The math that lets stormwater pipes think before it rains

Somewhere beneath a city street, a stormwater system is deciding what to do with the rain. It is not a glamorous place to think about artificial intelligence and control theory, but it is where a quiet revolution is already happening. Rather than pouring concrete and digging new tunnels every time rain gets heavier, a growing number of cities are teaching their existing pipes, reservoirs, and gates to react intelligently — opening valves before a storm hits, holding water back from one catchment while releasing it from another.

The mathematics behind this kind of "smart water" system is genuinely difficult, and a new paper from researchers at the University of Michigan and a control-theory group now attacks a piece of the problem that had been left unsolved. Jonathan Shell, Sepehr Moalemi, Branko Kerkez, and Jeff Scruggs have extended a framework called performance-guaranteed control (PGC) so that it can do something its predecessor could not: track a target — like a desired water level in a reservoir — rather than merely fight off disturbances (Shell et al., 2026). The catch is that the very physics of these systems imposes a strange nonlinear constraint, and the authors have found a way to guarantee performance despite it. The work, accepted to the 65th IEEE Conference on Decision and Control, is only six pages, but it closes a gap that stood between theory and a lot of real-world infrastructure.

The Science

To understand what these researchers did, you first have to appreciate the peculiar challenge of controlling water infrastructure. A stormwater system is a network of basins, pumps, and gates that shunts rainfall away from streets and basements. In a smart system, sensors measure water levels, and actuators — adjustable gates, controllable pumps — respond automatically. The goal sounds simple: keep water levels below flood thresholds, or hold them at some preferred operating point.

But there is a twist in the physics. Many of these actuators are unidirectional in a very specific sense. Consider a gate that can only release water from a reservoir. It can push stored water out, but it can never pump water back in. Or consider a pump that moves water one way. This creates what the authors call a power directionality constraint: the control effort — the power delivered by the actuator — is constrained in sign. You can only add power in one direction; you cannot subtract it.

Formally, this means the control input $u(t)$ must satisfy something like $u(t)\ge 0$ at all times (or, for pumps, that the control effort has a one-sided bound). That single inequality makes the problem dramatically harder. Feedback synthesis — designing the rule that maps sensor readings to actuator commands — is normally a clean, linear affair when you can push and pull freely. The moment you clamp the sign of the control, the closed-loop system becomes nonlinear, and the standard toolbox of linear control theory stops applying cleanly.

The situation is even more subtle because of how power flows in these systems. In a hydraulic network, power is the product of pressure and flow, and directionality constraints on power mean that certain states are reachable in one direction but not the other. A reservoir can fill from rain and empty through a gate, but it cannot be refilled by the gate. The system is genuinely asymmetric.

The team's key move is to develop a stochastic control synthesis procedure — a way of designing the controller — that yields provable performance bounds on mean-square reference tracking for the whole class of problems exhibiting this power directionality. The word "provable" matters. It means the guarantee is mathematical, not merely empirical: regardless of the particular stochastic disturbance realization, the controller is guaranteed, in expectation, to keep the tracking error within a computed bound.

What They Found

The central result is an extension of the PGC framework. Originally, PGC was developed for disturbance rejection — keeping a system at a setpoint despite external noise. Here, the authors generalize it to reference tracking — following a time-varying target, which is a considerably harder objective.

The contributions unfold in a few distinct steps. First, the paper develops the general synthesis procedure for the class of systems with power directionality constraints, giving a formal performance guarantee on how well the closed-loop system tracks a stochastic reference signal. Second, it instantiates this theory on a concrete, meaningful test case: a flood mitigation scenario built on a numerical model of a real-world smart water system.

The flood example is where the abstraction becomes tangible. The system model involves a stormwater catchment where an actuator must hold a reservoir level near a reference — the "target" water level — while rain arrives as a stochastic disturbance. With only a one-sided control authority (the actuator can only draw water down, not push it up), the controller must anticipate the stochastic rainfall and pre-position the reservoir so it has room when the storm hits. The tracking error, checked against the analytical bound, stays within the guaranteed envelope even as the reference moves.

What makes this nontrivial is the coupling between the directionality constraint and the stochastic reference. When the reference itself varies in time, the controller must decide when to act — and the one-sided nature of the control means that a wasted opportunity (using the actuator when you didn't need to) cannot be undone. The provable bound captures this tension quantitatively.

