Identification of a Two-Input System: Variance Analysis

This paper examines the identification of a single-output two-input system. Motivated by an experiment design problem(should one excite the two inputs simultaneously or separately), we examine the effect of the (second) input signal on the variance of the various polynomial coefficients in the case of FIR, ARX, ARMAX, OE and BJ models. A somewhat surprising result is to show that the addition of a second input in an ARMAX model reduces the variance of all polynomials estimates

Identification of multi-input systems: variance analysis and input design issues

This paper examines the identification of multi-input systems. Motivated by an experiment design problem (should one excite the various inputs simultaneously or separately), we examine the effect of an additional input on the variance of the estimated coefficients of parametrized rational transfer function models, with special emphasis on the commonly used FIR, ARX, ARMAX, OE and BJ model structures. We first show that, for model structures that have common parameters in the input–output and noise models (e.g. ARMAX), any additional input contributes to a reduction of the covariance of all parameter estimates. We then show that the accuracy improvement extends beyond the case of common parameters in all transfer functions, and we show exactly which parameter estimates are improved when a new input is added. We also conclude that it is always better to excite all inputs simultaneously

Asymptotic Variance Expressions for Estimated Frequency Functions

Expressions for the variance of an estimated frequency function are necessary for many issues in model validation and experiment design. A general result is that a simple expression for this variance can be obtained asymptotically as the model order tends to infinity. This expression shows that the variance is inversely proportional to the signal-to-noise ratio frequency by frequency. Still, for low order models the actual variance may be quite different. This has also been pointed out in several recent publications. In this contribution we derive an exact expression for the variance, which is not asymptotic in the model order. This expression applies to a restricted class of models: AR-models, as well as fixed pole models with a polynomial noise model. It brings out the character of the simple approximation and the convergence rate to the limit as the model order increases. It also provides nonasymptotic lower bounds for the general case. The calculations are illustrated by numerical examples

Asymptotic Variance Expressions for Estimated Frequency Functions

Expressions for the variance of an estimated frequency function are necessary for many issues in model validation and experiment design. A general result is that a simple expression for this variance can be obtained asymptotically as the model order tends to infinity. This expression shows that the variance is inversely proportional to the signal-to-noise ratio frequency by frequency. Still, for low order models the actual variance may be quite different. This has also been pointed out in several recent publications. In this contribution we derive an exact expression for the variance, which is not asymptotic in the model order. This expression applies to a restricted class of models: AR-models, as well as fixed pole models with a polynomial noise model. It brings out the character of the simple approximation and the convergence rate to the limit as the model order increases. It also provides nonasymptotic lower bounds for the general case. The calculations are illustrated by numerical examples

Contributions to the Theory and Implementation of the LSCR Method

[ES] El método LSCR (Leave-out-Sign-dominant-Correlation-Regions) proporciona regiones de confianza para los parámetros de un sistema evaluando un conjunto de funciones de correlation calculadas a partir de los datos disponibles. Al confeccionar una aproximación para la región completa, el procedimiento debe repetirse para cada valor del vector de parámetros. Los atributos principales de LSCR son su validez para un conjunto de datos finitos y los escasos supuestos sobre el ruido. Sin embargo, el procedimiento necesita mucho esfuerzo computacional, lo que limita su aplicación a casos muy simples. En este trabajo se mejoran aspectos teóricos del método LSCR y se sugieren alternativas de implementación. También se lo compara, en términos del esfuerzo computacional, con Bootstrap, otra forma de obtener regiones de confianza.[EN] The LSCR method (Leave-out-Sign-dominant-Correlation-Regions) provides confidence regions for the parameters of a system by evaluating a set of correlation functions calculated for the available data. To do the approximation for the whole region, the procedure must be repeated for each value of the parameter vector. The main attributes of LSCR are its validity for a finite set of data and the scarce asumptions on the noise. However, the procedure needs much computational effort, which limitates its application to very simple cases. In this work some theoretical aspects of the LSCR method are improved and some implementation altenatives are suggested. It is also compared, in terms of computational effort, with Bootstrap, another way to obtain confidence regions.Este trabajo ha sido realizado parcialmente gracias al financiamiento de la Universidad Técnica Federico Santa María del Proyecto USM 23.06.24.Ramírez, J.; Rojas, CR.; Jarur, JC.; Rojas, RA. (2010). Aportes a la Teoría y la Implementación del Método LSCR. Revista Iberoamericana de Automática e Informática industrial. 7(3):83-94. https://doi.org/10.1016/S1697-7912(10)70045-5OJS839473Campi, M. C., & Weyer, E. (2002). Finite sample properties of system identification methods. IEEE Transactions on Automatic Control, 47(8), 1329-1334. doi:10.1109/tac.2002.800750Campi, M. C., & Weyer, E. (2005). Guaranteed non-asymptotic confidence regions in system identification. Automatica, 41(10), 1751-1764. doi:10.1016/j.automatica.2005.05.005Gordon, L. (1974). Completely Separating Groups in Subsampling. The Annals of Statistics, 2(3), 572-578. doi:10.1214/aos/1176342719Hartigan, J. A. (1969). Using Subsample Values as Typical Values. Journal of the American Statistical Association, 64(328), 1303-1317. doi:10.1080/01621459.1969.10501057Jarur, J.C. (2008). Cálculo de Regiones de Confianza Paramétricas con LSCR: Análisis, Experiencias y Aplicaciones. Tesis de Magíster, Universidad Técnica Federico Santa María, Valparaíso, Chile.Lennart Ljung, & Zhen-Dong Yuan. (1985). Asymptotic properties of black-box identification of transfer functions. IEEE Transactions on Automatic Control, 30(6), 514-530. doi:10.1109/tac.1985.1103995Ljung, L. (1985). Asymptotic variance expressions for identified black-box transfer function models. IEEE Transactions on Automatic Control, 30(9), 834-844. doi:10.1109/tac.1985.1104093Ninness, B., Hjalmarsson, H., & Gustafsson, F. (1999). The fundamental role of general orthonormal bases in system identification. IEEE Transactions on Automatic Control, 44(7), 1384-1406. doi:10.1109/9.774110Ninness, B., & Hjalmarsson, H. (2004). Variance Error Quantifications That Are Exact for Finite-Model Order. IEEE Transactions on Automatic Control, 49(8), 1275-1291. doi:10.1109/tac.2004.832202Ninness, B., & Hjalmarsson, H. (2005). On the frequency domain accuracy of closed-loop estimates. Automatica, 41(7), 1109-1122. doi:10.1016/j.automatica.2005.03.005Ninness, B., & Hjalmarsson, H. (2005). Analysis of the variability of joint input–output estimation methods. Automatica, 41(7), 1123-1132. doi:10.1016/j.automatica.2005.03.006Tjärnström, F. (2000). Quality Estimation of Approximate Models. Licenciate thesis 810, Department of Electrical Engineering, Linköping University, Suecia.Weyer, E., Williamson, R. C., & Mareels, I. M. Y. (1999). Finite sample properties of linear model identification. IEEE Transactions on Automatic Control, 44(7), 1370-1383. doi:10.1109/9.774109Weyer, E., & Campi, M. C. (2002). Non-asymptotic confidence ellipsoids for the least-squares estimate. Automatica, 38(9), 1539-1547. doi:10.1016/s0005-1098(02)00064-xLiang-Liang Xie, & Ljung, L. (2001). Asymptotic variance expressions for estimated frequency functions. IEEE Transactions on Automatic Control, 46(12), 1887-1899. doi:10.1109/9.97547