Nonlinear output regulation: a reasoned overview and new - - PowerPoint PPT Presentation

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Lorenzo Marconi -

Nonlinear output regulation: a reasoned overview and new developments

Lorenzo Marconi

C.A.SY. / D.E.I.S. - University of Bologna - Italy

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Outline

The framework of output regulation A reasoned overview of recent results:

the “Asymptotic Internal Model Property’’ as unifying property

A recent result: output regulation without immersion Conclusive remarks and future developments Practical design of the regulator:

Uniform practical output regulation

Recent applications Relationship between design of nonlinear internal models

and nonlinear observers: immersion vs. observability

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The framework of output regulation

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The framework exosystem Plant Reference and/or disturbance generator Regulated outputs Measured output

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The framework Plant

invariant

Reference and/or disturbance generator

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The framework Plant

invariant

Reference and/or disturbance generator Lorenz attractor

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The framework

Boundedness of closed-loop trajectories (Uniform) converge of the error to zero

for any initial condition in known compact sets

(semi-global) invariant

Plant The problem of semiglobal output regulation

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The conceptual formulation

worst case disturbance (Hoo control): too pessimistic exact knowledge of disturbance/reference (inversion-based control): too optimistic 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:

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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)

The exosystem

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Applications

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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

• f the form

with unknown frequency amplitude and phase.

• C. Bonivento, A. Isidori, L. Marconi, A. Paoli, AUTOMATICA, 2003

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Application 2: Shunt Active Filters

Control Problem: Control the inverter switches in order to inject currents into the mains to compensate for the load higher harmonics 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

• L. Marconi, F. Ronchi, A. Tilli, AUTOMATICA, 2006

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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

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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.

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Application 4: Automatic landing on a oscillating ship

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Application 5: Pursuit-evasion in urban environment pursuer evader

Prototype of urban canyon

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.

• A. Isidori, L. Marconi, NOLCOS, 2004

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A reasoned overview of recent results

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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 Nonlinear Systems globally diffeomorphic to normal forms (well-defined relative degree) In this talk: Furthermore, in the spectrum of possible stabilization techniques proposed so-far in literature, we are particularly interested to high-gain techniques

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Of course, this is our choice. A number of others class

• f 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

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Zero dynamic Chain of integrator High frequency gain Normal form – relative degree= r

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Intention: to adopt high-gain arguments as customary stabilization tool Minimum phase requirement Normal form r.d.= 1 h.f.g.= 1

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A “weak” minimum-phase requirement = 0 Byrnes, Isidori TAC 03 The trajectories of this system originating from are uniformly bounded

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Steady state for nonlinear systems Byrnes - Isidori TAC 03 A fundamental step in the solution of the regulator problem is the notion of steady state for nonlinear systems 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

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Boundedness of trajectory

• riginating from

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Boundedness of trajectory

• riginating from

Existence of a compact Invariant set which uniformly attracts the trajectories from

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Boundedness of trajectory

• riginating from

Existence of a compact Invariant set which uniformly attracts the trajectories from Precisely, the omega limit set

• f

(Hale, Magalhaes, Oliva, Dynam. in Infinite Dimen. Syst., Springer Verlag)

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Steady state set (locus): Steady state: trajectories of = 0 In our framework it makes sense to define (Isidori-Byrnes TAC03)

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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

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= 0 Internal Model Property

Internal model property: capability of reproducing the

steady state input needed to keep the regulated error to zero

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+

• Internal Model Property

Internal model property: capability of reproducing the

steady state input needed to keep the regulated error to zero

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Stabilizability property: capability of stabilizing the closed-

loop system on a compact attractor on which the error is zero. Stabilizability Property

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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

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The asymptotic internal model property related (i) ASYMPTOTIC INTERNAL MODEL PROPERTY if there exists a The triplet is said to have the function such that :

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(i) yields +

• ASYMPTOTIC INTERNAL MODEL PROPERTY if there exists a

The triplet is said to have the function such that : The asymptotic internal model property

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(ii) In the composite system The set is LAS The asymptotic internal model property ASYMPTOTIC INTERNAL MODEL PROPERTY if there exists a The triplet is said to have the function such that :

