for t e [to, oo) and subject to r] e S,r Replacing (64)-(70) in (84) one obtains Q{r),\,Ui ) =

Noiv using (85) together with the projected adaptive laws u^ = Proy J^-j ^ deduced from (72)-(78) with (79), the function (83) can be bounded on the right by

Using the theorem statement that considers smooth parametric changes and considering (64)-(70), it is valid that u*.(t) does exist, and that is uniformly continuous and satisfy u* e C,x n £00 (and/or JZ2 n Cco). Thai u*.—>0 for t—» co in t e [fo, 00) (see Ioannou & Sun, 1996, Lemma 3.2.5, pp. 76). On the other side, it is valid u* e £00. Thai the projected adaptive law (79) aisures that all u,-, be bounded in t e [to, <»). Finally, integrating (86) with all these suppositions, one achieves lim / f—r] Kpr]—\KV\\ dr < ^lim / —c(t)Q (t, r], v) dr

/to where c(t) and c0 are positive real values satisfying max z(t) =

with Xj(.) representing the eigenvalue j of the matrix indicated in parenthesis. Using the Lemma of Barbalat (Ioannou & Sun, 1996), it follows lim (r](t)—rir(t)) = 0. In consequence, with (56) it is also valid

This proves that the path errors go to zero asymptotically as stated in the theorem.

For proving that the Ui's are bounded, one employes (88), (59) and (62) together with the fact that M,

7], v, r\ and J are bounded. Moreover it is valid lim M, CCl,..., Cre, Dj, Dqi, ..., Dqe, Bi, #2 = 0.

So one yields first Jt°° v dr < oo and ft°° v dr < oo. Then, from (72)-(77) one obtains

\Uj\ < c.j v rfr < dc. for some constant c.; > 0. (92)

Similarly, from (78) it is valid

and in this way it is concluded that the matrices Ui's are also bounded. Finally, the boundness ofrc in (63) is demonstrated from the boundness of rj and v, d in (62), J and the proved boundness of the Ui's. u

Teorema II (Asymptotic convergence for piecewise-constant time-varying dynamics) Let the statement of Theorem I be valid with the difference that the physical system matrices M, CCi, Bi and B 2 are constant for all t e [fo, 00)/ Stk and that they have finite changes for a finite sequence of isolated time points 4 e Stk k = 1, ..., n, while the system matrices Dj and Dqi are constant. Then, for every initial condition r/(io) e and v(fo) e Sw the path tracking problem for given smooth reference trajectories nr(t) and vr(t) is achieved asymptotically with null error and additionally the boundness of all variables of the adaptive control loop is ensured if the condition r](f) e S,t is fulfilled for all t > to. Proof:

As Q(t) in (84) is continuous within [4_i, 4) with k = 1, ..., n and additionally in the period [t„, oo), and that r/(i) e S,f it is valid (see (86) with constant matrices in (64)-(69) and [7*7=0, Theorem I)

for t e [tk-k, tk) and t e [t„, x). Employing the solution v(t) in [tn-1, tn) and according to (43) and (56) one obtains

Since r](t„) nnd v(fn) nre in the attraction domain of the error system in the ODEs, it yields and by Lemma of Barbalat lim Q = 0 and consequently also the errors v and rj tend asymptotically to zero.

To demonstrate the boundness of TJ\, one notes first that the solutions r\ are uniformly continuous while the solutions v are only bounded in t < t„. Thus, it is valid Jt" v \dt < oo and ft resulting | Ui | < <» in this period. Then, from tn < <» in advance, both solutions are bounded and continuous, and so the boundness of the matrices Ui's in [tn, <»] is guaranteed. Finally, tc in (63) is also bounded as it can be deduced from the boundness of r] and v, of d in (62)

and J together with the proved boundness of the Ui's. ■

Corollary I (Asymptotic convergence for staggered-continuous time-varying dynamics) Let the statement of Theorem I be valid with the difference that the physical system matrices M, CCi, Bt and B2 are piecewise continuous and that Dl and D% are continuous for all t e [t, <»]. The sudden bounded changes occur at ti e Stk k = 1, ..., n. Additionally, all changes are jCa n £oo or Li n £oo. Then the asymptotic convergence of the tracking errors and the boundness of the adaptive control variables are ensured if the condition r/(i) e St] is satisfied for all t > to. Proof:

The proof of this corollary rises directly from the combination of the results in Theorem I and II in intervals of continuity [tu-i, h) and then, after the last step of the trajectory v(t), by considering the asymptotic disappearance of the parameter variation. ■

Now the static characteristic of the actuators in (46) is considered in the convergence analysis (see also Fig. 2). It is remembering from the end part of Section 3 that by neglected thruster dynamics it is valid n = nr and f = fideal.

Corollary II (Asymptotic performance of the adaptive control with static thruster characteristic) The adaptive control system employed in the path tracking problem for time-varying dynamics with piecewice continuous system parameters preserves the asymptotic properties and the boundness of all variables in the control loop, when the thrusters are described by (46) and (52). Proof:

As v(t) is piecewise continuous, so are va(t) in (46), Tt(t) and fideal (t) in (52). Thus, considering the static characteristic, it is deduced that by solving (46) for nr(t) with fdeal (t) and va(t), there may exist one or two solutions at t = ti e Stk, each one producing a propulsion that fulfills {¡deal - f = 0 in all t > to. Therefore, adopting some criterion in the case of multiplicity 0/11, (4), one obtains from the last instant t„ of sudden change in advance that v(f) will evolve continuously from v(fn) to zero asymptotically when t^rn. Moreover, due to the dynamic projection, all variables in the adaptive control loop result bounded in the intervals of continuity and also at tk, independent of the multiple solutions for nr(tk) and of the temporal variation rate of the parameters so long as they vanish in time.

5.2 Transient performance

The last results has concerned the asymptotic performance of the adaptive control system. However, nothing could be concluded about the control performance in short terms, it is, 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 fi e Stk and depending on the rate laws of there exists a time point tt > ti 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 s> 0. Proof:

At fi e St., the kinematic reference trajectory fulfills (ifc)AMi) lim v(i) (see

(43)). Take this vector value as initial condition for the next piece of trajectory of v and consider (86) for t > ti. Then it yields v(t,v,v,Ui) < -ck{

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