Alejandro Donaire

Brief paper: On the addition of integral action to port-controlled Hamiltonian systems

A technique that provides closed loop integral action depending on the passive outputs of port-controlled Hamiltonian systems is already available. This paper addresses a new method that allows us to add integral action also on system variables having relative degree higher than one, while still preserving the Hamiltonian form and, thus, closed loop stability. The new approach is applied to design speed regulation controllers for the permanent magnet synchronous motor. Closed loop stability and asymptotic rejection of unknown piecewise constant load torques are formally proved. This theoretically predicted control system performance is illustrated via simulation experiments, which also show that the properties hold under parameter uncertainties. This is in line with the usual practice of including integral action in a controller with the aim of improving its closed loop robustness. The fact that the method enhances the range of possible integral actions in the controller, enriched with this robustness property, allows us to assess it as a practically important complement to the well-known interconnection and damping assignment techniques developed in the framework of port-controlled Hamiltonian systems.

Robust energy shaping control of mechanical systems

The problem of robustness improvement, vis a vis external disturbances, of energy shaping controllers for mechanical systems is addressed in this paper. First, it is shown that, if the inertia matrix is constant, constant disturbances (both, matched and unmatched) can be rejected simply adding a suitable integral action—interestingly, not at the passive output. For systems with non-constant inertia matrix, additional damping and gyroscopic forces terms must be added to reject matched disturbances and, moreover, enforce the property of integral input-to-state stability with respect to matched disturbances. The stronger property of input-to-state stability, this time with respect to matched and unmatched disturbances, is ensured with further addition of nonlinear damping. Finally, it is shown that including a partial change of coordinates, the controller can be significantly simplified, preserving input-to-state stability with respect to matched disturbances.

Brief paper: Dynamic positioning of marine craft using a port-Hamiltonian framework

Dynamic positioning of marine craft refers to the use of the propulsion system to regulate the vessel position and heading. This type of motion control is commonly used in the offshore industry for surface vessels, and it is also used for some underwater vehicles. In this paper, we use a port-Hamiltonian framework to design a novel nonlinear set-point-regulation controller with integral action. The controller handles input saturation and guarantees internal stability, rejection of unknown constant disturbances, and (integral-)input-to-state stability.

Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations

Control of underactuated mechanical systems via energy shaping is a well-established, robust design technique. Unfortunately, its application is often stymied by the need to solve partial differential equations (PDEs). In this technical note a new, fully constructive, procedure to shape the energy for a class of mechanical systems that obviates the solution of PDEs is proposed. The control law consists of a first stage of partial feedback linearization followed by a simple proportional plus integral controller acting on two new passive outputs. The class of systems for which the procedure is applicable is identified imposing some (directly verifiable) conditions on the systems inertia matrix and its potential energy function. It is shown that these conditions are satisfied by three benchmark examples.

Passivity-based control for multi-vehicle systems subject to string constraints

In this paper, we show how heterogeneous bidirectional vehicle strings can be modelled as port-Hamiltonian systems. Analysis of stability and string stability within this framework is straightforward and leads to a better understanding of the underlying problem. Nonlinear local control and additional integral action is introduced to design a suitable control law guaranteeing l 2 string stability of the system with respect to bounded disturbances.

Energy shaping, interconnection and damping assignment, and integral control in the bond graph domain

This paper presents a methodology to perform energy shaping and interconnection and damping assignment in the bond graph domain, and addresses a new result on integral action control which improves the robustness of these passivity based control methods, which were first introduced on the port-controlled Hamiltonian systems with dissipation formalism. The methods perform expressing the desired closed-loop energy, interconnection and damping properties on a so-called target bond graph, which is built as follows on the plant bond graph: (i) adding virtual storage elements enforces a minimum on the target bond graph energy at a prespecified stable closed-loop equilibrium state, (ii) changing the R-field and its interconnection with the rest of the graph assigns a new dissipation function, and (iii) suitably inserting of power conserving elements (bonds and other structural BG-elements) among junctions yields the desired power conserving interconnection structure. The control law is then determined developing physically based heuristics and formal techniques on the target and plant bond graphs, which deliver a set of partial differential equations to be solved. The method to achieve integral control consists in adding to the target bond graph virtual elements representing the integral action and a change of variables, and then computing the integral control law with standard BG equation-reading procedures. These BG heuristics and prototyping allow to provide integral action on outputs of relative degree greater than one, a contribution of this work not previously available in the literature. This shows that the physical properties of bond graphs are beneficial not only to expediently perform methods contributed by control theory, but also to derive new theoretical results.

Simultaneous interconnection and damping assignment passivity-based control of mechanical systems using dissipative forces

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 Batlle et al. (2009) and Gomez-Estern and van der Schaft (2004), 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 Chang (2014), we propose to consider the inclusion of dissipative forces, going beyond the gyroscopic ones used in standard IDA-PBC. The contribution of our work is the proof that the addition of these two elements provides a non-trivial extension to the basic IDA-PBC methodology. It is also 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.

Robustifying energy shaping control of mechanical systems

The problem of robustness improvement, vis à vis external disturbances, of energy shaping controllers for mechanical systems is addressed in this paper. First, it is shown that-if the inertia matrix is constant-constant disturbances (both, matched and unmatched) can be rejected simply adding a suitable integral action. For systems with non-constant inertia matrix, additional damping and gyroscopic forces terms must be added to reject matched disturbances and, moreover, enforce the property of integral input-to-state stability with respect to matched disturbances. Finally, the stronger property of input-to-state stability, this time with respect to matched and unmatched disturbances, is ensured with further addition of nonlinear damping.

Energy Shaping of Mechanical Systems via PID Control and Extension to Constant Speed Tracking

In a recent contribution, it was shown that a class of mechanical systems, which contains many practical examples, can be stabilized via energy shaping without solving partial differential equations. The proposed controller consists of two terms, a partial linearizing state-feedback and a linear PID loop around two new passive outputs. In this brief note we prove that the first, admittedly non-robust, step can be obviated leaving only the linear PID. A second contribution of the note is to propose a slight modification to the controller to go beyond regulation tasks-being able to follow ramp references in the actuated coordinates.

Disturbance rejection via control by interconnection of port-Hamiltonian systems

In this paper we present a new result on rejection of unmatched external disturbances on port-Hamiltonian systems using Control by Interconnection (CbI). The PHS structure is used to design a controller that rejects unmatched constant disturbances from non-passive outputs. In the PHS framework, the disturbance rejection problem has been addressed adding integral action and using a change of coordinates. In our approach, we avoid a change of coordinates keeping the original state vector, which contains variables with physical interpretation. The methodology proposed in this paper is illustrated on an electrical circuit and on a permanent magnet synchronous motor. Simulation of the later example shows the performance of the control design.