Speeding up solving of differential matrix Riccati equations using GPGPU computing and MATLAB

In this work, we developed a parallel algorithm to speed up the resolution of differential matrix Riccati equations using a backward differentiation formula algorithm based on a fixed‐point method. The role and use of differential matrix Riccati equations is especially important in several applications such as optimal control, filtering, and estimation. In some cases, the problem could be large, and it is interesting to speed it up as much as possible. Recently, modern graphic processing units (GPUs) have been used as a way to improve performance. In this paper, we used an approach based on general‐purpose computing on graphics processing units. We used NVIDIA © GPUs with unified architecture. To do this, a special version of basic linear algebra subprograms for GPUs, called CUBLAS, and a package (three different packages were studied) to solve linear systems using GPUs have been used. Moreover, we developed a MATLAB © toolkit to use our implementation from MATLAB in such a way that if the user has a graphic card, the performance of the implementation is improved. If the user does not have such a card, the algorithm can also be run using the machine CPU. Experimental results on a NVIDIA Quadro FX 5800 are shown. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  A. J. Laub,et al.  Efficient matrix-valued algorithms for solving stiff Riccati differential equations , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[2]  C. Kenney,et al.  Numerical integration of the differential matrix Riccati equation , 1985 .

[3]  Jacinto Javier Ibáñez,et al.  A family of BDF algorithms for solving Differential Matrix Riccati Equations using adaptive techniques , 2010, ICCS.

[4]  Edward J. Davison,et al.  The numerical solution of the matrix Riccati differential equation , 1973 .

[5]  M. Sorine,et al.  Superposition laws for solutions of differential matrix Riccati equations arising in control theory , 1985 .

[6]  D. Vaughan,et al.  A negative exponential solution for the matrix Riccati equation , 1969 .

[7]  Enrique Ponsoda,et al.  Non-autonomous Riccati-type matrix differential equations: existence interval, construction of continuous numerical solutions and error bounds , 1995 .

[8]  Peter Benner,et al.  BDF Methods for Large-Scale Differential Riccati Equations , 2004 .

[9]  Enrique S. Quintana-Ortí,et al.  Exploiting the capabilities of modern GPUs for dense matrix computations , 2009, Concurr. Comput. Pract. Exp..

[10]  Hamid Laga,et al.  CUDA (Computer Unified Device Architecture) , 2009 .

[11]  Demetrios Lainiotis.  Generalized Chandrasekhar algorithms: Time-varying models , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[12]  Susan Ostrouchov,et al.  LAPACK Working Note 41: Installation Guide for LAPACK , 1992 .

[13]  Jack Dongarra,et al.  LAPACK Working Note 81: Quick Installation Guide for LAPACK on UNIX Systems , 1994 .

[14]  Luca Dieci,et al.  Numerical integration of the differential Riccati equation and some related issues , 1992 .

[15]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[16]  J. Demmel,et al.  Sun Microsystems , 1996 .

[17]  D. W. Rand,et al.  Nonlinear superposition principles: a new numerical method for solving matrix Riccati equations , 1984 .

[18]  J. M. Sanz-Serna,et al.  Symplectic integrators for Hamiltonian problems: an overview , 1992, Acta Numerica.

[19]  L. Fox,et al.  Initial-value methods for boundary-value problems , 1987 .

[20]  Enrique S. Quintana-Ortí,et al.  Exploiting the capabilities of modern GPUs for dense matrix computations , 2009 .

[21]  N. Higham.  Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics) , 2008 .

[22]  Jack Dongarra,et al.  ScaLAPACK Users' Guide , 1987 .

[23]  Alan J. Laub,et al.  Efficient algorithms for solving stiff matrix-valued riccati differential equations , 1988 .

[24]  Jacinto Javier Ibáñez,et al.  A fixed point-based BDF method for solving differential Riccati equations , 2007, Appl. Math. Comput..