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 freely downloadable software packages that implement the proposed identification methods.
- Release of free data sets supporting the development and benchmarking of new methods for nonlinear system identification.

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 25 and the paper 19 solves 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 25. 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 and applied in 19-25.

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 16-18 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 16-18. However, as is shown in 15 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 12.

A refined output error RPEM algorithm is presented in 6, 7, 10. That algorithm uses a more accurate numerical integration algorithm than the Euler algorithm applied in the previous work. The refined algorithm is expected to give more accurate results. The paper 10 discusses the modifications of the discrete time model and gradient that follow from the application of the midpoint integration scheme which is also known as one of the RK-integration methods.

All the proposed algorithms are 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 23 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 17 in figure 1, shows that the effects on the conditioning of the identification problem can be very significant.

The paper 10 discusses the counterparts to the above three results for the midpoint integration based scheme. The convergence of the scheme is proved in 3 and 4, building on the methodology of 15. Recently, the algorithms of 6 and 19 have been applied also to selective catalythic aftertreatment for diesel engines with good results in the paper 5.

*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.*

The most advanced achievement of the project is definately the result of a convergence analysis of the output error RPEM that is presented in the paper 15. The analysis is based on the averaging analysis that Professor Lennart Ljung developed in the early and mid 70´s, using associated differential equations. In 15, it is first formally proved that the proposed output error RPEM fits in the framework of the general nonlinear model defined in the well known technical report of Ljung from 1975. By a sequential verification of regularity conditions, it is then first shown that two fundamental theorems hold for the output error RPEM. With these results avialable, it is then straightforward to prove the following main result:

**Theorem:** Assume that the regularity conditions defined in 15 hold for the output error RPEM. Then, provided that the Hessian is positive definite, the output error RPEM is globally convergent to one of the minimum points of the criterion function, or to the boundary of the model set. The convergence point may depend on the initialization.

Software for identification and plotting are available for free download (7, 11, 13, 14, 21, 24) together with a manual. Currently, this software implements the output error RPEM and a corresponding algorithm for identification of a static non-linear function. The software version 7 also includes the new initialization algorithm, whereas the refined RPEM using midpoint integration is included in revisions 8 and 10. The software will be further updated whenever new algorithms are added.

To support the further development in the field of nonlinear system identification, two sets of downloadable recorded data are discussed in 1, 2, 8 and 9.

1. 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.

2. 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.

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

4. 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.

5. 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.

6. 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.

7. T. Wigren, L. Brus and S. Tayamon, "MATLAB Software for Recursive Identification and Scaling Using a Structured Nonlinear Black-box Model - Revision 6", available at http://www.it.uu.se/research/publications/reports/2010-062/NRISoftwareRev6.zip , Uppsala University, Uppsala, Sweden, September, 2010.

8. 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.

9. T. Wigren, "Input-output data sets for development and benchmarking in nonlinear identification", available at http://www.it.uu.se/research/publications/reports/2010-020/NonlinearData.zip, Uppsala University, Uppsala, Sweden, August, 2010.

10. 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.

11. T. Wigren, L. Brus and S. Tayamon, "MATLAB Software for Recursive Identification and Scaling Using a Structured Nonlinear Black-box Model - Revision 5", available at http://www.it.uu.se/research/publications/reports/2010-002/NRISoftwareRev5.zip , Uppsala University, Uppsala, Sweden, January, 2010.

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

13. T. Wigren and L. Brus, "MATLAB Software for Recursive Identification and Scaling Using a Structured Nonlinear Black-box Model - Revision 4", available at http://www.it.uu.se/research/publications/reports/2008-007/NRISoftwareRev4.zip , Uppsala University, Uppsala, Sweden, March, 2008.

14. T. Wigren and L. Brus, "MATLAB Software for Recursive Identification and Scaling Using a Structured Nonlinear Black-box Model - Revision 3", available at http://www.it.uu.se/research/publications/reports/2007-013/NRISoftwareRev3.zip , Uppsala University, Uppsala, Sweden, April, 2007

15. L. Brus, "Convergence analysis of a recursive identification algorithm for nonlinear ODE models with a restricted black-box parameterization", 46th IEEE Conference on Decision and Control, New Orleans, LA, U.S.A. Dec. 12-14 2007.

16. 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.

17. 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.

18. 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.

19. 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.

20. L. Brus, "Nonlinear identification of an anaerobic digestion process", in Proceedings of IEEE Conference on Control Applications, Toronto, Canada, pp. 137-142, Aug. 2005.

21. T. Wigren, "MATLAB Software for Recursive Identification and Scaling Using a Structured Nonlinear Black-box Model - Revision 2", available at http://www.it.uu.se/research/publications/reports/2005-022/NRISoftwareRev2.zip , Uppsala University, Uppsala, Sweden, August, 2005

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

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

24. T. Wigren, "MATLAB Software for Recursive Identification and Scaling Using a Structured Nonlinear Black-box Model - Revision 1", available at http://www.it.uu.se/research/publications/reports/2005-002/NRISoftware.zip , Uppsala University, Uppsala, Sweden, January, 2005.

25. T. Wigren, "Recursive Prediction Error Identification of Nonlinear State Space Models", Technical Reports from the department of Information Technology 004-2004, Uppsala University, Uppsala, Sweden, January, 2004.