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5. Nonlinear Dynamic Models (Local Model Networks with OBF and FIR)

Dynamic Representations


Disadvantages of Traditional Nonlinear Dynamic Models

  • One-step-ahead prediction error (series-parallel configuration) is optimized (usually).
  • Tuning in simulation (parallel configuration) possible → Complex nonlinear problem.
  • Simulation can even become unstable → Unreliable.
  • Online adaptation thus is very risky.



  • Problems with stability due to external output feedback.
  • Very sensitive w.r.t. correct order and dead time.
  • Interpolation → Strange effects.
  • Optimization of equation error (in series-parallel). Drawbacks:
    - emphasis on high frequencies
    - one common denominator A(z) → identical dynamics for all inputs



  • No problems with stability because no feedback.
  • Optimization of simulation error → All NARX difficulties are gone.
  • New Drawback: Huge orders/dimensions necessary.



  • No problems with stability because only local feedback.
  • Optimization of simulation error → All NARX difficulties are gone.
  • New Drawback: Selection of filter pole(s)?

What's Bad With NARX?  Interpolation of 2 ARX Models


What's Bad With NARX?  Extrapolation


  • Extrapolation behavior in the y(k‒i)-axes determines dynamics
    - slope ≈ 0 → static
    - |slope| < 1 → stable dynamics
    - |slope| > 1 → unstable dynamics
  • Big advantage for all linearly extrapolating model architectures like local linear ...
  • Extremely dangerous for polynomials.
  • Most other model architectures extrapolate constantly.

What's Bad With NARX?  MISO Models


  • Each input → output: Individual dynamics.
  • One common denominator A(z).
  • Multivariate ARX model requires high order = sum of individual orders.
  • Approximate zero/pole cancellations.
  • Static inputs: Bi (z) needs to cancel A(z).
  • Popular solution in linear case → Subspace identification in state space(FIR-based approach)

But Even With NOE?  MISO Models

  • OE: Linear case, each input:
    - individual transfer function
    - individual feedbacks
    - individual states
    - no. of state variables  =  no. of inputs  x  order 


  • NOE: Nonlinear case, each input:
    - one feedback independent of inputs
    - one set of states
    - no. of state variables  =  order 



Regularized FIR Estimation – The Linear Case

Recent Progress for the FIR Identification Case

  • Usage of regularization for the estimation of linear FIR models (proposed by Pillonetto et. al. 2011)

Modified Loss Function for FIR Estimation


  • Regularized least squares problem
  • Choice of the regularization matrix
    - “Tuned correlated” (TC) kernel, penalizes an exponentially decaying version of the first order derivative of the system
    - Interpretation as RKHS or Gaussian Process Model
    - Nonlinear optimization of the hyperparameters

Regularized FIR


  • Loss function
  • P specifies the parameter covariances
  • P should decrease exponentially with α (α is the pole)
  • P also should decrease exponentially with distances k-l.
    This is the TC kernel:

3 Priors ~ Gaussian Process f64_b01 3 Realizations each, order m = 20

with: f64_b02

a) white, bk independent, non-decreasing
b) white, bk independent, exp. decreasing
c) correlated, bk dependent, exp. decreasing


Example: Identification of Damped Second Order System


Advantages of Regularized FIR (Blue Curve)

  • High variance of FIR is suppressed
  • Compared to ARX the stability of the impulse response is a priori guaranteed

Extension to the Nonlinear Case – Concept



  • Use regularized FIR models as local Models
  • Validity functions z can depend on delayed inputs and outputs either


  • Complete model trained for simulation error
  • Stability of the identified model is guaranteed
  • Model order estimation is not required


  • Hyperparameter search for every local model

Further aspects

  • Consideration of the LMN offset in the models
  • Interpretation of model error as a composition of nonlinearity and noise error

Example: Saturation Type Wiener System


Estimated Results with LOLIMOT and Different Local Model Types



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