Immersion and Invariance-based Disturbance Observer and Its Application to Safe Control

TL;DR

This work tackles disturbance handling in nonlinear control-affine systems by introducing an Immersion and Invariance-based Disturbance Observer (IIDOB) that circumvents PDE solvability and disturbance relative degree limitations. By augmenting the system, defining an estimated disturbance $\hat{d}=\xi+\beta$, and approximately solving a PDE via a regulator $\beta$ tied to a PDE surrogate, IIDOB achieves a globally UUB disturbance estimation error $e_d$ with compensable approximation error. The authors then couple IIDOB with a filter-based safe control strategy within the Control Barrier Function (CBF) framework to yield IIDOB-CBF-QP controllers that guarantee safety (i.e., $h(x(t))\ge 0$) under disturbances, while preserving nominal tracking as much as possible. Theoretical guarantees accompany practical parameter-tuning guidance, and simulations on nonlinear dynamics and a planar robot demonstrate accurate disturbance estimation and superior safety-preserving performance compared to robust CBF approaches. Overall, the approach offers a scalable, computation-friendly solution for safe control in the presence of unknown disturbances in nonlinear systems.

Abstract

When the disturbance input matrix is nonlinear, existing disturbance observer design methods rely on the solvability of a partial differential equation or the existence of an output function with a uniformly well-defined disturbance relative degree, which can pose significant limitations. This note introduces a systematic approach for designing an Immersion and Invariance-based Disturbance Observer (IIDOB) that circumvents these strong assumptions. The proposed IIDOB ensures the disturbance estimation error is globally uniformly ultimately bounded by approximately solving a partial differential equation while compensating for the approximation error. Furthermore, by integrating IIDOB into the framework of control barrier functions, a filter-based safe control design method for control-affine systems with disturbances is established where the filter is used to generate an alternative disturbance estimation signal with a known derivative. Sufficient conditions are established to guarantee the safety of the disturbed systems. Simulation results demonstrate the effectiveness of the proposed method.

Figures (4)

• Figure 1: Configuration of the proposed IIDOB-CBF-QP method that consists of three components: (i) an IIDOB used for disturbance estimation, (ii) a filter that can generate an alternative disturbance estimation signal with a known derivative, and (iii) an IIDOB-CBF-QP-based safe controller that can ensure safety of the closed-loop system.
• Figure 2: Simulation results of the proposed IIDOB and IIDOB-based tracking controller for system \ref{['sim:numericalsystem']}. (top) Trajectories of the states $x_1,x_2$ and the reference signals $x_{1d},x_{2d}$; (bottom) trajectories of the total disturbances $d_1,d_2$ and the estimated total disturbances $\hat{d}_1,\hat{d}_2$.
• Figure 3: Simulation results of the IIDOB-CBF-QP-based controller \ref{['cbfQP']} and the robust CBF-based controller proposed in jankovic2018robust. Both controllers can ensure safety, but controller \ref{['cbfQP']} has better tracking performance inside the safe region.
• Figure 4: Simulation results of the proposed IIDOB-CBF-QP-based safe control for the two-linked planer robot \ref{['eleqn']}. One can see that the disturbance estimation is accurate and the safety is ensured. Moreover, the tracking performance of the nominal controller is well preserved inside the safe region.

Theorems & Definitions (4)

• Proof 1
• Proof 2
• Proof 3
• Proof 4