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All sourcesNASA NTRSDOE OSTI
NASA NTRSDate not supplied

Mapping Aerosol Lidar Ratios Over Ocean Using Constrained Retrievals and A Global Aerosol Model

The current NASA Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) algorithms assign one lidar ratio (i.e., extinction-to-backscatter ratio; LR) value globally for each of the seven tropospheric aerosol types. In a future data products release, the CALIPSO project aims to improve these algorithms through the development of regional and seasonal LR climatologies. In this work, aerosol LRs are inferred through Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) backscatter profiles constrained by collocated aerosol optical depth (AOD) from Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) data. This analysis is subsampled for those profiles that are cloud-free and contain only one CALIOP-classified aerosol type (e.g., marine). The CALIOP profiles are then collocated with aerosol volume fractions obtained through Goddard Chemistry Aerosol Radiation and Transport (GOCART) model simulations. In this presentation, we show that the twelve-year (June 2006-August 2018) mean spatial distributions of inferred aerosol LRs for CALIOP-classified marine aerosols correspond inversely with patterns of GOCART sea salt volume fraction (SSVF). For example, smaller SSVFs (< 65%) and larger LRs (> 55 sr), are found near land masses. This is indicative of the influence of over-land aerosols (e.g., pollution and biomass burning smoke). In the remote oceans (i.e., regions likely less impacted by non-sea salt aerosols), larger SSVFs (> 95%) and smaller LRs (< 25 sr) are found. The developed relationship between the GOCART SSVFs and MODIS AOD constrained LRs (polynomial fit intersect values of ~56 sr for SSVF of 0% and ~21 sr for SSVF of 100%) is used to produce model-assisted climatological LR maps on seasonal scales. Additionally, we show maps of inferred LRs from constrained retrievals using the CALIPSO Ocean Derived Column Optical Depth (ODCOD) product and comparisons with those from the MODIS analyses. The technique demonstrated in this study not only benefits CALIPSO in the near-term, but similar methods can be applied to future spaceborne elastic backscatter lidars with collocated passive sensors, such as those associated with the upcoming NASA Atmosphere Observing System (AOS).

Travis Toth↗
NASA NTRSDate not supplied

Ice-Over-Water Cloud Identification in an Artificial Neural Network Approach

An artificial neural network (ANN) algorithm, employing several Aqua MODIS channels, the retrieved cloud phase and total cloud visible optical depth, and temperature and humidity vertical profiles is trained to detect multilayer (ML) ice-over-water cloud systems identified by matched 2008 CloudSat and CALIPSO (CC) data. The trained MLANN was applied to 2009 MODIS data resulting in combined ML and single layer detection accuracies of 87% (89%) and 86% (89%) for snow-free (snow-covered) regions during the day and night, respectively. When corrected for the viewing-zenith-angle dependence of each parameter, the ML fraction detected is relatively invariant across the swath. Compared to the CC ML variability, the MLANN is robust seasonally and interannually, and produces similar distribution patterns over the globe, except in the polar regions. Additional research is needed to conclusively evaluate the VZA dependence and further improve the MLANN accuracy. This approach should greatly improve the monitoring of cloud vertical structure using operational passive sensors.

MODIS↗
NASA NTRSDate not supplied

Mapping Aerosol Lidar Ratios Over Ocean using MODIS AOD Constrained Retrievals and GOCART Model Simulations

