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Changzhu Wei - One of the best experts on this subject based on the ideXlab platform.
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optimal h robust output feedback control for satellite formation in arbitrary elliptical Reference Orbits
Advances in Space Research, 2014Co-Authors: Changzhu Wei, Sangyoung Park, Chandeok ParkAbstract:Abstract A two degree-of-freedom signal-based optimal H ∞ robust output feedback controller is designed for satellite formation in an arbitrary elliptical Reference Orbit. Based on high-fidelity linearized dynamics of relative motion, uncertainties introduced by non-zero eccentricity and gravitational J2 perturbation are separated to construct a robust control model. Furthermore, a distributed robust control model is derived by modifying the perturbed robust control model of each satellite with the eigenvalues of the Laplacian matrix of the communication graph, which represent uncertainty in the communication topology. A signal-based optimal H ∞ robust controller is then designed primarily. Considering that the uncertainties involved in the distributed robust control model have a completely diagonal structure, the corresponding analyses are made through structured singular value theory to reduce the conservativeness. Based on simulation results, further designs including increasing the degrees of freedom of the controller, modifying the performance and control weighted functions, adding a post high-pass filter according to the dynamic characteristics, and reducing the control model are made to improve the control performance. Nonlinear simulations demonstrate that the resultant optimal H ∞ robust output feedback controller satisfies the robust performance requirements under uncertainties caused by non-zero eccentricity, J2 perturbation, and varying communication topology, and that 5 m accuracy in terms of stable desired formation configuration can be achieved by the presented optimal H ∞ robust controller. In addition to considering the widely discussed uncertainties caused by the Orbit of each satellite in a formation, the optimal H ∞ robust output feedback control model presented in the current work considers the uncertainties caused by varying communication topology in the satellite formation that works in a cooperative way. Other new improvements include adopting a new method to more accurately describe and analyze the effects of the higher-order J2 perturbation, combining all the uncertainties into a diagonal structure, and utilizing a structured singular value to synthesize and analyze the controller.
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linearized dynamics model for relative motion under a j2 perturbed elliptical Reference Orbit
International Journal of Non-linear Mechanics, 2013Co-Authors: Changzhu Wei, Sangyoung Park, Chandeok ParkAbstract:Abstract A method for directly establishing a linearized dynamics model of relative motion for a satellite formation flying on an arbitrary elliptical Reference Orbit is presented. The proposed linearized dynamics model of relative motion is intuitive and favorable to be utilized for designing formation control system, implementing guidance algorithm, and analyzing optimization problems etc. An analytical solution for the radius of a Reference Orbit affected by the J 2 geopotential disturbance is used to approximate the actual Reference Orbit rather than directly use the unperturbed standard Orbit. The accuracy of the analytical solution directly affects the accuracy of the linear relative motion model of a satellite formation. Thus, emphasis is placed on deducing an accurate analytical solution for a perturbed Reference Orbit radius, which is analyzed by adding perturbed motion in the radial direction to the corresponding unperturbed Reference Orbit radius. Furthermore, by considering time - varying angular velocity of an elliptical Reference Orbit, the analytical solution of a perturbed Orbit radius is obtained in the true anomaly domain. In order to better match the actual force conditions of a satellite formation, a perturbed true anomaly and perturbed argument of perigee are adopted and substituted into the gradient of the J 2 disturbance force. Simulation results indicate the analytical solution for the radius of a perturbed elliptical Reference Orbit is accurate, and the proposed linear dynamics model of relative motion can track the high - fidelity simulated relative motion more accurately than previously proposed dynamics models, even for an elliptical Reference Orbit with high eccentricity.
H E Moolenaar - One of the best experts on this subject based on the ideXlab platform.
