# Recursive identification of non-linear systems using differential equation models

The identification of non-linear systems has received an increasing interest recently. Noting that most methods for nonlinear controller design are based on continuous time ordinary differential equation (ODE) models, the present project is focused on

- Development of recursive identification algorithms based on black-box ODE models on state space form.
- Development and analysis of new strategies for scaling of the resulting identification problems.
- Analysis of the convergence properties of the developed algorithms.
- Development of software that implement the proposed identification methods.

Initially the parameterization problem was studied. The reason for this is that many physical models are built up from sub-models. This typically results in ODEs where the right hand side structure is quite complicated. However, since the main idea was to come up with a general black-box parameterization of the right hand side of the ODE, the problem with overparameterization seemed to be in conflict with the need for generality of the right hand side model.

The report 27 and the paper 22 solve this problem, by the use of a restricted black box model that models only one right hand side component of the ODE. The remaining model states are constructed from a chain of integrators. The solution to the indicated problem is then obtained by a theorem that proves that the proposed restricted black-box model is locally capable of modeling systems with arbitrary right hand side structure. The proof exploits the inverse function theorem and can be found in detail in the report 27. The restricted black box model is then parameterized as a multi-variable polynomial in the states and inputs. This leads to the output error RPEM algorithm described in 22-27 and applied in 9, 10, and 19.

As is well known, output error algorithms have the disadvantage that they can get stuck in local sub-optimal minima of the criterion function. Initiation is hence a central topic. The papers 19-21 therefore discuss an initiation algorithm where approximations of the model states are generated by differentiation of the measured output signal. As expected, this approach is sensitive to noise, see 19-21. However, as is shown in 19, the combined use of the initiation algorithm and the RPEM significantly improves the identification performance. The successful identification and control of a solar cooling plant at the University of Seville, Spain, illustrate such combined use further. A comprehensive description of the algorithms and the solar cooling application appears in the thesis 16.

The algorithms of 11 and 22 have also been applied also to selective catalythic aftertreatment for diesel engines with good results in the paper 9. A comparison to state of the art methods is available in 10. Very good results were also obtained in the papers 3 and 4 where an identified nonlinear model was used for feedback linearization control. That paper also used the identification algorithm for adaptive feedback linearizing control with very promising results.

A refined output error RPEM algorithm is presented in 7, 8, 11, 12, and 14. That algorithm uses a more accurate numerical integration algorithm than the Euler algorithm applied in the previous work. The refinement is expected to give more accurate results. The convergence of the scheme is proved in 7 and 8, building on the methodology of 14.

Recently, the original RPEM was generalized to include joint recursive identification of both the state space ODE and a similarly parametrized output equation, see 1. That paper also presents a complete analysis of the convergence properties of the algorithm.

All the proposed algorithmsare prepared to exploit a novel method for scaling of the identification problem. This new scheme uses a single parameter to scale the sampling period, which appears explicitly in the identification schemes. The following three results were proved in 22 for the Euler based algorithms:

- The scaling of the sampling period results in an exponential scaling of the model states.
- The scaling of the sampling period results in a scaling of the estimated parameters. A simple formula, depending on the scaling parameters, is derived. This formula retrieves the un-scaled continuous time parameters from the identified scaled parameters.
- A simple formula that describes the effect of the scaling on the Hessian of the identification problem is derived. The numerical results taken from 22 in Fig. 1, shows that the effects on the conditioning of the identification problem can be very significant.

*Figure 1: The effect of scaling (alpha) on the condition number of the Hessian. The solid line is the prediction by the derived formula and the circles are samples obtained from different identification runs. Note that there appears to be an optimal choice of scaling in this case.*

Software for identification and plotting are available (2, 12, 15, 17, 18, 23, 26) together with manuals. Currently, this software implements the output error RPEM and a corresponding algorithm for identification of a static non-linear function. The recent SW 2 adds the RPEM with an unknown output equation to the set of recursive identification algorithms. The software revision 17 includes the initialization algorithm, whereas the revisions 12 and 15 add also the output error RPEM based on the midpoint integration method. Live data is discusses in 5 , 6, and 13.

Finally, an example of the convergence of the parameter estimates for the simulated example of 1 appear in Fig. 2

*Figure 2: The convergence of the parameter estimates of the RPEM of 1, generated by 2.*

Recently, the above algorithms have been generalized to include an algorithm for joint identification of the ODE and a nonlinear output equation 1.

### References

1. T. Wigren, "Recursive identification of a nonlinear state space model", Int. J. Adaptive Contr. Signal Processing, vol. 37, no. 2, pp. 447-473, 2023.

2. T. Wigren, "MATLAB software for recursive identification and scaling using a structured nonlinear black-box model – revision 7", Technical Reports from the Department of Information Technology, 2021-008, Uppsala University, Uppsala, Sweden, December, 2021.

3. S. Tayamon and T. Wigren, "Control of selective catalytic reduction systems using feedback linearization", Asian J. Contr., vol. 18, no. 3, pp. 802-816, 2016. DOI: 10.1002/asjc.1164.

4. S. Tayamon, T. Wigren and B. Carlsson "NOx control for SCR systems using feedback linearisation", ERNSI 2014, Ostend, Belgium, September 21-24, 2014.

5. T. Wigren and J. Schoukens, "Three free data sets for development and benchmarking in nonlinear system identification", Proc. ECC 2013, Zurich, Switzerland, pp. 2933-2938, July 17-19. 2013.

6. T. Wigren and J. Schoukens, "Data for benchmarking in nonlinear system identification", Technical Reports from the department of Information Technology 2013-006, Uppsala University, Uppsala, Sweden, March, 2013.

7. S. Tayamon, T. Wigren and J. Schoukens, "Convergence analysis and experiments using an RPEM based on nonlinear ODEs and midpoint integration", Proc. CDC 2012, Maui, HI, pp. 2858-2865, December 10-13, 2012.

8. S. Tayamon and T. Wigren, "Convergence analysis of a recursive prediction error method", Proc. SYSID 2012, Brussels, Belgium, pp. 1496-1501, July 11-13, 2012.

9. S. Tayamon, D. Zambrano, T. Wigren and B. Carlsson, "Nonlinear black box identification of a selective catalytic reduction system", pp. 11845-11850, 18:th IFAC world congress, Milan, Italy, August 28-September 2, 2011.

10. D. Zambrano, S. Tayamon, B. Carlsson and T. Wigren, "Identification of a discrete-time nonlinear Hammerstein-Wiener model for a selective catalytic reduction system", in Proc. ACC 2011, San Fransisco, U.S.A., pp. 78-83, June 29-July 1, 2011.

11. S. Tayamon and T. Wigren, "Recursive identification and scaling of non-linear systems using midpoint numerical integration", Technical Reports from the department of Information Technology 2010-025, Uppsala University, Uppsala, Sweden, October, 2010.

12. T. Wigren, L. Brus and S. Tayamon, "MATLAB software for recursive identification and scaling using a structured nonlinear black-box Model - Revision 6", Technical Reports from the department of Information Technology 2010-022, Uppsala University, Uppsala, Sweden, September, 2010.

13. T. Wigren, "Input-output data sets for development and benchmarking in nonlinear identification", Technical Reports from the department of Information Technology 2010-020, Uppsala University, Uppsala, Sweden, August, 2010.

14. S. Tayamon and T. Wigren, "Recursive prediction error identification and scaling of non-linear systems with midpoint integration", in Proc. ACC 2010, Baltimore, MD, U.S.A., pp. 4510-4515, June 30-July 02, 2010.

15. T. Wigren, L. Brus and S. Tayamon, "MATLAB software for recursive identification and scaling using a structured nonlinear black-box Model - Revision 5", Reports from the department of Information Technology 2010-002, Uppsala University, Uppsala, Sweden, January, 2010.

16. L. Brus, "Nonlinear identification and control with solar energy applications," Ph.D. dissertation, Department of Information Technology, Uppsala University, Uppsala, Sweden, April 25, 2008.

17. T. Wigren and L. Brus, "MATLAB software for recursive identification and scaling using a structured nonlinear black-box Model - Revision 4", Technical Reports from the department of Information Technology 2008-007, Uppsala University, Uppsala, Sweden, March, 2008.

18. T. Wigren and L. Brus, "MATLAB software for recursive identification and scaling using a structured nonlinear black-box Model - Revision 3", Technical Reports from the department of Information Technology 2007-013, Uppsala University, Uppsala, Sweden, April, 2007.

19. L. Brus, T. Wigren and B. Carlsson, Initialization of a nonlinear identification algorithm applied to laboratory plant data, IEEE Trans. Contr. Sys. Tech., vol. 16, no. 4, pp. 708-716, 2008.

20. L. Brus, T. Wigren and B. Carlsson, Kalman filtering for black-box identification of nonlinear ODE models, Reglermöte 2006, Stockholm, Sweden, May 30-31, 2006.

21. L. Brus and T. Wigren, Constrained ODE modeling and Kalman filtering for recursive identification of nonlinear systems, in Proceedings of the 14th IFAC Symposium on System Identification, SYSID 2006 , Newcastle, Australia, pp. 997-1002, March 29-31, 2006.

22. T. Wigren, Recursive prediction error identification and scaling of nonlinear state space models using a restricted black box parameterization, Automatica, vol. 42, no. 1, pp. 159-168, 2006.

23. T. Wigren, "MATLAB software for recursive identification and scaling using a structured nonlinear black-box Model - Revision 2", Technical Reports from the department of Information Technology 2005-022, Uppsala University, Uppsala, Sweden, August, 2005

24. T. Wigren, Recursive identification based on nonlinear state space models applied to drum-boiler dynamics with nonlinear output equations, in Proc. IEEE ACC 2005 , Portland, Oregon, U.S.A., pp. 5066-5072, June 8-10, 2005.

25. T. Wigren, Scaling of the sampling period in nonlinear system identification, in Proc. IEEE ACC 2005, Portland, Oregon, U.S.A., pp. 5058-5065, June 8-10, 2005.

26. T. Wigren, "MATLAB software for recursive identification and scaling using a structured nonlinear black-box Model - Revision 1", Technical Reports from the department of Information Technology 2005-002, Uppsala University, Uppsala, Sweden, January, 2005.

27. T. Wigren, "Recursive prediction error identification of nonlinear state space models", Technical Reports from the department of Information Technology 2004-004, Uppsala University, Uppsala, Sweden, January, 2004.