UTFSM'2014

Energy based modelling and control of physical systems

In the last decade a powerful control theory based on physical balance equations and conservation laws has been developed for electrical, mechanical and electro-mechanical systems. This control theory is based on the principle of conservation of energy providing a clear physical interpretation of control design problems. Particular interest has been given to systems modeled as Hamiltonian control systems. Traditionally these systems arise from the Euler-Lagrange equations of motion, however they have been extended to deal with network models of general physical systems by using the framework of port-Hamiltonian systems (PHS). In network models the system is considered as the interconnection of energy storing elements via basic physical interconnection laws (e.g. Newtons third law or Kirchhof's law) together with energy dissipating elements. PHS theory formalizes the basic interconnection laws together with the power-conserving elements by a geometric (interconnection) structure, and defines the Hamiltonian function as the total energy stored in a system. Thus, PHS have direct physical interpretation since the the physical balance equations are directly derived from the interconnection structure and the physical energy of the system. Models based on energy approaches are very useful tools for engineers, since they are defined in terms of energy, which is a fundamental concept common to all engineering domains. They are also related with other network models, such as bond-graphs.

In this course we study the basic concepts of energy based modelling and their use for the modelling and control of simple and complex physical systems.

• In the first part (2 lectures), we will study models arising from physical balance equations. The port-Hamiltonian model is derived for classes of (linear and non-linear) mechanical, electrical and hydraulic applications and it is shown how models of complex physical systems can be constructed systematically via the interconnecting of simple sub-systems. We will also introduce how this approach may be used for the modelling of multi-energy or multi-physical systems (i.e., systems arising from the interconnection of different physical domains).
• In a second part (2 lectures) we will study how the particular structure of PHS can be use for energy based control design. For this purpose we will study the notion of passive system and specialize it to PHS. We will show how this approach is specially useful when dealing with the control of non-linear systems and how the physical energy is the base for the closed-loop stability analysis (passivity based control).

Pre-requisites: Undergraduate courses in linear dynamical systems. Some basic notions on control are useful but not necessary.

Bibliography:

[1] A.J. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, Lect. Notes in Control and Information Sciences, Vol. 218, Springer-Verlag, Berlin, 1996, p. 168, 2nd revised and enlarged edition, Springer-Verlag, London, 2000 (Springer Communications and Control Engineering series), p. xvi+249.

[2] Brogliato, B., Lozano, R., Maschke, B., and Egeland, O. (2007). Dissipative Systems Analysis and Control. Communications and Control Engineering Series. Springer Verlag, London, 2nd edition.

Lectures

Each lecture is split into blocks of 45 minutes, with a 15 minute break in between.

• Monday January 13/01 [Slides]

1. Balance equations, conservation laws, and passive systems. [10:15 - 12:00]

2. Port-Hamiltonian systems. [14:00 - 15:45]

• Tuesday January 14/01 [Slides]

3. Passivity based control [14:00 - 15:45]

• Thursday January 16/01 [Slides]

4. Passivity based control of port-Hamiltonian systems. [10:15 - 12:00]

Final project

The final project is available here.

Contact

• Hector Ramirez, UFC, FEMTO-ST, Besançon France. e-mail: hector.ramirez@femto-st.fr
• Juan Yuz, UTFSM, Valparaiso Chile. e-mail: juan.yuz@usm.cl

Héctor Ramirez, Juan Yuz and Thomas Schön at the UTFSM