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Identification of Wiener models using support vector regression

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Identification of Wiener models using support vector regression

Stefan Tötterman and Hannu Toivonen

Process Control Laboratory

Åbo Akademi University

Finland

Process Control Laboratory - Åbo Akademi University

Wiener models

- Output error identification
- The dynamic linear part F consist of an orthonormal filter
- The static nonlinear part N consists of a support vector model

Process Control Laboratory - Åbo Akademi University

- insensitive loss function

y:observation

yest: estimated function

y

y-yest

x

Process Control Laboratory - Åbo Akademi University

A set of basis functions:

Estimation of y is expanded in basis functions

Minimization of L and norm of the weight (smoothness, robustness)

Support Vector Regressionwhere w is a weight parameter

C is a weight

Process Control Laboratory - Åbo Akademi University

Support Vector Regression

- The optimization problem is transformed to a dual convex optimization problem and the approximation function is given by
- Most of the factors (αi - αi’) will be zero, the input vectors corresponding to the nonzero factors forms the so-called support vectors (correspond to observations outside the ε-tube)
- K(xi,xj) is the inner-product kernel, commonly RBF

Lagrange multipliers

Process Control Laboratory - Åbo Akademi University

Support Vector Regression

- SVMs can be seen as a network, where all the important network parameters are computed automatically.

bias, b

K(x,x1)

x1

(1-1’)

yest

K(x,x2)

(2-2’)

x2

SV

Most of the weights (i-’i) will be zero, the other will define the support vectors

(m1-m1’)

K(x,xm1)

xN

Input layer

RBF with centers x1,...,xm1

Process Control Laboratory - Åbo Akademi University

Some properties of SVR

- No need to compute (xi), enough to compute the kernel values directly (kernel trick).
- Convex optimization.
- Robust algorithm when using L.
- Optimal model complexity is obtained automatically as a part of the solution.
- Efficient optimization methods exist (high memory requirements).
- Hard to involve prior knowledge about the task.
- and C must be chosen simultaneously by the user.

Process Control Laboratory - Åbo Akademi University

Dynamic linear part

- Introducing orthonormal filters to the dynamic linear part have been found useful.
- Usually Laguerre or Kautz filter-types are used.
- Laguerre filters with a single real-valued pole are well suited for modelling well damped systems.
- Kautz filters with a pair of complex-valued poles are suitable for systems which have oscillatory behaviour.

Process Control Laboratory - Åbo Akademi University

Dynamic linear part

- Laguerre filters

q-1 is the backward-shift operator and || 1. Outputs are calculated for k = 1, 2, ..., l where l is the filter order.

Process Control Laboratory - Åbo Akademi University

Dynamic linear part

- The filter output xk can be derived from the previous filter output xk-1

Process Control Laboratory - Åbo Akademi University

Wiener models

- General Wiener model
- Wiener model in this identification method

Process Control Laboratory - Åbo Akademi University

Identification of Wiener models

- Design parameters:
- Dynamic linear part:
- (filter pole)
- l (filter order)

- Dynamic linear part:

- The identified systems dynamics are unknown

- Static nonlinear part:
- (insensitivity margin)
- C (weight)
- γ (RBF kernel)

Process Control Laboratory - Åbo Akademi University

Example – Control valve model*

- The input u(t) is a pneumatic control signal
- The output y(t) is a flow through a valve
- The simulated model is described by the following equations

e(t) is white gaussian measurement noise, standard deviation 0.05

- *T. Wigren, Recursive prediction error identification using the nonlinear Wiener model, Automatica 29(4) (1993)
- *A Hagenblad, Aspects of the Identification of Wiener Models, Linköping Studies in Science and Technology, Thesis No. 793, 1999

Process Control Laboratory - Åbo Akademi University

Example – Control valve model This choice of parameters results in a model consisting of 146 support vectors

- Laguerre filter of order l = 5 and with the pole = 0.4 was found to be a proper choice
- Optimal SVR parameters
- γ = 0.1
- = 0.08
- C = 2000

- RMSE 0.0541 (train)
- RMSE 0.0556 (test)

Process Control Laboratory - Åbo Akademi University

Example – Control valve model

- Last 100 samples of the test data set

Measured output (solid)

Model output (dashed)

Noisefree output (dotted)

Process Control Laboratory - Åbo Akademi University

Example – Control valve model

- Output errors (test data)

y-ŷ

yNF-ŷ

Samples

Process Control Laboratory - Åbo Akademi University

Example – Control valve model

- Laguerre filter parmeter sensitivity table

Process Control Laboratory - Åbo Akademi University

Example – Control valve model

- SVR parmeter sensitivity table

Process Control Laboratory - Åbo Akademi University

Conclusions

- This identification method works well for Wiener model identification and gives accurate models
- The model is determined by solving a convex quadratic minimization problem (global optimum is always obtained)
- Robust performance w.r.t. new data is achieved since SVR is based on structural risk minimization
- It is straightforward to extend this method to MIMO systems

Process Control Laboratory - Åbo Akademi University

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