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At least 541 records · Page 30

Ambient Performance Testing of the CubeSat Infrared Atmospheric Sounder (CIRAS)

Hyperspectral infrared measurements of Earth’s atmosphere from space have proven their value for weather forecasting, climate science and atmospheric composition. The CubeSat Infrared Atmospheric Sounder (CIRAS) instrument will demonstrate a fully functional infrared temperature, water vapor and carbon monoxide sounder in a CubeSat sized volume for at least an order of magnitude lower cost than legacy systems. Design for a CubeSat significantly reduces cost of access to space and enables flight in a constellation to reduce revisit time and enable new measurements including 3D winds. A technology demonstration of CIRAS is currently under development at JPL. The effort has completed integration and ambient testing of a high fidelity brassboard, complete with the flight configured optics assembly developed by Ball Aerospace with a JPL Immersion Grating and Black Silicon Entrance Slit. The brassboard includes a flight-configured High Operating Temperature Barrier Infrared Detector (HOT-BIRD) mounted in an Integrated Dewar Cryocooler Assembly (IDCA), enabling testing in the ambient environment. Ambient testing included radiometric testing of the system to characterize the instrument operability and NEdT. Spatial testing was performed to characterize the system line spread function (LSF) in two axes and report FWHM of the LSF. Spectral testing involved an air path test to characterize the spectral/spatial transformation matrix, and an etalon was used to measure the Spectral Response Functions (SRFs). Results of the testing show the CIRAS performs exceptionally well and meets the key performance required of the system. The end result of testing is the CIRAS instrument now meets TRL 4 with confidence in a brassboard configuration ready for thermal vacuum (TVac) testing necessary to achieve TRL 5 for the system.

Wilson, Robert C.↗

Integrated Systems Health Management for Sustainable Habitats (Using Sustainability Base as a Testbed)

Habitation systems provide a safe place for astronauts to live and work in space and on planetary surfaces. They enable crews to live and work safely in deep space, and include integrated life support systems, radiation protection, fire safety, and systems to reduce logistics and the need for resupply missions. Innovative health management technologies are needed in order to increase the safety and mission-effectiveness for future space habitats on other planets, asteroids, or lunar surfaces. For example, off-nominal or failure conditions occurring in safety-critical life support systems may need to be addressed quickly by the habitat crew without extensive technical support from Earth due to communication delays. If the crew in the habitat must manage, plan and operate much of the mission themselves, operations support must be migrated from Earth to the habitat. Enabling monitoring, tracking, and management capabilities on-board the habitat and related EVA platforms for a small crew to use will require significant automation and decision support software.Traditional caution and warning systems are typically triggered by out-of-bounds sensor values, but can be enhanced by including machine learning and data mining techniques. These methods aim to reveal latent, unknown conditions while still retaining and improving the ability to provide highly accurate alerts for known issues. A few of these techniques will briefly described, along with performance targets for known faults and failures. Specific system health management capabilities required for habitat system elements (environmental control and life support systems, etc.) may include relevant subsystems such as water recycling systems, photovoltaic systems, electrical power systems, and environmental monitoring systems. Sustainability Base, the agency's flagship LEED-platinum certified green building acts as a living laboratory for testing advanced information and sustainable technologies that provides an opportunity to test novel machine learning and controls capabilities. In this talk, key features of Sustainability Base that make it relevant to deep space habitat technology and its use of these kinds of subsystems previously listed will be presented. The fact that all such systems require less power to support human occupancy can be used as a focal point to serve as a testbed for deep space habitats that will need to operate within finite energy budgets.

Systems Health Management↗

Scientific Discovery with Physics-Informed System Identification (Abbreviated Report)

My fellowship research focused on making physics-based simulations faster and more useful through machine learning. Many problems in science and engineering are governed by partial differential equations, but high-fidelity simulations are often too expensive to run repeatedly. I worked on improving Latent Space Dynamics Identification (LaSDI), a reduced-order modeling framework that compresses large simulation data sets into a smaller representation and then learns how that representation evolves over time. The motivation was to develop reduced models that remain accurate for more challenging systems, especially when predictions must remain reliable over long time intervals or when the underlying dynamics are more complicated than standard methods can easily handle. I also contributed to related work on Quandary, a high-performance software effort for simulation and control of open quantum systems, before focusing primarily on Latent Space Dynamics Identification methods. The main outcomes of the fellowship were two new algorithms (both of which were published), Rollout-LaSDI and Higher-Order LaSDI, together with supporting work on multi-stage Latent Space Dynamics Identification. Rollout-LaSDI improved long-term prediction by training the model to stay accurate over extended time horizons, and Higher-Order LaSDI broadened the method so it could model systems with higher-order time dynamics. My contributions to multistage Latent Space Dynamics Identification also helped show that its later training stages could be simplified without losing effectiveness, and that this behavior held across different model architectures and training strategies. Taken together, these advances improved the accuracy, flexibility, and practical value of reduced-order modeling tools for computational science.

