# A Control Lyapunov Perspective on Episodic Learning via Projection to State Stability

## TL;DR

This paper introduces Projection to State Stability (PSS), a new concept for analyzing the robustness of control Lyapunov functions in learned control systems, enabling better handling of uncertainties in episodic learning.

## Contribution

It proposes PSS as a novel framework to analyze and bound uncertainties in control Lyapunov functions derived from learned data, enhancing robust control synthesis.

## Key findings

- PSS effectively characterizes uncertainty in projected dynamics.
- Episodic learning combined with PSS improves robustness in control.
- Bounded uncertainty leads to more reliable control strategies.

## Abstract

The goal of this paper is to understand the impact of learning on control synthesis from a Lyapunov function perspective. In particular, rather than consider uncertainties in the full system dynamics, we employ Control Lyapunov Functions (CLFs) as low-dimensional projections. To understand and characterize the uncertainty that these projected dynamics introduce in the system, we introduce a new notion: Projection to State Stability (PSS). PSS can be viewed as a variant of Input to State Stability defined on projected dynamics, and enables characterizing robustness of a CLF with respect to the data used to learn system uncertainties. We use PSS to bound uncertainty in affine control, and demonstrate that a practical episodic learning approach can use PSS to characterize uncertainty in the CLF for robust control synthesis.