Solitary-wave propagation and interactions for the ‘good’ Boussinesq equation

A third-order rational approximant in a three-time level reccurence relation is applied successfully to the ‘good’ Boussinesq equation, already known in the literature. The resulting nonlinear finite-difference scheme, which is analysed for stability, is solved using a predictor–corrector (P–C) scheme, in which the predictor and corrector are both explicit schemes. This P–C scheme is accelerated by a modifed P–C (MPC) in which the already evaluated values are used for the corrector. The behaviour of both the P–C and MPC schemes is tested numerically on the single- and double-soliton waves, and the results from the experiments are compared with that in the literature.     

A fourth order numerical scheme for the one-dimensional sine-Gordon equation

A numerical scheme arising from the use of a fourth order rational approximants to the matrix-exponential term in a three-time level recurrence relation is proposed for the numerical solution of the one-dimensional sine-Gordon (SG) equation already known from the bibliography. The method for its implementation uses a predictor–corrector scheme in which the corrector is accelerated by using the already evaluated corrected values modified predictor–corrector scheme. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. Both the predictor and the corrector schemes are analysed for stability. The predictor–corrector/modified predictor–corrector (P-C/MPC) schemes are tested on single and soliton doublets as well as on the collision of breathers and a comparison of the numerical results with the corresponding ones in the bibliography is made. Finally, conclusions for the behaviour of the introduced MPC over the standard P-C scheme are derived.     

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.     

A conservative exponential time differencing method for the nonlinear cubic Schrödinger equation

A mass and energy conservative exponential time differencing scheme using the method of lines is proposed for the numerical solution of a certain family of first-order time-dependent PDEs. The resulting nonlinear system is solved with an unconditionally stable modified predictor–corrector method using a second-order explicit scheme. The efficiency of the method introduced is analyzed and discussed by applying it to the nonlinear cubic Schrödinger equation. The results arising from the experiments for the single, the double soliton waves and the system of two Schrödinger equations are compared with relevant known ones.     

Turing models in the biological pattern formation through spectral meshless radial point interpolation approach

In the present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of pattern formation in nonlinear reaction diffusion systems. Firstly, we obtain a time discrete scheme by approximating the time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The effect of parameters and conditions are studied by considering the well-known Schnakenberg model.

     

Numerical solution of nonlinear delay differential equations of fractional variable-order using a novel shifted Jacobi operational matrix

This paper presents the generalized nonlinear delay differential equations of fractional variable-order. In this article, a novel shifted Jacobi operational matrix technique is introduced for solving a class of multi-terms variable-order fractional delay differential equations via reducing the main problem to an algebraic system of equations that can be solved numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical experiments are presented to demonstrate the efficiency, generality, accuracy of proposed scheme and the flexibility of this method. The numerical results compared it with other existing methods such as fractional Adams method (FAM), new predictor–corrector method (NPCM), a new approach, Adams–Bashforth–Moulton algorithm and L1 predictor–corrector method (L1-PCM). Comparing the results of these methods as well as comparing the current method (NSJOM) with the exact solution, indicating the efficiency and validity of this method. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated.

     

From linear to nonlinear MPC: bridging the gap via the real-time iteration

ABSTRACT

Linear model predictive control (MPC) can be currently deployed at outstanding speeds, thanks to recent progress in algorithms for solving online the underlying structured quadratic programs. In contrast, nonlinear MPC (NMPC) requires the deployment of more elaborate algorithms, which require longer computation times than linear MPC. Nonetheless, computational speeds for NMPC comparable to those of MPC are now regularly reported, provided that the adequate algorithms are used. In this paper, we aim at clarifying the similarities and differences between linear MPC and NMPC. In particular, we focus our analysis on NMPC based on the real-time iteration (RTI) scheme, as this technique has been successfully tested and, in some applications, requires computational times that are only marginally larger than linear MPC. The goal of the paper is to promote the understanding of RTI-based NMPC within the linear MPC community.     

Predictor–corrector Obreshkov pairs

The combination of predictor–corrector (PEC) pairs of Adams methods can be generalized to high derivative methods using Obreshkov quadrature formulae. It is convenient to construct predictor–corrector pairs using a combination of explicit (Adams–Bashforth for traditional PEC methods) and implicit (Adams–Moulton for traditional PEC methods) forms of the methods. This paper will focus on one special case of a fourth order method consisting of a two-step predictor followed by a one-step corrector, each using second derivative formulae. There is always a choice in predictor–corrector pairs of the so-called mode of the method and we will consider both PEC and PECE modes. The Nordsieck representation of Adams methods, as developed by C. W. Gear and others, adapts well to the multiderivative situation and will be used to make variable stepsize convenient. In the first part of the paper we explain the basic approximations used in the predictor–corrector formula. Those can be written in terms of Obreshkov quadrature. Next section we discuss the equations in terms of Nordsieck vectors. This provides an opportunity to extend the Gear Nordsieck factorization to achieve a variable stepsize formulation. Numerical tests with the new method are also discussed. The paper will present Prothero–Robinson and Kepler problem to illustrate the power of the approach.     

