Stability Analysis of Nonlinear Discrete-Time Adaptive Control Systems with Large Dead-Times - Theory and a Case Study

MARIO A. JORDÁN; JORGE L. BUSTAMANTE

Advances in Discrete Time

Intech

Lugar: Rijeka; Año: 2011;

Nonlinear discrete-time adaptive control systems have found a fruitful field of application in complex nonlinear systems for a long time. However, the most techniques encountered in the praxis, result from pure translations of existing continuous-time based algorithms for controller design into the discrete-time domain. Recently, there has been theoretical evidence that the direct controller designs in the discrete-time domain are more robust than digital counterparts obtained by translations from the continuous time-domain (indirect approach). Particularly, the important role played by the sampling time has been proved to be a destabilizing factor when it is chosen relatively large in both approaches. Relations between physical parameters of the system with the sampling time could also be globally given for an application in order to understand this phenomenon of instability more clearly. A highly important aspect in the design and implementation of digital adaptive control systems is the presence of dead times, which may be inherent to the process dynamics to be steered or, more frequently, they appear inevitably in the communication process between digital sensors and controller. It is well known that such dynamics are difficult to stabilize with simple controllers, after all by complex multivariable dynamics with oscillating behaviour. In this sense more refined performance requires unavoidably a model of the process. Due to the usual difficulties to perform the modelling process successfully and the appearance of uncertainties in the model, adaptive controllers emerge as an ideal approach to overcome these scenarios and to give satisfactory results with significant much little effort in the modelling. Another useful property of the adaptive systems is related to the auto-tuning property that enables the control system to acquire a high performance after a short commissioning phase in the presence of an unknown complex dynamics. Increasingly spreading by complex digital control systems is the implementation of a sensorics in where all sensors and actuators are interconnected through one or more communication buses with the controller. The dynamics of the whole system may be characterized usually by one or more dominant delays, which also may be variable in time. The analysis of features in such control systems with adaptive properties is an actual concern in the specialized community. Since many digital control systems are approximate approaches of their continuous-time counterparts, or they are built up onto sampled information of the dynamics, they are in consequence potentially unstable, after all for relatively large sampling times. They are also other influencing parameters such as delays and dynamics uncertainties. So the analysis of stability must include the construction of sensitivity functions of the performance for parameters of the ordinary differential equation system (ODES) on which the underlying dynamics is based. In the case of adaptive controllers, these influencing parameters are selected for analysis only, because many of them in the ODES are no transparent in the implementation. This analysis is crucial in order to gain a scope of what parameters may destabilize the control if their ranges of magnitude are not taken into account properly at the moment the approach is set up. In this chapter we face the analysis of digital adaptive controller for complex multivariable dynamics with dead times in the both control goals: tracking and regulation controls. Therein, the two approaches mentioned above, namely the direct and indirect designs are contrasted comparatively. The present description deals with Lyapunov-based techniques in where energies of errors in path tracking position and prescribed kinematics are influenced by control actions and adaptive laws in order to reduced them continuously in some norm after the appearance of parametric perturbations and to minimize them within certain bounds. In order to take pure delays into account, predictions of dynamics states are necessary in order to be able to construct positional and kinematic tracking errors. These predictions are developed with help of the adaptive laws, in where no model information is necessary. The predictor shows the property of converge in time to a bounded region in the state space. Also the boundness of the attraction domain of convergence is proved in terms of a set of influencing parametric variables. Particularly the loss of boundness is traced for the dependence of the state trajectories with the relation dead-time/samplingtime. Additionally some temporal changes of the dead time are taken into account and introduced in the analysis later. Finally, the whole development carried out in this chapter is globally oriented to the class of mechanic and hydrodynamic systems. Herein, autonomous underwater vehicles are taken as a case study of this class. Simulation results pretend to illustrate the developed results of the analysis.