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Results for “Optimal control, Observer design, Setting Control”

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At least 19 records

Disturbance observer-based dynamic optimal setting control

This paper deals with optimal operational control problems using a novel approach where a performance index is optimized by controller design together with set-points reselection. Firstly, the operational indices are obtained by employing some optimization algorithms. The controller is then designed for each closed loop to ensure the controlled variables can track the set points in the ideal situation. Taking advantage of disturbance observers, the set-points are reselected dynamically to minimize the effect caused by unexpected noises or disturbances. In the proposed method, the controller structure or parameters are never changed no matter what kind of disturbances affect the whole system. Finally, simulations on a 3 degrees of freedom robotic arm model are given to demonstrate the effectiveness and convenience of the proposed strategy.

Optimal control, Observer design, Setting Control↗

A method for optimal actuator and sensor placement for large flexible structures

A method of finding the optimal sensor and actuator locations for the control of flexible structures has been developed. The method is based on the orthogonal projection of structural modes into the intersection subspace of the controllable and observable subspaces. The controllability and observability grammians are then used to weight the projections to reflect the degrees of controllability and observability. This method produces a three-dimensional design space wherein sets of optimal actuator and sensors may be selected.

Lim, K. B.↗

Automated experimental design of safe rampdowns via probabilistic machine learning

Abstract Typically the rampdown phase of a shot consists of a decrease in current and injected power and optionally a change in shape, but there is considerable flexibility in the rate, sequencing, and duration of these changes. On the next generation of tokamaks it is essential that this is done safely as the device could be damaged by the stored thermal and electromagnetic energy present in the plasma. This works presents a procedure for automatically choosing experimental rampdown designs to rapidly converge to an effective rampdown trajectory. This procedure uses probabilistic machine learning methods paired with acquisition functions taken from Bayesian optimization. In a set of 2022 experiments at DIII-D, the rampdown designs produced by our method maintained plasma control down to substantially lower current and energy levels than are typically observed. The actions predicted by the model significantly improved as the model was able to explore over the course of the experimental campaign.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Nonlinear control of magnetic bearings

In this paper we present a variety of nonlinear controllers for the magnetic bearing that ensure both stability and robustness. We utilize techniques of discontinuous control to design novel control laws for the magnetic bearing. We present in particular sliding mode controllers, time optimal controllers, winding algorithm based controllers, nested switching controllers, fractional controllers, and synchronous switching controllers for the magnetic bearing. We show existence of solutions to systems governed by discontinuous control laws, and prove stability and robustness of the chosen control laws in a rigorous setting. We design sliding mode observers for the magnetic bearing and prove the convergence of the state estimates to their true values. We present simulation results of the performance of the magnetic bearing subject to the aforementioned control laws, and conclude with comments on design.

Pradeep, A. K.↗

Optimal placement of actuators and sensors in control augmented structural optimization

A control-augmented structural synthesis methodology is presented in which actuator and sensor placement is treated in terms of (0,1) variables. Structural member sizes and control variables are treated simultaneously as design variables. A multiobjective utopian approach is used to obtain a compromise solution for inherently conflicting objective functions such as strucutal mass control effort and number of actuators. Constraints are imposed on transient displacements, natural frequencies, actuator forces and dynamic stability as well as controllability and observability of the system. The combinatorial aspects of the mixed - (0,1) continuous variable design optimization problem are made tractable by combining approximation concepts with branch and bound techniques. Some numerical results for example problems are presented to illustrate the efficacy of the design procedure set forth.

Sepulveda, A. E.↗

LMI-Based Multiobjective Optimization and Control of Flexible Aircraft Using VCCTEF

This paper considers the control of coupled aeroelastic aircraft model with Variable Camber Continuous Trailing Edge Flap (VCCTEF) system. The relative motion between two adjacent flaps is constrained and this actuation constraint problem is converted into an output covariance constraint problem, and therefore can be formulated using linear matrix inequalities (LMIs). A set of LMI conditions is derived for the design of an observer-based dynamic output feedback controller for VCCTEF configured aeroelastic aircraft model. The proposed controller is then applied to the NASA Generic Transport Model (GTM) for simulation, and the results demonstrate the efficacy of the proposed approach.

