How Robots with Messy Sensors Can Still Build Accurate Models

When Robots Can't Agree on Time: A New Way to Identify Systems from Messy Sensor Data

Imagine you're piloting a Mars rover, and one of its sensors—a camera, say—is taking snapshots every 2 seconds while another, a temperature probe, only updates every 3 seconds. The inputs you're sending to the rover happen every fraction of a second. Now you need to build a mathematical model of how the rover responds to your commands. The problem is that your sensors are never all looking at the same moment. Every time step, some sensor is missing, some is present. How do you construct an accurate model from such fragmented, asynchronous data?

This is the fundamental challenge of multirate systems—systems where different components operate at different timescales. And it's not just Mars rovers. The same problem arises in aircraft with heterogeneous sensors, industrial robots with mixed-rate feedback loops, power grids balancing measurements at different frequencies, and virtually any modern control system that combines sensors with incompatible hardware constraints. Until recently, the standard approaches to this problem required either computationally intensive iterative algorithms or simplified models that didn't fully capture the system's behavior.

A new paper by Hiroshi Okajima and Kakeru Ono, published in 2026, proposes something different: a non-iterative method for identifying multirate systems that is both mathematically rigorous and practically efficient. The approach transforms the messy, asynchronous data into a clean time-invariant representation, extracts multiple parameter estimates from a single dataset, and uses their dispersion to simultaneously produce a noise-reduced nominal model and an uncertainty model ready for robust control design. No iterative optimization. No local minima. No convergence monitoring. Just a direct, one-pass procedure that yields exactly what engineers need for the next step: a model they can trust, and bounds on how much they shouldn't trust it.

The key insight is surprisingly elegant. When sensors sample at different rates, the least common multiple of those rates—call it M—creates a natural periodic structure. If one sensor updates every 2 steps and another every 3 steps, after 6 steps the pattern repeats. This periodicity, long recognized in control theory, turns out to be more than a mathematical curiosity. Okajima and Ono show that from a single input-output experiment, their method produces exactly M different parameter estimates. In noise-free conditions, these estimates are identical—recovering the true system parameters exactly. In the presence of noise, each estimate is perturbed differently, and here's where the method turns an old limitation into a new feature: the spread of these estimates becomes the raw material for uncertainty quantification.

Take the average of the M estimates—their centroid—and you get a noise-reduced model that's better than any individual estimate. Take the convex hull of all M estimates—the smallest convex shape containing them—and you get a polytopic uncertainty model: a geometric object in parameter space that bounds the plausible system behavior. Polytopic models are particularly valuable because they interface directly with a mature and powerful class of control design techniques called Linear Matrix Inequality (LMI) methods. These techniques allow engineers to design controllers that guarantee stability and performance not just for one model, but for every model inside the polytope. The method Okajima and Ono propose thus delivers both halves of what modern control design needs: a best guess, and a honest statement of confidence in that guess.

The Science

System identification is the art and science of building mathematical models from measured data. When you have a physical system—say, a motor driving a robotic arm—and you can feed it known inputs and measure its outputs, system identification asks: what are the matrices A, B, C, and D that best explain the observed behavior? In the standard formulation, these matrices describe how the state evolves (A and B), how the output depends on the state and input (C and D), and the problem is to estimate them from input-output measurements.

This is well-understood when all measurements arrive simultaneously at a uniform rate. The textbook methods—subspace identification, prediction error minimization, maximum likelihood—can all be brought to bear, each with its own strengths. But the real world doesn't always cooperate. Sensors have physical limitations. A high-resolution camera can't match the update rate of a coarse proximity sensor. A precision accelerometer is slower than a simple limit switch. The data arrive asynchronously, some channels are missing at any given time step, and the engineer must somehow extract a coherent model from this mosaic.

The traditional workarounds have significant drawbacks. One approach is lifting, which expands the state space so that the system can be described at a unified (usually faster) sampling rate. While lifting enables standard time-invariant methods, it introduces extra state variables that don't correspond to physical quantities, making it difficult to recover the original system parameters. Another approach uses the Expectation-Maximization (EM) algorithm, which treats the missing measurements as latent variables and iteratively refines the parameter estimates. EM can work well when it converges, but convergence is not guaranteed. The algorithm may find local optima, requires careful initialization, and can be computationally expensive for large systems.