After 17 years, the NASA Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission ceased science operations in August 2023. For the final CALIPSO data products release (Version 5), the CALIPSO project seeks to improve the accuracy of its aerosol extinction by advancing knowledge of aerosol lidar ratios (i.e., extinction-to-backscatter ratios; LRs) for various aerosol types. The current algorithm assigns one LR value globally for each of the seven tropospheric aerosol types. The CALIPSO team aims to improve the retrieval algorithm through the development of regional and seasonal LR climatologies for the same aerosol types. In this study, aerosol LRs are inferred through Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) backscatter profiles constrained by collocated aerosol optical depth (AOD) from Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) data over oceans during daytime. This analysis is subsampled for those profiles that are cloud-free and contain only one CALIOP-classified aerosol type. The CALIOP profiles are then collocated with aerosol volume fractions obtained through Goddard Chemistry Aerosol Radiation and Transport (GOCART) model simulations. This presentation will reveal findings that the 12-year (June 2006-August 2018) mean spatial distributions of inferred aerosol LRs for CALIOP-classified marine and dusty marine aerosols correspond inversely with patterns of GOCART sea salt volume fraction (SSVF). For example, smaller SSVFs (< 65%) and larger LRs (> 55 sr), are found near land masses (Fig. 1). This indicates the influence of advected anthropogenic aerosols (e.g., pollution and biomass burning smoke). In the remote oceans (i.e., regions likely less impacted by non-sea salt aerosols), the SSVFs are larger (> 95%) and the LRs are smaller (< 25 sr) (Fig. 1). A polynomial fit of the MODIS AOD constrained LRs to the corresponding GOCART SSVFs (intersect values of ~58 sr for SSVF of 0% and ~21 sr for SSVF of 100%) is further used to produce model-assisted climatological LR maps on seasonal scales. Additionally, we will show results of a LR validation analysis for which we compare the revised CALIPSO AODs obtained by applying the seasonal/regional constrained LRs against CALIPSO Version 4.51 Ocean Derived Column Optical Depth (ODCOD). While the majority of the presentation will focus on LRs for CALIOP-classified marine and dusty marine aerosols, an overview of LR results will show preliminary results for other aerosol types over ocean, such as dust and elevated smoke. The technique demon-strated in this study highlights the benefits not only to the final planned CALIPSO data release in 2025, but similar methods can be applied to future spaceborne elastic backscatter lidars with collocated passive sensors (e.g., such as those associated with NASA’s proposed Atmosphere Observing System).

Travis D Toth↗
NASA NTRSDate not supplied

Spaceborne Lidar Retrievals of PM2.5 for Air Quality Studies and Applications

Fine particulate matter (PM2.5) substantially contributes to air pollution and negatively affects human health. While many studies have investigated the use of passive column-integrated aerosol optical depth to infer surface PM2.5, the use of lidar observations for air quality characterization is not nearly as extensive. Lidar measurements are critical, however, due to the vertical aerosol information they provide, including near the surface. In this presentation, we first provide an overview of various lidar-based approaches for estimating PM2.5 concentrations and then discuss how lidar measurements can assist other air quality applications. For example, estimates of PM2.5 have been obtained in a physics-based approach through CALIOP near-surface aerosol extinction retrievals, assumptions on the mass extinction efficiency, and incorporating other parameters (an aerosol hygroscopic growth factor and PM2.5/PM10 ratio). Application of this algorithm over the contiguous United States (CONUS) from 2006 to 2018 yielded larger PM2.5 values over the eastern and western CONUS (~10-15 μg/m³) and lower PM2.5 levels in the central CONUS (~5 μg/m³). These spatial patterns were similar to those from gridded PM2.5 concentrations obtained through in situ measurements at ground stations operated by the US Environmental Protection Agency. In another approach, the Cloud Aerosol Transport System (CATS) lidar was used with the Goddard Earth Observing System (GEOS) model in a 1D ensemble-based variational technique to obtain PM2.5 over the US and Europe, and the spatial patterns of the CATS/GEOS based PM2.5 concentrations generally captured those from surface stations (with corresponding hourly EPA PM2.5 vs CATS PM2.5 statistics of R=0.4 and bias=1.5 μg/m³). In our recent work, as part of the Models, In situ, and Remote sensing of Aerosols (MIRA) Working Group, we have applied both the CALIOP and CATS/GEOS based approaches over the highly polluted country of India during the post-monsoon season (September-October 2016). We derived elevated levels of two-month mean PM2.5 (~100 μg/m³) in northern India, especially near New Delhi. These high PM2.5 concentrations in the Indo-Gangetic plain are driven in large part from the seasonal burning of crop residue and meteorological conditions typical at this time of the year, such as low wind speeds and a shallow boundary layer. While the satellite-derived PM2.5 moderately replicates (R = ~0.7-0.9) the spatial variability in the two-month mean of surface in situ PM2.5 from monitoring sites operated by the Central and State Pollution Control Boards, we show results from specific scenes for which there are large deviations between the satellite-derived PM2.5 and in situ measurements. Other current work on this topic focuses on developing PM2.5 estimates using airborne high spectral resolution lidar measurements through machine learning regression algorithms and involves several parameters (e.g., aerosol extinction, color ratio, lidar ratio). Application of this method over major metropolitan areas in the US and Asia have resulted in high correlations (R = 0.93) with surface measurements. This airborne lidar approach can be adapted to spaceborne lidar measurements, and all three of these approaches can be applied to ESA’s EarthCARE Atmospheric Lidar instrument, setting the stage for the future Cloud Aerosol Lidar for Global Scale Observations of the Ocean-Land Atmosphere System (CALIGOLA) mission. Ultimately, beyond estimates of PM2.5, the aerosol vertical distribution from lidars can benefit studies involving passive sensor approaches for PM2.5 proxies (including from geostationary satellites), wildfire smoke plume injection heights, volcanic emissions (e.g., ash height retrievals), and aerosol/air quality model assimilation, evaluation, and forecasts.

