Properties of the reachable set of control systems

We investigate some properties of the reachable set of a control system. Representing the system as a differential inclusion and using proximal Hamilton–Jacobi equation we describe its graph. We work in infinitely dimensional Hilbert space and use one sided Lipschitz approach. The funnel equation is considered in the last section. That equation describes the reachable set in arbitrary Banach space. We consider also the autonomous case and prove the existence of a limit of the reachable set.     

Case studies on the application of the stable manifold approach for nonlinear optimal control design

This paper presents application results of a recently developed method for approximately solving the Hamilton–Jacobi equation in nonlinear control theory. The method is based on stable manifold theory and consists of a successive approximation algorithm which is suitable for computer calculations. Numerical approach for this algorithm is advantageous in that the computational complexity does not increase with respect to the accuracy of approximation and non-analytic nonlinearities such as saturation can be handled. First, the stable manifold approach for approximately solving the Hamilton–Jacobi equation is reviewed from the computational viewpoint and next, the detailed applications are reported for the problems such as swing up and stabilization of a 2-dimensional inverted pendulum (simulation), stabilization of systems with input saturation (simulation) and a (sub)optimal servo system design for magnetic levitation system (experiment).     

Optimal control using an algebraic method for control-affine non-linear systems

A deterministic optimal control problem is solved for a control-affine non-linear system with a non-quadratic cost function. We algebraically solve the Hamilton–Jacobi equation for the gradient of the value function. This eliminates the need to explicitly solve the solution of a Hamilton–Jacobi partial differential equation. We interpret the value function in terms of the control Lyapunov function. Then we provide the stabilizing controller and the stability margins. Furthermore, we derive an optimal controller for a control-affine non-linear system using the state dependent Riccati equation (SDRE) method; this method gives a similar optimal controller as the controller from the algebraic method. We also find the optimal controller when the cost function is the exponential-of-integral case, which is known as risk-sensitive (RS) control. Finally, we show that SDRE and RS methods give equivalent optimal controllers for non-linear deterministic systems. Examples demonstrate the proposed methods.     

Spatially adaptive techniques for level set methods and incompressible flow

Since the seminal work of [Sussman, M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 1994;114:146–59] on coupling the level set method of [Osher S, Sethian J. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 1988;79:12–49] to the equations for two-phase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 1994;114:146–59] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both of its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as Hamilton–Jacobi WENO [Jiang G-S, Peng D. Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J Sci Comput 2000;21:2126–43], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [Enright D, Fedkiw R, Ferziger J, Mitchell I. A hybrid particle level set method for improved interface capturing. J Comput Phys 2002;183:83–116] and the coupled level set volume of fluid method [Sussman M, Puckett EG. A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J Comput Phys 2000;162:301–37], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [Losasso F, Gibou F, Fedkiw R. Simulating water and smoke with an octree data structure, ACM Trans Graph (SIGGRAPH Proc) 2004;23:457–62].     

ISE-optimal nonminimum-phase compensation for nonlinear processes

This work concerns the optimal regulation of single-input–single-output nonminimum-phase nonlinear processes. The problem of calculation of an ISE-optimal, statically equivalent, minimum-phase output for nonminimum-phase compensation is formulated using Hamilton–Jacobi theory and the normal form representation of the nonlinear system. A Newton–Kantorovich iteration is developed for the solution of the pertinent Hamilton–Jacobi equations, which involves solving a Zubov equation at each step of the iteration. The method is applied to the problem of controlling a nonisothermal CSTR with Van de Vusse kinetics, which exhibits nonminimum-phase behaviour.     

On optimal predefined‐time stabilization

This paper addresses the problem of optimal predefined‐time stability. Predefined‐time stable systems are a class of fixed‐time stable dynamical systems for which the minimum bound of the settling‐time function can be defined a priori as an explicit parameter of the system. Sufficient conditions for a controller to solve the optimal predefined‐time stabilization problem for a given nonlinear system are provided. These conditions involve a Lyapunov function that satisfies a certain differential inequality for guaranteeing predefined‐time stability. It also satisfies the steady‐state Hamilton–Jacobi–Bellman equation for ensuring optimality. Furthermore, for nonlinear affine systems and a certain class of performance index, a family of optimal predefined‐time stabilizing controllers is derived. This class of controllers is applied to optimize the sliding manifold reaching phase in predefined time, considering both the unperturbed and perturbed cases. For the perturbed case, the idea of integral sliding mode control is jointly used to ensure robustness. Finally, as a study case, the predefined‐time optimization of the sliding manifold reaching phase in a pendulum system is performed using the developed methods, and numerical simulations are carried out to show their behavior. Copyright © 2017 John Wiley & Sons, Ltd.     

