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Sampled-Data Primal-Dual Gradient Dynamics in Model Predictive Control

Ryuta Moriyasu, Sho Kawaguchi, Kenji Kashima

TL;DR

This work addresses fast, reliable MPC for linear systems by transitioning from continuous-time PDG controllers to a discrete-time PDG controller that accounts for sampling. It introduces a tunable gain $\zeta$ and a time-varying step-size $\gamma_k$ to guarantee Lyapunov decrease and stability in the resulting sampled-data system, complemented by a projection-based mechanism to improve constraint fulfillment. The authors derive discrete-time stability conditions within a dissipativity framework and demonstrate the approach on a DC motor and a diesel engine, achieving microsecond-scale computation while maintaining stability and improved performance. The method offers a practical route to ultra-fast MPC on resource-constrained hardware, with clear pathways to enhance constraint satisfaction further through projections or barrier-based methods.

Abstract

Model Predictive Control (MPC) is a versatile approach capable of accommodating diverse control requirements that holds significant promise for a broad spectrum of industrial applications. Noteworthy challenges associated with MPC include the substantial computational burden, which is sometimes considered excessive even for linear systems. Recently, a rapid computation method that guides the input toward convergence with the optimal control problem solution by employing primal-dual gradient (PDG) dynamics as a controller has been proposed for linear MPCs. However, stability has been ensured under the assumption that the controller is a continuous-time system, leading to potential instability when the controller undergoes discretization and is implemented as a sampled-data system. In this paper, we propose a discrete-time dynamical controller, incorporating specific modifications to the PDG approach, and present stability conditions relevant to the resulting sampled-data system. Additionally, we introduce an extension designed to enhance control performance, that was traded off in the original. Numerical examples substantiate that our proposed method, which can be executed in only 1 $μ$s in a standard laptop, not only ensures stability with considering sampled-data implementation but also effectively enhances control performance.

Sampled-Data Primal-Dual Gradient Dynamics in Model Predictive Control

TL;DR

This work addresses fast, reliable MPC for linear systems by transitioning from continuous-time PDG controllers to a discrete-time PDG controller that accounts for sampling. It introduces a tunable gain and a time-varying step-size to guarantee Lyapunov decrease and stability in the resulting sampled-data system, complemented by a projection-based mechanism to improve constraint fulfillment. The authors derive discrete-time stability conditions within a dissipativity framework and demonstrate the approach on a DC motor and a diesel engine, achieving microsecond-scale computation while maintaining stability and improved performance. The method offers a practical route to ultra-fast MPC on resource-constrained hardware, with clear pathways to enhance constraint satisfaction further through projections or barrier-based methods.

Abstract

Model Predictive Control (MPC) is a versatile approach capable of accommodating diverse control requirements that holds significant promise for a broad spectrum of industrial applications. Noteworthy challenges associated with MPC include the substantial computational burden, which is sometimes considered excessive even for linear systems. Recently, a rapid computation method that guides the input toward convergence with the optimal control problem solution by employing primal-dual gradient (PDG) dynamics as a controller has been proposed for linear MPCs. However, stability has been ensured under the assumption that the controller is a continuous-time system, leading to potential instability when the controller undergoes discretization and is implemented as a sampled-data system. In this paper, we propose a discrete-time dynamical controller, incorporating specific modifications to the PDG approach, and present stability conditions relevant to the resulting sampled-data system. Additionally, we introduce an extension designed to enhance control performance, that was traded off in the original. Numerical examples substantiate that our proposed method, which can be executed in only 1 s in a standard laptop, not only ensures stability with considering sampled-data implementation but also effectively enhances control performance.
Paper Structure (15 sections, 36 equations, 6 figures, 4 tables, 1 algorithm)

This paper contains 15 sections, 36 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries
  4. Assumptions
  5. Instant MPC
  6. Additional Considerations
  7. Method
  8. Sampled-data System
  9. Assumptions
  10. Controller Design
  11. Improving constraint fulfillment
  12. Numerical Experiments
  13. Control performance
  14. Computational efficiency
  15. Conclusion

Figures (6)

  • Figure 1: Schematics of feedback systems. $\mathcal{P},\mathcal{C},\mathcal{S},\mathcal{H}$ represent a plant, a controller, a sampler, and a holder, respectively. The subscript $\rm c,d$ indicate continuous-time and discrete-time, respectively.
  • Figure 2: Results of Case 1 ($\zeta=1, \Delta t =$ 1 ms)
  • Figure 3: Results of Case 2 ($\zeta=10, \Delta t =$ 1 ms)
  • Figure 4: Results of Case 3 ($\zeta=100, \Delta t =$ 1 ms)
  • Figure 5: Results of Case 5 ($\zeta=1000, \Delta t =$ 0.1 ms)
  • ...and 1 more figures