A tutorial account of variable structure control with sliding mode is presented in this paper. The purpose is to introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control system design approach. This approach is particularly attractive for the control of nonlinear systems. Prominent characteristics such as invariance, robustness, order reduction, and control chattering are discussed in detail. Methods for coping with chattering are presented. Both linear and nonlinear systems are considered. Future research areas are suggested and an extensive list of references is included. CONVENTIONS FOR SYMBOLS Capital, italic letters represent matrices, e.g., A and K. Boldface, roman lowercase letters represent vectors, brief discussions about its historical development are presented. A. The Basic Notion of VSC second-order system [431, [ 1541 similar to the following The basic idea of VSC was originally illustrated by a X = y y = 2 y-x + U U =-*x (l a) where (1b) * = 4 when s (x , y) > 0 =-4 when s (x , y) < 0 e.g., a and k. and k. respectively. s (x , y) = X U , u = 0. 5 x + y. (IC) Lowercase , italic letters represent scalars, e.g., a and dim-x and d i m 4 stand for dimensions of x and B, A block diagram of the system is shown in Fig. l(a). The variable s(x, y) in (1.14 is the product of two functions

Motor systems are very important in modern society. They convert almost 60% of the electricity produced in the U.S. into other forms of energy to provide power to other equipment. In the performance of all motor systems, bearings play an important role. Many problems arising in motor operations are linked to bearing faults. In many cases, the accuracy of the instruments and devices used to monitor and control the motor system is highly dependent on the dynamic performance of the motor bearings. Thus, fault diagnosis of a motor system is inseparably related to the diagnosis of the bearing assembly. In this paper, bearing vibration frequency features are discussed for motor bearing fault diagnosis. This paper then presents an approach for motor rolling bearing fault diagnosis using neural networks and time/frequency-domain bearing vibration analysis. Vibration simulation is used to assist in the design of various motor rolling bearing fault diagnosis strategies. Both simulation and real-world testing results obtained indicate that neural networks can be effective agents in the diagnosis of various motor bearing faults through the measurement and interpretation of motor bearing vibration signatures.

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