Block Triangular Structure Research Articles

A Geometric Approach for Analyzing Parametric Biological Systems by Exploiting Block Triangular Structure

A Geometric Approach for Analyzing Parametric Biological Systems by Exploiting Block Triangular Structure

Solving Boolean polynomial systems by parallelizing characteristic set method for cyber‐physical systems

SummaryMany cyber‐attach schemes and coding models established by algebra tools are build to address the problem of security of cyber‐pysical systems (CPS). As an important field of algebra computing, Boolean Polynomial System Solving (PoSSo) problem plays a very important role in many algebra applications. In this article, we propose an efficient Parallel Boolean Characteristic Set method (PBCS) under the high‐performance computing environment to improve the efficiency of solving Boolean polynomial systems. The PBCS is implemented based on the state‐of‐the‐art Boolean Characteristic Set method (BCS). It adopts a master‐slave parallel pattern, and distributes tasks based on the polynomial sets after initial zero decomposition. We design a strategy of dynamically reallocating tasks to ameliorate load imbalance, which is caused by dynamical zero decomposition of polynomials. Furthermore, we improve its performance by optimizing the parameter settings of PBCS, including the maximum number of polynomial branches that trigger the dynamic allocation policy and the scheduling time. Experimental results with solving several Boolean polynomial systems confirm that PBCS is efficient and scalable, especially for the equations generating from stream ciphers that have block triangular structure. Moreover, the method also has good scalability. It shows a stable speedup as well even extending to the size of thousands of CPU cores.

A Priori Error Analysis for Time-Stepping Discontinuous Galerkin Finite Element Approximation of Time Fractional Optimal Control Problem

In this paper a priori error analysis for time-stepping discontinuous Galerkin finite element approximation of optimal control problem governed by time fractional diffusion equation is presented. A time-stepping discontinuous Galerkin finite element method and variational discretization approach are used to approximate the state and control variables respectively. Regularity of the optimal control problem is discussed. Since the time fractional derivative is nonlocal, in order to reduce the computational cost a fast gradient projection algorithm is designed for the control problem based on the block triangular Toeplitz structure of the discretized state equation and adjoint state equation. Numerical examples are carried out to illustrate the theoretical findings and fast algorithm.

Robust Decentralized Fault Estimation for Loss of Actuator Effectiveness of Multi-motor Web-winding System

This paper develops a method estimating the actuator power loss faults for the multi-motor web-winding system. Firstly, the web-winding system is regarded as a synthetic system with several dynamic subsystems and their dynamic models are given. Then, the nonlinear coordinate transformation is introduced to obtain a transformed system with block triangular structure and the interconnections among subsystems are allowed. Next, a decentralized adaptive high-gain observer with sliding-mode is designed for the obtained transformed system to estimate the system actuator power loss faults. Meanwhile, the estimation error dynamics can be obtained. Sufficient conditions of asymptotic stability for estimation error system are derived by Lyapunov theory, and the sliding-mode gain and fault estimation law are achieved. After that, the fault estimation observer for original multi-motor web-winding system is designed by inverse coordinate transformation. Finally, the simulations and analysis are implemented on the three-motor web-winding system to verify the effectiveness of the proposed method.

Nonsymmetric Reduction-Based Algebraic Multigrid

Algebraic multigrid (AMG) is often an effective solver for symmetric positive definite (SPD) linear systems resulting from the discretization of general elliptic PDEs or the spatial discretization of parabolic PDEs. However, convergence theory and most variations of AMG rely on $A$ being SPD. Hyperbolic PDEs, which arise often in large-scale scientific simulations, remain a challenge for AMG, as well as other fast linear solvers, in part because the resulting linear systems are often highly nonsymmetric. Here, a novel convergence framework is developed for nonsymmetric, reduction-based AMG, and sufficient conditions derived for $\ell^2$-convergence of error and residual. In particular, classical multigrid approximation properties are connected with reduction-based measures to develop a robust framework for nonsymmetric, reduction-based AMG. Matrices with block-triangular structure are then recognized as being amenable to reduction-type algorithms, and a reduction-based AMG method is developed for upwind discretizations of hyperbolic PDEs, based on the concept of a Neumann approximation to ideal restriction ($n$AIR). $n$AIR can be seen as a variation of local AIR ($\ell$AIR) introduced in previous work, specifically targeting matrices with triangular structure. Although less versatile than $\ell$AIR, setup times for $n$AIR can be substantially faster for problems with high connectivity. $n$AIR is shown to be an effective and scalable solver of steady state transport for discontinuous, upwind discretizations, with unstructured meshes, and up to 6th-order finite elements, offering a significant improvement over existing AMG methods. $n$AIR is also shown to be effective on several classes of “nearly triangular” matrices resulting from curvilinear finite elements and artificial diffusion.

