Salt Manyetik Eylemeli Uydunun Yönelme Hareketinin Bütünsel Ve Dayanıklı Kontrolü
Salt Manyetik Eylemeli Uydunun Yönelme Hareketinin Bütünsel Ve Dayanıklı Kontrolü
Dosyalar
Tarih
2016-01-14
Yazarlar
Sofyalı, Ahmet
Süreli Yayın başlığı
Süreli Yayın ISSN
Cilt Başlığı
Yayınevi
Fen Bilimleri Enstitüsü
Institute of Science And Technology
Institute of Science And Technology
Özet
Tez çalışmasının konusu, Dünya’nın yörüngesindeki bir uydunun kendi kütle merkezi etrafındaki hareketinin, yani yönelmesinin uydunun üç gövde ekseni etrafında kontrol edilmesidir. Eyleyici olarak yalnızca manyetik moment üreticileri kullanılmaktadır. Ele alınan problemin birinci zorluğu, gövde eksenleri doğrultusunda birbirine dik olarak yerleştirilerek kullanılan üç manyetik eyleyicinin ürettiği manyetik momentin Dünya’nın manyetik alanıyla etkileşmesi sonucunda elde edilen manyetik kontrol torkunun, daima uydunun bulunduğu konumdaki jeomanyetik alan vektörüne dik olmasıdır. Bu nedenle, salt manyetik kontrol sisteminde jeomanyetik alan vektörü doğrultusunda kontrol torku üretilemez. Eğer uydunun yörünge düzlemi Dünya’nın manyetik ekvator düzlemiyle çakışık değilse, uydu yörüngesinde ilerlerken, jeomanyetik alan vektörünün gövde eksenleri, yani eyleyicilerin manyetik moment ürettikleri eksenler doğrultusundaki bileşenleri neredeyse periyodik bir değişim gösterir. Bu sayede, kontrol torkunun üretilemediği doğrultu gövde eksen takımına göre her anda farklı olur. Bu olgu, şimdiye dek, birçok küp uydunun yanı sıra, geride kalan yirmi yıl içinde biri yaklaşık 1 tonluk ve diğer ikisi birkaç 10 kg’lik üç uydunun, etkin (active) olarak yalnızca manyetik torkla kontrollü olarak görevlerini yerine getirmesini sağlamıştır. Söz konusu üç uydunun, çevresel bozucu etkiler altında edilgin (passive) kararlılık sahibi olacak şekilde tasarlanmış olduğunu not düşmek gerekmektedir; etkin manyetik kontrol sistemleri bu şekilde desteklenmiştir. Bu arada, salt manyetik yönelme kontrolü yönteminin mühendislik uygulamasının mümkün olmasını kuramsal olarak açıklayan önemli bir çalışma on yıl önce literatürde yerini bulmuştur. Manyetik alan vektörü sistemin hareket denklemlerinde yer almaktadır. Bu vektör yörünge üzerinde zamanla değişken (time varying) olduğundan dolayı sistem de zamanla değişkendir. Ele alınan problemin ikinci zorluğu budur, çünkü zamanla değişken (otonom olmayan) (non-autonomous) kontrol sistemleriyle ilgili olarak bir kararlılık sonucuna varılması, zamanla değişken olmayan (otonom) (autonomous) sistemlere göre daha güçtür. Tez çalışmasının birinci amacı, ele alınan bu zor probleme bütünsel (global) açıdan yaklaşmak, yani kontrol edilecek sistemi doğrusal olmayan asıl haliyle ele almak ve ardından sisteme bütünsel düzgün (uniform) asimptotik kararlılık kazandıracak bir kontrolör tasarlamaktır. İkinci amaçsa, elde edilen kararlılığın ideal olmayan koşullar altındaki sistem için dahi geçerli olacağını göstermektir. Literatürde ilk amaç doğrultusunda geliştirilmiş az sayıda çözüm bulunmaktadır. İkinci amaca yönelik çalışmaların sayısıysa daha da azdır. Tez çalışmasının, bu çözümlerin sunulduğu literatür araştırması bölümü literatürdeki ilgili mevcut boşluklara dikkat çekilerek tamamlanmıştır. Hipotez bölümünün sonundaysa, tez çalışmasının hipotezinin çıkarım ve sonuçlarla doğrulanması durumunda doldurulmuş olacağı savunulan boşluklara işaret edilmiştir. Yalnızca manyetik torkla kontrol edilen doğrusal olmayan yönelme hareketi denklemleri, önce sadece çevresel bozucular, ardından da uydunun modelini temsil eden eylemsizlik momentinin belirsizliği hesaba katılarak yazılmıştır. Çevresel bozucular dört farklı tork olarak sisteme girmektedir. Elemanlarının bilinen alt ve üst sınır değerler arasındaki herhangi bir değere sahip olabileceği varsayılarak modellenen belirsiz eylemsizlik matrisinin neden olduğu bozucu etki, sisteme etki eden ayrı bir bozucu torku olarak ifade edilebilmiştir. Kontrol sisteminin, anlık olarak eksik eylenme (instantaneous underactuation) ve zamanla değişkenlik dışında, bozucularının eşleşik (matched) olmaması ve düzenli (regular) biçimde olma özelliklerine de sahip olduğu saptanmıştır. Benzetim ortamının gerçeğe uygun olmasına önem verilmiştir. Dört çevresel bozucu torku ve model belirsizliği kaynaklı bozucu torku gerçeğe yakın matematiksel modeller tarafından hesaplanmaktadır. Bu beş bozucu torkunun bileşenlerinin alttan ve üstten sınırlandırılmış olduğundan hareketle, bu sınırların kontrol sistemi tasarımı sırasında hesaplanabilmesini sağlayan formüller çıkarılmıştır. Uydunun yörüngesi ile kütlesel ve boyutsal özellikleri biliniyorsa, dört çevresel bozucu torkuna ait sınır değerleri kontrol süreci başlamadan önce, daha açık deyişle, kontrol sisteminin tasarımı sırasında elde edilebilmektedir. Model