# Recursive identification of nonlinear Wiener systems

The Wiener model consists of a linear dynamic block in cascade with a static nonlinear function, see Figure 1.

*Figure 1: Block diagram of the nonlinear Wiener model.*

Despite its simplicity the Wiener model has been successfully used to describe a number of systems, some important ones being

- Joint mixing and chemical reaction processes in the chemical process industry. Various types of pH-control processes constitute typical examples.
- Biological processes, including e.g. vision.
- Effects of drugs in the human body, e.g. in Anaesthesia.

What is less well known is that the Wiener model is also useful for description of a number of situations where the measurement of the output of a linear system is highly nonlinear and non-invertible. Important examples include

- Output measurement saturation.
- Output measurement dead zones.
- Output measurememhts insensitive to sign, e.g. pulse counting angular rate sensors.
- Output quantization. This case has received a considerable interest recently with the emerging techniques for network control systems.
- Blind adaptation. This follows since the blind adaptation problem can sometimes be cast into the form of a Wiener system.

Wiener models have also been successfully used for extremum control. A main motivation for the use of Wiener models is that the dynamics is linear, a fact that simplifies the handling of properties like statistical stationarity and stability, as compared to when a general nonlinear model is applied.

With the above in mind, it is clear that identification of non-linear Wiener systems is an important topic. The present project is focused on recursive identification of Wiener systems using parametric methods. A key property is that all algorithms allow the non-linearity to be non-invertible. Many results have been obtained on

- The design of algorithms and software (3-6, 8, 9, 12-17).
- Analysis and validation w.r.t. under-modeling in non-linear system identification (2, 3, 11).
- The convergence properties of algorithms (5, 6, 8, 9, 13-17).
- The importance of the amplitude contents of the input signals for the success of nonlinear identification (2, 3, 16, 17).

Regarding applications, the papers 12 and 13 provide algorithms for blind adaptation that solve the so called ill-convergence problem noted by Professor Rick Johnson. The normalized algorithm of 9, identifying finite impulse response dynamics using coarsely quantized output measurements, is proved to be convergent w.p.1. to a parameter setting where the input-output prediction error equals zero. Hence, the convergence properties resemble those of the NLMS-algorithm. Numerical results displaying this property are illustrated by Table 1.

*Table 1: Convergence of the algorithm for compensation for quantization. The true parameter vector was (1 .000 -0.700 4.000 -2.800)^T and the quantizer had 5 levels. The left part of the table represent 16 initial values, whereas the right part represent the convergence points obtained after processing of 4000 samples. *

The normalized algorithm was also shown to be applicable to removal of PCM quantization effects in adaptive echo cancellation loops. For infinite impulse response models local convergence to the true parameter vector is proved in 14. The technique of the paper 13 was also applied to exhaust control by identification of air-fuel mixing dynamics in 7.

As a further illustration of the achieved results, the following figure illustrates the application of one of the circle criteria of 10 that are sufficient for convergence of a certain class of recursive algorithms that use approximations of gradients.

*Figure 2: Application of one of the circle criteria derived in 10 and 14.*

### References

1. T. Wigren, Discussion on: Subspace Identification of Multivariable Hammerstein and Wiener Models, European Journal of Control, vol. 11, no. 2, pp. 148-149, 2005.

2. T. Wigren, User choices and model validation in system identification using nonlinear Wiener models, Proc. 13:th IFAC Symposium on System Identification, Rotterdam, The Netherlands, pp. 863-868, August 27-29, 2003.

3. A. E. Nordsjö and T. Wigren, On estimation of model errors caused by nonlinear undermodeling in system identification, International Journal of Control , vol. 75, no. 14, pp. 1100-1113, 2002.

4. T. Wigren and A. E Nordsjö, Compensation of the RLS algorithm for output nonlinearities, IEEE Trans. Automat. Contr. , vol. 44, no. 10, pp. 1913-1918, 1999.

5. T. Wigren, Adaptive filtering using quantized output measurements, IEEE Trans. Signal Processing , vol. 46, no. 12, pp. 3423-3426, 1998.

6. T. Wigren, Output error convergence of adaptive filters with compensation for output nonlinearities, IEEE Trans. Automat. Contr., vol. 43, no. 7, pp. 975-978, 1998.

7. T. Wigren and B. Carlsson, On the use of quantized lambda-sensor measurements in recursive identification of air-fuel mixing dynamics, Preprints of the IFAC Workshop on Intelligent Components for Vehicles , Seville, Spain, pp. 77-82, March 23-24, 1998.

8. T. Wigren, Avoiding ill-convergence of finite dimensional blind adaptation schemes excited by discrete symbol sequences, Signal Processing, vol. 62, no. 2, pp. 121-162, 1997.

9. T. Wigren, ODE analysis and redesign in blind adaptation, IEEE Trans. Automat. Contr. , vol. 42, no. 12, pp. 1742-1747, 1997.

10. T. Wigren, Circle criteria in recursive identification, IEEE Trans. Automat. Contr. , vol. 42, no. 7, pp. 975-979, 1997.

11. A. E. Nordsjö, B. M. Ninness and T. Wigren, Quantifying model errors caused by nonlinear undermodeling in linear system identification, Preprints 13th World Congress of IFAC, San Francisco, California, U.S.A., vol. 1, pp. 145--149, 1996.

12. T. Wigren and P. Händel, Harmonic signal modeling using adaptive nonlinear function estimation, Proc. ICASSP, Atlanta, Georgia, U.S.A., pp. 2952-2955, 1996.

13. U. Lindgren, T. Wigren and H. Broman, On local convergence of a class of blind separation algorithms, IEEE Trans. Signal Processing, vol. 43, no. 12, pp. 3054-3058, 1995.

14. T. Wigren, Approximate gradients, convergence and positive realness in recursive identification of a class of nonlinear systems, Int. J. Adaptive Contr. Signal Processing, vol. 9, no. 4, pp. 325-354, 1995.

15. T. Wigren, Convergence analysis of recursive identification algorithms based on the nonlinear Wiener model, IEEE Trans. Automat. Contr., vol. AC-39, no. 11, pp. 2191-2206, 1994.

16. T. Wigren, Recursive prediction error identification using the nonlinear Wiener model, Automatica, vol. 29, no. 4, pp. 1011-1025, 1993.

17. T. Wigren, Recursive identification based on the nonlinear Wiener model, Ph.D. thesis, Acta Universitatis Upsaliensis, Uppsala Dissertations from the Faculty of Science 31, Uppsala University, Uppsala, Sweden, December, 1990.