A fast gradient projection algorithm for time fractional optimal control problem

In this paper a fast gradient projection algorithm for optimal control problems governed by time fractional diffusion equation is developed. The state equation is discretized by piecewise linear FE for space variable and L1 scheme for time variable, while the control variable is approximated by variational discretization. Based on the block triangular Toeplitz structure of the coefficient matrix of the discretized state equation and adjoint state equation, a fast gradient projection algorithm is designed for the control problem. Numerical examples are carried out to illustrate the effectiveness of the algorithms.

Robust observer design for multi-motor web-winding system

This paper develops a novel observer design method for multi-motor web-winding system. Firstly, the multi-motor web-winding system is regarded as a synthetic system with several subsystems, where the dynamic model for each subsystem is given. Then, the nonlinear diffeomorphism transformation is introduced to obtain a transformed system with block triangular structure and the interconnections among the subsystems are allowed. Next, a decentralized high-gain observer with sliding mode is designed for the transformed system, based on which the estimation error dynamics can be got. Sufficient condition of asymptotic stability for estimation error dynamics is derived by the Lyapunov stability theory and the observer gain is obtained. After that, the observer for original multi-motor web-winding system is achieved via inverse transformation. Finally, the simulation and analysis are performed in the three-motor web-winding system to verify the effectiveness of the proposed observer.

Structural Analysis Methods for Differential Algebraic Equations via Fixed-Point Iteration

Motivated by Pryce’s structural analysis method for differential algebraic equations (DAEs), we show the complexity of the fixed-point iteration algorithm (FPIA) and propose a fixed-point iteration method with parameters. It leads to a block fixed-point iteration method (BFPIM) which can be applied to immediately calculate the crucial canonical offsets for large-scale (coupled) DAE systems with block-triangular structure, and its complexity analysis is also given in this paper. Moreover, preliminary numerical experiments show that the time complexity of BFPIM can be reduced by at least O(l) compared to the FPIA.

Iterative structure of finite loop integrals

In this paper we develop further and refine the method of differential equations for computing Feynman integrals. In particular, we show that an additional iterative structure emerges for finite loop integrals. As a concrete non-trivial example we study planar master integrals of light-by-light scattering to three loops, and derive analytic results for all values of the Mandelstam variables $s$ and $t$ and the mass $m$. We start with a recent proposal for defining a basis of loop integrals having uniform transcendental weight properties and use this approach to compute all planar two-loop master integrals in dimensional regularization. We then show how this approach can be further simplified when computing finite loop integrals. This allows us to discuss precisely the subset of integrals that are relevant to the problem. We find that this leads to a block triangular structure of the differential equations, where the blocks correspond to integrals of different weight. We explain how this block triangular form is found in an algorithmic way. Another advantage of working in four dimensions is that integrals of different loop orders are interconnected and can be seamlessly discussed within the same formalism. We use this method to compute all finite master integrals needed up to three loops. Finally, we remark that all integrals have simple Mandelstam representations.

Structured [formula omitted]-optimal control for nested interconnections: A state-space solution

If imposing general structural constraints on controllers, it is unknown how to design optimal H∞-controllers by convex optimization. Under the so-called quadratic invariance condition on the generalized plant, the Youla parametrization allows to translate the structured synthesis problem into an infinite dimensional convex program. Nested interconnections that are characterized by a block-triangular structure of the standard plant’s control channel and of the controller fall into this class. Recently it has been shown how to design optimal H2-controllers for such nested structures in the state-space by solving algebraic Riccati equations. In the present paper we provide a state-space solution of the corresponding output-feedback H∞-synthesis problem without any counterpart in the literature. We argue that a solution based on Riccati equations is–even for state-feedback synthesis–not feasible and we illustrate our results by means of a simple numerical example.

A new class of superregular matrices and MDP convolutional codes

This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a sufficiently large finite field F. Such construction works for any given choice of characteristic of the field F and code parameters (n,k,δ) such that (n−k)|δ. We also discuss the size of F needed so that the proposed matrices are superregular.

ISS interval observers for nonlinear systems transformed into triangular systems

Designing an interval observer with stability properties for nonlinear systems, which are not cooperative and not globally Lipschitz, is an open problem. This paper studies a general canonical structure for which interval observers with input to state stability (ISS) properties can be derived. This canonical block triangular nonlinear structure is rather general and may result from a change of coordinates or an output injection. We provide a general method for explicitly constructing framers for systems for which can be given such a structure. We also construct ISS interval observers when additional properties are satisfied. The systems we consider are in general not cooperative and not globally Lipschitz. We illustrate the constructions by designing a framer and an ISS interval observer for two models of bioreactors. Copyright © 2012 John Wiley & Sons, Ltd.

Adaptive neural control for MIMO nonlinear systems with state time-varying delay

In this paper, adaptive neural control is proposed for a class of multi-input multi-output (MIMO) nonlinear unknown state time-varying delay systems in block-triangular control structure. Radial basis function (RBF) neural net- works (NNs) are utilized to estimate the unknown continuous functions. The unknown time-varying delays are compensated for using integral-type Lyapunov-Krasovskii functionals in the design. The main advantage of our result not only efficiently avoids the controller singularity, but also relaxes the restriction on unknown virtual control coefficients. Boundedness of all the signals in the closed-loop of MIMO nonlinear systems is achieved, while The outputs of the systems are proven to converge to a small neighborhood of the desired trajectories. The feasibility is investigated by two simulation examples.

