Time-Certified and Efficient NMPC via Koopman Operator

Two key requirements for deploying MPC in production are achieving (i) low average execution time, and (ii) a worst-case execution time that remains below the sampling period. While most works focus on improving average execution time Ferreau et al. (2014); Stellato et al. (2020); Wu and Bemporad (2023b,a), the more critical requirement is certifying worst-case execution time, as reflected in the standard assumption that each MPC optimization must be solved before the next sampling instant. where the denominator depends primarily on the embedded processor technology. Determining the worst-case total [flops] requires knowing the worst-case number of iterations of the optimization algorithm used in the NMPC, provided that each iteration has a known and uniform [flops].

Certifying the number of iterations of an iterative optimization algorithm is an open challenge in online parametric NMPC. Generally, the required number of iterations depends on the convergence speed, and the distance between the optimal point and the initial point. The latter is hard to bound in advance in parametric NMPC scenarios, where the problem data varies with the feedback states at each sampling time. For example, NMPC schemes that reformulate the NLP as a sequence of QPs, via successive linearization or real-time iteration Gros et al. (2020), result in a Hessian matrix that varies over time.

Execution time certificate problem has recently become an active research topic in MPC field, see (Richter et al., 2011;Giselsson, 2012;Patrinos and Bemporad, 2013;Cimini and Bemporad, 2017;Arnström and Axehill, 2019;Cimini and Bemporad, 2019;Arnström et al., 2020;Arnström and Axehill, 2021;Okawa and Nonaka, 2021;Wu and Braatz, 2025a,b). Among these works, our prior studies (Wu and Braatz, 2025a,b) provide iteration bounds that are both simple to compute and data-independent. Thus, we applied them for certifying execution times of inputconstrained NMPC problems, see Wu et al. (2024a,b), where the Koopman framework and RTI scheme are used, respectively. But, this is limited to input-constrained NMPC problems because the execution-time-certified algorithm in Wu and Braatz (2025a,b) is only for boxconstrained quadratic programs (BoxQP). Later, Ref. (Wu et al., 2025a) provided execution-time certification and infeasibility detection for general QPs, though its execution time certificate imposes a notable loss in practical speed.

Our latest work (Wu et al., 2025b) developed a predictorcorrector interior-point method (IPM) based BoxQP algorithm, which preserves execution time certificate and comparable computation efficiency at the same time. Based on that, this paper develops a time-certified and efficient NMPC solution via Koopman operator.

To certify execution times for state-input constrained NMPC, this paper makes the following contributions:

i) learns a linear high-dimensional model via Koopman operator; ii) proposes a dynamic-relaxation construction approach that puts ℓ 2 norm of the multi-step prediction model into the objective (rather than handled as equality constraints), thereby yielding a structured BoxQP rather than a general QP; iii) exploits the structure of BoxQP based on the algorithm framework (Wu et al., 2025b), where the dimension of the linear system solved in each iteration is reduced from 5(n u +n x )N p to n u N p (where n u , n x , N p denote the number of inputs, states, and length of prediction hoirzon), yielding substantial speedups (when n x ≫ n u , as in PDE control).

This article considers a nonlinear MPC problem (NMPC) for tracking, as shown in (1),

Nonlinear MPC: (Korda and Mezić, 2018a), which transforms NMPC (1) into a convex QP problem via data-driven Koopman approximations, allows the use of execution-time-certified and computationally efficient QP algorithms for real-time applications.

The Koopman operator (Koopman, 1931) provides a globally linear representation of nonlinear dynamics. In practice, the infinite-dimensional Koopman operator is truncated and approximated using data-driven Extended Dynamic Mode Decomposition (EDMD) methods (Williams et al., 2015(Williams et al., , 2016;;Korda and Mezić, 2018b). In EDMD specifically, the set of extended observables is designed as the “lifted” mapping, ψ(x) u( 0)

, where u(0) denotes the first component of the sequence u and ψ

sis function, e.g., Radial Basis Functions (RBFs) used in Korda and Mezić (2018a), instead of directly solving for them via optimization. In particular, the approximate Koopman operator identification problem is reduced to a least-squares problem, which assumes that the sampled data {(x j , u j ), (x + j , u + j )} (where j denotes the index of data samples and the superscript +