Skip to content
Snippets Groups Projects
Commit efcbb3fc authored by Tommi Nylander's avatar Tommi Nylander
Browse files

Added fixed arx file

parent 4e67025e
No related branches found
No related tags found
No related merge requests found
Pipeline #762 canceled
export toeplitz, getARXregressor, find_na, arx, bopl_confidence, bopl_confidence!
## Helper functions
rms(x) = sqrt(mean(x.^2))
sse(x) = sum(x.^2)
fit(y,yh) = 100 * (1-rms(y-yh)./rms(y-mean(y)));
aic(x,d) = log(sse(x)) + 2d/length(x)
"""
toeplitz(c::AbstractArray,r::AbstractArray)
Returns a Toeplitz matrix where `c` is the first column and `r` is the first row.
"""
function toeplitz(c::AbstractVector{T},r::AbstractVector{T}) where T
nc = length(c)
nr = length(r)
A = zeros(T, nc, nr)
A[:,1] = c
A[1,:] = r
for i in 2:nr
A[2:end,i] = A[1:end-1,i-1]
end
A
end
"""
getARXregressor(y::AbstractVector,u::AbstractVecOrMat, na, nb)
Returns a shortened output signal `y` and a regressor matrix `A` such that the least-squares ARX model estimate of order `na,nb` is `y\\A`
Return a regressor matrix used to fit an ARX model on, e.g., the form
`A(z)y = B(z)f(u)`
with output `y` and input `u` where the order of autoregression is `na` and
the order of input moving average is `nb`
# Example
Here we test the model with the Function `f(u) = √(|u|)`
```julia
A = [1,2*0.7*1,1] # A(z) coeffs
B = [10,5] # B(z) coeffs
u = randn(100) # Simulate 100 time steps with Gaussian input
y = filt(B,A,u)
yr,A = getARXregressor(y,u,3,2) # We assume that we know the system order 3,2
x = A\\yr # Estimate model polynomials
plot([yr A*x], lab=["Signal" "Prediction"])
```
For nonlinear ARX-models, see [BasisFunctionExpansions.jl](https://github.com/baggepinnen/BasisFunctionExpansions.jl/)
"""
function getARXregressor(y::AbstractVector,u::AbstractVecOrMat, na, nb)
@assert(length(nb) == size(u,2))
m = max(na+1,maximum(nb))
n = length(y) - m+1
offs = m-na-1
A = toeplitz(y[offs+na+1:n+na+offs],y[offs+na+1:-1:1])
y = copy(A[:,1])
A = A[:,2:end]
for i = 1:length(nb)
offs = m-nb[i]-1
A = [A toeplitz(u[nb[i]+offs:n+nb[i]+offs-1,i],u[nb[i]+offs:-1:1+offs,i])]
end
return y,A
end
"""
find_na(y::AbstractVector,n::Int)
Plots the RMSE and AIC For model orders up to `n`. Useful for model selection
"""
function find_na(y::AbstractVector,n::Int)
error = zeros(n,2)
for i = 1:n
w,e = ar(y,i)
error[i,1] = rms(e)
error[i,2] = aic(e,i)
print(i,", ")
end
println("Done")
scatter(error, show=true)
end
"""
Gtf, Σ = arx(h,y, u, na, nb; λ = 0)
Fit a transfer Function to data using an ARX model.
`nb` and `na` are the orders of the numerator and denominator polynomials.
"""
function arx(h,y::AbstractVector{Float64}, u::AbstractVector{Float64}, na, nb; λ = 0)
na -= 1
y_train, A = getARXregressor(y,u, na, nb)
if λ == 0
w = A\y_train
else
w = (A'A + λ*I)\A'y_train
end
a,b = params2poly(w,na,nb)
model = tf(b,a,h)
Σ = parameter_covariance(y_train, A, w, λ)
return model, Σ
end
"""
a,b = params2poly(params,na,nb)
"""
function params2poly(w,na,nb)
a = [1; -w[1:na]]
b = w[na+1:end]
a,b
end
"""
Σ = parameter_covariance(y_train, A, w, λ=0)
"""
function parameter_covariance(y_train, A, w, λ=0)
σ² = var(y_train .- A*w)
iATA = if λ == 0
inv(A'A)
else
ATA = A'A
ATAλ = factorize(ATA + λ*I)
ATAλ\ATA/ATAλ
end
iATA = (iATA+iATA')/2
Σ = σ²*iATA + sqrt(eps())*I
end
"""
bodeconfidence(arxtf::TransferFunction, Σ::Matrix; ω = exp10.(LinRange(0,3,200)))
Plot a bode diagram of a transfer function estimated with [`arx`](@ref) with confidence bounds on magnitude and phase.
"""
bodeconfidence
@userplot BodeConfidence
@recipe function BodeConfidence(p::BodeConfidence; ω = exp10.(LinRange(-2,3,200)))
arxtfm = p.args[1]
Σ = p.args[2]
L = cholesky(Hermitian(Σ))
am, bm = -reverse(denpoly(arxtfm)[1].a[1:end-1]), reverse(arxtfm.matrix[1].num.a)
wm = [am; bm]
na,nb = length(am), length(bm)
mc = 100
res = map(1:mc) do _
w = L.L*randn(size(L,1)) .+ wm
a,b = params2poly(w,na,nb)
arxtf = tf(b,a,arxtfm.Ts)
mag, phase, _ = bode(arxtf, ω)
mag[:], phase[:]
end
magmc = hcat(getindex.(res,1)...)
phasemc = hcat(getindex.(res,2)...)
mag = mean(magmc,dims=2)[:]
phase = mean(phasemc,dims=2)[:]
uppermag = getpercentile(magmc,0.95)
lowermag = getpercentile(magmc,0.05)
upperphase = getpercentile(phasemc,0.95)
lowerphase = getpercentile(phasemc,0.05)
layout := (2,1)
@series begin
subplot := 1
title --> "ARX estimate"
ylabel --> "Magnitude"
yscale --> :log10
xscale --> :log10
fillalpha --> 0.2
label --> ""
linewidth --> 0
fillrange --> lowermag[:]
linealpha --> 0.0
ω, uppermag[:]
end
@series begin
subplot := 1
title --> "ARX estimate"
ylabel --> "Magnitude"
yscale --> :log10
xscale --> :log10
label --> "ARX Magnitude"
ω, mag
end
@series begin
subplot := 2
ylabel --> "Phase [deg]"
xlabel --> "Frequency [rad/s]"
yscale --> :identity
xscale --> :log10
fillalpha --> 0.2
label --> ""
linewidth --> 0
fillrange --> lowerphase[:]
linealpha --> 0.0
ω, upperphase[:]
end
@series begin
subplot := 2
ylabel --> "Phase [deg]"
xlabel --> "Frequency [rad/s]"
label --> "ARX Phase"
yscale --> :identity
xscale --> :log10
ω, phase
end
nothing
end
function getpercentile(mag,p)
uppermag = mapslices(mag, dims=2) do magω
sort(magω)[round(Int,length(magω)*p)]
end
end
0% Loading or .
You are about to add 0 people to the discussion. Proceed with caution.
Please register or to comment