Bernhard Maschke

Putting energy back in control

Energy is one of the fundamental concepts in science and engineering practice, where it is common to view dynamical systems as energy-transformation devices. This perspective is particularly useful in studying complex nonlinear systems by decomposing them into simpler subsystems that, upon interconnection, add up their energies to determine the full systems behavior. The action of a controller may also be understood in energy terms as another dynamical system. The control problem can then be recast as finding a dynamical system and an interconnection pattern such that the overall energy function takes the desired form. This energy-shaping approach is the essence of passivity-based control (PBC), a controller design technique that is very well known in mechanical systems. Our objectives in the article are threefold. First, to call attention to the fact that PBC does not rely on some particular structural properties of mechanical systems, but hinges on the more fundamental (and universal) property of energy balancing. Second, to identify the physical obstacles that hamper the use of standard PBC in applications other than mechanical systems. In particular, we show that standard PBC is stymied by the presence of unbounded energy dissipation, hence it is applicable only to systems that are stabilizable with passive controllers. Third, to revisit a PBC theory that has been developed to overcome the dissipation obstacle as well as to make the incorporation of process prior knowledge more systematic. These two important features allow us to design energy-based controllers for a wide range of physical systems.

Dissipative Systems Analysis and Control: Theory and Applications

This is an up-dated addendum/erratum to the second edition of the book Dissipative Systems Analysis and Control, Theory and Applications, Springer-Verlag London, 2nd Edition, 2007.

Port-controlled Hamiltonian systems: modelling origins and systemtheoretic properties

It is shown that the network representation (as obtained through the generalized bond graph formalism) of non-resistive physical systems in interaction with their environment leads to a well-defined class of (nonlinear) control systems, called port-controlled Hamiltonian systems. A first basic feature of these systems is that their internal dynamics is Hamiltonian with respect to a Poisson structure determined by the topology of the network and to a Hamiltonian given by the stored energy. Secondly the network representation provides automatically (intrinsically to the notation) to every port-control variable (input) a port-conjugated variable as output. This definition of port-conjugated input and output variables, based on energy considerations, is shown to have important consequences for the observability and controllability properties, as well as the external characterization of port-controlled Hamiltonian systems.

On the Hamiltonian formulation of nonholonomic mechanical systems

A simple procedure is provided to write the equations of motion of mechanical systems with constraints as Hamiltonian equations with respect to a “Poisson” bracket on the constrained state space, which does not necessarily satisfy the Jacobi identity. It is shown that the Jacobi identity is satisfied if and only if the constraints are holonomic.

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.

Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators

Associated with a skew-symmetric linear operator on the spatial domain

Geometric scattering in robotic telemanipulation

[a,b]

An intrinsic Hamiltonian formulation of the dynamics of LC-circuits

we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated with this Dirac structure is an infinite-dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated with this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko beam.

Hamiltonian discretization of boundary control systems

In this paper, we study the interconnection of two robots, which are modeled as port-controlled Hamiltonian systems through a transmission line with time delay. There will be no analysis of the time delay, but its presence justifies the use of scattering variables to preserve passivity. The contributions of the paper are twofold: first, a geometrical, multidimensional, power-consistent exposition of telemanipulation of intrinsically passive controlled physical systems, with a clarification on impedance matching, and second, a system theoretic condition for the adaptation of a general port-controlled Hamiltonian system with dissipation (port-Hamiltonian system) to a transmission line.

Bond graph modelling for chemical reactors

First, the dynamics of LC-circuits are formulated as a Hamiltonian system defined with respect to a Poisson bracket which may be degenerate, i.e., nonsymplectic. This Poisson bracket is deduced from the network graph of the circuit and captures the dynamic invariants due to Kirchhoffs laws. Second, the antisymmetric relations defining the Poisson bracket are realized as a physical network using the gyrator element and partially dualizing the network graph constraints. From the network realization of the Poisson bracket, the reduced standard Hamiltonian system as well as the realization of the embedding standard Hamiltonian system are deduced. >