belirsizliği kaynaklı torksa, uydunun gövde eksen takımının referans eksen takımına göre açısal durumuna kuvvetli şekilde bağlıdır. Bu nedenle karşılık gelen sınır da değişkendir ve kontrol süreci içerisinde anlık olarak hesaplanmaktadır. Bu hazırlıkların ardından, tez çalışmasının asıl içeriği dördüncü bölümde sunulmuştur. Önce değişken yapılı kontrol kısaca tanıtılmış, ardından salt manyetik yönelme kontrolü problemine kayma kipli kontrol yönteminin uygulanmasıyla ilgili literatürde yer alan sayılı çalışmanın üzerinden geçilmiştir. Tez çalışmasında kullanılan yöntem geleneksel (klasik) kayma kipli kontrol yöntemi ve uygulanan yaklaşım eşdeğer kontrol yaklaşımıdır. Bu nedenle, manyetik kayma kipli kontrolörün tasarımına eşdeğer kontrol yönteminin çıkarım ve uygulama adımları verildikten sonra geçilmiştir. Momentum aktarımı ya da itki yoluyla sisteme doğrudan etki edecek şekilde üretilebilen tork ile yönelme kontrolü probleminde kullanılagelen kayma manifoldu sergilenmiştir. Bu manifolda eklenen iki integral terimi, yerel jeomanyetik alana bağlı bir manifold elde edilmesini sağlamıştır. Önerilen bu zamanla değişken manifold manyetik alan bilgisini içermesi sayesinde, probleme eşdeğer kontrol yönteminin uygulanmasının önündeki engellerin kalkmasını sağlamıştır. Bu manifoldda bir kayma kipinin var olup olmadığına ilişkin çözümleme de göstermiştir ki, tasarlanan manyetik kayma kipli kontrolör durum değişkenlerinin yörüngelerini manifolda sonlu zamanda (in finite time) eriştirerek yönelme hareketini kayma kipine sokmaktadır. Bu önemli sonuç, önerilen manifoldda, katı bir uydunun salt manyetik eylemeyle kontrol edilen yönelme hareketi için bir kayma kipinin var olduğu anlamına gelmektedir. Sonraki adımda, kayma kipindeki hareketin referansta son bulduğundan emin olunması için bu kipteki indirgenmiş mertebeli hareket çözümlenmiştir. Varılan sonuç, kayma kipine ait durum değişkenlerinin monoton bir şekilde referans değerlerine yakınsadığıdır. Böylelikle, tasarlanmış olan kontrolörün salt manyetik yönelme kontrolü problemine bütünsel düzgün asimptotik kararlı bir çözüm getirmiş olduğu anlaşılmıştır. Uygulanan yöntemin dayanıklı (robust) bir kontrol yöntemi olması sayesinde, problem aynı zamanda kararlılık dayanıklılığı (stability robustness) açısından da çözülmüş olmuştur. Kontrolör tasarımı, manifolda sonlu zamanda erişme koşulu çevresel ve model belirsizliği kaynaklı bozucular etkinken dahi sağlanacak şekilde gerçekleştirilmiştir. Öyleyse, elde edilmiş olan kararlılık bozucu etkilere dayanıklı bir kararlılıktır. Kuramsal sonuçların sayısal olarak doğrulanıp doğrulanmadığı sınanmıştır. Gerçekleştirilen benzetimlerde, iki farklı erişme yasası kullanılarak tasarlanan iki farklı manyetik kayma kipli kontrolör tarafından kontrol edilen yönelme durumunun, birbirinden uzak başlangıç ve referans durumları arasında ve ideallikten uzak koşullar altında kuramsal sonuçların işaret ettiği şekilde değiştiği görülmüştür. Son olarak, önerilen kayma manifoldu kullanılarak probleme integral kayma kipli kontrol yöntemi uygulanmıştır. Bu yolla, katı bir uydunun salt manyetik eylemeyle kontrol edilen yönelme hareketi için aynı zamanda bir integral kayma kipinin de var olduğu gösterilmiştir. Söz konusu kip kontrol sürecinin başlangıcından itibaren geçerli olduğundan dolayı, bu yöntem sayesinde, kontrol edilen harekete geleneksel kayma kipli kontrol yöntemi tarafından sağlanan kararlılık, bozuculara karşı tüm kontrol süresince dayanıklı kılınabilmiştir.
The topic of the thesis work is to control an Earth orbiting satellite’s angular motion around its center of mass, namely its attitude around satellite’s three body axes. The sole actuators used are magnetic torquers. The first difficulty of the considered problem is that the magnetic control torque, which is obtained by the interaction between the magnetic moment produced by the magnetic torquer triad and the Earth’s magnetic field, is always perpendicular to the geomagnetic field vector at the position of the satellite. Therefore, a purely magnetic control system is unable to produce control torque along the geomagnetic field vector. If the satellite’s orbital plane does not coincide with the Earth’s magnetic equator plane, the components of the geomagnetic field vector along the body axes, which are also the directions of the moments produced by the actuators, vary almost periodically while the satellite moves in its orbit. By this means, the direction along which no torque can be produced is different at each moment with respect to the body axes system. This fact has enabled in last twenty years the accomplishment of missions of a 1 ton satellite and two other satellites with mass of a few 10 kg’s besides many CubeSat’s as actively