When Should a Robot Open Its Eyes? The Math That Decides

The Smart Sensor That Decides When to Look

Imagine you're flying a drone over a disaster zone. Your battery is at 12 percent. You have minutes left, maybe less. Every time your sensors ping the environment—every LiDAR burst, every sonar pulse—you're burning power you cannot replace. The question isn't just how to fly. It's when to look.

This is the problem that Dung Tran, Tri Ngo, and Tuhin Das at the University of Central Florida have tackled. In a new paper published in Advanced Control for Applications, they ask a question that control theory has largely ignored: What if sensing itself costs something? What if the act of measuring a system is not free, not instantaneous, not unlimited—but something you must budget for, trade off against, and optimize?

Their answer is a mathematical framework that treats sensing as a first-class citizen in the optimal control problem. Not an afterthought. Not a constraint to be patched on later. A variable to be optimized alongside control effort and system stability. The result, in theory, is a controller that knows when to open its eyes—and when to keep them closed.

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The Science

Optimal control is one of the pillars of modern engineering. The classic formulation—known as LQR, for Linear Quadratic Regulator—asks: given a system governed by known dynamics, what control input should you apply to minimize some combination of state error and control effort? The math is elegant. The solution is well-understood. The assumptions are clean.

But these assumptions hide a cost that engineers have always known is real: the cost of knowing the state.

In textbook LQR, the system state is assumed to be available. Full stop. No energy budget. No bandwidth limit. No computational overhead. The controller simply knows where the system is, and computes the appropriate response.

Reality is different. A battery-powered UAV burning through its reserves with every sensor ping. An underwater robot navigating by sonar in the deep ocean, where communication is expensive and power is measured in days. A wireless sensor network in a remote environment, where each transmission costs energy that cannot be harvested fast enough. A planetary rover on Mars, where dust storms can blind instruments and every command sequence must be pre-computed with ruthless efficiency.

In all these cases, the question is not just "how do I control?" but "how do I control when looking at the system costs something?"

The researchers formalized this by introducing a switching variable, denoted z, that controls whether sensing is active. When z = 1, sensors are on. The controller has accurate state information and can compute an optimal response. When z = 0, sensors are off. The controller must operate on outdated information or rely on prediction. Each state comes with a price tag s, which the paper calls the "sensing-cost weight."

The system itself is a Linear Time-Invariant (LTI) system—a common mathematical model where the dynamics don't change over time and the relationship between inputs and outputs is linear. This isn't as restrictive as it sounds. Many real-world systems can be approximated as LTI, and even those that can't can often be controlled by linear controllers designed for their linearized approximations.

The dynamics take the form:

$$\dot{X}=AX+Bu$$

where X is the state vector, u is the control input, and A and B are matrices that describe the system dynamics. The goal is to minimize a cost functional:

$$J={\int }_{t_0}^{t_f}L(X,u,z)dt$$

where L includes not just the usual state-error and control-effort penalties, but a term proportional to the sensing cost s times z—accumulating cost whenever sensors are active.

To solve this, the researchers turned to Pontryagin's Minimum Principle. Developed in the 1950s by Soviet mathematician Lev Pontryagin, this principle provides necessary conditions for optimality in control problems with constraints. It works by introducing "co-state" variables—mathematical ghosts that encode the future cost of current decisions—and requiring that the Hamiltonian (a combined function of state, control, and co-state) be minimized at every moment.

The twist here is that z is not a continuous variable. It can only be 0 or 1. Sensing is either on or off. This makes the problem a hybrid optimal control problem—a system that switches between different modes of operation based on a decision variable. To handle this, the researchers "relaxed" the constraint, allowing z to vary continuously between 0 and 1, solving the resulting problem, and then showing that the optimal z collapses back to the discrete values 0 or 1.

The researchers developed three distinct formulations, corresponding to different assumptions about how the system behaves when sensors are off:

Case 1 assumes that when sensing is disabled, the control input is held constant (its derivative becomes zero). This is a "freeze" strategy—the controller continues applying whatever it was applying, without updates. The augmented state includes both X and u, doubling the system's dimension.

Case 2 assumes that when sensing is disabled, the control input drops to zero. This is a "coast" strategy—the system runs without active control, relying on its natural dynamics to carry it toward the goal. The state equation simplifies accordingly.

Case 3 assumes that when sensing is enabled, an optimal control is computed rather than using a predefined feedback law. This is the most general formulation, but also the most computationally demanding.

Each case yields a different switching condition—a rule that tells the controller when to turn sensing on and when to turn it off. These conditions are not constants. They depend on the current state, the co-state, and the system parameters. They must be evaluated continuously (or at least frequently) as the system evolves.

