Nonlinear Control Sy.. - Free

8.7. EXERCISES 221
Notes and References
This chapter introduced the concept of passivity in its purest (input-output) form. Our
presentation closely follows Reference [21], which is an excellent source on the input-output
theory of systems in general, and passivity results in particular. Early results on stability of
passive systems can be found in Zames [98]. Strictly positive real transfer functions and the
Kalman-Yakubovich lemma 8.1, play a very important role in several areas of system theory,
including stability of feedback systems, adaptive control, and even the synthesis of passive
networks. The proof of the Kalman-Yakubovich lemma can be found in several works.
See for example Narendra and Anaswami [56], Vidyasagar [88] or Anderson [1]. Reference
[1], in particular, contains a very thorough coverage of the Kalman-Yakubovich lemma and
related topics. See References [35] and [78] for a detailed treatment of frequency domain
properties of SPR functions. Example 8.4 is based on unpublished work by the author in
collaboration with Dr. C. Damaren (University of Toronto).