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

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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 <formula formulatype="inline"><tex Notation="TeX">$m(n+1)$</tex></formula>, where <formula formulatype="inline"> <tex Notation="TeX">$n$</tex></formula> is the system order and <formula formulatype="inline"> <tex Notation="TeX">$m$</tex></formula> 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 <formula formulatype="inline"><tex Notation="TeX">$i\rightarrow\infty$</tex> </formula>, where <formula formulatype="inline"><tex Notation="TeX">$i$</tex> </formula> 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 <formula formulatype="inline"> <tex Notation="TeX">$i$</tex></formula>, 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. </para>