Mian Ilyas Ahmad

Preconditioned Multishift BiCG for

We propose the use of a multishift biconjugate gradient method (BiCG) in combination with a suitable chosen polynomial preconditioning, to efficiently solve the two sets of multiple shifted linear ...

\mathcal{H}_2

We consider model order reduction for bilinear descriptor systems using an interpolatory projection framework. Such nonlinear descriptor systems can be represented by a series of generalized linear descriptor systems (also called subsystems) by utilizing the Volterra-Wiener approach (Rugh, 1981). Standard projection techniques for bilinear systems utilize the generalized transfer functions of these subsystems to construct an interpolating approximation. However, the resulting reduced-order system may not match the polynomial parts of the generalized transfer functions. This may result in an unbounded error in terms of H 2 or H ∞ norms. In this paper, we derive an explicit expression for the polynomial part of each subsystem by assuming a special structure of the bilinear system which reduces to an index-1 linear descriptor system or differential algebraic equation (DAE) if the bilinear terms are zero. This allows us to propose an interpolatory technique for bilinear DAEs which not only achieves interpolation, but also retains the polynomial parts of the bilinear systems. The approach extends the interpolatory technique for index-1 linear DAEs (Beattie and Gugercin, 2009) to bilinear DAEs. Numerical examples are used to illustrate the theoretical results.

-Optimal Model Reduction

Abstract In this paper a nonlinear model of an underactuated quad rotor aerial robot is derived, based on Newton-Euler formalism, and backstepping based PID control strategy is implemented for the derived model. Model derivation comprises determining equations of motion of the quad rotor in three dimensions and seeking to approximate actuation forces through modeling of aerodynamic coefficients and electric motor dynamics. The derived MIMO model, constituted of translational and rotational subsystem, is dynamically unstable. A nonlinear control strategy is therefore implemented for the quad rotor aerial robot. The control strategy includes integral backstepping control for the translational subsystem and backstepping based PID control for the rotational subsystem. The stability of the control design is ensured by Lyapunov stability theorem. The performance of the nonlinear control strategy is evaluated using nonlinear simulation. The simulation results, obtained from backstepping based PID, are compared with conventional optimized PID controller. For the conventional PID controller, the optimization algorithm used is to minimize the Integral of Absolute Error (IAE). Results of comparison validate effectiveness of the backstepping based PID control strategy for the underactuated aerial robot near quasi stationary flight.

Krylov subspace-based model reduction for a class of bilinear descriptor systems

We propose a model reduction technique for quadratic-bilinear descriptor systems. The approach involves approximating the system by a bilinear descriptor system using Carleman bilinearization [1]. It is shown that, by assuming a particular structure of the matrix pencil, the bilinearization process preserves the structure of the matrix pencil in the bilinearized system. Further, we extend the use of the bilinear iterative rational Krylov algorithm (B-IRKA) [2] to descriptor systems to identify a locally ℋ2-optimal reduced-order system for the bilinearized system under the assumption that the ℋ2 norm of the system exists. Applications to the simulation of a nonlinear RC circuit and a lid-driven cavity flow are presented to illustrate the proposed methodology.

Backstepping Based PID Control Strategy for an Underactuated Aerial Robot

Standard interpolatory subspaces for model reduction of linear descriptor systems may produce unbounded ℋ2 or ℋ∞ error. In this paper we investigate this issue and discuss modified interpolatory subspaces based on spectral projectors that ensure bounded errors. In the special case of index-3 descriptor systems, we show how to transform the system to an equivalent system that enables the use of standard interpolatory subspaces for model reduction with bounded errors, but without the explicit computation of spectral projectors. The approach can also be used to update interpolation points in the framework of ℋ2-norm approximation, thus extending the Iterative Rational Krylov Algorithm (IRKA) to index-3 descriptor systems. Also it is shown that the index-3 structure of the actual system can be preserved in the reduced order interpolating approximation.

