Denizaltı dikey düzlem modeli için uyarlamalı model öngörülü kontrolör tasarımı
Adaptive model predictive control design for underwater vehicle vertical plane model
Yazar: HARUN TOPBAŞ
Danışman: DOÇ. DR. YAPRAK YALÇIN
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Kontrol ve Otomasyon Mühendisliği Ana Bilim Dalı / Kontrol ve Otomasyon Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2021
Konu: Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrol = Computer Engineering and Computer Science and Control ; Elektrik ve Elektronik Mühendisliği = Electrical and Electronics Engineering
Dizin: Parametre optimizasyonu = Parameter optimization ; Tahmin hataları = Estimation errors ; Uyarlamalı kontrol =   ; Uyumlu kontrol teknikleri = Adaptive control techniques

Özet:
1800' lü yıllardan bugüne denizaltılar keşif, gözlem, askeri operasyonlar olmak üzere birçok misyona hizmet etmektedir. Özellikle 20. Yüzyılın ortalarından itibaren denizaltılar, kıyı ülkeleri için vazgeçilmez araçlar olduğu iyice anlaşılmıştır. Bu doğrultuda denizcilik uygulamaları alanlarında faaliyet gösteren birçok firma, kurum veya araştırmacılar tarafından çalışma alanı olarak tercih edilip önemi günden güne artmaktadır.
Denizaltı araçları, mühendislik bazında nispeten karmaşık olan sualtı çalışma ortamı, doğrusal olmayan dinamikleri, birtakım fiziksel büyüklüklerinin analitik olarak zor hesaplanması gibi zorluklarından ötürü araştırmacılar için geniş bir çalışma alanı vadetmektedir. Bu çalışma alanlarının içerisinde uyarlamalı kontrol yaklaşımları ve parametre kestirim yöntemleri önde gelen konulardan bazılarıdır.
Denizaltıların hidrodinamik sönümleme veya sürtünme katsayıları gibi birtakım fiziksel büyüklerinin hesaplanmasında kestirim uygulamalarına literatürde çokça rastlanmaktadır. Bu tez ise Daldırma ve Değişmezlik (I&I) tabanlı çevrimiçi anlık parametre kestirimi göz önünde bulundurulmuştur. Ayrıca parametre kestirimi esnasında regresör matrisinin tekil olması problemine bir çözüm önerisi getirilmiştir.
Bu çalışmada bilinmeyen denizaltı dikey düzlem modeli için derinlik pozisyonu ve yunuslama açısı referans takibi problemi için model öngörülü kontrolör tasarlanmıştır. İlk olarak model öngörülü kontrolün avantajları, çalışma prensibi ve tasarım aşamalarından bahsedilmiştir. Ardından öngörü denklemleri elde edilip setpoint ile öngörülen çıkışlar arasındaki hata baz alınarak bir maliyet fonksiyonu oluşturulmuştur. Denizaltı aracı için güvenlik hususu gereği yunuslama açısı için durum kısıtı, kontrol yüzeyi olan irtifa dümeni için yapısal olarak bir limiti olduğu varsayılıp maliyet fonksiyonuna entegre edilmiştir. Maliyet fonksiyonunun çözümünün daha hızlı yapılabilmesi için alternatif olarak "Fast Gradient Method (Nesterov's Method)" yöntemi önerilmiştir. 
Bu tez çalışmasında son olarak Daldırma ve Değişmezlik tabanlı çevrimiçi parametre kestirim yöntemi ile uyarlamalı model öngörülü kontrol uygulamasının bir benzetim çalışması yapılmıştır. Bu benzetim çalışması kestirilen parametrelerin öngörü denklemlerinde kullanılarak model öngörülü kontrol işareti hesaplanması fikrine dayandırılmıştır. Öncelikle parametre kestiricisinin tasarım parametrelerinin değişminin yakınsama karakteristiğine etkileri incelenmiştir. Ardından kısıtların dahil olmadığı durum için model öngörülü kontrolün tasarım parametresi olan öngörü ufkunun değişiminin referans setpoint takibi karakteristiğine etkileri grafiklerle yorumlanıp güvenlik esasları gereği yüksek değerde tutulması önerilmiştir. Ek olarak alternatif bir optimizasyon çözüm yöntemi ile İç Nokta Optimizasyonu (MATLAB / quadprog()) arasında kıyas yapılmıştır. Kıyas sonucunda çözüme ulaşma süresi bakımından alternatif önerilen yöntemin daha cazip olduğu grafiklerle gösterilmiştir.

Abstract:
Over the past century, underwater vehicles have been used in many missions, such as exploration of the deep seas and oceans, as well as as a military deterrent. Underwater vehicles, which are understood to be indispensable for coastal countries, have become the focus of many researchers and engineers over the years.
