# ControlSystemIdentification

System identification for ControlSystems.jl. Examples in the form of jupyter notebooks are provided here.

# LTI state-space models

There exist two methods for identification of statespace models, `n4sid`

and `pem`

. `n4sid`

uses subspace-based identification whereas `pem`

solves the prediction-error problem using an iterative optimization method (from Optim.jl). If unsure which method to use, try `n4sid`

first.

## Subspace based identification using n4sid

```
d = iddata(y,u)
sys = n4sid(d, :auto; verbose=false)
# or use a robust version of svd if y has outliers or missing values
using TotalLeastSquares
sys = n4sid(d, :auto; verbose=false, svd=x->rpca(x)[3])
```

Estimate a statespace model using the n4sid method. Returns an object of type `N4SIDResult`

where the model is accessed as `sys.sys`

.

#### Arguments:

`d`

: Identification data object, created using`iddata(y,u)`

.`y`

: Measurements N×ny`u`

: Control signal N×nu`r`

: Rank of the model (model order)`verbose`

: Print stuff?`i`

: Algorithm parameter, generally no need to tune this`γ`

: Set this to a value between (0,1) to stabilize unstable models such that the largest eigenvalue has magnitude γ.

### Filtering and simulation

Models can be simulated using `lsim`

from ControlSystems.jl and using `simulate`

. You may also convert the model to a `KalmanFilter`

from LowLevelParticleFilters.jl by calling `KalmanFilter(sys)`

, after which you can perform filtering and smoothing etc. with the utilities provided for a `KalmanFilter`

.

## PEM

A simple algorithm for identification of discrete-time LTI systems on state-space form:

```
x' = Ax + Bu + Ke
y = Cx + e
```

is provided. The user can choose to minimize either prediction errors or simulation errors, with arbitrary metrics, i.e., not limited to squared errors.

The result of the identification is a custom type `StateSpaceNoise <: ControlSystems.LTISystem`

, with fields `A,B,K`

, representing the dynamics matrix, input matrix and Kalman gain matrix, respectively. The observation matrix `C`

is not stored, as this is always given by `[I 0]`

(you can still access it through `sys.C`

thanks to `getproperty`

).

This package also supports estimating models on the form

```
Ay = Bu + Cw
```

through pseudo-linear regression. Estimation of the more general model form

```
Ay = B/F u + C/D w
```

or any of its other special cases is not supported. Since those models are also LTI systems, estimating a state-space model is in some sense equivalent.

Transfer-function estimation through spectral methods is supported through the functions `tfest`

and `coherence`

.

### Usage example

Below, we generate a system and simulate it forward in time. We then try to estimate a model based on the input and output sequences.

```
using ControlSystemIdentification, ControlSystems, Random, LinearAlgebra
function ⟂(x)
u,s,v = svd(x)
u*v
end
function generate_system(nx,ny,nu)
U,S = ⟂(randn(nx,nx)), diagm(0=>0.2 .+ 0.5rand(nx))
A = S*U
B = randn(nx,nu)
C = randn(ny,nx)
sys = ss(A,B,C,0,1)
end
Random.seed!(1)
T = 1000 # Number of time steps
nx = 3 # Number of poles in the true system
nu = 1 # Number of control inputs
ny = 1 # Number of outputs
x0 = randn(nx) # Initial state
sim(sys,u,x0=x0) = lsim(sys, u', 1:T, x0=x0)[1]' # Helper function
sys = generate_system(nx,nu,ny)
u = randn(nu,T) # Generate random input
y = sim(sys, u, x0) # Simulate system
d = iddata(y,u)
sysh,x0h,opt = pem(d, nx=nx, focus=:prediction) # Estimate model
yh = predict(sysh, d, x0h) # Predict using estimated model
plot([y; yh]', lab=["y" "ŷ"]) # Plot prediction and true output
```

We can also simulate the system with colored noise, necessitating estimating also noise models.

```
σu = 0.1 # Noise variances
σy = 0.1
sysn = generate_system(nx,nu,ny) # Noise system
un = u + sim(sysn, σu*randn(size(u)),0*x0) # Input + load disturbance
y = sim(sys, un, x0)
yn = y + sim(sysn, σy*randn(size(u)),0*x0) # Output + measurement noise
dn = iddata(yn,un)
```

The system now has `3nx`

poles, `nx`

for the system dynamics, and `nx`

for each noise model, we indicated this to the main estimation function `pem`

:

```
sysh,x0h,opt = pem(dn,nx=3nx, focus=:prediction)
yh = predict(sysh, dn, x0h) # Form prediction
plot([y; yh]', lab=["y" "ŷ"]) # Compare true output (without noise) to prediction
```