LMI control↗

BICEP2. II. Experiment and three-year data set

We report on the design and performance of the BICEP2 instrument and on its three-year data set. BICEP2 was designed to measure the polarization of the cosmic microwave background (CMB) on angular scales of 1°-5°(ℓ = 40-200), near the expected peak of the B-mode polarization signature of primordial gravitational waves from cosmic inflation. Measuring B-modes requires dramatic improvements in sensitivity combined with exquisite control of systematics. The BICEP2 telescope observed from the South Pole with a 26 cm aperture and cold, on-axis, refractive optics. BICEP2 also adopted a new detector design in which beam-defining slot antenna arrays couple to transition-edge sensor (TES) bolometers, all fabricated on a common substrate. The antenna-coupled TES detectors supported scalable fabrication and multiplexed readout that allowed BICEP2 to achieve a high detector count of 500 bolometers at 150 GHz, giving unprecedented sensitivity to B-modes at degree angular scales. After optimization of detector and readout parameters, BICEP2 achieved an instrument noise-equivalent temperature of 15.8 μK√s. The full data set reached Stokes Q and U map depths of 87.2 nK in square-degree pixels (5.'2 μK) over an effective area of 384 deg{sup 2} within a 1000 deg{sup 2} field. These are the deepest CMB polarization maps at degree angular scales to date. The power spectrum analysis presented in a companion paper has resulted in a significant detection of B-mode polarization at degree scales.

79 ASTRONOMY AND ASTROPHYSICS↗

Loop transfer recovery for general nonminimum phase discrete time systems. II - Design

The authors consider the design of controllers for the recovery of target loop transfer function or sensitivity and complementary sensitivity functions for general nonminimum phase discrete time systems. The necessary design constraints and the available design freedom are reviewed. In view of the available freedom, possible specifications on the eigenstructure of the observer dynamic matrix are formulated. Three different types of controllers which are respectively based on prediction, current, and reduced-order estimators are considered. For each one of those controllers, three different design techniques are developed. The first one is an eigenstructure assignment scheme, while the other two are optimization-based designs. The eigenstructure assignment method yields a controller design which achieves any chosen recovery error matrix among a set of admissible recovery error matrices. On the other hand, one of the optimization-based design methods leads to a controller that achieves a recovery error matrix having the infimum H-infinity norm, while the other does the same except it achieves a recovery error matrix having the infimum H2 norm.

Chen, Ben M.↗

Robust control of systems with real parameter uncertainty and unmodelled dynamics

During this research period we have made significant progress in the four proposed areas: (1) design of robust controllers via H infinity optimization; (2) design of robust controllers via mixed H2/H infinity optimization; (3) M-delta structure and robust stability analysis for structured uncertainties; and (4) a study on controllability and observability of perturbed plant. It is well known now that the two-Riccati-equation solution to the H infinity control problem can be used to characterize all possible stabilizing optimal or suboptimal H infinity controllers if the optimal H infinity norm or gamma, an upper bound of a suboptimal H infinity norm, is given. In this research, we discovered some useful properties of these H infinity Riccati solutions. Among them, the most prominent one is that the spectral radius of the product of these two Riccati solutions is a continuous, nonincreasing, convex function of gamma in the domain of interest. Based on these properties, quadratically convergent algorithms are developed to compute the optimal H infinity norm. We also set up a detailed procedure for applying the H infinity theory to robust control systems design. The desire to design controllers with H infinity robustness but H(exp 2) performance has recently resulted in mixed H(exp 2) and H infinity control problem formulation. The mixed H(exp 2)/H infinity problem have drawn the attention of many investigators. However, solution is only available for special cases of this problem. We formulated a relatively realistic control problem with H(exp 2) performance index and H infinity robustness constraint into a more general mixed H(exp 2)/H infinity problem. No optimal solution yet is available for this more general mixed H(exp 2)/H infinity problem. Although the optimal solution for this mixed H(exp 2)/H infinity control has not yet been found, we proposed a design approach which can be used through proper choice of the available design parameters to influence both robustness and performance. For a large class of linear time-invariant systems with real parametric perturbations, the coefficient vector of the characteristic polynomial is a multilinear function of the real parameter vector. Based on this multilinear mapping relationship together with the recent developments for polytopic polynomials and parameter domain partition technique, we proposed an iterative algorithm for coupling the real structured singular value.

