Nonlinear output regulation: a reasoned overview and new developments Lorenzo Marconi C.A.SY. / D.E.I.S. - University of Bologna - Italy 1 Lorenzo Marconi - MTNS – Kyoto 2006
Outline � The framework of output regulation � Recent applications � A reasoned overview of recent results: the “Asymptotic Internal Model Property’’ as unifying property � Relationship between design of nonlinear internal models and nonlinear observers: immersion vs. observability � A recent result: output regulation without immersion � Practical design of the regulator: Uniform practical output regulation � Conclusive remarks and future developments 2 Lorenzo Marconi - MTNS – Kyoto 2006
The framework of output regulation 3 Lorenzo Marconi - MTNS – Kyoto 2006
The framework Reference and/or exosystem disturbance generator Regulated outputs Plant Measured output 4 Lorenzo Marconi - MTNS – Kyoto 2006
The framework invariant Reference and/or disturbance generator Plant 5 Lorenzo Marconi - MTNS – Kyoto 2006
The framework invariant Reference and/or disturbance generator Plant Lorenz attractor 6 Lorenzo Marconi - MTNS – Kyoto 2006
The framework invariant Plant The problem of semiglobal output regulation � Boundedness of closed-loop trajectories � (Uniform) converge of the error to zero for any initial condition in known compact sets (semi-global) 7 Lorenzo Marconi - MTNS – Kyoto 2006
The conceptual formulation The peculiarity of the framework is the characterization of the class of all possible exogenous inputs (disturbances/ references) as the set of all possible solutions of a fixed (finite-dimensional) differential equation (Set Point Control) The exosystem-generated disturbances/references seems to be the right tradeoff between: � worst case disturbance (H oo control): too pessimistic � exact knowledge of disturbance/reference (inversion-based control): too optimistic 8 Lorenzo Marconi - MTNS – Kyoto 2006
The exosystem � The issue of robustness implicitly considered in the framework : � Case of particular interest: linear exosystem � Extension (unknown frequencies � Nonlinear exosystem) Lightly nonlinear ( Non linear Adaptive Regulation ) 9 Lorenzo Marconi - MTNS – Kyoto 2006
10 Lorenzo Marconi - Applications MTNS – Kyoto 2006
Application 1: asymmetries compensation in control of rotating machines Mechanical and/or electrical asymmetries in rotating machines (resulting from wear, damage, construction defect) can be modeled – in many cases – as periodically varying disturbances. Example: � rotor faults in induction motors Problem: design feedback controllers to offset disturbances of the form with unknown frequency amplitude and phase. C. Bonivento, A. Isidori, L. Marconi, A. Paoli, AUTOMATICA, 2003 11 Lorenzo Marconi - MTNS – Kyoto 2006
Application 2: Shunt Active Filters The Problem: To reduce the “ Harmonic Pollution ” in the electric grids due to nonlinear loads which Determine power losses and the risk of equipment damnage Control Problem: Control the inverter switches in order to inject currents into the mains to compensate for the load higher harmonics L. Marconi, F. Ronchi, A. Tilli, AUTOMATICA, 2006 12 Lorenzo Marconi - MTNS – Kyoto 2006
Application 3: Disturbance compensation in left ventricular assist devices (project steered by Brad Paden) Left Ventricular Assist Devices are implanted to cooperate with the natural heart in pumping blood. New generation pumps are feedback-controlled magnetically-levitated pumps 13 Lorenzo Marconi - MTNS – Kyoto 2006
Application 3: Disturbance compensation in left ventricular assist devices (project steered by Brad Paden) The control challenge is to magnetically levitate and rotate a pump impeller in the blood stream while minimizing pump size, blood damage, battery size and system weight. One of the main problems is the fact that the pump of the natural heart creates quasi-periodic load on the LVAD levitation system. 14 Lorenzo Marconi - MTNS – Kyoto 2006
Application 4: Automatic landing on a oscillating ship 15 Lorenzo Marconi - MTNS – Kyoto 2006
Application 5: Pursuit-evasion in urban environment The control problem: the pursuer (an helicopter) must track an evader (say, a car ) which follows a trajectory of this kind. Headings, turning points, radii of curvature, velocity, acceleration are not known, and must be estimated in real time. pursuer evader Prototype of urban canyon A. Isidori, L. Marconi, NOLCOS, 2004 16 Lorenzo Marconi - MTNS – Kyoto 2006
