Dynamic Systems

Dynamic systems are those processes or physical phenomena whose states (instantaneous description) change over time and their rate of change depends on their values.  These systems in general contain some type of internal memory so that the current values of their states depend on their past values. The mathematical descriptions of such processes are called dynamic system models.  These models are used to describe many dynamic processes in financial, economic, environmental, medical, engineering, manufacturing and many other applications for their analysis, forecasting and control.

Dynamic systems, depending on the type of signals they process and the way they process them can be categorized as continuous, discrete, linear, nonlinear, time varying, time-invariant, deterministic, and stochastic among others.  Their models are generally described by either differential equations or difference equations.  Response of dynamic systems can be predicted by finding the solution of their differential (or difference) equation models.  Here, we only consider continuous, linear time-invariant (LTI) dynamic systems, modeled by LTI differential equations. 

Example

An electric RC circuit, as shown in the Figure, is an example of a dynamic system.

Accoring to Kirkhoff’s voltage law (KVL) one has:

In addition, the current passing through both R and C is given by . Substituting this into the above equation, results in:

Defining  as the time constant of the circuit, and letting , and denoting , the above equation can be written as:

which defines a first-order linear time invariant (LTI) differential equation for the system model.