A Not-a-Knot meshless method using radial basis functions and predictor–corrector scheme to the numerical solution of improved Boussinesq equation

A numerical simulation of the improved Boussinesq (IBq) equation is obtained using collocation and approximating the solution by radial basis functions (RBFs) based on the third-order time discretization. To avoid solving the nonlinear system, a predictor–corrector scheme is proposed and the Not-a-Knot method is used to improve the accuracy in the boundary. The method is tested on two problems taken from the literature: propagation of a solitary wave and interaction of two solitary waves. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the new scheme presented in this paper.     

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

A parallel numerical simulation algorithm is presented for fractional-order systems involving Caputo-type derivatives, based on the Adams–Bashforth–Moulton predictor–corrector scheme. The parallel algorithm is implemented using several different approaches: a pure MPI version, a combination of MPI with OpenMP optimization and a memory saving speedup approach. All tests run on a BlueGene/P cluster, and comparative improvement results for the running time are provided. As an applied experiment, the solutions of a fractional-order version of a system describing a forced series LCR circuit are numerically computed, depicting cascades of period-doubling bifurcations which lead to the onset of chaotic behavior.

     

A P-Stable Linear Multistep Method For Solving General Third Order Ordinary Differential Equations

A P-stable linear multistep method for solving general third order initial value problems of ordinary differential equations without first reducing the problems into a system of first order equations is considered. The approach for the development of this method is essentially based on collocation of the differential system generated from a basis function. A predictor for the evaluation of $y_{n+k}$ for an odd $k\ge 3$ in the main method is also proposed. The two resulting methods, the corrector and the predictor are P-stable for $k = 3$ . These as a block are tested on a number of problems to show their efficiency. When the methods (the corrector and the predictor) are evaluated at $x = x_{n+3}$ identical schemes are obtained as special cases of the methods, while the set of first and second derivatives obtained from the corrector are different from those obtained from the predictor.     

A modified predictor-corrector scheme for the Klein–Gordon equation

A numerical method based on a three-time level finite-difference scheme has been proposed for the solution of the two forms of the Klein–Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor-corrector scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector is modified by considering, as known, the already evaluated corrected values instead of the predictor ones. This modified scheme is applied to problems possessing periodic, kinks and soliton waves. The accuracy as well as the long-time behaviour of the proposed scheme is discussed and comparisons with the relevant known in the bibliography schemes are given.     

Using hybrid automata for set-membership state estimation with uncertain nonlinear continuous-time systems

This paper deals with set-membership state estimation for continuous-time systems from discrete-time measurements, in the unknown but bounded error framework. The classical predictor–corrector approach to state estimation uses interval Taylor methods for solving the prediction phase, which are known to have poor performance in presence of large model or input uncertainty. In this paper, we show how to derive more efficient predictors by using a nonlinear hybridization method which builds hybrid automata to characterize the boundaries of reachable sets. The derived continuous–discrete set-membership predictor–corrector estimator is then tested with simulated data from a bioreactor. Our method is compared to classical continuous-time interval observers and is shown to have promising performance.     

Fast convergence pirkn-type pc methods with adams-type predictors ∗

This paper discusses predictor–corrector iteration schemes (PC iteration schemes) based on direct collocatio–based Runge–Kutt–Nyström corrector methods (RKN corrector methods) for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y′′(t) = f(y(t)) Our approach is to regard the well-known parallel-iterated RKN methods (PIRKN methods) as PC iteration processes in which the simple, low-order last step value predictors are replaced with the high-order Adams-type predictors. Moreover, the param-eters of the new direct collocation-based RKN corrector methods are chosen in such a way that the convergence rate of the considered PC iteration processes is optimized. In this way, we obtain parallel PC methods with fast convergence and high-accurate predictions. Application of the resulting parallel PC methods to a few widely-used test problems reveals that the sequential costs are very much reduced when compared with the parallel and sequential explicit RKN methods from the literature.     

Stochastic model predictive control with joint chance constraints

ABSTRACT

This article investigates model predictive control (MPC) of linear systems subject to arbitrary (possibly unbounded) stochastic disturbances. An MPC approach is presented to account for hard input constraints and joint state chance constraints in the presence of unbounded additive disturbances. The Cantelli–Chebyshev inequality is used in combination with risk allocation to obtain computationally tractable but accurate surrogates for the joint state chance constraints when only the mean and variance of the arbitrary disturbance distributions are known. An algorithm is presented for determining the optimal feedback gain and optimal risk allocation by iteratively solving a series of convex programs. The proposed stochastic MPC approach is demonstrated on a continuous acetone–butanol–ethanol fermentation process, which is used in the production of biofuels.     

