Mathematical Description of Continuous-Time Systems

Based on the phenomenological and physical principles, relevant to describing a particular dynamic system, the equations that characterize the system are carried out in a number of ways, some of which are in the time domain, and others written in a transformed domain. In the time domain the methods for the analysis of the response of the dynamic system are ordinary differential equations (ODEs) of order n, sets of n first-order ordinary differential equations, partial differential equations (PDEs), the superposition integral, the convolution integral, and so on. Solving these equations can be done using numerical methods, based on suitable mathematical models, while more and more indispensable tools for advanced systems analysis and synthesis are in use, as well as for computer-aided engineering design. In conjunction with an experimental verification method, the numericalsimulation results of the suitable mathematical model can be proved. Moreover, the stability analysis of dynamic systems are quite useful when designing optimal control systems that are stable. For this purpose one has to know whether the roots of the system will be located near the equilibrium point or not. Stability analysis can be done, for example in the time domain, by means of the Routh Hurwitz criterion in conjunction with the differential equations relating the response to the excitation. It has to be said that this method is restricted to linear systems.