Application of SVD in the Regularization of the Control Law for GPC-based Tracking Systems

Generalized predictive control (GPC) has become one of the most studied and popular control approaches. The GPC control law requires the estimation of the Hessian matrix, which requires a matrix inversion procedure. However, depending on the plant model and the GPC parameters, the aforementioned procedure may be ill-conditioned: a slight variation in the parameters may generate a more significant variation in the Hessian matrix value. In that case, the noise or quantization effect reduces the GPC robustness. The process of solving ill-conditioned problems is called regularization. This paper proposes the Singular Value Decomposition (SVD) application to regularize the matrix inversion procedure used to get the Hessian matrix. SVD decomposes a matrix based on the concept of singular values. Only the most significant singular values are used in the SVD regularization technique to calculate a matrix inverse, as the smallest singular values produce ill-conditioned problems. A methodology to define the singular values used in matrix inversion is explained in this work. The proposed approach was used in a GPC- based resonant controller, using 16 bits fixed-point numbers. Simulation and experimental tests using a FPGA show that the proposed approach allows getting an accurate and robust GPC response for the tracking of sinusoidal references.