Identification of Bilinear Systems With White Noise Inputs: An Iterative Deterministic-Stochastic Subspace Approach

• Published in: IEEE transactions on control systems technology 2009-09, Vol.17 (5), p.1145-1153
• Main Authors: dos Santos, P.L., Ramos, J.A., de Carvalho, J.L.M.
• Format: Article
• Language: English
• Subjects: Algorithms; Applied sciences; Bilinear Kalman filtering; bilinear systems; Biological system modeling; Computer science; control theory; systems; Control systems; Control theory. Systems; Exact sciences and technology; Hankel matrices; Iterative algorithms; Iterative methods; Linear systems; Mathematical analysis; Mathematical models; Matrices; Matrix methods; Modelling and identification; Nonlinear systems; Parameter estimation; State-space methods; Stochasticity; subspace identification; Subspaces; System identification; White noise
• ISSN: 1063-6536; 1558-0865
• DOI: 10.1109/TCST.2008.2002041

In this technical brief, a new subspace state space system identification algorithm for multi input multi output bilinear systems driven by white noise inputs is introduced. The new algorithm is based on a uniformly convergent Picard sequence of linear deterministic stochastic state space subsystems which are easily identifiable by any linear deterministic stochastic subspace algorithm such as MOESP, N4SID, CVA, or CCA. The key to the proposed algorithm is the fact that the bilinear term is a second order white noise process. Using a standard linear Kalman filter model, the bilinear term can be estimated and combined with the system inputs at each iteration, thus leading to a linear system with extended inputs of dimension m ( n + 1), where n is the system order and m is the dimension of the inputs. It is also shown that the model parameters obtained with the new algorithm converge to those of the true bilinear model. Moreover, the proposed algorithm has the same consistency conditions as the linear subspace identification algorithms when i ¿ ¿, where i is the number of block rows in the past/future block Hankel data matrices. Typical bilinear subspace identification algorithms available in the literature cannot handle large values of i , thus leading to biased parameter estimates. Unlike existing bilinear subspace identification algorithms whose row dimensions in the data matrices grow exponentially, and hence suffer from the ¿curse of dimensionality,¿ in the proposed algorithm the dimensions of the data matrices are comparable to those of a linear subspace identification algorithm. A case study is presented with data from a heat exchanger experiment.