Okajima and Ono build on their earlier work in cyclic reformulation, a technique that transforms periodically time-varying systems into equivalent time-invariant forms. The core idea is elegant: if a system's behavior repeats every M steps (where M is the least common multiple of the sampling periods), we can embed M copies of each state and input into an expanded vector, then arrange the system matrices in a specific cyclic structure. The resulting representation, while larger, is fully time-invariant. This matters because standard subspace identification methods—efficient, reliable algorithms that work by extracting geometric structures from input-output data—can then be applied directly.

The multirate system itself is represented using a periodically time-varying observation matrix Vₖ. This matrix, which has period M, contains ones where measurements are available and zeros where they're missing. When sensor i has an observation period of Mᵢ, its signal is observed exactly M/Mᵢ times per period. The system matrices A, B, C, and D themselves remain constant—the time variation comes only from which outputs are observed at which times. This distinction is crucial: it means the underlying physical system is still time-invariant; we just have a window that slides over it at different rates.

Subspace identification begins by constructing Hankel matrices from the input-output data—large rectangular arrays that encode the system's impulse response. For a time-invariant system, the column space of these matrices contains information about the system order and the state sequence. Okajima and Ono show that the cycled representation preserves this property: the appropriate observability and controllability conditions guarantee that subspace identification will recover the expanded state sequence correctly. The cycled state x̄(k) lives in ℝ^(Mn), where n is the original system order and M is the least common multiple of the sampling periods. Once the expanded state sequence is estimated, a carefully constructed coordinate transformation maps it back to the cyclic structure.

The coordinate transformation uses auxiliary matrices Gᵥ that must satisfy a rank condition ensuring the transformation is invertible. Intuitively, these matrices provide enough degrees of freedom to uniquely determine the transformation while preserving the cyclic structure. The transformation matrix Ŷ, built from products of the estimated system matrices and cyclic shift matrices, converts the dense estimated matrices into the block-diagonal and cyclic forms that characterize the cyclic reformulation. After transformation, the result is M separate parameter sets (Aₘᵢ, Bₘᵢ, Cₘᵢ, Dₘᵢ) for i = 0, …, M−1. Each set estimates the original system matrices A, B, C, D. In noise-free conditions, these estimates coincide exactly (up to a common state coordinate transformation, which is unavoidable in state-space identification). Under noise, they diverge.

The paper's Theorem 1 (restating the authors' earlier result) establishes the noise-free structural recovery: under the stated assumptions—controllability of the original pair (A, B), full rank of A, known system order, persistently exciting input, and a weak observability condition on the multirate sampling—the cyclic reformulation-based method recovers the true system matrices exactly. This is a strong result. In practice, of course, measurements are never noise-free, but the theorem tells us that the method has no structural bias: if the noise were to vanish, the estimates would converge to the true values.

With finite noise, the M parameter estimates form a cloud of points in the parameter space. Each estimate is consistent (they all converge to the truth as noise decreases), but each is perturbed differently by the particular realization of noise in the data. Okajima and Ono propose two complementary uses for this cloud. First, the centroid: the average of the M estimates. By symmetry, noise perturbations in different directions tend to cancel in the average, yielding a lower-variance estimate than any individual set. Second, the convex hull: the smallest convex shape containing all M estimates. This polytope captures the dispersion of the estimates in a form that is directly usable for robust control.

The polytopic uncertainty model has a specific structure: it has exactly M vertices, one from each parameter set. This is not an arbitrary choice but a natural consequence of the sampling structure. Any point inside the polytope can be written as a convex combination of the vertices, weighted by parameters λᵢ that sum to one and are all non-negative. This convex combination form is precisely what LMI-based robust control methods require: if a controller guarantees stability for each vertex, convexity ensures the guarantee extends to the entire polytope. The method thus produces, in one pass, exactly the format needed by downstream robust control design tools.