97 MATHEMATICS AND COMPUTING↗

Active Wireless Temperature Sensors for Aerospace Thermal Protection Systems

Health diagnostics is an area where major improvements have been identified for potential implementation into the design of new reusable launch vehicles in order to reduce life-cycle costs, to increase safety margins, and to improve mission reliability. NASA Ames is leading the effort to advance inspection and health management technologies for thermal protection systems. This paper summarizes a joint project between NASA Ames and Korteks to develop active wireless sensors that can be embedded in the thermal protection system to monitor sub-surface temperature histories. These devices are thermocouples integrated with radio-frequency identification circuitry to enable acquisition and non-contact communication of temperature data through aerospace thermal protection materials. Two generations of prototype sensors are discussed. The advanced prototype collects data from three type-k thermocouples attached to a 2.54-cm square integrated circuit.

Milos, Frank S.↗

Pathfinder Lander Rover Recharge System, and MARCO POLO Controls and ACME Regolith Feed System Controls and Integration

This project stems from the Exploration, Research, and Technology Directorate (UB) Projects Division, and one of their main initiatives is the "Journey to Mars". Landing on the surface of Mars which is millions of miles away is an incredibly large challenge. The terrain is covered in boulders, deep canyons, volcanic mountains, and spotted with sand dunes. The robotic lander is a kind of spacecraft with multiple purposes. One purpose is to be the protective shell for the Martian rover and absorb the impact from the landing forces; another purpose is to be a place where the rovers can come back to, actively communicate with, and recharge their batteries from. Rovers have been instrumental to the Journey to Mars initiative. They have been performing key research on the terrain of the red planet, trying to unlock the mysteries of the land for over a decade. The rovers that will need charging will not all have the same kind of internal battery either. RASSOR batteries may differ from the PbAC batteries inside Red Rover's chassis. NASA has invested heavily in the exploration of the surface of Mars. A driving force behind further exploration is the need for a more efficient operation of Martian rovers. One way is to reduce the weight as much as possible to reduce power consumption given the same mission parameters. In order to reduce the mass of the rovers, power generation, communication, and sample analysis systems currently onboard Martian rovers can be moved to a stationary lander deck. Positioning these systems from the rover to the Lander deck allows a taskforce of smaller, lighter rovers to perform the same tasks currently performed by or planned for larger rovers. A major task in transferring these systems to a stationary lander deck is ensuring that power can be transferred to the rovers.

Internship↗

Lightweight Nonmetallic Thermal Protection Materials Technology

To fulfill President George W. Bush's "Vision for Space Exploration" (2004) - successful human and robotic missions to and from other solar system bodies in order to explore their atmospheres and surfaces - the National Aeronautics and Space Administration (NASA) must reduce the trip time, cost, and vehicle weight so that the payload and scientific experiments' capabilities can be maximized. The new project described in this paper will generate thermal protection system (TPS) product that will enable greater fidelity in mission/vehicle design trade studies, support risk reduction for material selections, assist in the optimization of vehicle weights, and provide materials and processes templates for use in the development of human-rated TPS qualification and certification plans.