controlled by solely magnetic torque. It is necessary to note that those three satellites were designed as passively stable under environmental effects; their active magnetic control systems were aided by this way. Meanwhile, an important work theoretically explaining the possibility of the engineering application of the purely magnetic attitude control method was added to the literature ten years ago. The magnetic field vector is included in the system’s equations of motion. Because that vector is time varying along the orbit, the system is time varying, too. This is the second difficulty of the considered problem because it is more difficult to derive a stability conclusion for non-autonomous systems than it is for autonomous systems. The first aim of the thesis work is to approach the problem with a global perspective, i.e. to deal directly with the nonlinear system to be controlled, and then to design a controller that provides the system with global uniform asymptotic stability. The second aim is to show that the obtained stability is valid even for the system under non-ideal conditions. In literature, there are limited solutions developed with the first aim. The works with the second aim are even less. The literature survey section of the thesis work, which presents those solutions, was concluded by pointing the related gaps in literature out. At the end of the hypothesis section, the gaps that are to be filled if the hypothesis of the thesis work is verified by derivations and results were indicated. The nonlinear equations of the attitude motion controlled by solely magnetic torque were written by taking first environmental disturbances and then uncertainty in inertia matrix, which represents the model of the satellite, into account. Environmental torques enter the system as four distinct torques. The perturbing effect caused by the uncertain inertia matrix, which is modelled with elements that may have any value between known lower and upper bounds, could be expressed as another disturbance torque acting on the system. It was determined that the control system has also properties such as having unmatched disturbances and being in regular form except being instantaneously underactuated and time varying. The fidelity of the simulation environment was paid importance. The four environmental torques and the disturbance torque due to the model uncertainty were calculated by realistic mathematical models. Regarding the fact that those five disturbance torques’ components are lower- and upper-bounded, formulas that make calculation of those bounds possible during the control system design were derived. If the satellite’s orbit together with its mass and dimensional properties are known, the bounds on the four environmental torques can be obtained before the control process starts, more clearly, during the control system design. However, the torque due to the model uncertainty depends strongly on the angular orientation of the body axis system with respect to the reference axis system. Thus, the corresponding bound is varying and was calculated during the control process instantaneously. After these preparations, the main content of the thesis work was presented in the fourth chapter. First, variable structure control was briefly introduced, then limited works dealing with the application of the sliding mode control method to the purely magnetic attitude control problem were covered. The used method in the thesis work is the conventional (classical) sliding mode control method, and the applied approach is the equivalent control approach. Therefore, the design of the magnetic sliding mode control was started after giving the derivation and application steps of the equivalent control method. The sliding manifold that has been employed in the attitude control problem, in which a torque that is produced by momentum exchange or thrust directly acts on the system, was showed. Two integral terms added to that manifold led to a manifold that is dependent on the local geomagnetic field. That proposed time varying manifold including information on magnetic field enabled overcoming the obstacles for the application of the equivalent control method to the problem. The analysis on existence of a sliding mode along that manifold indicated that the designed magnetic sliding mode controller carries the attitude motion into sliding mode by making the state variables’ trajectories reach the sliding manifold in finite time. This important result means that there is a sliding mode along the proposed manifold in the attitude motion of a rigid satellite controlled by purely magnetic actuation. In the next step, the reduced order motion in the sliding mode was analysed to make sure that