Jesús Peinado

A family of BDF algorithms for solving Differential Matrix Riccati Equations using adaptive techniques

Abstract Differential Matrix Riccati Equations play a fundamental role in control theory, for example, in optimal control, filtering and estimation, decoupling and order reduction, etc. One of the most popular codes to solve stiff Differential Matrix Riccati Equations (DMREs) is based on Backward Differentiation Formula (BDF). In previous papers the authors of this paper showed two algorithms for solving DMREs based on an iterative Generalized Minimum RESidual (GMRES) approach and on a Fixed-Point approach. In this paper we present two contributions to improve the above algorithms. Firstly six variants of previous algorithms are carried out by using one of above algorithms in the first step and another algorithm to carry out the other steps until reaching convergence. Numerous tests on four case studies have been done comparing both precision and computational costs of MATLAB implementations of the above algorithms. Experimental results show that in some cases these algorithms improve on the speed and convergence of the original algorithms. Secondly, using the previous experimental results and since all algorithms have a similar structure and there is no best algorithm to solve all problems, two general-purpose adaptive algorithms have been designed for selecting the most appropriate algorithm, which can be chosen using a parameter that indicates the stiffness of the DMRE to be solved.

A GMRES-based BDF method for solving differential Riccati equations

Abstract Differential Riccati equations play a fundamental role in control theory, for example, optimal control, filtering and estimation, decoupling and order reduction, etc. The most popular codes to solve stiff differential Riccati equations use backward differentiation formula (BDF) methods. In this paper, a new approach to solve differential Riccati equations by means of a BDF method is described. In each step of these methods an algebraic Riccati equation is obtained, which is solved by means of Newton’s method. In the standard approach, this system is transformed into a Sylvester equation, which could be solved by means of the well-known Bartels–Stewart method. In our code, we obtain a system of linear equations, defined from a Kronecker product of matrices related to coefficient matrices of the differential Riccati equation, that is solved by means of the iterative generalized minimum residual (GMRES) method. We have also implemented an efficient matrix–vector product in order to reduce the computational and storage cost of the GMRES method. The above approach has been applied in the development of an algorithm to solve differential Riccati equations. The accuracy and efficiency of this algorithm has been compared with the BDF algorithm that uses the Bartels–Stewart method. Experimental results show the advantages of the new algorithm.

A fixed point-based BDF method for solving differential Riccati equations

Abstract This paper describes an approach for solving differential Riccati equations (DRE), by means of the backward differentiation formula (BDF) and resolution of the corresponding implicit equation using Newton’s method with a fixed point approach. The role and use of DRE is especially important in several applications such as optimal control, filtering, and estimation. The goodness of this new method is compared with respect to the so called Dieci method [L. Dieci, Numerical integration of the differential Riccati equation and some related issues, SIAM J. Numer. Anal. 29 (3) (1992) 781–815].

A New Parallel Approach to the Toeplitz Inverse Eigenproblem Using Newton-like Methods

In this work we describe several portable sequential and parallel algorithms for solving the inverse eigenproblem for Real Symmetric Toeplitz matrices. The algorithms are based on Newtons method (and some variations), for solving nonlinear systems. We exploit the structure and some properties of Toeplitz matrices to reduce the cost, and use Finite Difference techniques to approximate the Jacobian matrix. With this approach, the storage cost is considerably reduced, compared with parallel algorithms proposed by other authors. Furthermore, all the algorithms are efficient in computational cost terms. We have implemented the parallel algorithms using the parallel numerical linear algebra library SCALAPACK based on the MPI environment. Experimental results have been obtained using two different architectures: a shared memory multiprocessor, the SGI PowerChallenge, and a cluster of Pentium II PCs connected through a Myrinet network. The algorithms obtained show a good scalability in most cases.

Efficient and accurate algorithms for computing matrix trigonometric functions

Trigonometric matrix functions play a fundamental role in second order differential equations. This work presents an algorithm based on Taylor series for computing the matrix cosine. It uses a backward error analysis with improved bounds. Numerical experiments show that MATLAB implementations of this algorithm has higher accuracy than other MATLAB implementations of the state of the art in the majority of tests. Furthermore, we have implemented the designed algorithm in language C for general purpose processors, and in CUDA for one and two NVIDIA GPUs. We obtained a very good performance from these implementations thanks to the high computational power of these hardware accelerators and our effort driven to avoid as much communications as possible. All the implemented programs are accessible through the MATLAB environment.

A Covariant Spacetime Approach to Transformation Acoustics

Transformation acoustics focuses on the design of advanced acoustic devices by employing sophisticated mathematical transformation techniques for engineering acoustic metamaterials—materials artificially fabricated with extraordinary acoustic properties beyond those encountered in nature. We present differential-geometric methods together with a variational principle and show how they form the basis for a powerful framework to control acoustic waves in industrial applications. We conclude with a practical example and implement the acoustic wave equation within a uniform accelerating rigid frame (UAF). As expected, an acoustic event horizon emerges, i.e., a boundary in spacetime beyond which events cannot acoustically affect any outside observer.

Two algorithms for computing the matrix cosine function

The computation of matrix trigonometric functions has received remarkable attention in the last decades due to its usefulness in the solution of systems of second order linear differential equations. Several state-of-the-art algorithms have been provided recently for computing these matrix functions. In this work, we present two efficient algorithms based on Taylor series with forward and backward error analysis for computing the matrix cosine. A MATLAB implementation of the algorithms is compared to state-of-the-art algorithms, with excellent performance in both accuracy and cost.

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

Adams-Bashforth and Adams-Moulton methods for solving differential Riccati equations

Differential Riccati equations play a fundamental role in control theory, for example, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper several algorithms for solving differential Riccati equations based on Adams-Bashforth and Adams-Moulton methods are described. The Adams-Bashforth methods allow us explicitly to compute the approximate solution at an instant time from the solutions in previous instants. In each step of Adams-Moulton methods an algebraic matrix Riccati equation (AMRE) is obtained, which is solved by means of Newtons method. Nine algorithms are considered for solving the AMRE: a Sylvester algorithm, an iterative generalized minimum residual (GMRES) algorithm, a fixed-point algorithm and six combined algorithms. Since the above algorithms have a similar structure, it is possible to design a general and efficient algorithm that uses one algorithm or another depending on the considered differential matrix Riccati equation. MATLAB versions of the above algorithms are developed, comparing precision and computational costs, after numerous tests on five case studies.

A new efficient and accurate spline algorithm for the matrix exponential computation

Abstract In this work an accurate and efficient method based on matrix splines for computing matrix exponential is given. An algorithm and a MATLAB implementation have been developed and compared with the state-of-the-art algorithms for computing the matrix exponential. We also developed a parallel implementation for large scale problems. This implementation allowed us to get a much better performance when working with this kind of problems.