Neuro-Fuzzy Processor

Transcription

1 An Introuction to Fuzzy State Automata L.M. Reyneri Dipartimento i Elettronica - Politecnico i Torino C.so Duca Abruzzi, Torino - ITALY e.mail reyneri@polito.it; phone ; fax Abstract This paper introuces Fuzzy State Automata for control applications. They are erive from the integration of traitional nite state automata an neuro-fuzzy systems, where a nite state automaton tracks the state of the plant uner control an moies the characteristic of the neuro-fuzzy system accoringly. The main ierence with respect to existing systems is that the states of the automata are ientie by fuzzy, instea of crisp variables, therefore state transitions an the corresponing controller characteristic are smoother an easier to train or tune. 1 Introuction Fuzzy logic [2] an neural networks [1] are currently use in several control applications [3] ue to their interesting performance. They allow either to inclue \human knowlege" into a controller esign, or to esign learning controllers which can be traine \by examples", or both of them (neuro-fuzzy integration). Neuro-fuzzy systems are also applie in several other els, such as pattern classication, function approximation, an many others. Unfortunately, all fuzzy systems an most neural networks are fee-forwar systems, with no feeback, an therefore with no memory. This limits the applicability of neuro-fuzzy systems, especially in the el of control, where most controllers must have memory. In particular, recurrent neural networks [1] o have memory, but they are icult to train an they cannot use the human knowlege irectly (this is vali for any neural network), unlike fuzzy system which can learn from human experience but which are selom use in a recurrent architecture, ue to the iculty to eal with feeback in fuzzy logic. Furthermore, feeback in recurrent networks is often limite to simple variables (for instance, spee, acceleration, etc.). This paper introuces Fuzzy State Automata, a metho to esign recurrent neuro-fuzzy systems an to eal in a neuro-fuzzy fashion with sequences of states an events. Section 2 rst escribes how neuro-fuzzy systems can be interface with traitional Finite State Automata, an then introuces the theory of Fuzzy State Automata. Section 3 escribes their operating principles, while section 4 escribes few applications. Throughout the paper, FFSA are escribe bearing in min their application to control systems, but the propose theory can also be applie to other omains. 2 State of the Art Many plants may have many ierent states, accoring to operating conitions. Examples are: the ierent strokes of strongly non-linear trajectories, control of robot arms with ierent loa conitions; forwar an backwar steps of walking robots; optimal operation of tooling machines with ierent tools; etc.

2 Parameters of specialize neuro-fuzzy controllers Sensors Neuro-Fuzzy Controllers Neuro-Fuzzy Processor Plant Selector Actuators Finite State Automaton Plant iscrete states Figure 1: Interaction of neuro-fuzzy controllers with FSA in control applications. Neuro-fuzzy control methos can be use in conjunction with traitional Finite State Automata (FSA) to eal with the ierent plant states, for the following reasons: 1. fuzzy logic allows to inclue the available \human knowlege" into the solution of a problem [2], but is not well suite to eal with sequences of events; 2. neural networks may eal with time-epenent signals (with an appropriate training), but they cannot easily acquire the human knowlege; 3. FSA are ieally suite to eal with sequences of events, but they are not goo at hanling continuous signal an human knowlege; 4. fuzzy systems are mostly fee-forwar systems, without feeback, as fuzzy logic is mostly associate with memoryless systems. This is because no metho has been foun so far to escribe eciently recurrent fuzzy systems; 5. mixing neuro-fuzzy systems with FSA allows to inclue human knowlege an training capabilities into recurrent systems (therefore, with memory) which can therefore eal with sequence of events, ierent plant states, etc. Neuro-fuzzy controllers interacting with FSA have alreay been use in control applications [5, 6]. The basic iea is sketche in g. 1: a neuro-fuzzy system controls a plant through a set of sensors an actuators, in a quite well known fashion. A traitional FSA tracks the iscrete plant states an moies the controller characteristic accoringly, by selecting one out of many weight matrices (or knowlege bases, for fuzzy controllers). A ierent weight matrix (or knowlege base) is associate with each state of the FSA. Plants operating with a set of well ene states may either be controlle by a single controller, or by a set of simpler controllers interacting with a FSA. The avantages of the secon solution are the following: 1. the overall controller is subivie into a set of simpler controllers; each of these may often be as simple as a linear controller; 2. each controller will be traine only over a limite subset of the state space; 3. each controller has a reuce size an can be implemente in an optimal way, also by using ierent paraigms for each of them;

