There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Guidance And Control Of Ocean Vehicles

The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. The belief that “something” should be minimized was in fact a long-standing conviction of natural philosophers who felt that God had constructed the universe to operate in the most efficient manner—but how that efficiency was to be assessed was subject to interpretation. However, Fermat (1657) had already invoked such a principle successfully in declaring that light travels through a medium along the path of least time of transit. Indeed, it was by recognizing that the brachistochrone should give the least time of transit for light in an appropriate medium that Johann Bernoulli “proved” that it should be a cycloid in 1697. (See Problem 1.1.) And it was Johann Bernoulli who in 1717 suggested that static equilibrium might be characterized through requiring that the work done by the external forces during a small displacement from equilibrium should vanish. This “principle of virtual work” marked a departure from other minimizing principles in that it incorporated stationarity—even local stationarity—(tacitly) in its formulation. Efforts were made by Leibniz, by Euler, and most notably, by Lagrange to define a principle of least action (kinetic energy), but it was not until the last century that a truly satisfactory principle emerged, namely, Hamilton’s principle of stationary action (c. 1835) which was foreshadowed by Poisson (1809) and polished by Jacobi (1848) and his successors into an enduring landmark of human intellect, one, moreover, which has survived transition to both relativity and quantum mechanics. (See [L], [Fu] and Problems 8.11 8.12.)

Variational Principles in Mechanics

A survey of dynamic positioning control systems

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

https://www.steffi-knorn.de/wp-uploads/knorn2014.pdf

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

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.

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.

Alejandro Donaire

Papers

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

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

Robust energy shaping control of mechanical systems

Robust energy shaping control of mechanical systems

Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations