Numerical Ill-conditioning Research Articles

A delta operator approach to discrete-time H ∞ control

To increase the performance of digitally implemented control laws, it is desirable to directly synthesize the digital control law. Hence, this paper considers the direct synthesis of discrete-time, H ∞ control laws. Previously, two approaches have been proposed to this problem. The most common method uses the bilinear transformation to convert the discrete-time problem into an equivalent continuous-time problem, allowing the controller to be synthesized with software developed for continuous-time systems. This method is often effective but requires unnecessary steps and is not applicable in all circumstances. The second approach is to synthesize the discrete-time, H ∞ control law by solving discrete-time, H ∞ Riccati equations. Due to numerical ill-conditioning, this approach can fail at sufficiently high sample rates. This numerical ill-conditioning can be eliminated by representing the discrete-time system and H ∞ controller using the delta operator. Hence, the H ∞ design equations for direct synthesis using the delta operator are developed. These equations reduce to standard continuous-time H ∞ design equations when the sample period is set to zero. The results are illustrated with a numerical example which also shows that the central continuous-time H ∞ controller of the continuous-time system obtained by a bilinear transformation of the original discrete-time system does not necessarily correspond to the central discrete-time controller.

An algorithm for coprime matrix fraction description using sylvester matrices

Several algorithms for the computation of coprime matrix fraction descriptions have been proposed in the past. Here we explore some of the properties of Sylvester matrices and develop a different approach to the problem which is based on singular value decompositions and therefore avoids problems of numerical ill-conditioning.

A group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell

The accurate prediction of the buckling load of thin shell structures is an important yet elusive goal. It is particularly important in the aerospace industry where thin shell members are commonly used as structural elements. Due to a lack of adequate analytical results, current practices in industry put heavy reliance on experimental testing and empirical data to supplement theoretical analysis (see D. Bushnell, Computerized Buckling Analysis of Shells, Martinus Nijhoff, Dordrecht, 1985). This paper focuses on recent results of the group theoretic approach to a numerical, global postbuckling analysis of a perfect, axially compressed cylindrical shell with “built-in” end conditions. The “built-in” end conditions obviate the existence of a “trivial” membrane solution branch. The example of an axially compressed cylindrical shell was chosen because it is well known that for thin shells, the primary axisymmetric solution branch is riddled with closely spaced, symmetry-breaking bifurcation points. In a numerical arc-length continuation scheme, the close proximity of the bifurcation points on the primary path manifests itself in severe ill-conditioning of the tangent stiffness matrix. Group theory helps one systematically find an “optimal” set of basis vectors, or symmetry modes, which reflect the symmetry of a given solution path. The immediate payoff in using these symmetry modes as basis vectors is that the tangent stiffness matrix block-diagonalizes and the numerical ill-conditioning is avoided. Thus, an efficient and accurate technique for computing solution branches of a specific type and a subsequent diagnosis for symmetry-breaking bifurcations is made relatively simple. Understanding the global behavior of the perfect structure is crucial in identifying critical imperfections and will ultimately diminish the heavy reliance on expensive experimental verification.

Exponential Characteristic Spatial Quadrature for Discrete Ordinates Radiation Transport in Slab Geometry

The exponential characteristic spatial quadrature for discrete ordinates neutral particle transport in slab geometry is derived and compared with current methods. It is similar to the linear characteristic (or, in slab geometry, the linear nodal) quadrature but differs by assuming an exponential distribution of the scattering source within each cell, S(x) = a exp(bx), whose parameters are root-solved to match the known (from the previous iteration) average and first moment of the source over the cell. Like the linear adaptive method, the exponential characteristic method is positive and nonlinear but more accurate and more readily extended to other cell shapes. The nonlinearity has not interfered with convergence. We introduce the “exponential moment functions,” a generalization of the functions used by Walters in the linear nodal method, and use them to avoid numerical ill-conditioning. The method exhibits O(Δx4) truncation error on fine enough meshes; the error is insensitive to mesh size for coarse meshes. In a shielding problem, it is accurate to 10% using 16-mfp-thick cells; conventional methods err by 8 to 15 orders of magnitude. The exponential characteristic method is computationally more costly per cell than current methods but can be accurate with very thick cells, leading to increased computational efficiency on appropriate problems.

Near-Optimal Incentive Stackelberg Strategies for Singularly Perturbed Systems

When the systems contain clow and fast modes. the Stackelberg games are numerical stiffness and ill-conditioning. It is necessary to find some methods to alleviate this ill-conditioning and to reduce the amount of computation. In this paper, we propose a method to treat linear-quadratic Stackelberg games for singularly perturbed systems. To obtain a series of well-conditioned and reduced-order problems, we assume, at first, that the follower responds to the leader’s incentive strategy with composite strategy. A near team-optimal incentive strategy, which contains the follower's composite strategy as information, is constructed by solving these reduced-order problems. And then we prove that the near-team-optimal incentive strategy is well-posed in the sense that the resulting value of the cost function for the leader will have the same limit with the full-order team value of the leader’s cost function as the small singular perturbation parameter e tends to zero, even if the follower responds to the leader’s incentive streategy with full-order strategy.