Travis D Toth↗
DOE OSTI2022

All-Ceramic Passive Wireless Temperature Sensor Realized by Tin-Doped Indium Oxide (ITO) Electrodes for Harsh Environment Applications

In this work, an all-ceramic passive wireless inductor–capacitor (LC) resonator was presented for stable temperature sensing up to 1200 °C in air. Instead of using conventional metallic electrodes, the LC resonators are modeled and fabricated with thermally stable and highly electroconductive ceramic oxide. The LC resonator was modeled in ANSYS HFSS to operate in a low-frequency region (50 MHz) within 50 × 50 mm geometry using the actual material properties of the circuit elements. The LC resonator was composed of a parallel plate capacitor coupled with a planar inductor deposited on an Al2O3 substrate using screen-printing, and the ceramic pattern was sintered at 1250 °C for 4 h in an ambient atmosphere. The sensitivity (average change in resonant frequency with respect to temperature) from 200–1200 °C was ~170 kHz/°C. The temperature-dependent electrical conductivity of the tin-doped indium oxide (ITO, 10% SnO2 doping) on the quality factor showed an increase of Qf from 36 to 43 between 200 °C and 1200 °C. The proposed ITO electrodes displayed improved sensitivity and quality factor at elevated temperatures, proving them to be an excellent candidate for temperature sensing in harsh environments. The microstructural analysis of the co-sintered LC resonator was performed using a scanning electron microscope (SEM) which showed that there are no cross-sectional and topographical defects after several thermal treatments.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1987

Edge passivated charge-coupled device image sensor

A charge-coupled device (CCD) image sensor includes in a substrate of single crystalline silicon of one conductivity type an array of a plurality of spaced, parallel channel regions of the opposite conductivity type extending along one major surface of the substrate. A plurality of parallel conductive gates are over the one major surface of the substrate and extend transversely across the channel regions. The outermost channel regions of the array are positioned adjacent edges of the substrate so that a plurality of the image sensors can be mounted in edge-to-edge relation with the channel regions of the various sensors being close together. The sensor includes passivating means between each outermost channel region and the adjacent edge to prevent charge carriers generated by the edge from being injected into the outermost channel region. The passivating means includes a highly conductive drain region of a conductivity type opposite to that of the substrate within the substrate and extending along the one major surface between the outermost channel region and the edge. Also, a highly conductive region of the same conductivity type as the substrate is on the substrate and extends along the one major surface directly at the edge.

