2.29.  SUMMARY

Many mathematical techniques have been presented in this chapter for use by the control-system engineer. Starting with complex-variable theory, we then developed the Fourier transform and Laplace transform. The transfer function, block diagram, and signal-flow graphs were then presented. It was pointed out that these concepts were not applicable to the more general multivariable inputs and outputs, nonlinear, time-varying systems. For this class of systems, the state-variable concept was then presented. Matrix algebra was reviewed, and the state-variable signal-flow graph and the state transition matrix were presented. It is reasonable for the reader at this point to ask which methods he or she should use.

There are no hard and fast guidelines, but reasonable rules of thumb can be outlined. In general, if the problem is one of analysis, if the system has one input and one output, and if its differential equation can be described by a linear differential equation having constant coefficients, then the engineer can use the simple Laplace-transform/transfer-function/block-diagram approach or the state-variable method, each technique complementing the other. On the other hand, if the analysis problem involves nonlinearities, time-varying characteristics and/or multivariable inputs and outputs, then the state-variable approach should be used. If the problem is one of synthesis involving optimal control theory which is discussed in Chapter 11, then again the state-variable approach has to be used.

Because the main purpose of this book is pedagogical, both the Laplace transform/transfer-function/block-diagram and state-variable concepts will be used wherever possible. For example, in the following chapter, where it is desired to represent mathematically various linear physical components, both approaches are used. The reader should also develop this dual capability and be able to handle a problem from either point of view whenever possible.