Quaternion-Based Attitude Stabilization via Discrete-Time IDA-PBC

Abstract

In this paper, we propose a new sampled-data controller for stabilization of the attitude dynamics at a desired constant configuration. The design is based on discrete-time interconnection and damping assignment (IDA) passivity-based control (PBC) and the recently proposed Hamiltonian representation of discrete-time nonlinear dynamics. Approximate solutions are provided with simulations illustrating performances.

Passivity Techniques and Hamiltonian Structures in Discrete Time

“…Further on, the fundamental characteristics of port-Hamiltonian structures as the qualifying closeness property under power-preserving interconnection [40,38], are validated. As a consequence, these forms are efficient for the design of average passivity based control strategies for complex and networked discrete-time dynamics as illustrated in [41,22,23,38]. Port-Hamiltonian structures are discussed in Section 4 while dedicated studies are in [39,41,38].…”

“…Definition 5 properly states that the discrete gradient function satisfying (23) describes the rate of change of this function between two states. It is not uniquely defined and different methods to solve the equality can be worked out [11,24,12].…”

“…Specifying the result in Theorem 4 to port-Hamiltonian structures, one gets preservation of the port-Hamiltonian structure under power-preserving interconnection; a qualifying property to discuss energy management based control schemes. Examples in this direction are developed in [41,43,42,22,23,38].…”

“…In the proposed differential algebraic framework, discrete-time Port-controlled Hamiltonian structures that validate the usually required energy balance properties are described in the proposed differential algebraic framework. Accordingly, the basic stabilizing techniques behind energy-based control strategies are revisited to confirm the open perspectives regarding control design through energy management along the lines developed in [41,22,23,38]. All the material discussed in this paper regards a purely discrete time setting but it can be specified to the sampled-data context.…”

A gradient descent algorithm built on approximate discrete gradients

“…This work is inspired by the recent contribution of the authors [16], [17] where an exact sampled-data equivalent model to a gradient dynamics has been computed. It results that the discrete-time dynamics is implicitly defined via the discrete-gradient function [18]- [20]. Since such a dynamics represents exactly the dynamics of the gradient at the sampling instants (corresponding to the step size), one can naturally raise the question: is it possible to define an optimization algorithm based on a suitably computed approximation of the discrete equivalent dynamics?…”