Layered Multirate Control of Constrained Linear Systems

TL;DR

This work proposes an efficient control design that guarantees the lower-layer system tracks the output of the higher-layer system with computable precision and derives conditions and presents a method for propagating the constraints of the lower-layer system to the higher-layer system.

Abstract

Layered control architectures have been a standard paradigm for efficiently managing complex constrained systems. A typical architecture consists of: i) a higher layer, where a low-frequency planner controls a simple model of the system, and ii) a lower layer, where a high-frequency tracking controller guides a detailed model of the system toward the output of the higher-layer model. A fundamental problem in this layered architecture is the design of planners and tracking controllers that guarantee both higher- and lower-layer system constraints are satisfied. Toward addressing this problem, we introduce a principled approach for layered multirate control of linear systems subject to output and input constraints. Inspired by discrete-time simulation functions, we propose a streamlined control design that guarantees the lower-layer system tracks the output of the higher-layer system with computable precision. Using this design, we derive conditions and present a method for propagating the constraints of the lower-layer system to the higher-layer system. The propagated constraints are integrated into the design of an arbitrary planner that can handle higher-layer system constraints. Our framework ensures that the output constraints of the lower-layer system are satisfied at all high-level time steps, while respecting its input constraints at all low-level time steps. We apply our approach in a scenario of motion planning, highlighting its critical role in ensuring collision avoidance.

Figures (8)

• Figure 1: We observe that although the higher-layer system maintains a safe trajectory, the trajectory of the lower-layer system deviates from it sometimes, entering the unsafe area. These deviations arise due to input constraints and additional dynamics in the lower-layer system.
• Figure 2: Two-layer multirate control architecture for linear systems with output and input constraints. $\Sigma$ is the system that we want to control, while $\bm\bar{\Sigma}$ provides a simplified description of $\Sigma$'s dynamics.
• Figure 3: We propose a joint design of tracking controllers $u_L$ and planning constraint sets $\bm\bar{\mathcal{X}}_p$ and $\bm\bar{\mathcal{U}}_p$. The sets $\bm\bar{\mathcal{X}}_p$ and $\bm\bar{\mathcal{U}}_p$ result from propagating the constraints $\mathcal{Y}$ and $\mathcal{U}$ of the lower-layer system to the higher-layer system and combining them with the given constraint sets $\bm\bar{\mathcal{X}}$ and $\bm\bar{\mathcal{U}}$ of the latter.
• Figure 4: Tracking precision of $\bm\bar{\Sigma}_L$ by $\Sigma_L$ and maximum input norm for $\bm\bar{\Sigma}$ across different sampling frequencies $1/T_L$. We observe that higher sampling frequencies improve tracking precision and lead to more restricted inputs for the higher-layer system $\bm\bar{\Sigma}$.
• Figure 5: Output trajectories of the lower-layer system $\Sigma$ and the higher-layer system $\bm\bar{\Sigma}$. The orange disk represents the region that $\Sigma$ aims to reach. The output of $\Sigma$ remains within the safe area at all times, despite temporarily leaving the output constraint set of $\bm\bar{\Sigma}$ when $\bm\bar{\Sigma}$ changes direction.

Theorems & Definitions (9)

• Definition 1: Discrete-Time Simulation Functions and Corresponding Tracking Controllers
• Lemma 1: Tracking Precision Guarantee
• Lemma 2
• Theorem 1: Output Tracking using Discrete-Time Simulation Functions
• Remark 1
• Proposition 1: Conditions for Constraint Propagation
• Remark 2
• Remark 3
• Theorem 2: Layered Multirate Control of Constrained Linear Systems