A.J. van der Schaft

L2-Gain and Passivity Techniques in Nonlinear Control

1 Input-Output Stability.- 2 Small-gain and Passivity of Input-Output Maps.- 3 Dissipative Systems Theory.- 4 Hamiltonian Systems as Passive Systems.- 5 Passivity by Feedback.- 6 Factorizations of Nonlinear Systems.- 7 Nonlinear H? Control.- 8 Hamilton-Jacobi Inequalities.

L/sub 2/-gain analysis of nonlinear systems and nonlinear state-feedback H/sub infinity / control

Previously obtained results on L2-gain analysis of smooth nonlinear systems are unified and extended using an approach based on Hamilton-Jacobi equations and inequalities, and their relation to invariant manifolds of an associated Hamiltonian vector field. On the basis of these results a nonlinear analog is obtained of the simplest part of a state-space approach to linear H/sub infinity / control, namely the state feedback H/sub infinity / optimal control problem. Furthermore, the relation with H/sub infinity / control of the linearized system is dealt with. >

On a state space approach to nonlinear H ∞ control

We study the standard H∞ optimal control problem using state feedback for smooth nonlinear control systems. The main theorem obtained roughly states that the L2-induced norm (from disturbances to inputs and outputs) can be made smaller than a constant γ > 0 if the corresponding H∞ norm for the system linearized at the equilibrium can be made smaller than γ by linear state feedback. Necessary and sufficient conditions for the latter problem are by now well-known, e.g. from the state space approach to linear H∞ optimal control. Our approach to the nonlinear H∞ optimal control problem generalizes the state space approach to the linear H∞ problem by replacing the Hamiltonian matrix and corresponding Riccati equation as used in the linear context by a Hamiltonian vector field together with a Hamiltonian-Jacobi equation corresponding to its stable invariant manifold.

Dynamics and control of a class of underactuated mechanical systems

This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable.

An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators

The aim of this paper is to provide an intrinsic Hamiltonian formulation of the equations of motion of network models of non-resistive physical systems. A recently developed extension of the classical Hamiltonian equations of motion considers systems with state space given by Poisson manifolds endowed with degenerate Poisson structures, examples of which naturally appear in the reduction of systems with symmetry. The link with network representations of non-resistive physical systems is established using the generalized bond graph formalism which has the essential feature of symmetrizing all the energetic network elements into a single class and introducing a coupling unit gyrator. The relation between the Hamiltonian formalism and network dynamics is then investigated through the representation of the invariants of the system, either captured in the degeneracy of the Poisson structure or in the topological constraints at the ports of the gyrative type network structure. This provides a Hamiltonian formulation of dimension equal to the order of the physical system, in particular, for odd dimensional systems. A striking example is the direct Hamiltonian formulation of electrical LC networks.

Stabilization of Hamiltonian systems

1. HAMILTONIAN SYSTEMS IN THIS paper we will be concerned with the stabilization by feedback of Hamiltonian systems. In order to facilitate our discussions (especially when applying Lyapunov’s second method) we will restrict ourselves to a particular, although natural, subclass of Hamiltonian systems given in the following way [l]. Let Q be an n-dimensional smooth manifold, denoting the configuration space, and let T*Q be the cotangent bundle, denoting the phase space or srufe space. Furthermore there is a smooth m-dimensional output manifold Y (m < n) and a smooth output map C : Q- Y. (Smooth will mean C” or Ck, with k sufficiently big, although we shall restrict ourselves in the second part of Section 2 to analytic data.) For simplicity we take C to be submersive. so rank dC(q) = m. We assume that the system on T*Q has an internal energy which is the sum of a kinetic energy K and a porentiul energy V. This means that there exists a Riemannian metric (,) on Q, in local coordinates (q,, . . . , q,J for Q given by

On realization of nonlinear systems described by higher-order differential equations

