**1. Introduction**

[149] Wang, S. H., & Davison, E. (1973). On the stabilization of decentralized control sys‐

[150] Davison, E. J., & Chang, T. N. (1990). Decentralized stabilization and pole assignment for general proper systems. *IEEE Transactions on Automatic Control*, 35, 652-664.

[151] Du, X., Xi, Y., & Li, S. (2001). Distributed model predictive control for large-scale sys‐

[152] Camponogara, E., Jia, D., Krogh, B. H., & Talukdar, S. (2002). Distributed model pre‐

[153] Dunbar, W. B. (2007). Distributed receding horizon control of dynamically coupled nonlinear systems. *IEEE Transactions on Automatic Control*, 52(7), 1249-1263.

[154] Dunbar, W. B., & Murray, R. M. (2006). Distributed receding horizon control for mul‐

[155] Li, S., Zhang, Y., & Zhu, Q. (2005). Nash-optimization enhanced distributed model predictive control applied to the Shell benchmark problem. *Information Sciences*,

[156] Richards, A., & How, J. P. (2007). Robust distributed model predictive control. *Inter‐*

[157] Venkat, A. N., Rawlings, J. B., & Wright, S. J. (2007). Distributed model predictive control of large-scale systems. In: Proceedings of the assessment and future direc‐ tions of nonlinear model predictive control. Berlin Heidelberg, Springer, 591-605.

[158] Liu, J., Muñoz la, Peña. D., Christofides, P. D., & Davis, J. F. (2009). Lyapunovbased model predictive control of nonlinear systems subject to time-varying measurement

[159] Muñoz la, Peña. D., & Christofides, P. D. (2008). Lyapunov-based model predictive control of nonlinear systems subject to data losses. *IEEE Transactions on Automatic*

[160] Franco, E., Magni, L., Parisini, T., Polycarpou, M. M., & Raimondo, D. M. (2008). Co‐ operative constrained control of distributed agents with nonlinear dynamics and de‐ layed information exchange: a stabilizing receding-horizon approach. *IEEE*

[161] Liu, J., de la Peña, D. M., & Christofides, P. D. (2010). Distributed model predictive control of nonlinear systems subject to asynchronous and delayed measurements.

delays. *International Journal of Adaptive Control and Signal Processing*, 23-788.

tems. In: Proceedings of the American control conference , 4, 3142-3143.

tems. *IEEE Transactions on Automatic Control*, 18, 473-478.

dictive control. *IEEE Control Systems Magazine*, 22, 44-52.

ti-vehicle formation stabilization. *Automatica*, 42(4), 549-558.

*national Journal of Control*, 80(9), 1517-1531.

*Transactions on Automatic Control*, 53, 324-338.

170(2-4), 329-349.

116 Advances in Discrete Time Systems

*Control*, 53, 2076-2089.

*Automatica*, 46, 52-61.

Accordingly to modern digital technology in control communications and optic fiber, unmanned underwater vehicles (UUV) are usually controlled digitally. In the case of remotely operated vehicles (ROV), partial control systems are useful, for instance for the roll-pitch stabilization, as long as the main modes of motion are performed basically by teleoperation. In the case of autonomous underwater vehicles (AUV) on the contrary, the control of modes is complete, it means, the advance, the immersion and the roll-pitch stabilization are carried out automatically with a large degree of self-decision both in guidance and control.

In all cases, digital communications between the controller, the guide system and the navigation system are often affected by a pure delay in the control action, which correctively manipulates the vehicle behavior with lateness through their thrusters. Generally speaking, the delay is variable because of the commonly sophisticated nature of protocols involved in the usual communication standards, for instance in the well-widespread protocols RS422/485. In many cases, the delay problem is much more complex and embraces a pure delay for the sensor instruments and other quite different ones for the controller and guidance communications. Moreover, sensors may have different delays each one, due to different hardware and baud rates in data transmission.

Pure delays can influence significantly the stability of UUV's, principally in fast motions like in the modes of pitch and roll, causing, in extreme fall, the capsize of the vehicle. Additionally, the well-known strong interaction among the modes in the dynamics of UUV's may cause large oscillations of the pitch modes that are induced by acelerations in the advance mode. Thus, complex control systems for the whole 6-degrees-of freedom (DOF) dynamics are much more preferable than many single-mode controllers.

