## Linear Systems

The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics modules and provide opportunities for further study. The theory of linear systems is engineering mathematics.

In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.

Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The module shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables students to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.

The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilise a (A,B,C,D) system.