Modeling and Control of Physical Systems: an Approach Based on Energy and Interconnection∗

Modeling and control of physical systems: an approach based on energy and interconnection∗

Bernhard Maschke.1/, Romeo Ortega.2/, Arjan van der Schaft.3/ .1/Laboratoire d’Automatisme Industriel, CNAM, 21 rue Pinel, 75013 Paris, France .2/Lab. des Signaux et Syst`emes, Sup´elec, Plateau de Moulon, 91192 Gif sur Yvette, France .3/Fac. of Mathematical Sciences, University of Twente, P.O. Box 217, 7500AE Enschede, The Netherlands [email protected],[email protected] [email protected]

Abstract implications for regulation and tracking, has undergone a re- markable revival. This last development was also spurred by It is discussed how network modeling of lumped-parameter work in robotics on the possibilities of shaping by feedback physical systems naturally leads to a geometrically defined the physical energy in such a way that it can be used as a suit- class of systems, called port-controlled Hamiltonian systems able Lyapunov function for the control purpose at hand, see with dissipation. The structural properties of these systems e.g. the influential paper [38]. This has led to what is some- are investigated, in particular the existence of Casimir func- times called passivity-based control, see e.g. [23, 29, 10]. tions and their implications for stability. It is shown how a Many other important developments have taken place, and power-conserving interconnection of port-controlled Hamil- much attention has been paid to special subclasses of systems tonian systems defines another port-controlled Hamiltonian like mechanical systems with nonholonomic constraints. All system, and how this may be used for design and for control this has resulted in a very lively research in nonlinear control, by shaping the internal energy. Extensions to implicit system with many actual and potential applications. descriptions (constraints, no a priori input-output structure) In this mini-course we want to stress the importance of and distributed parameter systems are indicated. modeling for nonlinear control. Of course, this is com- keywords: nonlinear control, modeling, passivity, Hamil- mon practice for working engineers, but a general theoreti- tonian systems, interconnection cal framework for modeling is also of utmost importance for the development of nonlinear control theory. We limit our- 1 Introduction selves to lumped-parameter physical systems, such as mechanical and electrical systems as well as electro-mechanical Nonlinear systems and control theory has witnessed tremen- systems, although in principle the framework can be ex- dous developments over the last three decades, see for ex- tended to distributed-parameter systems, see [14]. We dis- ample the textbooks [9, 22]. Especially the introduction of cuss how network modeling of such systems naturally leads geometric tools like Lie brackets of vector fields on mani- to a geometrically defined class of systems called port- folds has greatly advanced the theory, and has enabled the controlled Hamiltonian systems with dissipation (PCHD sys- proper generalization of many fundamental concepts known tems). These systems are determined by an internal intercon- for linear control systems to the nonlinear world. While the nection structure, a Hamiltonian defined as the total stored emphasis in the eighties has been primarily on the structural energy, and a resistive structure. Historically, the Hamilto- analysis of smooth nonlinear dynamical control systems, in nian approach has its roots in analytical mechanics and starts the nineties this has been combined with analytic techniques from the principle of least action, via the Euler-Lagrange for stability, stabilization and robust control, leading e.g. to equations and the Legendre transform, towards the Hamil- backstepping techniques and nonlinear H∞- control. More- tonian equations of motion. On the other hand, the network over, in the last decade the theory of passive systems, and its approach stems from electrical engineering, and constitutes a cornerstone of systems theory. While most of the analysis of ∗ This paper is an adapted and expanded version of the paper “Port- physical systems has been performed within the Lagrangian controlled Hamiltonian systems: towards a theory for control and design and Hamiltonian framework, the network modelling point of of nonlinear physical systems”, by A.J. van der Schaft, which is going to appear in the Journal of the Society of Instrument and Control Engineers view is prevailing in modelling and simulation of (complex) (SICE) of Japan. Part of this material can be also found in [29]. physical systems. The framework of PCHD systems com- bines both points of view, by associating with the intercon- equations nection structure of the network models a geometric struc- q˙ = @H .q; p/.=M−1.q/p/ ture given by a Poisson structure, or more generally a Dirac @p (4) structure. Dirac structures encompass Poisson structures, but p˙=−@H.q;p/+− they allow a considerable generalization since they also de- @q scribe systems with constraints as arising from the intercon- where nection of sub-systems. Furthermore, Dirac structures allow H.q; p/ = 1 pT M−1.q/ p + P.q/ to extend the Hamiltonian description of distributed param- 2 (5) . = 1 ˙T . / ˙ + . // eter systems by including boundary conditions, leading to a 2 q M q q P q non-zero energy balance at the boundary of the spatial do- is the total energy of the system. The equations (4) are called main, see [14]. the Hamiltonian equations of motion, and H is called the In the present paper we will restrict ourselves for simplicity Hamiltonian. The following energy balance immediately to PCHD systems that are defined with respect to a Poisson follows from (4): structure. The structural properties of these PCHD systems d @T H @T H H = @ .q; p/q˙+ @ .q; p/p˙ are investigated through geometric tools stemming from the dt q p (6) = @T H . ; /− =˙T−; theory of Hamiltonian systems. It is indicated how the inter- @p q p q connection of PCHD systems leads to another PCHD system, expressing that the increase in energy of the system is equal and how this may be exploited for control and design. In par- to the supplied work (conservation of energy). ticular, we investigate the existence of Casimir functions for If the potential energy is bounded from below,thatis∃C> the feedback interconnection of a plant PCHD system and a −∞ such that P.q/ ≥ C, then it follows that (4) with in- controller PCHD system, leading to a reduced PCHD system puts u = − and outputs y =˙qis a passive (in fact, lossless) on invariant manifolds with shaped energy. We thus provide state space system with storage function H.q; p/ − C ≥ 0 an interpretation of passivity-based control from an intercon- (see e.g. [39, 8, 29] for the general theory of passive and dis- nection point of view. sipative systems). Since the energy is only defined up to a constant, we may as well take as potential energy the func- 2 Port-controlled Hamiltonian systems tion P.q/ − C ≥ 0, in which case the total energy H.q; p/ becomes nonnegative and thus itself is the storage function. 2.1 From the Euler-Lagrange and Hamiltonian equa- System (4) is an example of a Hamiltonian system with col- tions to port-controlled Hamiltonian systems located inputs and outputs, which more generally is given in the following form Let us briefly recall the standard Euler-Lagrange and Hamil- @ tonian equations of motion. The standard Euler-Lagrange ˙ = H . ; /; . ; / = . ;:::; ; ;:::; / q @ q p q p q1 qk p1 pk equations are given as p @

