Tt Bf

where B is the matrix containing the vectors bi as columns, with i = 1, ..., n. The thruster dynamics is characterized as (see Healey, 1995; Pinto, 1996; Fossen, 1994)

where n is the shaft rate vector, nr is the reference of the rpm rate for n, va is the vector with the axial velocities of the thrusters, ua is the vector of the armature voltages for each DC motor, ni, n2 are auxiliary vectors, the operation (x.y) represents a new vector obtained by an element-by-element product, and similarly | n | represents a vector with elements \nt\. The factors K1 and K2 in (46) are diagonal gain matrices describing the non-linear characteristic (see Fig. 2), G1 and G2 represents diagonal matrices with strictly proper Laplace transfer functions, and similarly, Gpid is a diagonal matrix with Laplace transfer functions representing an usual tachometric PID control loop of the DC motors. Finally, the relation between the thruster output n and its reference nr and the propulsion f is n =

Fig. 2. Static Thruster characteristic

As Tt is actually the desired control action calculated by some guidance controller, the relation of it with the actuator thrust is also a desired propulsion referred to as fideal = BT (BBT) Tt

In order to reach fideai, the true f is provided by inputting nr in (51) to the actuators. This will require the knowledge of n which is supposed here a non-measurable vector. So, an observer for nr, n and f will be employed later that uses fideai and the knowledge of the thruster model to perform the estimations.

Sometimes, the thruster dynamics can be modelled as parasitics in comparison to the dominant vehicle dynamics and consequently be simplified. A typical case of parasitics is described by vehicles with large inertia for instance. In this case, making s ^ 0 in (51) one accomplishes n = nr and clearly the thruster is described only by its static characteristic (46) of the Fig. 2. Additionally, using fideal and va in (46), nr is determined if K and K2 are known. So, no observer is needed in this case to estimate f since f = fideal.

4. Servo-tracking problem 4.1 Asymptotic path tracking

The path tracking problem for the navigation system is introduced for two reference trajectories nr(t) and vr(t) = J'1(nr) nr, which are uniformly continuous. Moreover let n(t) and v(f) be measured vectors that fulfills r] e St] = { x, y, z, cp, £ TZ1,1 0 I < tt/2} and v e Sv c TZ . The specifications for the servo-tracking system are r](t)—r]r(t) -> 0, for t oo

for arbitrary initial conditions r/(0) e <S,(/ v(0) e Sv. The sets iS,( and Sv describe working regions of stability. Particularly, <S,( characterizes the region where J(rf) is non singular (cf. (13) and description below it).

The way to keep spacial and kinematic vehicle trajectories close to their references is achieved by manipulating conveniently the thrust f by means of a control system. As introduced before, we focus in this work the design of an adaptive control system to achieve this goal.

To this end, let us define first a convenient expression to take account of the positioning and kinematic errors as (Conte & A. Serrani, 1999)

with the gain matrix Kp = Kj > 0. Clearly, from (55)-(56) if r) is zero with (25), it is valid Accordingly, from (24)-(25) with (55)-(56) one obtains the path error system

Now, we will employ a speed-gradient technique to solve the path tracking problem asymptotically.

4.2 Controller design

The design of speed-gradient controllers starts with the definition of an energy cost functional of the state error vectors r) and v, which must be constructed as a radially unbounded and nonnegative scalar function (Fradkov et al., 1999). Accordingly, we propose

where Q results piecewise continuous due to the properties of Min time (see Section 2.3). For an asymptotic stable controlled dynamics it is aimed that for every initial vectors r/(0) e Sn, v(0) e Sv, it is valid from (59) and (53)-(54)

According to the speed-gradient method (SG) (Fradkov et al., 1999), the manipulated variable Tt has to be build up in that way that Q be smooth, bounded and radially decreasing in the error space S^ x <Sy, and convex in the space of the controller parameters. For the time-varying conditions of the dynamics stated here, Q (t) does not fulfills the conditions of continuity because of M(t) in (40).

So, we can analyze Q(t) in the periods of continuity of the trajectory, it is for f e [fo, cc)\Stk-Combining (57)-(58), (55)-(56) with (59) and stating the first derivative of Q(t) one obtains

With the criterion of eliminating terms with undefined signs in (61) and so achieving the desired properties for Q (t), it can be deduced that the following control action is optimal in this sense

where the U{s are matrices of the controller, CVi are the system matrices indicated in (32), gi and g2 are the vectors defined in (36) and (37), and finally Kv is a new design matrix involving in the energy of the kinematic errors that accomplishes Kv= KJ > 0. It is noticing that the cable force Tc here is supposed to be measurable and employed directly in the compensation in (63). Moreover, it is important to stress that the U/s will have the same structure of null and non-null elements as Cci, ..., Cc, (Dl- Mc /2), Dq1, ..., Dq„ By B2, M, and Mc have, respectively for i = 1, ..., 17. Thus, the number of elements in Ui can be reduced to a minimum considering the system matrix structure. In fact, one simple fixed controller (denoted by U*) could be designed subject to the knowledge of a true dynamics model as

Besides, the vector function candidate Tt (Ui) in (63) must be such one that Q result convex in the controller parameters in the U/s. It can be verified from (61)-(62) with (63) that this requirement is in fact satisfied.