3 4. controllers can be traine inepenently of each other, therefore training one of them oes not aect any of the others. Unfortunately, traitional FSA interacting with neuro-fuzzy controllers have few major rawbacks: 1. The overall controller characteristic may change abruptly when the FSA has a state transition. This may often cause very high accelerations an jerks which may increase mechanical stresses, vibrations, an often reuce comfort an possibly also the lifetime of the plant. 2. To reuce iscontinuities, there is the nee for either an aitional \smoothing" subsystem (but this often worsens the overall system performance), or the iniviual controllers shoul be esigne to take care of state changes (but this increases the complexity an the uration of training) The following sections escribe Fuzzy State Automata (FFSA), which are an alternative solution to the problems liste above. They are erive from the tight integration of neurofuzzy systems an more traitional FSA, but they have better performance an are more suite to esign smooth characteristics an to eal with sequences of events. 2.1 Fuzzy State Automata The main ierence of FFSA with respect to traitional FSA is that transitions in the automata are not triggere by crisp events but by fuzzy variables, an state transitions are fuzzy as well. It immeiately results that, at any time, the whole system is not necessarily in one an only one well-ene state, but it may well be in more states at the same time, each one associate with its own membership value. State transitions are therefore smoother an slower, even if the controllers activate by each state are not esigne to smooth the transitions. As a consequence, all the iniviual controllers may become as simple as a traitional PID, each one esigne for a ierent target specic of that state. The FFSA then takes care of smoothing iscontinuities between partial local characteristics, but the overall system is much simpler to esign an tune than a traitional controller with similar characteristics. Unfortunately FFSA have a rawback, that is, as the automata are often in more than one state, they have to process more than one controller at any time. This increase of computing time often counterbalances the higher spee that can be achieve through the use of simpler controllers. In practice, FFSA achieve approximately the same spee of traitional FSA controlling neuro-fuzzy controllers, but they are easier to esign (especially for complex systems) an prouces smoother trajectories. So far, the theory of FFSA has been teste on some applications in the el of control, for instance to esign the Motion an Leg Coorination Controls of a walking hexapo [6], where FFSA have been use as a way to specify the cooperation between iniviual legs. This example is shortly escribe in section 4. 3 Operation of Fuzzy State Automata This section escribes theory an operation of FFSA, an in particular how FFSA can be converte to fuzzy escriptions an the corresponing mathematical moels. The theory will be escribe by means of the example shown in g. 2. This is nothing but a simple example with only one input variable, but the iagram coul well be part of a more complex FFSA. Furthermore, the theory applies also to FFSA with more than one input.

4 x is NOT A x is B SA CA x is A SB CB x is C x is D SC CC SD CD Figure 2: An example of an FFSA. An FFSA looks very similar to a traitional FSA, in the sense that it can be represente as a collection of fuzzy states S j (namely, the circles), connecte by fuzzy transitions (namely, the arrows). Each fuzzy transition is labele by a fuzzy expression which can be as complex as esire (A, B, C, D are traitional fuzzy sets). As in a traitional FSA, a state represents univocally the operating conitions of an FFSA, at any given instant. But, unlike FSA, a system nee not be in only one state at a time. Each state S j is therefore associate with a fuzzy state activity Sj 2 [0; 1], which represents \how much" the system is in that particular state. The state activity is somehow equivalent to the egree of membership of a fuzzy variable, yet note that the state in an FFSA is not a fuzzy variable. If we associate a state activity to states in a traitional FSA, it woul be: Sj = 1; for the active state 0; for all other states (1) with the constraint that the total activity: X j Sj =1; (2) meaning that the system is always in one an only one well ene state S j (the one with Sj = 1) an in no other state. Constraint (2) applies also to FFSA, therefore activity can be istribute among several states an can partially move from one state to another (possibly more than one), but the total activity shall always be constant an equal to 1. FFSA can be of ierent types: time-inepenent an time-epenent; the former can either be synchronous or asynchronous, which somewhat reect synchronous an asynchronous FSA, respectively. They ier only in the way state activity moves from one state to another, accoring to the egrees of membership of the fuzzy transitions. Roughly we coul say that: 1. in asynchronous, time-inepenent FFSA, state transitions may take place as soon as inputs vary (yet transitions are not so abrupt as in traitional FSA); 2. in synchronous, time-inepenent FFSA, state transitions are compute in a way similar to asynchronous FFSA, but they are applie only at the next clock cycle; 3. in time-epenent FFSA, there is always an intrinsic elay (usually larger than the clock perio, if any) between input variations an the corresponing state transitions.