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What They Found

The core result is a set of switching conditions that determine when the cost of sensing outweighs its benefits.

For Case 1, the switching condition is:

$$z^{\ast }=\{\begin{array}{ll}0& \text{if }{\lambda }_u^{T}K(AX+Bu)<s\\ 1& \text{if }{\lambda }_u^{T}K(AX+Bu)\ge s\end{array}$$

Here, λᵤ is the co-state associated with the control input, K is the predefined feedback gain, and s is the sensing-cost weight. When the quantity λᵤᵀK(AX + Bu) falls below s, the Hamiltonian is minimized by turning sensing off. When it rises above s, sensing should be activated.

For Case 2, the condition simplifies to:

$$z^{\ast }=\{\begin{array}{ll}0& \text{if }{\lambda }^{T}BKX<s\\ 1& \text{if }{\lambda }^{T}BKX\ge s\end{array}$$

This formulation avoids the computational overhead of augmenting the state. When sensors are off, the system runs open-loop, with control set to zero.

For Case 3, where optimal control is computed when sensing is on, the condition becomes:

$$z^{\ast }=\{\begin{array}{ll}0& \text{if }0.25{\lambda }^{T}B{R}^{-1}{B}^T\lambda <s\\ 1& \text{if }0.25{\lambda }^{T}B{R}^{-1}{B}^T\lambda \ge s\end{array}$$

This expression depends only on the co-state and the control-weight matrix R, not on the current state X. The researchers note that this formulation does not permit "singular intervals"—periods where the switching condition hovers at equality—making it potentially more tractable.

To validate these conditions, the researchers solved the resulting Two-Point Boundary Value Problems (TPBVPs) numerically. A TPBVP is a mathematical problem where the solution must satisfy boundary conditions at two different points in time (here, the initial time and the final time). The initial state is known; the final co-state is typically zero; and the intermediate trajectory must satisfy the system dynamics and optimality conditions simultaneously.

They used the "shooting method," which guesses the unknown initial co-state, propagates the system forward, checks the final boundary condition, and iteratively refines the guess until everything lines up. This is computationally intensive but accurate.

The numerical results for a first-order system (a single state variable) are illustrated in

and

, which compare the system's behavior under different sensing-cost weights. When s is low, the controller senses frequently—paying the small cost to maintain accurate state information. When s is high, the controller conserves energy, sensing only when the state drifts too far from optimal.

**

** shows the states and co-states for two scenarios: a low sensing-cost weight and a high sensing-cost weight. Notice that for the low-cost case, sensing remains active more consistently, while for the high-cost case, the co-state trajectory shows extended periods where sensing is disabled. The switching variable z jumps between 0 and 1 according to the derived condition.

**

** provides a direct comparison of the control input and sensing state across these scenarios. The control effort is visibly more aggressive and oscillatory when sensing is continuous (low s) and smoother when sensing is intermittent (high s). The trade-off is explicit: cheaper sensing allows tighter control.

The researchers also derived closed-form solutions for specific cases. For an infinite-horizon first-order system—one that runs forever, with the goal of stabilization rather than reaching a specific target at a specific time—the optimal sensing interval can be expressed analytically. In this limit, the system exhibits a single switching instant: it senses initially to establish an accurate state estimate, then runs open-loop until the state drifts enough to warrant re-sensing.

The infinite-horizon multi-dimensional case with a single switching point yields a reduced-form expression that the researchers present in their paper. While the full derivation involves matrix algebra and Riccati-like equations, the intuition is straightforward: the optimal strategy is to sense at the beginning, then wait as long as possible, then sense again—creating a periodic schedule with sensing bursts separated by open-loop coasting.

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A Practical Test: Wastewater Treatment

Theory is only as valuable as its ability to survive contact with the real world. The researchers knew this. So they applied their framework to a wastewater treatment plant—an application domain chosen for its practical importance and its genuine sensing costs.

A wastewater treatment plant is, in control terms, a biological reactor. Microorganisms consume pollutants. Aeration tanks bubble oxygen into the mixture to keep the microbes alive and active. Sensors measure the concentrations of various compounds—ammonia, nitrate, organic matter—to monitor the process and adjust aeration rates.

But sensors are expensive. Online sensors for biological oxygen demand or nutrient concentrations can cost thousands of dollars and require regular maintenance. More importantly, they consume power. In a large facility, the sensor network's energy footprint is not trivial. And the computational cost of processing continuous sensor streams—filtering noise, running estimators, updating models—adds further overhead.

The researchers modeled a reduced-order wastewater treatment system with two state variables: the concentration of biodegradable organic matter (X₁) and the concentration of ammonia (X₂). The control input is the aeration rate in the reactor, which affects oxygen transfer and thus the biological processes driving pollutant removal.

