Nonlinear Control Systems

[Kelsi Corsi]

Nonlinearcontrol theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dynamical systems with inputs, and how to modify the output by changes in the input using feedback, feedforward, or signal filtering. The system to be controlled is called the "plant". One way to make the output of a system follow a desired reference signal is to compare the output of the plant to the desired output, and provide feedback to the plant to modify the output to bring it closer to the desired output.

Control theory is divided into two branches. Linear control theory applies to systems made of devices which obey the superposition principle. They are governed by linear differential equations. A major subclass is systems which in addition have parameters which do not change with time, called linear time invariant (LTI) systems. These systems can be solved by powerful frequency domain mathematical techniques of great generality, such as the Laplace transform, Fourier transform, Z transform, Bode plot, root locus, and Nyquist stability criterion.

Nonlinear control theory covers a wider class of systems that do not obey the superposition principle. It applies to more real-world systems, because all real control systems are nonlinear. These systems are often governed by nonlinear differential equations. The mathematical techniques which have been developed to handle them are more rigorous and much less general, often applying only to narrow categories of systems. These include limit cycle theory, Poincar maps, Lyapunov stability theory, and describing functions. If only solutions near a stable point are of interest, nonlinear systems can often be linearized by approximating them by a linear system obtained by expanding the nonlinear solution in a series, and then linear techniques can be used.[1] Nonlinear systems are often analyzed using numerical methods on computers, for example by simulating their operation using a simulation language. Even if the plant is linear, a nonlinear controller can often have attractive features such as simpler implementation, faster speed, more accuracy, or reduced control energy, which justify the more difficult design procedure.

An example of a nonlinear control system is a thermostat-controlled heating system. A building heating system such as a furnace has a nonlinear response to changes in temperature; it is either "on" or "off", it does not have the fine control in response to temperature differences that a proportional (linear) device would have. Therefore, the furnace is off until the temperature falls below the "turn on" setpoint of the thermostat, when it turns on. Due to the heat added by the furnace, the temperature increases until it reaches the "turn off" setpoint of the thermostat, which turns the furnace off, and the cycle repeats. This cycling of the temperature about the desired temperature is called a limit cycle, and is characteristic of nonlinear control systems.

Control design techniques for nonlinear systems also exist. These can be subdivided into techniques which attempt to treat the system as a linear system in a limited range of operation and use (well-known) linear design techniques for each region:

An early nonlinear feedback system analysis problem was formulated by A. I. Lur'e.Control systems described by the Lur'e problem have a forward path that is linear and time-invariant, and a feedback path that contains a memory-less, possibly time-varying, static nonlinearity.

Non-linear control systems have gained importance in many industrial areas and research has undergone significant developments recently. There are significant challenges in various fields of nonlinear control. The broad non-linear control community address these challenges with its focus on latest developments in theory and applications as well as related areas of research and engineering.

Acknowledged as the major international gathering of leading experts in industry and academia in the field of nonlinear control, NOLCOS aims at strengthening contacts between academia and industry to build up new networks and cultivate existing relations. High-level speakers will present the global spectrum of nonlinear control systems, state-of-the-art applications and developing directions.

Other series of events are sponsored by this TC as MICNON, ACNDC, LHMNCS, and TFMST. See the left menu for the list of sponsored and co-sponsored conferences. Left menu contains also the members list and all documents related to the TC activities (in particular on the TC meetings).

Isidori's book is essential for anyone preparing for serious reading or basic research in the differential geometric approach to control theory and will not disappoint those mathematically trained. I have observed its use in the hands of two teachers other than the author; the students enjoyed it and made good use of it later. There is no universal solvent for nonlinear control problems, but the methods presented here are powerful.

Alberto Isidori was born in Rapallo, Italy. He graduated in electrical engineering from the University of Rome in 1965. In 1969 he obtained a degree equivalent to a doctorate in automatic control from the University of Rome.

Since 1975, he has been Professor of Automatic Control at the University of Rome "La Sapienza". Since 1989, he has also held a position of rofessor (on a half-time basis) at the Department of Systems Science and Mathematics, Washington University, St. Louis, Missouri. He has held visiting positions at several academic institutions, including the University of Illinois (Urbana, Il.), the University of California (Berkeley, Ca.) and the ETH (Zurich, Switzerland).

