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Barrier Integral Control for Global Asymptotic Tracking of Uncertain Nonlinear Systems under State and Input Constraints

Christos K. Verginis

TL;DR

The paper tackles global asymptotic tracking for high-order MIMO nonlinear systems with unknown dynamics under state and input constraints. It introduces Barrier Integral Control (BRIC), which fuses a reciprocal barrier with error-integral terms to confine the state in a predefined funnel and guarantee $\,\lim_{t\to\infty} e(t)=0$ from any initial condition, without requiring explicit dynamic bounds or approximations. An extension incorporates input saturation via reference modification and anti-windup, yielding a continuous nominal and a bounded-input regime that preserve stability and convergence. Simulations on a coupled inverted-pendulum system demonstrate superior steady-state accuracy and feasible control inputs, validating the method’s practical viability for constrained, uncertain nonlinear systems.

Abstract

This paper addresses the problem of asymptotic tracking for high-order control-affine MIMO nonlinear systems with unknown dynamic terms subject to input and transient state constraints. We introduce Barrier Integral Control (BRIC), a novel algorithm designed to confine the system's state within a predefined funnel, ensuring adherence to the transient state constraints, and asymptotically drive it to a given reference trajectory from any initial condition. The algorithm leverages the innovative integration of a reciprocal barrier function and error-integral terms, featuring smooth feedback control. We further develop an extension of the algorithm, entailing continuous feedback, that uses a reference-modification technique to account for the input-saturation constraints. Notably, BRIC operates without relying on any information or approximation schemes for the (unknown) dynamic terms, which, unlike a large class of previous works, are not assumed to be bounded or to comply with globally Lipschitz/growth conditions. Additionally, the system's trajectory and asymptotic performance are decoupled from the uncertain model, control-gain selection, and initial conditions. Finally, comparative simulation studies validate the effectiveness of the proposed algorithm.

Barrier Integral Control for Global Asymptotic Tracking of Uncertain Nonlinear Systems under State and Input Constraints

TL;DR

The paper tackles global asymptotic tracking for high-order MIMO nonlinear systems with unknown dynamics under state and input constraints. It introduces Barrier Integral Control (BRIC), which fuses a reciprocal barrier with error-integral terms to confine the state in a predefined funnel and guarantee from any initial condition, without requiring explicit dynamic bounds or approximations. An extension incorporates input saturation via reference modification and anti-windup, yielding a continuous nominal and a bounded-input regime that preserve stability and convergence. Simulations on a coupled inverted-pendulum system demonstrate superior steady-state accuracy and feasible control inputs, validating the method’s practical viability for constrained, uncertain nonlinear systems.

Abstract

This paper addresses the problem of asymptotic tracking for high-order control-affine MIMO nonlinear systems with unknown dynamic terms subject to input and transient state constraints. We introduce Barrier Integral Control (BRIC), a novel algorithm designed to confine the system's state within a predefined funnel, ensuring adherence to the transient state constraints, and asymptotically drive it to a given reference trajectory from any initial condition. The algorithm leverages the innovative integration of a reciprocal barrier function and error-integral terms, featuring smooth feedback control. We further develop an extension of the algorithm, entailing continuous feedback, that uses a reference-modification technique to account for the input-saturation constraints. Notably, BRIC operates without relying on any information or approximation schemes for the (unknown) dynamic terms, which, unlike a large class of previous works, are not assumed to be bounded or to comply with globally Lipschitz/growth conditions. Additionally, the system's trajectory and asymptotic performance are decoupled from the uncertain model, control-gain selection, and initial conditions. Finally, comparative simulation studies validate the effectiveness of the proposed algorithm.
Paper Structure (6 sections, 3 theorems, 59 equations, 4 figures)

This paper contains 6 sections, 3 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Barrier Integral Control
  4. Input-constrained Barrier Integral Control
  5. Simulation Results
  6. Conclusion

Key Result

Lemma 1

Assume that $|s_{k_j}(t)|\leq \bar{s}$, for a constant $\bar{s} > 0$ and all $t \geq 0$, $j\in\{1,\dots,n\}$. Then, there exist positive constants $\bar{e}_{i,1}$, $\bar{e}_{i,2}$, and $\lambda_0 \in (0,\lambda)$ such that, for all $i\in\{1,\dots,k-1\}$, $j\in\{1,\dots,n\}$, $t\geq 0$,

Figures (4)

  • Figure 1: The errors $s_2(t)$ (top), $e_1(t)$ (middle), and control inputs $u(t)$ (bottom) for BRIC (left) and PPC (right), along with the barrier functions $\phi(t)$ and $\rho(t)$ (top).
  • Figure 2: The integral signals $\hat{d}_1(t)$ and $\hat{d}_2(t)$.
  • Figure 3: The errors $s_2(t)$ (top), $e_1(t)$ (middle), and control inputs $u(t)$ (bottom) for BRIC (left) and PPC (right), along with the barrier functions $\phi(t)$ and $\rho(t)$ (top).
  • Figure 4: The errors $s_2(t)$ (top left), required and saturated control inputs $u(t)$, $\overline{\textup{sat}}(u(t))$ (top right), and adaptation variables $\hat{d}_1(t)$, $\hat{d}_2(t)$ for $q_R > 0$ and references and dynamics that do not satisfy Assumption \ref{['ass:F = 0']}.

Theorems & Definitions (9)

  • Lemma 1
  • Theorem 1
  • Remark 1: Approx.-free asymptotic tracking
  • Remark 2: Measurement noise
  • proof
  • Theorem 2
  • Remark 3: Relation among $u_\textup{sat}$ and $u_{\textup{sat},P}$
  • proof
  • Remark 4: Robustness of $\hat{d}_1$, $\hat{d}_2$