Valentine, Peter G.↗

"Electrostructural Phase Changes" In Charged Particulate Clouds: Planetary and Astrophysical Implications

There is empirical evidence that freely-suspended triboelectrostatically charged particulate clouds of dielectric materials undergo rapid conversion from (nominally) monodispersed "aerosols" to a system of well-defined grain aggregates after grain motion or fluid turbulence ceases within the cloud. In United States Microgravity Laboratory Space Shuttle experiments USML-1 and USML-2, it was found that ballistically-energized grain dispersions would rapidly convert into populations of filamentary aggregates after natural fluid (air) damping of grain motion. Unless continuously disrupted mechanically, it was impossible to maintain a non-aggregated state for the grain clouds of sand-size materials. Similarly, ground- based experiments with very fine dust-size material produced the same results: rapid, impulsive "collapse" of the dispersed grains into well-defined filamentary structures. In both ground-based and microgravity experiments, the chains or filaments were created by long-range dipole electrostatic forces and dipole-induced dielectric interactions, not by monopole interactions. Maintenance of the structures was assisted by short-range static boundary adhesion forces and van der Waals interactions. When the aggregate containers in the USML experiments were disturbed after aggregate formation, the quiescently disposed filaments would rearrange themselves into fractal bundles and tighter clusters as a result of enforced encounters with one another. The long-range dipole interactions that bring the grains together into aggregates are a product of randomly-distributed monopole charges on the grain surfaces. In computer simulations, it has been shown that when the force vectors of all the random charges (of both sign) on a grain are resolved mathematically by assuming Coulombic interaction between them, the net result is a dipole moment on individual grains, even though the grains are electrically neutral insofar as there is no predominance, on their surface, of one charge sign over another. The random charges of both sign derive from natural grain-to-grain interactions that produce triboelectrification via charge exchange every time grain surfaces make contact with one another. The conversion from a random distribution of grains (upon which there are randomly distributed charges) into an organization of electrostatically-ordered aggregates, can be regarded (within the framework of granular-material science) as an "electrical or Coulombic phase change" of the particulate cloud. It is not totally dissimilar from the more normal phase-change concept in which, for example, a gas with long free-path-molecules suddenly becomes a solid as a result of structural ordering of the molecules (notably, also the result of electronic forces, albeit at a different scale). In both the gas-to-solid case, and the aerosol-to-aggregate case, the same materials and charges are present before and after the phase change, but their arrangement now has a higher degree of order and a lower-energy configuration. An input of energy into the system is required to reverse the situation. The aggregates in the USML experiments were observed to undergo at least two phase changes as noted above. The point about phase changes, and by implication, the "electrostructural" reorganizations in particulate clouds, is the following: (a) they can occur very rapidly, almost spontaneously, above a critical cloud density, (b) in going from a higher energy state to a lower energy state, they convert to a denser system, (c) energy must be required to reverse the situation, implying that energy is released during the high-to-low energy phase change. In applying this information to natural particulate clouds, some inferences can be made (it is stressed that reference is still to dielectric materials attracted by dipole forces). There are several natural settings to which the USML observations apply, and to which the phase-change implications likewise apply. Dense clouds of triboelectrically-charged, kinetically-energized grains are to be found in volcanic eruptions (particularly on earth), aeolian dust storms (particularly on Mars), meteorite impact ejecta curtains (on all planets), in "immature" debris rings around planets (e.g., that from which our own Moon may have condensed), and in gravitationally collapsing protoplanetary dust/planetesimal debris disks where dielectric granules are being increasingly brought into collisional relationships with one another (increasing both charge exchange and physical proximity). It is noted that in many of these cases, the degree of electrical charging on the grains is likely to be much higher than that in the USML experiments where charging was not enhanced above the "normal", naturally encountered level for the particular materials (quartz, glass, and various silicate minerals). Application of the phase-change concept suggests that volcanic, aeolian, and impact debris clouds may, under certain circumstances, undergo rapid, impulsive, or even catastrophic collapse into a denser state that will lead to rapid precipitation or fall-out of suspended particulates. Although this idea has been suggested previously by the author , the phase change concept possibly permits some new insights into cloud-system behavior. For example, in a protoplanetary debris disk, the work of gravity may suddenly be enhanced by electrostatically driven collapse of the system when materials reach a critical intergranular spacing or grain number density. This might reduce the rate of planet formation by orders of magnitude (considering the ratio of g-forces to electrostatic ones for very small grains in close proximity), and indeed, it might drive the collapse into a system configuration that would perhaps not be created by gravity alone. Additional information is contained in the original.