Some of the difficulties of underwater vehicle modeling and control studies are the excess of nonlinear dynamic terms, the difficulty of calculating model parameters, and the complex and variable underwater working environment. Because of these difficulties, there are many adaptive control applications and estimation methods of unknown model parameters in the literature of modeling and control of underwater vehicles. Generally, these methods are based on the idea of using underwater vehicle navigation data with analytical methods based on hydrodynamic and geometrical properties of underwater vehicles. 
In this thesis, a solution to the problem of unknown underwater model parameters has been provided with the online nonlinear parameter estimation approach based on the Immersion and Invariance (I&I) method. The I&I estimation method, which is considered a new method in the adaptive control literature, was first introduced by Astolfi and Ortega. Later, the I&I method has been used and recommended by many researchers working in adaptive control and parameter estimation.
Thanks to this method, parameter estimation can be performed asymptotically for the unknown underwater vehicle model by using stable system dynamics, as well as the ability to work online with the controller by obtaining sufficient estimation speed. In addition, thanks to the online estimator, resistance to possible slow changes in model coefficients has been gained in the complex underwater environment.
In the second chapter of this thesis, the vertical axis model was obtained by using the non-linear model of the 6-degree of freedom underwater vehicle in Fossen's study. Then, linearization process was performed around constant velocity in the forward direction, ignoring the effect of small angular movements.
In the next chapter, the I&I method was formulated for a nonlinear discrete-time linear parameterized system model, using the work of Yalçın and Astolfi. In this formulation, the unknown model parameters are assigned to a vector, and the regressor matrix of appropriate dimensions is obtained from the states related to the unknown parameters. All other terms of the system model are written in a matrix of appropriate dimensions, and the control rule and its related terms are kept in a vector. It is assumed that all dynamics remain within a limited value range under the control rule and the states converge to the zero equilibrium point asymptotically. Under this assumption, estimation error dynamic is obtained first. Then, the parameter updating rule and the design function for the I&I method were determined to stabilize the estimation error dynamic. In order to verify this claim, a lyapunov function has been determined. Then, by showing that the lyapunov function is a decreasing function, it is proved that the designed parameter update rule stabilizes the estimation error dynamic. 
Subsequently, the formulation mentioned above has been applied parametrically for the underwater vehicle discrete time state space model. However, it was seen that the rank of the regressor matrix was lower than the number of unknown parameters. In this case, no solution could be found for the parameter estimation calculation in each sampling time. In order to overcome this problem, the vertical axis linear model of the considered underwater vehicle has been extended by combining it with the state equations corresponding to one sampling period previous values. Thus, the online estimation process is made possible with the newly expanded regressor matrix whose rank is increasing. 
It is also assumed that the system is performed so that the system states related to unknown parameters change at each sampling time. Otherwise, the system states in the rows of the regressor matrix in the extended system model are the same as the previous values. In this case, there is no unique solution for each sampling time analytically, and the rank of the regressor matrix for the model will decrease. For this reason, in this thesis, the reference tracking problem, which constantly changes states associated with unknown parameters, is investigated.
In the fourth chapter, a model predictive controller is designed for the reference tracking problem of the unknown model of the underwater vehicle. First of all, the advantages of model predictive control over other methods, working principle and design stages were presented. Then, the model predictive control structure was designed for the depth and pitch reference tracking problem of the adapted underwater vehicle model. 
Prediction equations were obtained first in the model predictive control design stages. Afterwards, a cost function was created in order to minimize the difference between the reference signal and the predicted outputs. Some control signals and state constraints have been determined for the cost function created. Finally, MATLAB / quadprog() function and "Nesterov's Fast Gradient Method" were used to solve the optimization problem of the cost function under the determined constraints.
The last chapter includes the simulation study of the adaptive model predictive control application for the underwater vehicle vertical plane model. The adaptive model predictive control application was created with the idea of using the estimated parameters, obtained by design method given in the third chapter, in the prediction equations. 
Firstly, the performance of the parameter estimator designed for the unknown underwater vehicle vertical plane model and the effects of the change of the design parameters on the convergence characteristic were investigated. After that, the reference setpoint tracking performance is presented and interpreted with graphics for situations such as the change of design parameters in the closed loop of the adaptive model predictive control design and the addition of constraints. Finally, for the optimization problem of the cost function, MATLAB / quadprog() function and "Fast Gradient Method (Nesterov's Method)" are compared.
Under the assumption that constraints are not included, the unit step reference response is examined according to the change in the prediction horizon, which is the design parameter of the model predictive control structure. The closed-loop response becomes more aggressive as the predictive horizon decreases. It is seen that the pitch angle exceeds 90 degrees. In underwater vehicle missions, the pitch angle should always be kept as low as possible for safety reasons. Therefore, it is important to keep the prediction horizon high for a relatively conservative closed-loop response.
In addition, in this study, the safety of the underwater vehicle was taken into consideration and the inequality constraint was determined as 5 degrees for the pitch angle. It is also assumed that the control surface is structurally limited to 20 degrees.
As a result, the unknown model parameters were successfully estimated online, and the model predictive control rule, calculated with the estimated parameters, enabled the system model to successfully tracking the depth position and pitch angle reference.