Simultaneous Placement of Actuators and Sensors
The norm of a flexible structure with multiple actuators and sensors is a root-mean-square sum of norms of structures equipped with single sensor and single actuator.
mass↗
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Results for “sum of squares”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Search for Transient Gravitational-wave Signals Associated with Magnetar Bursts during Advanced LIGO's Second Observing Run
We present the results of a search for short- and intermediate-duration gravitational-wave signals from four magnetar bursts in Advanced LIGO's second observing run. We find no evidence of a signal and set upper bounds on the root sum squared of the total dimensionless strain ( h rss ) from incoming intermediate-duration gravitational waves ranging from 1.1 × 10 −22 at 150 Hz to 4.4 × 10 −22 at 1550 Hz at 50% detection efficiency. From the known distance to the magnetar SGR 1806–20 (8.7 kpc), we can place upper bounds on the isotropic gravitational-wave energy of 3.4 × 10 44 erg at 150 Hz assuming optimal orientation. This represents an improvement of about a factor of 10 in strain sensitivity from the previous search for such signals, conducted during initial LIGO's sixth science run. The short-duration search yielded upper limits of 2.1 × 10 44 erg for short white noise bursts, and 2.3 × 10 47 erg for 100 ms long ringdowns at 1500 Hz, both at 50% detection efficiency.
B. P. Abbott↗
Assessment of ICESat-2’s Horizontal Accuracy Using Precisely Surveyed Terrains in McMurdo Dry Valleys, Antarctica
This article presents an assessment of the horizontal accuracy and precision of the laser altimetry observations collected by NASA’s Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2) mission. We selected the terrain-matching method to determine the position of laser altimeter profiles within a precisely known surface, represented by a digital elevation model (DEM). We took this classical approach a step further, approximated the DEM by planar surfaces, and calculated the optimal position of the laser profile by minimizing the square sum of the elevation differences between reference DEMs and ICESat-2 profiles. We found the highly accurate DEMs of the McMurdo Dry Valleys (DV), Antarctica, ideal for this research because of their stable landscape and rugged topography. We computed the 3-D shift parameters of 379 different laser altimeter profiles along two reference ground tracks collected within the first two years of the mission. Analyzing these results revealed a total geolocation error (mean +1 σ ) of 4.93 m for version 3 and 4.66 m for version 4 data. These numbers are the averages of the six beams, expressed as mean +1 σ and lie well within the mission requirement of 6.5 m.
calibration↗
Combined Experimental and Modeling Study of the Interactions of Acid Gas with Common Spacecraft Surfaces for Fire Safety Applications
A fire in a spacecraft poses detrimental consequences and risks mission success in addition to crew safety. This is compounded during long-duration missions when the crew has limited options to recover from a fire. A common spacecraft fire concern is the smoldering of wire insulation, typically made from Polyvinyl chloride (PVC) or Polytetrafluoroethylene (PTFE). This creates acid gases such as Hydrogen Chloride (HCl), Hydrogen Fluoride (HF) and Hydrogen Cyanide (HCN). These poisonous gases are hazardous to the crew. They also interact with common surfaces within the spacecraft more than dominant combustion products such as CO2 and H2O. This makes them more difficult to track for potential fire detection techniques, or for postfire clean-up. It is imperative to be able to understand and predict the fate of these poisonous species in a microgravity environment in order to design a safe vehicle. HCl interacts with a number of materials inside a spacecraft. Primary among these materials is aluminum, which is abundantly used due to its strong and light weight nature. Aluminum has a natural oxide layer that protects it from corrosion but is typically treated to enhance this oxide layer. Among these treatments is a chromate conversion coating (CCC), which provides a thin enough protective oxide layer to still conduct electricity, and a traditional anodized material that has a thicker oxide layer that does not conduct electricity. Nomex is another common material found inside a spacecraft. It is a flame-resistant woven polymer that is related to nylon. This commercially available material is used for cargo storage bags and as a fire barrier. Physics-based models were developed to predict the uptake of HCl by these materials. The ultimate objective of these models is to predict the fate of HCl within the spacecraft so that sensors can be placed in meaningful locations in future missions based on the model predictions. To support these modeling efforts, experiments were performed in a cast acrylic test cell that measured the difference between the inlet and outlet concentration of HCl after inserting a sample rod of the test material. Different uptake capacities were realized for each type of sample tested. A computational fluid dynamics model (CFD) model of the reactor was then constructed that used a one-step global reaction rate with calibratable reaction (or kinetic) constants. These constants were calibrated to match the HCl uptake on the CCC aluminum samples, and the same kinetic constants were then tested for the stock and anodized aluminum samples. Model predictions matched the experimental data for the stock aluminum, and to a