Overview
Research proposes a novel Model Predictive Control (MPC) formulation tailored for nonlinear systems. This formulation diverges from conventional MPC by not incorporating a terminal penalty or a dedicated stabilizing terminal set. Instead, it implements a heterogeneous approach to enforce state constraints across its prediction horizon. Specifically, near-term predictions are certified safe via a control-invariant set, while subsequent predictions are constrained by a less restrictive set, with the expectation of re-certification in future updates.
Research Context
The motivation for this MPC structure stems from the requirements of safety-critical formulations. Examples provided include Control Barrier Function (CBF) based MPC, collision avoidance systems, and robotic receding-horizon planning. In these contexts, immediate future states necessitate certification for safety, whereas predictions further along the horizon can accommodate a less stringent constraint set, as their safety status can be re-evaluated and confirmed in subsequent control iterations.
Approach
The proposed MPC formulation employs a dual-tier constraint strategy. A control-invariant set is used to certify near-term safety. For predictions that extend further into the future, a distinct, less restrictive set is applied. This less restrictive set allows for the expectation that later predictions will undergo re-certification during subsequent updates of the control system.
The methodology includes a value-function-difference analysis. This analysis differentiates the impact of two key factors on closed-loop performance bounds: the selection of the constraint set and the length of the prediction horizon. This analytical framework aims to generate implicit suboptimality certificates.
Additionally, the research introduces an upper-bound certificate. This certificate accounts explicitly for several factors: distinct constraint sets, their associated decay rates, and prediction horizons. This is contingent on the assumption of cost-controllability.
A lower-bound certificate for the closed-loop cost is also provided. This certificate extends beyond the finite-horizon open-loop cost. For the admissible parameter range, this lower-bound certificate is noted as not being weaker than the standard finite-horizon bound.
Findings
The proposed MPC formulation, which employs multiple constraint horizons and omits terminal penalties or dedicated stabilizing terminal sets, offers a structured approach for nonlinear systems, particularly in safety-critical applications.
- The value-function-difference analysis successfully separates the effects of constraint-set selection and prediction-horizon length on closed-loop performance bounds, yielding implicit suboptimality certificates.
- An explicit upper-bound certificate, developed under the assumption of cost-controllability, accounts for distinct constraint sets, their decay rates, and horizons.
- A lower-bound certificate for the closed-loop cost was provided, extending beyond the finite-horizon open-loop cost, which for the admissible parameter range, was not weaker than the standard finite-horizon bound.
- Simulations were conducted on both linear and nonlinear safety-critical systems. These simulations demonstrated the proposed certificates, both a priori and a posteriori.