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Analysis of the Unscented Transform Controller for Systems with Bounded Nonlinearities

Siddharth A. Dinkar, Ram Padmanabhan, Anna Clarke, Per-Olof Gutman, Melkior Ornik

TL;DR

The paper analyzes the Unscented Transform Controller (UTC) for discrete-time systems composed of a linear part with a bounded nonlinear term $f(x)$, establishing a Lyapunov-based stability result that ensures the joint state-input vector converges to a bounded region whose radius scales with the nonlinearity bound. It presents both 1-step and $N$-step UTC formulations, detailing sigma-point propagation around the input and a UKF-like update with gain $\mathbf{K}$ to achieve reference tracking. Theoretical results show that, despite nonlinearities, the closed-loop behavior remains confined within a calculable bound $R = \bar{D}[\|\mathbf{Z}\|p_{max} + \sqrt{\|\mathbf{Z}\|^2 p_{max}^2 + p_{max}}]$, with $\bar{D}$ depending on $\bar{f}$ and horizon $N$. Empirical demonstrations on an ADMIRE fighter jet and a nonlinear quadcopter illustrate satisfactory regulation and tracking, while highlighting trade-offs between transient performance and long-term accuracy as the prediction horizon increases. The work provides a rigorous stability baseline for UTC and informs horizon selection, with avenues for extending to broader nonlinearities, continuous-time systems, and dual controllers for other estimation frameworks.

Abstract

In this paper, we present an analysis of the Unscented Transform Controller (UTC), a technique to control nonlinear systems motivated as a dual to the Unscented Kalman Filter (UKF). We consider linear, discrete-time systems augmented by a bounded nonlinear function of the state. For such systems, we review 1-step and N-step versions of the UTC. Using a Lyapunov-based analysis, we prove that the states and inputs converge to a bounded ball around the origin, whose radius depends on the bound on the nonlinearity. Using examples of a fighter jet model and a quadcopter, we demonstrate that the UTC achieves satisfactory regulation and tracking performance on these nonlinear models.

Analysis of the Unscented Transform Controller for Systems with Bounded Nonlinearities

TL;DR

The paper analyzes the Unscented Transform Controller (UTC) for discrete-time systems composed of a linear part with a bounded nonlinear term , establishing a Lyapunov-based stability result that ensures the joint state-input vector converges to a bounded region whose radius scales with the nonlinearity bound. It presents both 1-step and -step UTC formulations, detailing sigma-point propagation around the input and a UKF-like update with gain to achieve reference tracking. Theoretical results show that, despite nonlinearities, the closed-loop behavior remains confined within a calculable bound , with depending on and horizon . Empirical demonstrations on an ADMIRE fighter jet and a nonlinear quadcopter illustrate satisfactory regulation and tracking, while highlighting trade-offs between transient performance and long-term accuracy as the prediction horizon increases. The work provides a rigorous stability baseline for UTC and informs horizon selection, with avenues for extending to broader nonlinearities, continuous-time systems, and dual controllers for other estimation frameworks.

Abstract

In this paper, we present an analysis of the Unscented Transform Controller (UTC), a technique to control nonlinear systems motivated as a dual to the Unscented Kalman Filter (UKF). We consider linear, discrete-time systems augmented by a bounded nonlinear function of the state. For such systems, we review 1-step and N-step versions of the UTC. Using a Lyapunov-based analysis, we prove that the states and inputs converge to a bounded ball around the origin, whose radius depends on the bound on the nonlinearity. Using examples of a fighter jet model and a quadcopter, we demonstrate that the UTC achieves satisfactory regulation and tracking performance on these nonlinear models.

Paper Structure

This paper contains 9 sections, 1 theorem, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Unscented Transform Controller
  3. 1-Step UTC
  4. N-Step UTC
  5. Stability Analysis
  6. Simulation Examples
  7. Example 1: ADMIRE Fighter Jet Model
  8. Example 2: Quadcopter
  9. Conclusions

Key Result

Proposition 1

Under the system dynamics and controller in eq:closed-loop which describe the $N$-step UTC applied to a linear system affected by a nonlinearity with norm bound $\bar{f}$, the vector of states and inputs $a_k$ with the dynamics eq:lyap_system converges to a bounded ball of radius $R$, where and $p_{max}$ is the maximum eigenvalue of the matrix $\mathbf{P}$ satisfying $\mathbf{Z}^T\mathbf{P}\mathb

Figures (4)

  • Figure 1: Performance of the $1$-step UTC in stabilizing states of the ADMIRE model to the origin.
  • Figure 2: Output regulation in a quadcopter model using a $3$-step UTC.
  • Figure 3: Output regulation in a quadcopter model using a $5$-step UTC.
  • Figure 4: Tracking performance of the UTC applied to a nonlinear quadcopter model, in terms of the norm of the tracking error over time.

Theorems & Definitions (1)

  • Proposition 1