Applied Nonlinear Control by Jean-Jacques Slotine, Weiping Li

By Jean-Jacques Slotine, Weiping Li

Covers in a innovative type a few research instruments and layout ideas at once acceptable to nonlinear regulate difficulties in excessive functionality structures (in aerospace, robotics and car areas).

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4. 5 is devoted to the question of how to use these theorems to study the stability of particular classes of nonlinear systems. 6 sketches how the direct method can be used as a powerful way of designing controllers for nonlinear systems. 1 Nonlinear Systems and Equilibrium Points Before addressing the main problems of defining and determining stability in the next sections. let us discuss some relatively simple background issues. 1) where f is a nx 1 nonlinear vector function, and x is the nxl state vector.

The following example illustrates this point. \', - = X-, One easily shows that the system has an equilibrium point at the origin. 8, system trajectories starting from any non-zero initial states all asymptotically approach a limit cycle. 4). This implies instability of the origin. Thus, even though the state of the system does remain around the equilibrium point in a certain sense, it cannot stay arbitrarily close to it. This is the fundamental distinction between stability and instability.

In practice, however, system properties often change very slowly, and we can neglect their time variation without causing any practically meaningful error. It is important to note that for control systems, the above definition is made on the closed-loop dynamics. Since a control system is composed of a controller and a plant (including sensor and actuator dynamics), the non-autonomous nature of a control system may be due to a time-variation either in the plant or in the control law. , if u = g(x, t).

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