ROMEO ORTEGA (Laboratoire des Signaux et Systèmes, LSS/CNRS/Supélec, France)

Passivity-Based Control of Physical Systems: Control by Interconnection and State-Feedback Laws (6 hours)

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As vividly illustrated by the quintessential Watt's governor a natural procedure to modify the behavior of a dynamical system is to interconnect it with another dynamical system. Examples of this approach abound in modern high-performance practical applications and are proven to be very robust and reliable. These include, among many others, mechanical suspension and flapper systems, flotation devices, damping windings and impedance matching filters in electrical systems. (It may be even argued that biological and medical applications, of great current interest, are best studied invokinginterconnection principles instead of simplistic causeeffect preconceptions.)

Adopting the interconnection perspective allows us to formulate the control problem in terms of the physical properties of the systems like energy-shaping and damping injection, it furthermore underscores the role of interconnection to achieve these objectives. This should be contrasted with the classical actuator-plant-sensor paradigm that leads to a signal-processing view of control in which the systems physical properties are di±cult to incorporate.

In our previous works we have proposed a mathematical framework to design controllers using the aforementioned systems interconnection perspective that we called Control by Interconnection (CbI). Towards this end we restricted ourselves to systems described by Port-Hamiltonian (PH) models, which suitably describe the dynamics of many physical processes, and where the importance of the energy function, the interconnection pattern and the dissipation of the system is highlighted. In CbI the controller is another PH system connected to the plant (through a power-preserving interconnection) to add up their energy functions. In spite of the conceptual appeal of formulating the control problem as the interaction of dynamical systems, the current version of CbI imposes a severe restriction on the plant dissipation structure that stymies its practical application.

The purpose of this course is to propose some extensions to the CbI method to make it more widely applicable|in particular, to overcome the dissipation obstacle [1]. Furthermore, we establish the connections between CbI and Standard PassivityBased Control (PBC). Standard PBC, where energy shaping is achieved via static (or dynamic) state feedback, is one of the most successful controller design techniques [2] [3]. However, the control law is usually derived either from an uninspiring and non-intuitive "passive output generation" viewpoint or from an, equally restrictive, model matching perspective-where a quadratic storage function is assigned to the error dynamics. We prove in this talk that Standard PBC is obtained restricting CbI to a suitable subset of the state space|providing a nice geometric interpretation to Standard PBC.

The application of the methods is illustrated with the following practical examples: electromechanical systems (double-fed induction machines, synchronous motors and micro-electromechanical systems), power system stabilizers, switched power converters and underactuated mechanical systems.

References

[1] R. Ortega, A. van der Schaft, F. Casta~nos and A. Astolfi, Control by state{modulated interconnection of port{Hamiltonian systems, IEEE Trans. Automat. Contr., Vol. 53, No. 11, pp. 2527{2542, 2008.

[2] R. Ortega, A. Loria, P. J. Nicklasson and H. Sira{Ramirez, Passivity-Based Control of Euler{Lagrange Systems, Springer-Verlag, Berlin, Communications and Control Engineering, 1998.

[3] A. Astolfi, D. Karagiannis and R. Ortega, Nonlinear and Adaptive Control with Applications, Springer-Verlag, Berlin, Communications and Control Engineering, 2007.