{Ay(r,)}. As the vectors u| (t) e Ci fi Coo or C.2 n £oo, then these start to decrease after expiring some period referred to as Tij. Thus, for a given s> 0 arbitrarily small, there exists some instant tt > max (T.^ ) + t\, certain minimum values oJ'ck and cr from which on V (t, rj, v, Ui) < 0 and the previous inequality satisfies

for t > tt. Clearly, this result is maintained for all t > tt if no new sudden change of M occurs any

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 static characteristic according to (46) and (52). Proof:

Since f - fideal is identically zero for all t > t0, the true propulsion of the vehicle can exactly be generated according to Tt (t) by the adaptive control system. So the same conditions of Theorem III are satisfy and the same results are valid for the energy of the path error. ■

It is seen that the adaptive control system stresses the path tracking property by proper setup of the matrices T/s, not only in the selftuning modus but also in the adaptation phase for time-varying dynamics. This can occur independently of the set of Kp and Kv, whose function is more related to the asymptotic control performance, it is when | u|. (f) | = 0. Moreover, it is noticing that in absence of time-varying parameters the dynamic projection on the adaptive laws (79) does not alter the properties of the adaptive control system since more. ■

the terms (u^ — u* (t))TVi 1u*(i) in (86) are null. The reason for the particular employment of a projection with a smoothness property on the boundary is just the fact that by time-invariant dynamics the control action will result always smooth.

6. State/disturbance observer

The last part of this work concerns the inclusion of the thruster dynamics together with its static characteristic according to (46)-(51) to complete the vehicle dynamics. By the computation of Tt (t) with a suitable selection of the design matrices Kp, Kv and r/s, it is expected that the controlled vehicle response acquires a high performance in transient and steady states. However, as supposed previously, the thruster dynamics (51) has to be considered as long as this does not look as parasitics in comparison with the achievable closed-loop dynamics. There exist approaches to deal with the inclusion of the thruster dynamics in a servo-tracking control problem that takes fideal (or the related n) as reference to be followed by f under certain restrictions or linearizations of the whole dynamics. A comparative analysis of common approaches is treated in Whitcomb et al., 1999, see also Da Cunha, et al., 1995. Though the existing solutions have experimentally proved to give some acceptable accuracy and robustness, they do not take full advantage of the thruster dynamics and its model structure to reach high performance.

In this work a different solution is aimed that employs the inverse dynamics of the thrusters. In this case, the calculated thrust fideal in (52) will be used to be input a state/disturbance observer embedded in the adaptive control system to finally estimate the reference nr to the shaft rate vector of the actuators that asymptotically accomplishes the previous goal of null tracking errors stated in (53)-(54), see Fig. 3 for the proposed control approach.

Fig. 3. Extended adaptive control system for the vehicle with thruster dynamics

As start point for an observer design, consider first one element of G2GPjD(s) in (49)- (50) corresponding to one thruster, and a state space description for this dynamics n2 — n — ni = c x

with ( A, b, c) a minimal set of a minimal description, x the state of this component and g3(s) a low pass filter to smooth sudden changes of n. Then, let

a differential equation for a state estimation x, with k„2 a gain vector for the shaft rate error and

with n2 and ec estimations of n2 and of the thruster control error ( nT - n ), respectively, and kn, kn and kf suitable gains for the components of ec. The function n2 and its first derivative 712 can be deduced and calculated analytically from U2= g?,n - Hi with Hi = gi(s)fideat and (100). Also with (46) and Fig. 2 with n* = ^va, it is valid

I Slgn(n) fideal h

k2 2klVa

On the other side, with (99), (102), (103) and (101), the state error vector x = x—x satisfies

x=(.A - k„2cr)x+b ec + k„2(l - g3(s))n, with ec = (nr —n) — ec. Using (99)-(100) one gets which combined with (104) and (101) it yields It is noticed that there exist particular values of the gains k^, kn and kn in (108) that fulfills

For the state space description in the observer canonical form one has cT = [ 1, 0, ..., 0] and bT = [bm-1, ..., bo]. Thus, with the thruster dynamics having a relative degree equal to one (i.e., bm-1 # 0), which is, on the other side, physically true, the observer conditions (109) turns into k Ï =

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