Control Theory

Control is the helmsman on this bridge. The captain (guidance) gives the order, and the navigator (navigation, all the Kalman filter stuff done up to this point) gives the captain the data needed for his decision, and the helmsman what he needs in order to control. Let's see if we can take this crew analogy right up to the edge. The helmsman and captain are blind, and as the navigator figures things out, he reads them aloud so the helmsman and captain can hear. The helmsman cares mostly about the current heading, which he gets when the navigator reads the compass. He also cares about the desired heading, which he gets from the captain. The captain cares about all the navigation stuff, such as latitude and longitude, as well as current heading, and he figures out the desired heading and tells the helmsman. Is that too far?

Error
Anyway, this control loop is given by navigation a current state element, and by guidance a desired value for that state element. The control loop then figures out the correct setting for the control that acts on that state element. To make it concrete, let's think about pitch. The guidance program knows that to maintain a certain speed, a certain pitch is required. It calculates that pitch (actually the sine of that pitch) and passes it as a commanded value to the control loop. The navigation program knows the current quaternion. It transforms a vector pointing towards the nose in the body frame into the inertial frame. The vertical component of this vector is the sine of the pitch. It will be positive if the nose is up, and negative conversely. This is the actual value. The job of the control system is to make the commanded and actual values match. When the values match, the error value is zero. When they are different, the error value is the difference between the two. So, to make them match, the control system tries to drive the error to zero.


Stability
Some systems are naturally stable. The classic stable system is a marble in a bowl. "An airplane wants to fly." Left to itself, it will largely fly straight and level. Any small disturbance like a gust of wind may knock the plane off of straight and level, but it will naturally correct itself. The aerodynamics designers work long and hard to give an airframe this stability, but once its there, the plane is stable largely without any effort by the pilot, who is then free to navigate and guide the plane. The airframe itself acts as the control system.

Some systems are unstable. The classic unstable system is a marble on a dome or a pencil balanced on its point. Any small disturbance in this system will be amplified, with the system falling farther and farther off balance. The Wright Flyer I was unstable, intentionally, and required a well-trained pilot as the control system to keep it nose into-the-wind. Anyone who has balanced a broomstick on end, or ridden a bicycle, has been a control system for an unstable system.

Some systems are neutrally stable. The classic example is a marble on a flat table top. If it is disturbed, it will not depart from its original condition, but will not return either. Steering in a car is neutrally stable.

One way to think about a control system is as a modification to a system to make it stable, and in particular stable around the desired value. You can think of it this way, that the control system is like a bunch of springs attached to the marble surrounding the desired point. These springs are strong enough to push the marble back to the desired point and overwhelm any instability in the system. The new system is stable, and stable at the point you choose.

The PID controller

The helicopter I have is stable in roll, neutrally stable in yaw and altitude, and unstable in pitch. There is no control for roll, and since it is stable in roll, none is necessary. The system will then run three independent control loops for main rotor average thrust to control altitude, main rotor differential torque to control yaw and tail rotor thrust to control pitch. These three variables are largely decoupled, and so can be handled independently.

The controller will be a PID controller, standing for Proportional, Integral, and Derivative. The control loop will look at the difference between the actual and commanded state, and generate a control signal proportional to the error. It will look at the total integrated error over the past, and generate a control signal proportional to this integral. It will look at the current rate of change of the error, and generate a signal proportional to this differential. It will then add them all up and use that control signal to drive the correct control.

The proportional part is obvious. If there is an error, you want to correct it, and drive the error back to zero. So, the proportional part is just the current value of the error multiplied by a coefficient. If the error is positive, you generally want to push negative, to push the error down, and conversely for negative error, so this proportional coefficient is negative.

The derivative part is there to keep the system from oscillating. If you have ever worked with differential equations, you have probably learned all about simple and damped oscillators. A simple oscillator is like a weight on a spring, or a pendulum. The restoring force is always proportional to the distance from equilibrium, so it is like the proportional control term. However, a system with just a proportional term will just keep bouncing around, always oscillating around the desired point with the same amplitude and frequency forever. To prevent this, we have a damping term. We drive the control system opposite to the direction of motion of the system, proportional to the speed. With enough of this term, any oscillations, either natural or driven by the proportional system, are damped out to zero.

The integral part is there for two reasons. The first is this: The proportional and derivative terms are somewhat at crossed purposes. The proportional part is trying to move the system back to the control value, while the derivative part is trying to stop the system completely. These two parts could balance out, leaving the system in equilibrium but not at the chosen point. The integral term is proportional to the integrated error in the system. Even if the error is constant, because the proportional and derivative terms have balanced out, the integral of the error will get larger and larger. So we attach a control to this integral error, proportional and opposite as usual.  This way, a constant error will lead to a larger and larger integral term, which will eventually break the deadlock between P and D

Secondly, if a system is not just unstable but actively pushed, a pure proportional controller will not be able to drive the system to zero error. Imagine a ball on an inclined plane. There is a disturbing force proportional to gravity on the ball. When the ball is in position, the P and D terms are zero, there is no control force, and the disturbing force pushes the ball away. The system will come to rest when the proportional term and disturbing term are equal, but this can only happen when the error is nonzero, since the proportional term is proportional to the error. The system will come to rest at a nonzero error. Wikipedia calls this phenomenon droop. Droop is bad.

In both cases, the integral term exists to destroy nonzero equilibrium error. It will grow larger and larger as needed. It can't be bargained with. It can't be reasoned with. It doesn't feel pity, or remorse, or fear. And it absolutely will not stop, ever, until the error is dead.

In our case, the pitch is unstable, because the center of gravity of the vehicle is not under the main rotors. If it were, pitch would be neutrally stable. This could in principle be achieved by shifting the components, or by adding ballast weight. But it would have to be totally accurate, and I don't think I can achieve that. So, PID it is.