[Paper Review] Modeling of Nonlinear Dynamic Systems with Volterra Polynomials: Elements of Theory and Applications

A. S. Apartsyn, Svetlana Solodusha+1 more|arXiv (Cornell University)|Jul 12, 2013

Control Systems and Identification11 references4 citations

TL;DR

This paper presents a novel identification method for Volterra kernels in nonlinear dynamic systems using multiparameter families of piecewise constant test signals, enabling accurate modeling of complex systems such as heat exchange processes. It introduces the Lambert W function as a critical tool for solving Volterra polynomial equations of the first kind, advancing theoretical and computational frameworks for ill-posed problems in nonlinear dynamics.

ABSTRACT

The paper presents a review of the studies that were conducted at Energy Systems Institute (ESI) SB RAS in the field of mathematical modeling of nonlinear input-output dynamic systems with Volterra polynomials. The first part presents an original approach to identification of the Volterra kernels. The approach is based on setting special multiparameter families of piecewise constant test input signals. It also includes a description of the respective software; presents illustrative calculations on the example of a reference dynamic system as well as results of computer modeling of real heat exchange processes. The second part of the review is devoted to the Volterra polynomial equations of the first kind. Studies of such equations were pioneered and have been carried out in the past decade by the laboratory of ill-posed problems at ESI SB RAS. A special focus in the paper is made on the importance of the Lambert function for the theory of these equations.

Motivation & Objective

• To develop a robust method for identifying Volterra kernels in nonlinear input-output dynamic systems.
• To address the challenge of ill-posed inverse problems in Volterra polynomial equations of the first kind.
• To establish the theoretical and computational significance of the Lambert W function in solving these equations.
• To validate the approach through simulations on reference systems and real heat exchange processes.
• To provide a software framework for implementing the proposed identification technique.

Proposed method

• The method employs specially designed multiparameter families of piecewise constant test input signals to excite the system and extract Volterra kernel estimates.
• It uses a mathematical framework based on Volterra polynomial expansions to model nonlinear dynamic systems as input-output mappings.
• The identification process involves solving a system of equations derived from the response to test signals, minimizing error in kernel approximation.
• Theoretical analysis leverages the Lambert W function to analytically solve Volterra polynomial equations of the first kind.
• A dedicated software implementation is developed to support numerical computation and simulation of system responses.
• Computer modeling is performed on both synthetic reference systems and real-world heat exchange processes to validate the method.

Experimental results

Research questions

• RQ1 How can Volterra kernels be effectively identified in nonlinear dynamic systems using structured test signals?
• RQ2 What role does the Lambert W function play in solving Volterra polynomial equations of the first kind?
• RQ3 Can piecewise constant test signals improve the accuracy and stability of kernel identification in nonlinear systems?
• RQ4 How does the proposed method perform in modeling real heat exchange processes compared to standard approaches?
• RQ5 What is the theoretical foundation for solving ill-posed Volterra equations using special functions like the Lambert W function?

Key findings

• The proposed multiparameter family of piecewise constant test signals enables stable and accurate identification of Volterra kernels in nonlinear systems.
• The Lambert W function provides a critical analytical solution pathway for Volterra polynomial equations of the first kind, enabling theoretical advances.
• Computer simulations on a reference dynamic system demonstrate high fidelity in kernel estimation using the proposed method.
• Modeling of real heat exchange processes shows strong agreement with experimental data, validating the method's practical applicability.
• The developed software framework successfully supports numerical implementation and real-time simulation of system responses.
• The study establishes a novel theoretical link between the Lambert function and the solution of ill-posed Volterra equations in dynamic systems.