Design of nonlinear regulators for linear plants

A procedure for designing stable nonlinear control laws for linear plants is presented. The design is based on an inverse optimal control problem with the closed form controller being directly given by the solution of a set of linear algebraic equations. The nonlinear controllers can be used to produce saturation effects on particular states, and an example is included to illustrate the results.     

Discrete-time optimal control for stochastic nonlinear polynomial systems

This paper presents a solution to the discrete-time optimal control problem for stochastic nonlinear polynomial systems over linear observations and a quadratic criterion. The solution is obtained in two steps: the optimal control algorithm is developed for nonlinear polynomial systems by considering complete information when generating a control law. Then, the state estimate equations for discrete-time stochastic nonlinear polynomial system over linear observations are employed. The closed-form solution is finally obtained substituting the state estimates into the obtained control law. The designed optimal control algorithm can be applied to both distributed and lumped systems. To show effectiveness of the proposed controller, an illustrative example is presented for a second degree polynomial system. The obtained results are compared to the optimal control for the linearized system.     

Optimal control of nonlinear power systems by an imbedding method

Realistic models of power systems involve high-order non-linear differential equations. The application of optimal control theory in optimizing such systems involves a high-order nonlinear 2-point boundary-value problem whose solution is very cumbersome. In this paper the imbedding method of solving a free-end, fixed-time optimal control problem is applied to a sixth-order nonlinear power system. The optimal designs for both open and closed loop plants are presented. The results of this application are compared with those by the hybrid linearization method considered by other authors. The results indicate that the 2 methods follow quite closely.     

Neural networks for feedback feedforward nonlinear control systems

This paper deals with the problem of designing feedback feedforward control strategies to drive the state of a dynamic system (in general, nonlinear) so as to track any desired trajectory joining the points of given compact sets, while minimizing a certain cost function (in general, nonquadratic). Due to the generality of the problem, conventional methods are difficult to apply. Thus, an approximate solution is sought by constraining control strategies to take on the structure of multilayer feedforward neural networks. After discussing the approximation properties of neural control strategies, a particular neural architecture is presented, which is based on what has been called the "linear-structure preserving principle". The original functional problem is then reduced to a nonlinear programming one, and backpropagation is applied to derive the optimal values of the synaptic weights. Recursive equations to compute the gradient components are presented, which generalize the classical adjoint system equations of N-stage optimal control theory. Simulation results related to nonlinear nonquadratic problems show the effectiveness of the proposed method.     

Construction of optimal feedback control for nonlinear systems via Chebyshev polynomials

A method is proposed to determine the optimal feedback control law of a class of nonlinear optimal control problems. The method is based on two steps. The first step is to determine the open-hop optimal control and trajectories, by using the quasilinearization and the state variables parametrization via Chebyshev polynomials of the first type. Therefore the nonlinear optimal control problem is replaced by a sequence of small quadratic programming problems which can easily be solved. The second step is to use the results of the last quasilinearization iteration, when an acceptable convergence error is achieved, to obtain the optimal feedback control law. To this end, the matrix Riccati equation and another n linear differential equations are solved using the Chebyshev polynomials of the first type. Moreover, the differentiation operational matrix of Chebyshev polynomials is introduced. To show the effectiveness of the proposed method, the simulation results of a nonlinear optimal control problem are shown.     

逐点线性化后退区间最优控制在飞机非线性控制中的应用

常规线性飞控系统针对推力矢量飞机这样的多控制冗余、非线性MIMO系统,无法实现非线性控制．本文针对推力矢量飞机非线性系统,阐述了一种逐点线性化后退区间最优控制算法满足飞行品质要求．首先将作动器,飞行品质和逐点线性化的飞机线性模型综合实现在线建模,然后以飞行状态与预测状态之间的误差、作动器的位置限制和速率限制作为最优指标,最后以此为基础,根据最优控制原理计算当前时刻飞机最优控制指令,实现飞机非线性控制．采用国内某型号飞机气动数据验证此算法的鲁棒性和稳定性．     

Galerkin approximationwith Legendre polynomials for a continuous-time nonlinear optimal control problem

