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1.
In this paper, we consider the use of nonlinear networks towards obtaining nearly optimal solutions to the control of nonlinear discrete-time (DT) systems. The method is based on least squares successive approximation solution of the generalized Hamilton-Jacobi-Bellman (GHJB) equation which appears in optimization problems. Successive approximation using the GHJB has not been applied for nonlinear DT systems. The proposed recursive method solves the GHJB equation in DT on a well-defined region of attraction. The definition of GHJB, pre-Hamiltonian function, HJB equation, and method of updating the control function for the affine nonlinear DT systems under small perturbation assumption are proposed. A neural network (NN) is used to approximate the GHJB solution. It is shown that the result is a closed-loop control based on an NN that has been tuned a priori in offline mode. Numerical examples show that, for the linear DT system, the updated control laws will converge to the optimal control, and for nonlinear DT systems, the updated control laws will converge to the suboptimal control.  相似文献   

2.
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.  相似文献   

3.
In this paper, we consider the use of nonlinear networks towards obtaining nearly optimal solutions to the control of nonlinear discrete-time (DT) systems. The method is based on least squares successive approximation solution of the generalized Hamilton-Jacobi-Bellman (GHJB) equation which appears in optimization problems. Successive approximation using the GHJB has not been applied for nonlinear DT systems. The proposed recursive method solves the GHJB equation in DT on a well-defined region of attraction. The definition of GHJB, pre-Hamiltonian function, HJB equation, and method of updating the control function for the affine nonlinear DT systems under small perturbation assumption are proposed. A neural network (NN) is used to approximate the GHJB solution. It is shown that the result is a closed-loop control based on an NN that has been tuned a priori in offline mode. Numerical examples show that, for the linear DT system, the updated control laws will converge to the optimal control, and for nonlinear DT systems, the updated control laws will converge to the suboptimal control.  相似文献   

4.
In this paper, a new iterative method is proposed to solve the generalized Hamilton-Jacobi-Bellman (GHJB) equation through successively approximate it. Firstly, the GHJB equation is converted to an algebraic equation with the vector norm, which is essentially a set of simultaneous nonlinear equations in the case of dynamic systems. Then, the proposed algorithm solves GHJB equation numerically for points near the origin by considering the linearization of the non-linear equations under a good initial control guess. Finally, the procedure is proved to converge to the optimal stabilizing solution with respect to the iteration variable. In addition, it is shown that the result is a closed-loop control based on this iterative approach. Illustrative examples show that the update control laws will converge to optimal control for nonlinear systems.   相似文献   

5.
近似动态规划方法求解非线性系统最优控制, 需要迭代无限步才能得到最优控制律. 本文提出了一种ε–近似最优控制算法, 选择ε误差限, 通过自适应迭代不断逼近哈密顿– 雅可比– 贝尔曼(HJB)方程的解, 应用神经网络实现在有限步迭代后得到带ε误差限的近似最优控制律. 计算机仿真结果表明了该算法的有效性.  相似文献   

6.
A finite-dimensional asymptotic observer is derived on the basis of a Calcrkin approximation for a class of distributed parameter systems. The systems are described by a partial differential equation of parabolic type. The measured outputs are assumed to be obtained through a finite number of sensors located in the interior. The sensor influence functions are added to the usual basis for the Galerkin approximation and Schmidt's orthogonalization is performed to yield a new basis. The Galerkin approximate solution is sought in terms of this basis. By this procedure the observation spillover problem is overcome. Moreover, in view of the fact that it is generally difficult to obtain the closed-form expressions for the eigenfunctions of the equation, the method is useful for practical substantiation of the observer. The uniform convergence of the Galerkin approximate sequence for the partial differential equation is proved and used to ensure the convergence of the estimated state in a somewhat stronger sense. A numerical example is given which illustrates the power of the method.  相似文献   

7.
为连续非线性系统提出了一种有效的最优控制设计方法. 广义模糊双曲模型(Generalized fuzzy hyperbolic model, GFHM)首次作为逼近器用来估计 HJB (Hamilton-Jacobi-Bellman)方程的解 (值函数,即它是状态与代价函数之间的映射), 然后,利用该近似解获得最优控制. 本文方法只需要一个GFHM估计值函数. 首先, 阐述了对于连线非线性系统最优控制的设计过程; 然后,证明了逼近误差是一致最终有界的 (Uniformly ultimately bounded, UUB); 最后, 一个数值例子验证了本文方法的有效性. 另一个例子通过与神经网络自适应动态规划的方法作比较, 演示了本文方法的优点.  相似文献   

8.
自适应动态规划综述   总被引:24,自引:14,他引:10  
自适应动态规划(Adaptive dynamic programming, ADP)是最优控制领域新兴起的一种近似最优方法, 是当前国际最优化领域的研究热点. ADP方法 利用函数近似结构来近似哈密顿--雅可比--贝尔曼(Hamilton-Jacobi-Bellman, HJB)方程的解, 采用离线迭代或者在线更新的方法, 来获得系统的近似最优控制策略, 从而能够有效地解决非线性系统的优化控制问题. 本文按照ADP的结构变化、算法的发展和应用三个方面介绍ADP方法. 对目前ADP方法的研究成果加以总结, 并对这 一研究领域仍需解决的问题和未来的发展方向作了进一步的展望.  相似文献   

9.
The Hamilton-Jacobi-Bellman (HJB) equation corresponding to constrained control is formulated using a suitable nonquadratic functional. It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS). The value function of this HJB equation is solved for by solving for a sequence of cost functions satisfying a sequence of Lyapunov equations (LE). A neural network is used to approximate the cost function associated with each LE using the method of least-squares on a well-defined region of attraction of an initial stabilizing controller. As the order of the neural network is increased, the least-squares solution of the HJB equation converges uniformly to the exact solution of the inherently nonlinear HJB equation associated with the saturating control inputs. The result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line.  相似文献   

10.
The purpose of this paper is to describe the application of the notion of viscosity solutions to solve the Hamilton-Jacobi-Bellman (HJB) equation associated with an important class of optimal control problems for quantum spin systems. The HJB equation that arises in the control problems of interest is a first-order nonlinear partial differential equation defined on a Lie group. Hence we employ recent extensions of the theory of viscosity solutions to Riemannian manifolds in order to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used develop a finite difference approximation method for numerically computing the solution to such problems. The convergence of these approximations is proven using viscosity solution methods. In order to illustrate the techniques developed, these methods are applied to an example problem.  相似文献   

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