Infinite fuzzy logic controller and maximum entropy principle Danilo Rastovic,Control systems group Nehajska 62,10000 Zagreb, Croatia Abstract. The infinite fuzzy logic controller, based on Bayesian learning rules, is used for the simulation of a discrete countable controlled Markov chain, with adaptive techniques. Key Words. Infinite fuzzy logic controller, maximum entropy principle, Bayesian methods 1.Introduction Fuzzy logic becames tool for solving many control problems. In the situations of chaos we may see that there is a difference between deterministic and stochastic attractors. So, we must take different kinds of deffuzifications for simulating of chaotic behavior. In the case of determinism it is center of area defuzzification method, and in the case of stochastics we can make differencies between different scenarious of controlled stochastic process in order to have maximal spatial and temporal entropy.we have reasoning with Bayesian strategy since in the case of stochastics the complete scenarious is essential and not how near we are to the state of system at some given moment in some branch of the tree. In the paper [1] the model of infinite fuzzy logic controller has been introduced. This model holds true only for the case of Lebesque measure ( see [ 2 ], [ 3 ] ). In the paper [ 4] is found the counterexample with some probabilistic, for example point mass measure, for which the results of the paper [1] are not valid. In this note we present the one approach for constructions of infinite fuzzy logic controller which is valid for the some case of probabilistic measure. We use Bayesian learning rules and Jaynes maximum entropy principle ( see [5] ). We can consider as example the combinations of the signals of the type y = A sin ( b t + c ), where b is a fixed number, c and A are stochastic numbers. 2. Stationary Markov process This part is devoted to the exposition of the theory of countable controlled Markov processes with discrete time parameter. Controlled Markov process is given by the following elements :
The sets of the countable state spaces x i, i=0,1, 2,..., the sets of the action spaces a i, i=0,1, 2,...,the probability distributions called the transitions functions and the probability distribution called the initial distribution. Our aim is to find a control procedure under which the appropriate mathematical expectation of the appropriate path L= x 0 a 0 x 1 a 1...x n a n..is as large as possible. In the case of pattern recognition, it corresponds to the ability see the complete pictures with avoiding some visual noises. In the choice of the optimal path, at each step we need to take account not only of the point where we find ourselves, but also of how many steps there remain to be done ( see [6] ). The usual way to solve this problem is to compute the relative conditional expectations. For a conditional if A then B i.e. B A and for P(A) > 0 we define P(B A)= P(AB)/P(A). It will be necessary to contruct a distribution P which for given probabilities P i and for conditionaly B i A i satisfies P(B i A i ) = P i. Let P 0 be a prior distribution. This expresses the prior knowledge about all dependencies between the variables. Consider furthemore a set R = { B i A i [P i ], i= 1,2,...} of conditionals with desired probabilities P i. Such expressions we call ( probabilistic ) rules. From P 0 and R we derive the unique distribution P* which solves the optimisation problem: Maximising the entropy H(Q) = - Σ Q(i) ln Q(i), where Q is a distribution subject R, that is Q( B i A i ) = P i for all i= 1,2,.... This is precisely the Bayesian learning scenario. The question that Jaynes ( see [ 7 ] ) posed is, if we have information about some quantities, what are the best predictions that we make about some other quantities? A standard tool to have answer this question is a variational principle using the method of Lagrange multipliers. New interpretation to the concept of entropy in his formulation S = - Σ i=0 P i ln P i where P i represents the probability that a certain i-th macrostate will be realized. In other words, it must learn an optimal decision policy, which is a state-action mapping that maximizes the performance measure.the basic learning scenario assumed is as follows. At each time step i the learner observes the current environment state x i, selects and performs an action a i and then observes the consequences of this act. Let { x t, t > 0 } be an irreducible Markov chain on general state space X with σ -algebra B (X). Haken suggested to apply the maximum information calibre to Markov processes. If x assumes the value x i at time i, the its value at slightly later time point i + τ will be
