Solving Linear Equations With Separable Problem Data Over Directed Networks

In learning systems, high operation precision is often a desirable objective for the algorithm design. Though centralized algorithms are generally adopted, they are subjected to restrictive hypotheses on the learning systems. To overcome this challenging problem, we aim to propose some distributed learning algorithms that focus specifically on achieving the perfect tracking tasks for iterative learning control (ILC) systems. By noting the equivalent relation between the perfect tracking problem of ILC systems and the solving problem of linear algebraic equations (LAEs), we first present an observer-based distributed learning algorithm to solve LAEs, where a multiagent system is constructed with every agent being only required to access some partial information for LAEs. The distributed learning algorithm benefits from integrating both observer-based design and consensus-based design ideas such that for any solvable LAE, all agents can agree on a common solution of it under any initial conditions of agents, regardless of whether it has a unique solution or not. Then, with the distributed learning algorithm for LAEs, we further develop two classes of distributed learning control algorithms for ILC systems, which establish the perfect tracking objective in the presence of the trackable desired output even without using the basic relative degree condition that is generally imposed for conventional ILC.

Chapter 2 - Simultaneous linear algebraic equations

In this study, an advanced numerical simulation tool has been developed for the rotor-floater-mooring coupled dynamic analysis of a Multiple Unit Floating Offshore Wind Turbine (MUFOWT) in the time domain including aero-blade-multiple tower dynamics and control, mooring dynamics, and platform motion. In particular, the developed numerical tool is extended from the single turbine analysis tool FAST, which has been developed by the National Renewable Energy Laboratory (NREL) for years. For the linear and nonlinear hydrodynamics of a floating platform and the generalized coordinate-based Finite Element Method (FEM) mooring line dynamics, CHARM3D is used. Therefore, the entire dynamics of a floating offshore wind turbine were obtained by coupling FAST-CHARM3D in the time domain, which has been further extended to include additional coupled dynamics among multiple turbines. The global coefficient matrix, including one floating platform and a number of turbines, is built at each time step of the simulation and is solved simultaneously to obtain the responses of the entire degrees of freedom (DOFs) of the MUFOWT system. To investigate the dynamic coupling effects between the platform and turbines, a five-turbine semisubmersible is modeled and simulated. Then 1P excitation from one turbine is intentionally generated by the partial breaking of a blade, and the loads transferred to the platform and other turbines are investigated. The analysis shows that the dynamic load imbalance of one turbine in the MUFOWT may induce appreciable changes in the performance of the other turbines or the floating platform.

Polynomial basis conversion made stable by truncated singular value decomposition

Application of the conjugate projected gradient method to large-scale contact problems

A novel dynamic event-triggered mechanism for dynamic average consensus

Distributed dynamic average consensus for nonlinear multi-agent systems in the presence of external disturbances over a directed graph

In this paper, we study the unconstrained optimization problem in a distributed way over directed strongly connected communication graphs. We propose an algorithm, which combines techniques of both gradient descent (GD) and finite-time exact ratio consensus (FTERC). Different from the techniques of average or dynamic average consensus with asymptotic convergence or techniques of finite-time “approximate” consensus with inexact accuracy in the literature, with the help of FTERC for gradient tracking, our proposed distributed FTERC based GD algorithm has a faster convergence rate related to the optimization iteration number and a larger step-size upper bound compared with other algorithms, as demonstrated in the simulations.

In this paper, the receding horizon near-optimal tracking control problem about a class of continuous-time nonlinear systems with fully unknown dynamics is considered. The main challenges of this problem lie in two aspects: 1) most existing systems only restrict their considerations to the state feedback part while the input channel parameters are assumed to be known. This paper considers fully unknown system dynamics in both the state feedback channel and the input channel and 2) the optimal control of nonlinear systems requires the solution of nonlinear Hamilton-Jacobi-Bellman equations. Up to date, there are no systematic approaches in the existing literature to solve it accurately. A novel model-free adaptive near-optimal control method is proposed to solve this problem via utilizing the Taylor expansion-based problem relaxation, the universal approximation property of sigmoid neural networks, and the concept of sliding mode control. By making approximation for the performance index, it is first relaxed to a quadratic program, and then, a linear algebraic equation with unknown terms. An auxiliary system is designed to reconstruct the input-to-output property of the control systems with unknown dynamics, so as to tackle the difficulty caused by the unknown terms. Then, by considering the property of the sliding-mode surface, an explicit adaptive near-optimal control law is derived from the linear algebraic equation. Theoretical analysis shows that the auxiliary system is convergent, the resultant closed-loop system is asymptotically stable, and the performance index asymptomatically converges to optimal. An illustrative example and experimental results are presented, which substantiate the efficacy of the proposed method and verify the theoretical results.

