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A nonlinear small gain theorem for the analysis of control systems with saturation. (English) Zbl 0863.93073
Tools are developed for analyzing the dynamical behavior of interconnected systems of the form $\dot x_1 = f_1 \bigl(x_1, (y_2,d_1) \bigr),\;y_1 = h_1 \bigl(x_1, (y_2, d_1) \bigr),$
$\dot x_2 = f_2 \bigl( x_2, (y_1,d_2) \bigr),\;y_2= h_2 \bigl( x_2, (y_1, d_2) \bigr)$ where $$x_1$$ and $$x_2$$ are the states of the two systems, $$y_1$$ and $$y_2$$ their corresponding outputs and $$d_1$$ and $$d_2$$ are external disturbances. The stability of the interconnected system (containing saturation elements) is analyzed by means of a nonlinear small gain theorem. The theorem can be used in an iterative fashion for the control of systems in feedforward form. The results are applied to different classes of systems including (saturated) linear systems, globally asymptotically stable systems cascaded with certain linear systems and a few illustrative examples.

##### MSC:
 93D21 Adaptive or robust stabilization 93C10 Nonlinear systems in control theory
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