Over the past ten years, the focus in the area of control theory and engineering has shifted from linear to nonlinear systems, providing control algorithms for systems that are both more general and more realistic. Nonlinear control dominates control conferences and has strong presence in academic curricula and in industry.

This tutorial will teach the participants the design tools for control of nonlinear systems with uncertain models that every graduate student or control practitioner today must know. The tutorial will also put research-oriented participants on a fast track towards some of the open problems in the area. The tutorial content consists of essentials scattered in some 20 textbooks and monographs published in the last decade on nonlinear and adaptive control. Familiarity with the basic Lyapunov stability concept is the only prerequisite required to follow the tutorial. The tutorial topics have been class tested in the instructor's one year long graduate sequence on nonlinear and adaptive control at University of California, San Diego, and at similar workhsops he has presented at conferences in the United States and several other countries. The past participants of these workshops include many of the leading researchers in the control area.

Backstepping is a design approach whose significance for nonlinear control can be compared to root locus or Nyquist's method for linear systems. Its roots are in the theory of feedback linearization of the 1980's. With its added flexibility to handle modeling nonlinearities and some classes of systems that are not feedback linearizable, backstepping is the most important ingredient in the nonlinear control advances of the last decade.

We first present algorithms for adaptive nonlinear control. Lyapunov-based tools such as the tuning functions method are covered in detail, which allow simultaneous design of adaptive controllers and parameter estimators, and as a result lead to the strongest performance properties among the methods for adaptive control. This is followed by modular methods that recover a linear-like controller/estimator separation principle---but with a twist. Special controller strengthening allows the use of any parameter estimator, even in the presence of rapidly growing nonlinearities.

Robust nonlinear control is covered next in the form of disturbance attenuation. It is explained how other forms of uncertainties (functionally uncertain nonlinearities, unmodeled dynamics, etc.) can be handled using a nonlinear extension of the small gain theorem. A general framework is presented for achieving a strong form of distrurbance attenuation called input-to-state stability. Backstepping is used to develop control algorithms for disturbance attenuation in a specific, broad class of systems. Nonlinear systems with two types of disturbances are treated: deterministic and stochastic. The stochastic case will particularly reveal the power of the backstepping method, which, in combination with elements of Ito's calculus, turns some notoriously difficult nonlinear stochastic problems into easy exercises.

Optimal control formulations for nonlinear systems produce solutions (based on Hamilton-Jacobi nonlinear partial differential equations) that are seldom feasible, even numerically. It will be shown how the backstepping approach, in turn, produces closed-form solutions to inverse optimal control problems. These controllers have familiar gain and phase robustness marings, the classical benefit of optimality discovered by Kalman for linear systems. Using backstepping one can even obtain explicit control laws that solve the nonlinear H-infinity problem.

Nonlinear observers and output feedback are a tool of great practical necessity and a major open research problem. Output feedback design is covered both in the context of adaptive control and disturbance attenuation.

Forwarding is a procedure applicable to a class of systems dual to those tractable by backstepping. A compact presentation of forwarding will be given, fully exploiting the analogy with backstepping, which will introduce the participants to this procedure in a manner much clearer, simpler, and more accessible than in any other resource available.

In terms of industrial and applications impact over the past decade, backstepping and nonlinear control rank at the top among control theoretic design tools. Examples of uses of backstepping will be presented from among the long list of applications that have emerged recently: marine vehicles, satellites and airplanes, automotive problems, electric machines and robotics, jet engines, fluid and thermal problems, bioreactors, and even fusion reactors.

Tutorial participants will receive typed, book-like notes specially prepared for this tutorial. A literature summary for further self-study will be included.

Miroslav Krstic is Professor and Vice Chair in the Department of Mechanical and Aerospace Egineering at University of California, San Diego. Prior to moving to UCSD, he was on the faculty of the Department of Mechanical Engineering and the Institute of Systems Research at University of Maryland.

Krstic got his PhD in Electrical Engineering from University of California at Santa Barbara, under Petar Kokotovic as his advisor. His dissertation received the UCSB Best Dissertation Award.

Krstic is a recipient of several paper prize awards, including the George S. Axelby Outstanding Paper Award of IEEE Transactions on Automatic Control, and the O. Hugo Schuck Award for the best paper at American Control Conference. He has also received the National Science Foundation Career Award, Office of Naval Research Young Investigator Award, and is the only recipient of the Presidential Early Career Award for Scientists and Engineers (PECASE) in the area of control theory.

Krstic is a co-author of the books Nonlinear and Adaptive Control Design, Wiley, 1995 and Stabilization of Nonlinear Uncertain Systems, Springer-Verlag, 1998 and over 120 refereed papers. He has two patents on control of aeroengine compressor and combustion instabilities.

He has served as Associate Editor for the IEEE Transactions on Automatic Control, International Journal of Adaptive Control and Signal Processing, Systems and Control Letters, and Journal for Dynamics of Continuous, Discrete, and Impulsive Systems. Krstic served for three years as the Chair of the committee for the IEEE Conference on Decision and Control Student Best Paper Award. He is a member of the Board of Governors of the IEEE Control Systems Society. Krstic's research interests include nonlinear, adaptive, robust, and stochastic control theory for finite dimensional and distributed parameter systems. He teaches graduate courses in all of these areas at UCSD. Krstic is active in control applications to propulsion systems, fluid flows, marine and aerospace vehicles, bioengineering, and fusion/plasma engineering.