Date of final exam: 26/03/1999

E-mail: lmarconi@deis.unibo.it

Tutor: Prof.  C. Bonivento, Università di Bologna

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Tracking and Regulation of Linear and Nonlinear Systems   ___________________________________________________________________________________________________________

Advisor:

Prof.  C. Bonivento, Università di Bologna

Summary of the thesis

The problem of controlling the output of a dynamical system in order to achieve desired objectives is, without doubt, one of the arguments most addressed in the control literature. The investigation of this control problem is motivated by the existence of several real applications which can profit by the development of a congruent and easy-to-apply theory. The problem of tracking a profile with a machine tool, tracking a route with an aircraft or a trajectory with a robot are just simple examples which justify the above sentence.
The overall work is divided in four Chapters organized as follows. In the first Chapter the problem of right invertibility of linear and nonlinear systems is addressed. As far as linear systems are concerned, typical geometric tools, like the maximum controlled invariant subspace contained in a given subspace and its dual the minimum conditioned invariant subspace containing a given subspace, are adopted in order to introduce simple new algorithms useful to test the right invertibility and to compute the generalized relative degree. This (apparently not justified) attempt aiming to reformulate results which are known from several years is indeed justified, besides by the computational aspects testifyed by the introduction of easy-to-implement algorithms, also by the fact that this new perspective allows a meaningful insight with respect to the problem of perfect tracking. As far as nonlinear systems are concerned, after having just sketched an extension of the theory developed for linear systems in a nonlinear framework, it is shown how recently introduced input-output normal forms (called triangular generalized normal forms) can be helpful for constructively inverting a particular class of nonlinear systems.
The results presented in the previous Chapter are completed in Chapter 2 by addressing also stability issues. It is shown how a stable right inverse can be computed for nonminimum-phase nonlinear systems by resorting noncausal techniques. After the introduction of the concepts of preaction and preview which arise in the problem of perfectly tracking a generic sufficiently smooth reference trajectory, the case in which the latter is provided by an autonomous system (exosystem) is developed. It is shown how, by means of nonlinear geometric tools like the concept of zero dynamics submanifold of a nonlinear system, it is possible to set up an elegant and a meaningful geometric framework where the solution of the problem can be provided. Furthermore, by concentrating only on linear systems, the problem of perfect tracking with stability in a discrete-time fashion is also solved. The solution which is provided draws inspiration by the nonlinear analysis previously developed so that a compact theory for continuous and discrete-time case can be established.
The third Chapter is devoted in the first part to present the main results concerning with the problem of asymptotic tracking. The solution is presented by introducing the well-known generalization of the internal model principle in a nonlinear setting. The geometric interpretation behind the solution of the problem will be stressed by comparing the results obtained by solving the asymptotic and the perfect tracking problem. By taking advantage of the theory presented in the previous Chapters, and in particular by the computation of the inverse in case of systems in triangular generalized normal forms, a constructive solution of the regulator is presented. The second part of the Chapter is focused on the problem of asymptotic tracking in presence of unmodelled matched disturbances. By means of (discontinuous) sliding mode techniques, the problem is solved by employing, in sintony with the smooth solution, the powerful center manifold theory.
All the theory previously developed flows in the last Chapter in order to formulate and solve the Asymptotically Robust Perfect Tracking Problem. Again by strongly using geometric tools, the effectiveness of the control scheme in which an inversion-based feedforward action (designed according to the nominal model of the plant) is injected in an internal model-based feedback loop is investigated. It is shown that in case of systems/exosystems which exhibit a nonlinear internal model, this control scheme fails to solve the Asymptotically Robust Perfect Tracking Problem in the sense that, in case the real model of the plant is different from the nominal one, then the asymptotic robustness usually guaranteed by the feedback loop is lost. To avoid this unpleaseant situation a different solution, which is based on a suitable initialization of the internal model embedded in the regulator and on a slight modification of the feedforward signal, is proposed.
 

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