A. Modeling the Plant Dynamics
A subsystem block diagram of the plant is shown in Figure IV.A.1. The plant consists of the solenoids and payload. The linear dynamics of the payload moving under the influence of the solenoids are expressed as
The dot notation refers to the time derivative of a quantity. The net input force is expressed as
The motion plant requires two separate forces: the force exerted by the left solenoid and that exerted by the right solenoid. The net force is the difference of the individual forces from each solenoid. If the linearization module is to control each solenoid individually, it must calculate two different current values. These two solenoid currents are the control inputs for the nonlinear motion plant.
Partitioning the Plant / A Strategy for Compensator Design
Most basic techniques for designing compensators for control systems assume that the plant is a linear, time-invariant system. A linear system is one where
Looking at the solenoid force equation,
it is obvious that our motion plant is time-invariant but non-linear. Using a non-linear compensator will not just be more complicated in design but also require more memory, consume more CPU cycles in its computations, and might not produce high-quality performance. The goal is to use a basic linear control system for this motion plant. So, why don't we examine the motion plant a little more in-depth by first removing the non-linear solenoids?
Linear Dynamics
Figure IV.A.2 shows the linear portion of the motion plant. Once again, the differential equation that describes this system is
We can use a linear compensator to control the payload. We will need a linear observer, as we can only measure the displacement but not the force or acceleration.
Figure IV.A.3: Nonlinear portion of plant
Nonlinear dynamics
Figure IV.A.3 shows the nonlinear portion of the motion plant. The two blocks labeled "Solenoid_Force" will calculate the force according to the nonlinear equation.
If the nonlinear portion of a plant is limited to a single area at the input or output of the plant, it might be possible to use a technique called feedback linearization. This technique involves inverting any non-linear dynamics of the plant before treating it as a standard linear time-invariant system. This is possible because we also have an equation for the solenoid current that the microprocessor can compute.