Marshall, J. R.↗

Rotorcraft Flight Dynamics and Controls Research at NASA

In recent years, NASA has invested in key activities in the areas of flight controls, handling qualities and operations of rotorcraft for civilian applications. More specifically, the flight dynamics and control discipline has focused on analyzing the unique flight control and handling qualities challenges of large rotary wing vehicles anticipated for future passenger service, and examining the effect of control system augmentation on handling qualities for current civilian helicopters in order to improve safety and reduce accident rates. This paper highlights two recent research efforts in these areas. The first is an examination of flight control and handling qualities aspects of large rotorcraft. A series of experiments were performed in the large-motion Vertical Motion Simulator at NASA Ames Research Center to quantify the effects of vehicle size on flight control requirements and piloted handling qualities. These experiments used a large tilt-rotor concept (~100 passengers) to also investigate the control augmentation required to obtain Level 1 handling qualities for a vehicle of this size. The second is an examination of the effect of control system augmentation on handling qualities for current civil rotorcraft, like those currently used for Emergency Medical Service type operations. Many current civilian helicopters have rate response type control systems and little or no control system augmentation, although current technologies allow helicopters to be fitted with stability augmentation systems, either as standard equipment or aftermarket options. A simulation experiment was conducted in the Vertical Motion Simulator to quantify the effects of advanced control modes available with a partial authority stability augmentation system on task performance and handling qualities in both good and degraded visual conditions. In addition to providing an overview of the rotary wing flight dynamics and controls research at NASA, this paper will provide an overview of these two research activities along with key results and conclusions.

Research at NASA↗

Rotorcraft Flight Dynamics and Controls Research at NASA

In recent years, NASA has invested in key activities in the areas of flight controls, handling qualities and operations of rotorcraft for civilian applications. More specifically, the flight dynamics and control discipline has focused on analyzing the unique flight control and handling qualities challenges of large rotary wing vehicles anticipated for future passenger service, and examining the effect of control system augmentation on handling qualities for current civilian helicopters in order to improve safety and reduce accident rates. This paper highlights two recent research efforts in these areas. The first is an examination of flight control and handling qualities aspects of large rotorcraft. A series of experiments were performed in the large-motion Vertical Motion Simulator at NASA Ames Research Center to quantify the effects of vehicle size on flight control requirements and piloted handling qualities. These experiments used a large tilt-rotor concept (~100 passengers) to also investigate the control augmentation required to obtain Level 1 handling qualities for a vehicle of this size. The second is an examination of the effect of control system augmentation on handling qualities for current civil rotorcraft, like those currently used for Emergency Medical Service type operations. Many current civilian helicopters have rate response type control systems and little or no control system augmentation, although current technologies allow helicopters to be fitted with stability augmentation systems, either as standard equipment or aftermarket options. A simulation experiment was conducted in the Vertical Motion Simulator to quantify the effects of advanced control modes available with a partial authority stability augmentation system on task performance and handling qualities in both good and degraded visual conditions. In addition to providing an overview of the rotary wing flight dynamics and controls research at NASA, this paper will provide an overview of these two research activities along with key results and conclusions.

Rotorcraft↗

Nonlinear dynamics of quadratic gravity in spherical symmetry

We present the first numerically stable nonlinear evolution for the leading-order gravitational effective field theory (quadratic gravity) in the spherically-symmetric sector. The formulation relies on (i) the harmonic gauge to cast the evolution system into quasilinear form (ii) the Cartoon method to reduce to spherical symmetry in keeping with the harmonic gauge, and (iii) order reduction to first order (in time) by means of introducing auxiliary variables. The well posedness of the respective initial-value problem is numerically confirmed by evolving randomly perturbed flat-space and black-hole initial data. In conclusion, our study serves as a proof-of-principle for the possibility of stable numerical evolution in the presence of higher derivatives.

79 ASTRONOMY AND ASTROPHYSICS↗

Multiscale modeling high-order methods and data-driven modeling

Projection-based reduced-order models (ROMs) comprise a promising set of data-driven approaches for accelerating the simulation of high-fidelity numerical simulations. Standard projection-based ROM approaches, however, suffer from several drawbacks when applied to the complex nonlinear dynamical systems commonly encountered in science and engineering. These limitations include a lack of stability, accuracy, and sharp a posteriori error estimators. This work addresses these limitations by leveraging multiscale modeling, least-squares principles, and machine learning to develop novel reduced-order modeling approaches, along with data-driven a posteriori error estimators, for dynamical systems. Theoretical and numerical results demonstrate that the two ROM approaches developed in this work - namely the windowed least-squares method and the Adjoint Petrov - Galerkin method - yield substantial improvements over state-of-the-art approaches. Additionally, numerical results demonstrate the capability of the a posteriori error models developed in this work.

