150 research outputs found
Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking
This paper addresses the sensitivity analysis for hybrid systems with
discontinuous (jumping) state trajectories. We consider state-triggered jumps
in the state evolution, potentially accompanied by mode switching in the
control vector field as well. For a given trajectory with state jumps, we show
how to construct an approximation of a nearby perturbed trajectory
corresponding to a small variation of the initial condition and input. A major
complication in the construction of such an approximation is that, in general,
the jump times corresponding to a nearby perturbed trajectory are not equal to
those of the nominal one. The main contribution of this work is the development
of a notion of error to clarify in which sense the approximate trajectory is,
at each instant of time, a firstorder approximation of the perturbed
trajectory. This notion of error naturally finds application in the (local)
tracking problem of a time-varying reference trajectory of a hybrid system. To
illustrate the possible use of this new error definition in the context of
trajectory tracking, we outline how the standard linear trajectory tracking
control for nonlinear systems -based on linear quadratic regulator (LQR) theory
to compute the optimal feedback gain- could be generalized for hybrid systems
Trajectory tracking in switched systems: an internal model principle approach: the elliptical billiard system as a benchmark for theory
Sistemi dinamici caratterizzati dall'interazione tra dinamiche continue e discrete sono detti sistemi ibridi. Un sistema switched è un particolare sistema
ibrido costituito da una famiglia di sottosistemi a tempo continuo e da una
legge che ne regola le transizioni. Questi sistemi hanno numerose applicazioni
nel controllo di sistemi meccanici, nell'industria automobilistica e aeronautica,
nel controllo del traffico aereo, nell'elettronica di potenza, etc.
Questa tesi sarà incentrata sul problema dell'inseguimento asintotico di traiettoria per sistemi switched. Nella prima parte, il problema di inseguimento è stato propriamente definito e risolto prendendo in esame il sistema biliardo
ellittico. Al fine di definire una classe di traiettorie di riferimento ammissibili per il sistema biliardo un problema di pianificazione di traiettoria è stato
approntato e risolto attraverso l'utilizzo di risultati della teoria dei polinomi non negativi e tecniche LMI. Il problema di inseguimento in presenza di incertezze
nei parametri del sistema è stato considerato e risolto sia nel caso di feedback
dallo stato che dalla sola posizione. Nella seconda parte della tesi i risultati
ottenuti per il sistema biliardo sono stati generalizzati per una classe di sistemi
switched con dinamica lineare in ogni modo operazionale, mappe di reset lineari
e dimensione dello spazio di stato possibilmente variabile tra i vari modi. In tutti
i casi la strategia di controllo proposta è basata su una versione discontinua del
principio del modello interno.Dynamical systems that are described by an interaction between continuous
and discrete dynamics are called hybrid systems. Their evolution is generally
given by equations of motion containing mixtures of logic, discrete-valued or digital dynamics, and continuous-variable or analog dynamics. A switched system
is a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. These systems
have numerous applications in control of mechanical systems, automotive industry, aircraft and air traffic control, switching power converters, and many
others.
This thesis will focus on the problem of asymptotic trajectory tracking for
switched systems. First, the tracking control problem is properly stated and
solved for a controlled elliptical billiard system. In order to find an admissible
class of reference trajectories inside the billiards a motion planning problem has
been solved by using results from the theory of non-negative polynomials and
LMIs techniques. The trajectory tracking problem in presence of uncertainties
on the plant parameters has been also considered and solved in both cases of
state-feedback and output-feedback. In the second part, the results obtained for
the billiard system are generalized for a class of switched systems having linear
dynamics in each operating mode, linear reset maps and possible nonuniform
state space among the different modes. In all cases the proposed control strategy
is based on a dynamic compensator, whose state is subject to discontinuities and
whose structure is based on a nonsmooth version of the internal model principle
Nonlinear H_inf -Control of Mechanical Systems under Unilateral Constraints on the Position
6 pagesNational audienceThe work focuses on the study of hybrid mechanical systems under unilateral constraints on the position. The problem of robust control of mechanical systems is addressed under unilateral constraints by designing a nonlinear H-infinity -controller developed in the nonsmooth setting, covering impact phenomena. Performance issues of the nonlinear H-infinity-tracking controller are illustrated in a numerical simulation
Comments on "Control of a Planar Underactuated Biped on a Complete Walking Cycle"
International audienceThe above paper [1] possesses several approximations and flaws, which we try to explain. Roughly, the topic concerns the problem of trajectory tracking for a class of mechanical Lagrangian systems subject to unilateral constraints on the generalized position (q) 0, (q) 2 IR m. Such multibody mechanical systems also involve a com-plementarity relation between the constraint and a Lagrange multiplier 0 (q) ? 0 (1), and generalized velocity jumps (impacts). The complementarity relations and the velocity jump law, form a specific contact model. A contact model is necessary for the chosen model to be meaningful from a mechanical point of view. When dealing with systems of rigid bodies, the complementarity conditions are the simplest way to deal with the contact dynamics: they state that adhesion or magnetic forces are excluded from the model. Such nonsmooth mechanical systems form a special class of complementarity systems, but other formalisms exist [7]. It is worth noting that the complementarity conditions are not included in the model presented in [1], which is therefore incomplete. Specifically, the authors deal with a particular biped robots model that fits within a class of impulsive ODEs, or measure differential equations. We will come back on this later in this note. The tracking problem is examined when the system undergoes an infinity of cycles, each cycle being composed of three phases of motion: single-support phase, double-support phase, and the impact when the feet hit the ground. Apart from possible underactuation, the problem is quite similar to what is tackled in [2]–[5], that concerns fully actuated Lagrangian systems undergoing cycles which consist of free motion phases, constrained motion phases, and transition phases with impacts. The effects of the impacts and of the complementarity relations do not change from one problem to the other one. This is why it is worth understanding the simplest case before tackling more sophisticated control problems (underactuated systems, flexible joint manipulators, to cite a few). It is worth noting that the infinity of cycles (and consequently Manuscript 1 The symbol ? means that (q) and have to be orthogonal one to each other. Since they are both non-negative, this is equivalent to the componentwise relation 0 (q), 0 , (q) = 0 for all 1 i m
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