We can have a look at the singular values of a balanced system Gramian:

```
s = ss(sysh) # Convert to standard state-space type
s2,G = balreal(s) # Form balanced representation (obs. and ctrb. Gramians are the same
diag(G) # Singular values of Gramians
# 9-element Array{Float64,1}:
# 3.5972307807882844
# 0.19777167699663994
# 0.0622528285731599
# 0.004322765397504325
# 0.004270259700592557
# 0.003243449461350837
# 0.003150873301312319
# 0.0005827927965893053
# 0.00029732262107216666
```

Note that there are 3 big singular values, corresponding to the system poles, there are also 2×3 smaller singular values, corresponding to the noise dynamics.

The estimated noise model can be extracted by `noise_model(sys)`

, we can visualize it with a bodeplot.

`bodeplot(noise_model(sysh), exp10.(range(-3, stop=0, length=200)), title="Estimated noise dynamics")`

See the example notebooks for these plots.

### Call signature

`sys, x0, opt = pem(d; nx, kwargs...)`

#### Arguments:

`d`

: Identification data object, created using`iddata(y,u [,sampletime=nothing])`

`y`

: Measurements, either a matrix with time along dim 2, or a vector of vectors`u`

: Control signals, same structure as`y`

`nx`

: Number of poles in the estimated system. This number should be chosen as number of system poles plus number of poles in noise models for measurement noise and load disturbances.`focus`

: Either`:prediction`

or`:simulation`

. If`:simulation`

is chosen, a two stage problem is solved with prediction focus first, followed by a refinement for simulation focus.`metric`

: A Function determining how the size of the residuals is measured, default`sse`

(e'e), but any Function such as`norm`

,`e->sum(abs,e)`

or`e -> e'Q*e`

could be used.`regularizer(p) = 0`

: function for regularization of the parameter vector`p`

. The structure of`p`

is detailed below. L₂ regularization, for instance, can be achieved by`regularizer = p->sum(abs2, p)`

`solver`

Defaults to`Optim.BFGS()`

`kwargs`

: additional keyword arguments are sent to`Optim.Options`

.

`p`

Structure of parameter vector ```
A = size(nx,ny)
B = size(nx,nu)
K = size(nx,ny)
x0 = size(nx)
p = [A[:];B[:];K[:];x0]
```

### Return values

`sys::StateSpaceNoise`

: identified system. Can be converted to`StateSpace`

by`convert(StateSpace, sys)`

or`ss(sys)`

, but this will discard the Kalman gain matrix, see`innovation_form`

.`x0`

: Estimated initial state`opt`

: Optimization problem structure. Contains info of the result of the optimization problem

### Functions

`pem`

: Main estimation function, see above.`predict(sys, d, x0=zeros)`

: Form predictions using estimated`sys`

, this essentially runs a stationary Kalman filter.`simulate(sys, u, x0=zeros)`

: Simulate the system using input`u`

. The noise model and Kalman gain does not have any influence on the simulated output.`innovation_form`

: Extract the noise model from the estimated system (`ss(A,K,C,0)`

).

### Internals

Internally, Optim.jl is used to optimize the system parameters, using automatic differentiation to calculate gradients (and Hessians where applicable). Optim solver options can be controlled by passing keyword arguments to `pem`

, and by passing a manually constructed solver object. The default solver is `BFGS()`

# ARX and PLR estimation

Basic support for ARX/ARMAX model estimation, i.e. a model on any of the forms

```
Ay = Bu + w
Ay = Bu + Cw
```

is provided. The ARX estimation problem is convex and the solution is available on closed-form. Usage example:

```
N = 2000 # Number of time steps
t = 1:N
Δt = 1 # Sample time
u = randn(N) # A random control input
G = tf(0.8, [1,-0.9], 1)
y = lsim(G,u,t)[1][:]
yn = y
d = iddata(y,u,Δt)
na,nb = 1,1 # Number of polynomial coefficients
Gls = arx(d,na,nb,stochastic=false) # set stochastic to true to get a transfer function of MonteCarloMeasurements.Particles
@show Gls
# TransferFunction{ControlSystems.SisoRational{Float64}}
# 0.8000000000000005
# --------------------------
# 1.0*z - 0.8999999999999997
```

As we can see, the model is perfectly recovered. In reality, the measurement signal is often affected by noise, in which case the estimation will suffer. To combat this, a few different options exist:

```
e = randn(N)
yn = y + e # Measurement signal with noise
d = iddata(yn,u,Δt)
na,nb,nc = 1,1,1
Gls = arx(d,na,nb, stochastic=true) # Regular least-squares estimation
Gtls = arx(d,na,nb, estimator=tls) # Total least-squares estimation
Gwtls = arx(d,na,nb, estimator=wtls_estimator(y,na,nb)) # Weighted Total least-squares estimation
Gplr, Gn = plr(d,na,nb,nc, initial_order=20) # Pseudo-linear regression
@show Gls; @show Gtls; @show Gwtls; @show Gplr; @show Gn;
# TransferFunction{ControlSystems.SisoRational{MonteCarloMeasurements.Particles{Float64,500}}}
# 0.824 ± 0.029
# ---------------------
# 1.0*z - 0.713 ± 0.013
# Gtls = TransferFunction{ControlSystems.SisoRational{Float64}}
# 1.848908051191616
# -------------------------
# 1.0*z - 0.774385918070221
# Gwtls = TransferFunction{ControlSystems.SisoRational{Float64}}
# 0.8180228878106678
# -------------------------
# 1.0*z - 0.891939152690534
# Gplr = TransferFunction{ControlSystems.SisoRational{Float64}}
# 0.8221837077656046
# --------------------------
# 1.0*z - 0.8896345125395438
# Gn = TransferFunction{ControlSystems.SisoRational{Float64}}
# 0.9347035105826179
# --------------------------
# 1.0*z - 0.8896345125395438
```

We now see that the estimate using standard least-squares is heavily biased and it is wrongly certain about the estimate (notice the ± in the transfer function coefficients). Regular Total least-squares does not work well in this example, since not all variables in the regressor contain equally much noise. Weighted total least-squares does a reasonable job at recovering the true model. Pseudo-linear regression also fares okay, while simultaneously estimating a noise model. The helper function `wtls_estimator(y,na,nb)`

returns a function that performs `wtls`

using appropriately sized covariance matrices, based on the length of `y`

and the model orders. Weighted total least-squares estimation is provided by TotalLeastSquares.jl. See the example notebooks for more details.

Uncertain transfer function with `Particles`

coefficients can be used like any other model. Try, e.g., `nyquistplot(Gls)`

to get a Nyquist plot with confidence bounds.

See also function `arma`

for estimation of signal models without inputs.

## Functions

`arx`

: Transfer-function estimation using closed-form solution.`ar`

: Estimate an AR model.`arma`

: Estimate an ARMA model.`plr`

: Transfer-function estimation using pseudo-linear regression`getARXregressor/getARregressor`

: For low-level control over the estimation See docstrings for further help.

## Model-based spectral estimation

The model estimation procedures can be used to estimate spectrograms. This package extends some methods from DSP.jl to accept a estimation function as the second argument. To create a suitable such function, we provide the function `model_spectrum`

. Usage is illustrated below.

```
using ControlSystemIdentification, DSP
T = 1000
fs = 1
s = sin.((1:1/fs:T) .* 2pi/10) + 0.5randn(T)
S1 = spectrogram(s,window=hanning, fs=fs) # Standard spectrogram
estimator = model_spectrum(ar,fs,6)
S2 = spectrogram(s,estimator,window=rect, fs=fs) # Model-based spectrogram
plot(plot(S1,title="Standard Spectrogram"),plot(S2,title="AR Spectrogram")) # Requires the package LPVSpectral.jl
```

# Transfer-function estimation using spectral techniques

Non-parametric estimation is provided through spectral estimation. To illustrate, we once again simulate some data:

```
T = 100000
h = 1
sim(sys,u) = lsim(sys, u, 1:T)[1][:]
σy = 0.5
sys = tf(1,[1,2*0.1,0.1])
ωn = sqrt(0.3)
sysn = tf(σy*ωn,[1,2*0.1*ωn,ωn^2])
u = randn(T)
y = sim(sys, u)
yn = y + sim(sysn, randn(size(u)))
d = iddata(y,u,h)
dn = iddata(yn,u,h)
```

We can now estimate the coherence function to get a feel for whether or nor our data seems to be generated by a linear system:

```
k = coherence(d) # Should be close to 1 if the system is linear and noise free
k = coherence(dn) # Slightly lower values are obtained if the system is subject to measurement noise
```

We can also estimate a transfer function using spectral techniques, the main entry point to this is the function `tfest`

, which returns a transfer-function estimate and an estimate of the power-spectral density of the noise (note, the unit of the PSD is squared compared to a transfer function, hence the `√N`

when plotting it in the code below):