much lesser extent, the anodized aluminum. The model was additionally validated at different flow rates, sample surface areas, and inlet concentrations, and showed good agreement for all stock and CCC samples. The model did not accurately predict the HCl uptake in the anodized samples compared to the other two types of aluminum. Adjusting the kinetic constants and transport properties did little to improve the prediction. X-Ray Photoelectron Spectroscopy (XPS) was used to determine that the oxide layer thickness of anodized aluminum is approximately 5,000 nm, compared to 250 nm for CCC and 50 nm for stock. XPS also revealed presence of chlorine further down in the aluminum oxide layer in anodized samples than CCC and stock samples after the samples were saturated with HCl, indicating that accounting for diffusion of HCl into the oxide layer is important for accurate prediction of HCl uptake onto anodized aluminum. Consequently, a multi-scale model was developed and tested. First, a single pore inside the anodized aluminum oxide layer was modeled and is referred to as the pore-scale model. In this model, HCl diffused through the pore and reacted with the aluminum oxide pore wall to create aluminum chloride. The sample was then saturated when the mass transfer resistance through the growing aluminum chloride layer became too large for the HCl to reach the aluminum oxide wall and continue the reaction. This pore-scale model was coupled to the reactor-scale model using a concentration-dependent diffusion coefficient, resulting in much more accurate predictions (approximately half the sum square error of the aforementioned reactor-scale model that produced good agreement for stock and CCC) for a variety of operating conditions. The amount of water vapor or relative humidity (RH) in the flow during a reactor experiment was determined to influence HCl uptake. Experiments were performed to understand the interaction of gaseous HCl with aluminum surfaces in the presence of water vapor. The results show that increasing levels of RH increased the capacity of aluminum to adsorb HCl but decreased the capacity of Nomex to uptake HCl. A series of tests were performed on individual aluminum samples after they had been saturated with a fixed concentration of HCl in dry air conditions with the goal of determining how their HCl uptake capacity changes after various treatments with water relative to the original saturation tests. HCl-saturated aluminum samples subjected to a second dry air flow at the same HCl concentration as the original test had an uptake of 23.5% of the original sample with no treatment in between. Saturated aluminum samples subjected to an air flow with a RH of 90% in between tests had an uptake of 35.6% of the original. Saturated aluminum samples submerged in distilled water for 12 hours in between tests had an uptake of 82.2% of the original sample. Previously saturated aluminum tested with HCl and a 50% RH air flow resulted in similar uptake characteristics in multiple repeated tests. The experiments show the profound effect water vapor has on HCl uptake onto aluminum surfaces. In the samples subjected to water vapor or liquid water, capillary condensation and capillary diffusion alters the transport of HCl significantly. A model was proposed that developed a relationship between RH and the coefficient of HCl diffusion in aluminum chloride. This produced an “S-shaped” curve with diffusion coefficient as a function of RH, with 45% RH represented as the point where the diffusion coefficient is halfway between no water saturation and 100% water saturation in the aluminum chloride product layer. No difference in uptake characteristics for the experiment or model were realized between 50% and 62% RH. The results from the large-scale microgravity experiment, Saffire, are discussed as they pertain to the fate of HCl throughout a spacecraft. HCl was released, both as a standalone event, and in concurrence with the burning of a structured cloth. These events only produced a small response in the far field HCl sensor, while a PMMA burn that did not produce HCl had a significantly greater response. A ground-based large-scale facility was constructed to flow acid gas at the scale and configuration realized in the Saffire experiments. A CFD model of this duct was constructed to test kinetic parameters developed in this work at a larger scale and different geometric configuration and to predict the results of the large-scale facility. The models developed in this work were used to interpret the results of the microgravity tests and lead the discussion on what further experiments and models are needed in order to predict the fate of acid gas in a spacecraft environment. To summarize, the major contributions of this work are as follows: the capacity to uptake HCl, with and without the presence of water vapor, was measured for a variety of real spacecraft surfaces. Several different models (single reactor-scale, multiscale, spacecraft-scale) were developed and with the aid of modeling, the rate of uptake for those surfaces was also predicted and validated. The kinetic parameters determined from the small-scale reactor experiments and models were used to predict large-scale and microgravity tests. Conclusions from this research will be used in the design of spacecraft vehicles and large-scale microgravity fire safety experiments. The models built by this work will aid designers in sensor placement and could be used to predict acid gas transport from fires in partial gravity, as would be seen in Lunar and Martian habitats.
