Dynamical system learning using extreme learning machines with safety and stability guarantees

Abstract

Summary This article presents a continuous dynamical system model learning methodology that can be used to generate reference trajectories for the autonomous systems to follow, such that these trajectories are invariant to a given closed set and uniformly ultimately bounded with respect to an equilibrium point inside the closed set. The autonomous system dynamics are approximated using extreme learning machines (ELM), the parameters of which are learned subject to the safety constraints expressed using a reciprocal barrier function, and the stability constraints derived using a Lyapunov analysis in the presence of the ELM reconstruction error. This formulation leads to solving a constrained quadratic program (QP) that includes a finite number of decision variables with an infinite number of constraints. Theorems are developed to relax the QP with infinite number of constraints to a QP with a finite number of constraints which can be practically implemented using a QP solver. In addition, an active sampling methodology is developed that further reduced the number of required constraints for the QP by only evaluating the constraints at a smaller subset of points. The proposed method is validated using a motion reproduction task on a seven degree‐of‐freedom Baxter robot, where the task space position and velocity dynamics are learned using the presented methodology.

Learning Discrete-Time Uncertain Nonlinear Systems With Probabilistic Safety and Stability Constraints

“…The parameters a p , b p and U of ELM are computed using intrinsic plasticity (IP) or batch intrinsic plasticity (BIP) algorithms. See [36] and [14] for preliminaries of ELM.…”

“…The active sampling strategy introduced in [14] is also used to select informative points for the constraint computations of the optimization problem in (54a)-(54c). For all the three examples, the risk tolerance of p k = 0.8, for ν k the covariance of Σ = 0.02I, and regularization parameter µ W = 0.01 are chosen.…”

“…BC presents many advantages such as increased sample efficiency compared to reinforcement learning and can be useful in many robotics and engineering applications [6]- [11]. Dynamic system learning-based BC methods have incorporated stability [8], [9], [11]- [13] and safety [5], [10], [14] constraints in learning linear or nonlinear continuous-time system models. However, incorporating these constraints for uncertain discrete-time nonlinear systems represented using neural networks (NN) with noise poses technical challenges for constrained learning of such system models.…”

“…Machine learning techniques, such as NN and Gaussian processes (GPs), have been used for nonlinear system identification [11], [29]. Model uncertainties, in the form of a NN approximation error, are accounted for in [14] along with stability and safety constraints using reciprocal BFs, where the uncertainty is bounded by a constant. In the context of Bayesian learning, GP state-space models are learned with stability constraints in [11].…”