Grey-box system identification

Question 1

Standard use of Kalman filter

Answer 1

It’s a SW algorithm that gives an estimation of the state of a system, based on some measured noisy outputs.
It’s mainly a variable estimation tool, but it can be extended to identify parameters.

Question 2

Extension of Kalman filter for grey-box parameter identification

Answer 2

Starting from the non-linear system model S:

x(t+1) = f(x(t); θ) + g(u(t); θ) + v1(t)
y(t) = h(x(t); θ) + v2(t)

where v1 is the model noise and v2 is the measurement noise
The goal is to estimate the state x^(t) and the parameter vector θ^.

The solution is obtained by extending the system, with the unknown parameters transformed into new state variables:
x(t+1) = f(x(t); θ) + g(u(t); θ) + v1(t)
θ(t+1) = θ(t) + vθ(t)
y(t) = h(x(t); θ) + v2(t)

• > extended state vector xE(t) = [x(t) θ(t)]’

Question 3

new equation

Answer 3

θ(t+1) = θ(t) + vθ

• fictitious equation (not physical)
• it represents a constant dynamic, as it is expected by the unknown parameters
• it represents a simply stable system, but this is not a problem since KF can deal also with unstable systems
• the noise vθ is needed to force the KF to modify the value of θ: if no noise is present, the algorithm maintains the initial value

Question 4

how to define vθ

Answer 4

vθ = WN(0, Vθ)
where Vθ is the covariance matrix of vθ (vector of white noise)
- assumption vθ is not correlated with v1 and v2
- vθ(t) is made by a set of independent white noises, all with the same variance:
Vθ = diag(λθ^2, λθ^2, … , λθ^2)

• parameter λθ^2 is a tuning parameter, and its definition is a critical design choice: tradeoff between speed of convergence and variance of the result
• the tuning of λθ^2 is done iteratively until a good compromise is obtained, depending on the application

Question 5

Applicability of this method

Answer 5

In principle, it can work with any number of unknown parameters, for example 3 sensors, 10 states, 20 parameters;
In practise, it works only with a limited number of parameters, like 3 sensors, 5 states, 2 parameters

Question 6

Simulation Error Method

Answer 6

SEM is a grey-box parametric identification method

1) Data collection
input {u(1), …, u(N)}
output {y(1), …, y(N)}

2) model class (grey-box approach):
y(t) = m(u(t); θ_; θ)
where
- m is a Mathematical model of the systems, written from first principles, which depends on
- input u
- vector of known parameters θ_
- vector of unknown parameters θ, usually with defined validity bounds

3) performance index
JN(θ) = 1/N * sum(t=1,N) (y(t) - m(u(t); θ_; θ))^2
where y(t) is the measured output
y(t) - m(u(t); θ_; θ) is the simulation error

4) optimization
θ^N = argminθ{JN(θ)}
in this case, JN(θ) is highly non-linear, and its minimmization requires iterative methods with different initial conditions, each of them is a complete simulation of the dynamics for t=1,…,N

Question 7

PEM vs SEM

Answer 7

pros of SEM

• more general, it can be applied to all types of modelling problems, black/grey box, linear/non-linear
• robust to noise

cons of SEM
- computationally much more expensive

• > SEM is usually used in case of grey-box identification with just a few unknown parameters