the sliding motion ends at the reference. The conclusion derived is that the state variables of the sliding motion converge to their reference values monotonously. Hereby, it appeared that the designed controller provides the purely magnetic attitude control problem with a globally uniformly asymptotically stable solution. Thanks to the fact that the applied method is a robust control method, the problem was also solved in terms of stability robustness. The controller design was carried out so that the condition of finite time reaching to the manifold is satisfied even under disturbances originating from environmental effects and model uncertainty. Then the obtained stability is such kind of stability that it is robust against perturbing effects. It was examined if the theoretical results were verified numerically. In simulations carried out, it was seen that the attitude controlled by two different magnetic sliding mode controllers, which were designed by two different reaching laws, varied between distant initial and reference states and under conditions that are far from ideality as the theoretical results indicated. Lastly, the integral sliding mode control method was applied to the problem by using the proposed sliding manifold. It was shown that there is also an integral sliding mode in the attitude motion of a rigid satellite controlled by purely magnetic actuation. Because the reaching phase was eliminated by that method, the stability of the controlled motion was made robust against perturbations for the whole control process.
The topic of the thesis work is to control an Earth orbiting satellite’s angular motion around its center of mass, namely its attitude around satellite’s three body axes. The sole actuators used are magnetic torquers. The first difficulty of the considered problem is that the magnetic control torque, which is obtained by the interaction between the magnetic moment produced by the magnetic torquer triad and the Earth’s magnetic field, is always perpendicular to the geomagnetic field vector at the position of the satellite. Therefore, a purely magnetic control system is unable to produce control torque along the geomagnetic field vector. If the satellite’s orbital plane does not coincide with the Earth’s magnetic equator plane, the components of the geomagnetic field vector along the body axes, which are also the directions of the moments produced by the actuators, vary almost periodically while the satellite moves in its orbit. By this means, the direction along which no torque can be produced is different at each moment with respect to the body axes system. This fact has enabled in last twenty years the accomplishment of missions of a 1 ton satellite and two other satellites with mass of a few 10 kg’s besides many CubeSat’s as actively controlled by solely magnetic torque. It is necessary to note that those three satellites were designed as passively stable under environmental effects; their active magnetic control systems were aided by this way. Meanwhile, an important work theoretically explaining the possibility of the engineering application of the purely magnetic attitude control method was added to the literature ten years ago. The magnetic field vector is included in the system’s equations of motion. Because that vector is time varying along the orbit, the system is time varying, too. This is the second difficulty of the considered problem because it is more difficult to derive a stability conclusion for non-autonomous systems than it is for autonomous systems. The first aim of the thesis work is to approach the problem with a global perspective, i.e. to deal directly with the nonlinear system to be controlled, and then to design a controller that provides the system with global uniform asymptotic stability. The second aim is to show that the obtained stability is valid even for the system under non-ideal conditions. In literature, there are limited solutions developed with the first aim. The works with the second aim are even less. The literature survey section of the thesis work, which presents those solutions, was concluded by pointing the related gaps in literature out. At the end of the hypothesis section, the gaps that are to be filled if the hypothesis of the thesis work is verified by derivations and results were indicated. The nonlinear equations of the attitude motion controlled by solely magnetic torque were written by taking first environmental disturbances and then uncertainty in inertia matrix, which represents the model of the satellite, into account. Environmental torques enter the system as four distinct torques. The perturbing effect caused by the uncertain inertia matrix, which is modelled with elements that may