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processes
BallAndBeam.jl
Commits
efcbb3fc
Commit
efcbb3fc
authored
5 years ago
by
Tommi Nylander
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Added fixed arx file
parent
4e67025e
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#762
canceled
5 years ago
Stage: test
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src/arx.jl
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View file @
efcbb3fc
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
))
+
2
d
/
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
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