[Paper Review] An Efficient MPC Algorithm For Switched Nonlinear Systems with Minimum Dwell Time Constraints

Yutao Chen, Mircea Lazar|arXiv (Cornell University)|Feb 22, 2020

Advanced Control Systems Optimization38 references4 citations

TL;DR

This paper proposes an efficient suboptimal MPC algorithm for switched nonlinear systems with minimum dwell time constraints (MTC) by decomposing the on-line optimization into two nonlinear programs (NLPs) and a rounding step, embedding MTC via move blocking in one NLP. The method ensures recursive feasibility using an l-step control invariant set and achieves faster online computation with guaranteed integer approximation error bounds, validated through numerical studies.

ABSTRACT

This paper presents an efficient suboptimal model predictive control (MPC) algorithm for nonlinear switched systems subject to minimum dwell time constraints (MTC). While MTC are required for most physical systems due to stability, power and mechanical restrictions, MPC optimization problems with MTC are challenging to solve. To efficiently solve such problems, the on-line MPC optimization problem is decomposed into a sequence of simpler problems, which include two nonlinear programs (NLP) and a rounding step, as typically done in mixed-integer optimal control (MIOC). Unlike the classical approach that embeds MTC in a mixed-integer linear program (MILP) with combinatorial constraints in the rounding step, our proposal is to embed the MTC in one of the NLPs using move blocking. Such a formulation can speedup on-line computations by employing recent move blocking algorithms for NLP problems and by using a simple sum-up-rounding (SUR) method for the rounding step. An explicit upper bound of the integer approximation error for the rounding step is given. In addition, a combined shrinking and receding horizon strategy is developed to satisfy closed-loop MTC. Recursive feasibility is proven using a $l$-step control invariant ($l$-CI) set, where $l$ is the minimum dwell time step length. An algorithm to compute $l$-CI sets for switched linear systems off-line is also presented. Numerical studies demonstrate the efficiency and effectiveness of the proposed MPC algorithm for switched nonlinear systems with MTC.

Motivation & Objective

• Address the challenge of efficiently solving on-line model predictive control (MPC) problems for switched nonlinear systems under minimum dwell time constraints (MTC), which are critical for system stability and physical feasibility.
• Overcome the computational difficulty of mixed-integer optimal control (MIOC) problems with MTC by avoiding combinatorial MILP formulations and instead using a decomposition into NLPs with move blocking.
• Ensure recursive feasibility of the closed-loop system under MTC by introducing an l-step control invariant (l-CI) set, where l is the minimum dwell time.
• Provide a theoretical upper bound on the integer approximation error introduced by the rounding step in the algorithm.
• Develop a combined shrinking and receding horizon strategy to enforce MTC in the closed-loop control implementation.

Proposed method

• Decompose the on-line MPC optimization into a sequence of two nonlinear programs (NLPs) and a rounding step, avoiding the need for mixed-integer linear programming (MILP) with combinatorial constraints.
• Embed the minimum dwell time constraint (MTC) directly into one of the NLPs using move blocking, enabling efficient use of modern NLP solvers and reducing computational burden.
• Apply a simple sum-up-rounding (SUR) method for the rounding step to convert continuous variables into discrete switching signals, with a provable upper bound on the resulting integer approximation error.
• Utilize a shrinking and receding horizon strategy to enforce MTC in the closed-loop system, ensuring that switching events do not violate the minimum dwell time.
• Compute l-step control invariant (l-CI) sets for switched linear systems off-line to guarantee recursive feasibility of the MPC algorithm.
• Leverage recent advancements in move blocking algorithms for NLPs to accelerate online computation, particularly in the NLP phase where MTC are embedded.

Experimental results

Research questions

• RQ1 How can the computational complexity of MPC for switched nonlinear systems with minimum dwell time constraints be reduced while maintaining closed-loop stability?
• RQ2 Can move blocking be effectively used to embed MTC in a nonlinear program rather than relying on combinatorial MILP formulations?
• RQ3 What is the theoretical bound on the integer approximation error introduced by the rounding step in the proposed decomposition?
• RQ4 How can recursive feasibility be guaranteed under MTC using a shrinking and receding horizon strategy?
• RQ5 Is it possible to compute l-step control invariant sets off-line for switched linear systems to support recursive feasibility in MPC?

Key findings

• The proposed MPC algorithm achieves faster on-line computation by replacing combinatorial MILP formulations with a two-NLP decomposition and move blocking, significantly reducing computational burden.
• The sum-up-rounding (SUR) step introduces a provable upper bound on the integer approximation error, ensuring that the rounding does not compromise system feasibility or performance.
• Recursive feasibility of the closed-loop system under MTC is guaranteed using an l-step control invariant (l-CI) set, where l is the minimum dwell time step length.
• The algorithm successfully enforces MTC in closed-loop operation through a combined shrinking and receding horizon strategy, ensuring that switching events do not violate the minimum dwell time.
• Numerical studies demonstrate the efficiency and effectiveness of the proposed MPC algorithm for switched nonlinear systems with MTC, showing practical viability and computational advantages over traditional approaches.