Kosonocky, Walter F.↗
NASA NTRS2012

Satellite Observation Systems for Polar Climate Change Studies

The key observational tools for detecting large scale changes of various parameters in the polar regions have been satellite sensors. The sensors include passive and active satellite systems in the visible, infrared and microwave frequencies. The monitoring started with Tiros and Nimbus research satellites series in the 1970s but during the period, not much data was stored digitally because of limitations and cost of the needed storage systems. Continuous global data came about starting with the launch of ocean color, passive microwave, and thermal infrared sensors on board Nimbus-7 and Synthetic Aperture Radar, Radar Altimeter and Scatterometer on board SeaSat satellite both launched in 1978. The Nimbus-7 lasted longer than expected and provided about 9 years of useful data while SeaSat quit working after 3 months but provided very useful data that became the baseline for follow-up systems with similar capabilities. Over the years, many new sensors were launched, some from Japan Aeronautics and Space Agency (JAXA), some from the European Space Agency (ESA) and more recently, from RuSSia, China, Korea, Canada and India. For polar studies, among the most useful sensors has been the passive microwave sensor which provides day/night and almost all weather observation of the surface. The sensor provide sea surface temperature, precipitation, wind, water vapor and sea ice concentration data that have been very useful in monitoring the climate of the region. More than 30 years of such data are now available, starting with the Scanning Multichannel Microwave Radiometer (SMMR) on board the Nimbus-7, the Special Scanning Microwave/Imager (SSM/I) on board a Defense Meteorological Satellite Program (DMSP) and the Advanced Microwave Scanning Radiometer on board the EOS/ Aqua satellite. The techniques that have been developed to derive geophysical parameters from data provided by these and other sensors and associated instrumental and algorithm errors and validation techniques will be discussed. An important issue is the organization and storage of hundreds of terabytes of data collected by even just a few of these satellite sensors. Advances in mass storage and computer technology have made it possible to overcome many of the collection and archival problems and the availability of comprehensive satellite data sets put together by NASA's Earth Observing System project will be discussed.

Comiso, Josefino C.↗
NASA NTRS1987

Millimeter-wave imaging sensor data evaluation

A passive 3-mm radiometer system with a mechanically scanned antenna was built for use on a small aircraft or an Unmanned Aerial Vehicle to produce real near-real-time, moderate-resolution (0.5) images of the ground. One of the main advantages of this passive imaging sensor is that it is able to provide surveillance information through dust, smoke, fog and clouds when visual and IR systems are unusable. It can also be used for a variety of remote sensing applications, such as measurements of surface moisture, surface temperature, vegetation extent and snow cover. It is also possible to detect reflective objects under vegetation cover.

Wilson, William J.↗
NASA NTRS1980

Fiber-optic sensing in cryogenic environments

Passive optical sensors using fiber-optic signal transmission to a remote monitoring station are explored as an alternative to electrical sensors used to monitor the status of explosive propellants. The designs of passive optical sensors measuring liquid level, pressure, and temperature in cryogenic propellant tanks are discussed. Test results for an experimental system incorporating these sensors and operating in liquid nitrogen demonstrate the feasibility of passive sensor techniques and indicate that they can serve as non-hazardous replacements for more conventional measuring equipment in explosive environments.

Sharma, M.↗
NASA NTRS2000

Passive Night Vision Sensor Comparison for Unmanned Ground Vehicle Stereo Vision Navigation

One goal of the "Demo III" unmanned ground vehicle program is to enable autonomous nighttime navigation at speeds of up to 10 m.p.h. To perform obstacle detection at night with stereo vision will require night vision cameras that produce adequate image quality for the driving speeds, vehicle dynamics, obstacle sizes, and scene conditions that will be encountered. This paper analyzes the suitability of four classes of night vision cameras (3-5 micrometer cooled FLIR, 8-12 micrometer cooled FLIR, 8-12 micrometer uncooled FLIR, and image intensifiers) for night stereo vision, using criteria based on stereo matching quality, image signal to noise ratio, motion blur and synchronization capability. We find that only cooled FLIRs will enable stereo vision performance that meets the goals of the Demo III program for nighttime autonomous mobility.

Owens, Ken↗
NASA NTRS2003

Development of Structural Health Management Technology for Aerospace Vehicles

As part of the overall goal of developing Integrated Vehicle Health Management (IVHM) systems for aerospace vehicles, NASA has focused considerable resources on the development of technologies for Structural Health Management (SHM). The motivations for these efforts are to increase the safety and reliability of aerospace structural systems, while at the same time decreasing operating and maintenance costs. Research and development of SHM technologies has been supported under a variety of programs for both aircraft and spacecraft including the Space Launch Initiative, X-33, Next Generation Launch Technology, and Aviation Safety Program. The major focus of much of the research to date has been on the development and testing of sensor technologies. A wide range of sensor technologies are under consideration including fiber-optic sensors, active and passive acoustic sensors, electromagnetic sensors, wireless sensing systems, MEMS, and nanosensors. Because of their numerous advantages for aerospace applications, most notably being extremely light weight, fiber-optic sensors are one of the leading candidates and have received considerable attention.

