Periodic Systems - Filtering and Control

The purpose of this volume is to provide a comprehensive view of the theory of periodic systems, by presenting the fundamental analytical tools for their analysis and the state of the art in the associated problems of filtering and control. This book is intended to be useful as a reference for all scientists, students, and professionals who have to investigate periodic phenomena or to study periodic control systems. For teaching purposes, the volume can be used as a textbook for a graduate course, especially in engineering, physics, economics and mathematical sciences. The treatment is self-contained; only a basic knowledge of input-output and state-space representations of dynamical systems is assumed. Periodic Systems Ordinary differential equations with periodic coefficients have a long history in physics and mathematics going back to the contributions of the 19th century by Faraday [134], Mathieu [230], Floquet [145], Rayleigh [250] and [251], Hill [184], and many others. As an intermediate class of systems bridging the time-invariant realm to the time-varying one, periodic systems are often included as a regular chapter in textbooks of differential equations or dynamical systems, such as [123, 175, 237, 256, 303]. In the second half of the 20th century, the development of systems and control theory has set the stage for a renewed interest in the study of periodic systems, both in continuous and in discrete-time, see e.g., the books [136, 155, 228, 252, 312] and the survey papers [29, 42]. This has been emphasized by specific application demands, in particular in industrial process control, see [1, 43, 76, 266, 267, 299], communication systems, [119, 144, 282], natural sciences, [225] and economics, [148, 161]. ix x Preface Periodic Control The fact that a periodic operation may be advantageous is well-known to mankind since time immemorial. All farmers know that it is not advisable to always grow the same crop over the same field since the yield can be improved by adopting a crops rotation criterion. So, cycling is good. In more recent times, similar concepts have been applied to industrial problems. Traditionally, almost every continuous industrial process was set and kept, in presence of disturbances, at a suitable steady state. It was the task of the designer to choose the optimal stationary regime. If the stationary requirement can be relaxed, the question arises whether a periodic action can lead to a better performance than the optimal stationary one. This observation germinated in the field of chemical engineering where it was seen that the performance of a number of catalytic reactors improved by cycling, see the pioneer contributions [17, 153, 189]. Suitable frequency domain tests have been developed to this purpose in the early 1970s, [25, 62, 162, 163, 172, 173, 275]. Unfortunately, as pointed out in [15], periodic control was still considered "too advanced" in the scenario of industrial control, in that "the steady-state operation is the norm and unsteady process behavior is taboo". Its interest was therefore confined to advanced applications, such as those treated in [274] and [257]. However, in our days, the new possibilities offered by the control technology, together with the theoretical developments of the field, opened the way for a wide use of periodic operations. For example, periodic control is useful in a variety of problems concerning under-actuated systems, namely systems with a limited number of control inputs with respect to the degrees of freedom. In this field, control is often performed by imposing a stable limit cycle, namely an isolated and attractive periodic orbit, [97, 150, 179]. Another example comes from non-holonomic mechanical systems, where in some cases stabilization cannot be achieved