Introduction to Nonlinear Systems.


Rafael Wisniewski (office C4-202)



Virtually all physical systems are nonlinear in nature. Sometimes it is possible to describe the operation of a physical system by a linear model, such as set of ordinary linear differential equations. This is the case, for example, if the mode of operation of the physical system does not deviate too much form the nominal set of operating conditions. Thus the analysis of linear systems occupies an important place in system theory. But in analyzing the behavior of any physical system, one often encounters situations where the linearized model is inadequate or inaccurate; that is the time when concepts of this course may prove useful.


Here is what Rudolf E. Kalman once said about nonlinear control

Whenever the controlled variable y(t) is allowed to have large deviations from the steady state, the linear constant-coefficient model will cease to represent the plant accurately because of intrinsic nonlinearities involved in the description of most natural dynamical phenomena. Equally important are intentionally introduced nonlinearities that result from reasons of economy, simplicity and reliability of engineering or from ignorance of the fact that the savings achieved by nonlinear control devices may be negated by factors resulting from the greater intrinsic difficulty of control. A good example of unavoidable nonlinearities in the object to be controlled is a space vehicle whose rockets, at the current state of technology, can be controlled only by turning them on or off. Continuously variable control action (which is desirable) in this case is technologically impossible; the high cost of rocket engines, however, justifies extremely sophisticated computer-based technology to achieve optimal control. The purpose of the computer is then simply one of switching the rockets on or off; the difficulty of the control lies in the extreme precision with which the time must be determined when this takes place.

When the basic mathematical data can be stated in the conventional optimization framework, effective methods are available for solving the optimal control problem even in the nonlinear case. The methods for doing this are an extension of the classical methods of the calculus of variations. These methods, however, have yielded little theoretical insight, and their straightforward application becomes prohibitively expensive for large-scale systems.

Even less satisfactory is the status of the nonlinear optimal filtering and state estimation problem, which, as has been noted, is a critical part of the general solution of the control problem. Because of nonlinearity, the Duality Principle no longer applies; even the formulation of the problem is controversial.

Nonlinear systems represent not a special case but simply everything that is not subjected to the special assumption of linearity; in a sense “nonlinear” is synonymous with “unknown.” Scientific progress will undoubtedly occur by singling out special classes of systems subject to restrictive structural assumptions other than linearity.


The nonlinear control is important, fun but difficult. In order to tackle these difficulties with success we shall need some mathematical background in the first part of the course (5 mini-modules). The second part in turn will pay off with a number of beautiful and powerful nonlinear control techniques.  So be patient. We think the highlights of the course will be Lyapunov stability and feedback linearization. The Lyapunov stability theorems are used for analysis of a nonlinear system. We answer the questions: Is it asymptotically stable, is it locally stable/unstable. Later on, we look at synthesis problems in Nonlinear Control Methods (5 extra mini-modules). Feedback linearization is a pure geometrical method which helps you to find a certain nice map which translates a nonlinear system into a linear one. Remark this is not a linearization; we merely will change the coordinates such that your system appears linear.


We shall use my notes and   H.K. Khalil, Nonlinear Systems, Prentice 2002. It is available in the campus bookshop.  The exercises are compulsory to be delivered on written a week after the lecture. The examination will be oral. You will be allowed to bring maximum one slide for each topic. I will pose the list of topics on the web page in due time.

MM1: Introduction to Nonliner Control

Written material: pages 1-3 in my notes




MM5: Stability of Nonlinear Systems

MM6: Backstepping

MM7: Potpourri in Stability of Nonlinear Systems

MM8: Sliding Mode Control