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Invariance of and (i)

+

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(ii) LAS

+

• Invariance of

and (i)

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+

• In summary:

AIMP Ability to reproduce asymptotically all the

• utput behaviors of

the -system restricted on

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Asymptotic internal model property and output regulation The reason why it is crucial to have a triplet fulfilling these properties, is clear by the following Result Suppose the triplet has the

• A. Internal Model Property. Then there exists

such that the regulator solves the problem of O.R. Proof: Extension of high-gain stabilization techniques to the case of compact attractors

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Achieving the Asymptotic Internal Model Property A number of regulator design techniques proposed so far in literature can be recast in terms of the previous vision, namely as the attempt of designing the regulator so as to achieve the asymptotic internal model property. This has been pursued by adopting different tools and

• strategies. Very often without a common vision.

The need of fulfilling the AIMP has motivated, in all the past literature, the requirement of “ad hoc” assumption

• n the system:

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The immersion assumption

In terms of the previous framework, a number of (I would say the majority) immersion assumptions proposed so far in literature can be given a common interesting interpretation presented in the next slides

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Asymptotic Internal Model Property and Nonlinear Observers Crucial intuition: Achieving the asymptotic internal model property is not so different from designing asymptotic non linear observers Ability of designing an observer Ability of fulfilling AIMP

+

• Observed system

Observer Innovation term

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According to this: Immersion Observability of

(Huang TAC ‘92 and majority of the works until ~ 2 years ago)

Immersed into Linear-observable

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Uniform nonlinear observability form (Gautier-Kupcka 1999) According to this: Immersion Observability of

(Byrnes-Isidori, TAC 2003)

Immersed into

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Adaptive observability form According to this: Immersion Observability of

(Delli Priscoli, Marconi, Isidori, SICON 2005, MTNS tomorrow)

Immersed into + Persistence of excitation conditions

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Asymptotic Internal Model Property and Nonlinear Observers All the previous approaches, in the attempt to fulfill the AIMP, are guided by a “certainty equivalence idea”. The replica of the output behavior of -system by means of the

• system is achieved by explicitly estimating

+

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Dropping the immersion condition

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Crucial Additional Intuition: Observability of the system Dropping the immersion condition is not (in principle) necessary to achieve the AIMP, as the goal is to reproduce the output behavior of the system and not necessarily to estimate the full state Intuitively: if part of the state is not “observable” by the

• utput, no problem as this part does not play any role in

achieving the internal model property Not !!

^ ^ ^

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Linked to the fact that regulation does not mean estimation

• f the exosystem's state but rather of the control input needed

to enforce zero regulation error. Crucial Additional Intuition: Observability of the system Dropping the immersion condition is not (in principle) necessary to achieve the AIMP, as the goal is to reproduce the output behavior of the system and not necessarily to estimate the full state Not !!

^ ^ ^

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How to make this intuition rigorous The result takes a major source of inspiration from a recent observer design philosophy

Kazantzis – Kravaris, S&CL 1998 Kresseilmeier – Engel, TAC 2003 Krener – Xiao, SICON 2004 Andrieu – Praly, SICON 2005 Praly-Marconi-Isidori, MTNS tomorrow

• L. Marconi, L. Praly, A. Isidori, SICON, 2006 (Accepted)

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How to make this intuition rigorous Consider the candidate triplet namely the candidate controller Hurwitz, controllable dim and

• and

to be chosen

• Result: , (continuous) s.t.

this triplet has the AIMP dim

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In fact

• +

with

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• Hurwitz

solution of the PDE is such that is invariant and LAS (LES)

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if

Without observability (immersion) assumption! dim dim satisfies a Partial Injectivity Condition the function

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Under the partial injectivity condition, there exists a C0

such that (Tietze's extension Theorem) +

• n

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In summary the design of the regulator amounts to

Choose Compute solution of the PDE

Good luck!

Compute so that

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Remark The previous discussions/results reflect and “exalt” the historical difference between adaptive control and output regulation which, quite often, have been wrongly confused.

adaptive control based on certainty equivalence principle.

Explicit estimation of uncertainties (in the previous framework ) instrumental to compute the steady state control law

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Remark The previous discussions/results reflect and “exalt” the historical difference between adaptive control and output regulation which, quite often, have been wrongly confused.