They simulated three scenarios with different sensing-cost weights:

• Low sensing cost (s = 0.001): The controller senses frequently, maintaining tight control over both concentrations. The aeration rate tracks the optimal trajectory closely.
• Intermediate sensing cost (s = 0.1): The controller balances accuracy and economy, sensing periodically rather than continuously.
• High sensing cost (s = 0.5): The controller minimizes sensing, allowing greater oscillations in the state variables but reducing sensor activation.

The simulation results are shown in **

**. The left column shows the true concentrations of the two state variables under each sensing-cost scenario. The right column shows the sensing state z(t)—when it's 1, sensors are active; when it's 0, the controller operates open-loop.

The striking finding is that even when sensing is infrequent, the controller maintains surprisingly good performance. The concentrations oscillate more when s is high—there's more "slop" in the system—but they remain within acceptable bounds. The biological process doesn't collapse. The microbes don't die. The treatment continues.

This is the key insight: sensing-cost awareness doesn't just save energy. It changes the control architecture. Instead of designing a controller that happens to use sensors, you design a controller that explicitly budgets for sensor usage. The result is a system that is aware of its own information needs—and rations observation accordingly.

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The Shrinking Horizon: Handling Uncertainty

Every model is wrong. Some are useful.

The infinite-horizon solutions derived in the paper assume perfect knowledge of the system dynamics. But real systems deviate from their models. Parameters drift. External disturbances occur. Unmodeled nonlinearities creep in.

To address this, the researchers introduced a Shrinking Horizon method. The idea is simple: instead of solving one big optimal control problem over the entire future, solve a series of shorter problems, each receding horizon shorter than the last.

At each time step, the controller:

1. Acquires the current state estimate (if sensors are on).
2. Solves an optimal control problem for a finite horizon extending into the future.
3. Applies only the first control action.
4. Repeats.

This is a well-established technique in model predictive control (MPC), and its applicability here is one of the paper's practical contributions. The shrinking horizon approach has two advantages: it handles model uncertainty by continually re-optimizing, and it reduces the computational burden of solving long-horizon TPBVPs.

**

** illustrates the shrinking horizon scheme. At each step, a new optimization problem is formulated over a horizon that ends at the (shrinking) final time. The solution provides a trajectory for the next N steps; only the first step is executed. Then the horizon shrinks, the problem is re-solved, and the cycle continues.

The researchers tested this approach on a first-order system with explicit model uncertainties. They introduced parametric errors—δa = 0.2 and δb = 0.1—representing a 20% error in the A parameter and a 10% error in the B parameter. The results are shown in **

**.

The comparison between the full TPBVP solution (which assumes perfect knowledge) and the shrinking horizon implementation (which corrects for errors over time) shows that the shrinking horizon controller is more robust. When the model is wrong, the TPBVP solution drifts. The shrinking horizon controller corrects course with each new measurement.

The difference is visible in the state trajectories. The TPBVP solution, optimized for the wrong model, maintains its prescribed path—wrong as that path may be. The shrinking horizon controller, refreshing its optimization with each step, gravitates toward the true optimal behavior despite the modeling errors.

This is not a minor result. In real engineering systems, model uncertainty is the norm, not the exception. A controller that degrades gracefully under uncertainty is more valuable than one that assumes perfect knowledge and collapses when reality disagrees.

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Why This Changes Things

The paper's most important contribution is conceptual, not mathematical. It reframes the optimal control problem.

Classical optimal control asks: What is the best control input, given the current state?

This paper asks: What is the best observation strategy, given that observation costs something?

The difference matters. In the classical formulation, sensing is a prerequisite—always available, always accurate, always free. The controller's job is to translate information into action. In the sensing-cost formulation, sensing is a choice. The controller must decide not only what to do, but whether it is worth looking first.

This framing has implications far beyond the mathematical details.

Battery-powered systems are the most obvious beneficiaries. Drones, underwater robots, remote sensors, implantable medical devices—all operate under strict energy budgets. Every joule spent on sensing is a joule not available for actuation, locomotion, or survival. By explicitly optimizing the sensing schedule, designers can maximize mission duration without sacrificing performance.

Networked control systems face a related constraint. When sensors communicate over wireless networks, each transmission consumes bandwidth and power. In large-scale sensor arrays—environmental monitoring networks, industrial Internet-of-Things deployments—the aggregate communication cost can be substantial. A sensing-cost-aware controller reduces transmission frequency without proportional loss of control performance.

