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By Jean-Jacques Slotine, Weiping Li

Covers in a revolutionary type a couple of research instruments and layout recommendations without delay acceptable to nonlinear regulate difficulties in excessive functionality structures (in aerospace, robotics and automobile areas).

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Although Lyapunov's direct method is originally a method of stability analysis, it can be used for other problems in nonlinear control. One important application is the design of nonlinear controllers. The idea is to somehow formulate a scalar positive function of the system states, and then choose a control law to make this function decrease. A nonlinear control system thus designed will be guaranteed to be stable. ) 16 Nonlinear Systems Analysis Part I in robotics and adaptive control. The direct method can also be used tc estimate the performance of a control system and study its robustness.

If the singular point of interest is not at the origin, by defining the difference between the original state and the singular point as a new set of state variables, one can always shift the singular point to the origin. 1) with a singular point at 0. 1b) can be rewritten as where g and g2 contain higher order terms. 9. 8, one observes that the system has an unstable node at the origin. Furtherml)re. there is a closed curve in the phase portrait. Trajectories inside the curve and thclje outside the curve all tend to this curve, while a motion started on this curve will sta\ o I it forever, circling periodically around the origin.

Question: will a trajectory perturbation due to a gust cause a significant deviatioi~in the later flight trajectory? Here, the desired operating point of the system is the flight trajectory in the absence of disturbance. Every control system, whe:he- linear or nonlinear, involves a stability problem which should be carefully studied. The most useful and general approach for studying the stability ot' nonlinear control systems is the theory introduced in the late 1 9 ' ~century by tl-e Russian mathematician Alexandr Mikhailovich Lyapunov.

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