Immersion and invariance yaklaşımıyla doğrusal olmayan mekanik sistemlerde ayrık-zamanlı bozucu bastırma ve uygulamaları
Discrete-time disturbance attenuation for nonlinear mechanical systems with immersion and invariance approach and its applications
Yazar: FATİH ADIGÜZEL
Danışman: DOÇ. DR. YAPRAK YALÇIN
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Mekatronik Mühendisliği Ana Bilim Dalı / Mekatronik Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2018
Konu: Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrol = Computer Engineering and Computer Science and Control ; Elektrik ve Elektronik Mühendisliği = Electrical and Electronics Engineering
Dizin: Doğrusal olmayan kontrol = Nonlinear control

Özet:
Bu yüksek lisans çalışmasında, I&I yaklaşımı kullanılarak doğrusal olmayan mekanik sistemler için yeni bir ayrık zamanlı bozucu kestirim tasarımı önerilmiştir, tam eyleyici ve eksik eyleyici doğrusal olmayan sistemler için kestirilen bozucu değerleri sabit bozucuların bastırılması için ayrık zamanlı geri adımlı kontrol tasarımında kullanılmıştır. 

Bölüm 1'de literatür incelemesinde ele alınan konular şu şekilde sıralanabilir; doğrusal olmayan mekanik sistemler, tam eyleyicili ve eksik eyleyicili mekanik sistemler, geri beslemeli doğrusallaştırma ve kısmi geri besleme ile doğrusallaştırma, ayrık zamanlı geri adımlı kontrol, I&I uyarlamalı kontrol yöntemi, 2-link düzlemsel robot kol modelleme ve kontrol yöntemleri, araç sarkaç sistemi modelleme ve kontrol yöntemleri.
 
Bölüm 2'de, bu çalışmada kullanılan konuların matematiksel alt yapı bilgileri özetlenmiştir. İlk olarak Euler-Lagrange yaklaşımı ile tam eyleyicili ve eksik eyleyicili doğrusal olmayan mekanik sistemler için dinamik denklemlerin türetilmesi sunulmuştur. Daha sonra, geri besleme ile doğrusallaştırma ve kısmi geri besleme ile doğrusallaştırma yöntemleri anlatılmıştır. Ardından,  tam / eksik eyleyicili mekanik sistemlerin bu çalışmada ele alınan biçimiyle bozucu içeren denklemleri  ve bu sürekli-zamanlı denklemlerin ileri-Euler yaklaşımı kullanılarak ayrıklaştırılması verilmiştir. Ayrıca, bilinmeyen parametreye sahip doğrusal olmayan sistemler için ayrık-zamanlı I&I uyarlamalı kontrol teorisi verilmiştir. Son olarak, kesin geri beslemeli formda genel sistemler için ayrık zamanlı geri adımlamalı kontrol prosedürü sunulmuştur.

Bölüm 3'te, geri besleme ile doğrusallaştırma ve kısmi geri besleme ile doğrusallaştırma, tam / eksik eyleyicili ayrık zamanlı mekanik sistemler için verilmiştir.

Bölüm 4'te, sistem girişinde bozucu içeren genel doğrusal olmayan mekanik sistemler için ayrık zamanlı dinamik denklemler verilmiştir. Daha sonra, hem tam eyleyicili hem de eksik eyleyicili doğrusal olmayan mekanik sistemler için önerilen I&I bozucu kestirim algoritması en genel formatta sunulmuştur. Bozucu kestirim dinamiği için kararlılık analizi Lyapunov yaklaşımı ile yapılmıştır.

Bölüm 5 ve 6'da, önerilen yöntem tam eyleyicili sistemlerin bir örneği olarak 2-link düzlemsel robot sistemi ve eksik eyleyicili sistemlerin bir örneği olarak da araç sarkaç sistemine uygulanmaktadır. Bozucu kestirimi, her iki örnekte de hem tam eyleyicili hem de eksik eyleyicili sistem koordinatları üzerinde etkili olan bozucuları başarılı bir şekilde tahmin etmiştir. Kestirim bozucu bilgisine sahip olan geri adımlı kontrol ünitesi, tam eyleyici sistem örneğindeki bozucuları başarılı bir şekilde bastırmıştır. Eksik eyleyicili sistem için, eyleyicili koordinat üzerinde etki eden bozulma tamamen bastırılmış, ancak bozucunun eksik eyleyicili koordinata etkisi, bozucu bilgisinin bilinmesine rağmen tamamen yok edilememiştir.

Son bölümde elde edilen sonuçlar özetlenmiş ve bu çalışmanın katkıları vurgulanmıştır.

Abstract:
In this graduate study, a novel discrete-time disturbance estimator design is proposed for underactuated nonlinear mechanical systems using Immersion and Invariance approach, the estimated disturbance values are used in discrete-time back-stepping control design in order the attenuate the constant disturbances in considered applications for fully actuated and underactuated non-linear mechanical systems.

In Section 1, a literature review is organized for the topics; underactuated nonlinear mechanical systems, fully actuator and under actuator mechanical systems, linearization with feedback and linearization with partial feedback, a discrete-time back-stepping control, I&I adaptive control method, 2-link planar robot arm modelling and control methods, cart-pendulum system modelling and control methods.
 