©2012 Jordan et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 A. Jordan et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 A. Jordan et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Dealing with pure delays in the controller design, a Smith-predictor solution stays in mind at the first place. However, the needs of a precise model to construct the predictor upon the usually uncertain and potentially unstable dynamics of an UUV, makes this alternative quite unfeasible (Leonard and Abba, 2012). On the other way, simple controllers like for instance PID controllers, are viable to be tuned with less dynamics information but generally can not counteract by itself the undesirable effects of relatively large pure delays. Adaptive controllers had proved to work properly in these scenarios with many advantages (Jordán and Bustamante, 2011). Nevertheless, the roll played by pure delays in the stability and performance of control systems for underwater vehicles is much less dealt with in the specialized literature.

From now on, brackets are employed to indicate functional dependence and parenthesis to denote common factor. Besides vectors are indicated in bold, variables in italics and matrices

Stability Analysis of Nonlinear Discrete-Time Adaptive Control Systems with Large Dead-Times

Moreover, we will concentrate henceforth on disturbed measures *η<sup>δ</sup>* and *vδ*, and not on exogenous perturbations *τ<sup>c</sup>* and *vc*, so we have set *τc*=*vc*=0 throughout the Chapter. For more explanations about the influence of *τc* and *vc* on adaptive guidance systems see (Jordán

For the continuous-time dynamics there exists an associated exact sampled-data dynamics

times *ti* with a sampling rate *h*. When measures are affected with noise values {*δηtn* ,*δvtn* },

Since the pure delay period is supposed to be originated in the control communication

which is saying that *d* is a variable integer, while *d*<sup>0</sup> is a constant positive integer and *δd* a sign-undefined integer representing a perturbation that fulfills *d*<sup>0</sup> ≥ |*δd*| ≥ 0 and *d* can range between 0 and 2*d*0. Another feature of the communication hardware is that the sample times *ti* of the instrument (from Gyro and DVL for instance) are indicated together with the samples, and so, when these are transmitted to the controller, the calculation of *d* is possible.

*p*[*η*,*v*]+*M*<sup>−</sup><sup>1</sup>

with *p* and *q* being Lipschitz vector functions located at the right-hand memberships of the (1) and (2), respectively. Here no exogenous perturbation was considered as agreed above. For further analysis, we can state a model-based predictor for one step or more steps ahead. To this end it might employ high order approximators like Adams-Bashforth types, though for perturbed states a simple Euler approximator is more convenient (Jordán and Bustamante,

system, we will assume in some particular scenarios that *Td* in (1), satisfies

*<sup>v</sup>*˙ <sup>=</sup> *<sup>M</sup>*<sup>−</sup><sup>1</sup>

*<sup>v</sup>n*+<sup>1</sup> = *<sup>v</sup>tn*+*δvtn*+*hM*−<sup>1</sup>

where *ηn*+<sup>1</sup> and *vn*+<sup>1</sup> are one-step-ahead predictions. Herein it is valid with (1)-(2)

= {*ηtn*+*δηtn* ,*vtn*+*δvtn* } instead.

,*vti* } for the states *η*[*t*] and *v*[*t*] at sample

http://dx.doi.org/10.5772/52224

119

*τ* (4)

(6)

*Td* = *d h* = *d*<sup>0</sup> *h* + *δd h*, (3)

*η*˙ = *q*[*η*,*v*], (5)

*pδtn* +*τn*−*<sup>d</sup>*

*ηn*+<sup>1</sup> = *ηtn*+*δηtn*+*hqδtn* , (7)

Notice from (1) the nonlinear dependence of *C*, *D* and *g* with the states *v* and *η.*

and Bustamante, 2008; Jordán and Bustamante 2007), respectively.

described by the set of sequences {*η*[*ti*], *v*[*ti*]} = {*ηti*

Now, let us rewrite the ODE (1)-(2) in a more compact form

 *ηδt i* ,*vδ<sup>t</sup> i* 

in capital letters.

**3. Sampled-data behavior**

we use {*ηδ*[*ti*],*vδ*[*ti*]} =

2009b). So

In this Chapter we focus the design of a 6-degrees-of-freedom adaptive controller for UUV's directly in the sample-time domain. The controller pertains to the class of speed gradient adaptive control systems. This controller was developed in (Jordán and Bustamante, 2011) and shows clearly advantages in stability over the same digital speed-gradient controller which is first designed in continuous time and finally translated to discrete time. One starts from the fact that the UUV dynamics, together with the control communication link in the feedback, involves a considerable delay. In this work, we depart from a hypothesis that an optimal sampling time for the control stability should have some upper limit for the stability in the presence of pure delays, disturbances in the samples and rapid desired maneuvers. To support our hypothesis, we close the analysis with a classification of certain influence variables on the stability and performance of the proposed digital adaptive control system.