Chang, Bor-Chin↗

Advanced Energy Harvesting Control Schemes for Marine Renewable Energy Devices

The Advanced Energy Harvesting Control Schemes for Marine Renewable Energy Devices (Project) investigated, analyzed and modeled advanced turbine control schemes with the objective of increasing the energy harvested by hydrokinetic turbines in turbulent flow. Ocean Renewable Power Company (ORPC) implemented and validated a feedforward controller to increase power capture; and applied and tested the controls on ORPC’s RivGen® Power Systems in Igiugig, Alaska. Assessments of performance improvements were made for the RivGen® in the Igiugig environment and for ORPC’s TidGen® Power System in a reference tidal environment. Annualized Energy Production (AEP) and Levelized Cost of Energy (LCOE) improvements associated with implementation of the recommended control methodology were made for the TidGen® Power System in the DOE reference tidal environment. System Performance Advancement (SPA) goals were selected for the project. SPA targets were to improve Power to Weight Ratio (PWR) and system Availability, with the intention of reducing Levelized Cost of Electricity (LCOE). This project focused primarily reducing in PWR. Reductions in PWR of 25.5% were achieved. Reductions of 20.3% in LCOE were achieved. This project evaluated four types of controllers which were tested in simulation, emulation, a laboratory flume, and the field. The adaptive Kω2 controller performs similarly to the non-adaptive version of the same controller and may be useful in tidal channels where the mean velocity is continually evolving. Trends in simulation were largely verified through experiments, which also provided the opportunity to test assumptions about turbine responsiveness and control resilience to varying scales of turbulence. Laboratory experiments provided an essential stepping stone between simulation and implementation on a field-scale turbine. Experiments also demonstrated that using “energy loss” as a metric to differentiate between well-designed controllers operating at an optimal tip-speed ratio set-point is difficult, which anticipated the outcome from field experiments. The clear message is that the feedforward Kω2 controller out-performs the feedback controllers in almost all aspects and modes of evaluation. The controllers proved a substantial improvement over the baseline performance of the TidGen® turbine, in terms of energy capture. The effects of noise-contaminated angular velocity signals were investigated and validated by simulation as an explanation for the performance limitations observed for TidGen® turbine operations in Eastport, Maine. Measurements of loads performed as part of the laboratory testing indicate that there are limited differences in average axial thrust force between control architectures. This suggests that none of the control strategies are likely to substantially affect loads on the turbine support structure. Velocity measurements during the ORPC RivGen® turbine deployment at Igiugig, Alaska, in 2014 were used to assess the variability of the river flow. Results suggest that the river flow is approximately steady, in the mean sense, at any particular location in the river, with random turbulent fluctuations that are around 10% of the mean flow. The mean flow in the center channel of the river is 2.5 m/s, with reductions near the riverbanks and in the shallows. Spectral analysis and lagged correlation results indicate that temporal fluctuations at a given point are dominated by large scale fluctuations, such that measurements at the turbine location are just as useful for inflow control implementation as upstream measurements. At this site, and likely at many other river sites, flow is generally steady at a given location, but flow varies dramatically between locations, particularly laterally across the river. The primary result is that a lateral change in position of a few meters results in changes to flow speed that far exceed the turbulence fluctuations at any given location. The turbulence is dominated by long time scales. Following final system tests, the RivGen® device was submerged and each evaluated controller was tested across a range of gain/set point values and filter configurations for a minimum of 5 minutes, with longer runs attempted for well-performing cases. In addition to testing controllers during the 2015 deployment season, LGL Alaska Research Associates Inc. (LGL) performed a fish monitoring study in compliance with Alaska Department of Fish and Game fisheries habitat permit for the project. During the season, LGL reviewed 10% of 179 one-hour blocks of footage (6 minutes on the hour) and documented a total of over 1200 fish in the vicinity of the RivGen® device, including over 800 salmon smolt and over 350 adult salmon. No evidence of adverse effects including passage delay by upstream migrating salmon was noted. This is an important result for future deployments and has a direct impact on commercial system designs.

