Pre-requisite: 70320
To apply control to any 'real' problem, it is first necessary to express the system to be controlled in mathematical terms. The 'state space' approach is taught both for expressing the system dynamics and for analysing stability both before and after feedback is applied. These concepts involve revision and extension of matrix manipulation and the solution of differential equations. By defining a time-step to be small, these state equations give a means of simulating the system and its controller for both linear and nonlinear cases. Many of the implementations of on-line control now involve a computer, which applies control actions at discrete intervals of time rather than continuously. It is shown that discrete-time state equations can be derived which have much in common with the continuous ones. Simulation does not then rely on a very small time step. The operator 'z' is first introduced with the meaning of 'next', resulting in a higher order difference equation to represent the system, then shown to be a parameter in the infinite series which is summed to form a 'z- transform'. It is shown that the discrete-time transfer function in z can be derived from the Laplace transform of the continuous system, with additional terms to represent the zero order hold of the DAC. Analysis of stability in terms of the roots of a characteristic equation are seen to parallel the continuous methods and techniques of pole assignment and root locus are also seen to correspond. Techniques are presented for synthesising transfer functions by means of a few lines of computer code, to make stable control possible for systems which would be unstable with simple feedback.