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagin's maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.     

Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density

In the present paper, an innovative procedure for designing the feedback control of multi-degree-of-freedom (MDOF) nonlinear stochastic systems to target a specified stationary probability density function (SPDF) is proposed based on the technique for obtaining the exact stationary solutions of the dissipated Hamiltonian systems. First, the control problem is formulated as a controlled, dissipated Hamiltonian system together with a target SPDF. Then the controlled forces are split into a conservative part and a dissipative part. The conservative control forces are designed to make the controlled system and the target SPDF have the same Hamiltonian structure (mainly the integrability and resonance). The dissipative control forces are determined so that the target SPDF is the exact stationary solution of the controlled system. Five cases, i.e., non-integrable Hamiltonian systems, integrable and non-resonant Hamiltonian systems, integrable and resonant Hamiltonian systems, partially integrable and non-resonant Hamiltonian systems, and partially integrable and resonant Hamiltonian systems, are treated respectively. A method for proving that the transient solution of the controlled system approaches the target SPDF as t→∞ is introduced. Finally, an example is given to illustrate the efficacy of the proposed design procedure.     

Viscosity solutions and optimal control problems with integral constraints

We discuss optimal control problems with integral state-control constraints. We rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system, and prove that the value function, although possibly discontinuous, is the unique viscosity solution of the constrained boundary value problem for the corresponding Hamilton–Jacobi equation. The state constraint is the epigraph of the minimal solution of a second Hamilton–Jacobi equation. Our framework applies, for instance, to systems with design uncertainties.     

一类混沌系统的非线性反馈控制

基于微分动力学的不稳定流形定理,针对二维离散混沌动力系统,用不稳定流形的了函数逼;近系统。采取微扰控制方式,得出非线性反馈控制规律,稳定双曲平衡点。与ＯＧＹ方法相比,增大了控制收敛区域,减少了迭代次数,并以Ｈｅｎｏｎ映射为例验证了所提出方法的有效性。     

一类强非线性二阶微分方程的多模态近似解析解研究

利用自治力学系统的哈密顿函数为守恒量的性质,提出一种求非线性二阶微分方程多模态近似解析解的方法,称为哈密顿函数法.首先,介绍哈密顿函数法求多模态近似解的基本理论.其次,以质点在旋转的抛物线上运动为模型建立强非线性二阶微分方程.最后,用哈密顿函数法求得在给定初始条件和参数下强非线性二阶微分方程的三模态近似解析解表达式,作出三模态近似解析解的解曲线,并与直接用Mathematica软件作出的解曲线进行比较,讨论三模态近似解析解的精确性.结果表明：用哈密顿函数法求得的三模态近似解析解的解曲线与直接用Mathematica软件作出的解曲线十分吻合.     

Solutions of the Cahn–Hilliard equation

In this paper, we found some exact solutions of the Cahn–Hilliard equation and the system of the equations by considering a modified extended tanh function method. A numerical solution to a Cahn–Hilliard equation is obtained using a homotopy perturbation method (HPM) combined with the Adomian decomposition method (ADM). The comparisons are given in the tables.     

H∞-norm and invariant manifolds of systems with state delays

The problem of finding bounds on the H_∞-norm of systems with a finite number of point delays and distributed delay is considered. Sufficient conditions for the system to possess an H_∞-norm which is less or equal to a prescribed bound are obtained in terms of Riccati partial differential equations (RPDE’s). We show that the existence of a solution to the RPDE’s is equivalent to the existence of a stable manifold of the associated Hamiltonian system. For small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary differential equations that govern the flow on the slow manifold of the Hamiltonian system. This leads to an algebraic, finite-dimensional, criterion for systems with small delays.     

周期时变不确定性线性系统的MINIMAX控制方法

应用约束最优化方法和微分对策理论,讨论周期时变不确定性线性系统在范数有界外 部干扰情况下的MINIMAX控制和参数摄动情况下的MINIMAX控制.问题可解的充分条件 是一类Riccati微分方程具有稳定化解,且关于最坏扰动的某个附加条件满足相应的MINIMAX 控制恰为一个线性状态反馈.此外,还给出了闭环系统的性能指标的保证值.     

