Here we obtain the optimal

quadratic performance index consisting of tracking errors and control input.

According to the dynamic model of the AUVs heading control system (6), we select the following average quadratic performance index:

The optimal control problem is to search for a control law [u.sup.*](t) for system (6), which makes the value of the average quadratic performance index (16) minimum.

Given a continuous-time switched system whose dynamics are governed by (1) and (2) for a fixed time interval [[t.sub.0],[t.sub.j]], the objective is to find the continuous control u* and the switching instants [t.sup.*.sub.k] that minimize the

quadratic performance indexThen the state-dependent Riccati equation was presented for an affine nonlinear system and finite horizon

quadratic performance index. Finally the designed performance controller has been measured by defining various scenarios.

To minimize both the state and control signals of the feedback control system, a

quadratic performance index is minimized:

A unique tuning strategy that makes use of a weighted

quadratic performance index to compute controller output change is used to help achieve improved control performance.

We introduce the

quadratic performance index function for system (1):

The problem would be to find an optimal control w(t) satisfying (14) while minimizing the

quadratic performance index as follows:

The design process was simplified by introducing a

quadratic performance index with corresponding input delay in [24].

In the case when G is asymptotically stable, the

quadratic performance index is

For (17), let v = [v.sub.0] and [v.sub.0] can minimize the

quadratic performance index as follows: