Computationally efficient simultaneous policy update algorithm for nonlinear H∞ state feedback control with Galerkin's method

The main bottleneck for the application of H_∞ control theory on practical nonlinear systems is the need to solve the Hamilton–Jacobi–Isaacs (HJI) equation. The HJI equation is a nonlinear partial differential equation (PDE) that has proven to be impossible to solve analytically, even the approximate solution is still difficult to obtain. In this paper, we propose a simultaneous policy update algorithm (SPUA), in which the nonlinear HJI equation is solved by iteratively solving a sequence of Lyapunov function equations that are linear PDEs. By constructing a fixed point equation, the convergence of the SPUA is established rigorously by proving that it is essentially a Newton's iteration method for finding the fixed point. Subsequently, a computationally efficient SPUA (CESPUA) based on Galerkin's method, is developed to solve Lyapunov function equations in each iterative step of SPUA. The CESPUA is simple for implementation because only one iterative loop is included. Through the simulation studies on three examples, the results demonstrate that the proposed CESPUA is valid and efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

The solution of the poisson-boltzmann equation between two spheres: Modified iterative method

We present a modification on the successive overrelaxation (SOR) method and the iteration of the Green's function integral representation for the solution of the (nonlinear) Poisson-Boltzmann equation between two spheres. In comparison with other attempts, which approximate the geometry or the nonlinearity, the computations here are done for the full problem and compared with those done by the finite element method as a typical method for such problems. For the parameters of general interest, while the SOR method does not work, and the iteration of the integral representation is limited in its convergence, our modification to these iterative schemes converge. The modified SOR surpasses both methods in simplicity and speed; it is about 100 times faster than the modified iteration of the integral representation, with the latter being still simpler and faster than the finite element method. These two examples further illustrate the advantage of our recent modification to iterative methods, which is based on an analytical fixed point argument.     

非线性重复运动系统的双迭代优化学习控制

本文针对一类具有非参数不确定性和输出约束的非线性系统,提出一种双迭代优化学习控制策略,将复杂的迭代学习过程简化为两个相对简单的迭代控制器.首先引入一类饱和非线性函数不仅可以满足系统的位置约束,同时能够保证系统跟踪误差收敛于给定的邻域,然后针对每次迭代初始误差设计参考轨迹自修正策略,在每个迭代周期上设置一个固定的调整时间域,根据上次迭代的输出调整下一次迭代的参考轨迹.双迭代的控制结构可以同时更新两个迭代控制器的参数,来处理系统的非参数不确定性.进一步利用Barrier复合能量函数证明双迭代控制策略的收敛性和稳定性,并给出收敛条件.最后,通过一个算例证明了该控制策略的有效性.     

The modulus-based nonsmooth Newton’s method for solving a class of nonlinear complementarity problems of <Emphasis Type="Italic">P</Emphasis>-matrices

By applying the Newton’s iteration to the equivalent modulus equations of the nonlinear complementarity problems of P-matrices, a modulus-based nonsmooth Newton’s method is established. The nearly quadratic convergence of the new method is proved under some assumptions. The strategy of choosing the initial iteration vector is given, which leads to a modified method. Numerical examples show that the new methods have higher convergence precision and faster convergence rate than the known modulus-based matrix splitting iteration method.     

Supercloseness of Continuous Interior Penalty Methods on Shishkin Triangular Meshes and Hybrid Meshes for Singularly Perturbed Problems with Characteristic Layers

A fast preconditioned penalty method is developed for a system of parabolic linear complementarity problems (LCPs) involving tempered fractional order partial derivatives governing the price of American options whose underlying asset follows a geometry Lévy process with multi-state regime switching. By means of the penalty method, the system of LCPs is approximated with a penalty term by a system of nonlinear tempered fractional partial differential equations (TFPDEs) coupled by a finite-state Markov chain. The system of nonlinear TFPDEs is discretized with the shifted Grünwald approximation by an upwind finite difference scheme which is shown to be unconditionally stable. Semi-smooth Newton’s method is utilized to solve the finite difference scheme as an outer iterative method in which the Jacobi matrix is found to possess Toeplitz-plus-diagonal structure. Consequently, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis. With the above-mentioned preconditioning technique via FFT, under some mild conditions, the operation cost in each Newton’s step can be expected to be \(\mathcal{O}(N\mathrm{log}N)\), where N is the size of the coefficient matrix. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed fast preconditioned penalty method.     

On the solution of a rational matrix equation arising in G-networks

We consider the problem of solving a rational matrix equation arising in the solution of G-networks. We propose and analyze two numerical methods: a fixed point iteration and the Newton–Raphson method. The fixed point iteration is shown to be globally convergent with linear convergence rate, while the Newton method is shown to have a local convergence, with quadratic convergence rate. Numerical experiments show the effectiveness of the proposed methods.     

Formation control for a class of nonlinear multiagent systems using model‐free adaptive iterative learning

In this paper, the problem of formation control is considered for a class of unknown nonaffine nonlinear multiagent systems under a repeatable operation environment. To achieve the formation objective, the unknown nonlinear agent's dynamic is first transformed into a compact form dynamic linearization model along the iteration axis. Then, a distributed model‐free adaptive iterative learning control scheme is designed to ensure that all agents can keep their desired deviations from the reference trajectory over the whole time interval. The main results are given for the multiagent systems with fixed communication topologies and the extension to the switching topologies case is also discussed. The feature of this design is that formation control can be solved only depending on the input/output data of each agent. An example is given to demonstrate the effectiveness of the proposed method.     

On options for interdisciplinary analysis and design optimization

The interdisciplinary optimization of engineering systems is discussed from the standpoint of the computational alternatives available to the designer. The analysis of such systems typically requires the solution of coupled systems of nonlinear algebraic equations. The solution procedure is necessarily iterative in nature. It is shown that the system can be solved by fixed point iteration, by Newton's method, or by a combination of the two. However, the need for sensitivity analysis may affect the choice of analysis solution method. Similarly, the optimization of the system can be formulated in several ways that are discussed in the paper. It is shown that the effect of the topology of the interaction between disciplines is a key factor in the choice of analysis, sensitivity and optimization methods. Several examples are presented to illustrate the discussion.     

Hierarchical Newton and least squares iterative estimation algorithm for dynamic systems by transfer functions based on the impulse responses

This paper develops a parameter estimation algorithm for linear continuous-time systems based on the hierarchical principle and the parameter decomposition strategy. Although the linear continuous-time system is a linear system, its output response is a highly nonlinear function with respect to the system parameters. In order to propose a direct estimation algorithm, a criterion function is constructed between the response output and the observation output by means of the discrete sampled data. Then a scheme by combining the Newton iteration and the least squares iteration is builded to minimise the criterion function and derive the parameter estimation algorithm. In light of the different features between the system parameters and the output function, two sub-algorithms are derived by using the parameter decomposition. In order to remove the associate terms between the two sub-algorithms, a Newton and least squares iterative algorithm is deduced to identify system parameters. Compared with the Newton iterative estimation algorithm without the parameter decomposition, the complexity of the hierarchical Newton and least squares iterative estimation algorithm is reduced because the dimension of the Hessian matrix is lessened after the parameter decomposition. The experimental results show that the proposed algorithm has good performance.     

On the computer implementation of feasible direction interior point algorithms for nonlinear optimization

The paper discusses the computer implementation of a class of interior point algorithms for the minimization of nonlinear functions with equality and inequality constraints. These algorithms consist of fixed point iterations to solve KKT firstorder optimality conditions. At each iteration a descent direction is defined by solving a linear system. Then, the linear system is perturbed in such a way as to deflect the descent direction and obtain a feasible descent direction. A line search is finally performed to obtain a new interior point with a lower objective. Newton, quasi-Newton, or first-order versions of the algorithm can be obtained. This paper is mainly concerned with the solution of the internal linear systems, the algorithms that are employed for the constrained line search and also with the quasi-Newton matrix updating. Some numerical results obtained with a quasi Newton algorithm are also presented. A set of test problems were solved very efficiently with the same values of the internal parameters.     

Two-Level Newton’s Method for Nonlinear Elliptic PDEs

A combination method of Newton’s method and two-level piecewise linear finite element algorithm is applied for solving second-order nonlinear elliptic partial differential equations numerically. Newton’s method is to find a finite element solution by solving $m$ Newton equations on a fine mesh. The two-level Newton’s method solves $m-1$ Newton equations on a coarse mesh and processes one Newton iteration on a fine mesh. Moreover, the optimal error estimates of Newton’s method and the two-level Newton’s method are provided to justify the efficiency of the two-level Newton’s method. If we choose $H$ such that $h=O(|\log h|^{1-2/{p}}H^2)$ for the $W^{1,p}(\Omega )$ -error estimates, the two-level Newton’s method is asymptotically as accurate as Newton’s method on the fine mesh. Meanwhile, the numerical investigations provided a sufficient support for the theoretical analysis. Finally, these investigations also proved that the proposed method is efficient for solving the nonlinear elliptic problems.     

Fixed point iteration in identifying bilinear models

Inspired by fixed point theory, an iterative algorithm is proposed to identify bilinear models recursively in this paper. It is shown that the resulting iteration is a contraction mapping on a metric space when the number of input–output data points approaches infinity. This ensures the existence and uniqueness of a fixed point of the iterated function sequence and therefore the convergence of the iteration. As an application, one class of block-oriented systems represented by a cascade of a dynamic linear (L), a static nonlinear (N) and a dynamic linear (L) subsystems is illustrated. This gives a solution to the long-standing convergence problem of iteratively identifying LNL (Winer–Hammerstein) models. In addition, we extend the static nonlinear function (N) to a nonparametric model represented by using kernel machine.     

Optimal Newton–Secant like methods without memory for solving nonlinear equations with its dynamics

We construct two optimal Newton–Secant like iterative methods for solving nonlinear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.     

A generalisation of the interval Newton single-step method for nonlinear systems of equations

In this paper interval iteration methods for solving large nonlinear systems of equations are considered. Already well-known methods are combined to a new one, whose enclosing properties are better than those of previous methods. Convergence of this new method is shown, based on a new convergence proof for the interval Newton single-step method. The central concept in this case is the fixpoint inverse of an interval matrix. Practical tests with nonlinear systems of equations arising from discretisation of certain elliptic partial differential equations show the efficiency of the new method.     

A novel optimal tracking control scheme for a class of discrete-time nonlinear systems using generalised policy iteration adaptive dynamic programming algorithm

In this paper, a novel iterative adaptive dynamic programming (ADP) algorithm, called generalised policy iteration ADP algorithm, is developed to solve optimal tracking control problems for discrete-time nonlinear systems. The idea is to use two iteration procedures, including an i-iteration and a j-iteration, to obtain the iterative tracking control laws and the iterative value functions. By system transformation, we first convert the optimal tracking control problem into an optimal regulation problem. Then the generalised policy iteration ADP algorithm, which is a general idea of interacting policy and value iteration algorithms, is introduced to deal with the optimal regulation problem. The convergence and optimality properties of the generalised policy iteration algorithm are analysed. Three neural networks are used to implement the developed algorithm. Finally, simulation examples are given to illustrate the performance of the present algorithm.     

求解非线性方程的蛛网-迭代算法

用蛛网迭代算法求解非线性方程,只要求函数在定义域内存在反函数;由定理及其证明可知,不动点迭代是该迭代方法的特殊情况;通过数值实验进一步证明了该方法的有效性和实用性。     

On a nonlinear iterative method in applied mechanics, part II

In Part I of this paper, a particular iterative method used in applied mechanics is shown to belong to a general class of methods termed SOR-Newton-(m_k) iterative methods. The purpose here is to make a case study of the method and compare its performance with Newton iteration and Newton-SOR iteration. The numerical experiments are designed to examine the range of convergence of the m_k = 1 step method, rates of convergence, and the effect of relaxation.     

Policy Iteration for Optimal Control of Discrete-Time Time-Varying Nonlinear Systems

Aimed at infinite horizon optimal control problems of discrete time-varying nonlinear systems, in this paper, a new iterative adaptive dynamic programming algorithm, which is the discrete-time time-varying policy iteration (DTTV) algorithm, is developed. The iterative control law is designed to update the iterative value function which approximates the index function of optimal performance. The admissibility of the iterative control law is analyzed. The results show that the iterative value function is non-increasingly convergent to the Bellman-equation optimal solution. To implement the algorithm, neural networks are employed and a new implementation structure is established, which avoids solving the generalized Bellman equation in each iteration. Finally, the optimal control laws for torsional pendulum and inverted pendulum systems are obtained by using the DTTV policy iteration algorithm, where the mass and pendulum bar length are permitted to be time-varying parameters. The effectiveness of the developed method is illustrated by numerical results and comparisons.      

Variable Schrittweitensteuerungen für die Homotopiemethode bei adäquaten nichtlinearen Ausgleichsproblemen

For adequate models of nonlinear regression the homotopy method is shown to enlarge the region of convergence of the ordinary Gauss-Newton-Method theoretically as well as practically. Two methods to control the stepsize are proposed: the first one is in analogy to the empirically based ?doubling-/bisecting-” damping strategy for the relaxed Newton Method and the other one is based upon more local information of the function. Both methods are shown to be effective with practical problems and improve the old method of a constant stepsize; in addition the second method can follow the homotopy path and can even detect it's singularities. In case of a system of nonlinear equations both methods reduce to a stepsize control for the embedded Newton method.     

SINGULARITY COMPUTATION FOR ITERATIVE CONTROL OF NONLINEAR AFFINE SYSTEMS

This paper considers a gradient type of iterative algorithm applied to the open loop control for nonlinear affine systems. The convergence of the algorithm relies on the control signal in each iteration be nonsingular. We present an algorithm for computing the singular control for a general class of nonlinear affine systems. Various nonlinear mechanical systems, including nonholonomic systems, are included as examples.