5 3.1 Time-inepenent automata In time-inepenent FFSA, the activity moves from each state to one or more other states, as a function of the present state activity an the egrees of membership of the state transitions, but inepenently of time (hence the name). State activity may \move" only along state transitions. In more etails, a time-inepenent FFSA (either synchronous or asynchronous) can be translate into a more traitional fuzzy escription mae of one rule per each transition, where each rule is in the form: IF (STATE is olstate) AND (fuzzyexpression) THEN (STATE is newstate) (3) where the term (STATE is S j ) has, by enition, a egree of membership equal to the state activity Sj, whereas the AND, OR an NOT implicators have the traitional meaning of any fuzzy system [2]. Two rules with the same consequent are suppose to be connecte by anor implicator. Transitions starting from an ening into the same state must also be taken into account. For instance, for the example shown in g. 2, the equivalent time-inepenent fuzzy escription is mae of ve rules: IF (STATE is SA) AND (x is NOT A) THEN (STATE is SA) IF (STATE is SA) AND (x is A) THEN (STATE is SB) IF (STATE is SB) AND (x is B) THEN (STATE is SB) IF (STATE is SB) AND (x is C) THEN (STATE is SC) IF (STATE is SB) AND (x is D) THEN (STATE is SD) (4) A constraint has to be place on the membership functions i (x) associate with all the transitions exiting from a state, to let the FFSA operate properly (not proven here): X i2tj i (x) 1 8x 2U x ; 8j (5) where T j is the set of transitions exiting from state S j an U x is the universe of iscourse of the input variable x. For the example of g. 2, the above ientity hols if an only if B = NOT (C OR D). In traitional FSA, the transitions exiting from a state must be both exhaustive (meaning that the logic sum of all transition labels must always be true) an mutually exclusive (meaning that no two transitions must be active at the same time). Similarly, in FFSA, constraint (5) is equivalent to exhaustivity, but there is no equivalent ofmutual exclusion. As any other fuzzy system, the fuzzy equivalent of an FFSA also has a mathematical moel. Supposing to use the min an max operators as AND an OR implicators, respectively, the fuzzy system (4) results into the following mathematical expression, vali only for synchronous FFSA (one equation per state): 0 SA(t +t) = maxfminf SA (t); 1, A (x)g; :::g 0 SB(t +t) = maxfminf SA (t); A (x)g; minf SB (t); B (x)g :::g 0 SC(t +t) = maxfminf SB (t); C (x)g; :::g 0 SD(t +t) = maxfminf SB (t); D (x)g; :::g (6) where the ots stan for any other transition reaching the corresponing states (not shown in g. 2). The values 0 S j (t +t)represent the new state activities at time (t +t), where t is the sampling perio.

6 state activity (u) ua uc usa usb state activity (u) ua uc usa usb a) time (t) b) time (t) Figure 3: a) a ealock in an FFSA; b) correct behavior of an asynchronous time-inepenent FFSA. Simulation results with D (x) 0 an B (x) (1, C (x)) This might seem a straightforwar implementation of an FFSA, but it oes not guarantee that total activity remains constant. In practice, it suers from the problem epicte in g. 3.a: suppose that, at t =0,the FFSA is mostly in state SA (namely, SA 1, SB ; SC ; SD 0), A (x); C (x); D (x) 0, therefore B (x) 1 (from (5)). In this case equations (6) simplify an, as soon as A (x) begins to monotonically increase: 0 SA(t +t) = minf SA (t); (1, A (x))g = (1, A (x)) 0 SB(t +t) = maxfminf SA (t); A (x)g; SB (t)g = = maxfminf(1, A (x)); A (x)g; SB (t)g (7) Therefore, as long as A (x) 0:5, the more A (x) increases, the more the activity moves from SA to SB; in particular, the activity of SB increases an tracks A (x), an that of SA ecreases accoringly, thus total activity remains constant. The problems arises when A (x) > 0:5. From that point, activity of SA keeps ecreasing but that of SB stops an gets stuck atavalue of 0:5 (from (7)), as shown in g. 3.a. This problem is solve by using constraint (2) to normalize the state activities given by equation (6), therefore: Sj (t +t)= 0 S j (t+t) (8) Pk 0 S k (t+t) By oing so, the total activity is guarantee to remains equal to 1. Instea, asynchronous time-inepenent FFSA use an alternative criterion to upate Sj ; the mathematical moel of asynchronous FFSA can be obtaine by moifying formulae (6) an (8) as, respectively: 8 > < > : 0 S j (t) =f j (::: i (x):::;::: Sk (t):::) Sj (t)= P S 0 j (t) k 0 S k (t) where f j () are appropriate functions obtaine by analyzing the state iagram (for instance, formulae (6)). The ierence with respect to the synchronous FFSA is that formulae (9) give the value of Sj at time t (an not (t +t)), therefore they are a set of non-linear equations in the variables Sj (which appear in both sies of (9)) an must be solve appropriately (for instance, by means of relaxation methos). Also in this case, total activity remains equal to 1; see g. 3.b, which shows how the activity of SB keeps tracking A (x) also when larger than 0.5. (9)