Output regulation

direct estimation of the “friend” without explicit uncertainties estimation (“essential controller”)

adaptive control based on certainty equivalence principle.

Explicit estimation of uncertainties (in the previous framework ) instrumental to compute the steady state control law

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Uniform Practical Output Regulation Practical regulation: design controller achieving arbitrarily small (but not zero) asymptotic regulation error (for engineering purposes this is what is needed) Motivation: approximate design of the regulator to overtake the difficulties in the design of the function !!

• L. Marconi, L. Praly, “Nonlinear practical regulation without

high-gain”, next CDC. Journal version in preparation.

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Previous approaches to practical output regulation A.J. Krener/ C.I. Byrnes - F. Delli Priscoli - A. Isidori /

• J. Huang - W.J. Rugh

Drawback: explosion of the internal model dimension (in general)

polynomial approximation and/or power series expansion

• f the so-called regulator equations to identify an approximation

Of , with a degree of accuracy depending on the bound

• f the residual error, which can be dynamically reproduced by

means of a linear regulator.

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high-gain error feedback (indeed no internal model is used)

• A. Ilchmann and others

Drawback: typical problems linked to high-gain control structures, such as sensitivity to measurement noise and minimum-phase constraints Previous approaches to practical output regulation

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The idea behind the new theory is to address a problem

• f practical output regulation in which:

the dimension of the regulator (internal model) the value of the regulator gain nearby the zero error manifold

are UNIFORM with respect to the asymptotic error bound Practical regulation obtained by only “playing” with the design of

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Uniform practical output regulation It is possible to set-up a constructive framework to design the approximate regulator. In particular

Numerical algorithms to approximate :

approx of order

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Uniform practical output regulation It is possible to set-up a constructive framework to design the approximate regulator. In particular

Numerical algorithms to approximate :

approx of order if Numerical approx of a PDE

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Uniform practical output regulation It is possible to set-up a constructive framework to design the approximate regulator. In particular

Approximate expressions of :

Integral-based approximation (Kresselmeier-Engel TAC 2003) Covering of by means of balls of radius

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Uniform practical output regulation It is possible to set-up a constructive framework to design the approximate regulator. In particular

Approximate expressions of :

Covering of by means of balls of radius Optimization-based approximation (McShane, Bull Amer. Math Soc. 1934)

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Example 1 (Adaptive triangle disturbance compensation in linear systems) Unknown in amplitude, phase and frequency Regulator: “Small”

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−1.5 −1 −0.5 0.5 1 1.5 −2 2 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 w2 w1

τ2(w1,w2,1) τ1(w1,w2,1)

Approximate steady state (first two components)

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Errors

5 10 15 20 25 30 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Time (s) n(w) 5 10 15 20 25 30 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 Time (s) (z,y)

no

5 10 15 20 25 30 −0.2 −0.15 −0.1 −0.05 0.05 0.1 Time (s) (z,y) 5 10 15 20 25 30 −1.5 −1 −0.5 0.5 1 1.5 Time (s) u

Control input with

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Example 1 (Lorenz dynamics compensation) Problem: to design so that

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Example 1 (Lorenz dynamics compensation) Problem: to design so that Lorenz attractor

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−20 −15 −10 −5 5 10 15 20 −40 −20 20 40 5 10 15 20 25 30 35 40 45 z1 z2 z3 200 400 600 800 1000 1200 1400 −40 −30 −20 −10 10 20

τ1 τ2 τ3 τ4

200 400 600 800 1000 1200 1400 −35 −30 −25 −20 −15 −10 −5 5 10

τ5 τ6 τ7 τ8

PDE approximation on a particular trajectory

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5 10 15 20 25 30 35 40 45 50 −4 −2 2 4 6 8 10 12 Time (sec) y(t)

Output with and without internal model

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Conclusive remarks and developments

• Beyond output regulation: “killing the junk” in dynamical

interconnection

Practical Uniform output regulation theory. A way for

addressing in a practical way a huge number of O.R. problems in a way which is meaningful from an engineering viewpoint

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

Dropping the immersion assumption