fire safety↗
Australian tidal currents – assessment of a barotropic model (COMPAS v1.3.0 rev6631) with an unstructured grid
While the variations of tidal range are large and fairly well known across Australia (less than 1 m near Perth but more than 14 m in King Sound), the properties of the tidal currents are not. We describe a new regional model of Australian tides and assess it against a validation dataset comprising tidal height and velocity constituents at 615 tide gauge sites and 95 current meter sites. The model is a barotropic implementation of COMPAS, an unstructured-grid primitive-equation model that is forced at the open boundaries by TPXO9v1. The mean absolute error (MAE) of the modelled M2 height amplitude is 8.8 cm, or 12 % of the 73 cm mean observed amplitude. The MAE of phase (10°), however, is significant, so the M2 mean magnitude of vector error (MMVE, 18.2 cm) is significantly greater. The root sum square over the eight major constituents is 26 % of the observed amplitude. We conclude that while the model has skill at height in all regions, there is definitely room for improvement (especially at some specific locations). For the M2 major axis velocity amplitude, the MAE across the 95 current meter sites, where the observed amplitude ranges from 0.1 to 156 cm s −1 , is 6.9 cm s −1 , or 22 % of the 31.7 cm s −1 observed mean. This nationwide average result is encouraging, but it conceals a very large regional variation. Relative errors of the tidal current amplitudes on the narrow shelves of New South Wales (NSW) and Western Australia exceed 100 %, but tidal currents are weak and negligible there compared to non-tidal currents, so the tidal errors are of little practical significance. Looking nationwide, we show that the model has predictive value for much of the 79 % of Australia's shelf seas where tides are a major component of the total velocity variability. In descending order this includes the Bass Strait, the Kimberley to Arnhem Land, and southern Great Barrier Reef regions. There is limited observational evidence to confirm that the model is also valuable for currents in other regions across northern Australia. We plan to commence publishing “unofficial” tidal current predictions for chosen regions in the near future based on both our COMPAS model and the validation dataset we have assembled.
Tidal currents↗
The cross section for double Compton scattering
Employing elementary methods in nonrelativistic quantum electrodynamics, the cross section for gamma sub 0 + e yields e + gamma + gamma is computed for arbitrary energy in the spectrum of the outgoing photons. The final result is given, differential in the energy of one of these photons, for the case where the incident photon is unpolarized and has energy E sub 0 much less than mc-squared, a polarization sum and angular integration being performed for the final-state photons. The cross section has a simple algebraic form resulting from contributions from the sum of squared direct and exchange amplitudes; interference terms from these amplitudes do not contribute to the angular-integrated cross section.
Gould, R. J.↗
Oceanographic and meteorological research based on the data products of SEASAT
Reservations were expressed concerning the sum of squares wind recovery algorithm and the power law model function. The SAS sum of squares (SOS) method for recovering winds from backscatter data leads to inconsistent results when V pol and H pol winds are compared. A model function that does not use a power law and that accounts for sea surface temperature is needed and is under study both theoretically and by means of the SASS mode 4 data. Aspects of the determination of winds by means of scatterometry and of the utilization of vector wind data for meteorological forecasts are elaborated. The operational aspect of an intermittent assimilation scheme currently utilized for the specification of the initial value field is considered with focus on quantifying the absolute 12-hour linear displacement error of the movement of low centers.
