A statistical learning approach to subspace identification of dynamical systems

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A statistical learning approach to subspace
identification of dynamical systems

• Tijl De Bie
• John Shawe-Taylor
• ECS, ISIS, University of Southampton

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Warning!work in progress
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Overview

• Dynamical systems and the state space
  representation
• Subspace identification for linear systems
• Regularized subspace identification
• Kernel version for nonlinear dynamical systems
• Preliminary experiments

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Dynamical systems

• Dynamical systems
• Accepts inputs ut 2 ltm
• Generates outputs yt 2 ltl
• Is affected by noise
• E.g. yt (a1yt-1 a2yt-2) (b0ut b1ut-1)
  nt

u0, u1, , ut,
y0, y1, , yt,
Dynamical system
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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,

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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 ltn of the system speed of car /
  mixture composition
• This leads to the state space representation

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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 xt1 fstate(xt,ut,wt)
Output equation yt foutput(xt,ut,vt)
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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 xt1 Axt But wt
Output equation yt Cxt Dut vt
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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

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Overview

• Dynamical systems and the state space
  representation
• Subspace identification for linear systems
• Regularized subspace identification
• Kernel version for nonlinear systems
• Preliminary experiments

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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

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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

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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

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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

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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
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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

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Overview

• Dynamical systems and the state space
  representation
• Subspace identification for linear systems
• Regularized subspace identification
• Kernel version for nonlinear systems
• Preliminary experiments

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Regularized subspace identification

• Introduce regularization
• Solution generalized eigenvalue problem

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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)

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Overview

• Dynamical systems and the state space
  representation
• Subspace identification for linear systems
• Regularized subspace identification
• Kernel version for nonlinear systems
• Preliminary experiments

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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)

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Kernel version for nonlinear systems

• Use representer theorem
• where vector containing
  (with the training samples)
• and containing dual variables

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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

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Overview

• Dynamical systems and the state space
  representation
• Subspace identification for linear systems
• Regularized subspace identification
• Kernel version for nonlinear systems
• Preliminary experiments

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Preliminary experiments

• Experiments with system (200 samples)
• Random Gaussian inputs with std1, Gaussian
  noise

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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

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Thanks!

• Questions?