Model reduction of quadratic-bilinear descriptor systems via Carleman bilinearization

In this paper we consider the ℋ2 optimal model reduction problem which has important applications in system approximation and has received considerable attention in the literature. An important link between special cases of this problem and rational interpolation using Krylov projection methods has been recently established. We use this link to derive a solution in the special case when the original system is single input single output (SISO) and the approximation has first order. We also indicate extensions to higher order approximations. Finally, we give a few examples to illustrate our procedure.

Interpolatory Model Reduction Techniques for Linear Second-Order Descriptor Systems

The development of efficient interior point methods has greatly enlarged the range of control problems with feasible numerical solution. These methods are nevertheless difficult to solve for large-scale problems. In this paper we suggest the use of a Krylov subspace technique for the efficient low-rank approximate solution to large-scale Sylvester equations. The suggested method is a novel restart scheme which improves further the computation efficiency and storage requirements of the standard Krylov subspace methods for the solution of large-scale Sylvester equations.

ℋ 2 Optimal model reduction of linear dynamical systems

In this paper a nonlinear model of a 6-DOF quad rotor aerial robot is derived, based on Newton-Euler formalism, and backstepping based PID flight control strategy is implemented for motion control of the derived model. The derivation comprises determining equations of motion of quad rotor aerial robot in three dimensions and seeking to approximate actuation forces through modeling of the aerodynamic coefficients and electric motor dynamics. The derived MIMO model, constituted of translational and rotational subsystem, is dynamically unstable. A nonlinear control strategy that includes integrator backstepping control for the translational subsystem and backstepping based PID control for the rotational subsystem is implemented for the quad rotor aerial robot. The stability of the control design is ensured by Lyapunov global stability theorem. The performance of the nonlinear control algorithm is evaluated using nonlinear simulation. Results from nonlinear simulation validate effectiveness of the designed control strategy for quad rotor aerial robot near quasi stationary (hover or near hover) flight.

Krylov subspace restart scheme for solving large-scale Sylvester equations

PurposeInternationally adopted variant interpretation guidelines from the American College of Medical Genetics and Genomics (ACMG) are generic and require disease-specific refinement. Here we developed CardioClassifier (http://www.cardioclassifier.org), a semiautomated decision-support tool for inherited cardiac conditions (ICCs).MethodsCardioClassifier integrates data retrieved from multiple sources with user-input case-specific information, through an interactive interface, to support variant interpretation. Combining disease- and gene-specific knowledge with variant observations in large cohorts of cases and controls, we refined 14 computational ACMG criteria and created three ICC-specific rules.ResultsWe benchmarked CardioClassifier on 57 expertly curated variants and show full retrieval of all computational data, concordantly activating 87.3% of rules. A generic annotation tool identified fewer than half as many clinically actionable variants (64/219 vs. 156/219, Fisher’s P = 1.1  ×  10−18), with important false positives, illustrating the critical importance of disease and gene-specific annotations. CardioClassifier identified putatively disease-causing variants in 33.7% of 327 cardiomyopathy cases, comparable with leading ICC laboratories. Through addition of manually curated data, variants found in over 40% of cardiomyopathy cases are fully annotated, without requiring additional user-input data.ConclusionCardioClassifier is an ICC-specific decision-support tool that integrates expertly curated computational annotations with case-specific data to generate fast, reproducible, and interactive variant pathogenicity reports, according to best practice guidelines.

Backstepping based Nonlinear Flight Control Strategy for 6 DOF Aerial Robot

Abstract In this paper we consider the H 2 optimal model reduction problem which has important applications in system approximation and has received considerable attention in the literature. An important link between special cases of this problem and rational interpolation using Krylov subspace projection methods has been recently established. We use this link to derive a solution of the second order optimal H 2 approximation problem that involves the computation of all simultaneous solutions to two bivariate polynomials. We show that this is equivalent to the simultaneous solution of two 2-dimensional eigenvalue problems. To illustrate our procedure we give a few numerical examples