Pierson, W. J., Jr.↗
Designing a parallel Feel-the-Way clustering algorithm on HPC systems
This paper introduces a new parallel clustering algorithm, named Feel-the-Way clustering algorithm, that provides better or equivalent convergence rate than the traditional clustering methods by optimizing the synchronization and communication costs. Our algorithm design centers on how to optimize three factors simultaneously: reduced synchronizations, improved convergence rate, and retained same or comparable optimization cost. To compare the optimization cost, we use the Sum of Square Error (SSE) cost as the metric, which is the sum of the square distance between each data point and its assigned clusters. Compared with the traditional MPI k-means algorithm, the new Feel-the-Way algorithm requires less communications among participating processes. As for the convergence rate, the new algorithm requires fewer number of iterations to converge. As for the optimization cost, it obtains the SSE costs that are close to the k-means algorithm. In the paper, we first design the full-step Feel-the-Way k-means clustering algorithm that can significantly reduce the number of iterations that are required by the original k-means clustering method. Next, we improve the performance of the full-step algorithm by adopting an optimized sampling-based approach, named reassignment-history-aware sampling. Our experimental results show that the optimized sampling-based Feel-the-Way method is significantly faster than the widely used k-means clustering method, and can provide comparable optimization costs. More extensive experiments with several synthetic datasets and real-world datasets (e.g., MNIST, CIFAR-10, ENRON, and PLACES-2) show that the new parallel algorithm can outperform the open source MPI k-means library by up to 110% on a high-performance computing system using 4,096 CPU cores. In addition, the new algorithm can take up to 51% fewer iterations to converge than the k-means clustering algorithm.
97 MATHEMATICS AND COMPUTING↗
Year-round measurements of ozone at 66 deg S with a visible spectrometer
In March 1990, a zenith-sky UV-visible spectrometer of the design 'Systeme Automatique d'Obervation Zenithal' (SAOZ) was installed at Faraday in Antarctica (66.3 deg S, 64.3 deg W). SAOZ records spectra between 290 and 600 nm during daylight. Its analysis program fits laboratory spectra of constituents, at various wavelengths, to the differential of the ratio of the observed spectrum and a reference spectrum. The least-squares fitting procedure minimizes the sum-of-squares of residuals. Ozone is deduced from absorption in its visible bands between 500 and 560 nm. The fortunate colocation of this SAOZ with the well-calibrated Dobson at Faraday has allowed us to examine the calibration of the zero of the SAOZ, difficult at visible wavelengths because of the small depth of absorption. Here we describe recent improvements and limitations to this calibration, and discuss SAOZ measurements of ozone during winter in this important location at the edge of the Antarctic vortex.
Roscoe, Howard K.↗
Higher-Order Corrections to Optimisers based on Newton's Method
The Newton, Gauss–Newton and Levenberg–Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some directions than others and the space of possible improvement in sum of squares becomes a long narrow ellipsoid in the linear model. This means that even a small amount of nonlinearity in the problem parameters can cause a proposed point far down the long axis of the ellipsoid to fall outside of the actual curved valley of improved values, even though it is quite nearby. This paper presents a differential equation that ‘follows’ these valleys, based on the technique of geodesic acceleration, which itself provides a 2 nd order improvement to the Levenberg–Marquardt iteration step. Higher derivatives of this equation are computed that allow n th order improvements to the optimisation methods to be derived. These higher-order accelerated methods up to 4 th order are tested numerically and shown to provide substantial reduction of both number of steps and computation time.
43 PARTICLE ACCELERATORS↗
On the probability density function of the squared envelope of a sum of random phase vectors
Weber et al. (1980) have found that a generic problem arises in the study of multiuser FHMA/MFSK systems. According to this problem, the presence of the other channel users acts as interference to reception of the desired user's transmission. The present investigation is concerned with a recursive solution in which the recursion parameter is the number of vectors in the sum. A mathematical characterization of such a solution is presented, taking into account the simple two-vector case in which a well-known closed form result is possible. Some numerical results are also discussed. It is pointed out that the considered recursive solution provides a break in the bottleneck associated with the exact computation of the error probability performance of multiuser FHMA/MFSK systems and single access MFSK systems corrupted by multitone jamming.
