1. Tijl De Bie John Shawe-Taylor ECS, ISIS, University of Southampton
A statistical learning approach to subspace identification of dynamical systems
Tijl De Bie
John Shawe-Taylor
ECS, ISIS, University of Southampton

2. Warning! work in progress…

3. PASCAL workshop - Bohinj
Overview
Dynamical systems and the state space representation
Subspace identification for linear systems
Regularized subspace identification
Kernel version for nonlinear dynamical systems
Preliminary experiments
PASCAL workshop - Bohinj

4. PASCAL workshop - Bohinj
Dynamical systems
Dynamical systems:
Accepts inputs ut 2 <m
Generates outputs yt 2 <l
Is affected by noise…
E.g.: yt = (a1yt-1 + a2yt-2) + (b0ut + b1ut-1) + nt
u0, u1, …, ut, …
y0, y1, …, yt, …
Dynamical system
PASCAL workshop - Bohinj

5. PASCAL workshop - Bohinj
Dynamical systems
Practical examples:
Car:
inputs = position of steering wheel, force applied to wheel, clutch position, road conditions,…
outputs = position of the car
Chemical reactor:
inputs = inflow of reactants, warmth added,…
outputs = temperature, pressure,…
Other: bridges, vocal tract, electrical systems,…
PASCAL workshop - Bohinj

6. State space representation
Next position of car (output) depends on the current inputs, and on the current position and speed
Temperature and pressure (outputs) depend on the current inputs and on the current composition of the mixture, temperature and pressure
Summarize total effect of past inputs in the state xt 2 <n of the system: speed of car / mixture composition
This leads to the state space representation:
PASCAL workshop - Bohinj

7. State space representation
State space representation (SSR):
Memory, stored in the state vector: xt
Summarizes the past
u0, u1, …, ut, …
y0, y1, …, yt, …
State update equation: xt+1 = fstate(xt,ut,wt)
Output equation: yt = foutput(xt,ut,vt)
PASCAL workshop - Bohinj

8. State space representation
Linear state space model:
Interpretation: the states are latent variables with Markov dependency…
Note: even simple systems like the car are nonlinear, but often linear is a good approximation around working point
State update equation: xt+1 = Axt + But + wt
Output equation: yt = Cxt + Dut + vt
PASCAL workshop - Bohinj

9. State space representation
Advantages of the SSR:
Intuitive, often close to ‘first principles’
State observer (such as Kalman filter) allows to estimate states based on input and outputs  optimal control based on these Kalman states. This is, if the system (i.e and ) is known…
If the system is not specified: algorithms for system identification exist, often identifying a SSR
PASCAL workshop - Bohinj

10. PASCAL workshop - Bohinj
Overview
Dynamical systems and the state space representation
Subspace identification for linear systems
Regularized subspace identification
Kernel version for nonlinear systems
Preliminary experiments
PASCAL workshop - Bohinj

11. Subspace identification for linear systems
System identification:
Given:
and
Noise unknown but iid
… (other technical conditions)
Determine: system parameters, i.e. the system matrices
Classical system identification focuses on asymptotic unbiasedness (sometimes consistency) of the estimators  ok for large samples
Regularization issues are at most a side-note (+/- 10 pages in the 600 pages reference work by Ljung)
Explanation: often datasets are relatively low dimensional and large
PASCAL workshop - Bohinj

12. Subspace identification for linear systems
Fact:
If an estimate for the state
sequence is known, determining the system matrices is a (least squares) regression problem:
PASCAL workshop - Bohinj

13. Subspace identification for linear systems
Estimate state sequence first !
Two observations:
We can estimate state based on past inputs
and past outputs (and of initial state x0)
So, for any vector there are vectors and such that:
PASCAL workshop - Bohinj

14. Subspace identification for linear systems
Future outputs are depending on the
current state and future inputs
So, for any vector there are vectors and such that:
PASCAL workshop - Bohinj

15. Subspace identification for linear systems
Use notation:
Similarly past and future outputs and
State sequence:
‘Past inputs’ as columns, time shifted
‘Future inputs’ as columns, time shifted
PASCAL workshop - Bohinj

16. Subspace identification for linear systems
Then:
which is strongly related to CVA (Canonical Variate Analysis), a well known subspace identification method…
Solution: generalized eigenvalue problem (like CCA); as many significantly nonzero eigenvalues as the dimensionality of the state space
Reconstruct state sequence as:
However: generalization problems with high-dimensional input space and small sample sizes…
PASCAL workshop - Bohinj

17. PASCAL workshop - Bohinj
Overview
Dynamical systems and the state space representation
Subspace identification for linear systems
Regularized subspace identification
Kernel version for nonlinear systems
Preliminary experiments
PASCAL workshop - Bohinj

18. Regularized subspace identification
Introduce regularization:
Solution: generalized eigenvalue problem
PASCAL workshop - Bohinj

19. Regularized subspace identification
Introduce regularization:
Notes:
Different regularization: if output space is high dimensional, also on the
and
Different constraints (e.g. if is omitted and without regularization, we get exactly CVA)
PASCAL workshop - Bohinj

20. PASCAL workshop - Bohinj
Overview
Dynamical systems and the state space representation
Subspace identification for linear systems
Regularized subspace identification
Kernel version for nonlinear systems
Preliminary experiments
PASCAL workshop - Bohinj

21. Kernel version for nonlinear systems
Hammerstein models (input nonlinearity):
Examples: in the car examples relations between forces, angles of the steering wheel,… involve sinus functions; also: saturation in actuators correspond to (‘soft’) sign functions
Using feature map on inputs , then:
(with B having potentially infinitely many columns)
PASCAL workshop - Bohinj

22. Kernel version for nonlinear systems
Use representer theorem:
where = vector containing (with the training samples)
and containing dual variables…
PASCAL workshop - Bohinj

23. Kernel version for nonlinear systems
Then:
 Kernel version of subspace identification algorithm; regularization is absolutely necessary here…
Solution: similar generalized eigenvalue problem (cfr. Kernel-CCA)
Extensions (further work): output nonlinearity (Wiener-Hammerstein systems); requires the inverse feature map…
PASCAL workshop - Bohinj

24. PASCAL workshop - Bohinj
Overview
Dynamical systems and the state space representation
Subspace identification for linear systems
Regularized subspace identification
Kernel version for nonlinear systems
Preliminary experiments
PASCAL workshop - Bohinj

25. Preliminary experiments
Experiments with system (200 samples):
Random Gaussian inputs with std=1, Gaussian noise
PASCAL workshop - Bohinj

26. PASCAL workshop - Bohinj
Further work
Conclusions:
Regularization ideas imported into system identification for large dimensionalities / small samples
Kernel trick allows for identification of Hammerstein nonlinear systems
Further work:
Motivation of our type of regularization with learning theory bounds
Wiener-Hammerstein models (also output nonlinearity)
Extension of notions like controllability to such nonlinear models
Design of Kalman filter and controllers based on this kind of Hammerstein systems
PASCAL workshop - Bohinj

27. PASCAL workshop - Bohinj
Thanks!
Questions?
PASCAL workshop - Bohinj