Layered Nonlinear Model Predictive Control for Robust Stabilization of Hybrid Systems

TL;DR

The paper addresses real-time robust stabilization of nonlinear hybrid systems, where directly solving the receding-horizon problem is prohibitively slow. It introduces a layered MPC architecture: a slow, high-level hybrid-MPC selects a domain/guard sequence, a fast fixed-mode MPC computes a robust trajectory, and a low-level tracker executes the plan; robustness is quantified via tracking-error tubes. Stability is established through a Lyapunov-based analysis and a tube-based robustness framework, including a virtual node to handle guards and tightened constraints across multiple domains. Simulation examples on a five-link biped and a controlled bouncing ball demonstrate improved robustness, particularly with more frequent re-planning and hybrid planning at contact events, highlighting the practical viability of the approach for legged robotics and other hybrid systems.

Abstract

Computing the receding horizon optimal control of nonlinear hybrid systems is typically prohibitively slow, limiting real-time implementation. To address this challenge, we propose a layered Model Predictive Control (MPC) architecture for robust stabilization of hybrid systems. A high level "hybrid" MPC is solved at a slow rate to produce a stabilizing hybrid trajectory, potentially sub-optimally, including a domain and guard sequence. This domain and guard sequence is passed to a low level "fixed mode" MPC which is a traditional, time-varying, state-constrained MPC that can be solved rapidly, e.g., using nonlinear programming (NLP) tools. A robust version of the fixed mode MPC is constructed by using tracking error tubes that are not guaranteed to have finite size for all time. Using these tubes, we demonstrate that the speed at which the fixed mode MPC is re-calculated is directly tied to the robustness of the system, thereby justifying the layered approach. Finally, simulation examples of a five link bipedal robot and a controlled nonlinear bouncing ball are used to illustrate the formal results.

Figures (4)

• Figure 1: The hybrid MPC layer determines a feasible path at a slow rate, and passes these modes to the fixed mode MPC. This MPC computes a robustly stabilizing solution at a higher rate. The trajectory and feed forward input is passed to the low level controller where it is tracked. We use properties of the low level controller to determine the size of the tube. The resulting control action is applied to the dynamics.
• Figure 2: Cartoon depiction of the additional node. The blue line depicts the guard, the solid green line depicts the flow of the system, and the dotted lines show the action of the reset map. Given that the tube at the next node is completely past the guard, at some point the true state will hit the guard.
• Figure 3: This plot shows the controlled bouncing ball simulated with three different controllers while subject to the same 10 disturbances. The large magenta circles show the initial position, the black lines are the guards, and the red circles indicate unstable trajectories. The orange line shows the trajectory without disturbances. On the left, the MPC is run slow and there is no hybrid re-planning which leads to two unstable trajectories. In the middle, the MPC is run twice as fast and all trajectories stabilize. On the right, the MPC is run slow but hybrid re-planning occurs at each contact; this controller stabilizes all the trajectories. This highlights how the different tools proposed in the paper can lead to a more robust controller.
• Figure 4: The five link biped traverses a set of stairs with varying lengths and heights. Fixed mode MPC is run at 10Hz and the hybrid MPC is recomputed at each contact. A low level PD controller tracks the MPC target. The biped chooses multiple steps on the longer stair and skips over the shorter stair. The torso joint $x$ and $y$ positions are plotted as well as the disturbance forces applied to the torso in the $x$ direction.

Theorems & Definitions (27)

• Definition 1: Hybrid System
• Definition 2: Discrete Time Lyapunov Function
• Definition 3: E-ISS
• Definition 4: Continuous Time E-ISS Lyapunov Function
• Remark
• Definition 5: Hybrid Stability
• Lemma 1
• proof
• Theorem 1: High Level Hybrid Stability
• proof