Solving high‐order linear differential equations by a Legendre matrix method based on hybrid Legendre and Taylor polynomials

A numerical method for solving the high‐order linear differential equations with variable coefficients under the mixed conditions is presented. The method is based on the hybrid Legendre and Taylor polynomials. The solution is obtained in terms of Legendre polynomials. Comparison of the present solution is made with the existing solution and excellent agreement is noted. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Legendre wavelets method for constrained optimal control problems

A numerical method for solving non‐linear optimal control problems with inequality constraints is presented in this paper. The method is based upon Legendre wavelet approximations. The properties of Legendre wavelets are first presented. The operational matrix of integration and the Gauss method are then utilized to reduce the optimal control problem to the solution of algebraic equations. The inequality constraints are converted to a system of algebraic equalities; these equalities are then collocated at the Gauss nodes. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2002 John Wiley & Sons, Ltd.     

Control parameterization enhancing transform for optimal control of switched systems

In this paper, a class of optimal switching control problems with prespecified order of the sequence of subsystems is considered, where the switching instants are included in the cost functional. Both the switching instants and the control function are to be chosen such that the cost functional is minimized. Through the discretization of the control space, each control component is approximated by a piecewise constant function. The partition points and the heights of each of these piecewise constant functions are taken as decision varibles. Using the control parameterization enhancing transform, we map both types of switching instants into preassigned knot points via the introduction of an additional control, known as the enhancing control. In this way, we construct a sequence of approximate optimal parameter selection problems with fixed switching time points. We then show that these approximate optimal parameter selection problems are solvable as mathematical programming problems. The convergence analysis of this approximation is investigated. Two examples are solved using the proposed method so as to demonstrate the effectiveness of the method proposed.     

A numerical method for hybrid optimal control based on dynamic programming

This contribution extends a numerical method for solving optimal control problems by dynamic programming to a class of hybrid dynamic systems with autonomous as well as controlled switching. The value function of the hybrid control system is calculated based on a full discretization of the state and input spaces. A bound for the error due to discretization is obtained from modeling the error as perturbation of the continuous dynamics and the cost terms. It is shown that the bound approaches zero and that the value function of the discretized variant converges to the value function of the original problem if the discretization parameters go to zero. The performance of a numerical scheme exploiting the discretized system is illustrated for two different examples treated previously in literature.     

Biorthogonal polynomials and the bordering method for linear systems

First, the problem of solving a system of linear equations is shown to be equivalent to the computation of biorthogonal polynomials. The bordering method is a procedure for solving recursively a sequence of linear systems with increasing dimensions and it gave rise to a recurrence relationship between two adjacent families of biorthogonal polynomials. Of course, one relation is not sufficient for computing two families. However, in some particular cases, a second recurrence relationship exists between these biorthogonal polynomials thus leading to procedures for solving recursively such linear systems with increasing dimensions. The cases of Hankel and Toeplitz matrices are treated in details. Conferenza tenuta da C. Brezinski il 12 ottobre 1993     

Convergence of the control parameterization Ritz method for nonlinear optimal control problems

This paper studies the local convergence properties of the control parameterization Ritz method in which the control variable is approximated over a finite-dimensional subspace. The nonlinear free-endpoint optimal control problem is considered, and error bounds are derived for both the cost functional and state-control convergence. Explicit error bounds are obtained for the particular case of approximations over spline spaces. On specializing the general results to the linear-quadratic regulator problem, global convergence results are obtained. Computational results supporting the theoretically derived error bounds are presented.This research was supported by the University Grants Committee of New Zealand.     

On relative stability of linear stationary feedback control systems with time delay

A method is presented whereby the absolute and relative stabilityof linear control systems containing transport lag can be determined.As a result feedback systems with variable time delay and loopgain, may be investigated in straightforward manner.     

A computational method for solving optimal control of a system of parallel beams using Legendre wavelets

The optimal control of transverse vibration of two Euler–Bernoulli beams coupled in parallel by discrete springs is considered. An index of performance is formulated which consists of a modified energy functional of two coupled structures at a specified time and penalty functions involving the point control forces. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. By use of the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of a linear time-invariant lumped-parameter systems. A computationally attractive method based on Legendre wavelets in time domain for solving the optimal control of the lumped parameter systems for any finite interval is proposed. Legendre wavelet integral operational matrix and the properties of a Kronecker product are used to find the approximated optimal trajectory and optimal law of the linear systems with respect to a quadratic cost function by only solving a linear system of algebraic equations. This method provides a straightforward and convenient approach for digital computation. A numerical example is provided to demonstrate the applicability and effectiveness of the proposed method.     

The Newton-Kleinman method for the optimal control of stable weakly regular linear systems

We give an algorithm to find the approximate solution of the linear-quadratic optimal control problem for stable weakly regular linear systems. This algorithm can be understood as a generalization of the Newton-Kleinman method known from the finite-dimensional theory. The central characteristic of our approach is the possibility to solve problems with unbounded control and observation operators, which is motivated by partial differential equations with boundary control and observation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A control parametrization algorithm for optimal control problems involving linear systems and linear 1

Computational schemes based on control parametrization techniques are known to be very efficient for solving optimal control problems. However, the convergence result is only available for the case in which the dynamic system is linear and without the terminal equality and inequality constraints. This paper is to improve this convergence result by allowing the presence of the linear terminal inequality. For illustration, an example arising in the study of optimally one-sided heating of a metal slab in a furnace is considered.     

