Ordinary Differential Equations

Ordinary Differential Equations The classic system of ordinary differential equations has N state variables y1,...yN each of which has an equation for its derivative of the form

In practice this equation holds from t=0 up to some final time of interest i.e for

In addition the initial value or starting value of each state variable must be known

It is convenient to express this in vector notation using y=[y1, y2,...yN]T

giving for

This is often known as an initial value problem. Solving or integrating the equations means determining how the state variables change as the time t progresses from 0 to tfin. The state variables defining the system change because:

• the starting values do not represent an equilibrium position,

• some external disturbances (changes) are introduced (duty cycle).

The equations that arise in modeling engineering systems vary enormously in their characteristics. One way of analyzing the local characteristics at some point in the solution is to evaluate the Jacobian matrix

and determine its eigenvalues λi. These are in general complex with the real parts normally negative. Of interest to the engineer are the corresponding time constants Ԏi which are defined as follows:

These give an idea of the damping of particular components of the solution. The corresponding imaginary parts of the eigenvalues give some idea of the local frequencies.

Two characteristics, which are particularly demanding for integration methods, are:

• Problems where the smallest time constant is much less than the simulation time e.g.

Such problems are called stiff and require very stable methods for efficient solution. The best-known and most successful algorithm for solving such problems is Gear’s method [1].

• Problems where the frequency is high compared with tfin and the damping is low. All normal methods are slow when applied to these problems.