Nonlinear Optimal Control Benchmark

Updated 28 July 2025

Nonlinear optimal control benchmark is a standardized set of complex test cases with defined metrics, used for validating and comparing nonlinear control algorithms across various domains.
It employs advanced methodologies such as sequential linear quadratic programming, Galerkin approximations, and symbolic abstractions to address challenges in switched systems and constrained dynamics.
The benchmarks enhance reproducibility and practical insights by quantifying performance metrics like cost, constraint satisfaction, and computational runtime in fields like robotics, power networks, and aerospace.

A nonlinear optimal control benchmark establishes standardized or canonical problems, methodologies, and performance metrics for the rigorous evaluation, comparison, and validation of nonlinear optimal control algorithms. Such benchmarks are of foundational importance for theoretical assessment, algorithmic development, and reproducible research in nonlinear control, estimation, and hybrid and switched systems. Benchmarks typically involve challenging model classes (e.g., nonlinear switched systems, systems with constraints, or high-dimensional power networks), well-defined cost and constraint structures, and standard simulation, convergence, or computational runtime metrics.

1. Problem Classes and Key Benchmark Structures

Nonlinear optimal control benchmarks have been developed across a spectrum of problem classes—switched/hybrid systems, PDE-based systems in Hilbert spaces, high-dimensional power networks, time-optimal trajectories, and robust and constrained settings. Some canonical representative structures include:

Nonlinear Switched Systems: Problems where the plant switches among nonlinear subsystems, with fixed or optimizable switching sequences and switching times, as in "Sequential Linear Quadratic Optimal Control for Nonlinear Switched Systems" (Farshidian et al., 2016).
Hilbert Space and PDE-Control Problems: Infinite-dimensional systems (e.g., semilinear heat equations, energy balance climate models) approximated with Galerkin techniques, providing rigorous theoretical benchmarks for value function and trajectory convergence (Chekroun et al., 2017).
Symbolic and Abstraction-Based Control: Discrete-time, continuous-state nonlinear systems in which state and input space abstractions yield symbolic feedback controllers with provable value function convergence (Reissig et al., 2017).
Robust, Constrained, and Economic MPC: Nonlinear systems with parametric uncertainties or constraints, using bi-level optimization, system-level synthesis, or economic MPC frameworks for periodic operation and economic performance (Köhler et al., 2020, Leeman et al., 2023, Abdul et al., 6 Apr 2025).
Time-Optimal and Bang-Bang Control: Formulations leveraging hierarchical least-squares programming and continuous approximations of switching functions to recover bang-bang profiles in nonlinear and hybrid settings (Pfeiffer et al., 2023).

These problem classes reflect typical real-world nonlinear control challenges, including mode scheduling, constraint satisfaction, robustness to uncertainties, and optimal system performance across diverse domains.

2. Methodological Foundations in Benchmark Design

Representative benchmarks crystallize methodological advances and enable algorithmic evaluation on a common footing. Key methodologies include:

Sequential Linear Quadratic (SLQ) Methods: These iterative algorithms linearize the nonlinear switched dynamics about nominal trajectories and apply quadratic cost approximations, enabling the use of dynamic programming (Riccati equations) and analytical gradients with respect to switching times for computationally efficient solution of high-dimensional switched OCPs (Farshidian et al., 2016).
Galerkin Projection and Approximation Theory: For evolution equations in Hilbert spaces, Galerkin approximations discretize the infinite-dimensional problem onto finite-dimensional subspaces (using, e.g., eigenfunction expansions). Convergence theorems under checkable assumptions link the approximation of state trajectories and value functions, providing rigorous performance benchmarks across nonlinear PDE-control problems (Chekroun et al., 2017).
Symbolic Abstractions and Hypo-Convergence: By discretizing state, input, and cost and forming abstractions, symbolic optimal control approaches offer a generic minimax framework guaranteeing that abstract value functions (and the performance of the resulting memoryless feedback) converge to the true value in the original continuous system (Reissig et al., 2017).
Bi-Level and System Level Synthesis Approaches: In robust nonlinear optimal control, decomposing the uncertain nonlinear system into nominal and error components enables the deployment of SLS-based feedback synthesis and convex overbounding, yielding tight, computationally tractable robust controllers with quantifiable feasibility/performance—illustrated in applications such as post-capture satellite stabilization (Leeman et al., 2023).
Time-Optimal and Hierarchical Least Squares Programming: Nonlinear hierarchical least-squares methods reformulate discrete time-optimal control as prioritized optimization layers. The introduction of differentiable Heaviside approximations with enhanced non-vanishing gradients enables the recovery of bang-bang controllers in both linear and nonlinear benchmarks (Pfeiffer et al., 2023).
Dynamic Programming and Iterative Linear Quadratic/DP Templates: Differentiable programming and dynamic programming templates enable the effective design of scalable, fast-converging iterative solvers (including DDP, Gauss–Newton, and Newton) for nonlinear systems, validated across benchmarks such as pendulum/cart and autonomous vehicle racing (Roulet et al., 2022).

