New Robust Tracking and Stabilization Methods for Significant Classes of Uncertain Linear and Nonlinear Systems

There exist many mechanical, electrical, electro-mechanical, thermic, chemical, biologicaland medical linear and nonlinear systems, subject to parametric uncertainties and nonstandard disturbances, which need to be efficiently controlled. Indeed, e.g. consider thenumerous manufacturing systems (in particular the robotic and transport systems,…) andthe 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), thefollowing 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 singlespecification;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 trackgeneric references, provided that sufficiently regular, e.g. the generally continuouspiecewise linear signals, easily produced by using digital technologies, new theoreticalresults are needful for the scientific and engineering community in order to design controlsystems with non standard references and/or disturbances and/or with ever harderspecifications.In the first part of this chapter, new results are stated and presented; they allow to design acontroller of a SISO process, without zeros, with measurable state and with parametricuncertainties, such that the controlled system is of type one and has, for all the possibleuncertain parameters, assigned minimum constant gain and maximum time constant orsuch that the controlled system tracks with a prefixed maximum error a generic referencewith limited derivative, also when there is a generic disturbance with limited derivative, hasan 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 (Serajand Tarokh, 1977), (Freeman and Kokotovic, 1995) and they are based on the choice of asuitable set of reference poles, on a proportionality parameter of these poles and on thetheory of externally positive systems (Bru and Romero-Vivò, 2009).The utility and efficiency of the proposed methods are illustrated with an attractive andsignificant 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 arestated which easily allow to determine robust control laws of the PD type, with a possiblepartial compensation, in order to force the output and its derivative to go to rectangular neighbourhoods (ofthe origin) with prefixed areas and with prefixed time constants characterizing theconvergence of the error. Clearly these results allow also designing control laws to take andhold a generic articulated system in a generic posture less than prefixed errors also in thepresence of parametric uncertainties and limited disturbances.Moreover the stated theorems can be used to determine simple and robust control laws inorder to force the considered class of systems to track a generic preassigned limited in“acceleration” trajectory, with preassigned majorant values of the maximum “positionand/or velocity” errors and preassigned increases of the time constants characterizing theconvergence of the error