• We can write a differential equation, and then manipulate it to put in terms of time steps of length ‘T’



• In review consider how we can approximate derivatives using two/three points on a line.



• Consider the example,




1.1.1 Getting a Discrete Equation


• First, consider the example of a simple differential equation,



• We can continue the example and use this equation to simulate the system. Here the ‘x’ values are given, and the first ‘y’ value is assumed to be 0. Assume T=0.5 and K=0.2.



• Find the discrete equation for the differential equation below. Then find values over time.



• We can expand this model by also including a ‘disturbance’. This can be used to model random noise, or changes in system loading, found in all systems.




1.1.2 First Order System Example


• The water tank below has a small outlet, and left alone the fluid level in the tank will drop until empty. There is a valve controlled flow of fluids into the tank to raise the height of the fluid.



• As long as the fluid levels in the tank are normal, the inlet and outlet flow rates are independent, We can model them both with simple differential equations,



• This difference equation can then be used to predict fluid height. If we change the valve position, this will also be reflected in the calculated values.





• Now try varying the input valve angle,




1.1.3 Second Order System Example


• Consider the second order example below,



• try the following problem.




1.1.4 Example of Dead (Delay) Time


• In a real system there is a distance between an actuator, and a sensor. This physical distance results to a lag between when we actuate something, and when the sensor sees a result.


• Assume we have a simple process where after a change in x there is a delay of ‘m’ time steps before the proportional change occurs in y. We can write the equation as,



• Consider the example,