[Paper Review] Simultaneous Interconnection and Damping Assignment Passivity-based Control of Mechanical Systems Using Generalized Forces

Alejandro Donaire, Roméo Ortega+1 more|arXiv (Cornell University)|Jun 25, 2015

Control and Stability of Dynamical Systems10 references4 citations

TL;DR

This paper introduces a generalized Simultaneous Interconnection and Damping Assignment Passivity-Based Control (SIDA-PBC) framework for mechanical systems by enabling simultaneous energy shaping and damping injection, and by incorporating generalized forces beyond standard gyroscopic terms. The key contribution is showing that several previously non-systematic controller designs actually conform to this extended SIDA-PBC class, thereby unifying and systematizing their design under a rigorous passivity-based methodology.

ABSTRACT

To extend the realm of application of the well known controller design technique of interconnection and damping assignment passivity-based control (IDA-PBC) of mechanical systems two modifications to the standard method are presented in this article. First, similarly to [1], it is proposed to avoid the splitting of the control action into energy-shaping and damping injection terms, but instead to carry them out simultaneously. Second, motivated by [2], we propose to consider the inclusion of generalised forces, going beyond the gyroscopic ones used in standard IDA-PBC. It is shown that several new controllers for mechanical systems designed invoking other (less systematic procedures) that do not satisfy the conditions of standard IDA-PBC, actually belong to this new class of SIDA-PBC.

Motivation & Objective

• To extend the applicability of Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) by relaxing constraints on force modeling and design procedure.
• To overcome the limitations of standard IDA-PBC, which splits energy shaping and damping injection into separate steps and restricts forces to gyroscopic forms.
• To demonstrate that several recent, non-systematic controller designs for mechanical systems actually belong to the new SIDA-PBC framework with generalized forces.
• To unify diverse controller designs under a single, systematic, and well-established control methodology.

Proposed method

• Proposes a simultaneous design procedure that merges energy shaping and damping injection into a single step, avoiding the loss of generality inherent in two-step IDA-PBC.
• Introduces generalized forces in the target dynamics, extending beyond skew-symmetric (gyroscopic) forces to include more general force structures.
• Derives matching equations that incorporate both generalized forces and simultaneous control actions, ensuring passivity and stability.
• Uses a port-Hamiltonian formulation to represent the system dynamics and defines a target Hamiltonian with modified kinetic and potential energy terms.
• Applies a Lyapunov-based stability analysis to verify that the closed-loop system achieves asymptotic stability at the desired equilibrium.
• Demonstrates the framework's generality by showing that existing controllers not fitting standard IDA-PBC do satisfy the new SIDA-PBC conditions with generalized forces.

Experimental results

Research questions

• RQ1 Can the standard two-step IDA-PBC methodology be generalized to allow simultaneous energy shaping and damping injection without loss of generality?
• RQ2 Does the inclusion of generalized forces—beyond the standard gyroscopic forces—broaden the class of mechanical systems amenable to IDA-PBC?
• RQ3 Are controller designs previously derived via ad hoc or non-systematic methods actually consistent with a more general SIDA-PBC framework?
• RQ4 Can the new SIDA-PBC framework with generalized forces unify and systematize diverse recent controller designs for underactuated mechanical systems?

Key findings

• The proposed SIDA-PBC with generalized forces successfully stabilizes the underactuated beam-ball system, with the controller (90) proven to satisfy the matching equation (75) and stability condition (74).
• The controller design from [17] for the beam-ball system, previously derived via non-systematic means, is shown to conform to the new SIDA-PBC framework with generalized forces.
• The matrix Λ(q, p) derived from the controller (90) is shown to satisfy the stability condition (74), with the first term being skew-symmetric and the second positive definite, ensuring asymptotic stability.
• The inclusion of generalized forces does not reduce the number of PDEs to solve, but significantly extends the realm of application of IDA-PBC, as demonstrated by the successful control of systems previously outside its scope.
• The framework unifies previously disparate controller designs under a single, systematic methodology, demonstrating that they are instances of the generalized SIDA-PBC approach.
• The method is validated through explicit construction of the matching equations and stability analysis, confirming that the closed-loop dynamics conform to the target port-Hamiltonian structure (25).