Bifurcation analysis in a discrete-time single-directional network with delays

In this paper, we consider a simple discrete-time single-directional network of four neurons. The characteristics equation of the linearized system at the zero solution is a polynomial equation involving very high-order terms. We first derive some sufficient and necessary conditions ensuring that all the characteristic roots have modulus less than 1. Hence, the zero solution of the model is asymptotically stable. Then, we study the existence of three types of bifurcations, such as fold bifurcations, flip bifurcations, and Neimark–Sacker (NS) bifurcations. Based on the normal form theory and the center manifold theorem, we discuss their bifurcation directions and the stability of bifurcated solutions. In addition, several codimension two bifurcations can be met in the system when curves of codimension one bifurcations intersect or meet tangentially. We proceed through listing smooth normal forms for all the possible codimension 2 bifurcations.     

Symbolic methods to construct exact solutions of nonlinear partial differential equations

Two straightforward methods for finding solitary-wave and soliton solutions are presented and applied to a variety of nonlinear partial differential equations. The first method is a simplied version of Hirota's method. It is shown to be an effective tool to explicitly construct. multi-soliton solutions of completely integrable evolution equations of fifth-order, including the Kaup-Kupershmidt equation for which the soliton solutions were not previously known. The second technique is the truncated Painlevé expansion method or singular manifold method. It is used to find closed-form solitary-wave solutions of the Fitzhugh-Nagumo equation with convection term, and an evolution equation due to Calogero. Since both methods are algorithmic, they can be implemented in the language of any symbolic manipulation program.     

Closed-loop balancing for a class of non-linear singularly perturbed systems

The aim of this article is to investigate the closed-loop balancing reduction method for a class of non-linear singularly perturbed systems. We show that the well-known two-stage strategy involved commonly within the singular perturbation theory can be used to derive an approximate closed-loop balancing. The proposed two-stage method avoids the difficult task of solving high dimensional and ill conditioned non-linear Hamilton–Jacobi equation due to the presence of the small perturbation parameter.     

A unification between nonlinear-nonquadratic optimal control and integrator backstepping

In this paper we develop an optimality-based framework for backstepping controllers. Specifically, using a nonlinear-nonquadratic optimal control framework we develop a family of globally stabilizing backstepping controllers parametrized by the cost functional that is minimized. Furthermore, it is shown that the control Lyapunov function guaranteeing closed-loop stability is a solution to the steady-state Hamilton–Jacobi–Bellman equation for the controlled system and thus guarantees both optimality and stability. The results are specialized to the cases of integrator backstepping and block backstepping for cascade systems with linear and nonlinear input subsystems. © 1998 John Wiley & Sons, Ltd.     

An imbedded initialization of Newton's algorithm for matrix Riccati equation

An effective numerical computation of the steady-state Riccati matrix is based on the successive solutions of a Lyapunov equation using Newton's method. The requirements of this algorithm are an initial stabilizing matrix and the numerical solution of the associated Lyapunov equation. Computationally, the first requirement is the more influencing factor in solving the Riccati equation with reasonable accuracy and speed. In this paper an initial matrix, based on the parameter imbedded solution of the Riccati equation, is introduced for the Newton's algorithm. The imbedding Newton algorithm has been applied to a variety of system, both stable and unstable as well as high-dimensional, A matrices, one of which is reported here. The proposed modification has improved the required CPU time of previous initialization schemes by as much as a factor of 6 times for the same order of accuracy.     

一般阻尼动力系统非奇异摄动法

借助矩阵摄动理论,将模态叠加法运用于一般阻尼矩阵的动力学方程求解结构的动响应是一种较为理想的方法.但当系统的外荷载激振频率接近于系统的固有频率时,直接将阻尼矩阵作为摄动矩阵,会使解产生奇异,并导致求解失败或误差过大,这是因为模态坐标下的动力学方程是无阻尼方程.为了解决这一问题,本文考虑在模态坐标的动力学方程中保留一定的阻尼.即将阻尼做分解,代入振动方程,得到不同阶次摄动方程,再将摄动方程变换到模态坐标,即采用非奇异摄动方法.最后通过数值算例,得到一阶、二阶摄动,将其与精确解进行比较.精度明显得到改善,基本趋于精确解.从而验证了本方法的精确性和有效性.