3. Metrics, Performance Measures, and Model Problems

Robust benchmarking depends on explicit, quantitative metrics, including:

Metric Type	Example Formulation	Notable Benchmarks
Cost Function	$J = \int_0^T l(x,u)\,dt + \Phi(x_T)$ or discrete sum	SLQ (Farshidian et al., 2016), PID/MPC (Dehghan et al., 2018)
Constraint Satisfaction	$c_i(x_k, u_k) \leq 0$	Robust/ALM approaches (Lv et al., 20 Mar 2025, Abdul et al., 6 Apr 2025)
Computational Runtime/Scalability	Wall-clock time, scaling with dimension	Switched systems (Farshidian et al., 2016), DDP/DP (Roulet et al., 2022)
Approximation/Convergence	$\sup_{t \in [0,T]} \\|y_n(t)-y(t)\\| \to 0$	Galerkin/PDE (Chekroun et al., 2017)
Benchmark Instances	Two-mode systems, pendulum swing-up, AGV tracking	Model problems in cited works

Systems used in benchmarking include switched-mode dynamical systems (e.g., 2D, 4D, and high-dimensional robotic or power networks), PDE-based climate models, energy balance systems, hybrid/impulsive models (bouncing ball, SLIP), AGV and mobile robot trajectory tracking, and time-optimal point-to-point manipulations.

Performance is assessed with a combination of:

Final value of the cost function (including tracking or economic indices)
Constraint violation rates
Computational effort (total/average solver time, iteration count)
Convergence rates (superlinear, linear, or otherwise)
Fidelity and consistency of value function approximation and optimal control law (e.g., compared to analytical or numerically precise solutions).

4. Scalability, Computational Feasibility, and Efficiency

A critical aspect for modern nonlinear optimal control benchmarks is the computational tractability and scalability of solution algorithms. Several benchmark studies report orders-of-magnitude improvements:

The SLQ-based OCS2 approach achieves CPU times reduced from 235s (baseline BVP) to 31s for a 2D switched problem and from ~1500s to 76s in a higher-dimensional setting, demonstrating linear scalability with problem size due to dynamic programming structure (Farshidian et al., 2016).
In benchmarking nonlinear economic MPC and robust SLS-based schemes, recursive feasibility and convex constraint structures enable practical real-time control in large-scale systems (Köhler et al., 2020, Leeman et al., 2023).
The explicit FBDE-based constrained optimal control algorithm outperforms standard fmincon/interior-point/SQP implementations by a factor of ~5, achieving update times of ~1.8 ms per iteration versus 8–20 ms in AGV path tracking (Lv et al., 20 Mar 2025).
In robust trajectory optimization under worst-case disturbances, successive convexification combined with robust inner-loop reformulations (solved via ADMM) enforce near-complete satisfaction of robust constraints, whereas non-robust approaches fail in >99% of Monte-Carlo scenarios (Abdul et al., 6 Apr 2025).

Such empirical metrics are reported alongside theoretical guarantees of uniform convergence, recursive feasibility, robustness, and constraint satisfaction, establishing a clear mathematically and practically motivated hierarchy of solution quality.