In this chapter, the receding-horizon near-optimal tracking control problem about a class of continuous-time nonlinear systems with fully unknown dynamics are considered. The main challenges of this problem lie in two aspects: (1) Most existing systems only restrict their considerations to the state feedback part while the input channel parameters are assumed to be known. This chapter considers fully unknown system dynamics in both the state feedback channel and the input channel. (2) The optimal control of nonlinear systems requires the solution of nonlinear Hamilton–Jacobi–Bellman equations. Up to today, there are no systematic approaches in the existing literature to solve it accurately. A novel model-free adaptive near-optimal control method is presented to solve this problem by utilizing the Taylor expansion based problem relaxation, the universal approximation property of sigmoid neural networks, and the concept of sliding mode control. By making an approximation for the performance index, it is first relaxed to a quadratic program, and then a linear algebraic equation with unknown terms. An auxiliary system is designed to reconstruct the input-to-output property of the control systems with unknown dynamics, so as to tackle the difficulty caused by the unknown terms. Then, by considering the property of the sliding mode surface, an explicit adaptive near-optimal control law is derived from the linear algebraic equation. Theoretical analysis shows that the auxiliary system is convergent, the resultant closed-loop system is asymptotically stable, and the performance index asymptomatically converges to optimal. An illustrative example and experimental results are presented, which substantiate the efficacy of the presented method and verify the theoretical results.

We present a high‐order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second‐order one‐dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well‐conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 307–349, 2016

The application of mathematics to natural phenomena often brings up the question : when a linear equation is changed slightly, how much of a change results in the solution? Here vector is used to include, for example, vectors in Hubert space and elements of (linear) function spaces; also it is assumedC) that the equation is uniquely solvable. We formulate and answer the question in §2 for a linear equation in a Banach space, then specialize to deduce perturbation limits for linear algebraic systems, infinite systems, and integral equations. Finally we obtain error limitations for certain approximate methods of solving infinite linear systems (method of segments) and integral equations (method of Goursat-Schmidt). It is hoped that these limitations may be useful to applied mathematicians. The methods and results of this paper unify and extend investigations concerning algebraic systems by F. R. Moulton [15](2), Etherington [6] and the author [13, 14]; and on integral equations of Fredholm type and second kind by Tricomi [21]. However, they do not cover perturbation questions associated with characteristic values and characteristic functions, which have been studied by Lord Rayleigh [16, p. 115], Courant [4, p. 296] and Mrs. Adams [l]. §1 is expository, containing as much about abstract spaces as is needed for §2. 1. Vector spaces. It will be useful to collect here some facts about normed linear spaces ispaces L), Banach spaces ispaces B), and linear transformations. A space L is linear: if xEL and a is any complex number, the product ax is defined and axEL; if also y EL, the sum x+y is defined and x+yEL. With each element x of L is associated a non-negative real number ||*||, its norm; ||*|| >0 unless x = @, the zero-element of L; |J@|| =0. The norm has properties of an absolute value: ||a*|| = \a -11*11, ||*+y|| a||*||+||y||(We have described a complex space L; in a real space, number a must be real.) A normed linear space is a Banach space 5 [2, p. 53] if it is furthermore complete: if {*„} is an infinite sequence of elements of 5, and if II*»—*»||—*0 as m, m—»oo, there exists a xo of 5 such that ||*„—*o||—»0 as n—+ oo (strong completeness). Examples of such spaces are listed in Banach's book [2, pp. 10-12, examples 3-10], and several occur in the remainder of this paper.

This paper addresses some of the issues concerning the use of variable accuracy optical processors to improve the processing time required to obtain a high accuracy solution to a set of Linear Algebraic Equations (LAEs). We begin with a standard error analysis of the Steepest Descent iterative algorithm used to find the solution to the LAEs. This results in an expression relating the accuracy of the solution to the number of iterations and the inherent system accuracy in each mode of operation, along with the eigen-structure of the matrix describing the LAE. The accuracy at any iteration is a combination of terms representing the ideal algorithmic improvement plus the degradation due to processor inaccuracies. An evaluation of the proper number of iterations for processing in both low and high accuracy modes can be inferred through an examination of the tradeoffs in accuracy between these two terms. The expression is evaluated for several sample problems obtained from the Adaptive Phased Array Radar field. These results are then interpreted with respect to specific optical processing architectures.

The possibility of application of the interval analysis for data processing in the field of spectral analysis is considered. It is assumed that the data have interval uncertainty; therefore the problem of finding unknown concentrations is posed as a linear interval tolerance problem. The incompatibility of the interval system of linear algebraic equations is shown for the initial data using the apparatus of the recognizing functional. The relevance of the topic is due to the need for regularization of inconsistent interval systems of linear equations. The idea of S. P. Shary of a combined method for correcting a linear tolerance problem has been implemented. A new method for managing the solution by changing the linear algebraic equations interval system matrix elements radii has been developed. The research results can be used for example, to calculate the substance’s concentrations by measurement of the characteristic X-ray radiation.

We consider networks of dynamic agents that execute cooperative, distributed control algorithms in order to coordinate themselves and to collectively achieve goals. The agents rely on consensus algorithms that are based on local interactions with their nearest neighbors in the communication graph. However, such systems are not robust to one or more malicious agents and there are no performance guarantees when one or more agents do not cooperate. Recent results in network science deal with this problem by requiring specific graph topological properties. Nevertheless, the required network topologies imply high connectivity levels, which may be difficult to achieve in systems that exhibit time-varying communication graphs. In this paper, we propose an approach that provides resilience for networks of dynamic agents whose communication graphs are time-varying. We show that in the case where the required connectivity constraints cannot be satisfied at all times, we can resort to a consensus protocol that guarantees resilience when the union of communication graphs over a bounded period of time satisfies certain robustness properties. We propose a control policy to attain resilient behavior in the context of perimeter surveillance with a team of robots. We provide simulations that support our theoretical analyses.