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In this paper we will examine the problem of learning a two layer hierarchical Fuzzy controller for the control of the inverted pendulum (with nonlinear dynamics). The fuzzy rules are learnt using cooperative co-evolution, whereby two distinct evolutionary populations are used: one defining the first fuzzy layer a n d the other defining the second fuzzy layer. We c o m p a r e the results from the eo-evolutionary algorithm with t h e results from a classical evolutionary algorithm. lntroductionThe control of the inverted pendulum or simulated pole-cart system has been undenaken using linear and nonlinear dynamics and include both classical and fuzzy logic control techniques. see for example [ I , 2. 3.1. 5. 6. 7. 81It has already heen demonstrated that ihis system can he controlled hy a fuzzy logic controller. the rules for which can he learnt hy a genetic algorithms. Our ohjeclivr in this paper is to consider a hierarchical two layer fuzzy control system and show the fuzzy rules can he learnt by coevolving two evolutionary populations. one defining the first layer and the other defining the second layer. We shall compare the results of this co-evolutionary learning with the learning of the rules in hoth layers by a classical evolutionary algorithm. Inverted Pendulum SystemThe nonlinear system to be controlled consists of the cart and a rigid pole hinged to the top of the can. The can is free to move left or right on a straight hounded track and the pole can swing in the venical plane determined by the track. It is modelled by: 2, = 2.2 i? = U + rnt(sin(rs)z: -ir cos(x:i))/(A1 + V I ) j . , = 2:4 gsin(z3) + cos(z3)(ri -m~z~s i i i (~~) ) / (~~ + m ) C(4/3 -7ncoS(x3]~/(~1/ + m)) = where z1 is the position ofthe cart, o'? is the velocity ofthe cart, :cg is the angle of the pole, .TJ is the angular velocity of the pole. U is the control force on the cart, m is the mass of the pole. ! \ I is the mass of the can. 1 is the length of the pole. and 9 is gravitational acceleration. The control force is applied to the cart to prevent the pole from falling while keeping [he cart within the specified bounds on the track. We takem = O.lk.9. 41 = lkg, 1 = 0.5m. {I = 9.81rn.s-'. with stale limits: -1.0 5 T I 5 1.0. & -s/G 5 ~' 3 5 s/[i.The goal is to determine fuzzy controllers necessary to stahilise the system ahout the uostahle reference position .I; = 0 as quickly as possible, whilst maintaining thesystem kith& the specified bounds given above. Hierarchical fuzzy systemFor the two level hierarchical fuzzy,system the first knowledge base IiBl has the two inputs, zI (position of the can) and o'? (velocity of the car( ) to produce as output a first approximation to the control U'. This U' together with S Q (the current angle of the pole) and 5 4 (the angular velocity o f the pole) are used as input in the second knowledge base IiBn to produce the final control output U . see Figure I Each domain region for .xi was divided into 5 overlapping ~ ~ ~~~ ~ ::% KB* , .

A general method is developed to generate fuzzy rules by using Genetic Algorithms (GAS) and a Fuzzy Logic controller (FLC). By using GAS as a learning procedure and a FLC as the system's performance evaluator, the proposed architecture can construct an input-output mapping in the form of fuzzy If-Then rules. The performance of the new architecture is compared with an artificial neural networks controller and pure limited-ryle fuzzy rule controller for the truck back-upper problem.

We consider the control of a rigid robotic manipulator using robust adaptive sliding mode tracking control. Physical state constraints are

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