3. Optimal control of fuzzy systems

#### 3.1 Fuzzy systems

min u tð Þ¼u∗ð Þ<sup>t</sup> lxt ð Þþ ð Þ u tð ÞTRu tð Þþ LfWxt ð Þþ ð Þ LgWxt ð Þ ð Þ u tð Þ

<sup>2</sup><sup>α</sup> LgWxt ð Þ ð Þ <sup>R</sup>�<sup>1</sup> LgWxt ð Þ ð Þ <sup>T</sup>

lxt ð Þ ð Þ . 0 for all x tð Þ 6¼ 0 and α≥ 1: This completes the proof.

approach which is referred to as the inverse optimal approach.

the control effort is viewed as being expensive.

to be the largest constant <sup>γ</sup> . 0 such that lim<sup>t</sup>!<sup>∞</sup> <sup>e</sup><sup>γ</sup><sup>t</sup>

structure.

the decay rate.

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<sup>¼</sup> min u tð Þ¼u∗ð Þ<sup>t</sup> �LfWxt ð Þþ ð Þ <sup>1</sup>

<sup>4</sup>α<sup>2</sup> LgWxt ð Þ ð Þ <sup>R</sup>�<sup>1</sup> LgWxt ð Þ ð Þ <sup>T</sup>

<sup>þ</sup> LfWxt ð Þ� ð Þ <sup>1</sup>

<sup>þ</sup> ð Þ <sup>α</sup> � <sup>1</sup>

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<sup>4</sup><sup>α</sup> LgWxt ð Þ ð Þ <sup>R</sup>�<sup>1</sup> LgWxt ð Þ ð Þ <sup>T</sup>

<sup>4</sup>α<sup>2</sup> LgWxt ð Þ ð Þ <sup>R</sup>�<sup>1</sup> LgWxt ð Þ ð Þ <sup>T</sup>

¼ 0, Wð Þ¼ 0 0,

k k x tð Þ <sup>2</sup> ¼ 0 holds for all trajec-

(8)

 þ 1

which implies that <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) and in (5) are solutions of the H-J-B equation in

In proposition 1, we see that the globally asymptotically stabilizing control law in

It is remarkable that as shown in (4, 5), we can adjust the penalty on the control input vector, which is R, and the penalty on the state vector, which is lxt ð Þ ð Þ , with the weight matrix R:Indeed, we can decrease the penalty on the control input vector and increase the penalty on the state vector with a weight matrix R having small values. In this condition, we can obtain a cheap optimal control law requiring a large control effort, and this cheap optimal control law makes the nonlinear dynamic system in (1) stable within a short period of time. Note that the term "cheap" refers to the fact that the control effort is viewed as being cheap. On the other hand, we can increase the penalty on the control input vector and decrease the penalty on the state vector with a weight matrix R having large values. In this condition, we can obtain an expensive optimal control law requiring a small control effort, and this expensive optimal control law makes the nonlinear dynamic system in (1) stable within a long period of time. Note that the term "expensive" refers to the fact that

As shown in (3), since the constant <sup>α</sup> of the optimal control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) plays the role of a feedback gain for <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> even though the weight matrix <sup>R</sup> is predetermined to impose the penalties on the control input and state vectors, it is also remarkable that the optimal control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) has a relaxed feedback gain

Now, if we consider a practical application, we know that any control law for dynamic systems must provide good convergence rates of state trajectories to an equilibrium point. For achieving this requirement in the design of a control law, the decay rate can be used as a design factor to dominate convergence rates of state trajectories to an equilibrium point. Note that the decay rate of a system is defined

tories of a system, where k k x tð Þ <sup>2</sup> denotes the Euclidean norm of x tð Þ: From the definition of decay rate, the convergence rate of the system trajectory to the equilibrium point can be controlled. Also, note that stability of dynamic systems corresponds to a positive decay rate. In the following, the author presents a theory about

(7). In addition, by (2) and the property of <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . <sup>0</sup>,lxt ð Þ ð Þ in (5) satisfies

(3) for the nonlinear dynamic system in (1) can be found without the task of numerically solving the H-J-B equation in (7) and the control law in (3) is optimal with respect to the cost function in (4). The key point of this work is that we posteriorly determine the penalty on the state vector, which is lxt ð Þ ð Þ , rather than we priorly choose it. Sepulcher, Janković, and Kokotović [30] proposed this

> The TS-type fuzzy model and the TS-type fuzzy control law for a system are given by the following IF-THEN fuzzy implications, respectively [15]:
