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add controller function #512

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May 28, 2021
Merged

add controller function #512

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3f15cab
add controller function
baggepinnen May 17, 2021
13f96fe
rename controller and predictor
baggepinnen May 28, 2021
5343eab
Merge branch 'dev' into controller
baggepinnen May 28, 2021
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1 change: 1 addition & 0 deletions src/ControlSystems.jl
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Expand Up @@ -43,6 +43,7 @@ export LTISystem,
prescale,
innovation_form,
predictor,
controller,
# Stability Analysis
isstable,
pole,
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22 changes: 20 additions & 2 deletions src/matrix_comps.jl
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Expand Up @@ -653,7 +653,25 @@ function predictor(sys::ST, R1, R2) where ST <: AbstractStateSpace
predictor(sys, K)
end

function ControlSystems.predictor(sys, K::AbstractMatrix)
function predictor(sys, K::AbstractMatrix)
A,B,C,D = ssdata(sys)
ss(A-K*C, [B K], C, [D zeros(size(D,1), size(K, 2))], sys.timeevol)
end
end

"""
cont = controller(sys, L::AbstractMatrix, K::AbstractMatrix)

Return the controller `cont` that is given by
`ss(A - B*L - K*C + K*D*L, K, L, 0)`

Such that `feedback(sys, cont)` produces a closed-loop system with eigenvalues given by `A-KC` and `A-BL`.

# Arguments:
- `sys`: Model of system
- `L`: State-feedback gain `u = -Lx`
- `K`: Observer gain
"""
function controller(sys, L::AbstractMatrix, K::AbstractMatrix)
A,B,C,D = ssdata(sys)
ss(A - B*L - K*C + K*D*L, K, L, 0, sys.timeevol)
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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)

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@albheim albheim May 18, 2021 •
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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.

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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.

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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.

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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?

end
23 changes: 22 additions & 1 deletion test/test_matrix_comps.jl
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Expand Up @@ -112,4 +112,25 @@ K = kalman(sys, I(2), I(1))
@test sysp.A == sys.A-K*sys.C
@test sysp.B == [sys.B K]

end

# Test controller
sys = ssrand(2,3,4)
Q1 = I(4)
Q2 = I(3)
R1 = I(4)
R2 = I(2)
L = lqr(sys, Q1, Q2)
K = kalman(sys, R1, R2)
cont = controller(sys, L, K)
syscl = feedback(sys, cont)

pcl = pole(syscl)
A,B,C,D = ssdata(sys)
allpoles = [
eigvals(A-B*L)
eigvals(A-K*C)
]
@test sort(pcl, by=LinearAlgebra.eigsortby) ≈ sort(allpoles, by=LinearAlgebra.eigsortby)
@test cont.B == K

end