A reasoned overview of recent results 17 Lorenzo Marconi - MTNS – Kyoto 2006
The class of systems Since the problem at hand includes, as a particular case, a problem of semiglobal stabilization (w= 0), it makes sense to restrict the attention on the class of systems on which well-established stabilization techniques have been developed In this talk: Nonlinear Systems globally diffeomorphic to normal forms (well-defined relative degree) Furthermore, in the spectrum of possible stabilization techniques proposed so-far in literature, we are particularly interested to high-gain techniques 18 Lorenzo Marconi - MTNS – Kyoto 2006
Of course, this is our choice. A number of others class of systems and associated stabilization techniques can be selected for design purposes. Besides others let us recall the class of systems diffeomorphic to systems in the adaptive nonlinear form and adaptive stabilization techniques for which a bunch of output regulation theory have been developed: • Marino, Santosuosso and Tomei • Ding • Serrani and Isidori • Huang • …. and others 19 Lorenzo Marconi - MTNS – Kyoto 2006
Normal form – relative degree= r Zero dynamic Chain of integrator High frequency gain 20 Lorenzo Marconi - MTNS – Kyoto 2006
Normal form r.d.= 1 h.f.g.= 1 Intention: to adopt high-gain arguments as customary stabilization tool Minimum phase requirement 21 Lorenzo Marconi - MTNS – Kyoto 2006
A “weak” minimum-phase requirement Byrnes, Isidori TAC 03 0 = 0 The trajectories of this system originating from are uniformly bounded 22 Lorenzo Marconi - MTNS – Kyoto 2006
Steady state for nonlinear systems A fundamental step in the solution of the regulator problem is the notion of steady state for nonlinear systems Byrnes - Isidori TAC 03 A possible notion has been given in based on the concept of omega limit set of a set Applied to our zero dynamics , under the weak minimum phase assumption, the notion can be explained as follows 23 Lorenzo Marconi - MTNS – Kyoto 2006
Boundedness of trajectory originating from 24 Lorenzo Marconi - MTNS – Kyoto 2006
Boundedness of trajectory originating from Existence of a compact Invariant set which uniformly attracts the trajectories from 25 Lorenzo Marconi - MTNS – Kyoto 2006
Boundedness of trajectory originating from Existence of a compact Invariant set which uniformly attracts the trajectories from Precisely, the omega limit set of (Hale, Magalhaes, Oliva, Dynam. in Infinite Dimen. Syst. , Springer Verlag) 26 Lorenzo Marconi - MTNS – Kyoto 2006
0 = 0 In our framework it makes sense to define (Isidori-Byrnes TAC03) Steady state set (locus): Steady state: trajectories of 27 Lorenzo Marconi - MTNS – Kyoto 2006
Two main issues: internal model and stabilizability property � Internal model property: capability of reproducing the steady state input needed to keep the regulated error to zero 28 Lorenzo Marconi - MTNS – Kyoto 2006
Internal Model Property � Internal model property: capability of reproducing the steady state input needed to keep the regulated error to zero 0 = 0 0 0 0 0 29 Lorenzo Marconi - MTNS – Kyoto 2006
Internal Model Property � Internal model property: capability of reproducing the steady state input needed to keep the regulated error to zero 0 + 0 - 0 0 30 Lorenzo Marconi - MTNS – Kyoto 2006
Stabilizability Property � Stabilizability property: capability of stabilizing the closed- loop system on a compact attractor on which the error is zero. 31 Lorenzo Marconi - MTNS – Kyoto 2006
Two main issues: internal model and stabilizability property The two properties are interlaced. � In particular the ability of achieving the stabilizability property is strongly affected by how the internal model property is achieved. � It turns out that it is possible to capture the essential properties which must be achieved in order to be able to design the regulator into the so-called: The asymptotic internal model property 32 Lorenzo Marconi - MTNS – Kyoto 2006
The asymptotic internal model property The triplet is said to have the ASYMPTOTIC INTERNAL MODEL PROPERTY if there exists a function such that : (i) related 33 Lorenzo Marconi - MTNS – Kyoto 2006
The asymptotic internal model property The triplet is said to have the ASYMPTOTIC INTERNAL MODEL PROPERTY if there exists a function such that : (i) + - yields 34 Lorenzo Marconi - MTNS – Kyoto 2006
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