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Numerical Algebra, Control & Optimization

2016 , Volume 6 , Issue 4

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A POD projection method for large-scale algebraic Riccati equations
Boris Kramer and  John R. Singler
2016, 6(4): 413-435 doi: 10.3934/naco.2016018 +[Abstract](87) +[PDF](583.9KB)
The solution of large-scale matrix algebraic Riccati equations is important for instance in control design and model reduction and remains an active area of research. We consider a class of matrix algebraic Riccati equations (AREs) arising from a linear system along with a weighted inner product. This problem class often arises from a spatial discretization of a partial differential equation system. We propose a projection method to obtain low rank solutions of AREs based on simulations of linear systems coupled with proper orthogonal decomposition. The method can take advantage of existing (black box) simulation code to generate the projection matrices. We also develop new weighted norm residual computations and error bounds. We present numerical results demonstrating that the proposed approach can produce highly accurate approximate solutions. We also briefly discuss making the proposed approach completely data-based so that one can use existing simulation codes without accessing system matrices.
Closed-form expression for the inverse of a class of tridiagonal matrices
Sigve Hovda
2016, 6(4): 437-445 doi: 10.3934/naco.2016019 +[Abstract](188) +[PDF](313.0KB)
Despite the simplicity of tridiagonal matrices, they have shown to be very resilient to closed-form solutions. We consider a class of tridiagonal stiffness matrices that stems from a variety of lumped element models in mechanical, acoustical and electrical systems. The computational efforts in such models are related to solving the generalized eigenvalue problem and finding the inverse of the stiffness matrix. To improve accuracy, it is desired to discretisize the problem as much as possible at the expense of growing matrices. This paper improves the efficiency of finding the inverse by a factor of at least three and the computational memory involved is at least halved. Moreover, the result provides an analytical expression for where the stable position is, which might be used in control systems. Surprisingly, it is the practical application itself that guides the proof.
Solving higher index DAE optimal control problems
Stephen Campbell and  Peter Kunkel
2016, 6(4): 447-472 doi: 10.3934/naco.2016020 +[Abstract](160) +[PDF](990.0KB)
A number of methods have been proposed for solving optimal control problems where the process being optimized is described by a differential algebraic equation (DAE). However, many of these methods require special circumstances to hold or the user to have special software. In this paper we go over many of these options and discuss what is usually necessary for them to be successful. We use a nonlinear index three control problem to illustrate many of our observations..
An approximation scheme for stochastic programs with second order dominance constraints
Yongchao Liu , Hailin Sun and  Huifu Xu
2016, 6(4): 473-490 doi: 10.3934/naco.2016021 +[Abstract](52) +[PDF](510.5KB)
It is well known that second order dominance relation between two random variables can be described by a system of stochastic semi-infinite inequalities indexed by $\mathcal R$, the set of all real number. In this paper, we show the index set can be reduced to the support set of the dominated random variable strengthening a similar result established by Dentcheva and Ruszczyński [9] for discrete random variables. Viewing the semi-infinite constraints as an extreme robust risk measure, we relax it by replacing it with entropic risk measure and regarding the latter as an approximation of the former in an optimization problem with second order dominance constraints. To solve the entropic approximation problem, we apply the well known sample average approximation method to discretize it. Detailed analysis is given to quantify both the entropic approximation and sample average approximation for various statistical quantities including the optimal value, the optimal solutions and the stationary points obtained from solving the sample average approximated problem. The numerical scheme provides an alternative to the mainstream numerical methods for this important class of stochastic programs.
2D system analysis via dual problems and polynomial matrix inequalities
Vladimir Pozdyayev
2016, 6(4): 491-504 doi: 10.3934/naco.2016022 +[Abstract](35) +[PDF](425.8KB)
Application of the Lyapunov method to 2D system stability and performance analysis yields algebraic systems that can be interpreted as either sum-of-squares problems for nontrivial matrix polynomials, or parameterized linear matrix inequalities that need to be satisfied for certain ranges of parameter values. In this paper we show that dualizing core inequalities in the latter forms allows converting these systems to conventional optimization problems on sets described by polynomial matrix inequalities. Methods for solving these problems include moment-based methods or the “atomic optimization” method proposed earlier by the author. As a result, we obtain necessary conditions for 2D system stability and lower bounds on system performance. In particular, we demonstrate respective results for discrete-discrete system stability and mixed continuous-discrete system $\mathcal{H}_\infty$ performance. A numerical example is provided.
Convergence analysis of a parallel projection algorithm for solving convex feasibility problems
Yazheng Dang , Fanwen Meng and  Jie Sun
2016, 6(4): 505-519 doi: 10.3934/naco.2016023 +[Abstract](62) +[PDF](356.4KB)
The convex feasibility problem (CFP) is a classical problem in nonlinear analysis. In this paper, we propose an inertial parallel projection algorithm for solving CFP. Different from the previous algorithms, the proposed method introduces a sequence of parameters and uses the information of last two iterations at each step. To prove its convergence in a simple way, we transform the parallel algorithm to a sequential one in a constructed product space. Preliminary experiments are conducted to demonstrate that the proposed approach converges faster than the general extrapolated algorithms.




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