Global complete observability and output-to-state stability imply the existence of a globally convergent observer

We consider systems which are globally completely observable and output-to-state stable. The former property guarantees the existence of coordinates such that the dynamics can be expressed in observability form. The latter property guarantees the existence of a state norm observer and therefore the possibility of bounding any continuous state functions. Both properties allow to conceptually build an observer from an approximation of an exponentially attractive invariant manifold in the space of the system state and an output driven dynamic extension. The proposed observer provides convergence to zero of the estimation error within the domain of definition of the solutions. The work of A. Astolfi is partly supported by the Leverhulme Trust.     

Asymptotic stabilization of some equilibria of an underactuated underwater vehicle

The stabilization problem for selected relative equilibria of an underactuated rigid body, modelling a simple underwater vehicle, moving in an ideal fluid is addressed. State feedback control laws achieving local asymptotic stability of a forward motion and of a diving/rising with forward/reverse motion are proposed. The control design exploits the Hamiltonian nature of the system to be controlled and it is based on the so-called interconnection and damping assignment (IDA) procedure. Simulation results complete the work.     

Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations Via Coordinate Changes

Total energy shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function - that qualifies as Lyapunov function for the desired equilibrium. The success of the method relies on the possibility of solving two PDEs which identify the kinetic and potential energy functions that can be assigned to the closed loop. Particularly troublesome is the partial differential equation (PDE) associated to the kinetic energy which is nonlinear and inhomogeneous and the solution, that defines the desired inertia matrix, must be positive-definite. In this note, we prove that we can eliminate or simplify the forcing term in this PDE by modifying the target dynamics and introducing a change of coordinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coordinates, the possibility of simplifying the PDEs is determined by the interaction between the Coriolis and centrifugal forces and the actuation structure. The examples of pendulum on a cart and Furuta's pendulum are used to illustrate the results.     

Some results on disturbance attenuation for Hamiltonian systems via direct discrete‐time design

The disturbance attenuation and robust disturbance attenuation problems for Hamiltonian systems in the discrete‐time setting are considered and some new results are presented. The new results are derived utilizing the recently presented dissipativity equality obtained by adding the dissipation rate function to the classical dissipativity inequality. A selection of the dissipation rate function yields new results. These results include a condition on the dissipation structure of the system to achieve the desired disturbance attenuation level and gives direct construction of optimal control laws for any desired disturbance attenuation level. The results remove the need to solve Hamilton–Jacobi–Isaacs inequalities. Copyright © 2014 John Wiley & Sons, Ltd.     

Global asymptotic stabilization of the attitude and the angular rates of an underactuated non-symmetric rigid body

The paper deals with the global stabilization of both the attitude and the angular velocities of an underactuated rigid body. First a stability theorem is proven for a class of systems; subsequently, the equations describing the physics of the rigid body are presented, showing that the rigid body belongs to the considered class of systems, and a sufficient condition for the application of the theorem to the stability of the rigid body equilibrium is pointed out. Finally, some simulation results are reported showing the effectiveness of the proposed methodology.     

Noninteracting control with stability for Hamiltonian systems

The problem of noninteraction with stability via dynamic state feedback is addressed and solved for a class of nonlinear Hamiltonian systems. A simple way to check necessary and sufficient conditions is proposed. It is well known that to decide if the problem is solvable, and which class of state feedback has to be used, the stability properties of some special dynamics are to be investigated. For this reason, on the way to the main result, it is shown that such dynamics are not necessarily Hamiltonian. Several examples, clarifying the role of different classes of state-feedback control laws (either static or dynamic) in the solution of the problem, are proposed     

Nonlinear control of feedforward systems with bounded signals

The stabilization problem for a class of nonlinear feedforward systems is solved using bounded control. It is shown that when the lower subsystem of the cascade is input-to-state stable and the upper subsystem not exponentially unstable, global asymptotic stability can be achieved via a simple static feedback having bounded amplitude that requires knowledge of the "upper" part of the state only. This is made possible by invoking the bounded real lemma and a generalization of the small gain theorem. Thus, stabilization is achieved with typical saturation functions, saturations of constant sign, or quantized control. Moreover, the problem of asymptotic stabilization of a stable linear system with bounded outputs is solved by means of dynamic feedback. Finally, a new class of stabilizing control laws for a chain of integrators with input saturation is proposed. Some robustness issues are also addressed and the theory is illustrated with examples on the stabilization of physical systems.     

Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one

Interconnection and damping assignment passivity-based control is a new controller design methodology developed for (asymptotic) stabilization of nonlinear systems that does not rely on, sometimes unnatural and technique-driven, linearization or decoupling procedures but instead endows the closed-loop system with a Hamiltonian structure with a desired energy function-that qualifies as Lyapunov function for the desired equilibrium. The assignable energy functions are characterized by a set of partial differential equations that must be solved to determine the control law. We prove in this paper that for a class of mechanical systems with underactuation degree one the partial differential equations can be explicitly solved. Furthermore, we introduce a suitable parametrization of assignable energy functions that provides the designer with a handle to address transient performance and robustness issues. Finally, we develop a speed estimator that allows the implementation of position-feedback controllers. The new result is applied to obtain an (almost) globally stabilizing scheme for the vertical takeoff and landing aircraft with strong input coupling, and a controller for the pendulum in a cart that can swing-up the pendulum from any position in the upper half plane and stop the cart at any desired location. In both cases we obtain very simple and intuitive position-feedback solutions.     

Construction of discrete-time models for port-controlled Hamiltonian systems with applications

The issues of constructing a discrete-time model for Hamiltonian systems are in general different from those for dissipative systems. We propose an algorithm for constructing an approximate discrete-time model, which guarantees Hamiltonian conservation. We show that the algorithm also preserves, in a weaker sense, the losslessness property of a class of port-controlled Hamiltonian systems. An application of the algorithm to port-controlled Hamiltonian systems with quadratic Hamiltonian is presented, and we use this to solve the stabilization problem for this class of systems based on the approximate discrete-time model constructed using the proposed algorithm. We illustrate the usefulness of the algorithm in designing a discrete-time controller to stabilize the angular velocity of the dynamics of a rigid body.