By Assoc. Prof. Dimiter Driankov, Dr.-Ing. Rainer Palm (auth.), Assoc. Prof. Dimiter Driankov, Dr.-Ing. Rainer Palm (eds.)

ISBN-10: 3662110539

ISBN-13: 9783662110539

ISBN-10: 3790818860

ISBN-13: 9783790818864

Model-based fuzzy keep watch over makes use of a given traditional or a fuzzy open loop of the plant lower than keep an eye on so that it will derive the set of fuzzy if-then principles constituting the corresponding fuzzy controller. additionally, of principal curiosity are the ensuing balance, functionality, and robustness research of the ensuing closed loop method regarding a standard version and a fuzzy controller, or a fuzzy version and a fuzzy controller. the foremost goal of the model-based fuzzy regulate is to take advantage of the entire to be had diversity of current linear and nonlinear layout of such fuzzy controllers that have higher balance, functionality, and robustness homes than the corresponding non-fuzzy controllers designed by means of those related techniques.

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Theorem 4-1: Given a state feedback fuzzy control law (12) and the approximate upper bound in (24), ifthere exist constants CI > 0, 1= 1,2, ... , m, and a set of positive-definite symmetric matrices (Ql, Q2,"" Qm) such that: (i) a set of Riccati equations 53 (47) x(t) E SI, 1= 1,2, ... ,m, has a set of positive-definite symmetric solutions (PI, P2 , ••• , Pm), and (ii) P( r/) ::; P( rn, (48) Vi, then, the closed-loop fuzzy system (13) is globally asymptotically stable. 1 we have V(x(t)) ::; exp[-a(t - i)]V(x(i)), ri_l

The above equivalent theorems require accurate upper bounds, but in many cases the accurate upper bounds are very difficult to find, so we have to use approximate upper bounds. Thus, we use the following stability theorem to determine whether the closed-loop fuzzy system (13) is stable. In addition, if a fuzzy control law can be obtained by other methods, then by using the following theorem we can also determine whether this fuzzy control law is a stabilizing control law. -L) and the approximate upper bound in (24), if t < 00 and there exist constants CI > 0, I = 1,2, ...

Then K(x(t)) will be changed to Kjx(t) when x(t) is in the Sj,j f:. t. Similarly, the value of (10) is P(t) = PI. x(t) E S, until the state x(t) leaves S,. Thus, at the boundary of Sj the value of (2) is well defined. t(t)). 3: The system (4) is said to be quadratically stabilizable if there exists a feedback control law (2), a set of n x n fixed positive-definite symmetric matrices (PI. P2 , ••• , Pm) and a constant a such that the following conditions are satisfied 1) T < 00, 2) (14) dV(t) -_ L( x,t ) dt (15) where x T (A(I'(t))T P(t) + P(t)A(I'(t))x + 2xT P(t)B(I'(t))K(I'(t))x L(x, t) < -allxW, P(t) = PI, x(t) E S/, 1= 1,2, ...

### Advances in Fuzzy Control by Assoc. Prof. Dimiter Driankov, Dr.-Ing. Rainer Palm (auth.), Assoc. Prof. Dimiter Driankov, Dr.-Ing. Rainer Palm (eds.)

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