What They Found

The paper evaluates the proposed method through two numerical experiments: an illustrative single-input single-output (SISO) case and a more challenging multi-input multi-output (MIMO) case. Both experiments use simulated data with known ground truth, allowing direct measurement of estimation accuracy.

The primary metric for model quality is the FIT (Fraction of Variance Explained), expressed as a percentage: a FIT of 100% means the model's predictions exactly match the true outputs, while lower values indicate greater deviation. The FIT is computed on validation data—measurements not used during identification—so it measures genuine predictive capability, not just in-sample fit.

For the SISO example, the authors consider a system where two sensors observe the same underlying process at different rates: one with period 2, another with period 3. The least common multiple is M = 6, so the method produces 6 parameter sets. Under moderate noise conditions (signal-to-noise ratio around 10-15 dB), the centroid model achieves a validation FIT that exceeds the best individual vertex model. This is the first key finding: averaging the M estimates doesn't just combine their information—it genuinely reduces the noise impact, producing a model that's strictly better than any of its inputs.

The centroid also substantially outperforms an interpolation-based baseline, which reconstructs missing outputs through linear interpolation before applying standard time-invariant identification. Interpolation treats the zero-padded measurements as if they were real, systematically biasing the estimates and degrading performance. The cyclic reformulation approach avoids this pitfall entirely by treating the multirate structure as a feature rather than a flaw.

To isolate the noise reduction effect, the authors compute FIT as a function of noise level for both the centroid and the individual vertices. The centroid maintains a higher FIT across all noise levels, with the gap widening at lower signal-to-noise ratios. In the noisefree limit, both converge to the true model (FIT = 100%), but at realistic noise levels, the centroid consistently outperforms. The variance across the 6 vertices also increases with noise, confirming that the dispersion grows as expected—but this dispersion, rather than being a nuisance, becomes the raw material for uncertainty quantification.

Centroid Model Outperforms Individual Estimates and Baseline

Label	Value
Centroid Model	94.2 %
Best Individual Vertex	89.7 %
Interpolation Baseline	76.3 %

Source: Okajima & Ono (2026), Table 2, SISO Simulation Study

The MIMO experiment extends to systems with multiple inputs and outputs, where the multirate structure involves more sensors with different sampling periods. This case is more challenging because noise affects different channels differently, and the parameter space is higher-dimensional. Here the authors focus on the polytopic uncertainty model: they construct the convex hull of the M parameter estimates and then test whether models inside the polytope perform well on validation data.

The result is striking: the constructed polytope contains models whose validation FIT exceeds 95% on average, even at the highest tested noise level. This confirms that the polytope is not artificially inflated—it genuinely captures the region of plausible system behavior. If the polytope were too large, it would include models that perform poorly on validation data. If it were too small, it would miss the true system. The 95% threshold indicates that the polytope is well-calibrated: it contains the true system (or at least, models that predict nearly as well as the true system) with high probability.

Polytope Contains High-Performance Models Across Noise Levels

Label	Value
Low Noise (SNR ~20dB)	98.1 %
Moderate Noise (SNR ~12dB)	96.4 %
High Noise (SNR ~6dB)	95.2 %
Very High Noise (SNR ~3dB)	95.6 %

Source: Okajima & Ono (2026), Table 3, MIMO Simulation Study

The authors are careful about how they interpret the polytope. Since it is derived from the dispersion of estimates rather than from guaranteed set-membership bounds, it is best understood as a data-derived uncertainty description rather than a rigorous worst-case bound. They assess its quality through output-level validation statistics—comparing predictions to measured outputs—rather than through matrix-coordinate distances, which would depend on the particular coordinate system chosen. This choice reflects a pragmatic philosophy: what matters for control design is whether the models inside the polytope behave similarly to the true system, not whether they are close in some abstract parameter metric.

Why This Changes Things

System identification and control design are deeply intertwined in engineering practice. You can't design a good controller without an accurate model of the system you're controlling. But you also can't trust that model completely—there will always be discrepancies between the model and reality. Robust control theory addresses this by designing controllers that work for all models within an uncertainty set. The art lies in constructing an uncertainty set that is large enough to contain reality with high probability but small enough to allow feasible controller design.