Prosser, W. H.↗
NASA NTRS2007

Precise Measurement of Carbon Dioxide Column by Passive Ground Based Sensor

Over the past four years we have developed a family of differential radiometers based upon the Fabry-Perot interferometers that exhibit very great sensitivity to changes in the atmospheric column of carbon dioxide, oxygen, and water vapor. Our instruments employ a solid Fabry-Perot etalon that is tuned to the proper wavelength by changing its temperature. The thickness of the etalon has been selected so that its multiple pass bands align with regularly space absorption features of the molecule under investigation. Using multiple absorption features improves the optical throughput of the instrument and improves the stability of the instrument response with respect to environmental changes. We are presently working to extend this technique to the carbon 13 isotope of carbon dioxide and to methane. Our instruments are intrinsically rugged and can be fabricated in a small package at relatively low cost.. As such they hold promise for widespread use in ground based networks for calibration and validation of satellite instruments such as OCO and GOSAT. Results will be presented for long term ground based operations of these systems. The effects of atmospheric scattering, pointing errors, pressure broadening and temperature effects will be discussed with regard to achieving precision better than .5% required for validation of carbon dioxide column measured from space. Finally we will outline the approach for extension of this methodology to additional molecular species of interest.

Heaps, William S.↗
NASA NTRS2009

Application of TRMM PR and TMI Measurements to Assess Cloud Microphysical Schemes in the MM5 Model for a Winter Storm

Seen by the human eye, precipitation particles are commonly drops of rain, flakes of snow, or lumps of hail that reach the ground. Remote sensors and numerical models usually deal with information about large collections of rain, snow, and hail (or graupel --also called soft hail ) in a volume of air. Therefore, the size and number of the precipitation particles and how particles interact, evolve, and fall within the volume of air need to be represented using physical laws and mathematical tools, which are often implemented as cloud and precipitation microphysical parameterizations in numerical models. To account for the complexity of the precipitation physical processes, scientists have developed various types of such schemes in models. The accuracy of numerical weather forecasting may vary dramatically when different types of these schemes are employed. Therefore, systematic evaluations of cloud and precipitation schemes are of great importance for improvement of weather forecasts. This study is one such endeavor; it pursues quantitative assessment of all the available cloud and precipitation microphysical schemes in a weather model (MM5) through comparison with the observations obtained by National Aeronautics and Space Administration (NASA) s and Japan Aerospace Exploration Agency (JAXA) s Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR) and microwave imager (TMI). When satellite sensors (like PR or TMI) detect information from precipitation particles, they cannot directly observe the microphysical quantities (e.g., water species phase, density, size, and amount etc.). Instead, they tell how much radiation is absorbed by rain, reflected away from the sensor by snow or graupel, or reflected back to the satellite. On the other hand, the microphysical quantities in the model are usually well represented in microphysical schemes and can be converted to radiative properties that can be directly compared to the corresponding PR and TMI observations. This study employs this method to evaluate the accuracy of the simulated radiative properties by the MM5 model with different microphysical schemes. It is found that the representations of particle density, size, and mass in the different schemes in the MM5 model determine the model s performance when predicting a winter storm over the eastern Pacific Ocean. Schemes lacking moderate density particles (i.e. graupel), with snow flakes that are too large, or with excessive mass of snow or graupel lead to degraded prediction of the radiative properties as observed by the TRMM satellite. This study demonstrates the uniqueness of the combination of both an active microwave sensor (PR) and passive microwave sensor (TMI) onboard TRMM on assessing the accuracy of numerical weather forecasting. It improves our understanding of the physical and radiative properties of different types of precipitation particles and provides suggestions for better representation of cloud and precipitation processes in numerical models. It would, ultimately, contribute to answering questions like "Why did it not rain when the forecast says it would?"

