**9. Case study**

16 Advances in Discrete Time Systems

robust properties and finally to range them.

2) Select a sampling time *h* as small as possible,

The adaptive control algorithm can be summarized as follows.

4) Define the adaptive gain matrices Γ*<sup>i</sup>* (usually Γ*<sup>i</sup>* = *αiI* with *α<sup>i</sup>* > 0),

**8. Adaptive control algorithm**

<sup>∆</sup>*Qn* and <sup>∆</sup>*Q*<sup>∗</sup>

*Continuously at each sample point:*

turn unstable.

*Preliminaries:*

to reduce *f* <sup>∗</sup>

section),

respectively,

**Remark**

(Section 6),

*Long-term tuning:*

the control system:

trajectories to a residual set B<sup>∗</sup>

initial conditions and for *h* >> 0, or similarly for *Td* large, the adaptive control system may

In conclusion, when comparing two digital controllers, the sensitivity of the stability to *h* and indirectly the presence of large pure delays in the dynamics, is fundamental to draw out

1) Estimate a lower bound *M* , for instance *M* = *Mb* (Jordán & Bustamante, 2011),

3) Choose design gain matrices *Kp* and *Kv* according to (56)-(57), and simultaneously in order

5) Stipulate the desired sampled-data path references for the geometric and kinematic trajectories in 6 DOF´s: *ηrtn* and *vrtn* , respectively (see related commentary in previous

6) Calculate the control thrust *τ<sup>n</sup>* with components *τ*1*<sup>n</sup>* in (29) and *τ*2*<sup>n</sup>* (40) (or (60)),

For the present approach, we can summarize the different steps carried out in this Chapter after the control design in order to determine its convergence properties and performance of

a) Establishment of the adaptive laws for the designed controller using a lower bound of *M*

b) Stability and convergence analysis of the control system to a residual set dependent of the sign-undefinite terms *f*∆*Q*<sup>1</sup> and *f*∆*Q*<sup>2</sup> in (31) and (42), respectively, which depend on the pure dead-time *Td* (Section 7). Moreover, the conjoint incidence of local model errors *εηn*+<sup>1</sup> and

<sup>0</sup> is illustrated in (Section 7.1).

*M* in the the convergence of state

7) Calculate the adaptive controller matrices (44) with the lower bound *M* instead of *M*,

7) Redefine *Kp*, *Kv* and *h* in order to achieve optimal tracking performance.

*<sup>ε</sup>vn*<sup>+</sup><sup>1</sup> , measure noises *<sup>δ</sup>ηtn* and *<sup>δ</sup>vtn* , and the product *<sup>M</sup>*<sup>−</sup><sup>1</sup>

*tn* (see related commentary in previous section),

With the end of illustrating the features of our control system approach, we simulate a path-tracking problem in 6 DOF´s for an underwater vehicle in a planar motion with some sporadic immersions to the floor.

A continuous-time model of a fully-maneuverable underwater vehicle is employed for the numerical simulations and a pure dead time in the control transmission was included. Details of this dynamics are given in (Jordán & Bustamante, 2009a).

We present the simulation of the adaptive control algorithm summarized in the previous section for an immersion in the depth (motion in the heave modus *z*) with translations aside in the modes surge *x* and sway *y* simultaneously.

Fig. 2 illustrates the evolution of the six controlled modes of the UUV in the descending. The initial conditions of the UUV at *t*<sup>0</sup> were *η*(0)=[1.1; 1.1; 1.1; 0.1; 0.1; 0.1] and *v*(0)=[0.1; 0.1; 0.1; 0.1; 0.1; 0.1]. The design matrices were set in *Kp* = *Kv* = *I*. The sampling period was selected *h* = 0.1 s.

It is seen that after a short transient, which is not longer than 7 s, the UUV is positioned at the coordinates to the start point. Then it begins to the maneuver of descending. In the meantime the matrices *Ui*'s are adapted and the path tracking result asymptotically convergent with unappreciated stationary errors. The predictors are sufficiently accurate during the evolution of the states.

In the simulation, between 20 s and 30 s, it can be seen a high interaction between the traslational modes which are tracked (namely: *x*, *y*, *z*) and the rotational modes which are regulated about zero only (namely: *ϕ*, *θ*, *ψ*). In the last ones, it one observes a significant variation of these magnitudes. However, this is reasonable considering the dynamics with large dead-times we are dealing with.