H.;/+./; ∈ m; ˙ =− R d @L @L p @ qp Bqu u (7) .q;q˙/ − .q;q˙/ = −; (1) q dt @q˙ @q @ T H . ; /.=T m; y = B .q/ q p B.q/q˙/; y ∈ R T @p where q = .q1;:::;qk/ are generalized configuration coordinates for the system with k degrees of freedom, the La- Here B.q/ is the input force matrix, with B.q/u denoting the ∈ m grangian L equals the difference K − P between kinetic en- generalized forces resulting from the control inputs u R . T . ; / ergy K and potential energy P,and−=.−1;:::;−k/ is the The state space of (7) with local coordinates q p is usually vector of generalized forces acting on the system. Further- called the phase space. Normally m < k, in which case we @L speak of an underactuated system. more, @q˙ denotes the column-vector of partial derivatives of = T . / ˙ L.q; q˙/ with respect to the generalized velocities q˙1;:::;q˙k, Because of the form of the output equations y B q q we @L again obtain the energy balance and similarly for @q . In standard mechanical systems the kinetic energy K is of the form dH .q.t/; p.t// = uT .t/y.t/ (8) 1 dt K.q; q˙/ = q˙T M.q/q˙ (2) 2 and if H is bounded from below, any Hamiltonian system (7) is a lossless state space system. For a system-theoretic × . / where the k k inertia (generalized mass) matrix M q is treatment of Hamiltonian systems (7), we refer to e.g. [3, 26, symmetric and positive definite for all q. In this case the 27,4,22]. = . ;:::; /T vector of generalized momenta p p1 pk , defined A major generalization of the class of Hamiltonian systems = @L for any Lagrangian L as p @q˙ , is simply given by (7) is to consider systems which are described in local coordinates as p = M.q/q˙; (3)

