On subspace identification of cascade structured systems

In identification it is important to take a priori structural information into account in many applications, something that is difficult when using subspace methods. Here will study how to incorporate a special structure, a cascade structure with two subsystems. Two new methods are derived for estimating system with this structure. The problem when using subspace identification on cascade structured system is that the states from the first subsystem are mixed with states from the second subsystem via a unknown similarity transform. The first indirect method finds a similarity transform that takes the system back to a form such that the subsystems can be recovered. The second method uses the fact that the structure of the extended observability matrix is known for cascade systems. However, it only works when both subsystems have order one. In practice this is still a common case. The results of the two methods seem promising, as illustrated by applying the methods to a real process, the double tank process. The performance is comparable with state of the art methods. Finally the problem of optimal input design for cascade systems are introduced, and illustrated by a simple example. ©2010 IEEE.References: Ljung, L., (1999) System Identification: Theory for the User, 2nd Edition, , Upper Saddle River, New Jersey: Prentice Hall; Van Overschee, P., DeMoor, B., (1996) Subspace Identification of Linear Systems: Theory, Implementation, Applications, , Kluwer; Wahlberg, B., Jansson, M., Matsko, T., Molander, M., Experiences from subspace system identification - Comments from process industry users and researchers (2007) Modeling, Estimation and Control, Lecture Notes in Information and Control Sciences, pp. 315-327. , Springer, Oct; Wahlberg, B., Hjalmarsson, H., Mårtensson, J., Variance analysis for identification of cascade systems Decision and Control, 2008. CDC 2008. 47th IEEE Conference On, Dec. 2008, pp. 131-136; Wahlberg, B., Hjalmarsson, H., Mårtensson, J., Variance results for identification of cascade systems (2009) Automatica, 45, pp. 1443-1448. , Jun; Lyzell, C., Enqvist, M., Ljung, L., Handling certain structure information in subspace identification Proceedings of the 15th IFAC Symposium on System Identification, Apr. 2009; Van Gestel, T., Suykens, J., Van Dooren, P., De Moor, B., Identi- Fication of stable models in subspace identification by using regularization (2001) Automatic Control, IEEE Transactions on, 46 (9), pp. 1416-1420. , sep; Lacy, S., Bernstein, D., Subspace identification with guaranteed stability using constrained optimization (2003) Automatic Control, IEEE Transactions on, 48 (7), pp. 1259-1263. , july; Wahlberg, B., Sandberg, H., Cascade structural model approximation of identified state space model Proceedings IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 2008; Wahlberg, B., Hjalmarsson, H., Mårtensson, J., On identification of cascade systems 17th IFAC World Congress, Jul. 2008, pp. 5036-5040; Horn, R.A., Johnson, C.R., (1990) Matrix Analysis, , Cambridge University Press; Johansson, K., Horch, A., Wijk, O., Hansson, A., Teaching multivariable control using the quadruple-tank process (1999) Decision and Control, 1999. Proceedings of the 38th IEEE Conference on, 1, pp. 807-812. , vol.1; Hjalmarsson, H., System identification of complex and structured systems European Control Conference, Budapest, Hungary, Aug. 2009, pp. 3424-3452; Bombois, X., Scorletti, G., Gevers, M., Den Hof, P.V., Hildebrand, R., Least costly identification experiment for control (2006) Automatica, 42 (10), pp. 1651-1662. , http://www.sciencedirect.com/science/article/B6V21-4KR85M7-1/2/ 23969a3b1ef2e293b63428b4d3fc62a7, [Online]. AvailableQC 2012013