Safety-critical systems can benefit from the same trade-off. Consider a vehicle navigating in low-visibility conditions. Continuous sensing (radar, lidar, camera) provides accurate state estimation but consumes power and generates heat that might be detectable. Intermittent sensing reduces the system's signature but requires more aggressive prediction and compensation. The optimal strategy depends on the mission context—stealth vs. reliability—and the sensing-cost framework provides a principled way to navigate that trade-off.

Industrial processes like the wastewater treatment plant demonstrate that even systems with abundant power can benefit from sensing-cost awareness. The savings aren't energy; they're computational. By reducing the processing burden of continuous sensor fusion, the framework can simplify hardware requirements and reduce latency.

But perhaps the deepest implication is theoretical. By showing that sensing-cost optimal control is tractable—that it yields solvable TPBVPs and interpretable switching conditions—the paper opens a new subfield for investigation. Researchers can now ask: How does the optimal sensing schedule change with system dynamics? With sensor noise? With multi-sensor architectures? With nonlinear systems? Each question is a research program.

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What's Next

The paper is a foundation, not a finished edifice. The authors acknowledge several limitations and point the way toward future work.

Nonlinear systems are the most pressing extension. The paper's results are derived for LTI systems, which are linear and time-invariant. Real-world systems are rarely both. Extending the sensing-cost framework to nonlinear dynamics—using the Pontryagin principle with nonlinear system models—will require new mathematical tools and likely numerical methods that don't rely on closed-form solutions.

The control cost term was intentionally omitted from the first two formulations to simplify the analysis. Including a term proportional to uᵀRu in the cost functional—penalizing control effort, not just state error—complicates the TPBVP structure and affects convergence. The researchers note that this is an area for future investigation.

Singular intervals remain an open question. When the switching condition hovers at equality over a finite time interval, the optimal z might take an intermediate value between 0 and 1—not pure sensing or pure silence, but some combination. The paper shows that for scalar systems, this would require z = a/(bk), an unlikely value when a, b, and k are nonzero. But in higher-dimensional systems, singular arcs may exist that yield genuinely suboptimal solutions under the discrete switching constraint.

Experimental validation is the missing piece. The paper's simulations are compelling, but a physical implementation would demonstrate whether the framework holds up in the face of real sensor noise, actuator lag, and model errors. A drone flight test or a hardware-in-the-loop experiment would provide evidence that sensing-cost awareness translates from simulation to reality.

Computational efficiency matters for real-time implementation. The shooting method works for research problems, but industrial controllers need faster algorithms. The shrinking horizon method helps, but it still requires solving optimization problems at each time step. Fast model predictive control techniques—explicit MPC, online homotopy methods, neural network approximations—could make sensing-cost-aware control practical for high-speed systems.

Multi-sensor architectures introduce new dimensions. What happens when there are multiple sensor types with different costs, accuracies, and latencies? The framework would need to generalize to vector-valued sensing decisions—which sensor to activate, at what granularity, with what fusion strategy.

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The Broader Significance

There is a quiet revolution happening in control theory. For decades, the field assumed that information was free—that the controller could always know the state, if not exactly then at least to sufficient accuracy. This assumption was never true, but it simplified the mathematics and made the problems tractable.

As control systems have proliferated into resource-constrained environments—drones, implants, sensor networks, Mars rovers—the assumption has become untenable. Engineers have responded with heuristic fixes: duty cycling, event-triggered control, self-triggered strategies. These work, but they are not principled. They are rules of thumb dressed in technical language.

This paper takes a different approach. It takes the cost of information seriously and asks: What is the optimal way to budget for observation? Not a heuristic. A solution, derived from first principles, with mathematical rigor.

The answer is a controller that senses strategically—not constantly, not randomly, but when the value of information exceeds its cost. This is not a surprising insight philosophically. It is what any rational agent would do. But formalizing it, embedding it in optimal control theory, and deriving the conditions for optimality—this is new.

The wastewater treatment plant example hints at the practical stakes. A large treatment facility might have dozens of online sensors, each requiring power, calibration, and maintenance. If the sensing-cost framework can reduce sensor activation by 50% without degrading water quality, the savings—in energy, in hardware, in operator time—are real. Scale this across thousands of facilities, and the aggregate impact is not negligible.

For battery-powered systems, the implications are starker. A drone that can intelligently ration its sensing might fly 20% longer on the same battery. An underwater robot might complete a mission it would otherwise abort. A planetary rover might survive a dust storm.

The paper does not claim to solve these problems. It claims to provide a framework for thinking about them. That is enough. Frameworks shape the questions engineers ask, the designs they explore, and the systems they build.

The next time a drone hovers over a disaster zone, burning through its battery, it might thank Pontryagin.