i. Full State Feedback Control Law
The FSFB control law will decide the net input force that the solenoids should exert on the payload. To formulate it, this kinematics equation,
must be written in state-space form,
Remember, the dot notation means to differentiate with respect to time. Thus, we must model the plant as a series of first-order differential equations. The derivative of each state variable is expressed. Let's express the kinematics equation as two first-order differential equation. Our first state variable should be the payload displacement. It is the time derivative of the velocity.
Our second state variable should be the payload velocity, which, in turn, is the time derivative of the acceleration, which is force divided by mass.
The total force acting on the payload is the sum of two components:
Now, we are in a position to write out the state-space form of the motion plant's kinematics equation.
The FSFB control law is expressed as
The FSFB gains are decided offline by pole placement. If the B and E matrices are identical, then we can set the exogenous input gain coefficient to unity. From the state-space equation, we can obtain the characteristic equation:
A stable control system is one where, for any bounded input, every output and state variable will be bounded. Stable control systems are needed for precise, automatic control. An unstable system will, for some bounded input, produce an output that will shoot off into infinity. Unless you are building a nuke , you don't want your system to be unstable. You'll just end up damaging your plant. An asymptotically stable system is one that will bring the outputs and state variables to zero for any bounded input. We usually normalize the outputs and state variables to have an equilibrium position of zero. Our control system will be stable so long as the real part of s is negative. That is, the s values must lie on the left half of the complex plane. This leaves us with a lot of options, which we will need when we go to select the gain.