Parallel block pc methods with rkn-type correctors and adams-type predictors ∗

This paper describes the construction of block predictor - corrector methods based on Runge-Kutta-Nyström correctors. Our approach is to apply the predictor - corrector method not only at step point, but also at off-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. In the next step, these approximations are used to obtain a high-accurate predictions using Adams-type formulas. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictions using last step values. Since the block of approximations at the off-step points can be computed in parallel, the sequential costs of these block predictor - corrector methods are comparable with those of a conventional predictor - corrector method. Furthermore, by using Runge-Kutta-Nyström corrector methods, the computation of the approximation at each off-step point is also highly parallel. Application of the resulting block predictor -corrector methods to a few widely-used test problems reveals that the sequential costs are very much reduced when compared with the best parallel and sequential methods from the literature.     

基于多模糊模型的非线性预测控制

研究了基于多模糊模型的非线性预测控制问题 ,提出了基于多模型融合的非线性预测控制方法 .首先根据实际对象在不同运行点附近的状态建立了非线性系统的线性多模糊模型表示 ,然后给出了基于多模糊模型的预测控制原理结构框图 .非线性多模糊模型被用来作为预测模型 ,CSTR过程的仿真研究表明是一种有前景的非线性预测控制方法 .     

Linearly implicit predictor–corrector methods for space-fractional reaction–diffusion equations with non-smooth initial data

In this paper, linearly implicit predictor–corrector methods are proposed for solving space-fractional reaction–diffusion equations with non-smooth initial data. The methods are based on Matrix Transfer Technique for spatial discretization and are shown to be unconditionally stable. It is observed that the linearly implicit predictor–corrector method derived by using (1,1)-Padé approximation to matrix exponential function incurs oscillatory behavior for some time steps. These oscillations are due to high frequency components present in the solution and are diminished as the order of the space-fractional derivative decreases (slow diffusion). We present a priori reliability constraint to avoid these unwanted oscillations and generalize the constraints for all $(m,m)$-Padé approximants, $m\in {\mathbb{Z}}^{+}$, to the matrix exponential functions. These time stepping constraints are seen to be dependent on the order of the space-fractional derivative. The linearly implicit predictor–corrector method based on the (0,2)-Padé approximations to the matrix exponential function is shown to be oscillation-free for any time step. Error estimates are obtained for the methods and are theoretically shown to be second-order convergent. Computational complexity of the algorithms is discussed for solving multidimensional space-fractional reaction–diffusion systems. Several numerical experiments are performed to support our theoretical observations and to show the effectiveness, reliability, and efficiency of the methods.     

Input‐to‐state stability of min‐max MPC scheme for nonlinear time‐varying delay systems

This paper studies the robustness problem of the min–max model predictive control (MPC) scheme for constrained nonlinear time‐varying delay systems subject to bounded disturbances. The notion of the input‐to‐state stability (ISS) of nonlinear time‐delay systems is introduced. Then by using the Lyapunov–Krasovskii method, a delay‐dependent sufficient condition is derived to guarantee input‐to‐state practical stability (ISpS) of the closed‐loop system by way of nonlinear matrix inequalities (NLMI). In order to lessen the online computational demand, the non‐convex min‐max optimization problem is then converted to a minimization problem with linear matrix inequality (LMI) constraints and a suboptimal MPC algorithm is provided. Finally, an example of a truck‐trailer is used to illustrate the effectiveness of the proposed results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A data‐driven bilinear predictive controller design based on subspace method

In this paper, a new data‐driven model predictive control (MPC), based on bilinear subspace identification, is considered. The system's nonlinear behavior is described with a bilinear subspace predictor structure in an MPC framework. Thus, the MPC formulation results in a fixed structure objective function with constraints regardless of the underlying nonlinearity. For unconstrained systems, the identified subspace predictor matrices can be directly used as controller parameters. Therefore, we design optimization algorithms that exploit this feature. The open‐loop optimization problem of MPC that is nonlinear in nature is solved with series quadratic programming (SQP) without any approximations. The computational efficiency already demonstrated with the current formulation presents further opportunities to enable online control of nonlinear systems. These improvements and close integration of modeling and control also eliminate the intermediate design step, which provides a means for data‐driven controller design in generalized predictive controller (GPC) framework. Finally, the proposed control approach is illustrated with a verification study of a nonlinear continuously stirred tank reactor (CSTR) system. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society