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Closed-loop Analysis of ADMM-based Suboptimal Linear Model Predictive Control

Anusha Srikanthan, Aren Karapetyan, Vijay Kumar, Nikolai Matni

TL;DR

This work addresses real-time MPC under computational constraints by formulating a constrained LQR problem and solving it with a finite, fixed number of ADMM iterations per time step. By splitting the OCP into a dynamics-free feasibility step and a constraint-free LQR step, and employing warm-starts, the authors derive conditions under which the resulting suboptimal closed-loop is recursively feasible and locally asymptotically stable within a forward-invariant region, even with active state and input constraints. They establish Lipschitz continuity of the optimal solution map, an explicit forward-invariant ROA for the optimal MPC, and an ISS-type bound that leads to a computable lower bound on the required ADMM iterations, ell^*. Numerical simulations on a double integrator validate the theory and illustrate computation-performance tradeoffs. The results support deployment in compute-constrained layered control architectures and provide a foundation for extensions to nonlinear dynamics and tracking tasks.

Abstract

Many practical applications of optimal control are subject to real-time computational constraints. When applying model predictive control (MPC) in these settings, respecting timing constraints is achieved by limiting the number of iterations of the optimization algorithm used to compute control actions at each time step, resulting in so-called suboptimal MPC. This paper proposes a suboptimal MPC scheme based on the alternating direction method of multipliers (ADMM). With a focus on the linear quadratic regulator problem with state and input constraints, we show how ADMM can be used to split the MPC problem into iterative updates of an unconstrained optimal control problem (with an analytical solution), and a dynamics-free feasibility step. We show that using a warm-start approach combined with enough iterations per time-step, yields an ADMM-based suboptimal MPC scheme which asymptotically stabilizes the system and maintains recursive feasibility.

Closed-loop Analysis of ADMM-based Suboptimal Linear Model Predictive Control

TL;DR

This work addresses real-time MPC under computational constraints by formulating a constrained LQR problem and solving it with a finite, fixed number of ADMM iterations per time step. By splitting the OCP into a dynamics-free feasibility step and a constraint-free LQR step, and employing warm-starts, the authors derive conditions under which the resulting suboptimal closed-loop is recursively feasible and locally asymptotically stable within a forward-invariant region, even with active state and input constraints. They establish Lipschitz continuity of the optimal solution map, an explicit forward-invariant ROA for the optimal MPC, and an ISS-type bound that leads to a computable lower bound on the required ADMM iterations, ell^*. Numerical simulations on a double integrator validate the theory and illustrate computation-performance tradeoffs. The results support deployment in compute-constrained layered control architectures and provide a foundation for extensions to nonlinear dynamics and tracking tasks.

Abstract

Many practical applications of optimal control are subject to real-time computational constraints. When applying model predictive control (MPC) in these settings, respecting timing constraints is achieved by limiting the number of iterations of the optimization algorithm used to compute control actions at each time step, resulting in so-called suboptimal MPC. This paper proposes a suboptimal MPC scheme based on the alternating direction method of multipliers (ADMM). With a focus on the linear quadratic regulator problem with state and input constraints, we show how ADMM can be used to split the MPC problem into iterative updates of an unconstrained optimal control problem (with an analytical solution), and a dynamics-free feasibility step. We show that using a warm-start approach combined with enough iterations per time-step, yields an ADMM-based suboptimal MPC scheme which asymptotically stabilizes the system and maintains recursive feasibility.
Paper Structure (11 sections, 9 theorems, 62 equations, 3 figures)

This paper contains 11 sections, 9 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Suboptimal ADMM-MPC via Operator Splitting
  4. Optimal MPC Properties
  5. Closed-loop Stability Analysis
  6. Numerical Simulations and Discussion
  7. Conclusions
  8. Derivation of Proposition \ref{['prop:admm-convergence-thm6']}
  9. Detailed steps in the proof of Theorem \ref{['thm:asympt-stable-suboptimal-cl']}
  10. A summary of our main result in Theorem \ref{['thm:asympt-stable-suboptimal-cl']}
  11. Additional numerical comparisons

Key Result

Proposition 1

nishihara2015general Let Assumptions assum:riccati-assum:closed-mpc hold, and suppose at time $t$, ${\boldsymbol{\varphi}}_t^\ell= \left[ \boldsymbol{r}_{t}^{\ell\top}, \boldsymbol{y}_{t}^{\ell\top}\right]^\top$ is generated by running the optimization eq:linear_ocp_admm with $\alpha \in (0, 2)$ and where $F = \otimes I_{2s}$.

Figures (3)

  • Figure 1: Performance of ADMM-based suboptimal MPC for the system \ref{['eq:double_integrator']} for varying number of ADMM iterations $\ell$.
  • Figure 2: We provide a comparison of state trajectories in (a) and inputs in (b) computed by our approach and the algorithm from schulze2021closed for the double integrator example in Section V of our paper. The red dotted lines denote the boundaries of state and input constraints. The blue lines are from our approach, the green dotted lines are from the optimal MPC and the orange lines are from schulze2021closed. We run $\ell = 30$ ADMM iterations for our approach.
  • Figure 3: We provide a comparison of state trajectories in (a) and inputs in (b) computed by our approach and the algorithm from schulze2021closed for the double integrator example from schulze2021closed with $B = [0.5; 1]$. The red dotted lines denote the boundaries of state and input constraints. The blue lines are from our approach, the green dotted lines are from the optimal MPC and the orange lines are from schulze2021closed. We run $\ell = 30$ ADMM iterations for our approach.

Theorems & Definitions (19)

  • Proposition 1
  • Remark 1
  • Definition 1
  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 9 more