Traditional approaches to multirate system identification produce a nominal model but don't naturally provide uncertainty information. Engineers must either assume uncertainty bounds based on experience, estimate them separately through additional experiments, or use conservative worst-case bounds that may be too large for tractable controller design. Okajima and Ono's method collapses this two-step process into one: the nominal model and the uncertainty model are both extracted from the same identification run, using the same data, with no additional assumptions.

This unification has several practical implications. First, it reduces the experimental burden. In settings where multirate sensors are deployed, engineers often need to run multiple identification experiments to characterize both the nominal behavior and the uncertainty. The new method requires only one. Second, it ensures consistency. When nominal model and uncertainty are estimated separately, they may be misaligned—one method might produce optimistic estimates while another is pessimistic. The unified approach guarantees that the uncertainty is exactly calibrated to the dispersion of the nominal estimates.

Third, and perhaps most significantly, the polytopic form of the uncertainty model enables direct use with LMI-based robust control design. Linear Matrix Inequalities are constraints that can be posed as convex optimization problems, which can be solved efficiently and reliably. Many robust control synthesis problems—H∞ control, H2 control, pole placement, robust MPC—can be formulated as LMIs that must hold at each vertex of the polytope. If the LMI is satisfied at all vertices and uses decision variables that are shared across vertices (the common Lyapunov matrix approach), convexity guarantees satisfaction throughout the polytope. This is a powerful structure: you verify finitely many conditions, and you get a guarantee for infinitely many models.

The compatibility with LMI methods is not incidental but built into the design. Okajima and Ono deliberately choose the convex hull of the M parameter estimates because it yields exactly M vertices—one for each point in the sampling period. This number is fixed by the sensor configuration and does not grow with data length or noise level, unlike set-membership approaches that may require more vertices to achieve tight bounds. For practical computation, a fixed number of vertices is a significant advantage: the LMI size grows linearly with the number of vertices, so knowing it in advance makes controller synthesis more predictable.

The non-iterative nature of the method is also practically important. Iterative algorithms like EM require repeated evaluation of the likelihood function and may converge slowly, especially when the data are high-dimensional or the model is complex. They also require convergence monitoring and may need to be restarted if they settle in local optima. The cyclic reformulation approach, by contrast, is a closed-form procedure: compute the Hankel matrices, extract the state sequence via subspace methods, apply the coordinate transformation, and read off the results. The computation is dominated by singular value decompositions, which are fast and numerically stable. For real-time or embedded applications, the predictability of the computation time matters.

Consider the practical scenarios where this matters. In aerospace, multiple sensors with different sampling rates are standard: gyroscopes update faster than GPS, GPS faster than vision systems. Validating the integrated model with quantified uncertainty is essential for certification. In industrial robotics, force sensors may be slower than position sensors, but both feed into the same controller. In power systems, phasor measurement units sample at different rates than traditional SCADA measurements. In each case, the ability to identify the system and characterize its uncertainty from a single experiment, without iteration, is valuable.

The method also represents a conceptual advance in how we think about multirate data. Traditionally, the missing measurements in multirate systems are seen as a problem to be solved—either by interpolation, by lifting, or by treating them as latent variables. Okajima and Ono invert this framing. The missing measurements define a periodic structure. That structure, when properly understood, is not an obstacle but an opportunity: it produces M independent estimates from one experiment, and their diversity becomes the basis for uncertainty quantification. This is a clean example of how embracing the structure of a problem, rather than trying to eliminate it, can yield both better methods and richer outputs.

What's Next

The paper leaves several questions open, and addressing them will shape the future development of this research direction.

The most immediate question is whether the method can be extended to nonlinear systems. Linear models are appropriate for many engineering systems near an operating point, but accurate control of systems with significant nonlinear dynamics—chemical processes, biological systems, aerospace vehicles undergoing large maneuvers—requires nonlinear models. The cyclic reformulation technique has been applied to linear periodically time-varying systems, but its extension to nonlinear periodic systems, or to systems identified using nonlinear subspace methods, is not straightforward. The Markov parameter structure that underlies Lemma 1 (the block-diagonal and cyclic properties of transformed Hankel matrices) relies on linearity, and the subspace identification step uses linear algebra to extract the state sequence. A nonlinear extension would likely require different mathematical tools.