Han, Mei↗
DOE OSTI2020

Passive wireless pressure sensor for harsh environments

Methods and apparatuses for measuring static and dynamic pressures in harsh environments are disclosed. A pressure sensor according to one embodiment of the present invention may include a diaphragm constructed from materials designed to operate in harsh environments. A waveguide may be operably connected to the diaphragm, and an electromagnetic wave producing and receiving (e.g., sensing) device may be attached to the waveguide, opposite the diaphragm. A handle may be connected between the diaphragm and the waveguide to provide both structural support and electrical functionality for the sensor. A gap may be included between the handle and the diaphragm, allowing the diaphragm to move freely. An antenna and a ground plane may be formed on the diaphragm or the handle. Electromagnetic waves may be reflected off the antenna and detected to directly measure static and dynamic pressures applied to the diaphragm.

42 ENGINEERING↗
DOE OSTI2022

Evaluation of the Reliability of Passive Infrared (PIR) Occupancy Sensors for Residential Indoor Lighting

Solid-state lighting (SSL) technologies have penetrated the general illumination market in recent years, largely replacing conventional technologies such as incandescent and fluorescent lighting. Most of the initial excitement about light-emitting diode (LED) sources for SSL devices focused on their energy savings potential resulting from vast improvements in source efficiency and luminous efficacy compared with conventional illumination products. More recently, the focus has shifted toward other aspects of lighting application efficiency, namely intensity effectiveness and spectral efficiency, because of the ease of controlling both the light intensity of LEDs with drive voltage and the color properties of the LED-based illuminators. To capitalize on the energy savings of increased intensity effectiveness and spectral efficiency, a lighting control system (LCS) is often used. While the current penetration of LCSs is relatively modest, it is anticipated that lighting controls (i.e., connected lighting, controls and LED and conventional lighting) could have an installed penetration as higher as 46% by the year 2035, saving an additional 1.3 quads of energy. Despite the large energy savings that can be gained from using LCSs, standard test methods for evaluating sensors employed in LCSs and reliability data of the LCSs and their components are generally lacking. The National Electrical Manufacturers Association (NEMA) developed the only standard, NEMA WD 7-2011 (R2016), to test occupancy and motion sensor performance (herein referred to as “the NEMA protocol”). In a previous U.S. Department of Energy (DOE) effort, some concerns of the NEMA protocol were identified (e.g., strict height and weight limits on test subjects, large amounts of manual effort, sometimes inconsistent repeatability). During the current work, a consistent detection length test (DLT) and a Robotic Sensor Evaluation System (RoboSES) were developed to alleviate some of these concerns. RoboSES acts as a human surrogate by using thermal pads on a mannequin and a remote-controlled mobile base to test a sensor’s field of view (FOV). This report builds on the earlier DOE efforts to understand sensor technologies used in LCSs for general illumination. Specifically, this report describes the optimization of RoboSES, characterizes and establishes test methods to assess the reliability of multiple passive infrared (PIR) sensors, and reports the findings of robustness and reliability testing on two commercial PIR sensors intended for residential applications. The information presented in this report is gained from up to 4,000 hours (hrs) of accelerated stress tests (ASTs).

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
NASA NTRS1993

Status of the Fiber Optic Control System Integration (FOCSI) program

This report presents a discussion of the progress made in the NASA/NAVY Fiber Optic Control System Integration (FOCSI) program. This program will culminate in open-loop flight tests of passive optical sensors and associated electro-optics on an F-18 aircraft. Currently, the program is in the final stages of hardware fabrication and environmental testing of the passive optical sensors and electro-optics. This program is a foundation for future Fly-by-Light (FBL) programs. The term Fly-by-Light is used to describe the utilization of passive optical sensors and fiber optic data links for monitoring and control of aircraft in which sensor and actuation signals are transmitted optically. The benefits of this technology for advanced aircraft include the following: improved reliability and reduced certification cost due to greater immunity to EME (electromagnetic effects); reduced harness volume and weight; elimination of short circuits and sparking in wiring due to insulation deterioration; lower maintenance costs (fewer components); greater flexibility in data bus protocol and architecture; absence of ground loops; and higher operating temperatures for electrically passive optical sensors.

Baumbick, Robert J.↗
NASA NTRS1991

Robust controller design for structures with displacement sensors

The passivity theorem may be used to design robust controllers for structures with positive transfer functions. The authors obtain a similar result for systems with displacement sensors. In order to illustrate the ideas discussed, a specific second-order dynamic system is considered.

Morris, K. A.↗