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From the above two inequalites, we get E xT (t)x(t) ab−1 E T 0 xT (t)x(t) dt + b−1 V (ϕ(t), r0 ). Thus applying Gronwall-Bellman lemma to this inequality yields, E xT (t)x(t) a−1 (1 − exp(−ab−1 T ))V (ϕ(t), r0 ). As T → ∞, there exists c such that lim E t→∞ T 0 xT (t)x(t)|(ϕ(t), r0 ) c V (ϕ(t), r0 ) c sup −τ t 0 |ϕ(t)|2 . H∞ Fuzzy Control of Delayed MJFS 43 From (2), the closed loop MJFS defined in (6) is stochastically asymptotically stable. In order to cater the H∞ control problem, let us consider the H∞ performance index for given γ > 0 as ∞ J =E 0 z T (t)z(t) − γ 2 wT (t)w(t) dt (15) On the other hand, z T (t)z(t) − γ 2 wT (t)w(t) = r T μj (xT (t) GTjk + xT (t − τ (t))GTdjk + wT (t)Hwjk )× j=1 (Gjk x(t) + Gdjk x(t − τ (t)) + Hwjk w(t)) − γ 2 wT (t)w(t) +JV (xt , k, t) − JV (xt , k, t) under zero initial conditions we have J (jj ) ζ T (t)Ξku ζ(t), where ζ(t) = x(t − τ (t)) x(t) x(t − δ1 ) .

N, is ωk (ψ(t)). , ηk (θ(t)) = mωk (θ(t)) k=1 ωk (θ(t)) the membership function of the system with respect to the plant rule: k and r k ωk (θ(t)) = j=1 Mj (θ(t)); It is obvious that the fuzzy weighting functions n ωk (θ(t)) satisfy ωk (θ(t)) ≥ 0, k ∈ {1, 2, . . , n} and k=1 ηk (θ(t)) = 1, for all t ≥ 0. 3 Main Results In this section, we consider the stability criteria for stochastic TS fuzzy system with time varying delays. Further, this result is extended to obtain a stability condition for stochastic TS fuzzy system with constant time delays.

19–26, 2011. © Springer-Verlag Berlin Heidelberg 2011 20 A. V. Hote, and M. Rastogi In summary, the rest of this paper is organized as follows. Section 2 deals with the need for the controller tuning and various tuning techniques such as Z-N, CHR, and Genetic Algorithm (GA). In section 3, the proposed design of PID controller for an unstable electronic circuit is explained with the necessary calculations and results are compared graphically. Finally, conclusion is written in section 4. 2 Need for Controller Tuning Controller tuning means selecting the tuning parameters to ensure the best response of the controller.

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