16 TIDAL AND WAVE POWER↗

Characterization of Grain‐Size Distribution, Thermal Conductivity, and Gas Diffusivity in Variably Saturated Binary Sand Mixtures

Core Ideas Parameterized models are relevant to applications where different porous media mixtures are used. Models include a grain‐size distribution function to describe bimodal behavior for binary mixtures. Improved model describes observed thermal conductivity–saturation relations for binary mixtures. Combined Buckingham–Penman model used to describe observed gas diffusivity–air content relations. This work is relevant for proper simulation of mixed porous media. Characterization of differently textured porous materials, as well as different volumetric porous media mixtures, in relation to mass and heat transport is vital for many engineering and research applications. Functional relations describing physical (e.g., grain‐size distribution, total porosity), thermal, and gas diffusion properties of porous media and mixtures are necessary to optimize the design of porous systems that involve heat and gas transport processes. However, only a limited number of studies provide characterization of soil physical, thermal, and gas diffusion properties and the functional relationships of these properties under varying soil water contents, especially for soil mixtures, complicating optimization efforts. To better understand how mixing controls the physical, thermal, and gas diffusion properties of porous media, a set of laboratory experiments was performed using five volumetric mixtures of coarse‐ and fine‐grained sand particles. For each mixture, the grain‐size distribution (GSD), thermal conductivity, and gas diffusivity were obtained and parameterized using existing and suggested parametric models. Results show that the extended, two‐region Rosin–Rammler particle‐size distribution model proposed in this study could successfully describe the bimodal behavior of the GSD of binary mixtures. Further, the modified Côté and Konrad thermal conductivity model adequately described the thermal conductivity–water saturation relations observed in different mixtures. The proposed simple soil‐gas diffusivity descriptive model parameterized the upper limit, average, and lower limit behavior in gas diffusivity–air content relations in apparently texture‐invariant gas diffusivity data. Results further show a close analogy between gas diffusivity and thermal conductivity and their variation with saturation across different binary mixtures. Overall, the results of the study provide useful numerical insight into the physical, thermal, and gas transport characteristics of binary mixtures, with wide implications for future engineering and research applications that involve multicomponent porous systems.

Deepagoda, T. K. K. Chamindu↗

Antarctic ice sheet model comparison with uncurated geological constraints shows that higher spatial resolution improves deglacial reconstructions

Accurately reconstructing past changes to the shape and volume of the Antarctic ice sheet relies on the use of physically based and thus internally consistent ice sheet modeling, benchmarked against spatially limited geologic data. The challenge in model benchmarking against geologic data is diagnosing whether model-data misfits are the result of an inadequate model, inherently noisy or biased geologic data, and/or incorrect association between modeled quantities and geologic observations. In this work we address this challenge by (i) the development and use of a new model-data evaluation framework applied to an uncurated data set of geologic constraints, and (ii) nested high-spatial-resolution modeling designed to test the hypothesis that model resolution is an important limitation in matching geologic data. While previous approaches to model benchmarking employed highly curated datasets, our approach applies an automated screening and quality control algorithm to an uncurated public dataset of geochronological observations (specifically, cosmogenic-nuclide exposure-age measurements from glacial deposits in ice-free areas). This optimizes data utilization by including more geological constraints, reduces potential interpretive bias, and allows unsupervised assimilation of new data as they are collected. We also incorporate a nested model framework in which high-resolution domains are downscaled from a continent-wide ice sheet model. We highlight the application of this framework by applying these methods to a small ensemble of deglacial ice-sheet model simulations, and demonstrate that the nested approach improves the ability of model simulations to match exposure age data collected from areas of complex topography and ice flow. We develop a range of diagnostic model-data comparison metrics to provide more insight into model performance than possible from a single-valued misfit statistic, showing that different metrics capture different aspects of ice sheet deflation.

Geosciences↗

Local Estimators for Spacecraft Formation Flying

A formation estimation architecture for formation flying builds upon the local information exchange among multiple local estimators. Spacecraft formation flying involves the coordination of states among multiple spacecraft through relative sensing, inter-spacecraft communication, and control. Most existing formation flying estimation algorithms can only be supported via highly centralized, all-to-all, static relative sensing. New algorithms are needed that are scalable, modular, and robust to variations in the topology and link characteristics of the formation exchange network. These distributed algorithms should rely on a local information-exchange network, relaxing the assumptions on existing algorithms. In this research, it was shown that only local observability is required to design a formation estimator and control law. The approach relies on breaking up the overall information-exchange network into sequence of local subnetworks, and invoking an agreement-type filter to reach consensus among local estimators within each local network. State estimates were obtained by a set of local measurements that were passed through a set of communicating Kalman filters to reach an overall state estimation for the formation. An optimization approach was also presented by means of which diffused estimates over the network can be incorporated in the local estimates obtained by each estimator via local measurements. This approach compares favorably with that obtained by a centralized Kalman filter, which requires complete knowledge of the raw measurement available to each estimator.

