Large Linear Dynamic Systems Research Articles

Parallel GMRES solver for fast analysis of large linear dynamic systems on GPU platforms

In this paper, we propose an efficient parallel dynamic linear solver, called GPU-GMRES, for transient analysis of large linear dynamic systems such as large power grid networks. The new method is based on the preconditioned generalized minimum residual (GMRES) iterative method implemented on heterogeneous CPU–GPU platforms. The new solver is very robust and can be applied to power grids with different structures as well as for general analysis problems for large linear dynamic systems with asymmetric matrices. The proposed GPU-GMRES solver adopts the very general and robust incomplete LU based preconditioner. We show that by properly selecting the right amount of fill-ins in the incomplete LU factors, a good trade-off between GPU efficiency and convergence rate can be achieved for the best overall performance. Such tunable feature can make this algorithm very adaptive to different problems. GPU-GMRES solver properly partitions the major computing tasks in GMRES solver to minimize the data traffic between CPU and GPUs to enhance performance of the proposed method. Furthermore, we propose a new fast parallel sparse matrix–vector (SpMV) multiplication algorithm to further accelerate the GPU-GMRES solver. The new algorithm, called segSpMV, can enjoy full coalesced memory access compared to existing approaches. To further improve the scalability and efficiency, segSpMV method is further extended to multi-GPU platforms, which leads to more scalable and faster multi-GPU GMRES solver. Experimental results on the set of the published IBM benchmark circuits and mesh-structured power grid networks show that the GPU-GMRES solver can deliver order of magnitudes speedup over the direct LU solver, UMFPACK. The resulting multi-GPU-GMRES can also deliver 3–12×speedup over the CPU implementation of the same GMRES method on transient analysis.

Distributed Kalman Filtering Over Massive Data Sets: Analysis Through Large Deviations of Random Riccati Equations

This paper studies the convergence of the estimation error process and the characterization of the corresponding invariant measure in distributed Kalman filtering for potentially unstable and large linear dynamic systems. A gossip network protocol termed Modified Gossip Interactive Kalman Filtering (M-GIKF) is proposed, where sensors exchange their filtered states (estimates and error covariances) and propagate their observations via inter-sensor communications of rate $\overline{\gamma}$; $\overline{\gamma}$ is defined as the averaged number of inter-sensor message passages per signal evolution epoch. The filtered states are interpreted as stochastic particles swapped through local interaction. The paper shows that the conditional estimation error covariance sequence at each sensor under M-GIKF evolves as a random Riccati equation (RRE) with Markov modulated switching. By formulating the RRE as a random dynamical system, it is shown that the network achieves weak consensus, i.e., the conditional estimation error covariance at a randomly selected sensor converges weakly (in distribution) to a unique invariant measure. Further, it is proved that as $\overline{\gamma} \rightarrow \infty$ this invariant measure satisfies the Large Deviation (LD) upper and lower bounds, implying that this measure converges exponentially fast (in probability) to the Dirac measure $\delta_{P^*}$, where $P^*$ is the stable error covariance of the centralized (Kalman) filtering setup. The LD results answer a fundamental question on how to quantify the rate at which the distributed scheme approaches the centralized performance as the inter-sensor communication rate increases.

Modelling and Design of Large Linear Dynamic Systems by Repeated Decomposition-Aggregation

Modelling and Design of Large Linear Dynamic Systems by Repeated Decomposition-Aggregation

Dynamic response by large step integration

AbstractA family of unconditionally stable algorithms for the economical computation of large linear dynamic systems is described and applied. Possible application to divergent systems is considered and some of the difficulties of extending the use of the algorithms to non‐linear systems are discussed. In the Appendix a previously developed conditionally stable algorithm is applied to the non‐linear gust response of a prestressed cable roof structure over the Munich Olympic Stadium. The idealization involves 1,164 degrees of freedom.

Applications of Symbolic Computing Methods to the Dynamic Analysis of Large Systems

The dynamic analysis and control of very large systems yet remains a problem for the analyst. The tools currently available to him rely largely on matrix methods. It is recognized that most man-made systems, either physical or otherwise, tend to be not highly interactive. That is, most of the elements are connected only to other elements near to it rather than inter-connected with all other elements which compose the system. The use of matrix techniques to analyze such problems is found lacking because a large number of matrix elements are zero.A new approach lies in the use of symbolic computer methods applied directly to the system equations. Two specific techniques are applied to the analysis of large dynamic systems. Since the symbolic computing language is well suited to the operations with algebraic equations, both techniques use the transfer function concept as a tool for the analysis of large linear dynamic systems. Also, both techniques have been coded in the experimental symbolic computer language FORMAC. The first approach REDUCE I, establishes the techniques and a computer program to symbolically reduce arbitrary block diagrams associated with large systems for desired transfer functions. Symbolic closed form solutions are determined in several forms. The program has been applied to several research problems which include both lumped and distributed parameter systems.A second computer program, REDUCE II, is also based on the use of symbolic commuting methods and has been written to accommodate large engineering systems. REDUCE II symbolically calculates the transfer functions of any linear block diagram output variable to any or all input variables. The solution using this technique is presented in the compact form of a set of nested functions. The program can handle systems as large as 600 equations (of essentially any order) and is intended as a tool for the analysis of complex control and dynamic systems.The paper also projects other areas of application of the symbolic method.

Relationship between Stability and Connectedness of Non-linear Systems

Gardner and Ashby1 have studied the relation between the probability of stability p(S) and the extent of connectedness (“connectance”) C of large linear dynamical systems. They found a very interesting on–off-type (numerical) result: for reasonably large systems (number of components n ≥ 10) there is a critical value, Cc of C, below which the system is almost certainly stable, but above which it becomes almost certainly unstable.

A new method for simplifying large linear dynamic systems

A new method for simplifying large linear dynamic systems

The Identification of Parameters in Large Linear Dynamic Systems

ABSTRACT The identification of parameters in large dynamic systems is difficult because of the excessive computational time required to solve the large number of differential equations occurring. A method of solving these equations is presented which for the identification problem enables solutions to be quiekly obtained. The identification of parameters is then achieved by hill climbing on the unknown parameters in the mathematical model.