97 MATHEMATICS AND COMPUTING↗

Low-Cost, Light Weight, Thin Film Solar Concentrator

This research addresses a cost barrier towards achieving a solar thermal collector system with an installed cost of $75/sq m and meet the Department of Energy's (DOE's) performance targets for optical errors, operations during windy conditions and lifetime. Current concentrators can cost as much as 40-50% of the total installed costs for a CSP plant. In order to reduce the costs from current $200-$250/sq m, it is important to focus on the overall system. The reflector surface is a key cost driver, and our film-based polymer reflector will help significantly in achieving DOE's cost target of $75/sq m. The ease of manufacturability, installation and replacement make this technology a compelling one to develop. This technology can be easily modified for a variety of CSP options including heliostats, parabolic dishes and parabolic troughs.

Concentrating Solar Power (CSP)↗

An effective field theory for non-maximal quantum chaos

In non-maximally quantum chaotic systems, the exponential behavior of out-of-time-ordered correlators (OTOCs) results from summing over exchanges of an infinite tower of higher “spin” operators. We construct an effective field theory (EFT) to capture these exchanges in (0 + 1) dimensions. The EFT generalizes the one for maximally chaotic systems, and reduces to it in the limit of maximal chaos. The theory predicts the general structure of OTOCs both at leading order in the 1/N expansion (N is the number of degrees of freedom), and after resuming over an infinite number of higher order 1/N corrections. These general results agree with those previously explicitly obtained in specific models. We also show that the general structure of the EFT can be extracted from the large q SYK model.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Improvements to the Tendon-Actuated Lightweight In-Space MANipulator (TALISMAN)

Devices for manipulating and precisely placing payloads are critical for efficient space operations including berthing of spacecraft, in-space assembly, construction and repair. Key to the success of many NASA space activities has been the availability of long-reach crane-like devices such as the Shuttle Remote Manipulation System (SRMS) and the Space Station Remote Manipulation System (SSRMS). These devices have been used for many operations including berthing visiting spacecraft to the International Space Station, deployment of spacecraft, space station assembly, astronaut positioning, payload transfer, and spacecraft inspection prior to atmospheric re-entry. Retiring the Space Transportation System has led to the removal of the SRMS from consideration for in-space missions, thus creating a capability gap. Recognizing this gap, work was initiated at NASA on a new architecture for long-reach space manipulators. Most current devices are constructed by joining revolute joints with carbon composite tubes, with the joints accounting for the majority of the device mass. For example in the case of the SRMS, the entire device mass is 410 kg (904 lbm); the joint structure, motors, gear train, cabling, etc., accounts for the majority of the system mass because the carbon composite tubes mass is 46 kg (101 lbm). An alternate space manipulator concept, the Tendon-Actuated Lightweight In-Space MANipulator (TALISMAN) was created to address deficiencies in the current state-of-the-art in long-reach manipulators. The antagonistic tendon actuated joint architecture allows the motors actuating the joint to be removed from the joint axis, which simplifies the joint design while simultaneously providing mechanical advantage for the motors. The improved mechanical advantage, in turn, reduces the size and power requirements for the motor and gear train. This paper will describe recent architectural improvements to the TALISMAN design that: 1) improve the operational robustness of the system by enabling maneuvers not originally possible by varying the TALISMAN geometry; 2) enable efficient active antagonistic control of a joint while sharing cable between antagonistic tension networks; and 3) uses a unique arrangement of differential capstans to reduce motor torque requirements by an order of magnitude. The paper will also summarize recent efforts to enable autonomous deployment of a TALISMAN including the deployment concept of operations and associated hardware system design. The deployment forces are provided by the same motor systems that are used for articulation, thus reducing the mass associated with the deployment system. The deployment approach is being tested on a TALISMAN prototype which is designed to provide the same operational performance as a shuttle-class manipulator. The prototype has been fabricated and is operational in a new facility at NASA Langley Research Center that has a large area (15.2 m by 21.3 m [50 ft by 70 ft]) air-bearing floor.

