110th Anniversary: Generalized Singular Value Decomposition Reduced-Order Observers for Linear Time-Invariant Systems with Noisy Measurements

Abstract

The use of observers for improvement of output feedback process control is important for achieving accuracy and processing efficiency, especially for large multivariate process systems. Reduced-order observers are particularly advantageous for reducing computational complexity of estimating state variables. The processes considered herein can be modeled as a linear time-invariant continuous system with stochastic elements of modified white Gaussian noise accounting for both process disturbances and sensor inaccuracies. When process measurements are supernumerary to the process states with full rank observation matrix, noise contributions to output measurement are filtered by a soft sensor based on the generalized singular value decomposition (GSVD) solution of best linear unbiased estimation for state variable estimation. When only system observability can be guaranteed, this soft sensor is combined with a reduced-order Kalman-Bucy observer design for continuous estimation of unmeasured state variables at low computation cost. The proposed design procedure is evaluated on a biochemical continuous stirred tank reactor (CSTR) and a simplified Tennessee Eastman model around stable and unstable steady states. The observer is shown to outperform the ordinary Kalman filter design with noisy measurements that deviate from a Gauss-Markov model.

A Higher-Order Generalized Singular Value Decomposition for Rank Deficient Matrices

“…where rank A = 2. The matrix A 2 (A 3 ) has right singular vectors [1,0] T and [0, 1] T associated with singular values 0 and 1 (1 and 0), respectively. For each of the cases, the matrix S π from (1.2) is obtained as…”

“…Our HO-GSVD extension could be used to diagonalize a system with N > 2 actuator arrays, such as encountered in the cross-directional control of paper machines [14]. In [1], the GSVD has been used to design an observer. Our HO-GSVD extension could be used to simoultaneously design the observer and the controller.…”