Logical Predictive Control
A high order system, such as the roll channel of an aeroplane, can consist of a string of state variables where each is the integral of the one before it.
Twin fast models are used to obtain near-optimal control, with an input that switches between its limits.
The simulations linked here consist of three, four and five cascaded integrators.
At each step of the system to be controlled, the states of the fast models are set to correspond to its present state.
The fast models are then run, one with maximum positive input and the other with maximum negative.
At some future time, the position and all the state derivatives of each model will take the same sign as its input.
Until then, it can be said that the model is 'offside'.
Both models are run in parallel for as long as both models are offside. Then the run ceases.
One of the models will still be offside, and that sense of drive is applied to the system.
If both models have come onside together, the system drive is zero.
You can see the strategy in action in simulations of third, fourth and fifth order at
Predictive3twin.xhtml
Predictive4twin.xhtml and
Predictive5twin.xhtml
The fifth-order simulation shows some overshoot, but a technique I call 'slugging' can solve this.
In Predictive5slug.xhtml, this model derivative states have factors applied to them when being reset.
Here the factors are 1.35, 1.9, 1.9 and 1.35 respectively. There is no overshoot.