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Frequency domain techniques for H control of distributed parameter systems
Özbay H., Gümüssoy S., Kashima K., Yamamoto Y., SIAM-Society for Industrial and Applied Mathematics, Philadelphia, PA, 2018. 192 pp. Type: Book (978-1-611975-39-0)
Date Reviewed: Aug 23 2019

In traditional control theory, the object to be controlled (called a plant) is described by a differential equation (or by a system of such equations) in which the rate of change of the plant’s state depends on the current control value. Analyzing differential equations is in general not easy. In many practical situations, we can safely ignore terms that are quadratic (or of higher order) and thus conclude that the corresponding system is linear. In such a linear case, frequency domain techniques are very helpful; if we apply the Laplace transform to both sides of the corresponding linear differential equation, we get an easy-to-solve algebraic equation--since the Laplace transform of the derivative is simply s times the Laplace transform of the original function. These techniques help us solve many control problems, for example, compute the optimal control or, in situations when the system’s parameters are known with uncertainty, compute the robust control, that is, control under which the system remains stable for all possible values of these parameters.

In real life, plants are often more complicated. For example, the traditional approach assumes that a control leads to an immediate change in the system, while in reality there is usually some delay. Another complication is that plants are often distributed: to describe the state of such a plant, we need to describe the values of the corresponding quantities at each spatial location; a good example of such a plant is a beam--the main element of buildings and other structures. In such situations, we have partial differential equations instead of the ordinary ones. Finally, in some practical situations, a more adequate description is provided not by the usual differential equations, but by fractional differential equations with properly defined derivatives of fractional order (for example, the derivative of order 1/2 can be defined as an operator whose iteration leads to the usual differentiation). For such systems, a natural generalization of the usual stability is the following H property: there is a constant C such that the L2 norm of the output does not exceed C times the L2-norm of the input. In all such cases, as long as the system remains linear, it is possible to describe it in Laplace transform terms; however, the corresponding functions of s are no longer rational functions, so the traditional algorithms for generating control are no longer applicable.

In this book, the authors describe new techniques that provide efficient ways of computing control for such systems. Most of these techniques have been developed by the authors themselves in the last decade. These techniques are illustrated by several numerical examples, with a step-by-step analysis nicely illustrated by appropriate graphs. Many algorithms come with an explanation of how to implement the corresponding calculations in MATLAB. Application examples include more traditional control problems, as well as not so traditional ones: flow control in an oil/gas pipeline, control of investments, control of information flow via a router, control of a beam, control of cell population dynamics in acute leukemia, and so on. Most of the examples are for single-input, single-output (SISO) systems, but some multiple input, multiple output (MIMO) examples are provided too.

This book will be a useful tool for engineers and researchers interested in realistic control problems beyond the usual simplifying formulations. It can also serve as a textbook for a graduate course.

Reviewer:  V. Kreinovich Review #: CR146669 (1911-0383)
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