5. Application Domains and Practical Impact

Nonlinear optimal control benchmarks have direct relevance for diverse application areas:

Robotics: Quadruped locomotion (e.g., CoM tracking with switched stance phases), underactuated system swing-up (cart-pendulum, pendubot), robot arms, whole-body motion with contact and constraint management (Farshidian et al., 2016, Ansari et al., 2017, Pfeiffer et al., 2023).
Power Systems: High-dimensional, switched-mode regulation of electrical grids, including the prototypical IEEE 118-bus test case; assessment of real-time implementation feasibility for continental networks (Caldwell et al., 2017).
Climate and Geophysics: Optimal management of energy balance models and geoengineering strategies via PDE-based optimal control on manifolds (Chekroun et al., 2017).
Autonomous Vehicles: Trajectory optimization for AGV systems, including obstacles and state/input limits, evaluation in both simulated and experimental platforms (Lv et al., 3 Nov 2024, Lv et al., 20 Mar 2025).
Aerospace: Robust trajectory planning and closed-loop stabilization for uncertain spacecraft post-capture maneuvers (Leeman et al., 2023).
Machine Learning: Turnpike theory applied to deep neural ODEs, supporting architectural decisions and convergence analyses (Esteve-Yagüe et al., 2020).

Benchmarks thereby both motivate algorithmic advances and yield critical insight into the interplay of model structure, computational demands, and real-world constraints across modern control engineering domains.

6. Theoretical Impact: Convergence, Regularization, and Robustness

Nonlinear optimal control benchmarking is grounded in theoretical properties:

Convergence under Approximation: Uniform convergence results (e.g., for Galerkin approximations) and established error bounds set strict standards against which new methods (including RL, DP, or learning-based approaches) must be measured (Chekroun et al., 2017).
Robustness Guarantees: Frameworks such as robust SLS, trust-region successive convexification, and state/constraint convergences quantify guaranteed feasibility and output bounds over all relevant disturbances (Leeman et al., 2023, Abdul et al., 6 Apr 2025).
Regularization Strategies: Infinite-horizon problems are regularized via matched terminal cost design and free final-time splitting, ensuring both tractability and asymptotic stability (Mohamed et al., 2023).
Turnpike and Invariant Manifold Theory: Modern benchmarks rationalize long-time system behavior (exponential convergence to steady-state) and structure-preserving spectral approaches (via Koopman/symplectic theory), motivating deep links between optimal control and dynamical systems (Villanueva et al., 2020, Esteve-Yagüe et al., 2020).

This theoretical underpinning ensures that benchmarks are not ad hoc but are instead foundationally rigorous, influencing the discipline’s standards of validity and proper methodology.

7. Summary Table: Core Dimensions in Nonlinear Optimal Control Benchmarking

Dimension	Description / Example Papers
Model Class	Nonlinear switched (Farshidian et al., 2016), PDEs (Chekroun et al., 2017), hybrid (Ansari et al., 2017)
Solution Methodology	SLQ/DP (Roulet et al., 2022), Galerkin approx. (Chekroun et al., 2017), SLS/convexification (Leeman et al., 2023, Abdul et al., 6 Apr 2025)
Performance Evaluation	Cost, constraint violation, runtime, value function convergence
Application Domain	Robotics (Farshidian et al., 2016, Ansari et al., 2017), power systems (Caldwell et al., 2017), climate (Chekroun et al., 2017)
Scalability Considerations	Explicit reporting of CPU time, state/input dimension scaling
Robustness Properties	Modeling/linearization error treated explicitly (Leeman et al., 2023, Abdul et al., 6 Apr 2025)
Real-World Feasibility	Experimental validation (AGV, satellite, robot) (Lv et al., 3 Nov 2024, Lv et al., 20 Mar 2025, Leeman et al., 2023)

Benchmarks in nonlinear optimal control thus encapsulate formally precise problem statements, algorithmic frameworks grounded in mathematical control theory, empirically quantified computational and performance metrics, and relevance across a variety of high-impact control domains. They serve both as driving forces for innovation and as objective standards for the validation of new methodologies, ensuring that progress in nonlinear control remains both rigorous and reproducible.