A second contribution concerns the structure of the performance guarantee itself. The paper shows that the mean-square tracking error satisfies a bound that degrades gracefully as the power directionality constraint becomes more binding. In other words, the guarantee tells you exactly how much performance you give up by having a one-sided actuator versus an idealized two-sided one. This is practically valuable: an operator can look at the bound and decide whether a unidirectional actuator is good enough for the job or whether they need a more expensive bidirectional one.

Why This Changes Things

The broader significance is about how cities can expand stormwater capacity without pouring concrete. Building new infrastructure — bigger basins, deeper tunnels, more pumps — is enormously expensive and slow. Retrofitting existing infrastructure with sensors and controllers is far cheaper and can be deployed much faster. But that approach only works if the control theory behind it is sound. And for a long time, the theory had a hole.

That hole was precisely the reference tracking problem with power directionality. Some of the most useful operating strategies for smart stormwater systems are inherently tracking problems: hold reservoir X at 80% capacity; keep the outflow from catchment Y below a threshold; follow a schedule that pre-emptively lowers levels before forecast rain. These are not disturbance rejection problems — they are reference tracking problems. The earlier PGC framework, built for the former, could not certify the latter. This paper fixes exactly that.

The practical consequence is a certificate of safety. When a municipality deploys an automated gate that can only release water, the public agency needs to know, with confidence, that the closed-loop system will not let the reservoir overflow or the basements flood. Empirical tuning gives you hope; a provable bound gives you a guarantee. This is the difference between "we tested it in ten simulations" and "we can prove, in expectation, that the tracking error is bounded by this number regardless of the rain sequence."

There's also a broader principle worth noting. Power directionality constraints are not unique to water. They show up in a wide range of physical systems: one-way valves in heating and cooling, rectifiers in power electronics, unidirectional actuation in robotics and prosthetics, storage systems that can only charge or only discharge in a given phase. Whenever a system's control authority is sign-constrained, the same class of mathematical difficulty appears. The paper's general synthesis procedure is therefore not just a water paper. It is a contribution to the control theory of one-sided actuation, with potential reach into energy storage, building control, and beyond.

The extension matters conceptually too. Disturbance rejection treats the system as defending a fixed point. Reference tracking treats the system as an agent pursuing a moving target. The latter is a much richer — and much more human — way to think about infrastructure. A flood control system that tracks a reference is not just reacting; it is planning, positioning itself ahead of the event. That is closer to how a skilled human operator thinks than the older feedback paradigm.

What's Next

As with any good paper, the open questions are as interesting as the answered ones. The current work establishes the performance guarantee in the mean-square sense — a sum over time and expectation over randomness. A natural next step is to strengthen this to more stringent guarantees, such as bound with high probability, or transient/overshoot guarantees that make the controller's behavior more transparent to operators who are nervous about automated decisions.

Another direction is the deepening of the "directionality" concept itself. The paper treats a one-sided constraint on control power. But real systems often have partial, time-varying, or state-dependent directionality — a valve that can open freely but close slowly, a reservoir whose capacity shrinks as it fills, a pump with efficiency that varies with head. Extending the provable framework to these richer constraints would make the guarantees applicable to an even wider range of hardware.

There is also the question of multi-agent coordination. Real stormwater systems are networks of many reservoirs and actuators working together. This paper analyzes a single controlled subsystem. Extending the provable performance bounds to coupled networks — where the tracking of one reservoir affects the tracking of its neighbor — is a formidable but crucial step toward city-scale deployment.

Finally, there is the bridge from simulation to field. The flood example uses a numerical model of a real-world smart water system, which is an honest and rigorous way to test the theory. But the ultimate test is in the field, with real rain, real gates, and real operator nerves. The authors' framework — with its provable bounds — is precisely the kind of tool that makes field deployment safer and more palatable to risk-averse infrastructure agencies.

What this paper represents, in the end, is a shift in how we think about urban resilience. The default answer to "the pipes aren't big enough" has always been "build bigger pipes." But there is another answer: "teach the pipes to think." That answer is cheap, fast, and — as this work shows — increasingly backed by mathematics rather than hope. The next big storm will be the real exam, and the theory now has a much better chance of passing.