Adaptive output-feedback control of a general class of uncertain feedforward systems via a dynamic scaling approach

An adaptive output-feedback control design technique is proposed for a class of feedforward systems with input-to-state stable (ISS) appended dynamics in a block triangular (feedforward) structure. The system contains uncertain functions involving all states with the assumed bounds on uncertain functions allowed to contain crossproducts of unknown parameters and unmeasured states. The control design is based on the dual dynamic high-gain scaling technique and introduces a novel adaptation dynamics and scaling dynamics to globally stabilise the system using output feedback. The ISS appended dynamics are coupled with all the system states and the input and the ISS Lyapunov inequalities also feature crossproducts of unknown parameters and unmeasured states. The proposed control design does not require any knowledge of magnitude bounds on unknown parameters.

다입력 다출력 비선형 시스템의 관측기 설계 및 인덕션 모터에 응용

This paper presents an observer design method for a special class of multi input multi output(MIMO) nonlinear systems. First, we characterize the class of MIMO nonlinear systems with a block triangular structure. Also, the observability matrices for SISO nonlinear systems are extended to MIMO systems. By using the generalized observability matrices, it is shown that under the boundedness conditions of system state and input, the proposed observer guarantees the local exponential stability of error dynamics. Finally, its application to induction motor is given to verify the proposed method.

Adaptive Fuzzy Output Tracking Control of MIMO Nonlinear Uncertain Systems

In this paper, the adaptive fuzzy tracking control problem is discussed for a class of uncertain multiple-input-multiple-output (MIMO) nonlinear systems with the block-triangular structure. The fuzzy logic systems are used to approximate the unknown nonlinear functions. By using the backstepping technique, the adaptive fuzzy tracking control design scheme is developed, which has minimal learning parameterizations. The adaptive fuzzy tracking controllers guarantee that the outputs of systems converge to a small neighborhood of the reference signals and all the signals in the closed-loop system are semiglobally uniformly ultimately bounded. Two examples are used to show the effectiveness of the approach

Nonlinear observer based control of induction motors

This paper presents an observer design methodology for a special class of MIMO nonlinear systems. First, we characterize the class of MIMO nonlinear systems that consists of the linear observable part and the nonlinear part with a block triangular structure. Also, the similarity transformation that plays an important role in proving the convergence of the proposed observer is generalized to MIMO systems. we have considered the nonlinearity of the system in designing the gain of the observer and then, it is prevented that the observer gain is unnecessarily increased. Moreover, by using the generalized similarity transformation, it is shown that under some observability and boundedness conditions, the proposed observer guarantees the global exponential convergence to zero of the estimation error. Finally, the simulation results for induction motor are included to illustrate the validity of our design scheme.

A generalized Jacobian based Newton method for semismooth block-triangular system of equations

This paper will consider the problem of solving the nonlinear system of equations with block-triangular structure. A generalized block Newton method for semismooth sparse system is presented and a locally superlinear convergence is proved. Moreover, locally linear convergence of some parameterized Newton method is shown.

Adaptive neural control of uncertain MIMO nonlinear systems.

In this paper, adaptive neural control schemes are proposed for two classes of uncertain multi-input/multi-output (MIMO) nonlinear systems in block-triangular forms. The MIMO systems consist of interconnected subsystems, with couplings in the forms of unknown nonlinearities and/or parametric uncertainties in the input matrices, as well as in the system interconnections without any bounding restrictions. Using the block-triangular structure properties, the stability analyses of the closed-loop MIMO systems are shown in a nested iterative manner for all the states. By exploiting the special properties of the affine terms of the two classes of MIMO systems, the developed neural control schemes avoid the controller singularity problem completely without using projection algorithms. Semiglobal uniform ultimate boundedness (SGUUB) of all the signals in the closed-loop of MIMO nonlinear systems is achieved. The outputs of the systems are proven to converge to a small neighborhood of the desired trajectories. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters. The proposed schemes offer systematic design procedures for the control of the two classes of uncertain MIMO nonlinear systems. Simulation results are presented to show the effectiveness of the approach.

State observer for MIMO nonlinear systems

A state observer design for a special class of MIMO nonlinear systems which has a block triangular structure is presented. For this purpose an extension of the existing design for SISO triangular systems to MIMO cases is performed. Since the gain of the proposed observer depends on both the nonlinear and linear parts of the system, it improves the transient performance of the high gain observer. Also, by using a generalised similarity transformation for the error dynamics, it is shown that under a boundedness condition, the proposed observer guarantees the global exponential convergence of the estimation error. Finally, an illustrative example is included to show the validity of the design approach.