Comment on "discrete-time optimal fuzzy controller design: Global concept approach" - Reply

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286 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 2, APRIL 2005

III. A COUNTEREXAMPLE

Since the case in [1] is also presented and discussed for continuous systems in [2], we give a continuous system case to illustrate that the trajectories are different for ﬁnite-horizon and inﬁnite-horizon opti-mization problem.

Consider the linear time-invariant linear system deﬁned as follows: _x(t) = 0x(t) + u(t):

The costindex is deﬁned as

J = 0:5x(T ) + 0:5 T

0 [x

2_{(t) + u}2_{(t)] dt:}

The optimal control law is

u(t) = 0p(t)x(t)

wherep(t) is the solution of the following Riccati equation: _p(t) = 2p(t) + p2_{(t) 0 1} p(T ) = 1 : Then p(t) = ( p 2 0 1)(2 +p2) + (p2 + 1)(2 0p2)e2p2(t0T ) (2 + 2) 0 (2 0p2)e2p2(t0T ) : WhenT ! 1 lim T !1p(t) = 0:414:

The optimal trajectory is

x(t) = x(0) exp t

0 [01 0 p()] d:

It is obvious that the two trajectories, ﬁnite-horizon optimal trajec-tory in time interval[0; T] and inﬁnite-horizon optimal trajectory in time interval[0; T], are different.

IV. CONCLUSION

References [1] and [2] present some helpful researches in the optimal controller design for the fuzzy system. However, there are some issues, such as how to simplify the computation for the optimization problem of fuzzy system and how to ensure some characteristics of the closed-loop system, that need to be resolved.

REFERENCES

[1] S.-J. Wu and C.-T. Lin, “Discrete-time optimal fuzzy controller design: Global conceptapproach,” IEEE Trans. Fuzzy Syst., vol. 10, no. 1, pp. 21–38, Feb. 2002.

[2] , “Optimal fuzzy controller design in continuous fuzzy system: Global conceptapproach,” IEEE Trans. Fuzzy Syst., vol. 8, no. 6, pp. 713–728, Dec. 2000.

Authors’ Reply S. J. Wu and C. T. Lin

We would like to thank Drs. Song and Chai for their comments. The papers mentioned are based on the idea that the optimal decision is, in fact, a step-by-step on-going decision process. In other words, at any time state, sayingXi(k), the following two decisions are to be made. 1) Minimize J1(R(1)) = 1 k=k [Xt_{(k)L(k)X(k) + R}t (k)Wt_{(Y (k))W(Y (k))R(k)]}

regarding nonlinear system

X(k + 1) =H(X(k))A(k)X(k) + H(X(k))B(k)W(Y (k))R(k) Y (k) =C(k)X(k): 2) Minimize Ji 1(R(1)) = 1 k=k [Xt_{(k)LX(k) + R}t_(k)Wt iWiR(k)]

regarding linear system

X(k + 1) =HiAX(k) + HiBWiR(k)

Y (k) =CX(k):

With the aid of the dynamic decomposition algorithm (DDA), the non-linear system behavior can be captured by the non-linear system for all k 2 [ki

0; ki1 0 1] and for all i = 1; . . . ; N. We then know these

two decisions are the same for allk 2 [ki₀; ki₁0 1]. Hence, we have X3

1(k) = X1i3(k). We can now imagine the original optimal

trajec-tory as just being composed of linked segments, which are the starting sections of segmental inﬁnite trajectoriesX₁i3(k) for i = 1; . . . ; N. If each segmental inﬁnite trajectory is uniformly exponential stable, i.e.,

9 q < 1 s:t: kXi

1(k)k < q 8 k k0i

8 > 0; 9 an integer T () > 0 s:t: kXi

1(k)k

8 k T ()

then the original optimal trajectory is guaranteed to be exponentially stable.

This work is inferred from backward reasoning but written in a for-ward sense. Lemma 3 is used to connectX₁i (k) to Xi (k). However, we lost one condition in Lemma 3:X(ki₁) is free by equal in these two optimal issues, i.e., X3(k₁i) = X3(k₁i). This condition can compen-sate the defect in Lemma 3 and, hence, cancel out the misleading switch phenomena. For readability considerations, this paper is written not only in a forward-inference way, but also not to emphasize thatXi (k) is in fact from the starting section ofX₁i (k) for the inﬁnite-horizon issue orX_{[k ;k 01]}i (k), ﬁnite-horizon issue.

This work is correct in stability analysis and the inﬁnite-horizon so-lution is optimal positively. However, the ﬁnite-horizon soso-lution can be called suboptimal or near-optimal in large sense to denote the less op-timality of the last few segments.

Manuscriptreceived March 26, 2003; revised March 1, 2004.

The authors are with the Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]).