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Underwater Vehicles Part 8

Tham khảo tài liệu 'underwater vehicles part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Adaptive Control for Guidance of Underwater Vehicles 269 about how significant are the transients and how fast the guidance system can adapt the initial uncertainty as well as the temporary changes of the dynamics. Finally in the focus of future analysis there would be the rolls that ad-hoc design parameters in both the control loop i.e. Kp and Kv and in the adaptive loop i.e. the r s play in the control performance during adaptation transients. A powerful result is given in the following theorem. Theorem III Transient performance of the adaptive control system Let the statement of Corollary I be considered for a piecewise-continuous time-varying dynamics. Then after an isolated sudden change of M in t1 e 5 and depending on the rate laws of .11 . . if. D. . DI and ÍỈ Ị there exists a time point tt t1 such that from this on the adaptive control system with gains Kp Kv and ri that are selected sufficiently large can track any smooth reference trajectories n and vr with a path error energy that is lower than a certain arbitrarily small level e 0. Proof At t1 e 5 the kinematic reference trajectory fulfills v A V 1Í111 v see 43 . Take this vector value as initial condition for the next piece of trajectory of v and consider 86 for t t1. Then it yields v .T7.v.r J I V h Ã1 IZH K ucG IK z l- 97 i l j l with li omi Xị .x . and mill _ D - As the vectors ũ I e I n x or n x then these start to decrease after expiring some period referred to as Tij. Thus for a given e 0 arbitrarily small there exists some instant tt m ax J ij I 11. certain minimum values of cK and cr from which on V t TỊ V Ui 0 and the previous inequality satisfies 1 Ui -u 0 u i . 98 II II CKCr i i j i for t tt. Clearly this result is maintained for all t tt if no new sudden change of M occurs any more. Corollary III Transient performance of the adaptive control with static thruster characteristic The result of Theorem III is preserve if the dynamics model and its adaptive control involve actuators with a .