A data-based value iteration algorithm with the bidirectional approximation feature is developed for discounted optimal control. The unknown nonlinear system dynamics is first identified by establishing a model neural network. To improve the identification precision, biases are introduced to the model network. The model network with biases is trained by the gradient descent algorithm, where the weights and biases across all layers are updated. The uniform ultimate boundedness stability with a proper learning rate is analyzed, by using the Lyapunov approach. Moreover, an integrated value iteration with the discounted cost is developed to fully guarantee the approximation accuracy of the optimal value function. Then, the effectiveness of the proposed algorithm is demonstrated by carrying out two simulation examples with physical backgrounds.

开发了一种具有双向逼近特征的基于数据的值迭代算法，用于折扣最优控制。首先通过建立模型神经网络来识别未知的非线性系统动力学。为了提高识别精度，在模型网络中引入了偏差。带有偏差的模型网络通过梯度下降算法进行训练，其中更新所有层的权重和偏差。使用李雅普诺夫方法分析了具有适当学习率的统一极限有界稳定性。此外，开发了具有折扣成本的积分值迭代，以充分保证最优值函数的逼近精度。然后，通过两个具有物理背景的仿真实例证明了所提出算法的有效性。