Adaptive Tracking Stochastic Distribution Control for Two-step Neural Network Models

Abstract

In this chapter, two-step NN are applied to solve the SDC problem for non-Gaussian systems (see Figure 5.1 for details). The objective is to control the conditional PDF of the system output to follow a given target function. B-spline NN are used to approximate the PDF of the system output, and DNNs are applied to identify the nonlinear relationships between the control input and the weight vectors. To achieve the control objective, a dynamic adaptive controller is developed so that the weight dynamics can follow the outputs of a reference model, as the desired weight values. Stability analysis for both the identification and tracking errors is developed via the use of the Lyapunov stability criterion. The main contributions of this chapter include:

1. Two kinds of NN models are first applied at the same time to solve the above mentioned stochastic control problem. Under the new control framework, it is much more intuitionist to design the control input and analyze the rigorous stability than previous results (see [61, 63, 64, 157]).

2. We attempt to import DNNs with undetermined parameters to perform black box identification in SDC. This represents a significant extension to the previous results [31, 58, 61–69, 72, 73, 87, 88, 109, 146, 155–169, 172, 173, 181–184, 186– 188, 196].

3. Compared with other results related to DNNs [126, 131, 189], we consider the error term and construct the compensation term with a saturation function to guarantee the identification error convergence to zero.

4. Compared with references [63, 64, 157, 164], this chapter not only solves the dynamical tracking problem through the designed control input, but also identifies the dynamic trajectory of the weighting vectors related to the output PDFs.