Fathpour, Nanaz↗

Advanced HVAC Humidity Control for Hot-Humid Climates

During this project we develop and validate a cost-effective, integrated control solution to improve humidity control and comfort for energy-efficient homes in hot-humid climates. This study focuses on developing a strategy that is effective, field tested, and practical for builders to install with minimal disruption to standard practices. A successful solution would simplify the transition to high-performance humidity control and be the basis for design and installation guidance. By relying on the central system as a starting point, the strategy employed minimizes system complexity and cost for builders, while improving comfort and operating cost for homeowners. The solution strategy was to coordinate the cooling, dehumidification, and ventilation functions of central, ducted HVAC systems to better control indoor humidity, improve occupant thermal comfort, and capture energy savings. The primary strategic goals were to: (1) optimize dehumidification by the central air-conditioning system, particularly during part-load conditions, using conventional equipment with modified control settings and lower system airflows; (2) maximize ventilation during heating/cooling on-cycles, to “bank” and condition outdoor ventilation air, and minimize ventilation during off-cycles; (3) quantify the effectiveness and energy impact of the dehumidification and ventilation strategies, while identifying a metric that would be useful to evaluate latent effectiveness. For the test houses in our study, located in Richmond Hill, Georgia; Houston, Texas; and Monroe, Louisiana we observed: (1) the indoor humidity did not exceed 60% RH during the monitored cooling season for 99% of the time in Richmond Hill, 96% of the time in Houston, and 90% of the time in Monroe; (2) the dehumidification strategy improved the steady-state latent capacity of the HVAC system at design conditions by 16% to 49% at the Houston test house and by 28% to 71% at the Monroe test house, depending on which mode the system was operating in; and (3) the good results at the test houses were primarily due to the amount of time the air-conditioning system operated in ramping or dehumidification modes, or both, particularly during the early cooling season. This study demonstrates that air conditioners or heat pumps with a single-stage compressor can provide good humidity control without the need for a two-stage or variable-stage compressor system. The airflow and control settings for ramping and dehumidification modes are critical to control indoor humidity in hot-humid climates, particularly during part-load and shoulder season conditions. The dehumidification strategy used in this study did not jeopardize the mechanical reliability of the cooling equipment. The strategies used in this study are applicable across various equipment brands, models, and efficiency levels, and also applicable to a broad range of homes in hot-humid climates. Results will vary by specific equipment, location, and house configuration and construction.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

An Integrated Approach to Parameter Learning in Infinite-Dimensional Space

The availability of sophisticated modern physics codes has greatly extended the ability of domain scientists to understand the processes underlying their observations of complicated processes, but it has also introduced the curse of dimensionality via the many user-set parameters available to tune. Many of these parameters are naturally expressed as functional data, such as initial temperature distributions, equations of state, and controls. Thus, when attempting to find parameters that match observed data, being able to navigate parameter-space becomes highly non-trivial, especially considering that accurate simulations can be expensive both in terms of time and money. Existing solutions include batch-parallel simulations, high-dimensional, derivative-free optimization, and expert guessing, all of which make some contribution to solving the problem but do not completely resolve the issue. In this work, we explore the possibility of coupling together all three of the techniques just described by designing user-guided, batch-parallel optimization schemes. Our motivating example is a neutron diffusion partial differential equation where the time-varying multiplication factor serves as the unknown control parameter to be learned. We find that a simple, batch-parallelizable, random-walk scheme is able to make some progress on the problem but does not by itself produce satisfactory results. After reducing the dimensionality of the problem using functional principal component analysis (fPCA), we are able to track the progress of the solver in a visually simple way as well as viewing the associated principle components. This allows a human to make reasonable guesses about which points in the state space the random walker should try next. Thus, by combining the random walker's ability to find descent directions with the human's understanding of the underlying physics, it is possible to use expensive simulations more efficiently and more quickly arrive at the desired parameter set.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Error Analysis System for Spacecraft Navigation Using the Global Positioning System (GPS)

