Passivity-Based Control of Nonlinear Systems: A Tutorial - IEEE Xplore

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Gif-sur-Yvette. NSW 2006. FRANCE. AUSTRALIA rortega&lss.supelec.fr. {zjiang, davidh}Qee.usyd.edu.au. Abstract. In this paper we survey some recent resultsĀ ...
Proceedings of the American Control Conference Albuquerque, New Mexico June 1997 I 997 AACC 0-7a03-3a32-41971$10.00

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Passivity-Based Control of Nonlinear Systems: A Tutorial Romeo Ortega Laboratoire des Signaux et Systemes CNRS-SUPELEC Plateau de Moulon, 91192 Gif-sur-Yvette FRANCE

Zhong P. Jiang, David J. Hill Dept. Electrical Engineering Bldg J13 Sydney University NSW 2006 AUSTRALIA

rortega&lss.supelec.fr

{zjiang, davidh}Qee.usyd.edu.au

Abstract

in [37] and in [33] for cascaded nonlinear systems.

The purpose of this paper is to survey recent developments on passivity-based control of finitedimensional nonlinear dynamical systems. in the first part of the paper we treat general systems and develop a unified framework for passivity-based nonlinear control design. Exploiting the particular inherent structure of physical systems, we can reasonably expect to design a stabilizing controller with better performance. in the second part, we turn our attention to the practically important class of nonlinear systems described by Euler-Lagrange (EL) equations. Note This is an abridged version of the full paper which is available upon request to the authors.

In this paper we survey some recent results on stabilization of nonlinear systems using a passivity approach. In the first part of the paper we treat general systems and develop a unified framework for passivity-based nonlinear control design. In the second part we center our attention on systems described by Euler-Lagrange equations, with particular emphasis on mechanical systems, power converters and AC motors.

2 1

Introduction

State-space representations

We consider in this section nonlinear control systems with outputs :

It is well-known that passivity properties play a vix = f(.) G(z)u (2.1) tal role in designing asymptotically stabilizing controllers for nonlinear systems. At a theoretic level, Y = h(x) (2.2) a fairly complete theory has been set up for genwhere x E R", u , y E Rm, f : Rn -+ R",G : eral nonlinear feedback systems - see, for instance, R" + 72"'" and h : R" +- R". Assume that [la, 53, 191. Several testing tools including nonthese functions are locally Lipschitz with f ( 0 ) = 0 linear versions of the Kalman-Yacubovitch-Popov and h ( 0 ) = 0. (KYP) lemma [18] are available to tell when a finiteWe present here several recent results about feeddimensional nonlinear dynamical system is passive. back equivalence of a nonlinear system to a passive A recent breakthrough in the field of nonlinear syssystem via static state feedback or adaptive control tems is that an answer has been given to the longlaw.' standing open but fundamental question (posed by Willems [53]) of when a finite-dimensional nonlinear dynamical system can be made passive via state feed- 2.1 Feedback passive systems back (see [lo]). More importantly, together with geometric nonlinear control theory, the results of this Definition 1 A system (2.1)-(2.2) is said to be feedkind have been used systematically in [lo] to unify back (strictly) C"-passive if there exists a feedback a number of global stabilization results for intercon'We assume the reader is familiar with the basic definitions nected nonlinear systems in the past literature. It of passivity, the ubiquitous nonlinear KYP lemma [18],and the should be mentioned that the passivity idea for stabi- input-output and internal stability properties of EL systems lization was independently used for adaptive systems as presented in, e.g. [34].

+

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c

i fo(c)

Suppose = as GAS at = 0. It zs also as(2.3) sumed that { f , g , h } zs zero-state detectable and CT-passzve wzth a posztzve dejinzte and proper storsuch that the system (2.1)-(2.2)-(2.3) wzth new znput age functzon (r 2 1). Then system (2 6) zs GAS b y U as (strzctly) C"-passzve. smooth state feedback law

U

= a(.)

+ by.).

In [lo], Brynes et al. gave a sufficient and necessary condition for the longstanding open question: when is a finite-dimensional nonlinear system made passive via state feedback? More explicitly,

Theorem 1 [IU] Consider a nonlinear system (2.1)(2.2) having a global normal form:

Next, we show that the AFP property can be propagated through any minimum-phase nonlinear system with relative degree one. More specifically, consider an interconnected nonlinear system with linearly appearing parametric uncertainty:

6

= fio(J) + fi( 0, L $ ( Q P ) + C > 0 and that the external forces are due to dissipation (captured by the Rayleigh function F p ( j p ) and ) the

-+ R+ such that

A

V1(qp) = & ( q P ) f Vcz(qy) has a unique minimum at the desired qz = Mzq,, for instance,

terized by the triple C, = Cp{T,, V, , F,},which we A 1 + A1qy)TA2(qc + vc2(qY) call in the sequel the EL parameters, and the matrix v,(qcl q Y ) = M,. Actually, the design philosophy that we propose with A1 a full rank matrix and A2 > 0 , centers around the idea of modifying, via the control, these EL parameters. A.3 (Dissipation propagation) ( U 3 const and j y 5 0) =j 1imtAwM,I~, = 0,

z(q"

3.1

Stabilization with Euler-Lagrange Controllers

Output feedback stabilization problem Given an EL system of the form (3.1) with generalized coordinates qp E R"P,regulated and measurable outputs qz = M z q p , qy = M y q p ,respectively, and a desired 'Onstant reference q z . Design an output feedback controller that insures global asymptotic stability (GAS) of an equilibrium (q p , ip)= (ijp, 0) satisfying qz = M z qp .

U.t.c., the EL controller

where Q~

a~~4c11zf o r

~

> 0, solves the

output feedback global stabilization problem. The corollary below provides a theoretical justification to the standard practice of replacing the velocity measurement by its approximate differentiation.