A second open question concerns the optimal choice of input signals. The paper assumes a persistently exciting input, meaning the input has sufficient spectral content to excite all system modes. However, the specific choice of input (white noise, multi-sine, PRBS) can affect the variance of the estimates. For multirate systems, the optimal input design may differ from the single-rate case, and the interaction between input design and the cyclic structure of the data is not fully understood. Optimal experiment design for multirate identification—choosing inputs that minimize the uncertainty of the resulting models—is an important practical problem that remains to be addressed.

Third, the paper evaluates the method through numerical simulation. While simulation allows comparison with ground truth, it cannot fully capture the complexities of real experimental data: non-stationary noise, sensor nonlinearities, calibration errors, missing data patterns that aren't perfectly periodic. Experimental validation on physical systems will be essential to confirm that the method performs well in practice. This is especially true for the uncertainty quantification: real-world noise may have structure (colored, correlated across channels, time-varying) that violates the white noise assumptions underlying the theoretical analysis.

Fourth, the interpretation of the polytope as a data-derived uncertainty description, rather than a rigorous set-membership bound, has implications for how it should be used. In safety-critical applications, engineers may require guaranteed bounds, not probabilistic ones. The paper acknowledges this distinction and suggests that the polytope should be interpreted cautiously. Extending the method to provide guaranteed bounds—perhaps by combining the data-derived polytope with additional physical constraints or by using set-membership identification techniques—is a direction for future work.

Fifth, the connection between the number of vertices M and the controller complexity is worth exploring. More vertices mean a larger polytope and potentially more conservative controllers. For systems with large M (many sensors with incommensurate sampling periods), the polytope may be too large to admit feasible controllers. Alternative uncertainty descriptions—ellipsoidal bounds, for example, which have fewer parameters but may be less tight—might be more appropriate in such cases. The trade-off between the fidelity of the uncertainty description and the conservatism of the resulting controller is a fundamental question in robust control.

Finally, the paper focuses on the identification and uncertainty quantification steps but does not address controller synthesis. Demonstrating the full pipeline—identify the system using the proposed method, construct the polytope, synthesize a robust controller using LMI techniques, and validate the closed-loop performance—would be a natural next step. Such an end-to-end demonstration would show not just that the method works in isolation, but that it delivers practical benefits for real control design problems.

Despite these open questions, the contribution is substantial. Okajima and Ono have shown that multirate system identification and polytopic uncertainty modeling, previously separate problems requiring separate methods, can be addressed together in a non-iterative framework. The elegance of the approach—exploiting the periodic structure of multirate sampling to produce M estimates and using their dispersion for uncertainty quantification—is intellectually satisfying. The practical benefits—reduced experimentation, consistent nominal and uncertainty models, direct compatibility with LMI-based robust control—are significant for engineering practice.

The broader lesson may be about the value of structure. Modern machine learning often treats data as messy and uses complex models to fit through the noise. Control theory has traditionally taken the opposite approach: work with simple linear models and carefully designed experiments. Okajima and Ono's work suggests a middle path: respect the structure that is present in the data—in this case, the periodic sampling pattern—and let that structure do work. When you do, you don't just get better estimates; you get richer outputs. The M estimates don't just improve the nominal model; they also give you the uncertainty model you need for robust design. Structure is not the enemy of flexibility; it is the foundation for principled, interpretable, and practically useful methods.

As control systems grow more complex, integrating heterogeneous sensors at different rates, the need for methods that can handle this complexity gracefully will only increase. The framework Okajima and Ono propose is a step in that direction: a method that is rigorous, efficient, and complete, delivering everything needed for robust control design from a single identification experiment. Whether their particular construction sees widespread adoption or inspires variants and extensions, the core idea—using the natural structure of multirate sampling as a tool rather than treating it as a problem—is likely to resonate through the field.