add controller function

[baggepinnen]

This is yet another small PR that aims to break up the quite large #467 into smaller chunks.
This PR adds a function controller that takes state feedback and observer gains and returns the controller

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[olof3]

This seems like something that could occasionally be convenient.

I'm a little bit skeptical to using such a generic name for a function like this, though.
Wouldn't it be good with a more descriptive name, both to have a slightly better idea of what the function is about, and also if the user would like to introduce a variable called controller for example.

I'd say that sscontroller is slightly better, but there are probably even better alternatives.

Hopefully the corresponding predictor method could be renamed accordingly as well.

[baggepinnen]

The most descriptive name I can come up with is observer_statefeedback_controller, but perhaps this can be conveyed in a slighltly shorter form?
Some alternatives

• sf_obsv_controller
• innovation_form_controller

[olof3]

Yes, I think something like that would be alright. Doesn't one sometimes say "observer-based state feedback", so obsv_sf_controller?

[albheim]

Wouldn't this correspond to the output of the controller being -Lx?

Suggested change

	ss(A - B*L - K*C + K*D*L, K, L, 0, sys.timeevol)
	ss(A - B*L - K*C + K*D*L, K, -L, 0, sys.timeevol)

[albheim]

Ohh, you want the output to be Lx so that you can use negative feedback. Seems reasonable, but could maybe confuse more people than me. Maybe just state clearly that the output of the controller is Lx and not -Lx.

Or maybe that is not something that will commonly happen, since you give a tip on how to interconnect the systems it should be fine.

[olof3]

Mm, yeah I guess this is alright.

But what about the case when one wants the controller to treat the measurement and reference signals differently. I would say that this convention is still acceptable, although it would feel weird to add the measurement signal to the reference signal, instead of subtracting it.

[baggepinnen]

When you introduce reference signals as well it can get very messy. In RobustAndOptimalControl.jl I'm experimenting with using a type ExtendedStateSpace which is similar to, and should probably be merged with PartitionedStateSpace, such that one can defined the controller with references as

function extended_controller(K)
    nx,nu,ny = K.nx, K.nu, K.ny
    A,B,C,D = ssdata(K)
    ss(A, B, -B, zeros(0,nx), C, D21=D, D22=-D)
end

i.e., it has two inputs, r and y. I haven't landed on how to represent everything yet though, I have also experimented with systems with named signals, ideally this should be used as well to not rely on indices everywhere.

[olof3]

Sounds good, not sure either what makes the most sense, so good that you are thinking about it. Yes, it seems more like a PartitionedStateSpace. We could change PartitionedStateSpace, I don't think we really need to explicitly use such a type for the delay systems.

Perhaps it won't even be needed here either with appropriate name of the signals.

Note: In this case one would have to use positive feedback?

[baggepinnen]

I've tried to come up with a better name here. matlab has a simiar function that is named lqgreg
https://se.mathworks.com/help/control/ref/ss.lqgreg.html
They accept some kalman filter type into that which we don't do here. Also, a state feedback controller with observer does not have to be derived from an LQG problem.

There is really two aspects here, the controller has an internal observer with gain K, and a state feedback. Can we come up with a name that is not horribly long and captures both of those aspects? Perhaps obsever_controller is sufficient since an observer by definition observes the states and any controller using an observer would thus be a state feedback controller from observed states?

[olof3]

I think what you said, but with a small reverse, i.e., obsv_sf_controller is fine. https://www.google.com/search?q="observer+based+state+feedback"

[baggepinnen]

I changed both predictor and controller to

• observer_predictor
• observer_controller
  Looks nice and symmetric, and observer_controller kind of implies state feedback so the slight loss in descriptiveness should be fine, the docstring will clarify if any questions arise.

[olof3]

Good choices!