have any value between known lower and upper bounds, could be expressed as another disturbance torque acting on the system. It was determined that the control system has also properties such as having unmatched disturbances and being in regular form except being instantaneously underactuated and time varying. The fidelity of the simulation environment was paid importance. The four environmental torques and the disturbance torque due to the model uncertainty were calculated by realistic mathematical models. Regarding the fact that those five disturbance torques’ components are lower- and upper-bounded, formulas that make calculation of those bounds possible during the control system design were derived. If the satellite’s orbit together with its mass and dimensional properties are known, the bounds on the four environmental torques can be obtained before the control process starts, more clearly, during the control system design. However, the torque due to the model uncertainty depends strongly on the angular orientation of the body axis system with respect to the reference axis system. Thus, the corresponding bound is varying and was calculated during the control process instantaneously. After these preparations, the main content of the thesis work was presented in the fourth chapter. First, variable structure control was briefly introduced, then limited works dealing with the application of the sliding mode control method to the purely magnetic attitude control problem were covered. The used method in the thesis work is the conventional (classical) sliding mode control method, and the applied approach is the equivalent control approach. Therefore, the design of the magnetic sliding mode control was started after giving the derivation and application steps of the equivalent control method. The sliding manifold that has been employed in the attitude control problem, in which a torque that is produced by momentum exchange or thrust directly acts on the system, was showed. Two integral terms added to that manifold led to a manifold that is dependent on the local geomagnetic field. That proposed time varying manifold including information on magnetic field enabled overcoming the obstacles for the application of the equivalent control method to the problem. The analysis on existence of a sliding mode along that manifold indicated that the designed magnetic sliding mode controller carries the attitude motion into sliding mode by making the state variables’ trajectories reach the sliding manifold in finite time. This important result means that there is a sliding mode along the proposed manifold in the attitude motion of a rigid satellite controlled by purely magnetic actuation. In the next step, the reduced order motion in the sliding mode was analysed to make sure that the sliding motion ends at the reference. The conclusion derived is that the state variables of the sliding motion converge to their reference values monotonously. Hereby, it appeared that the designed controller provides the purely magnetic attitude control problem with a globally uniformly asymptotically stable solution. Thanks to the fact that the applied method is a robust control method, the problem was also solved in terms of stability robustness. The controller design was carried out so that the condition of finite time reaching to the manifold is satisfied even under disturbances originating from environmental effects and model uncertainty. Then the obtained stability is such kind of stability that it is robust against perturbing effects. It was examined if the theoretical results were verified numerically. In simulations carried out, it was seen that the attitude controlled by two different magnetic sliding mode controllers, which were designed by two different reaching laws, varied between distant initial and reference states and under conditions that are far from ideality as the theoretical results indicated. Lastly, the integral sliding mode control method was applied to the problem by using the proposed sliding manifold. It was shown that there is also an integral sliding mode in the attitude motion of a rigid satellite controlled by purely magnetic actuation. Because the reaching phase was eliminated by that method, the stability of the controlled motion was made robust against perturbations for the whole control process.
Açıklama
Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2015
Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2015
Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2015
Anahtar kelimeler
Uydu yönelme kontrolü,
Satellite attitude control,
robust control,
purely magnetic actuation,
nonlinear control,
dayanıklı kontrol,
;doğrusal olmayan kontrol,
salt manyetik eyleme