Comparisons of notions of stability for nonlinear control systems with outputs

Comparisons of several notions of stability are provided in this work. The properties addressed are generalizations of input to state stability to systems with outputs. We consider two roles for the output of a control system. The first is that of an error associated with the performance of the system. In this case we investigate stability of the output signal. The second role for an output is that of a measurement of the state. In this case we consider notions of detectability. We also deal with the case of a system with two outputs, one an error and the other a measurement. Here we formulate a notion of partial detectability (i.e. detectability of the error signal through the measurement signal).; Several preliminary results on nonlinear gain functions are presented. These are followed by some results on differential inclusions, most notably a relaxation result which complements the Filippov-Wažewski Relaxation Theorem, and a result on global stability of differential inclusions with compact values. Notions of stability for systems with outputs are then addressed. For systems with error outputs, input to state stability has been generalized to the property of input to output stability. A separation principle for this property is described, which provides a characterization in terms of more fundamental stability notions. Next, generalizations of input to state stability to systems with measurement outputs are considered. The resulting properties are notions of detectability. A Lyapunov characterization for one such notion is given. Finally, a notion of partial detectability for systems with two outputs is introduced. Again, a Lyapunov characterization is presented.