Decoupled State-Estimation in Energy Control Centres

Abstract STATE-ESTIMATION (SE) is the process of assigning a value to an unknown system state-variable based on measurements from that system according to some criteria. Usually, the process involves imperfect measurements inherent in the process and therefore estimating the system state is based on a statistical criteria that estimates the true value of the state-variable so as to minimize or maximize the selected criteria. A commonly used and familiar criteria is that of minimizing the sum of squares of the differences between the estimated and ‘true’ (i.e. measured) values. Power System Static State-Estimation (PSSE) is a proven technique for providing modern energy control centres with reliable real-time data bases for security monitoring and on-line control. The state-estimation processes real-time raw measurements to obtain optimal estimates of voltage magnitudes and angles at all buses, which determines the static state of the Power System (PS). The estimation problem of PSSE is usually formulated as a nonlinear least-square problem. Among state estimators based on the normal equations approach, the weighted least square (WLS) estimator provides reliable estimates but is slow, mainly due to its requirement for updating and factorization of the full-matrix in every iteration. The decoupled state-estimator requires less computation time per iteration and less storage requirement than WLS. The concept of p- δ, Q-V decoupling technique as applied in load flow studies is extended to state estimation technique also which reduces the storage requirement and also the time for convergence. The most common approach to the SE problem is to formulate it as a nonlinear problem usually solved by an iterative procedure based on successive linearization; in each iteration one has to solve a linear least-squares problem. Associated with WLS estimation is a set of redundant equations are to be solved to yield the solution. These equations are ‘Normal Equations’ and are solved using Cholesky factorization. The normal equations are very satisfactory with respect to preserving sparsity, but they may give rise to numerical ill-conditioning. Methods using orthogonal transformations, such as those of Householder have far better stability properties. The Method of Data Equations which uses the Householder orthogonal transformation is more efficient than the conventional method of normal equations. Also, it is less prone to computer round-off errors and is more numerically stable. The standard approach to the solution of the WLS state estimation in power system is the iterative normal equations method. Occassional ill-conditioning has been experienced with this method. Alternative approaches to state-estimation are based on ‘orthogonal transformation’. These methods have improved numerical behaviour, but generally their sparsity suffers when the measurement redundancy is high. In this paper the decoupled estimator based on transformation method is presented using different algorithms and the results agree with the method of normal equations. Measurements are transformed into new ‘measurements’ that are functions of the states and of the original measurements. In PSSE redundancy rate is an important aspect. Estimator obtained from various redundancy rates, by considering active and reactive redundancy seperately obtained by uniformly distributing the measurements over the system is presented. Line flows and bus injection measurements required for stateestimation studies, have been obtained from load-flow computations. Error (noise) terms (σ = 0.01 and 0.001) are added to these measurements. The error vector being simulated using a random number generator. Convergence tolerance of 0.001 PU and 0.001 are used for V and δ, respectively. Results obtained for various combinations of measurements; and with different order of processing these measurements have been presented. Measurements are classified as. Type 1, 2 : Power injections. Type 3, 4 : Power flows. Type 5 : Voltage magnitudes. Different cases analysed are. i) Case 1 : 1, 2, 3, 4 & 5 Type measurements. ii) Case 2 : 1, 3, 4 and 5 Type measurements. iii) Case 3 : 1, 2, 3 and 4 Type measurements. iv) Case 4 : 3 and 5 Type measurements. It is important that at least few voltage measurements (NBUS/ 10 to NBUS/5) must be included while estimating the states. When line-flows only are used as measurements, solution takes large number of iterations to converge. Comparative results obtained from ‘method of Normal equations’ and ‘Data equations’ have been presented. Illustrative examples considered are IEEE-14 bus system and 5-bus sytem.

Observability Analysis for Orthogonal Transformation Based State Estimation

Recently a triangular-factorization based observability analysis has been proposed. The method uses subroutines already existing in a state estimation software using the standard normal equations approach. The normal equations approach works well in most applications. However, occasional numerical ill-conditioning of the gain matrix has been observed in practice. Orthogonal transformation methods have been suggested to alleviate this problem. A hybrid approach combining the sparsity of the normal equations method and the numerical robustness of the orthogonal transformation methods has recently been developed. The method solves the normal equations using the orghogonal transformation of the Jacobian matrix. This paper extends the basic idea of the triangular-factorization based observability analysis to orthogonal transformation and hybrid state estimators. The observability analysis is performed during the process or orthogonal transformation (Householder or Givens) of the Jacobian matrix. The deailed algorithm is given. Test results are presented.

Perturbation theory for evaluation algorithms of arithmetic expressions

The paper presents the theoretical foundation of a forward error analysis of numerical algorithms under data perturbations, rounding error in arithmetic floating-point operations, and approximations in ’built-in’ functions. The error analysis is based on the linearization method that has been proposed by many authors in various forms. Fundamental tools of the forward error analysis are systems of linear absolute and relative a priori and a posteriori error equations and associated condition numbers constituting optimal bounds of possible accumulated or total errors. Derivations, representations, and properties of these condition numbers are studied in detail. The condition numbers enable simple general, quantitative definitions of numerical stability, backward analysis, well- and ill-conditioning of a problem and an algorithm. The well-known illustration of algorithms and their linear error equations by graphs is extended to a method of deriving condition numbers and associated bounds. For many algorithms the associated condition numbers can be determined analytically a priori and be computed numerically a posteriori. The theoretical results of the paper have been applied to a series of concrete algorithms, including Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis.

On joint/interface elements and associated problems of numerical ill-conditioning. Short communication : Pande, G N; Sharma, K G Int J Num Anal Meth Geomech, V3, N3, July–Sept 1979, P293–300

On joint/interface elements and associated problems of numerical ill-conditioning. Short communication : Pande, G N; Sharma, K G Int J Num Anal Meth Geomech, V3, N3, July–Sept 1979, P293–300