Analysis of Discrete and Hybrid Stochastic Systems by Nonlinear Contraction Theory

We investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) contracting stochastic system is upper-bounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete noisy interactions.Comment: 6 pages, 1 figur

Convex Optimization-based Controller Design for Stochastic Nonlinear Systems using Contraction Analysis

This paper presents an optimal feedback tracking controller for a class of Itô stochastic nonlinear systems, the design of which involves recasting a nonlinear system equation into a convex combination of multiple non-unique State-Dependent Coefficient (SDC) models. Its feedback gain and controller parameters are found by solving a convex optimization problem to minimize an upper bound of the steady-state tracking error. Multiple SDC parametrizations are utilized to provide a design flexibility to mitigate the effects of stochastic noise and to ensure that the system is controllable. Incremental stability of this controller is studied using stochastic contraction analysis and it is proven that the controlled trajectory exponentially converges to the desired trajectory with a non-vanishing error due to the linear matrix inequality state-dependent algebraic Riccati equation constraint. A discrete-time version of stochastic contraction analysis with respect to a state- and time-dependent metric is also presented in this paper. A simulation is performed to show the superiority of the proposed optimal feedback controller compared to a known exponentially-stabilizing nonlinear controller and a PID controller

Robust Controller Design for Stochastic Nonlinear Systems via Convex Optimization

This paper presents ConVex optimization-based Stochastic steady-state Tracking Error Minimization (CV-STEM), a new state feedback control framework for a class of Ito stochastic nonlinear systems and Lagrangian systems. Its strength lies in computing the control input by an optimal contraction metric, which greedily minimizes an upper bound of the steady-state mean squared tracking error of the system trajectories. Although the problem of minimizing the bound is nonlinear, its equivalent convex formulation is proposed utilizing state-dependent coefficient parameterizations of the nonlinear system equation. It is shown using stochastic incremental contraction analysis that the CV-STEM provides a sufficient guarantee for exponential boundedness of the error for all time with L₂-robustness properties. For the sake of its sampling-based implementation, we present discrete-time stochastic contraction analysis with respect to a state- and time-dependent metric along with its explicit connection to continuous-time cases. We validate the superiority of the CV-STEM to PID, H∞, and given nonlinear control for spacecraft attitude control and synchronization problems

Optimization-based Framework for Stability and Robustness of Bipedal Walking Robots

As robots become more sophisticated and move out of the laboratory, they need to be able to reliably traverse difficult and rugged environments. Legged robots -- as inspired by nature -- are most suitable for navigating through terrain too rough or irregular for wheels. However, control design and stability analysis is inherently difficult since their dynamics are highly nonlinear, hybrid (mixing continuous dynamics with discrete impact events), and the target motion is a limit cycle (or more complex trajectory), rather than an equilibrium. For such walkers, stability and robustness analysis of even stable walking on flat ground is difficult. This thesis proposes new theoretical methods to analyse the stability and robustness of periodic walking motions. The methods are implemented as a series of pointwise linear matrix inequalities (LMI), enabling the use of convex optimization tools such as sum-of-squares programming in verifying the stability and robustness of the walker. To ensure computational tractability of the resulting optimization program, construction of a novel reduced coordinate system is proposed and implemented. To validate theoretic and algorithmic developments in this thesis, a custom-built “Compass gait” walking robot is used to demonstrate the efficacy of the proposed methods. The hardware setup, system identification and walking controller are discussed. Using the proposed analysis tools, the stability property of the hardware walker was successfully verified, which corroborated with the computational results