.2 b is the best Consider an unsolvable system of FRE’s. Then the S-pseudo-solution R approximate solution in the space Rl : Rl = {R ∈ R | Ai ◦ R ⊆ Bi for all 1 ≤ i ≤ n} . under the ranking ≤l : R0 ≤l R00
iff Ai ◦ R00 ⊆ Ai ◦ R0 for all 1 ≤ i ≤ n .
b is the best approximate solution in the Remark: Similarly one can prove that R space Rl under the ranking ≤δ : R0 ≤δ R00 for δ ∗ (R) =
n ³ ^
Bi ≡∗ Ai ◦ R
i=1
iff
´ =
δ(R00 ) ≤ δ(R0 ) n ^ ^
(Bi (y) ↔ (Ai ◦ R)(y)) .
(11.1)
i=1 y∈Y
This index δ ∗ (R) is quite similar to the solvability degree δ(R) to be introduced later on in (12.1).
11.2 Optimality of the MA-Pseudo-Solution For the MA-pseudo-solution the situation is different, as was indicated in [7]. Proposition 11.3 There exist systems (3.3) of relation equations for which their MA-pseudo-solution RMA is an *u-approximate solution which is not optimal, i.e. is an approximate solution in the approximation space Ru : Ru = {R ∈ R | Ai ◦ R ⊇ Bi for all 1 ≤ i ≤ n} , but is not optimal in this set under the preorder ≤u : R0 ≤u R00
iff
Ai ◦ R0 ≤ Ai ◦ R00 for all 1 ≤ i ≤ n.
b to the situation for RMA is that The crucial difference of the optimality result for R b in the former case the solvable approximating system has its own (largest) solution S. But a solvable approximating system may fail to have his MA-pseudo-solution RMA as a solution. The last remark leads us to a partial optimality result w.r.t. the MA-pseudosolution. The proofs of the results which shall be mentioned now can be found in [13], or easily derived from the results given there.
14
Mathematical Fuzzy Control. A Survey of Some Recent Results
Definition 11.4 Let us call a system (3.3) of relation equations MA-solvable iff its MA-pseudo-solution RMA is a solution of this system. Proposition 11.5 If a system of FRE’s has an MA-solvable *u-approximating system Ai ◦ R = Bi∗ ,
i = 1, . . . , n
(11.2)
such that for the MA-pseudo-solution RMA of the original system of FRE’s one has
then one has
Bi ⊆ Bi∗ ⊆ Ai ◦ RMA ,
i = 1, . . . , n ,
Bi∗ = Ai ◦ RMA
i = 1, . . . , n .
for all
Corollary 11.6 If all input sets of (3.3) are normal then the system Ai ◦ R = Ai ◦ RMA ,
i = 1, . . . , n ,
(11.3)
is the smallest MA-solvable *u-supersystem for (3.3). This leads back to the iterated pseudo-solution strategies. Corollary 11.7 b be the S-pseudo-solution of (3.3), let be B ci = Ai ◦ R b for i = 1, . . . , n, and Let R suppose that the modified system ci , Ai ◦ R = B
i = 1, . . . , n ,
(11.4)
b is an optimal *lis MA-solvable. Then this iterated pseudo-solution RMA [Ak ◦ R] approximate solution of (3.3). Furthermore it is a best approximate solution of the original system in the space Rl under the ranking ≤l . Let us mention also the following result (cf. [12]). Theorem: Consider an unsolvable system of FRE’s such that all input fuzzy sets Ai , 1 ≤ i ≤ n, are normal and form a semi-partition of X. Then RMA (x, y) =
n _
(Ai (x) ∗ Bi (y))
i=1
is a best possible approximate solution in the space Ru = {R ∈ R | Ai ◦ R ⊇ Bi for all 1 ≤ i ≤ n} , under the preorder ≤u : R0 ≤u R00 iff Ai ◦ R0 ≤ Ai ◦ R00 for all 1 ≤ i ≤ n. These considerations can be further generalized. Consider some pseudo-solution strategy S, i.e. some mapping from the class of families (Ai , Bi )1≤i≤n of input-output data pairs into the class of fuzzy relations, which yields for any given system (3.3) of relation equations an S-pseudo-solution RS . Then the system (3.3) will be called S-solvable iff RS is a solution of this system.