denoted x i + τ. The joint probability distribution can be expressed by a product of the conditional probability P(x i + τ x i ) and the steady state probability distribution P s t. P ( x i + τ, x i ) = P ( x i + τ x i ) P s t ( x i ) Call a process on X recurrent if every state x is infinitely recurrent ( see [ 8] ). Call it transient if has a finite expected number of occurences. Every simple irreducible process is either reccurent or transient. A simple irreducible process, with an equilibrium state must be recurrent. We have the next theorem. Theorem 2.1. Let X be a recurrent,stationary strategy,countable Markov process. Then it can be simulated approximately by Bayesian learning rules over the tree of events. Proof.We consider a sequence of times t 0,t 1, t 2... t N at which the system is measured, with measured values of the state vector x i at time t i. We wish to make an imbiased estimate on the joint distrubution function P N = P ( x N, t N ; x N-1, t N-1 ; ; x 0, t 0 ). For Markovian case, the process can be split into P N = П N-1 i=0 P ( x i+1, t i+1 x i, t i ) p 0 (x 0,t 0 ). To this end we maximize the information S, or in the formulation of Jaynes, the calibre S = - Σ N i=0 P i ln P i, in time domain. We have the following Bayesian rules : IF P (x 0 ) is E 1( i) AND P(x i ) is E 2(i) THEN P(x i+1 x i ) is U(i) (i=0,1,2....) for some values E 1( i ), E 2( i), U ( i) of probabilistic measurable sets. We can choise such probabilities for which the maximum of information calibre is obtained by Poincare recurrence theorem ( see [9] ) and with the method of ordering. In other words, necessary condition for speaking about precise measurement is the possibility to repeat the observations. 3.Universal fuzzy controller The motivation for this part involves modeling the behavior of particles as they move within medium ( see [ 10 ] )Such particles motions are often well modeled by a Markov chain. For many problems involving complex material geometries, simulating a single particles history is nontrivial. We must use intelligently chosen dependent samples. The intelligent choise investigated here corresponds to allowing the process to learn and adapt at various stage, a notion which has been termed sequential or adaptive Monte Carlo. We investigate
the possibility of generalization of basic probabilistic concept to the case when the sample points are fuzzy. We consider fuzzy dynamics as an alternative to statistical mechanics. The dynamics laws in such systems are usually described by experts in the form «if the system is in the state x 1 then after a short time the state x 2 will be preferable than the state x 3 «.It means that we will take the action for transition to the first state instead for the second state.mostly two experts will be consistent on the preferability comparation between possible ways of dynamics. We have extensions of various concepts from classical measure theory to fuzzy subset. So, we obtain fuzzy σ - algebras and conditional entropy ( see [11] ) Let us consider the models with incomplete information. We suppose that the state of the system at the time t is described by a pair x t y t the first of these components becoming known to us and the second not. The actions a t and the observed states x t are connected as before by a projection J ( x t-1 = J (a t ) ).In this case we also calculate with some uncertainities. In order to define a measure on the space of trajectories, it is necessary to give the initial distribution P 0 and the strategy Π.In supposing that P 0 is known, we choose the Bayesian approach. Generally, Bayesian approach will work better than non-bayesian approach when uncertainty exists and the uncertainty is better understood by using previous observations, and helps decision maker to derive the corresponding optimal strategy better than the non-bayesian approach. We can note that actually all depends about the nature of chaotic attractors, i.e. whether deterministic or stochastic attractor arrives in the decision environment ( see [12], [13], [14]). To pass from the physical continuum to the mathematical continuum requires an idealization, namely that infinitely precise measurements are in principle, if not in fact, attainable. Suppose that our measuring apparatus has a finite resolution, i.e., it cannot distinguish between two points which are separated by a distance less that. We shall obtain average quantities over some appropriate distribution ( see [15] ).The probability distribution P( E) has been replaced by P (E ). In most practical, real-life problems the information about the objective is vague, imprecize and uncertain. Clearly, this kind of problem is the best match for the collective brain, whose low precision is compensated by a high degree of universality. The control experience is not usually formulated in terms of the natural language rules like «if x is small, then control must also be small». The methodology of translating these rules into an actual control strategy was proposed by Zadeh under the name of fuzzy control. Three steps are necessary to specify this translations: first, we must determine membership functions that correspond to all natural language terms ( like «small» or «big» ) that appear in the rules. Second, we must choose operations that correspond to & and V. As a result, we