In 1979, Alberto Isidori initiated a research program aimed at the extension of so-called "geometric theory" of multivariable linear systems, pioneered in the early 1970s by various authors,to linear systems. Linear algebra and linear geometric methods were replaced in nonlinear systems by the methods of differential geometry, whose usefulness in the study of controllability, observability, and minimality of nonlinear systems had been demonstrated in the early 70s. The main intuition of Isidori was to use differential geometric methods in the synthesis of feedback laws for nonlinear systems, more or less in the same way as linear geometric methods were used in the synthesis of feedback laws for linear systems. The result of this seminal work was the development of systematic methods addressing outstanding design problems like feedback linearization, noninteracting control, disturbance decoupling, and model matching.

From 1985 to 1990 Isidori's research concentrated on the development of the "nonlinear analogue" of the notion of the "zero" of a transfer function. Taking as a point of departure the "geometric" interpretation of this notion, the concept of nonlinear zero dynamics was introduced, studied, and applied. As a result, it was shown that most of the features of the notion of zeros of the transfer function of a linear system are actually manifestations of more general principles. Remarkable examples of application of this theory consisted in the study and the solution of the nonlinear equivalent of the so-called "servomechanism problem" of linear system theory and in the characterization of the conditions for feedback equivalence to a nonlinear passive system.

-Georgio Quazza Medal of IFAC, for ?pionering and fundamental contributions to the theory of nonlinear control,? 1996.

-Fellow of the Institute of Electrical and Electronic Engineers, for ?fundamental contributions to nonlinear control theory?, 1987.

-Outstanding Paper Award for papers published on IEEE Transactions on Automatic Control, in 1981 and in 1990.

-Outstanding Paper Award a paper published on Automatica in1991.

AEE 5803 Nonlinear Control SystemsCredit Hours: 3Includes nonlinear system fundamentals (stability and dynamic peculiarities, methods of nonlinear analysis); basic nonlinear control methods (sliding control and feedback linearization, multidimensional extension); advanced nonlinear control methods (adaptive control, neural networks); and nonlinear control applications.

Recommended: Background in control systems

From 1994 to 1999, he was a Fulbright Fellow with Washington University in St. Louis. From 2004 to 2007, he served as the Technical Leader of the Reusable Launch Vehicle area of the AFOSR/AFRL Collaborative Center for Control Sciences, The Ohio State University, Columbus, OH, USA. He held visiting positions at the University of Bologna, Bologna, Italy, and the University of Padova, Padua, Italy, and multiple summer faculty positions at the Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, including four AF-SFFP Fellowships. Since 2002, he has been with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus OH, where he is currently a Professor and the Chair of Graduate Studies. He has authored or co-authored over 150 articles in journals, proceedings of international conferences and book chapters, and has co-authored the book Robust Autonomous Guidance: An Internal Model Approach (Springer-Verlag). His research interests are at the intersection of methodological aspects of nonlinear, adaptive and geometric control theory with advanced applications in aerospace and marine systems, fluidic systems, robotics, and automotive engineering.

Dr. Serrani is a member of International Federation of Automatic Control and American Institute of Aeronautics and Astronautics. He received the Certificate of Outstanding Service as an Associate Editor (AE) of Automatica, was selected three times as an Outstanding Reviewer of the IEEE Transaction on Automatic Control, and three times as an Excellent Reviewer of the AIAA Journal of Guidance, Control, and Dynamics. He served, among other positions, as the Publications Chair of the 53rd and 52nd IEEE Conference on Decision and Control, and the 2009 American Control Conference. He has served as an Editor-at-Large of the 53rd and 51st IEEE CDC, and the 2017 ACC. He has been a Distinguished Lecturer of the IEEE Control System Society (CSS) and has delivered semi-plenary lectures at the 8th IFAC Symposium on Nonlinear Control Systems and the 2011 Chinese Control and Decision Conference. He is currently the Editor-in-Chief of the IEEE Transactions on Control Systems Technology and also serves in the IEEE CSS and IFAC Editorial Boards. He is a past AE of the IEEE Transactions on Control Systems Technology, Automatica, and the International Journal of Robust and Nonlinear Control. He will serve as the Program Chair of the 2019 ACC, and is currently serving as General co-Chair for the 2022 IEEE CDC.

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