Mathematical Fuzzy Control. A Survey of Some Recent Results 15 Definition 11.8 We shall say that the S-pseudo-solution RS depends isotonically (w.r.t. inclusion) on the output data of the system (3.3) of relation equations iff the condition if
Bi ⊆ Bi0 for all i = 1, . . . , n
then
RS ⊆ RS0
holds true for the S-pseudo-solutions RS of the system (3.3) and RS0 of an “outputmodified” system Ai ◦ R = Bi0 , i = 1, . . . , n. Definition 11.9 We understand by an S-optimal *u-approximate solution of the system (3.3) the S-pseudo-solution of an S-solvable *u-approximating system of (3.3) which has the additional property that no strongly better *u-approximating system of (3.3) is Ssolvable. Proposition 11.10 Suppose that the S-pseudo-solution depends isotonically on the output data of the systems of relation equations. Assume furthermore that for the S-pseudo-solution RS of (3.3) one always has Bi ⊆ Ai ◦RS (or always has Ai ◦RS ⊆ Bi ) for i = 1, . . . , n. Then the S-pseudo-solution RS of (3.3) is an S-optimal *u-approximate (or: *l-approximate) solution of system (3.3). It is clear that Corollary 11.6 is the particular case of the MA-pseudo-solution strategy. But also Proposition 11.1 is a particular case of this Proposition 11.10: the case of the S-pseudo-solution strategy (having in mind that S-solvability and solvability are equivalent notions).
12
Introducing the Solvability Degree
Following [4, 5] one may consider for a system of relation equations the (global) solvability degree n Y ξ = [[∃X (Ai ◦ X ≡ Bi )]] , (12.1) i=1
and for any fuzzy relation R their solution degree δ(R) = [[
n Y
(Ai ◦ R ≡ Bi )]] .
(12.2)
i=1
Q Here means the finite iteration of the strong conjunction connective &, and is defined in the standard way. The following result was first proved in [3], and has been further discussed in [4, 5]. Theorem 12.1 b ≤ ξ. ξ n ≤ δ(R) Of course, the n-th power here is again the iteration of the strong Q conjunction operation ∗, i.e. the semantical counterpart of the syntactic operation . Obviously this result can be rewritten in a slightly modified form which makes it more transparent that Theorem 12.1 really gives an estimation for the solvability degree ξ in terms of a particular solution degree.
16
Mathematical Fuzzy Control. A Survey of Some Recent Results
Corollary 12.2 b n ≤ ξ n ≤ δ(R). b δ(R) One has for continuous t-norms that they are ordinal sums of isomorphic copies of two basic t-norms, of the L Ã ukasiewicz t-norm tL given by tL (u, v) = max{u + v − 1, 0} and of the arithmetic product tP . (Sometimes also G¨odel’s t-norm min is allowed for these summands. However, this is unimportant because of the definition of an ordinal sum of t-norms.) Corollary 12.3 b and ξ always belong to In the case that ∗ is a continuous t-norm t, the values δ(R) the same ordinal t-summand. A further property is of interest for the case of t-norm based structures L. Proposition 12.4 For each continuous t-norm t and each 1 ≤ n ∈ N there exists n-th roots. Having this in mind, one can immediately rewrite Theorem 12.1 for this particular case in an even nicer form as we did it in Corollary 12.2. Proposition 12.5 For t-norms which have n-th roots one has the inequalities q b ≤ ξ ≤ n δ(R) b . δ(R) Using as in [14] the notation z(u) for the largest t-idempotent below u, this last result allows for the following slight modification. Corollary 12.6 For t-norms which have n-th roots one has the inequalities q n b b . z(δ(R)) ≤ ξ ≤ δ(R) b of the S-pseudoBesides these core results which involve the solution degree δ(R) solution of the system (3.3), the problem appears to determine the solution degree of the relation RMA . Proposition 12.7 If all input sets Ai are normal then Y^ δ ∗ (RMA ) = [[ (Ai ∩t Aj 6≡ ∅ → Bi j Bj )]] . i
j
This is a generalization of the former Klawonn criterion. We also find, as explained in [7], a second result which indicates that RMA [Ak ◦ b k ]] is at least sometimes as good a pseudo-solution as RMA . R[B Proposition 12.8 If all input sets Ai are normal and if one has b j Aj ◦ R b, |= Bi j Bj → Ai ◦ R then
b k ]]) . δ ∗ (RMA ) ≤ δ ∗ (RMA [Ak ◦ R[B
Mathematical Fuzzy Control. A Survey of Some Recent Results 17
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Received 13 October 2004.