@H m ˙ = . / . / + . / ; ∈ ; ∈ R x J x @x x g x u x X u T and by deﬁning the state vector .q1;:::;qk;p1;:::;pk/ (9) T @H m

the k second-order equations (1) transform into 2k first-order = . / . /; ∈ R y g x @x x y Here J.x/ is an n × n matrix with entries depending with H.Q;'1;'2/ := H1.'1/ + H2.'2/ + H3.Q/ the total smoothly on x, which is assumed to be skew-symmetric energy. Clearly the matrix J is skew-symmetric, and since J is constant it trivially satisfies (11). In [19] it has been . / =− T. /; J x J x (10) shown that in this way every LC-circuit with independent elements can be modelled as a port-controlled Hamiltonian and x = .x1;:::;xn/ are local coordinates for an n- dimensional state space manifold X . Because of (10) we eas- system, with the constant skew-symmetric matrix J being dH. . // = T. / . / solely determined by the network topology (i.e., Kirchhoff’s ily recover the energy-balance dt x t u t y t ,showing that (9) is lossless if H ≥ 0. We call (9) with J satisfying laws). Furthermore, also any LCTG-circuit with independent (10) a port-controlled Hamiltonian (PCH) system with struc- elements can be modelled as a PCH system, with J deter- ture matrix J.x/ and Hamiltonian H ([18, 13, 12]). mined by Kirchhoff’s laws and the constitutive relations of 2 As an important mathematical note, we remark that in many the transformers T and gyrators G. examples the structure matrix J will satisfy the “integrability” conditions Another important class of PCH systems are mechanical systems as arising from reduction by a symmetry group,suchas Xn @J @Jkj @Jji Euler’s equations for a rigid body. A third important class of J .x/ ik .x/ + J .x/ .x/+ J .x/ .x/ = 0 lj @x li @x lk @x systems is constituted by mechanical systems with kinematic l=1 l l l constraints. Consider as before a mechanical system with i; j; k = 1;:::;n (11) k degrees of freedom, locally described by k configuration In this case we may find, by Darboux’s theorem (see variables q = .q1;:::;qk/. Suppose that there are constraints ˙ e.g. [11]) around any point x0 where the rank of the on the generalized velocities q, described as matrix J.x/ is constant, local coordinates x˜ = .q; p; s/ = T .q1;:::;qk;p1;:::;pk;s1;:::sl/; with 2k the rank of J and A .q/q˙ = 0; (14) n = 2k + l, such that J in these coordinates takes the form   with A.q/ a r × k matrix of rank r everywhere (that is, there 0 Ik 0 are r independent kinematic constraints). Classically, the =  −  J Ik 00 (12) constraints (14) are called holonomic if it is possible to find 000 = . ;:::; / new configuration coordinates q q1 qk such that the The coordinates .q; p; s/ are called canonical coordinates, constraints are equivalently expressed as and J satisfying (10) and (11) is called a Poisson structure ˙ = ˙ =···= ˙ = ; matrix. In such canonical coordinates the equations (9) are qk−r+1 qn−r+2 qk 0 (15) very close to the standard Hamiltonian form (7). PCH systems arise systematically from network-type mod- in which case one can eliminate the configuration vari- ;:::; els of physical systems as formalized within the (general- ables qk−r+1 qk, since the kinematic constraints (15) are ized) bond graph language ([25, 2]). Indeed, the structure equivalent to the geometric constraints matrix J.x/ and the input matrix g.x/ may be directly as- = ;:::; = ; sociated with the network interconnection structure given by qk−r+1 ck−r+1 qk ck (16) the bond graph, while the Hamiltonian H is just the sum of ;:::; the energies of all the energy-storing elements; see our pa- for certain constants ck−r+1 ck determined by the ini- pers [13, 18, 15, 19, 32, 30, 20, 28]. This is most easily tial conditions. Then the system reduces to an uncon- exemplified by electrical circuits. strained system in the remaining configuration coordinates . ;:::; / q1 qk−r .Ifitisnot possible to find coordinates q such Example 2.1 (LCTG circuits) Consider a controlled LC- that (15) holds (that is, if we are not able to integrate the kine- circuit consisting of two parallel inductors with magnetic en- matic constraints as above), then the constraints are called ergies H1.'1/; H2.'2/ ('1 and '2 being the magnetic flux nonholonomic. linkages), in parallel with a capacitor with electric energy The equations of motion for the mechanical system with La- . / H3 Q (Q being the charge). If the elements are linear then grangian L.q; q˙/ and constraints (14) are given by the Euler- .' / = 1 '2 .' / = 1 '2 . / = 1 2 H1 1 2L 1, H2 2 2L 2 and H3 Q 2C Q .Fur- Lagrange equations (Neimark & Fufaev, [21]) 1 = 2 thermore let V u denote a voltage source in series with the first inductor. Using Kirchhoff’s laws one immediately @ @ d L − L = . /½ + . / ;½∈r; A q B q u R arrives at the dynamical equations dt @q˙ @q  

