Inverse Learning-Based Output Feedback Control of Nonlinear Systems with Verifiable Guarantees

TL;DR

This paper establishes a verifiable sufficient condition on the dataset under which the proposed controller guarantees practical output regulation, and demonstrates the effectiveness of the proposed controller in the presence of output measurement noise.

Abstract

In this paper, we present a data-driven output feedback controller for nonlinear systems that achieves practical output regulation, using noise-free input/output measurement data. The proposed controller is based on (i) an inverse model of the system identified via kernel interpolation, which maps a desired output and the current state to the corresponding desired control input; and (ii) a data-driven reference selection framework that actively chooses a suitable desired output from the dataset which has been used for the identification. We establish a verifiable sufficient condition on the dataset under which the proposed controller guarantees practical output regulation. Numerical simulations demonstrate the effectiveness of the proposed controller, with additional evaluations in the presence of output measurement noise to assess its robustness empirically.

Figures (5)

• Figure 1: Abstract illustration of $\mathcal{S}_\delta$ (gray area), $\mathcal{A}_\delta^{0}$ (cyan area), and $\mathcal{A}_\delta^{1}$ (orange area). For each $i\in\{1,2,3\}$, $\zeta_i^+$ is given by \ref{['eq:zeta+']} for $([y_i^+;\zeta_i],u_i)\in\mathcal{D}$ and $(i,r_i)\in\mathcal{I}(\mathcal{S}_\delta)$.
• Figure 2: Output trajectories generated by the proposed controller with initial conditions $[-1;-1;0]$ (blue solid line), $[-0.5;-0.5;0]$ (green dashed line), $[0;0;0]$ (red dotted line), $[0.5;0.5;0]$ (yellow dash-dotted line), and $[1;1;0]$ (violet dash-dot-dotted line).
• Figure 3: Two-dimensional projections of the three-dimensional trajectories $\zeta(t)=[y(t-1);y(t);u(t-1)]$ generated by the proposed controller with initial conditions $[-1;-1;0]$ (blue solid line), $[-0.5;-0.5;0]$ (green dashed line), $[0;0;0]$ (red dotted line), $[0.5;0.5;0]$ (yellow dash-dotted line), and $[1;1;0]$ (violet dash-dot-dotted line). The gray areas correspond to the set defined by \ref{['eq:sumSet']}. The initial conditions of each trajectory and the equilibrium point $\zeta^*=[0;0;u^*]$ (cyan star) are indicated by their respective markers.
• Figure 4: Inverted pendulum with torque input $\tau$ and parameters $m$ and $l$.
• Figure 5: Noise-free (top-left) and noisy (top-right) training trajectories generated by the PI controllers with different gains $(K_p,K_I)$ and initial condition $\zeta(0)=[a;a;0]$. The middle-left and middle-right subplots show the output trajectories from the proposed controller in the noise-free and noisy cases, respectively, from several initial conditions. The bottom-left and bottom-right subplots show the corresponding noise-free and noisy output trajectories generated by the baseline PI controller with gains $(K_p,K_I)=(15,0.01)$, respectively.

Theorems & Definitions (6)

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