numerical methods for calculating PID equivalent functions

Hi,

For a simplified 3dimensional sparsely programmed state space (filled from real world data) given input data -10,0,5, then doing the calculation on all rows of the state space table:

best match = minimize(diff(1) + diff(2) + diff(3))

This will give the closest state match to the input state. This is just a first step but each state in the state space can be labelled with a string identifier that gives a valid sequence (ie from recorded flight data) of a path through the state space that occurred from a real quadcopter flight for example. If the input state space matches closely enough to the reference state space, then the craft may be able to traverse the string sequence of state spaces. To keep it locked on to the state space string, or to traverse to another state space string, basic n-dimensional state space matrix operations could be done, from pre-recorded valid flight maneuvers, ie adjusting altitude or rotating the craft while holding other variables constant, to move the state space to another state space string or to keep it locked on to a string being traversed, or to branch a new string towards a new state space "goal". This all allows for a sparsely populated state space, but one problem could be the state space recorded data will keep accumulating and a lot of the common maneuvers may be essentially very similar data, so a separate classification algorithm could be used to bunch together muliple state space strings into a bundle to save rows in the state space table and make searching faster too.

cheers, Jamie

ions, and - for well-conditioned problems - can get to the best solution re markably quickly.

to > > get someplace that isn't all that interesting.

p of > the Washington Monument is the high point. One might start a 'hill climbing' > algorithm in the nearby reflecting pool (which is perfectly fl at and level), > and get nowhere.

Jamie M opened this thread with his example being " a lunar lander with yaw /pitch/thrust to land on a given X/Y coordinate from a given altitude" whic h is a rather better-conditioned problem than the frivolous example you've come up with.

dom > searching would work better for that problem, at least initially.

But this isn't remotely the kind of problem that's being talked about.

I've used a Marquardt-based non-linear multi-parameter least squares curve fitting package. When I came to write my own, the Fletcher-Powell approach was more attractive - it works much the same way, but IIRR it made for a sh orter and neater program in my application.

Certainly the Fletcher-Powell approach gave me a local model of the least-s quares fitting surface around the best fit solution. Had the slope and curv ature of the fitting surface been close to zero for any of the parameters I 'd been trying to fit I'd have known the problem was ill-conditioned, and m y program used this information to generate confidence limits of each of th e parameters being estimated. It was exactly this output that convinced me that I couldn't get a forth - thermal conductance - parameter out of single chunks of the reaction rate data I produced.

What you've identified as a bug is something that I'd already identified as a feature (when properly understood).

Listen, you asked one think, I answered. If you don't want to use the state-space control, no problem, is up to you. What you want to do with the lookuptable is IMHO completely crazy, it's not going to work, but, hey, feel free to try.

Last thing I tell you is: go read some literyture on control systems, and by literature I mean books, not wikipedia articles or 2 pages howto found with a 3 minutes search on google.

Bye Jack

On a sunny day (Tue, 03 Mar 2015 18:26:27 -0800) it happened Jamie M wrote in :

If I understand you right you are getting real close to a neural net. in the neural net the lookup 'string' is in fact in the weights between the neurons. And it can learn, you may like this:

There is simple free software for Linux to construct such a net.

Both complex programs and human neural nets have flown planes into the ground.

Hi,

Ya if the state space for a typical system that has lots of repeating or related input data was recorded for a long time, the state space could have an n-dimensional spiderweb type pattern maybe, the strings connecting with nodes. The nodes themselves would be bunches of strings that are close together, ie multiple strings diverging off a node one for each different type of behaviour in the system, or in different words, one or more strings converging to a similar state, and then diverging to multiple different states.

This looks like a neural network kind of as a string can be bundled (categorized together with other nearby strings) effectively making this behaviour encoded by the string more recognizable or common.

The nodes though are still just closely parallel bunched strings I think, so there aren't really any neurons or action potentials.

That means there aren't really that many calculations that have to be done too, since all behaviour paths are already precalculated, there isn't any real time computation required, future actions have to result from prerecorded strings or combinations of strings (ie string convolutions to make a new string)

The main calculations that would need to be done are correlating strings similarity and categorizing strings into bundles, and also combining strings to make new ones :D

cheers, Jamie

If this n-dimensional string spiderweb was used in a control algorithm system, and the spiderweb and its nodes completely described the system accurately, then as the spiderweb is traversed and a node bundle is reached, to decide which of the output strings to traverse, the sensor data from the system will correlate highest with the output string that has the same prerecorded sensor data. So in this type of situation there is a single correlation calculation required at each node junction during operation.

This is only true if the strings are quantized into a single bundle, if strings have enough spatial resolution to remain separate then they can be followed without any correlation calculation required, but if a single string being traversed branches, ie for a fully described system that relies on a random number generator, then depending on the state of the random number variable, the output branch string that correlates closest to the random number will be followed, so this means that a correlation calculation is required any time a choice between which string to traverse is required to be made (branches and quantized string bundles)

The correlation calculations input data would be a small subset of the overall state space data luckily (just the data related to the branch strings or the node strings) so wouldn't be that much computation.

cheers, Jamie

utions, and - for well-conditioned problems - can get to the best solution remarkably quickly.

ke a

top of > the Washington Monument is the high point. One might start a 'hil l climbing' > algorithm in the nearby reflecting pool (which is perfectly flat and level), > and get nowhere.

aw/pitch/thrust to land on a given X/Y coordinate from a given altitude" wh ich is a rather better-conditioned problem than the frivolous example you'v e come up with.

Maybe, but maybe not. The lander hardware includes actuators, and there ar e dead zones due to backlash in some of the mechanical systems involved. If you live l ong enough, a control-model problem will find a way to bite you.