   @   T m ./˙= ; ∈R ˙ H Aqq 0 u (17) Q 01−1 @Q  0 @ '˙  = −10 0 H  + 1u (13) 1  @'1  where B.q/u are the external forces (controls) applied to the '˙ @ 2 100 H 0 system, for some k × m matrix B.q/, while A.q/½ are the | {z } @'2 J constraint forces. Defining as before (cf. (3)) the general- @ ized momenta the constrained Euler-Lagrange equations (17) = H y (= current through first inductor) transform into constrained Hamiltonian equations (compare @'1 with (7)), has a direct physical interpretation. A very important prop- @H erty is the possible existence of dynamical invariants inde- q˙ = .q; p/ @p pendent of the Hamiltonian H. Consider the set of p.d.e.’s @H @T C p˙ =− .q;p/+A.q/½ + B.q/u .x/J.x/ = 0; x ∈ X ; (19) @q @x @H y = BT .q/ .q; p/ (18) in the unknown (smooth) function C : X → R. If (19) has a @p solution C then it follows that the time-derivative of C along @H the port-controlled Hamiltonian system (9) satisfies 0 = AT .q/ .q; p/ @p dC @TC @H @TC = @ .x/J.x/ @ .x/+ @ .x/g.x/u with H.q; p/ = 1 pT M−1.q/ p + P.q/ the total energy, and dt x x x (20) 2 = @TC . / . / A.q/½ the constraint forces. One way of proceeding with @x x g x u these equations is to eliminate the constraint forces, and to Hence, for the input u = 0, or for arbitrary input functions T reduce the equations of motion to the constrained state space if additionally @ C .x/g.x/ = 0, the function C.x/ remains ={. ; /| T. /@H. ; /= } @x Xc q p A q @p q p 0. In [31] it has been constant along the trajectories of the port-controlled Hamil- shown that this leads to a port-controlled Hamiltonian sys- tonian system, irrespective of the precise form of the Hamil- tem (9). Furthermore, the structure matrix Jc of the resulting tonian H: A function C : X → R satisfying (19) is called a PCH system satisfies the integrability conditions (11) if and Casimir function (of the structure matrix J.x/). only if the constraints (14) are holonomic. (In fact, if the con- It follows that the level sets LC := {x ∈ X |C.x/ = c} ; c ∈ R, straints are holonomic then the coordinates s as in (12) can of a Casimir function C are invariant sets for the autonomous be taken to be equal to the “integrated constraint functions” ˙ = . / @H . /; Hamiltonian system x J x @x x while the dynamics re- q − + ;:::;q of (16).) k r 1 k stricted to any level set LC is given as the reduced Hamilto- nian dynamics 2.2 Basic properties of port-controlled Hamiltonian sys- @H tems x˙ = J .x / C .x / (21) C C C @x C Recall that a port-controlled Hamiltonian system is defined with H and J the restriction of H; respectively J; to L : . ; ; / C C C by a state space manifold X endowed with a triple J g H . The existence of Casimir functions has immediate conse- . . /; . //; ∈ The pair J x g x x X , captures the interconnection quences for stability analysis of (9) for u = 0. Indeed, if structure of the system, with g.x/ modeling in particular the ; ··· ; dH = C1 Cr are Casimirs, then by (19) not only dt 0for ports of the system. Independently from the interconnection u=0, but structure, the function H : X → R defines the total stored energy of the system. d .H + Ha.C1;··· ;Cr// .x.t// = 0 (22) PCH systems are intrinsically modular in the sense that any dt