7 The ierence between synchronous an asynchronous FFSA is that the the latter is more complex to compute, but the activity of SB tracks perfectly A (x) for any value of it, while the former is faster to compute, but the activity of SB tracks A (x) > 0:5 with a slight time elay. 3.2 Time-epenent automata Like in time-inepenent FFSA, the activity moves from each state to one or more other states, as a function of the state activity an the egrees of membership of state transitions. But, unlikely time-inepenent FFSA, variations are slower an epen on time (hence the name). State activity \moves" only along transitions. In more etails, a time-epenent FFSA can be translate into a more traitional fuzzy escription mae of one rule per each state transition, where each rule is in the form: IF (STATE is olstate) AND (fuzzyexpression) THEN (STATE ows to newstate) (10) where all terms have the same meaning escribe in section 3.1, except for the new operator (STATE ows to newstate), whose mathematical moel is expresse by two ierential equations: is: t newstate(t) = 1 fuzzyexpression t olstate(t) =, t newstate(t) (11) For instance, for the example shown in g. 2, the equivalent time-epenent fuzzy escription IF (STATE is SA) AND (x is NOT A) THEN (STATE ows to SA) IF (STATE is SA) AND (x is A) THEN (STATE ows to SB) IF (STATE is SB) AND (x is B) THEN (STATE ows to SB) IF (STATE is SB) AND (x is C) THEN (STATE ows to SC) IF (STATE is SB) AND (x is D) THEN (STATE ows to SD) (12) which prouces to the following mathematical moel (two equations per rule): t SA(t) = 1 minf SA(t); 1, A (x)g ; t SB(t) = 1 minf SA(t); A (x)g ; t SB(t) = 1 minf SB(t); B (x)g ; t SC(t) = 1 minf SB(t); C (x)g ; t SD(t) = 1 minf SB(t); D (x)g ; t SA(t) =, t SA(t) (13) t SA(t) =, t SB(t) (14) t SB(t) =, t SB(t) (15) t SB(t) =, t SC(t) (16) t SB(t) =, t SD(t) (17) (18) where all terms t Sj with the same j, at the left sies of the above set of equations must be summe up together. It is clear from (13) an (15) that, unlikely time-inepenent FFSA, transitions starting from an ening into the same state have no eect at all, as they prouce two opposite terms which cancel each other out. Therefore such transitions nee not be taken into account.

8 state activity (u) ua uc usa usb time (t) Figure 4: Behavior of a time-epenent FFSA. The main ierence with respect to time-inepenent FFSA is that in time-epenent FFSA, state activity moves from one state to another with a time constant (which may also be ierent for each transition), therefore there is a time-lag between variations of inputs an of state activities, as shown in g Integration with neuro-fuzzy systems So far, FFSA alone have escribe in etails; this section explains how they can be integrate with neuro-fuzzy systems to prouce operational controllers to be use in practical applications. As for FFSA, the technique propose below can also be applie to other omains. To integrate FFSA with neuro-fuzzy systems(or traitional controllers, as well), it is necessary to associate each state S j with one controller C j, as shown in the example of g. 2. Controllers can be either linear, non-linear, neural, fuzzy, neuro-fuzzy, look-up tables, or any combination thereof. Furthermore, each controller can be implemente using the technique which best suits the requirements of that particular state, therefore the iniviual controllers nee not be of the same type. The only constraint is that all controllers must be functions of the same input variables an must prouce the same output variables. In vector form: ~Y j = C j ~X; t (19) where X ~ an Y ~ are the input an output vector, respectively. The overall characteristic, resulting from the integration of the controllers with X the FFSA becomes: ~Y = Sj (t) C j ~X; t (20) j As state activity Sj (t) moves smoothly from one state to another, when input conitions vary, the overall controller characteristic varies smoothly as well, removing all abrupt changes an variations. This has a lot of benets in control applications, as escribe in section 1. 4 Application Examples This section escribes shortly two application examples of FFSA in the el of control, an in particular to control the trajectory of two robots. The rst one is a simple metho to ene circular (or elliptical) trajectories, by means of an FFSA. Fig. 5.a shows an FFSA with four states an two inputs x an y, which are the