Simple robust technique using time delay estimation for the control and synchronization of Lorenz systems

This work presents two simple and robust techniques based on time delay estimation for the respective control and synchronization of chaos systems. First, one of these techniques is applied to the control of a chaotic Lorenz system with both matched and mismatched uncertainties. The nonlinearities in the Lorenz system is cancelled by time delay estimation and desired error dynamics is inserted. Second, the other technique is applied to the synchronization of the Lü system and the Lorenz system with uncertainties. The synchronization input consists of three elements that have transparent and clear meanings.Since time delay estimation enables a very effective and efficient cancellation of disturbances and nonlinearities, the techniques turn out to be simple and robust. Numerical simulation results show fast, accurate and robust performance of the proposed techniques, thereby demonstrating their effectiveness for the control and synchronization of Lorenz systems.     

A computational method for free terminal time optimal control problem governed by nonlinear time delayed systems

This paper considers a free terminal time optimal control problem governed by nonlinear time delayed system, where both the terminal time and the control are required to be determined such that a cost function is minimized subject to continuous inequality state constraints. To solve this free terminal time optimal control problem, the control parameterization technique is applied to approximate the control function as a piecewise constant control function, where both the heights and the switching times are regarded as decision variables. In this way, the free terminal time optimal control problem is approximated as a sequence of optimal parameter selection problems governed by nonlinear time delayed systems, each of which can be viewed as a nonlinear optimization problem. Then, a fully informed particle swarm optimization method is adopted to solve the approximate problem. Finally, two free terminal time optimal control problems, including an optimal fishery control problem, are solved by using the proposed method so as to demonstrate its applicability.     

Optimal persistent disturbance attenuation control for linear hybrid systems

In this paper, the disturbance attenuation properties for a class of linear hybrid systems are investigated, and a hybrid optimal persistent disturbance attenuation control problem is studied. First, a procedure is developed to determine the minimal l^∞$l^{\infty }$ induced gain of linear hybrid systems. However, for general hybrid systems, the termination of the procedure is not guaranteed. Then, the decidability issues are discussed. The termination of the procedure in a finite number of steps is shown for a subclass of hybrid systems with simplified discrete event dynamics, called switched linear systems. Finally, the optimal persistent disturbance attenuation controller synthesis problem is studied. It is shown that the optimal performance level can be achieved by a piecewise linear state feedback control law, and a systematic approach is proposed to design such feedback control.     

Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations

In this paper we study the mean-square (MS) stability of the Milstein method for linear stochastic delay integro-differential equations (SDIDE) with Markovian switching by extending the techniques of [Z. Wang, C. Zhang, An analysis of stability of Milstein method for stochastic differential equations with delay, Computers and Mathematics with Applications 51 (2006) 1445–1452; L. Ronghua, H. Yingmin, Convergence and stability of numerical solutions to SDDEs with Markovian switching, Applied Mathematics and Computation 175 (2006) 1080–1091]. It is established that the Milstein method is MS-stable for linear stochastic delay differential equations (Wang and Zhang (2006); in the above reference). Here we prove that it is MS-stable for linear SDIDE with Markovian switching also under suitable conditions on the integral term. A numerical example is provided to illustrate the theoretical results.     

A hybrid iterative method for symmetric indefinite linear systems

Hybrid iterative methods that combine a conjugate direction method with a simpler iteration scheme, such as Chebyshev or Richardson iteration, were first proposed in the 1950s. The ease with which Chebyshev and Richardson iteration can be implemented efficiently on a large variety of computer architectures has in recent years lead to renewed interest in iterative methods that use Chebyshev or Richardson iteration. This paper presents a new hybrid iterative method for the solution of linear systems of equations with a symmetric indefinite matrix. Our method combines the conjugate residual method with Richardson iteration. Special attention is paid to the determination of two real intervals, one on each side of the origin, that contain most of the eigenvalues of the matrix. These intervals are used to compute suitable iteration parameters for Richardson iteration. We also discuss when to switch between the methods. The hybrid scheme typically uses the Richardson method for most iterations, and this reduces the number of arithmetic vector operations significantly compared with the number of arithmetic vector operations required when only the conjugate residual method is used. Computed examples illustrate the competitiveness of the hybrid scheme.     

The parameterization method in optimal control problems and differential–algebraic equations

The paper is devoted to the explanation of the numerical parameterization method (PM) for optimal control (OC) problems with intermediate phase constraint and to its circumstantiation for classical calculus of variation (CV) problems that arise in connection with singular ODEs or DAEs, especially in cases of their essential degeneracy. The PM is based on a finite parameterization of control functions and on derivation of the problem functional with respect to control parameters. The first and the second derivatives are calculated with the help of adjoint vector and matrix impulses. Results of the solution to one phase constrained OC and two degenerate CV problems, connected with singular DAEs nonreducible to the normal form, are presented.     

Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models

A new efficient type of Chebyshev wavelet is used to find the optimal solutions of general linear, continuous-time, multi-delay systems with quadratic performance indices and also to obtain the responses of linear time-delay systems. According to the new definition of Chebyshev wavelets, the operational matrices of integration, product, delay and inverse time and the integration matrix are derived. Furthermore, new operational matrices as the piecewise delay operational matrix and the stretch operational matrix of the desired Chebyshev wavelets are introduced to analyze systems with, in turn, piecewise constant delays and stretched arguments or proportional delays. Two novel algorithms based on newly Chebyshev wavelet method are proposed for the optimal control and the analysis of delay models. Some examples are solved to establish that the accuracy and applicability of Chebyshev wavelet method in delay systems are increased.     

Stability of hybrid stochastic differential systems with switching and time delay

This article introduces a hybrid stochastic differential system with impulsive, switching and time-delay. Some stability criteria of p-moment global asymptotical stability, p-moment global exponential stability and mean square stability of this system are derived by using switching Lyapunov function approach, Itô formula, impulsive differential inequality method, and linear matrix equality techniques. Three examples are presented to demonstrate the efficiency of the obtained results.