4.3 Adaptive controller

Since the system dynamics is unknown, let the controller matrices U/s be defined adaptively based on speed-gradient laws. Thus, introducing (63) in (61)-(62), it is defined

with Tj = Tj > 0 constant gain matrices for tuning the adaptation speed in each component Ui of Tt. These laws allow to obtained U/s as integral solutions of (71). Particularly, for the Coriolis and centripetal matrix in its component Cc and its decomposition CCi. x Cv% as in (32), it is found for the U{s with i = 1,..., 6

Then, for the linear drag component Di in (33) the law is applied as

Analogously, for the quadratic drag component of Dq in (33) for the U/s with i = 8, ..., 13, one achieves

where vt is the i-th element of v.

Similarly, for the buoyancy matrices (38) and (39) in (35) it is obtained Next, for the inertia matrix M following law is assigned

where d is the auxiliary vector in (62).

Finally, for the unbounded Coriolis and centripetal component |Mc, one gets

The integration of the adaptive laws (72)-(78) with Ui(t0) e Su^ 9i6 6, for i = 1, S u being a compact set, provides a direct calculation of the control action for the path tracking problem without knowledge about the variable system dynamics.

4.4 Modified adaptive laws

With the goal of obtaining a-priori bounded and smooth matrices Ui(t)'s, we shall modify (71) with a smooth dynamic projection of the controller matrices (Pomet & Praly, 1992) on every column vector j of U referred to as Uj in the following. Thus, the new U/s are restrained to


where Proy(.) denotes the operator of the dynamic projection, additionally (.)j. refers to the column j of the matrix in parenthesis, V is a convex function in the parametric space of u,^ and is specified in the convex set Mu+e = {u^e TV/V(Ui^) < s} composed by the union Mu = [u^eTZ0/V(u,^) < 0} and the surrounding boundary MU+S\MU with tliickness s > 0, arbitrary small. So Mu is the interior of Mu, dMu is the contour of Mu and 3Mu+e is the external contour of Mu+e, both supposed smooth. Finally, A is a matrix that fulfills A = Ar >0.

For future developments of the adaptive controller, it is noticing that c(dMu) and c(3Mu+s). Generally, a good choice of 'Pfu¡j) is an hypersphere P(uj.)=11? u^—Mq — £ < 0/ with M0> £>0.

5. Performance and stability of the dominant vehicle dynamics

Let us consider the path tracking problem for the time-varying dynamics of the vehicle and its solution by the adaptive control system described previously. First, let us assume that the thruster dynamics can be neglected, not yet its static characteristic. The actuator parasitics will be considered in the next section. We introduce here the analysis of convergence of tracking error trajectories for the adaptively controlled time-varying dynamics in three steps, namely for system parameters varying: 1) in continuous form, 2) in piecewise-constant form and 3) in piecewise-continuous form. These results are stated by theorems.

5.1 Asymptotic performance

Theorem I (Asymptotic convergence for time-varying dynamics in continuous form) Consider the vehicle system (24)-(25), with bounded, pieceivise continuous parameters M, CCl, D;, D%, Bi and Bt, and rates M, CCi, Di, Dqi, B\, B2 £ Ci n£oo or n£oo- Let moreover 77 in (63) generated by the direct adaptive controller (72)-(78) with the dynamic projection (79). Assume the thrusters are only described by their nonlinear characteristic in (46). Then, for every initial condition r/(fo) e S,} and v(t0) e Sv, the path tracking problem for given smooth reference trajectories r/, (i) and vr(t) is achieved asymptotically with null error and the boundness of all variables of the adaptive control loop is ensured if the condition r/(i) e Stl is fulfilled for all t > t0. Proof:

Invoking the fact rj e S,}, eventual singularities of J(rf) are avoided and a Lipschitz condition is guaranteed for the right member of the ODE system (24)-(25) in St] x Sv. Then it is assumed the existence and uniqueness of solutions n(t) and v(t) for t e [t0, <»] and arbitrary initial conditions given in the domain of attraction.

Now, let us consider Q in (59) and the following candidate of Lyapunov function where

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