Differential neural network identifier with composite learning laws for uncertain nonlinear systems Alejandro Guarneros ∗ Iv´ an Salgado ∗ Manuel Mera ∗∗ Hafiz Ahmed ∗∗∗ ∗ CIDETC-Instituto Polit´ ecnico Nacional, Gustavo A. Madero, Mexico City (e-mail: isalgador@ipn.mx). ∗∗ ESIME Ticom´ an-Instituto Polit´ ecnico Nacional, Gustavo A. Madero, Mexico City, Mexico (e-mail: mlmerah@gmail.com) ∗∗∗ Coventry University, Coventry CV1 5FB, United Kindgdom (e-mail: hafiz.h.ahmed@ieee.org) Abstract: This manuscript describes the design and numerical implementation of a novel composite differential neural network aimed to estimate nonlinear uncertain systems. A differential neural network (DNN) with a composite feedback matrix approximates the structure of non-linear uncertain systems. The feedback matrix is assumed to belong to a convex set as well as the free parameters of the DNN (weights) at any instant of time. Therefore, l-different DNN works in parallel. A composite Lyapunov function finds the convex hull approximation of the set of DNN working together to improve the approximation capabilities of classical neural networks. The main result of this study shows the practical stability of the estimation error. Numerical simulations demonstrate the approximation capabilities of the composite DNN implemented in a Van Der Pol oscillator where the presence of high-frequency components makes difficult a classical DNN approximation. Keywords: Composite Lyapunov function, Differential neural networks, nonlinear systems, uncertain systems 1. INTRODUCTION The artificial neural networks (ANNs) are complex parallel structures that emulate how the human brain processes a large number of data (Poznyak et al., 2019). The ANNs have been successfully applied to solve the problem of non- parametric identification. In the case of dynamic systems oriented to identification, estimation, and control, differ- ential neural networks (DNNs) offer attractive features to deal with uncertain and perturbed systems. Among other characteristics, DNNs provide robustness and practical stability through the second Lyapunov’s stability method (Poznyak et al., 2001). Indeed, the learning laws are a consequence of this analysis. This method guarantees the stability of the equilibrium point for the identification or estimation error as well as the boundedness of the DNN’s weights. The outstanding results have brought about new structures with original learning processes. Some impor- tant applications of DNNs are found in delayed systems (Xu et al., 2019) and sliding mode based learning laws (Keighobadi et al., 2019), among others. However, for systems with high-frequency components, the DNN have problems to reproduce accurately the system dynamics. The elements of the basis used in classical theory limit the identification properties of the DNNs (Poznyak et al., 2001). One possible solution is the im- ⋆ Authors want to thank the financial support by SIP-Instituto Polit´ ecnico Nacional grant 20195253 plementation of several DNNs working in a parallel dis- tribution (Hunter and Wilamowski, 2011). The index to select which DNN is in the “ON” state can be obtained by training an additional neural network or a fuzzy logic approach (Cervantes et al., 2017). In this last methodology, the defuzzification stage in the fuzzy logic algorithm yields an appropriate selection of the DNN to apply. The concept of composite functions can expand the ap- proximation properties of a DNN. Even when there already exists in literature the concept of composite neural net- work (Meng and Karniadakis, 2019), in this manuscript, the objective is to propose a set of learning laws derived from a composite Lyapunov function (CLF) while the com- posite network in (Meng and Karniadakis, 2019) implies a cascade structure to signal processing. The CLF has the objective to make less conservative the estimations of invariant sets compared to ellipsoidal approximations (Hu and Lin, 2003). The CLFs have been applied for stability analysis, to estimate the regions of attraction for input saturated systems and to study the stability of piecewise linear and switched systems (Hu et al., 2008; Azhmyakov et al., 2019). A classical Lyapunov function is commonly defined by a quadratic structure. A CLF is composed of the convex hull of a group of individual Lyapunov equations generally selected as ellipsoids. These are, in general, larger sets than those corresponding to each Lyapunov quadratic function. One of the main advantages of working with CLFs applied Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020 Copyright lies with the authors 7995