Controller for Incremental Input-to-State Practical Stabilization of Partially Unknown systems with Invariance Guarantees

TL;DR

To address partially unknown nonlinear systems, the paper defines a new stability notion $\delta$-ISpS and shows a Lyapunov-based characterization via a $\delta$-ISpS-CLF. It integrates Gaussian process models to approximate the unknown drift with probabilistic bounds and uses a feedback linearization controller augmented by a safety filter derived from a Control Barrier Function to enforce invariance on a compact set. The main theoretical result proves that the closed-loop system is $\delta$-ISpS with respect to external inputs and remains safe with high probability $(1-\epsilon)^n$. A two-link manipulator case study demonstrates practical convergence of trajectories from different initial conditions and quantifies the safety guarantees.

Abstract

Incremental stability is a property of dynamical systems that ensures the convergence of trajectories with respect to each other rather than a fixed equilibrium point or a fixed trajectory. In this paper, we introduce a related stability notion called incremental input-to-state practical stability (δ-ISpS), ensuring safety guarantees. We also present a feedback linearization based control design scheme that renders a partially unknown system incrementally input-to-state practically stable and safe with formal guarantees. To deal with the unknown dynamics, we utilize Gaussian process regression to approximate the model. Finally, we implement the controller synthesized by the proposed scheme on a manipulator example

Figures (2)

• Figure 1: Evolution of the system under a inputs $\upsilon=[\sin(t),\cos(t)]$ and $\upsilon'=[\cos(t),\sin(t)]$ with the initial states $x_0=[-1.5,0.5,0.01,0.01]$ (blue line) and $x_0=[1.5,-0.5,-0.01,-0.01]$ (orange line).
• Figure 2: Distance between the trajectories of the controlled system under the inputs $\upsilon=[\sin(t),\cos(t)]$ and $\upsilon'=[\cos(t),\sin(t)]$ with the initial conditions $x_0=[-1.5,0.5,0.01,0.01]$ and $y_0=[1.5,-0.5,-0.01,-0.01]$ respectively.

Theorems & Definitions (11)

• Definition 2.1: $\delta$-ISpS
• Definition 2.2: Robustly Forward Invariant Set liu2019compositional
• Definition 2.3: $\delta$-ISpS Control Lyapunov Function
• Theorem 2.4
• proof
• Proposition 3.4
• proof
• Definition 3.5: jagtap2020control
• Theorem 4.2
• proof