ANDREW D. LEWIS

Queen’s University, Kingston, Canada

Controllability theory (5 hours)

Controllability is a fundamental problem in control theory, dealing with the problem of determining whether it is possible to steer a control system from one state to another.  The property of controllability underlies any control design methodology, and has deep connections to the theory of optimal control.  Despite its fundamental importance in control theory, the property of controllability remains somewhat poorly understood.

 These lectures will begin at a low level, introducing the problem of controllability, considering simple but illustrative examples, and considering the mechanisms for controllability.  Then some simple theorems concerning controllability will be stated and sometimes proved.

Basic theorems concerning the structure of the reachable set will be provided and discussed.

 What will then follow is a survey of known results, plus counterexamples illustrating the limits of applicability of these results.  The emphasis will be on the techniques and tools used in the study of controllability.

Material [PDF]

ANTHONY BLOCH

University Michigan, USA

Geometric Control of Mechanical and Nonholonomic Systems (7 hours)

This course will discuss the the geometry and dynamics of mechanical systems and various techniques for their control. The theory of geometric mechanics gives a general framework for analyzing aspects of mechanical systems theory, in particular notions of symmetry and conservation laws and stability. The theory extends in a natural fashion to nonlinear control theory where one is interested in controllability and stabilization of systems defined on manifolds. These lectures will cover the following topics: Key examples of mechanical and control systems; theory of geometric mechanics including the theory of Lagrangian systems, Hamiltonian systems and nonholonomic systems; topics in nonlinear control theory; controllability of nonholonomic systems, stability and stabilization of mechanical and nonholonomic systems including the energy momentum method and controlled Lagrangians; optimal control of mechanical systems and subRiemannian geometry.

A basic textbook is the book Nonholonomic Mechanics and Control Theory, by A Bloch, with J. Baillieul, P. Crouch and J.Marsden, but many other sources will be used as well.

Material [PDF]

VELIMIR JURDJEVIC

University of Toronto, Canada

Optimal Control on Lie Groups: Integrable Hamiltonian systems (5 hours)

 See PDF file

Passivity-Based Control of Physical Systems: Control by Interconnection

and State-Feedback Laws (6 hours)

PDF-file

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.