A lyapunov based stable online learning algorithm for nonlinear dynamical systems using extreme learning machines

Abstract

Abstract-Extreme Learning Machine (ELM) is a promising learning scheme for nonlinear classification and regression problems and has shown its effectiveness in the machine learning literature. ELM represents a class of generalized single hidden layer feed-forward networks (SLFNs) whose hidden layer parameters are assigned randomly resulting in an extremely fast learning speed along with superior generalization performance. It is well known that the online sequential learning algorithm (OS-ELM) based on recursive least squares [1] might result in ill-conditioning of the Hessian matrix and hence instability in the parameter estimation. To address this issue, the stability theory of Lyapunov is utilized to develop an online learning algorithm for temporal data from dynamic systems and time series. The developed algorithm results in parameter estimation that is globally asymptotically stable. Simulations results of the developed algorithm compared against online sequential ELM (OS-ELM) and the offline batch learning ELM (O-ELM) show that the online Lyapunov ELM algorithm can perform online learning at reduced computation, comparable accuracy and with a guarantee on the boundedness of the estimated system.

Barrier-Certified Adaptive Reinforcement Learning With Applications to Brushbot Navigation

“…Hence, we employ an adaptive filter with monotone approximation property, which shares similar ideas with stable online learning for adaptive control based on Lyapunov stability (c.f. [26]- [29], for example).…”

Robust identification of MISO neuro‐fractional‐order Hammerstein systems

“…whereŨ = [ũ 1 · · ·ũ n ] and y = [ 1 · · · n ] T are input and output filtered signals given in (11) and (12), respectively; e is the output prediction error (e =̂− ); P 2 and P 3 are arbitrary n × n -dimensional diagonal PD matrices; is a hyperparameter that is obtained from the statistical properties of the output signal y; and e − denotes the value of e at the previous time instant t − .…”

“…stands for the trace andŨ and y f are the filtered input and output signals given in (11) and (12), respectively, Lemma 2 can be utilized. By successive -order differentiating (19), it can be seen that, with d 1 = 1, u a i and a i (i ∈ {1, … , n}) given in (11) and (12), respectively, satisfy (19). By substituting (19) into (18) and following the similar procedure as that used in obtaining (18) from (16), it can be seen that a state-space realization of (18) is as follows:…”

“…Besides, since the state‐space modeling provides a deep insight into the nature of the physical systems, it has been a subject of recent research . Some of the distinguishing features of the state‐space‐based identification over the commonly used input‐output regression‐based one are as follows: (i) by the use of state‐space realization of a system, the well‐known Lyapunov stability theory can be conducted to achieve stable identification laws, which ensure boundedness of the estimated parameters; (ii) state‐space modeling eliminates the need for filtering techniques such as those used in the works of Wu and Chen and Cui et al arises in the continuous‐time identification, provided that the system states are available. However, a difficulty arises in that the states of the system are usually unmeasurable.…”

“…identification laws, which ensure boundedness of the estimated parameters 11,12 ; (ii) state-space modeling eliminates the need for filtering techniques such as those used in the works of Wu and Chen 13 and Cui et al 14 arises in the continuous-time identification, provided that the system states are available. However, a difficulty arises in that the states of the system are usually unmeasurable.…”