We investigate the use of an approximation method for obtaining near-optimal solutions to a kind of nonlinear continuous-time (CT) system. The approach derived from the Galerkin approximation is used to solve the generalized Hamilton-Jacobi-Bellman (GHJB) equations. The Galerkin approximation with Legendre polynomials (GALP) for GHJB equations has not been applied to nonlinear CT systems. The proposed GALP method solves the GHJB equations in CT systems on some well-defined region of attraction. The integrals that need to be computed are much fewer due to the orthogonal properties of Legendre polynomials, which is a significant advantage of this approach. The stabilization and convergence properties with regard to the iterative variable have been proved. Numerical examples show that the update control laws converge to the optimal control for nonlinear CT systems.     

State-constrained optimal control problems governed by coupled nonlinear wave equations with memory

This work considers the optimal control problems governed by coupled nonlinear wave equations with memory, where the state constraint is a type of integral. First, we investigate the existence of the optimal solution for the optimal control problem, and then we deduce the maximum principle for the optimal solution with state constraint. Moreover, we give a concrete example which is covered by the main result.     

Neighbouring extremals for nonlinear systems with control constraints

In this paper, we consider a class of nonlinear optimal control problems subject to control constraints. We assume that an initial condition for the dynamical system is specified. Then, we can easily compute an openloop optimal control using any convenient optimal control software package. Now, suppose the optimal trajectory is perturbed due to a change in initial conditions or uncertainty in the model equations. If the perturbations are not too large, it is known that the neighbouring extremal approach can be used to obtain a feedback law which adjusts the open-loop optimal control accordingly provided there are no bounds on the control variables. In this paper, our aim is to develop a computational method for the more general case of fixed-time problems with control constraints.     

System equivalence in a class of nonlinear optimal control problems

In a family of nonlinear optimal control problems, equivalence classes of problems that yield identical optimal closed-loop systems are delineated. The equivalence relations are established as one-to-one correspondences between the solutions of the Hamilton-Jacobi equations associated with the problems. Two such equivalence relations are presented. These allow an arbitrary problem in the family to be reduced to an equivalent basic standard form of problem. Also, a new formal procedure for solving this basic problem is given that allows the determination of coefficients in the series expansion of the optimal closed-loop system. The set of algebraic equations to be solved is considerably simpler than that derived by other methods.     

一类非线性奇异摄动系统的近似最优控制

针对一类非线性奇异摄动系统,基于自适应动态规划算法提出了一种新型的近似最优控制设计方法.该方法基于奇异摄动系统的快、慢Hamilton-Jacobi-Bellman(HJB)方程,从初始性能指标开始,通过神经网络的近似和控制律与性能指标的逐步更新迭代,最终收敛到最优的性能指标,而不用直接求解复杂的HJB方程.同时给出了...     

非线性系统近似最优PD动态补偿控制

本文研究了一类基于动态补偿的非线性系统的近似最优PD控制的问题.用微分方程的逐次逼近理论将非线性系统的最优控制问题转化为求解线性非齐次两点边值序列问题,并提供了从时域最优状态反馈到频域最优PD控制器参数的优化方法,从而获取系统最优的动态补偿网络,设计出最优PD整定参数,给出其实现算法.最后仿真示例将所提出的方法与传统的线性二次型调节器（LQR）逐次逼近方法相比较,表明该方法具有良好的动态性能和鲁棒性.     

Numerical problems involved in finding optimal control strategies by nonlinear programming techniques

When solving optimal control problem by nonlinear programming algorithms, the main tasks are the computation of ordinary differential equations and of definite integrals. It is shown how to make a best use of the very performing routine available to do those computations. Generally, the nonlinear programming algorithm will give a satisfactory value of the cost but a poor approximation of the optimal control function. However that approximation provides an efficient start for a two points boundary value problem solver.     