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testing a method for analyzing the effect of parameter change in climate driven ecological systems
Ecological Modelling, 2007Co-Authors: H E Moolenaar, Johan Grasman, Frank Selten, Maarten De GeeAbstract:In this study, we consider a herbivore-predator metapopulation model, consisting of two patches. Only the herbivores are allowed to migrate between the patches. Furthermore, the intrinsic growth rates of the herbivores depend on climatic fluctuations. Our objective is the conservation of the herbivores. We are interested in finding the parameter perturbations that, when applied, will reduce the extinction risk of one of the subpopulations the most. The risk of extinction is measured in terms of the fifth percentiles of the subpopulations. This is the value below which the subpopulation is found 5 out of 100 times in a series of values taken at fixed time intervals. We make use of the short-term behavior of the model by formulating the tangent linear equations. In the neighborhood of a short interval of a Reference Orbit the linear error growth can then be calculated. The parameter perturbation that causes the largest error growth, the so-called first singular vector, can be computed with the use of adjoint equations. The adjoint system acts as a backward integration. It turns out that at certain moments in time such a singular vector has a direction for which a parameter perturbation is likely effective in changing the dynamics in a long model simulation. The selection is based on a specific local behavior of the error growth. This adjoint method is compared with a method in which parameter perturbations are randomly chosen. Here we carry out a test for a model with only few parameters. However, it can be applied to models with a very large number of parameters, making the adjoint method an interesting alternative for the random method as then the required number of runs cannot be realized within a realistic computing time. A same objection holds for the use of a systematic method of finding an optimum, e.g. by computing the gradient in the fifth percentile with respect to the parameter values.
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parameter sensitivity of climate models and climate driven ecological systems
2006Co-Authors: H E MoolenaarAbstract:Uncertainty in the outcome of numerical models of physical and biological processes, such as the climate and ecological systems, is widely recognized. One contributing factor is uncertainty in model parameters. Because of this uncertainty, a range of model outcomes is usually given. This might obstruct policy making for topics such as the reduction of climate change and nature conservation management. Part of the estimation of uncertainty is a parameter sensitivity analysis. It is important to verify how small changes in parameters can affect the model outcome. Especially extreme deviations are of interest to gain an understanding of the variability of the result. We therefore need to identify the parameter perturbations the model is most sensitive to. Since the atmospheric circulation behaves as a chaotic system quantities that characterise the climate have to be computed froman integration over a large time interval. Perturbing a large set of parameters to analyse the variability of the outcome would require an enormous computing time. It would therefore be advantageous to select effective parameters a priori. In this thesis a method is described that selects these parameters in an efficient way. The short term behaviour of a nonlinear model is used to select parameter perturbations that are likely to cause a large change in the dynamics of the long term behaviour of the model. A short section of a Reference Orbit is calculated. Next the error growth from the parameter perturbation can be computed with the use of tangent linear equations. The adjoint of the model acts as a backward integration and can then be used to calculate the parameter perturbation that causes the largest error growth over this interval. This perturbation vector is more likely to be also an effective parameter perturbation for a long time integration simulating the climate than a randomly chosen one. More precisely, it turns out that not exactly at a point of the chaotic attractor with a large error growth but just a moment later when this growth has fallen back has to be selected. These points are found by analysing a succession of many short time intervals over each of which the tangent linear approximation holds. We apply this adjoint method to two climate models; the Lorenz 63 model and the atmospheric T21QG model, and a climate driven metapopulation model; the Rosenzweig-McArthur model coupled to the Lorenz 84 model. The success rate of drawing a parameter perturbation causing a large change should for the adjoint method be considerably higher than for a random search method. Climate change is defined in terms of changes in the occurrence and strength of different preferred atmospheric circulation patterns. In the context of metapopulation models and conservation management, the goal is to find perturbations in the biological parameters that lower the risk of extinction of herbivore subpopulations. It is found that in the simple models, where only 5 parameters are varied (Lorenz 63 and the Rosenzweig-McArthur model forced by Lorenz 84), the adjoint method has a significantly higher success rate in drawing effective parameter perturbations than a random search method does. In the more complex T21QG model drawing an effective parameter perturbation appears to be a much more strenuous task due to the large number of 1449 parameters that are varied. However, although hampered by this large parameter set and the required long time integration of this system with many degrees of freedom, the adjoint method comes much closer to selecting the parameter perturbation causing the largest climate change than the random method
Chandeok Park - One of the best experts on this subject based on the ideXlab platform.
