Local Stability and Convergence Analysis of Neural Network Controllers with Error Integral Inputs


This article investigates the local stability and local convergence of a class of neural network (NN) controllers with error integrals as inputs for reference tracking. It is formally proved that if the input of the NN controller consists exclusively of error terms, the control system shows a non-zero steady-state error for any constant reference except for one specific point, for both single-layer and multi-layer NN controllers. It is further proved that adding error integrals to the input of the (single- and multi-layers) NN controller is one sufficient way to remove the steady-state error for any constant reference. Due to the nonlinearity of the NN controllers, the NN control systems are linearized at the equilibrium points. We provide proof that if all the eigenvalues of the linearized NN control system have negative real parts, local asymptotic stability and local exponential convergence are guaranteed. Two case studies were explored to verify the theoretical results: a single-layer NN controller in a 1-D system and a four-layer NN controller in a 2-D system applied to renewable energy integration. Simulations demonstrate that when NN controllers and the corresponding generalized proportional-integral (PI) controllers have the same eigenvalues, all control systems exhibit almost the same responses in a small neighborhood of their respective equilibrium points.


Electrical and Computer Engineering

Publication Status

Early Access


This work was supported in part by the U.S. National Science Foundation under Grant CISE-MSI 2131214/2131175. The work of Donald C. Wunsch II was supported in part by Missouri University of Science and Technology Mary K. Finley Endowment and Intelligent Systems Center, in part by the National Science Foundation (NSF), in part by the Army Research Laboratory (ARL) and the Lifelong Learning Machines Program from the Defense Advanced Research Projects Agency (DARPA)/Microsystems Technology Office, under Cooperative Agreement W911NF-18-2-0260, and in part by Teledyne Scientific, LLC.

Keywords and Phrases

Artificial Neural Networks; Asymptotic Stability; Control Systems; Convergence; Eigenvalues and Eigenfunctions; Error Integral; Generalized Proportional-Integral (PI) Controller; Local Asymptotic Stability; Local Exponential Convergence; Mathematical Models; Neural Network (NN) Controller; Steady-State; Steady-State Error.

International Standard Serial Number (ISSN)

2162-2388; 2162-237X

Document Type

Article - Journal

Document Version


File Type





© 2021 Institute of Electrical and Electronics Engineers (IEEE), All rights reserved.

Publication Date

14 Oct 2021