nonlinear output regulation: a reasoned overview and new ...helton/mtnshistory/...the class of...

MTNS – Kyoto 2006 1Lorenzo Marconi -

Nonlinear output regulation: a reasoned overview and

new developments

Lorenzo Marconi

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


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 modelsand nonlinear observers: immersion vs. observability


The framework of output regulation


The framework

exosystem

Plant

Reference and/or disturbance generator

Regulated outputs

Measured output


The framework

Plant

invariantReference and/or disturbance generator


The framework

Plant

invariantReference and/or disturbance generator

Lorenz attractor


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


The conceptual formulation

worst case disturbance (Hoo control): too pessimisticexact 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:


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


Applications


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


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


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


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.


Application 4: Automatic landing on a oscillating ship


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


A reasoned overview of recent results


The class of systems

Since the problem at hand includes, as a particular case, a problem of semiglobal stabilization (w=0), it makes senseto 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 particularlyinterested to high-gain techniques


Of course, this is our choice. A number of others classof systems and associated stabilization techniques canbe selected for design purposes.

Besides others let us recall the class of systems diffeomorphicto systems in the adaptive nonlinear form and adaptive stabilization techniques for which a bunch of outputregulation theory have been developed:

• Marino, Santosuosso and Tomei• Ding• Serrani and Isidori• Huang• …. and others


Zero dynamic

Chain of integrator

High frequency gain

Normal form – relative degree=r


Intention: to adopt high-gain arguments as customary stabilization tool Minimum phase requirement

Normal formr.d.= 1h.f.g.=1


A “weak” minimum-phase requirement

=0

0

Byrnes, Isidori TAC 03

The trajectories of this system originating fromare uniformly bounded


Steady state for nonlinear systems

Byrnes - Isidori TAC 03

A fundamental step in the solution of the regulator problemis 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


Boundedness of trajectoryoriginating from



Existence of a compactInvariant set whichuniformly attracts the trajectories from



Existence of a compactInvariant set whichuniformly attracts the trajectories from

Precisely, the omega limit setof

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


Steady state set (locus):

Steady state: trajectories of

=0

0

In our framework it makes sense to define (Isidori-Byrnes TAC03)


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


=0

00

0

0 0

Internal Model Property



00

0

0+

-

Internal Model Property



Stabilizability property: capability of stabilizing the closed-loop system on a compact attractor on which the error is zero.

Stabilizability Property


Two main issues: internal model and stabilizability property

The two properties are interlaced.

In particular the ability of achieving the stabilizabilityproperty 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 todesign the regulator into the so-called:

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 :


(i)

yields

+ -






(ii) In the composite system

The setis LAS






Invariance of and

(i)

+-


(ii)

LAS

+-

Invariance of and

(i)


+-

In summary:

AIMP

Ability to reproduceasymptotically all the output behaviors of the -system restricted on


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


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, namelyas 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 pastliterature, the requirement of “ad hoc” assumption on the system:


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


Asymptotic Internal Model Property and Nonlinear Observers

Crucial intuition: Achieving the asymptotic internal modelproperty is not so different from designing asymptotic non linear observers

Ability of designing an observer

Ability of fulfilling AIMP

+-

Observed system Observer Innovation term


According to this:

Immersion Observability of

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

Immersed into

Linear-observable


Uniform nonlinear observability form (Gautier-Kupcka 1999)

According to this:


(Byrnes-Isidori, TAC 2003)

Immersed into


Adaptive observability form

According to this:


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

Immersed into

+ Persistence of excitation conditions


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

+-


Dropping the immersion condition


Crucial Additional Intuition: Observability of the system


is not (in principle) necessary to achieve the AIMP, as the goal is to reproduce the output behavior of the systemand not necessarily to estimate the full state

Intuitively: if part of the state is not “observable” by the output, no problem as this part does not play any role in achieving the internal model property

Not !!^ ^ ^


Linked to the fact that regulation does not mean estimationof the exosystem's state but rather of the control input needed to enforce zero regulation error.

Crucial Additional Intuition: Observability of the system


is not (in principle) necessary to achieve the AIMP, as the goal is to reproduce the output behavior of the systemand not necessarily to estimate the full state

Not !!^ ^ ^


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 2005Praly-Marconi-Isidori, MTNS tomorrow

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


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


In fact

- +

with


Hurwitz solution of the PDE

is such that is invariant and LAS (LES)


if

Without observability (immersion) assumption!

dim dim

satisfies a Partial Injectivity Condition

the function


Under the partial injectivity condition, there exists a C0

such that

(Tietze's extension Theorem)

+ -on


In summary the design of the regulator amounts to

Choose

Compute solution of the PDE

Good luck!

Compute so that


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


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


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 overtakethe difficulties in the design of the function

!!

L. Marconi, L. Praly, “Nonlinear practical regulation without high-gain”, next CDC. Journal version in preparation.


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 of the so-called regulator equations to identify an approximation Of , with a degree of accuracy depending on the bound of the residual error, which can be dynamically reproduced by means of a linear regulator.


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


The idea behind the new theory is to address a problem of 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


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




Numerical algorithms to approximate :

approx of order if

Numerical approx of a PDE




Approximate expressions of :

Integral-based approximation (Kresselmeier-Engel TAC 2003)

Covering of by means ofballs of radius




Approximate expressions of :

Covering of by means ofballs of radius

Optimization-based approximation (McShane, Bull Amer. Math Soc. 1934)


Example 1 (Adaptive triangle disturbance compensation in linear systems)

Unknown in amplitude, phase and frequency

Regulator:

“Small”


−1.5 −1 −0.5 0 0.5 1 1.5 −2

0

2

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

w2

w1

τ2(w

1,w

2,1)

τ1(w

1,w

2,1)

Approximate steady state (first two components)


Errors

0 5 10 15 20 25 30−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

n(w

)

0 5 10 15 20 25 30−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

(z,y

)

no

0 5 10 15 20 25 30−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Time (s)

(z,y

)

0 5 10 15 20 25 30−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

u

Control inputwith


Example 1 (Lorenz dynamics compensation)

Problem: to design so that


Example 1 (Lorenz dynamics compensation)

Problem: to design so that

Lorenz attractor


−20 −15 −10 −5 0 5 10 15 20−40

−20

0

20

405

10

15

20

25

30

35

40

45

z1

z2

z 3

0 200 400 600 800 1000 1200 1400−40

−30

−20

−10

0

10

20

τ1

τ2

τ3

τ4

0 200 400 600 800 1000 1200 1400−35

−30

−25

−20

−15

−10

−5

0

5

10

τ5

τ6

τ7

τ8

PDE approximation on a particular trajectory


0 5 10 15 20 25 30 35 40 45 50

−4

−2

0

2

4

6

8

10

12

Time (sec)

y(t)

Output withand withoutinternal model


Conclusive remarks and developments

Beyond output regulation: “killing the junk” in dynamicalinterconnection

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 modelsand nonlinear observers: immersion vs. observability

Dropping the immersion assumption

nonlinear output regulation: a reasoned overview and new ...helton/mtnshistory/...the class of...

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