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optimal h robust output feedback control for satellite formation in arbitrary elliptical Reference Orbits
Advances in Space Research, 2014Co-Authors: Changzhu Wei, Sangyoung Park, Chandeok ParkAbstract:Abstract A two degree-of-freedom signal-based optimal H ∞ robust output feedback controller is designed for satellite formation in an arbitrary elliptical Reference Orbit. Based on high-fidelity linearized dynamics of relative motion, uncertainties introduced by non-zero eccentricity and gravitational J2 perturbation are separated to construct a robust control model. Furthermore, a distributed robust control model is derived by modifying the perturbed robust control model of each satellite with the eigenvalues of the Laplacian matrix of the communication graph, which represent uncertainty in the communication topology. A signal-based optimal H ∞ robust controller is then designed primarily. Considering that the uncertainties involved in the distributed robust control model have a completely diagonal structure, the corresponding analyses are made through structured singular value theory to reduce the conservativeness. Based on simulation results, further designs including increasing the degrees of freedom of the controller, modifying the performance and control weighted functions, adding a post high-pass filter according to the dynamic characteristics, and reducing the control model are made to improve the control performance. Nonlinear simulations demonstrate that the resultant optimal H ∞ robust output feedback controller satisfies the robust performance requirements under uncertainties caused by non-zero eccentricity, J2 perturbation, and varying communication topology, and that 5 m accuracy in terms of stable desired formation configuration can be achieved by the presented optimal H ∞ robust controller. In addition to considering the widely discussed uncertainties caused by the Orbit of each satellite in a formation, the optimal H ∞ robust output feedback control model presented in the current work considers the uncertainties caused by varying communication topology in the satellite formation that works in a cooperative way. Other new improvements include adopting a new method to more accurately describe and analyze the effects of the higher-order J2 perturbation, combining all the uncertainties into a diagonal structure, and utilizing a structured singular value to synthesize and analyze the controller.
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linearized dynamics model for relative motion under a j2 perturbed elliptical Reference Orbit
International Journal of Non-linear Mechanics, 2013Co-Authors: Changzhu Wei, Sangyoung Park, Chandeok ParkAbstract:Abstract A method for directly establishing a linearized dynamics model of relative motion for a satellite formation flying on an arbitrary elliptical Reference Orbit is presented. The proposed linearized dynamics model of relative motion is intuitive and favorable to be utilized for designing formation control system, implementing guidance algorithm, and analyzing optimization problems etc. An analytical solution for the radius of a Reference Orbit affected by the J 2 geopotential disturbance is used to approximate the actual Reference Orbit rather than directly use the unperturbed standard Orbit. The accuracy of the analytical solution directly affects the accuracy of the linear relative motion model of a satellite formation. Thus, emphasis is placed on deducing an accurate analytical solution for a perturbed Reference Orbit radius, which is analyzed by adding perturbed motion in the radial direction to the corresponding unperturbed Reference Orbit radius. Furthermore, by considering time - varying angular velocity of an elliptical Reference Orbit, the analytical solution of a perturbed Orbit radius is obtained in the true anomaly domain. In order to better match the actual force conditions of a satellite formation, a perturbed true anomaly and perturbed argument of perigee are adopted and substituted into the gradient of the J 2 disturbance force. Simulation results indicate the analytical solution for the radius of a perturbed elliptical Reference Orbit is accurate, and the proposed linear dynamics model of relative motion can track the high - fidelity simulated relative motion more accurately than previously proposed dynamics models, even for an elliptical Reference Orbit with high eccentricity.
Peiliang Xu - One of the best experts on this subject based on the ideXlab platform.
