van der Arjan Schaft

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.

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

In this paper, we propose a constructive procedure to modify the Hamiltonian function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for nonzero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions can be easily constructed. Interestingly enough, for linear systems the resulting Lyapunov function is the incremental energy; thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are given.

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.

Nonlinear H ∞ almost disturbance decoupling

The L2-gain almost disturbance decoupling problem for SISO nonlinear systems is formulated. Sufficient conditions are identified for the existence of a parametrized state feedback controller such that the L2-gain from disturbances to output can be made arbitrarily small by increasing its gain. The controller is explicity constructed using a Lyapunov-based recursive scheme. Sufficient conditions for the solvability of the H∞ almost disturbance decoupling problem and the explicit construction of teh controller are given for a more restrictive class of nonlinear systems.

A receding-horizon approach to the nonlinear H∞ control problem

The receding-horizon (RH) methodology is extended to the design of a robust controller of

Physical Damping in IDA-PBC Controlled Underactuated Mechanical Systems

H_\infty

Equivalence of switching linear systems by bisimulation

type for nonlinear systems. Using the nonlinear analogue of the fake

Port Hamiltonian formulation of infinite dimensional systems I. Modeling

H_\infty

Port Hamiltonian formulation of infinite dimensional systems II. Boundary control by interconnection

H algebraic Riccati equation, we derive an inverse optimality result for the RH schemes for which increasing the horizon causes a decrease of the optimal cost function. This inverse optimality result shows that the input-output map of the closed-loop system obtained with the RH control law has a bounded

Inner-outer factorization for nonlinear noninvertible systems

L_2