r → R power-conserving interconnection of a number of PCH sys- for any function Ha : R . Hence, if H is not positive ∗ tems again defines a PCH system, with its overall intercon- definite at an equilibrium x ∈ X ,thenH+Ha.C1;··· ;Cr/ nection structure determined by the interconnection struc- may be rendered positive definite at x∗ by a proper choice of tures of the composing individual PCH systems together with Ha, and thus may serve as a Lyapunov function. This method their power-conserving interconnection, and the Hamiltonian for stability analysis is called the Energy-Casimir method, just the sum of the individual Hamiltonians (see [30, 28, 5]). see e.g. [11]. The only thing which needs to be taken into account is the + fact that a general power-conserving interconnection of PCH Example 2.2 (Example 2.1 continued) The quantity 1 systems not always leads to a PCH system with respect to a 2 is a Casimir function. Poisson structure J.x/ and input matrix g.x/ as above, since the interconnection may introduce algebraic constraints be- 2.3 Port-controlled Hamiltonian systems with dissipa- tween the state variables of the individual sub-systems. Nev- tion ertheless, also in this case the resulting system still can be Energy-dissipation is included in the framework of port- seen as a PCH system, which now, however, is defined with controlled Hamiltonian systems (9) by terminating some of respect to a Dirac structure, generalizing the notion of a Pois- the ports by resistive elements. In the sequel we concentrate son structure. The resulting class of implicit PCH systems on PCH systems with linear resistive elements u =−Sy has been studied e.g. in [30, 28, 5]. In the present paper we R R for some positive semi-definite symmetric matric S = ST ≥ restrict ourselves to explicit PCH systems (9). 0, where u and y are the power variables at the resistive From the structure matrix J.x/ of a port-controlled Hamilto- R R ports . This leads to models of the form nian system one can directly extract useful information about the dynamical properties of the system. Since the structure ˙ = . / − . / @H . / + . / x [J x R x ] @x x g x u matrix is directly related to the modeling of the system (cap- (23) turing the interconnection structure) this information usually = T . / @H . / y g x @x x where R.x/ is a positive semi-definite symmetric matrix, de- with p the momentum, R the resistance of the resistor, I the pending smoothly on x: In this case the energy-balancing current through the voltage source, and the Hamiltonian H property (7) takes the form being the total energy @T @ dH T H H 1 1 1 .x.t// = u .t/y.t/ − .x.t//R.x.t// .x.t// H.q; p; Q/ = p2 + k.q −¯q/2+ Q2; (26) dt @x @x 2m 2 2C.q/ ≤ uT.t/y.t/: (24) with q¯ denoting the equilibrium position of the spring. Note showing passivity if the Hamiltonian H is bounded from be- that Fq˙ is the mechanical power, and EI the electrical power low. We call (23) a port-controlled Hamiltonian system with applied to the system. In the application as a microphone the dissipation (PCHD system). Note that in this case two geo- voltage over the resistor will be used (after amplification) as metric structures play a role: the internal power-conserving a measure for the mechanical force F. interconnection structure given by J.x/, and an additional resistive structure given by R.x/: A rich class of examples of PCHD systems is provided by electro-mechanical systems such as induction motors, see F e.g. [24]. In some examples the interconnection structure J.x/ is actually varying, depending on the mode of opera- tion of the system, as is the case for power converters (see e.g. [6]) or for mechanical systems with variable constraints.

3 Control of port-controlled Hamiltonian sys- C tems with dissipation

The aim of this section is to discuss a general methodology for controlling PCH or PCHD systems which exploits R E their Hamiltonian properties in an intrinsic way. Since this exposition is based on ongoing recent research (see e.g. [16, 35, 17, 24, 29]) we only try to indicate its potential. An Figure 1: Capacitor microphone expected beneﬁt of such a methodology is that it leads to physically interpretable controllers, which possess inherent robustness properties. Future research is aimed at corrobo- Example 2.3 ([21]) Consider the capacitor microphone de- rating these claims. picted in Figure 1. Here the capacitance C.q/ of the ca- We have already seen that PCH or PCHD systems are pas- pacitor is varying as a function of the displacement q of the sive if the Hamiltonian H is bounded from below. Hence in right plate (with mass m), which is attached to a spring (with this case we can use all the results from the theory of passive spring constant k > 0 ) and a damper (with constant c > 0 systems, such as asymptotic stabilization by the insertion of ), and affected by a mechanical force F (air pressure aris- damping by negative output feedback, see e.g. [29]. The em- ing from sound). Furthermore, E is a voltage source. The phasis in this section is however on the somewhat comple- dynamical equations of motion can be written as the PCHD mentary aspect of shaping the energy of the system, which system directly involves the Hamiltonian structure of the system, as   opposed to the more general passivity structure. @H      @q   q˙ 010 00 0    ˙  =  − − @H 3.1 Control by interconnection p 100 0c 0 @p Q˙ 000 00R−1   @H Consider a port-controlled Hamiltonian system with dissipa-     @Q tion (23) regarded as a plant system to be controlled. Recall 0 0 the well-known result that the standard feedback intercon- +1F+0E (25) nection of two passive systems again is a passive system; a 0 R−1 basic fact which can be used for various stability and control purposes ([8, 23, 29]). In the same vein we consider the interconnection of the plant (23) with another port-controlled @H y = =˙q Hamiltonian system with dissipation 1 @p