Doggett, William R.↗

On-orbit systems identification of flexible spacecraft

The process of on-orbit systems identification of flexible spacecraft is examined in terms of the difficulties that are expected because of the potentially unmanageable number of unknown model parameters due to the very high order system models involved. A Jordon block canonical form and global model parameters are used to reduce the number of unknown parameters to manageable numbers. A Bayesian approach is discussed which enables the merging of theoretical models with ground or on-orbit test results by using Unconditional Maximum Likelihood Parameter Estimation. A Modified Newton-Raphson technique is proposed for determining the model parameter estimates and an expected fit error criterion is recommended for the determination of the model structure and order reduction.

Taylor, L. W., Jr.↗

Facsimile bandwidth compression using nonlinear analog processing.

A system is presented for reducing the number of picture elements in a scanned image by an order of magnitude while preserving the detail contrast. Starting with high-resolution data a 3 x 3 matrix of input elements is compressed into a single output element using a nonlinear algorithm. Optimum results are obtained by including the data on the outer periphery of the input sample in the processing algorithm thus requiring a 5 x 5 input matrix. Experimental results are presented using weather satellite photographs. This system has application in those cases where the sensor has higher resolution capability than the communication link can support.

Macovski, A.↗

A Multi-Center Space Data System Prototype Based on CCSDS Standards

Deep space missions beyond earth orbit will require new methods of data communications in order to compensate for increasing Radio Frequency (RF) propagation delay. The Consultative Committee for Space Data Systems (CCSDS) standard protocols Spacecraft Monitor & Control (SM&C), Asynchronous Message Service (AMS), and Delay/Disruption Tolerant Networking (DTN) provide such a method. However, the maturity level of this protocol stack is insufficient for mission inclusion at this time. This Space Data System prototype is intended to provide experience which will raise the Technical Readiness Level (TRL) of this protocol set. In order to reduce costs, future missions can take advantage of these standard protocols, which will result in increased interoperability between control centers. This prototype demonstrates these capabilities by implementing a realistic space data system in which telemetry is published to control center applications at the Jet Propulsion Lab (JPL), the Marshall Space Flight Center (MSFC), and the Johnson Space Center (JSC). Reverse publishing paths for commanding from each control center are also implemented. The target vehicle consists of realistic flight computer hardware running Core Flight Software (CFS) in the integrated Power, Avionics, and Power (iPAS) Pathfinder Lab at JSC. This prototype demonstrates a potential upgrade path for future Deep Space Network (DSN) modification, in which the automatic error recovery and communication gap compensation capabilities of DTN would be exploited. In addition, SM&C provides architectural flexibility by allowing new service providers and consumers to be added efficiently anywhere in the network using the common interface provided by SM&C's Message Abstraction Layer (MAL). In FY 2015, this space data system was enhanced by adding telerobotic operations capability provided by the Robot API Delegate (RAPID) family of protocols developed at NASA. RAPID is one of several candidates for consideration and inclusion in a new international standard being developed by the CCSDS Telerobotic Operations Working Group. Software gateways for the purpose of interfacing RAPID messages with the existing SM&C based infrastructure were developed. Telerobotic monitor, control, and bridge applications were written in the RAPID framework, which were then tailored to the NAO telerobotic test article hardware, a product of Aldebaran Robotics.

Rich, Thomas M.↗

Determination of robust stability margin for second-order systems

Robust stabilization of uncertain systems has been extensively investigated and the stability test for the whole set of uncertain parameters has been reduced to a finite number of test points, four points for the characteristic polynomial with independent coefficients. As a result the robust stability margin can be determined using a reasonable amount of computation. It is impossible to apply the results of the test to a practical system as the coefficients of the characteristic polynomial for a physical system are usually functions of uncertain parameters. However, many physical systems may be represented by a second-order mass-spring-damper system with a special multilinear form in its characteristic polynomial. This paper investigates second-order mass-spring-damper systems and the reduction of the number of test points. It is shown that such a system with arbritrary compensators always has a multilinear characteristic polynomial. It is also shown that a line in the two-dimensional parameter space forms the boundary after the mapping of a multilinear characteristic polynomial and this interior extreme line forms a conic curve in the complex plane. The boundary of uncertain domain for a multilinear polynomial with two uncertainty parameters can be determined analytically using this curve, and the four sides image of a square of the uncertain parameter. Therefore, the stability margin may be determined by checking the intersections of the boundary with the zero point. A similar procedure can be used for second-order systems with more than two uncertainty parameters when parameter optimization is used in determining the boundary.

Chuang, C.-H.↗