In Section 2, as background information summaries of the topics utilized in this study are given. First of all derivation of dynamic equations for fully actuated nonlinear mechanical systems and underactuated non-linear mechanical systems by Euler-Lagrange approximation is presented. Afterward, linearization with feedback and linearization with partial feedback, respectively, are specified for fully actuator nonlinear mechanical systems and underactuated nonlinear mechanical systems. After that, discretization of continuous dynamic equations using Forward-Euler approximation is given for nonlinear mechanical systems. On the other hand, discrete-time Immersion and Invariance adaptive control theory for the nonlinear systems with uncertainty is given. In the end, the discrete time back-stepping control procedure for general nonlinear mechanical systems in strict-feedback form has been presented. 

In Section 3, linearization with feedback and linearization with partial feedback, respectively, are given for discrete time fully actuated nonlinear mechanical systems and underactuated nonlinear mechanical systems.   

In Section 4, the discrete-time dynamic equations for general non-linear mechanical systems that include disturbance in the input is given. After that, for both fully actuated nonlinear mechanical systems and underactuated non-linear mechanical systems, the proposed Immersion and Invariance disturbance estimation algorithm has been presented in the most general format.  In the end, the stability analysis for the estimator dynamics has been done by the Lyapunov approach. 

In Section 5, attenuation of constant input disturbances in the control of robotic manipulators is considered in the discrete-time setting. First of all, a disturbance estimator design with Immersion and Invariance approach is presented for n-degree of freedom mechanical systems that comprises n-degree of freedom robotic manipulators. After that, a discrete time backstepping controller established that uses the estimated disturbance information and locally stability using Lyapunov's stability theorem, in an arbitrarily large compact set, of closed-loop system is shown. The method applied to 2-degree of freedom robotic manipulator and tested by simulations. 

For section 5 conclusions can be given as follows: a disturbance estimator design with Immersion and Invariance approach for n-degree of freedom nonlinear mechanical system with constant input disturbances which comprises n-degree of freedom robotic manipulators is presented and a backstepping controller construction made that uses the estimated disturbance information to attenuate the considered disturbances in the discrete-time setting. Firstly, discrete-time state-space equations in block form for n-degree of freedom mechanical systems with input disturbance from their well-known continuous-time Euler-Lagrange equations are formed and the obtained model is discretized using forward Euler approximation. After that disturbance estimator design established using this discrete-time model. The discrete-time model is not in a form that backstepping control can be directly applied, therefore first a feedback linearization controller obtained as the inner loop where the estimated disturbance term directly appears in the control rule. Afterwards, backstepping control design performed for the inner closed-loop system as the outer control loop. In the closed-loop system obtained with backstepping controller, a term with estimator error exist and therefore stability of the entire closed loop systems with estimator error dynamics analysed and local stability of the entire closed-loop system is shown. The proposed method is applied to 2-degree of freedom robotic manipulator. Its performance is tested by comparison with pure backstepping that does not use disturbance information. The simulation results demonstrate that Immersion and Invariance estimator successfully estimates the true disturbance values after a transient and thus the backstepping that uses this information successfully attenuate the disturbances whereas in backstepping without estimator step disturbances causes constant steady-state errors.

In Section 6, the stabilization of cart-pendulum system with a discrete-time backstepping controller and the attenuation of the disturbance affecting on the cart and pendulum are considered. In order to enable discrete-time backstepping design discrete-time partial feedback linearization performed and a novel discrete-time coordinate transformation is proposed. Before construction of backstepping controller, a nonlinear disturbance estimator design is proposed with Immersion and Invariance approach. The estimated disturbance values used in the partial feedback linearization and construction of backstepping controller. The performances of proposed algorithms are tested by simulations.

For section 6 conclusions can be given as follows: backstepping control and disturbance estimation for the cart-pendulum system with input disturbance and disturbance which is given the underactuated variable is considered in discrete time setting. After obtaining discrete-time dynamic equations in block form with forward Euler approximation, Immersion and Invariance disturbance estimator is designed. Then, the partial-feedback linearization controller constructed with the estimated disturbance term. Afterwards, for the partially linearized system a novel coordinate transformation is proposed and used to enable backstepping control design. Proposed methods are tested by simulations. Simulation results show that pure backstepping controller obtained with proposed coordinate transformation successfully stabilize the system in up-right position in the absence of the disturbance but when there is an input disturbance it can not attenuate as fast as backstepping with Immersion and Invariance estimation and in some cases it can not ever attenuate it, system becomes unstable where backstepping with Immersion and Invariance estimation successfully attenuate it. That is to say, the simulation results approve that proposed backstepping control together with Immersion and Invariance disturbance estimation successfully attenuate the input disturbance.

For section 5 and 6 summaries can be given as follows: the proposed method is applied to a 2-link planar robot as an example of fully actuated systems and for cart pendulum as an example of underactuated systems. The estimator successfully estimated the disturbances that effects both actuated and underactuated coordinates in both of the examples. The back-stepping controller with the estimated disturbance information successfully attenuated the disturbances in the fully actuated system example. In the case of the under actuator system, the disturbance that affects on the actuated coordinate is completely attenuated but the disturbance that affects on under-actuated coordinate cannot be completely attenuated even with the information of the disturbance.

In the last section, the obtained results are summarized and the contributions of this study are emphasized.