Low-Complexity Control for a Class of Uncertain MIMO Nonlinear Systems under Generalized Time-Varying Output Constraints (extended version)

TL;DR

This work addresses the challenge of enforcing multiple time-varying, potentially coupled output constraints in uncertain high-order MIMO nonlinear systems. It introduces a single consolidating constraint based on a smooth scalar metric $\alpha(t,x_1)$ that captures all constraints, and a low-complexity, model-free backstepping-style controller to guarantee convergence to and invariance within the feasible set in finite time. To handle infeasibility, the authors develop an online continuous-time optimization scheme to estimate $\alpha^*(t)$ and an adaptive lower-bound $\rho_{\alpha}(t)$ that yields a least-violating solution when the constraint set becomes empty for some interval. The approach extends existing funnel/PP C/TVBLF methods by accommodating coupling among constraints, without relying on online QP or full model knowledge, and is validated via trajectory and region-tracking simulations for a mobile robot. This framework offers a computationally tractable, robust pathway for enforcing complex spatiotemporal constraints in broad nonlinear control applications.

Abstract

This paper introduces a novel control framework to address the satisfaction of multiple time-varying output constraints in uncertain high-order MIMO nonlinear control systems. Unlike existing methods, which often assume that the constraints are always decoupled and feasible, our approach can handle coupled time-varying constraints even in the presence of potential infeasibilities. First, it is shown that satisfying multiple constraints essentially boils down to ensuring the positivity of a scalar variable, representing the signed distance from the boundary of the time-varying output-constrained set. To achieve this, a single consolidating constraint is designed that, when satisfied, guarantees convergence to and invariance of the time-varying output-constrained set within a user-defined finite time. Next, a novel robust and low-complexity feedback controller is proposed to ensure the satisfaction of the consolidating constraint. Additionally, we provide a mechanism for online modification of the consolidating constraint to find a least violating solution when the constraints become mutually infeasible for some time. Finally, simulation examples of trajectory and region tracking for a mobile robot validate the proposed approach.

Figures (8)

• Figure 1: Snapshots of $\bar{\Omega}(t)$ and its corresponding inner-approximation under \ref{['smooth_alph']} for three different examples.
• Figure 2: (a) Snapshot of the output constraint $\underline{\rho}(t) < h(x_1) = x_{1,1}^2 + x_{1,2}^2 < \overline{\rho}(t)$ for which its corresponding $\alpha(t,x_1)$ does not satisfy Assumption \ref{['assu:alpha_globalmax']} due to the existence of a local minimum at $x_1 = [0, 0]^\top$. (b) surface of $\alpha(t,x_1)$.
• Figure 3: The evolution of $\alpha(t,x_1(t;x(0)))$ under the consolidating constraint \ref{['eq:consuli_const']}, where $\rho_{\alpha}(t)$ is determined by \ref{['eq:rho_lower_optim_schme']}. The adaptation of $\rho_{\alpha}(t)$ (dashed line) based on the evolution of $\hat{\alpha} (t)$ in \ref{['eq:first_order_gradaccent']} (dotted line) allows for deviations of $\rho_{\alpha}(t)$ from the nominal lower bound function $\varrho(t)$ in \ref{['eq:alpha_lower_bound_nomi']}. Consequently, satisfaction of \ref{['eq:consuli_const']} during the time intervals when $\alpha^{\ast} (t) < 0$ (shaded intervals) results in a least violating solution. In this illustrative example, roughly after one second $\hat{\alpha} (t)$ provides a reliable estimate of $\alpha^{\ast} (t)$.
• Figure 4: Cascaded control architecture under the estimation scheme \ref{['eq:first_order_gradaccent']} and online computation of $\rho_{\alpha}(t)$ in \ref{['eq:rho_lower_optim_schme']}.
• Figure 5: Mobile robot.

Theorems & Definitions (25)

• Remark 1
• Remark 2
• Definition 1
• Example 1
• Example 2
• Example 3
• Lemma 1
• Lemma 2
• Remark 3
• Remark 4