• If our models include a device that is non linear we will need to linearize the model before we can proceed.
• A non-linear system can be approximated with a linear equation using the following method.
1. Pick an operating point or range for the component.
2. Find a constant value that relates a change in the input to a change in the output.
3. Develop a linear equation.
4. Use the linear equation in the analysis (Laplace or other)
• Consider the example below,
Figure 12.1 Linearizing non-linear elements
12.3.1 Time Variant
- system parameters vary as a function of time.
- system components turned on/off
- cables in tension/compression
- show an example where input conditions change
- give PWM (Pulse Width Modulation) example with ripple showing equivalent voltage. PWM is used to generate analog voltage equivalents. Show for a system with first order response with tau = 0.1s for a frequency of 1KHz, 10Hz and 1Hz. Point out the ripple and effective voltage.
- important to consider when doing system analysis
- Friction in all components
- costs money to reduce friction, so it is better to compensate in software
- small actuation signals not large enough to overcome friction
- This effect is normally known as ’stiction’, a combination of the words static and friction.
- Friction is common in less expensive motors, and when a motor is driving a mechanical system.
- In systems there are two type of friction that must be considered.
- The static friction, ’stiction’, will prevent initial motion. If the systems breaks free and starts turning, the kinetic friction will provide a roughly constant friction resistance.
- relationship in figure below.
- the region where the applied voltage has no effect is called the deadband.
Figure 12.2 Motor deadband for a bidirectional motor
- deadband compensation as shown in figure below.
Figure 12.3 Deadband approximation for a bidirectional motor
- equations for these are shown in figure
Figure 12.4 Deadband approximation for a bidirectional motor
- c-code below
Figure 12.5 Deadband Compensation Subroutine
Figure 12.6 Deadband Breakaway Subroutine
12.3.4 Saturation and Clipping
• Some devices have natural maximum values, such as voltage or pressure limitations caused by a regulated supply.
12.3.5 Hysteresis and Slip
- windup resulting from springiness and friction
- correct by tracking the previous motion direction and taking extra steps when reversing direction
- slip takeup can be done with the function below that adds a few steps when reversing direction.
Figure 12.7 Deadband Breakaway Subroutine
12.3.6 Delays and Lags
• Time delays are common in systems
• In the simplest form this is a period of time between when an event occurs and when the effect occurs.
• If an output delay is larger than the control system step time it may be necessary to predict future states and initiate outputs ahead of those.
• If an input delay is larger than the control system it might be necessary to slow the control action, or build it into the control law.