New robust tracking and stabilization methods for significant classes of uncertain linear and nonlinear systems

Introduction There exist many mechanical, electrical, electro-mechanical, thermic, chemical, biological and medical linear and nonlinear systems, subject to parametric uncertainties and non standard disturbances, which need to be efficiently controlled. Indeed, e.g. consider the numerous manufacturing systems (in particular the robotic and transport systems,…) and the more pressing requirements and control specifications in an ever more dynamic society. Despite numerous scientific papers available in literature (Porter and Power, 1970)-(Sastry, 1999), some of which also very recent (Paarmann, 2001)-(Siciliano and Khatib, 2009), the following practical limitations remain: 1. the considered classes of systems are often with little relevant interest to engineers; 2. the considered signals (references, disturbances,…) are almost always standard (polynomial and/or sinusoidal ones); 3. the controllers are not very robust and they do not allow satisfying more than a single specification; 4. the control signals are often excessive and/or unfeasible because of the chattering. By taking into account that a very important problem is to force a process or a plant to track generic references, provided that sufficiently regular, e.g. the generally continuous piecewise linear signals, easily produced by using digital technologies, new theoretical results are needful for the scientific and engineering community in order to design control systems with non standard references and/or disturbances and/or with ever harder specifications. In the first part of this chapter, new results are stated and presented; they allow to design a controller of a SISO process, without zeros, with measurable state and with parametric uncertainties, such that the controlled system is of type one and has, for all the possible uncertain parameters, assigned minimum constant gain and maximum time constant or such that the controlled system tracks with a prefixed maximum error a generic reference with limited derivative, also when there is a generic disturbance with limited derivative, has an assigned maximum time constant and guarantees a good quality of the transient. The proposed design techniques use a feedback control scheme with an integral action (Seraj and Tarokh, 1977), (Freeman and Kokotovic, 1995) and they are based on the choice of a suitable set of reference poles, on a proportionality parameter of these poles and on the theory of externally positive systems (Bru and Romero-Vivò, 2009). The utility and efficiency of the proposed methods are illustrated with an attractive and significant example of position control. In the second part of the chapter it is considered a significant class of uncertain pseudo-quadratic systems subject to additional nonlinearities and/or external signals. For this class of systems, including articulated mechanical systems, several theorems are stated which easily allow to determine robust control laws of the PD type, with a possible partial compensation, in order to force the output and its derivative to go to rectangular neighbourhoods (of the origin) with prefixed areas and with prefixed time constants characterizing the convergence of the error. Clearly these results allow also designing control laws to take and hold a generic articulated system in a generic posture less than prefixed errors also in the presence of parametric uncertainties and limited disturbances. Moreover the stated theorems can be used to determine simple and robust control laws in order to force the considered class of systems to track a generic preassigned limited in “acceleration” trajectory, with preassigned majorant values of the maximum “position and/or velocity” errors and preassigned increases of the time constants characterizing the convergence of the error.