9 x is NOT LEFT y is NOT DOWN UP x is LEFT LEFT Vx=-Vo;Vy=0 Vx=0;Vy=-Vo y is UP y is DOWN RIGHT x is RIGHT DOWN Vx=0;Vy=Vo Vx=Vo;Vy=0 a) y is NOT UP x is NOT RIGHT b) y x Figure 5: Circular trajectory controlle by a simple FFSA. The trajectory shown in g. 6.b looks more octagonal than circular; this is ue to sampling time in simulations. POWER LEGANGLE is BACK LIFT BACK FRONT C1 C2 LEGHEIGHT is DOWN LEGHEIGHT is UP DOWN UP LEGANGLE (egrees) CONTACT LEGANGLE is FRONT RETURN VERY HIGH C4 C3 LEGANGLE is MORELESS FRONT AND TIMEOUT LEGSPEED is STILL LEGHEIGHT is VERY HIGH LEGHEIGHT (mm) BACKSTEP C5 DELTA (LEGANGLE) is FEWDEGREES UPSTEP C6 STILL LEGSPEED (egrees/sec) Figure 6: FFSA of the Leg Control. coorinates of a moving spot (for instance, the robot's tip). Each fuzzy state activates one of four simple controllers, which move the spot towars either left, right, up, or own, at a constant spee. The four fuzzy sets LEFT, RIGHT, UP an DOWN are self-explanatory. Fig. 5.b shows the resulting trajectory, starting from a point external to the esire trajectory. The spee of the moving spot is almost constant an equal to V o (the target spee of the four controllers), while the iameter epens on the membership functions LEFT, RIGHT, UP an DOWN. When the four fuzzy sets are symmetrical with respect to axis origin, the trajectory is circular, but it may easily become elliptical, or its center can be move, by shifting the membership functions. Yet, spot spee remains constant. Another application is to ene the Motion an Leg Coorination Controls of a six-legge robot [6]. This is a quite tough task, which requires a tight interaction between ierent controllers. A much simpler approach can be obtaine using the FFSA shown in g. 6, which tracks the state of each leg of the robot (for instance, forwar step, backwar step, leg lift an contact phases). Each state is associate with a ierent controller, each one tailore to the specic system state. For instance, controllers C1, C2, C3 an C4 are simple PID controllers tune to, respectively: hol boy weight an move the leg backwars; rise the leg at maximum spee; return the leg forwar at maximum spee; lower the leg at reuce spee. The FFSA of g 6 has provisions for the normal leg step (namely, walking on smooth surfaces

10 without obstacles) an to recover from obstacles encountere uring leg movements. The actual FFSA is more complex, as it inclues provisions for: i) moication of hexapo attitue in presence of complex obstacles (e.g. to walk insie low tunnels), ii) moication of leg trajectory when walking sieways, iii) control of alternative paces for ynamic gaits (e.g. galloping, or climbing staircases); iv) other special cases a leg might have to face. 5 Conclusion The paper has introuce Fuzzy State Automata, as a metho which allows a tight integration between nite state automata an neuro-fuzzy systems. Fuzzy State Automata can be converte to more traitional fuzzy systems, therefore they can run on either traitional fuzzy processors, or on systems with fuzzy computing capabilities. Fuzzy State Automata are also a metho to escribe recurrent fuzzy systems. Two examples have been escribe in the el of control. References [1] S. Haykin, \Neural Networks: A Comprehensive Founation", Mc Millan College Publishing Company, New York, [2] L. Wang, \AaptiveFuzzy Systems an Control", Prentice Hall, Englewoo Clis, New Jersey, [3] D.A. White an D.A. Sofge, \Hanbook of Intelligent Control", Van Nostran Reinhol, [4] L.M. Reyneri, \Weighte Raial Basis Functions for Improve Pattern Recognition an Signal Processing", submitte to Neural Letters. [5] L.M. Reyneri, M. Chiaberge, L. Zocca, \CINTIA: A Neuro-Fuzzy Real Time Controller for Low Power Embee Systems", IEEE MICRO, special issue on Harware for Articial Neural Networks, June 1995, pp [6] F. Berari, M. Chiaberge, E. Mirana, L.M. Reyneri, \A Walking Hexapo Controlle by a Neuro- Fuzzy System", Proc. of MCPA'97, Pisa (I), February 1997, pp