Corollary 2 The "dirty derivative" controller

31n subsection 2.4 we will see an example where the control does not enter linearly in the dynamics and we have other external forces.

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bip M p u = -diag{ -}qy P ai

+

m 2 ( Y y ) - ___

n

q p ) the magnetic co-energy in the inductances, with p = $,a ; , b; > 0 , solves the output feedback 7,(qp, V p ( q p )the electric field energy of the capacitances, global stabilization problem f o r EL systems verifying Fp (&) the Rayleigh dissipation co-function due to A . 3 provided Vcz(qy)satisfies A . 1 . the presence of resistances, and Fp the forcing functions due to voltage sources. For example, for the 3.2 Saturated Inputs boost converter an average model has the EL parameters The methodology described above ca.n be easily modified to take into account input constraints, that is

( 3 . 3 ) where U E [O, 11 is the duty cycle, C , L , R , E > 0 are the capacitance, inductance, resistance and voltage The key observation is that, as it follows from (3.2), source, respectively. After the change of coordinates the control will be bounded if the derivative of V,(q,) z1 = QL and z2 = qc/C one obtains is saturated outside some specified ball, so that the control signal remains bounded. For instance, inD p iRpz zz Fp (3.4) A stead of quadratic functions, we could use f ( z ) = where ln{cosh(z)}, which is positive definite and its derivative f'(z)= tanh(z) has the desired saturation property. The resulting controllers will then automatically incorporate a suitable saturation function. and Fp = [E,0IT. The control problem is to design ?J E [0, 11 such that limt+m z2 = 22 with internal staProposition 4 [28] Consider the EL system (3.1) bility, where 2 2 > E is the desired constant voltage. with saturated inputs (3.31, measurable output q p , This is a challenging theoretical problem because the and a constant desired reference value tjp E % " p . Assystem (3.4) is nonlinear, nonminimum phase (with sume the system is fully actuated, that as M p = I . respect to z z ) , with uncertain parameters and satuChoose the potential energy of the EL controller as rated controls. Jup,J 5

u y , i = 11

" '

, 72U

+

In [44] we follow tjhe PBC approach and propose to preserve the EL structure but, modify the EL parameters to achieve the control objective. To this end, we choose the closed-loop EL parameters as

with

where Z = z - z d , with z d a desired value for z , which is to be defined later. Notice that we have chosen the desired EL parameters equal t o the open-loop bi > 0 , k z , ,suficiently small and mini kai > EL parameters, but in terms of the errors 2. Also, k e n , with k:ln some suitable constant. Then we have added dissipation with RI> 0. ( q p ,qp, q,, qc) = ( 0 , &,, 0 , 4,) is a GAS equilibrium point of the closed loop provided4 Proposition 5 The averaged P W M model (3.4) in closed loop with the controller 4P,

-B; sat (z;) dz;

[&I

CiZd ?J

3.3

Power Converters

=

1 -RZZd

= '2 2[d E -

+ &&[E-

-a

-2

Rl(zl - &)]

Rl(zl - &)]

has an equilbrium point

In a series of papers starting with [44] we have advocated the use of PBC for voltage regulation of dcto-dc converters. An averaged PWM model for these devices is given by the EL system (3.1) with gener- which is asymptotically stable. n alized coordinates qp = [q:, q:]* E R" the electric Remark charges in inductances and capacitances, repectively, The result above has been extended in several direc4Notice that the gradient of the systems potential energy is tions. We have incorporated an adaptation mechevaluated here u t the desired reference. anism that estima,tes the unknown load resistance,

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combined PBC with sliding mode control to reduce Proposition 6 [31] Assume the machine (3.5) satthe energy consumption, and treated the case of cas- isfies: caded converters. We refer the reader t o [14], where we carry out a comparative experimental study, for A , I (Decoupling) The n, xn,-dimensional (2,2) block is zero, and he rotor compoof the matrix Wl = the references of all these works. nents of the vector p are independent of qm.

A . 2 (Blondel-Park transformability) There exists a constant matrix U E R n e x n ~ solution of

AC Rotating Machines

3.4

For AC rotating machines the generalized coordinates are the charges at windings qei, i = 1, . . . , n e ,and the rotor angular position qm. If we assume that there is no magnetic saturation, flux X and current Qe are Under these conditions, there exists a dynamic output related by feedback P B C that solves the torque tracking problem 1De(qm)ie p(4m) above.

+

where De = DT > 0 is the inductance matrix of the windings, and p represents flux linkages due to permanent magnets. The Lagrangian is

Remark In [40] PBC was used t o solve the problem of global tracking of robot manipulators which are driven by AC machines.

Selected List of References where VP(ym) is the contribution of the permanent [lo] Byrnes, C., Isidori, A., Willems, J.C., Passivity, magnets to the potential energy. The equations of feedback equivalence, and the global stabilization of motion are minimum phase nonlinear systems, IEEE Trans Aut. Cont., Vol.36, no.11, pp. 1228-1240, 1991. [20] Z. P. Jiang, D. J . Hill and A. L. Fradkov, A passification approach to adaptive nonlinear stabilization, Systems and Control Letters, Vol. 28, No. 2, pp. 73-84, 1996; also see Proc. 35th IEEE Conf. Dec. Contr., Dec. 1996.

where

, . . . , R,,}

[21] M. Krstid, I. Kanellakopoulos and P. V. KokotoviC, Nonlinear and Adaptive Control Desagn. New York: John Wiley & Sons, 1995.

r~ is a load torque, and the generated torque is

A

For underactuated machines, MF = [2,s,0], n, n e ,else M e = Zne. In [31] we provide a solution to the following