Mathematical System Models
Given the dynamic underpinnings of real processes, the basic math description of system simulations includes (a) dependencies on the accumulated influence of process variables, and (b) dependencies on the differential of the variables.
These two features determine what is usually called the plant dynamics and demonstrate that the behavior of a real system cannot be modeled satisfactorily without including its past history and the response.
System models will typically be reduced to differential equations (continuous-time), difference equations (discrete-time), or a combination of them (hybrid or sample-data systems). The system models relate plant inputs to selected plant outputs , and they deal with a limited description of the system under design.
State Space Models
A very valuable and often employed method for plant modeling is a state-variable description. State variables form a set of inner variables which is a complete set. It is a complete set in the sense that, if these variables are known at an initial time, then any plant response, y(t), can be calculated, at all future times, as a function of the state variables and the present and future values of the inputs.