˙ @HC ¾ = [JC.¾/ − RC.¾/] @¾ .¾/ + gC.¾/uC 1@H C : ¾ ∈ XC (27) = = @ y2 I = T .¾/ HC .¾/ R@Q yC gC @¾ regarded as the controller system, via the standard feedback kc interconnection k e mc m u =−yC+e (28) = + uC y eC b with e, eC external signals inserted in the feedback loop. The closed-loop system takes the form Figure 2: Controlled mass " # ˙ J.x/ −g.x/gT .¾/ x = . C − ¾˙ .¾/ T . / .¾/   gC g x JC @ | {z } HC @1q . ;¾/     c    " Jcl #x " # ˙ @ 1qc 010  0 . / H . / @  R x 0 @x x     HC   / p˙c = −1−b1@  + 0 uC @  pc  HC ˙ 0 RC.¾/ .¾/ 1q 0−10   1 | {z } @¾ (29) @HC R .x;¾/ @1q cl @ . / y = HC + g x 0 e C @1q 0 gC.¾/ eC 1 ;1 " #" # where qc is the displacement of the spring kc qis the @ . / H. / displacement of the spring k,andpc is the momentum of y g x 0 @x x = the mass m : The plant Hamiltonian is H. p/ = 1 p2; and @HC c 2m yC 0 gC.¾/ @¾ .¾/ the controller Hamiltonian is given as HC .1qc; pc;1q/= 2 1 pc 2 2 . +k.1q/ + kc.1qc/ /: The variable b > 0 is the damp- 2 mc which again is a port-controlled Hamiltonian system with ing constant, and e is an external force. The closed-loop sys- dissipation, with state space given by the product space tem possesses the Casimir function X × XC, total Hamiltonian H.x/ + HC.¾/, inputs .e; eC/ and outputs .y; yC /. Hence the feedback interconnection of any C.q;1qc;1q/=1q−.q−1qc/; (31) two PCHD systems results in another PCHD system; just as in the case of passivity. This is a special case of a theorem implying that along the solutions of the closed-loop system ([29]), which says that any regular power-conserving inter- 1q = q − 1qc + c (32) connection of PCHD systems deﬁnes another PCHD system. It is of interest to investigate the Casimir functions of the with c a constant depending on the initial conditions. With closed-loop system, especially those relating the state vari- the help of LaSalle’s Invariance principle it can be shown ables ¾ of the controller system to the state variables x of that restricted to the invariant manifolds (32) the system is the plant system. Indeed, from a control point of view the asymptotically stable for the equilibria q = 1qc = p = pc = Hamiltonian H is given while HC can be assigned. Thus if 0: 2 we can ﬁnd Casimir functions Ci.¾; x/; i = 1; ··· ;r, relating ¾ to x then by the Energy-Casimir method the Hamiltonian As a special case (see [29] for amore general discussion) let us consider Casimir functions for (29) of the form H + HC of the closed-loop system may be replaced by the Hamiltonian H + HC + Ha.C1; ··· ;Cr/, thus creating the ¾i − Gi.x/;i=1;:::;dimXC = nC (33) possibility of obtaining a suitable Lyapunov function for the closed-loop system. That means that we are looking for solutions of the p.d.e.’s (with ei denoting the i-th basis vector)   Example 3.1 [34] Consider the “plant” system J.x/ − R.x/ −g.x/gT .¾/ @T G C − i . / T   = 0 @  x ei H @x .¾/ T . / .¾/ − .¾/ ˙ @q gC g x JC RC q = 01 + 0 ˙ − u p 10 @H 1 for i = 1;:::;nC, relating all the controller state variables   @p ¾ ;:::;¾ = @H (30) 1 nC to the plant state variables x. Denoting G @q .G ;:::;G /T this means ([29]) that G should satisfy y=01  1 nC @H @T G . / . / @G . / = .¾/ @p @x x J x @x x JC