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measurement based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry
arXiv: Geophysics, 2019Co-Authors: Peiliang XuAbstract:The numerical integration method has been routinely used to produce global standard gravitational models from satellite tracking measurements of CHAMP/GRACE types. It is implemented by solving the differential equations of the partial derivatives of a satellite Orbit with respect to the unknown harmonic coefficients under the conditions of zero initial values. From the mathematical point of view, satellite gravimetry from satellite tracking is the problem of estimating unknown parameters in the Newton's nonlinear differential equations from satellite tracking measurements. We prove that zero initial values for the partial derivatives are incorrect mathematically and not permitted physically. The numerical integration method, as currently implemented and used in satellite gravimetry and statistics, is groundless. We use three different methods to derive new local solutions to the Newton's nonlinear governing differential equations of motion with a nominal Reference Orbit. Bearing in mind that satellite Orbits can now be tracked almost continuously at unprecedented high accuracy, we propose the measurement-based perturbation theory and derive global uniformly convergent solutions to the Newton's nonlinear governing differential equations of motion. Since the solutions are global uniformly convergent, they are able to extract smallest possible gravitational signals from modern and future satellite tracking measurements for global high-precision, high-resolution gravitational models. By directly turning the nonlinear differential equations of satellite motion into the nonlinear integral equations, we reformulate the links between satellite tracking measurements and the global uniformly convergent solutions to the Newton's governing differential equations as a condition adjustment model with equality constraints.
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measurement based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry
Communications in Nonlinear Science and Numerical Simulation, 2018Co-Authors: Peiliang XuAbstract:Abstract The numerical integration method has been routinely used by major institutions worldwide, for example, NASA Goddard Space Flight Center and German Research Center for Geosciences (GFZ), to produce global gravitational models from satellite tracking measurements of CHAMP and/or GRACE types. Such Earth’s gravitational products have found widest possible multidisciplinary applications in Earth Sciences. The method is essentially implemented by solving the differential equations of the partial derivatives of the Orbit of a satellite with respect to the unknown harmonic coefficients under the conditions of zero initial values. From the mathematical and statistical point of view, satellite gravimetry from satellite tracking is essentially the problem of estimating unknown parameters in the Newton’s nonlinear differential equations from satellite tracking measurements. We prove that zero initial values for the partial derivatives are incorrect mathematically and not permitted physically. The numerical integration method, as currently implemented and used in mathematics and statistics, chemistry and physics, and satellite gravimetry, is groundless, mathematically and physically. Given the Newton’s nonlinear governing differential equations of satellite motion with unknown equation parameters and unknown initial conditions, we develop three methods to derive new local solutions around a nominal Reference Orbit, which are linked to measurements to estimate the unknown corrections to approximate values of the unknown parameters and the unknown initial conditions. Bearing in mind that satellite Orbits can now be tracked almost continuously at unprecedented accuracy, we propose the measurement-based perturbation theory and derive global uniformly convergent solutions to the Newton’s nonlinear governing differential equations of satellite motion for the next generation of global gravitational models. Since the solutions are global uniformly convergent, theoretically speaking, they are able to extract smallest possible gravitational signals from modern and future satellite tracking measurements, leading to the production of global high-precision, high-resolution gravitational models. By directly turning the nonlinear differential equations of satellite motion into the nonlinear integral equations, and recognizing the fact that satellite Orbits are measured with random errors, we further reformulate the links between satellite tracking measurements and the global uniformly convergent solutions to the Newton’s governing differential equations as a condition adjustment model with unknown parameters, or equivalently, the weighted least squares estimation of unknown differential equation parameters with equality constraints, for the reconstruction of global high-precision, high-resolution gravitational models from modern (and future) satellite tracking measurements.
Kyle T Alfriend - One of the best experts on this subject based on the ideXlab platform.