Simon, M. K.↗
Method and apparatus for frequency spectrum analysis
A method for frequency spectrum analysis of an unknown signal in real-time is discussed. The method is based upon integration of 1-bit samples of signal voltage amplitude corresponding to sine or cosine phases of a controlled center frequency clock which is changed after each integration interval to sweep the frequency range of interest in steps. Integration of samples during each interval is carried out over a number of cycles of the center frequency clock spanning a number of cycles of an input signal to be analyzed. The invention may be used to detect the frequency of at least two signals simultaneously. By using a reference signal of known frequency and voltage amplitude (added to the two signals for parallel processing in the same way, but in a different channel with a sampling at the known frequency and phases of the reference signal), the absolute voltage amplitude of the other two signals may be determined by squaring the sine and cosine integrals of each channel and summing the squares to obtain relative power measurements in all three channels and, from the known voltage amplitude of the reference signal, obtaining an absolute voltage measurement for the other two signals by multiplying the known voltage of the reference signal with the ratio of the relative power of each of the other two signals to the relative power of the reference signal.
Cole, Steven W.↗
Method of discrete modeling and its application to estimation of TF30 engine variables
A method of discrete modeling is presented that effectively isolates steady state model accuracy from dynamic model accuracy. The steady state model may be generated from the engine design equations with any desired degree of accuracy. The dynamic model is generated by applying a step disturbance of a manipulated variable to an open loop engine simulation. The sampled response of the variable is combined with the steady state model's response to form a set of weighting factors. These weighting factors are then used to weight past values of the manipulated variable, thus forming the dynamic model. The method is used to estimate various TF30-P-3 engine variables. A dynamic trim function is developed to compensate for the dynamic nonlinearities of the variables as well as for inaccuracies in dynamic definition. The trim function is shown to be realted to the square root of the sum of the squares of the weighting factors obtained at various engine operating conditions. Finally, the estimation of variables without dynamic modeling is discussed.
Arpasi, D. J.↗
SeaWiFS technical report series. Volume 32: Level-3 SeaWiFS data products. Spatial and temporal binning algorithms
The level-3 data products from the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) are statistical data sets derived from level-2 data. Each data set will be based on a fixed global grid of equal-area bins that are approximately 9 x 9 sq km. Statistics available for each bin include the sum and sum of squares of the natural logarithm of derived level-2 geophysical variables where sums are accumulated over a binning period. Operationally, products with binning periods of 1 day, 8 days, 1 month, and 1 year will be produced and archived. From these accumulated values and for each bin, estimates of the mean, standard deviation, median, and mode may be derived for each geophysical variable. This report contains two major parts: the first (Section 2) is intended as a users' guide for level-3 SeaWiFS data products. It contains an overview of level-0 to level-3 data processing, a discussion of important statistical considerations when using level-3 data, and details of how to use the level-3 data. The second part (Section 3) presents a comparative statistical study of several binning algorithms based on CZCS and moored fluorometer data. The operational binning algorithms were selected based on the results of this study.
