A new control method produced substantially lower tracking error than optimized static feedback in a numerical watershed simulation, according to a preprint. The performance-guaranteed controller’s running-average tracking-error norm was 0.33 feet, compared with 2.02 feet for the static controller — an 83.6% lower value in the reported simulation.
The result comes from mathematical analysis and a closed-loop SWMM/PySWMM simulation using modeled rainfall. It should therefore be read as evidence from the study’s numerical framework, rather than as a result established across multiple reported experiments.
A control problem built around uncertainty
The paper studies stationary mean-square reference tracking: measuring how closely a system’s output stays to a desired target in a steady statistical setting. In the watershed application, the outputs are basin depths, and the controller must respond while the model is driven by stochastic disturbances.
The proposed approach is designed for a setting in which the disturbance distribution does not have to be known in full. The algorithm assumes that only the disturbance’s first and second moments — quantities describing its average and variability — are known.
It also works under a power-directionality constraint. The theory assumes that this restriction can be represented as a sector, a bounded mathematical relationship that limits which directions of power transfer are feasible.
Two stages, with the harder calculation done first
The controller is built in two stages. First, the researchers find an optimized feasible linear time-invariant, or LTI, static-feedback controller. This provides the baseline against which the nonlinear design is assessed.
That first optimization is nonconvex, meaning the search has a difficult form and is not treated as a simple, single-step calculation. It is solved once offline, before the controller is used in the time-stepped simulation.
The second stage is a nonlinear full-state controller. At each time step, it minimizes a derived performance increment while remaining within the power-directionality feasibility conditions. The paper describes this per-time optimization as convex and suitable for efficient real-time solution.
The central theoretical result is a relative guarantee: under the paper’s assumptions, the proposed controller is stated to perform no worse than the optimized LTI static controller. That is a comparison with the selected baseline, not a claim that the method is best among every possible nonlinear causal controller.
A modeled watershed with three basins
To illustrate the method, the researchers used a high-fidelity numerical watershed model. It contains three catchment areas and three basins, with controllable links between the East and West basins and between the West basin and the outfall.
The simulation used a state-space model produced through N4SID subspace identification. It represented six latent-state coordinates, two realized valve-flow inputs, two passive outputs, and three basin-depth outputs.
The researchers then ran closed-loop SWMM simulations through PySWMM with NSRP rainfall. The comparison set the East Basin target at 4 feet and the West Basin target at 3 feet, allowing the two controller designs to be judged against the same stated reference levels.
The application also required an adjustment to the identified model. The input-output map was not exactly passive under the discrete-time KYP condition, so the researchers computed a nearby passive approximation with CVXPY while keeping the identified state dynamics fixed. The reported matrix differences were 0.171071 for Cv and 0.004230 for Du.
Closer to the stated water-depth targets
In the displayed simulation, the PGC controller’s average depths were closer to the stated targets than those under static feedback. For the output labeled y2, the reported average was 3.97 feet with PGC versus 2.57 feet with static control. For y3, the corresponding values were 2.95 feet and 2.36 feet.
The broader running-average error measure pointed in the same direction. The PGC value of 0.33 feet was lower than the 2.02-foot value for optimal static feedback, giving the reported 83.6% reduction in tracking-error norm.
The reported figures do not include a confidence interval, uncertainty estimate, statistical test, or summary across independent rainfall realizations. The supplied analysis also reports no replicate count or sample-size rationale.
A guarantee with clear boundaries
The guarantee depends on the study’s model and assumptions, including stationarity, passivity, sector feasibility, and controller feasibility. The watershed’s power-directionality relationship was conservatively approximated by a sector, so the result does not establish performance outside the stated sector approximation.
The passive-model step is another reason to treat the result as a test of the framework under an approximation. The identified map did not exactly satisfy the passivity condition and was replaced by the nearby passive model used in the controller design.
The authors explicitly stop short of claiming global optimality over all nonlinear causal controllers. The guarantee is relative to the optimized LTI static-feedback design, and the study does not show that this controller is the best possible choice across the full class of nonlinear strategies.
Questions left open include whether the reported improvement persists across independent rainfall realizations and whether the guarantee continues to hold beyond the conservative sector approximation. The authors also identify forecast-aware model predictive control as a direction for future work.
What readers should know
The document is a preprint, identified as arXiv:2608.20120v1 and dated 20 August 2026. The authors report support from NSF grants 2206018 and 2426817.
Taken together, the paper offers a mathematical performance bound and a numerical demonstration for stochastic reference tracking with passive valves. Its headline improvement is a result from the reported simulation, while the formal guarantee remains relative to optimized LTI static feedback and dependent on the stated assumptions.
Paper data and sources
Original title: Performance-Guaranteed Reference Tracking With Power Directionality Constraints: Application to Controlled Stochastic Watersheds
Authors: Jonathan Shell, Sepehr Moalemi, Branko Kerkez, Jeff Scruggs
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text