非线性动力学系统最优控制问题的保辛求解方法

基于对偶变量变分原理提出了求解非线性动力学系统最优控制问题的一种保辛数值方法.以时间区段一端状态和另一端协态作为混合独立变量,在时间区段内采用拉格朗日插值近似状态变量与协态变量,然后利用对偶变量变分原理并将非线性最优控制问题转化为非线性方程组的求解,最终得到求解非线性动力学系统最优控制问题的保辛数值方法.数值实验验证了本文算法在求解精度与求解效率上的有效性.     

Suppression of nonlinear regenerative chatter in milling process via robust optimal control

During the milling process, self-excited vibration or chatter adversely affects tool life, surface quality and productivity rate. In this paper, nonlinear cutting forces of milling process are considered as a function of chip thickness with a complete third order polynomial (instead of the common linear dependency). An optimal control strategy is developed for chatter suppression of the system described through nonlinear delay differential equations. Counterbalance forces exerted by actuators in x and y directions are the control inputs. For optimal control problem, an appropriate performance index is defined such that the regenerative chatter is suppressed while control efforts are minimized. Optimal control law is determined based on variation of extremals algorithm. Results show that under unstable machining conditions, regenerative chatter is suppressed effectively after applying the optimal control strategy. In addition, optimal controller guarantees robust performance of the process in the presence of model parametric uncertainties.     

非线性离散系统的近似最优跟踪控制

研究非线性离散系统的最优跟踪控制问题. 通过在由最优控制问题所导致的非线性两点边值问题中引入灵敏度参数, 并对它进行Maclaurin级数展开, 将原最优跟踪控制问题转化为一族非齐次线性两点边值问题. 得到的最优跟踪控制由解析的前馈反馈项和级数形式的补偿项组成. 解析的前馈反馈项可以由求解一个Riccati差分方程和一个矩阵差分方程得到. 级数补偿项可以由一个求解伴随向量的迭代算法近似求得. 以连续槽式反应器为例进行仿真验证了该方法的有效性．     

A technique for estimating the state of a nonlinear system

A technique for the estimation of the state of a general class of nonlinear systems is discussed. The technique involves linear optimal estimation of the state of a linear approximation to the nonlinear system. The linear approximation to the nonlinear system is derived solely from the nonlinear differential equations and the noise-corrupted observations of some of the components of the state vector of the nonlinear system. An example of optimal estimation of the current position and velocity of a low earth-orbit satellite, using range, elevation and azimuth observations from a ground tracking station, is discussed. Numerical results from a digital simulation of the example are presented.     

The control of large-scale nonlinear econometric systems

After pointing out the special characteristics of existing econometric models which form the basis of the application of optimal control techniques for macroeconomic policy analysis and formulation, this paper surveys the use of econometric models for the projection of the economic consequences of alternative policies and the formulation of optimal policies. It presents both the theory and the computational aspects of a particular approach of applying feedback control to study policy options using large-scale nonlinear systems of simultaneous econometric equations. It then illustrates this approach by solving an optimal control problem using the Michigan Quarterly Econometric Model Some topics of further research are suggested in the conclusion.     

A Chebyshev technique for solving nonlinear optimal controlproblems

A numerical technique for solving nonlinear optimal control problems is introduced. The state and control variables are expanded in the Chebyshev series, and an algorithm is provided for approximating the system dynamics, boundary conditions, and performance index. Application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the Chebyshev coefficients. The optimum condition is obtained by applying the method of constrained extremum. For linear-quadratic optimal control problems, the state and control variables are determined by solving a set of linear equations in the Chebyshev coefficients. Applicability is illustrated with the minimum-time and maximum-radius orbit transfer problems     

Optimal quadrature formula nonlinear estimators

This paper presents a method for the realization of nonlinear estimators based on an optimal quadrature approximation. The optimal quadrature formula is obtained by solving a set of nonlinear algebraic equations induced from a monospline subject to a set of interpolatory conditions. All the weights of the optimal quadrature formulas derived from monosplines which do not involve the derivatives of the integrand at the end points are positive. This guarantees the positiveness of the quantized density functions in numerical approximation of Bayesian recursive computations. The numerical errors associated with the optimal quadrature approximation to Bayesian recursive computations are discussed. Finally methods of quantizing and updating the prediction and filtering densities are derived.