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optimal formation design for magnetospheric multiscale mission using differential Orbital elements
Journal of Guidance Control and Dynamics, 2011Co-Authors: Christopher W T Roscoe, Srinivas R Vadali, Kyle T Alfriend, Uri DesaiAbstract:The Magnetospheric Multiscale Mission requires a formation of four satellites in a nearly regular tetrahedron throughout a region of interest defined near the apogee of a highly eccentric Reference Orbit. Previous papers have addressed the design of formations in Orbits of high eccentricity to maximize a quality factor in a region of interest, including the use of differential mean Orbital elements as design variables. In this paper, a robust optimization method is presented to improve formation performance in the presence of formation initialization errors. Several design methods are analyzed by applying differential semimajor axis errors, which have a strong effect on the long-term stability of spacecraft formations. It is shown that large formations can satisfy mission requirements for a longer time than smaller formations, when the same magnitude of errors are considered, and generally exhibit less variation in quality factors due to these errors. The robust optimization method is applied to these smaller formations and produces results that are much more stable when semimajor axis errors are included, at a cost of some performance in the nominal error-free case. The results are verified using the NASA General Mission Analysis Tool and are shown to be reasonably accurate, except in predicting very long-term behavior. A physical analysis of the geometry of several magnetospheric multiscale formation designs is provided, and eight distinct optimal tetrahedron orientations are identified (two configurations, in which the chief satellite can be placed at any of the four vertices).
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state transition matrix of relative motion for the perturbed noncircular Reference Orbit
Journal of Guidance Control and Dynamics, 2003Co-Authors: Dongwoo Gim, Kyle T AlfriendAbstract:A precise analytic solution that includes the effects of the Reference Orbit eccentricity and differential perturbations is needed for the relative motion of formation-flying satellites. As a result of the spherical Earthand circular Reference Orbit assumptions, Hill's equations, which have often been used for describing relative motion, are insufficient for the long-term prediction of the relative motion. A new approach, called the geometric method, is developed to obtain the state transition matrix for the relative motion that includes the effects caused by the Reference Orbit eccentricity and the differential gravitational perturbations. The geometric method uses the relationship between the relative states and differential Orbital elements to obtain the state transition matrix instead of directly solving the complex relative motion differential equations. The state transition matrices are derived for both mean and osculating elements with the primary gravitational perturbation that results from the equatorial bulge term J 2 . Although the results are based on the J 2 effects, the approach can be extended easily to include other perturbing forces.
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precise ephemeris reconstruction using the clohessy wiltshire frame and multiple sequential compressions
Journal of Guidance Control and Dynamics, 2003Co-Authors: Deok Jin Lee, Seokwoo Choi, Sangryul Lee, Hakjung Kim, Kyle T AlfriendAbstract:A precise onboard ephemeris propagation method that uses a multiple sequential compression is developed. A least-square regression is used to dee ne the Reference Orbit in the form of polynomial functions. Then the position and velocity residuals in terms of rectangular coordinates are generated by subtracting the Reference Orbit from a numerically generated preciseOrbit and expressed in the Clohessy ‐Wiltshire frame. The residuals are compressed using various kinds of basis functions to ree ect secular and periodic variations. Precise position and velocity reconstruction is achieved by adding the compressed residuals into the Reference Orbit. Numerical simulation experiments and performance comparisons with previous results support the validity of the proposed method.
