# TIME-VARYING AND NONLINEAR SYSTEMS

In the preceding sections we have presented the methods for analysis of linear/invariant systems. These systems are the simplest kind and the methods of analysis are in effect omnipotent, in that in principle they provide complete exact solutions for all such systems. Only sheer size provides limits to practical computation.

On the other hand, linear time-varying systems (linear systems with non­constant coefficients) and nonlinear systems present no such comfortable picture. Their characteristics are not simply classified and there are no general methods comparable in power to those of linear analysis. In the aerospace field, nonlinearities and time variation occur in several ways. The fundamental dynamical equations (see Chapter 6) are nonlinear in the inertia terms and in the kinematical variables. The external forces, especially the aerodynamic ones, may contain inherent nonlinearities. When the flight path is a transient, as in reentry, rocket launch, or a landing flare, the aero­dynamic coefficients are time-varying as well. In the automatic and powered control systems so widely used in aerospace vehicles, there commonly occur nonlinear control elements such as limiters, switches, dead-bands, and others. Finally, the human pilot, actively present in most flight-control situations, is the ultimate in time-varying nonlinear systems (see Chapter 12).

Although completely general methods, apart from machine computation of course, are not available for analyzing the performance and stability of time-varying and nonlinear systems, there are nevertheless many important particular methods suitable for particular classes of problems. This subject is much too large for a comprehensive treatment here. The reader is referred to refs. 3.8-3.10 for treatises devoted to the subject.

It should be pointed out that even when a flight vehicle system is essentially nonlinear, much may be learned about it by first carrying out a linear analysis of small disturbances from a reference steady state or reference transient. This normally provides a good base from which to extend the analysis to
include nonlinear effects, as well as a limiting check “point” for subsequent computation and analysis. Of the particular methods available for studying nonlinear systems, we consider two sufficiently relevant to flight dynamics to present brief introductions to them below.