Systematic Design of a Stable Type-2 Fuzzy Logic Controller Oscar Castillo Tijuana Institute of Technology Tijuana, Mexico [email protected]
Abstract: The concept of a type-2 fuzzy set was introduced by Prof. Zadeh, as an extension of the concept of an ordinary fuzzy set. A Fuzzy Logic System (FLS) described using at least one type-2 fuzzy set is called a type-2 FLS. Type-1 FLSs are unable to directly handle rule uncertainties, because they use type-1 fuzzy sets that are certain. On the other hand, type-2 FLSs, are very useful in circumstances where it is difficult to determine an exact membership function and the measurement of uncertainties is difficult or even impossible. Similar to a type-1 FLS, a type-2 FLS includes type-2 fuzzyfier, rule-base, inference engine and substitutes the defuzzifier by the output processor. The output processor includes a type-reducer and a type-2 defuzzyfier; it generates a type-1 fuzzy set output (from the type reducer) or a crisp number (from the defuzzyfier). A type-2 FLS is again characterized by IF-THEN rules, but its antecedent and/or consequent fuzzy sets are now of type-2. Type-2 FLSs, can be used when the circumstances are too uncertain to determine exact membership grades.
For our description we are going to consider the well-known problem of designing a stabilizing controller for the inverted pendu-
lum system. The state-variables are x1 = θ - the pendulum’s angle, and x 2 = θ - its angular velocity. The system’s actual dynamical equation, which we will assume unknown, are shown in (1)-(3): x1 = x 2 (1) x 2 = f ( x1 , x 2 ) + g ( x1 , x 2 ) u Where: mlx 22 cos x1 sin x1 mc + m f ( x1 , x 2 ) = 4 m cos 2 x1 l − m c + m 3 cos x1 mc + m g ( x1, x 2 ) = 4 m cos 2 x1 l − m c + m 3 9 . 8 sin x1 −
mc is the mass of the cart, m is the mass of the pole, 2l is the pole length, and u is the applied force (control). In the simulations that follow we chose: m c = 0.5kg , m = 0.2kg and l = 0.3m . To apply the fuzzy Lyapunov synthesis method, we assume that the exact equations are unknown ant that we have only the following partial knowledge about the plant: 1. The system has two degrees of freedom
θ and θ , referred to as x1 and x1 , respectively. Hence, x1 = x2 .
2. x 2 is proportional to u , that is, when u increases (decreases) x 2 increases (decreases). Our objective is to design the rule-base of a fuzzy controller u = u ( x1 , x 2 ) that will balance the inverted pendulum around its upright position x1 = x 2 = 0 . We choose: 1 (4) V ( x 1 , x 2 ) = ( x 12 + x 22 ) 2 as our Lyapunov function candidate. Clearly, V is positive-definite. Differentiating V , we have:
V = x1 x1 + x2 x 2 = x1 x2 + x2 x 2
hence, we require:
x1 x2 + x2 x 2 < 0
in some neighborhood of (0,0) . We can now derive sufficient conditions so that (6) will hold: If x1 and x 2 have opposite signs, then x1 x 2 < 0 and (6) will hold if
x 2 = 0 ; if x1 and x2 are both positive, then (6) will hold if x 2 < − x1 ; and if x1 and x2 are both negative, then (6) will hold if x 2 > − x1 .We can translate these conditions into the following fuzzy rules: • If x1 is positive and x 2 is positive Then x 2 must be negative big • If x1 is negative and x 2 is negative Then x 2 must be positive big • If x1 is positive and x 2 is negative Then x 2 must be zero • If x1 is negative and x 2 is positive Then x 2 must be zero
However, using our knowledge that x 2 is proportional to u , we can replace each x 2 with u to obtain the fuzzy rule-base for the stabilizing controller: • If x1 is positive and x 2 is positive Then u must be negative big •
If x1 is negative and x 2 is negative Then u must be positive big
If x1 is positive and x 2 is negative Then u must be zero
If x1 is negative and x 2 is positive Then u must be zero
It is interesting to note that the fuzzy partitions for x1 , x 2 , and u follow elegantly from expression (5). Because V = x2 ( x1 + x 2 ) , and since we require that V be negative, it is natural to examine the signs of x1 and x2 ; hence, the obvious fuzzy partition is positive, negative. The partition for x 2 , namely negative big, zero, positive big is obtained similarly when we plug the linguistic values positive, negative for x1 and x 2 in (6). To ensure that x 2 < − x1 ( x 2 > − x1 ) is satisfied even though we do not know
x1 ’s exact magnitude, only that it is positive (negative), we must set x 2 to negative big (positive big). Obviously, it is also possible to start with a given, pre-defined, partition for the variables and then plug each value in the expression for V to find the rules. Nevertheless, regardless of what comes first, we see that fuzzy Lyapunov synthesis transforms classical Lyapunov synthesis from the world of exact mathematical quantities to the world of computing with words. To complete the controller’s design, we must model the linguistic terms in the rule-base using fuzzy membership functions and determine an inference method. We can characterize the linguistic terms positive, negative, negative big, zero and positive big by the type-2 membership functions shown in Fig.1 for a Type-2 Fuzzy Logic Controller.
Fig. 1 Kind of type-2 membership functions: a) negative, b) positive, c) positive big, d) zero and e) negative big Our experiments were executed with Interval Type-2 Fuzzy Sets, this kind of Fuzzy Sets are ones in which the membership grade of every domain point is a crisp set whose domain is some interval contained in [0,1]. On Fig.1 are depicted some Interval Type-2 Fuzzy Sets, for each fuzzy set, the grey area is known as the Footprint Of Uncertain (FOU). In our experiments we increased and decreased the value of ε to determine how much can be extended or perturbed the FOU with out loss of stability in the FLC. Margaliot’s approach for the design of FLC is now proved to be valid for both, Type-1 and Type-2 Fuzzy Logic Controllers. On Type-2 FLC’s membership functions we can perturb or change the definition domain of the FOU without loss of stability of the controller while we give a proportional gain to the control loop gains. In our example of the inverted pendulum, the stability holds extending the FOU on the domain [0,1], now we must to look for an explanation for this domain, a first look can be that we used interval type-2 sets, and these ones are defined in the interval [0,1]. Short Bio: Prof. Oscar Castillo Prof. Castillo is a Professor of Computer Science in the Graduate Division, Tijuana Institute of Technology, Tijuana, Mexico. In addition, he is serving as Research Director of Computer Science and head of the research group on fuzzy logic and genetic algorithms. Currently, he is President of HAFSA (Hispanic American Fuzzy Systems Association) and VicePresident of IFSA (International Fuzzy Systems Association) in charge of publicity. Prof. Castillo is also Vice-Chair of the Mexican Chapter of the Computational Intelligence Society (IEEE). Prof. Castillo is also General Chair of the IFSA 2007 World Congress to be held in Cancun, Mexico. He also belongs to the Technical Committee on Fuzzy Systems of IEEE and to the Task Force on “Extensions to Type-1 Fuzzy Systems”. His research interests are in Type-2 Fuzzy Logic, Intuitionistic Fuzzy Logic, Fuzzy Control, Neuro-Fuzzy and Genetic-Fuzzy hybrid approaches. He has published over 50 journal papers, 5 authored books, 10 edited books, and 150 papers in conference proceedings. Address and Affiliation: Oscar Castillo Professor of Computer Science in the Graduate Division Tijuana Institute of Technology Tijuana, B. C., Mexico Mailing Address: P.O. Box 4207 Chula Vista CA 91909, USA Email: [email protected]
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