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formation flying accommodating nonlinearity and eccentricity perturbations
Journal of Guidance Control and Dynamics, 2003Co-Authors: S S Vaddi, Srinivas R Vadali, Kyle T AlfriendAbstract:Hill-Clohessy-Wiltshire equations describe the relative motion of one satellite with respect to another in a circular Reference Orbit. Initial conditions that generate periodic solutions to these equations have to be corrected to obtain bounded solutions in the presence of nonlinearity of the differential gravitational acceleration model and eccentricity of the Reference Orbit. The corrections to the initial conditions due to quadratic terms in the differential gravitational acceleration for circular Reference Orbits are established e rst by using a perturbation approach. These corrections are related to the period-matching constraint required for bounded relative motion. Next, the solution to the linear problem including the effect of eccentricity is determined, and a procedure for correcting the along-track bias is presented. The two solutions obtained are combined to produce an asymptotic solution for the quadratic eccentricity problem. The effects of nonlinearity and eccentricity on the relative Orbits are characterized as functions of their initial position in the formation. HE problem of relative motion dynamics of satellites has been of interest since the 1960s. Much of the work has been per- formed in the context of the rendezvous problem. Accurate mod- eling of the relative motion dynamics for initial conditions close to the target is important for the rendezvous problem. Formation e ying requires bounded relative motion. Therefore, the solutions of interest are restricted to a certain set of initial conditions that lead to bounded relative motion. One particular formation of interest is the relative Orbit that is circular when projected on the local hori- zontal plane. This solution is an exact solution to Hill- Clohessy- Wiltshire(HCW)equationsthatmodeltherelativemotiondynamics under the assumption of a circular Reference Orbit, spherical Earth, and linearized differential gravitational acceleration. Nonlinearity of the differential gravitational acceleration, eccentricity of the ref- erence Orbit,and the Earth' s oblateness are the three most important perturbations that breakdown the circular Orbit solutions to HCW equations. In this paper, we study the effects of nonlinearity and the eccentricity perturbations; the effects of J2 are ignored. The developments in this paper draw on several previous studies. Melton 1 developed a state transition matrix solution for the lin- earized relative motion dynamics by incorporating the effect of ec- centricity.Inalhan etal. 2 obtainedthe conditionforboundedrelative Orbit solutions to the linearized problem with nonzero eccentricity. The effects of including quadratic gravitational acceleration terms were studied in Refs. 3- 6. Karlgaard and Lutze 7 used the method of multiple timescalesto obtain aperturbationsolutiontothe quadratic problem formulated using spherical coordinates. Alfriend et al. 8 used a geometric approach to map relative motion coordinates to Orbital element differences. Mitchell and Richardson 9 developed an active nonlinear controller to accommodate quadratic nonlinearities in the zero-eccentricity problem. Broucke 10 has presented an exact state transition matrix solution for the linearized elliptic rendezvous problem. The solution is obtained by taking partial derivatives with
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spacecraft formation flying control using mean Orbit elements
2000Co-Authors: Hanspeter Schaub, Srinivas R Vadali, John L Junkins, Kyle T AlfriendAbstract:Two nonlinear feedback control laws are presented to reestablish a desired J2 invariant relative Orbit. Since it is convenient to describe the relative Orbit of a deputy satellite with respect to a chief satellite in terms of mean Orbit element differences, and because the conditions for a relative Orbit being J2 invariant are expressed in terms of mean Orbit elements, the first control law feeds back errors in terms of mean Orbit elements. Dealing with mean Orbit elements has the advantage that short period oscillations are not perceived as tracking errors; rather, only the long term tracking errors are compensated for. The second control law feeds back traditional cartesian position and velocity tracking errors. For both of the control laws, the desired Orbit is computed using mean Orbit elements. A numerical study compares and contrasts the two feedback laws. Introduction In recent years the challenging concept of spacecraft formation flying has been studied by various authors.1–6 These spacecraft may be in a simple along-track string formation or a more dynamic formation where several deputy spacecraft Orbit relative to a chief Reference spacecraft. With these formations, the purpose is to increase the baseline of scientific instruments placed on the spacecraft. These instruments could form a radio-telescope or surface-mapping radar array. One method to find natural relative Orbits about a Reference spacecraft is to use the Clohessy-Wiltshire (CW) equations.7 Here a circular Reference Orbit and spherical Earth model is assumed and the equations of motion of the Orbiting spacecraft are linearized relative to the rotating frame of the Reference spacecraft. These equations of motion are ∗Post-Doctoral Research Associate, Aerospace Engineering Department, Texas A&M University, College Station TX 77843. †Professor of Aerospace Engineering, Aerospace Engineering Department, Texas A&M University, College Station TX 77843. ‡George J. Eppright Distinguished Chair Professor of Aerospace Engineering, Aerospace Engineering Department, Texas A&M University, College Station TX 77843, Fellow AAS. §Department Head and Professor, Aerospace Engineering Department, Texas A&M University, College Station TX 77843.