Ordinary linear differential equations

The automatic control techniques employed in classical control require knowledge of the mathematical model of the physical system to be controlled. As has been shown in Chap. 2, these mathematical models are differential equations. The controller is designed as another differential equation that must be connected in closed-loop to the system to be controlled. This results in another differential equation representing the closed-loop control system. This differential equation is forced by the controller to possess the mathematical properties that ensure that its solution evolves as desired. This means that the controlled variable evolves as desired. Hence, it is important to know the properties of a differential equation determining how its solution behaves in time. Although several different approaches exist to solve differential equations, the use of the Laplace transform is the preferred method in classical control. This is the reason why the Laplace transform method is employed in this chapter, to study linear ordinary differential equations with constant coefficients.