add experimental covariance estimate to structured_pem

[baggepinnen]

Usage (experimental)

using ControlSystemIdentification, Optim, ControlSystemsBase
import LowLevelParticleFilters: SimpleMvNormal
using MonteCarloMeasurements
using Random, Test, LinearAlgebra

constructor = function (p)
    Mα, Mq, Mδe, Zα, Zq, Zδe = p
    V = 30.2
    A = [
        Mq Mα
        (1+Zq / V) Zα/V
    ]
    B = [Mδe; Zδe / V]
    return ss(A, B, I, 0)
end

# True Parameters
Mα_true = -83.17457223750834
Mq_true = -4.0485957994781785
Mδe_true = -80.71377329163104
Zα_true = -115.91677356408941
Zq_true = -0.573560626095269
Zδe_true = 32.13261325199936

Ts = 0.01
t = 0:Ts:10

p_true = [Mα_true, Mq_true, Mδe_true, Zα_true, Zq_true, Zδe_true]
p0 = [Mα_true, Mq_true, Mδe_true, Zα_true, Zq_true, Zδe_true] .* 1.2

P_true = constructor(p_true)
u = sign.(sin.((1 / 3 * t)')) .+ sign.(sin.((1 / 2 * t)'))
res_true = lsim(c2d(P_true, Ts, :tustin), u, t)
d = iddata(res_true)

nx = 2
function regularizer(p, _)
    sum(abs2, p.K)
end

d2 = deepcopy(d)
d2.y .+= randn.()

res = ControlSystemIdentification.structured_pem(d2, nx; focus=:simulation, p0, constructor, regularizer, show_trace = false, show_every = 1, iterations=300, method=:tustin)

H = res.Λ() # Inverse covariance matrix
Σ = inv(H) # Covariance matrix
dp = SimpleMvNormal(res.res.minimizer, Σ) # Parameter posterior distribution
pars = Particles(rand!(Xoshiro(), dp, zeros(length(dp), 100))') # Sample uncertain parameters
sysu, Ku, x0u = res.constructor(pars) # Construct uncertain system

w = exp10.(LinRange(-3, 2, 200))
unsafe_comparisons(true)
bodeplot(sysu, w, plotphase=false)
bodeplot!(res.sys, w, plotphase=false, legend=false)

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 88.36%. Comparing base (b30bfbe) to head (592d394).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #188      +/-   ##
==========================================
- Coverage   88.68%   88.36%   -0.32%     
==========================================
  Files          13       13              
  Lines        2404     2416      +12     
==========================================
+ Hits         2132     2135       +3     
- Misses        272      281       +9     

Flag	Coverage Δ
unittests	88.36% <100.00%> (-0.32%)	⬇️

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