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 me- chanical 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 co- ordinates 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 ki- netic 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 coor- dinates 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.