Backstepping Design for Incremental Input-to-State Stabilization of Unknown Systems

TL;DR

This work introduces and characterize a novel class of incremental Lyapunov functions, an incremental stability notion known as Incremental Input-to-State practical Stability ({\delta}-ISpS), and presents a backstepping control design scheme that provides state-feedback controllers that render the partially unknown control system {\delta}-ISpS.

Abstract

Incremental stability of dynamical systems ensures the convergence of trajectories from different initial conditions towards each other rather than a fixed trajectory or equilibrium point. Here, we introduce and characterize a novel class of incremental Lyapunov functions, an incremental stability notion known as Incremental Input-to-State practical Stability (δ-ISpS). Using Gaussian Process, we learn the unknown dynamics of a class of control systems. We then present a backstepping control design scheme that provides state-feedback controllers that render the partially unknown control system δ-ISpS. To show the effectiveness of the proposed controller, we implement it in two case studies.

Figures (4)

• Figure 1: Evolution of the states under a constant input $\hat{\upsilon}=200$ with the initial conditions $x_0=[1.5,0.5,7]$ (blue line) and $x_0=[2.5,-0.5,2]$ (red line).
• Figure 2: Distance between the trajectories (as computed using the distance metric in Theorem \ref{['theorem:SFControl']}, a.k.a closeness of trajectories) of the controlled system under a constant input $\hat{\upsilon}=200$ with the initial conditions $x=[1.5, 0.5, 7]$ and $x'=[2.5, -0.5, 2]$. The blue line denotes the closeness of the trajectories, and the red line is the probabilistic bound on the closeness, where c is computed with $\lVert\eta\rVert\lVert\Bar{\rho}\rVert=0.19$.
• Figure 3: Evolution of the system under a constant input $\hat{\upsilon}=[-1,2]$ with the initial conditions $x_0=[3,-2.65,1,1]$ (blue line) and $x_0=[-3,1.65,-1,-1]$ (orange line).
• Figure 4: Distance between the trajectories (as computed using the distance metric in Theorem \ref{['theorem:SFControl']}, a.k.a closeness of trajectories) of the controlled system under a constant input $\hat{\upsilon}=[-1,2]$ with the initial conditions $x=[3,-2.65,0.1,0.1]$ and $x'=[-3,1.65,-0.1,-0.1]$. Here, the blue line denotes the closeness of the trajectories, and the green line is the probabilistic bound on the closeness, where c is computed with $\lVert\eta\rVert\lVert\Bar{\rho}\rVert=0.19$.

Theorems & Definitions (18)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• proof
• Definition 3.3: GPBackStepping
• Lemma 3.5
• proof
• Remark 3.6
• Lemma 3.7