Stability criteria for nonlinear multivariable control systems.

The stability of some particular classes of control systems described by ordinary nonlinear differential equations is considered. As a means of introduction to the problem, systems containing a single nonlinearity in an otherwise linear, time-invariant closed loop are examined. Stability criteria based on the frequency-response of the linear part of the system are established by constructing a Liapunov function of a 'quadratic plus integral of non-linearity' form. The problem is extended to cover those classes of control systems which contain several such nonlinear functions (i.e. multivariable control systems) and frequency domain stability criteria are established by constructing a Liapunov function akin to that described above. It is also asserted that stability criteria less restrictive than those obtained previously for these multivariable systems may be achieved by placing certain additional restrictions on the nonlinear functions. Some classes of systems containing nonlinear functions of a most general nature are considered in later chapters of this thesis. Frequency-domain stability criteria are established with the aid of quadratic forms of Liapunov functions. Again, if the complexities of these nonlinearities are reduced it is seen that less restrictive criteria than obtained previously may be established for these classes of systems. Emphasis is laid throughout upon the development of a unified approach to the problem of stability of the classes of systems considered. The criteria, once formulated, can be applied in practice without any further reference to the Liapunov function used.