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An Execution-time-certified Riccati-based IPM Algorithm for RTI-based Input-constrained NMPC

Liang Wu, Krystian Ganko, Shimin Wang, Richard D. Braatz

TL;DR

This work addresses the challenge of providing execution-time certificates for RTI-based input-constrained NMPC. It introduces a time-certified feasible interior-point method that solves the condensed Box-QP arising from RTI NMPC, using a factorized Riccati recursion to compute the Newton step with complexity that scales linearly with the prediction horizon. Sensitivities from continuous-time dynamics are computed via RK4 to build the NMPC QP efficiently, and Hessian assembly in the preparation phase is avoided. The approach yields a fixed, data-independent iteration count bound and is validated on the chaotic Lorenz system, confirming the ability to meet strict sampling-time deadlines. This enables practical deployment of NMPC with tight temporal guarantees in real-time environments.

Abstract

Establishing an execution time certificate in deploying model predictive control (MPC) is a pressing and challenging requirement. As nonlinear MPC (NMPC) results in nonlinear programs, differing from quadratic programs encountered in linear MPC, deriving an execution time certificate for NMPC seems an impossible task. Our prior work \cite{wu2023direct} introduced an input-constrained MPC algorithm with the exact and only \textit{dimension-dependent} (\textit{data-independent}) number of floating-point operations ([flops]). This paper extends it to input-constrained NMPC problems via the real-time iteration (RTI) scheme, which results in \textit{data-varying} (but \textit{dimension-invariant}) input-constrained MPC problems. Therefore, applying our previous algorithm can certify the execution time based on the assumption that processors perform fixed [flops] in constant time. As the RTI-based scheme generally results in MPC with a long prediction horizon, this paper employs the efficient factorized Riccati recursion, whose computational cost scales linearly with the prediction horizon, to solve the Newton system at each iteration. The execution-time certified capability of the algorithm is theoretically and numerically validated through a case study involving nonlinear control of the chaotic Lorenz system.

An Execution-time-certified Riccati-based IPM Algorithm for RTI-based Input-constrained NMPC

TL;DR

This work addresses the challenge of providing execution-time certificates for RTI-based input-constrained NMPC. It introduces a time-certified feasible interior-point method that solves the condensed Box-QP arising from RTI NMPC, using a factorized Riccati recursion to compute the Newton step with complexity that scales linearly with the prediction horizon. Sensitivities from continuous-time dynamics are computed via RK4 to build the NMPC QP efficiently, and Hessian assembly in the preparation phase is avoided. The approach yields a fixed, data-independent iteration count bound and is validated on the chaotic Lorenz system, confirming the ability to meet strict sampling-time deadlines. This enables practical deployment of NMPC with tight temporal guarantees in real-time environments.

Abstract

Establishing an execution time certificate in deploying model predictive control (MPC) is a pressing and challenging requirement. As nonlinear MPC (NMPC) results in nonlinear programs, differing from quadratic programs encountered in linear MPC, deriving an execution time certificate for NMPC seems an impossible task. Our prior work \cite{wu2023direct} introduced an input-constrained MPC algorithm with the exact and only \textit{dimension-dependent} (\textit{data-independent}) number of floating-point operations ([flops]). This paper extends it to input-constrained NMPC problems via the real-time iteration (RTI) scheme, which results in \textit{data-varying} (but \textit{dimension-invariant}) input-constrained MPC problems. Therefore, applying our previous algorithm can certify the execution time based on the assumption that processors perform fixed [flops] in constant time. As the RTI-based scheme generally results in MPC with a long prediction horizon, this paper employs the efficient factorized Riccati recursion, whose computational cost scales linearly with the prediction horizon, to solve the Newton system at each iteration. The execution-time certified capability of the algorithm is theoretically and numerically validated through a case study involving nonlinear control of the chaotic Lorenz system.
Paper Structure (14 sections, 9 theorems, 34 equations, 1 figure, 3 algorithms)

This paper contains 14 sections, 9 theorems, 34 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. RTI-based input-constrained NMPC
  5. Computing the sensitivities $\{A_{t,k},B_{t,k},r_{t,k}\}_{k=0}^{N-1}$ from continuous-time systems
  6. $\text{QP}_{\text{NMPC}}$ construction
  7. Time-certified Riccati-based IPM Algorithm
  8. Strictly feasible initial point
  9. Newton direction
  10. Implementation of the Newton step via factorized Riccati recursion
  11. Iteration complexity and algorithm implementation
  12. Numerical Example: Chaotic Lorenz Stabilization
  13. Conclusion
  14. Acknowledgement

Key Result

Lemma 1

Let $m_f, m_{f_x},m_{f_u}$ denote the flops required by the evaluation of $f(\cdot),f_x(\cdot),f_u(\cdot)$, respectively. Then the flops required by the computation of the sensitivities $\{A_{t,k},B_{t,k},r_{t,k}\}_{k=0}^{N-1}$ is

Figures (1)

  • Figure 1: Closed-loop simulation of the chaotic Lorenz system with the RTI-based input-constrained MPC. The blue and red lines represent the results of uncontrolled and MPC controlled, respectively. Left: phase trajectory. Middle: time evolution of the three coordinates. Right: the three control inputs.

Theorems & Definitions (11)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2: Initialization strategy
  • Lemma 3: See frison2013efficient
  • Lemma 4: See wu2023direct
  • Lemma 5: See wu2023direct
  • Lemma 6: See wu2023direct
  • Lemma 7: See wu2023direct
  • Lemma 8: See wu2023direct
  • ...and 1 more