Modelling is about predicting the behaviour of a system. Often, need to know
- What is the output for a given input?
- Is the system stable?
- If the input changes quickly, how will the output change?
First order systems are those with only one energy store, and can be modelled by a first order differential equation.
For the general form of the equation , the solution for a step input at time , with : T is the time constant of the system.
- Free response:
- The response of a system to its stored energy when there is no input
- Zero Input
- Non-zero initial Conditions
- Homogenous differential equation
- Forced response:
- The response of a system to an input when there is no energy initially in the system
- Non-zero input
- Zero initial Conditions
- Non-homogeneous differential equation
- Total system response is a linear combination of the two
Different inputs can be used to determine characteristics of the system.
- A sudden increase of a constant amplitude input
- Can see how quickly the system responds
- Is there is any delay/oscillation?
- Is it stable?
- Can vary frequency and amplitude
- Shows frequency response of a system
- A spike of infinite magnitude at an infinitely small time step
- An input that starts increasing at a constant rate, starting at .
- The step response of the system is the output when given a step input
- System must have zero initial conditions
- Characteristics of a response:
- Final/resting value
- Rise time
- Oscillation (frequency & damping factor)
For a system with time constant , the response looks something like this:
The time constant of a system determines how long the system takes to respond to step input. After 1 time constant, the system is at about (63) % of its final value.
|Time (s)||% of final value|