We consider systems of smooth nonlinear differential and algebraic equations in which some of the variables are distinguished as “external variables.” The realization problem is to replace the higher-order implicit differential equations by first-order explicit differential equations and the algebraic equations by mappings to the external variables. This involves the introduction of “state variables.” We show that under general conditions there exist realizations containing a set of auxiliary variables, called “driving variables.” We give sufficient conditions for the existence of realizations involving only state variables and external variables, which can then be split into input and output variables. It is shown that in general there are structural obstructions for the existence of such realizations. We give a constructive procedure to obtain realizations with or without driving variables. The realization procedure is also applied to systems defined by interconnections of state space systems. Finally, a theory of equivalence transformations of systems of higher-order differential equations is developed.

Energy-shaping of port-controlled Hamiltonian systems by interconnection

Passivity-based control (PBC) has shown to be very powerful to design robust controllers for physical systems described by Euler-Lagrange (EL) equations of motion. The application of PBC in regulation problems of mechanical systems yields controllers that have a clear physical interpretation in terms of interconnection of the system with its environment. In particular, the total energy of the closed-loop is the difference between the energy of the system and the energy supplied by the controller. Furthermore, since the EL structure is preserved in closed-loop, PBC is robust vis a vis unmodeled dissipative effects. Unfortunately, these nice properties are sometimes lost when PBC is used in other applications, for instance, in electrical and electromechanical systems. In this paper we further contribute to develop a new PBC theory encompassing a broader class of systems, and preserving the aforementioned energy-balancing stabilization mechanism and the structure invariance, continuing upon our previous work. Towards this end we consider port-controlled Hamiltonian systems with dissipation (PCHD), which result from the network modeling of energy-conserving lumped-parameter physical systems with independent storage elements, and strictly contain the class of EL models.

Nonlinear State Space H ∞ Control Theory

Although the H ∞ control problem was originally formulated [55] as a linear design problem in the frequency domain (in fact, H ∞ stands for the Hardy space of complex functions bounded and analytic in the open righthalf complex plane), it can be naturally translated to the time-domain and extended to nonlinear state-space systems. Indeed, the standard H ∞ control problem can be equivalently formulated as the optimal attenuation of the L 2-induced norm from exogenous inputs (inputs with unknown power spectrum) to the to-be-controlled outputs, under the constraint of internal stability. Also, although early research in H ∞ control was conducted solely using frequency domain methods, a satisfactory state space solution to the linear H ∞ (sub-)optimal control problem was reached by the end of the eighties (see especially [12], [31], [17], [46], [16], [30], [48], [49], [47]). Moreover, this state space solution relies on tools familiar from LQ and LQG theory, in particular Riccati equations and Hamiltonian matrices. In the classical paper by Willems on LQ control [52] the relations of these tools with the underlying notion of dissipativity were being stressed; while in [53] dissipativity was defined for general nonlinear systems, encompassing notions of passivity of physical systems and input-output stability of nonlinear (feedback) systems. The resulting dissipation inequalities were fruitfully explored in e.g. [35], [20], [21], also linking them to the Hamilton-Jacobi equation from classical nonlinear optimal control (see also [36]).

Interconnection of port-Hamiltonian systems and composition of Dirac structures

Port-based network modeling of physical systems leads to a model class of nonlinear systems known as port-Hamiltonian systems. Port-Hamiltonian systems are defined with respect to a geometric structure on the state space, called a Dirac structure. Interconnection of port-Hamiltonian systems results in another port-Hamiltonian system with Dirac structure defined by the composition of the Dirac structures of the subsystems. In this paper the composition of Dirac structures is being studied, both in power variables and in wave variables (scattering) representation. This latter case is shown to correspond to the Redheffer star product of unitary mappings. An equational representation of the composed Dirac structure is derived. Furthermore, the regularity of the composition is being studied. Necessary and sufficient conditions are given for the achievability of a Dirac structure arising from the standard feedback interconnection of a plant port-Hamiltonian system and a controller port-Hamiltonian system, and an explicit description of the class of achievable Casimir functions is derived.