. / @G . / = = .¾/ with q the position and p being the momentum of the mass R x @x x 0 RC (34) m, in feedback interconnection .u =−yC+e;uC =y/with the controller system (see Figure 2) @T G . / . / = .¾/ T . / @x x J x gC g x In this case the reduced dynamics on any multi-level set (27). This fact has some favorable consequences. Indeed, it implies that the passivity-based control law deﬁned by (39) ={. ;¾/|¾ = . /+ ; = ;::: } LC x i Gi x ci i 1 nC (35) can be equivalently generated as the feedback interconnec- can be immediately recognized ([29]) as the PCHD system tion of the passive system (23) with another passive system (27). In particular, this implies an inherent invariance prop- @Hs erty of the controlled system: the plant system (23), the con- x˙ = [J.x/ − R.x/] .x/; (36) @x troller system (37), as well as any other passive system inter- with the same interconnection and dissipation structure as connected to (23) in a power-conserving fashion, may change in any way as long as they remain passive, and for any pertur- before, but with shaped Hamiltonian Hs given by bation of this kind the controlled system will remain stable. Hs.x/ = H.x/ + HC.G.x/ + c/: (37) For a further discussion of passivity-based control from this point of view we refer to [24]. In the context of actuated mechanical systems this amounts to the shaping of the potential energy as in the classical paper [38], see [29]. 4 Conclusions and future research A direct interpretation of the shaped Hamiltonian Hs in terms of energy-balancing is obtained as follows. Since RC.¾/ = 0 Clearly, the theory presented in this paper opens up the way dHC = T by (34) the controller Hamiltonian HC satisﬁes dt uCyC. for many control and design problems. Its potential for set- dHs = dH + point regulation has already received some attention (see Hence along any multi-level set LC given by (35) dt dt dHC = dH − T =− = : [16, 17, 24, 29]), while the extension to tracking problems is dt dt u y,sinceu yC and uC y Therefore, up to a constant, wide open. In this context we also like to refer to some recent Z work concerned with the shaping of the Lagrangian, see e.g. t T [1]. Also, the control of mechanical systems with nonholo- Hs.x.t// = H.x.t// − u .−/y.−/d−; (38) 0 nomic kinematic constraints can be fruitfully approached from this point of view, see e.g. [7], as well as the modelling and the shaped Hamiltonian H is the original Hamiltonian s and control of multi-body systems, see [15, 20, 36]. The H minus the energy supplied to the plant system (23) by framework of PCHD systems seems perfectly suited to theo- the controller system (27). From a stability analysis point of retical investigations on the topic of impedance control;see view (38) can be regarded as an effective way of generating already [34] for some initial results in this direction. Also the candidate Lyapunov functions H from the Hamiltonian H. s connection with multi-modal (hybrid) systems, correspond- ing to PCHD systems with varying interconnection structure 3.2 Passivity-based control of port-controlled Hamilto- [6], needs further investigations. Finally, our current research nian systems with dissipation is concerned with the formulation of distributed parameter In the previous section we have seen how under certain con- systems as port-controlled Hamiltonian systems, see [14], ditions the feedback interconnection of a PCHD system hav- and applications in tele-manipulation [37] and smart struc- ing Hamiltonian H (the “plant”) with another PCHD sys- tures [33]. tem with Hamiltonian HC (the “controller”) leads to a reduced dynamics given by (36) for the shaped Hamiltonian References Hs.Fromastate feedback point of view the dynamics (36) could have been directly obtained by a state feedback [1] A. Bloch, N. Leonard & J.E. Marsden, “Matching and u = Þ.x/ such that stabilization by the method of controlled Lagrangians”, @H .G.x/ + c/ in Proc. 37th IEEE Conf. on Decision and Control, g.x/Þ.x/ = [J.x/ − R.x/] C (39) @x Tampa, FL, pp. 1446-1451, 1998. Þ. / Indeed, such an x is given in explicit form as [2] P.C. Breedveld, Physical systems theory in terms of @ bond graphs, PhD thesis, University of Twente, Faculty Þ. / =− T. . /+ / HC. . / + / x gC G x c @¾ G x c (40) of Electrical Engineering, 1984

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