get a membership function for a control; then we need a method to transform this function into a single control value ( a defuzzification method ). For some cases of probabilistic measures we can not use the fuzzy weighted additive action rule a * ( x) = Σ N i=1 w i a i (x) / Σ N i=1 w i but Bayesian learning rule. The deterministic infinite fuzzy logic controller consists of rules considering fuzzy values of error e(k) between current set point s(k) and output y(k) and its first difference δ e(k) as input variables and the first difference of control variable δ u(k) as an output variable : IF e (k) is E 1(i ) AND δ e (k) is E 2( i) THEN δ u (k) is U (i) ( i=1,2,... ), where E 1( i), E 2( i) and U (i) are the fuzzy values of the relevant variables described by fuzzy sets, and the lower index i stands for the i-th fuzzy control rule. On this model is applied the center of area defuzzification method ( see [1]). Now, the problem is describe the simulation of nonstationary Markov chains ( see [16], [17]). It could be obtained with the techniques of ergodic subclasses of states ( see [18] ). We have the next theorem. Theorem 3.1. Let X be a recurrent, nonstationary strategy, countable Markov process with several equilibriums. Then, it can be simulated approximately with infinite fuzzy logic controller by Bayesian learning rules. Proof.We define the information entropy by S = - Σ i =0 Σ j=0 P i j ln P i j (3.1) The distribution functions P i j are considered as unknown variables still to be determined.the constraints f(k) are measurements of some quantities and can be seen as expectation values. This is done by requirement that (3.1) acquires a maximum value under given contraints Σ j=0 P i j f (k) i j = f(k) (3.2 ) and Σ j=0 P i j = 1 ( 3.3) for every step i. The maximization of (3.1) under the constraints (3.2) and (3.3) can be performed by the use of Lagrange multipliers λ k and λ-1 ( see [19] ) for S(i) is a finite sum over index j in expression for entropy. The procedure follows by applications of result in [19].We then have δ [S(i) - (λ - 1 ) Σ j P i j -Σ j k λ k P ij f (k) ij] = 0 (3.4), for every i. Performing the variation of (3.4) by differentiating the bracket with respect to p ij and puting the result equal to zero we obtain - ln p ij -1 - ( λ - 1 ) - Σ k λ k f (k) ij = 0 (3.5) or equivalently lnp ij = - λ - Σ k λ k f (k) ij which after putting both sides into the exponent of an exponential function yields the required result p ij = exp { - λ- Σ k λ k f (k) i j } (3.6)..It should be noted that we must still determine the Lagrange multipliers and that obtained
probabilitie depends crucially on the choice of constraints f(k) which we choose since these in turn define the variables f (k) i j in terms of which the probability distribution is expressed, what actually depends on appropriate actions. We arrive at maximum values of entropy. If on each i-th step the optimal result is obtained by maximum entropy method, then on whole time scale is obtained the global maximum. For controlled Markov chain x 0 a 0 x 1 a 1. a n we use the following infinite fuzzy logic controller with Bayesian learning rule x n IF P ( x i ) is E 1(i) AND P ( x i+1 x i ) is E 2(i) THEN a i is U (i) (i=0,1,2,...) where E 1(i), E 2(i), U (i) are the fuzzy values of the relevant variables described by fuzzy sets. Since fuzzy irreducible Markov process is, by assumptions of this theorem, also irreducible ( see [ 18] ) we can apply Theorem 2.1..The problem is actually converted from nonstationary strategy case to the stationary strategy case. As defuzzification on G i = E 1 (i ) E 2( i) U (i) we can take the following reason : δ u (k)=(a 0,a 1, a 2,...) is given on such a way that the maximum of information entropy S = -Σ i=0 Σ j=0 P ij ln P ij is obtained for : = Π N-1 i=0 P (x i+1,t i+1 x i,t i ) P (x 0, t 0 ), where P i =Σ j=0 p ij is possible clustering, for global optimization in time domain. P N It is a rather widespread assumption in uncertain reasoning, and one that we make that a piece of uncertain knowledge can be adequately captured by attaching a real number ( signifying the degree of uncertainty ) on some scale to some unequivocal statement or conditional, and that an intelligent agent's knowledge base consists of a large set of such expressions ( see [20], [21] ).As a real example, we can take the problem of walking through the labirint of several open doors, to reach some goal. When fuzzy set satisfies desired accuracy, we have desired result and so we can maximize information entropy over all doors. There is a motivation for the idea of control of Brownian motion and the Markov chain is possibly modelling. So with appropriate maximum entropy function we can easy make estimation thath the Brownian motion walked through all open doors. Such example arrives in transport theory, where the moving of ions via positions and velocities on each step is controlled by the action values of magnetic fields. Conclusion
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