The Flight Dynamics Division (FDD) at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC) is currently developing improved space-navigation filtering algorithms to use the Global Positioning System (GPS) for autonomous real-time onboard orbit determination. In connection with a GPS technology demonstration on the Small Satellite Technology Initiative (SSTI)/Lewis spacecraft, FDD analysts and programmers have teamed with the GSFC Guidance, Navigation, and Control Branch to develop the GPS Enhanced Orbit Determination Experiment (GEODE) system. The GEODE system consists of a Kalman filter operating as a navigation tool for estimating the position, velocity, and additional states required to accurately navigate the orbiting Lewis spacecraft by using astrodynamic modeling and GPS measurements from the receiver. A parallel effort at the FDD is the development of a GPS Error Analysis System (GEAS) that will be used to analyze and improve navigation filtering algorithms during development phases and during in-flight calibration. For GEAS, the Kalman filter theory is extended to estimate the errors in position, velocity, and other error states of interest. The estimation of errors in physical variables at regular intervals will allow the time, cause, and effect of navigation system weaknesses to be identified. In addition, by modeling a sufficient set of navigation system errors, a system failure that causes an observed error anomaly can be traced and accounted for. The GEAS software is formulated using Object Oriented Design (OOD) techniques implemented in the C++ programming language on a Sun SPARC workstation. The Phase 1 of this effort is the development of a basic system to be used to evaluate navigation algorithms implemented in the GEODE system. This paper presents the GEAS mathematical methodology, systems and operations concepts, and software design and implementation. Results from the use of the basic system to evaluate navigation algorithms implemented on GEODE are also discussed. In addition, recommendations for generalization of GEAS functions and for new techniques to optimize the accuracy and control of the GPS autonomous onboard navigation are presented.

Truong, S. H.↗

SU-F-J-16: Planar KV Imaging Dose Reduction Study

Purpose: IGRT has become an indispensable tool in modern radiotherapy with kV imaging used in many departments due to superior image quality and lower dose when compared to MV imaging. Many departments use manufacturer supplied protocols for imaging which are not always optimised between image quality and radiation dose (ALARA). Methods: Whole body phantom PBU-50 (Kyoto Kagaku ltd., Japan) for imaging in radiology has been imaged on Varian iX accelerator (Varian Medical Systems, USA) with OBI 1.5 system. Manufacturer’s default protocols were adapted by modifying kV and mAs values when imaging different anatomical regions of the phantom (head, thorax, abdomen, pelvis, extremities). Images with different settings were independently reviewed by two persons and their suitability for IGRT set-up correction protocols were evaluated. The suitable images with the lowest mAs were then selected. The entrance surface dose (ESD) for manufacturer’s default protocols and modified protocols were measured with RTI Black Piranha (RTI Group, Sweden) and compared. Image quality was also measured with kVQC phantom (Standard Imaging, USA) for different protocols. The modified protocols have been applied for clinical work. Results: For most cases optimized protocols reduced the ESD on average by a factor of 3(range 0.9–8.5). Further reduction in ESD has been observed by applying bow-tie filter designed for CBCT. The largest reduction in dose (12.2 times) was observed for Thorax lateral protocol. The dose was slightly increased (by 10%) for large pelvis AP protocol. Conclusion: Manufacturer’s default IGRT protocols could be optimised to reduce the ESD to the patient without losing the necessary image quality for patient set-up correction. For patient set-up with planar kV imaging the bony anatomy is mostly used and optimization should focus on this aspect. Therefore, the current approach with anthropomorphic phantom is more advantageous in optimization over standard kV quality control phantoms and SNR metrics.

60 APPLIED LIFE SCIENCES↗

Number and placement of control system components considering possible failures

One of the first questions facing the designer of the control system for a large space structure is how many components actuators and sensors - to specify and where to place them on the structure. This paper presents a methodology which is intended to assist the designer in making these choices. A measure of controllability is defined which is a quantitative indication of how well the system can be controlled with a given set of actuators. Similarly, a measure of observability is defined which is a quantitative indication of how well the system can be observed with a given set of sensors. Then the effect of component unreliability is introduced by computing the average expected degree of controllability (observability) over the operating lifetime of the system accounting for the likelihood of various combinations of component failures. The problem of component location is resolved by optimizing this performance measure over the admissible set of locations. The variation of this optimized performance measure with number of actuators (sensors) is helpful in deciding how many components to use.

Vander Velde, W. E.↗