```
G,N = tfest(dn)
bodeplot([sys,sysn], exp10.(range(-3, stop=log10(pi), length=200)), layout=(1,3), plotphase=false, subplot=[1,2,2], size=(3*800, 600), ylims=(0.1,300), linecolor=:blue)
coherenceplot!(dn, subplot=3)
plot!(G, subplot=1, lab="G Est", alpha=0.3, title="Process model")
plot!(√N, subplot=2, lab="N Est", alpha=0.3, title="Noise model")
```

The left figure displays the Bode magnitude of the true system, together with the estimate (noisy), and the middle figure illustrates the estimated noise model. The right figure displays the coherence function, which is close to 1 everywhere except for at the resonance peak of the noise `log10(sqrt(0.3)) = -0.26`

.

See the example notebooks for more details.

# Impulse-response estimation

The functions `impulseest(h,y,u,order)`

and `impulseestplot`

performs impulse-response estimation by fitting a high-order FIR model.

Example

```
T = 200
h = 1
t = h:h:T
sim(sys,u) = lsim(sys, u, t)[1][:]
sys = c2d(tf(1,[1,2*0.1,0.1]),h)
u = randn(length(t))
y = sim(sys, u)
d = iddata(y,u,h)
impulseestplot(d,50, lab="Estimate")
impulseplot!(sys,50, lab="True system")
```

See the example notebooks for more details.

# Validation

A number of functions are made available to assist in validation of the estimated models. We illustrate by an example

Generate some test data:

```
Random.seed!(1)
T = 200
nx = 2
nu = 1
ny = 1
x0 = randn(nx)
σy = 0.5
sim(sys,u) = lsim(sys, u', 1:T)[1]'
sys = tf(1,[1,2*0.1,0.1])
sysn = tf(σy,[1,2*0.1,0.3])
# Training data
u = randn(nu,T)
y = sim(sys, u)
yn = y + sim(sysn, randn(size(u)))
dn = iddata(yn,u)
# Validation data
uv = randn(nu,T)
yv = sim(sys, uv)
ynv = yv + sim(sysn, randn(size(uv)))
dv = iddata(yv,uv)
dnv = iddata(ynv,uv)
```

We then fit a couple of models, the flag `difficult=true`

causes `pem`

to solve an initial global optimization problem with constraints on the stability of `A-KC`

to provide a good guess for the gradient-based solver

`res = [pem(dn,nx=nx, iterations=100, difficult=true, focus=:prediction) for nx = [1,3,4]]`

After fitting the models, we validate the results using the validation data and the functions `simplot`

and `predplot`

(cf. Matlab sys.id's `compare`

):

```
ω = exp10.(range(-2, stop=log10(pi), length=150))
fig = plot(layout=4, size=(1000,600))
for i in eachindex(res)
(sysh,x0h,opt) = res[i]
simplot!( sysh,dnv,x0h; subplot=1, ploty=i==1)
predplot!(sysh,dnv,x0h; subplot=2, ploty=i==1)
end
bodeplot!(ss.(getindex.(res,1)), ω, plotphase=false, subplot=3, title="Process", linewidth=2*[4 3 2 1])
bodeplot!(innovation_form.(getindex.(res,1)), ω, plotphase=false, subplot=4, linewidth=2*[4 3 2 1])
bodeplot!(sys, ω, plotphase=false, subplot=3, lab="True", linecolor=:blue, l=:dash, legend = :bottomleft, title="System model")
bodeplot!(innovation_form(ss(sys),syse=ss(sysn)), ω, plotphase=false, subplot=4, lab="True", linecolor=:blue, l=:dash, ylims=(0.1, 100), legend = :bottomleft, title="Noise model")
display(fig)
```

In the figure, simulation output is compared to the true model on the top left and prediction on top right. The system models and noise models are visualized in the bottom plots. Both high-order models capture the system dynamics well, but struggle slightly with capturing the gain of the noise dynamics. The figure also indicates that a model with 4 poles performs best on both prediction and simulation data. The true system has 4 poles (two in the process and two in the noise process) so this is expected. However, the third order model performs almost equally well and may be a better choice.

# Other resources

- For estimation of linear time-varying models (LTV), see LTVModels.jl.
- For estimation of linear and nonlinear grey-box models in continuous time, see DifferentialEquations.jl (parameter estimation)
- Estimation of nonlinear black-box models in continuous time DiffEqFlux.jl and in discrete time Flux.jl
- For more advanced spectral estimation, cross coherence, etc., see LPVSpectral.jl
- This package interacts well with MonteCarloMeasurements.jl. See example file.