Hooker, Stanford B.↗
Audibility of Multiple, Low-Frequency Tonal Signals in Noise
The main purpose of this study is to examine the audibility of multiple, low-frequency tones that are placed in distinct auditory channels. Three experiments are described, the goals of which are to determine if the presence of sound in multiple channels results in enhanced audibility and to assess the applicability of the Statistical Summation Model (SSM) to this frequency range. This model predicts that for the case of multiple signals that are in separate auditory channels, implying statistical independence, each with sensitivity value d prime of i, the resulting total sensitivity is given by the square root of the sum of the squares of the individual d prime of i values. In common with previous studies conducted at higher frequencies, the signals are pure tones and the maskers are broadband noise. The requirement that low frequency tones be placed in separate auditory filters limited the number of tones to a maximum of three. The first of the three experiments measured the change in masked thresholds for two- and three-tone signals relative to the level of the equally-detectable single tones. The multiple tone signals were composed of combinations of 55, 120 and 200 Hz tones. The measured changes in thresholds exceeded those predicted by the SSM, although they did not differ statistically from the model predictions. The second experiment employed the same overall approach but acquired more data and concentrated on the three-tone signal. Once again, the measured changes in masked threshold exceeded the model predictions, this time to a statistically-significant degree. Two issues were postulated with the potential to yield inflated changes in masked threshold: interaction between tones resulting in perceptible intermodulation/difference tones, and the assumption that the tones were in distinct auditory filters and statistically independent of one another. The third experiment used two sets of three-tone signals to address these latter concerns. The first set of three tones was composed of harmonically related tone frequencies of 55, 110 and 165 Hz, which was an attempt to reduce effects of intermodulation difference tones. The second set of three tones was chosen to be 110, 220 and 330 Hz, again reducing effects of difference tones, but also providing greater separation between tones. Results for the first set of three tones compared to those of the earlier experiments indicated that intermodulation was not an important effect. The second set of three tones (110, 220, 330 Hz) yielded changes in masked thresholds that, on average, were in good agreement with the SSM, although intersubject variability was large and prohibited a definitive conclusion regarding the concern that tone spacing was inadequate. The results of the three experiments showed that the masked threshold of sounds with multiple (two or three) equally-detectable low frequency tones was lower than those of the single tones. In other words, it is clear that audibility is enhanced by the presence of signals in multiple auditory filters. This finding is consistent with most previous research conducted at higher frequencies. In contrast with previous research, test subjects were, on average, able to detect multitone sounds at lower levels than those predicted using the SSM. Analyses that included Monte Carlo simulations showed that normally distributed errors in the single tone thresholds result in biased estimates of the thresholds of multitone sounds. This phenomenon is likely responsible for at least a substantial fraction of the unexpected deviation of measurements from SSM predictions.
Rafaelof, Menachem↗
Open system magnetic evolution of the taos plateau volcanic field, Northern New Mexico. I - The petrology and geochemistry of the servilleta basalt
MULTIFIT, an embodiment of the conceptual structure needed in modeling multisource and multiprocess magmatic systems, is described. This program, which uses familiar materials balance methodology and the equilibrium form of the Rayleigh equations, links evolutionary arrays, which is turn collectively relate the starting and final compositions of a given magmatic system. Moreover, MULTIFIT incorporates variations within major element data arrays; the linkage between them can be tested using an extension of the least squares algorithm, which selects the best branch point according to the minimum-sum-of-squared-residuals criterion. Advantages and disadvantages of the materials balance approach used in this program are discussed, an example is provided, and equations utilized by MULTIFIT are summarized. While MULTIFIT may not be the best approach for poorly constrained models involving partial melting for complex mixing, it may ultimately prove useful for ascertaining trace element partition coefficients in magnetic systems.
Dungan, M. A.↗
Using Redundancy To Reduce Errors in Magnetometer Readings
A method of reducing errors in noisy magnetic-field measurements involves exploitation of redundancy in the readings of multiple magnetometers in a cluster. By "redundancy"is meant that the readings are not entirely independent of each other because the relationships among the magnetic-field components that one seeks to measure are governed by the fundamental laws of electromagnetism as expressed by Maxwell's equations. Assuming that the magnetometers are located outside a magnetic material, that the magnetic field is steady or quasi-steady, and that there are no electric currents flowing in or near the magnetometers, the applicable Maxwell 's equations are delta x B = 0 and delta(raised dot) B = 0, where B is the magnetic-flux-density vector. By suitable algebraic manipulation, these equations can be shown to impose three independent constraints on the values of the components of B at the various magnetometer positions. In general, the problem of reducing the errors in noisy measurements is one of finding a set of corrected values that minimize an error function. In the present method, the error function is formulated as (1) the sum of squares of the differences between the corrected and noisy measurement values plus (2) a sum of three terms, each comprising the product of a Lagrange multiplier and one of the three constraints. The partial derivatives of the error function with respect to the corrected magnetic-field component values and the Lagrange multipliers are set equal to zero, leading to a set of equations that can be put into matrix.vector form. The matrix can be inverted to solve for a vector that comprises the corrected magnetic-field component values and the Lagrange multipliers.
Kulikov, Igor↗
A Convex Data-Driven Approach for Nonlinear Control Synthesis
We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.
97 MATHEMATICS AND COMPUTING↗