Kontrollü lagrange yöntemleri ve uygulamaları
Controlled lagrangian methods and applications
Yazar: HÜSEYİN ALPASLAN YILDIZ
Danışman: PROF. DR. AFİFE LEYLA GÖREN
İstanbul Teknik Üniversitesi / Lisansüstü Eğitim Enstitüsü / Kontrol ve Otomasyon Mühendisliği Ana Bilim Dalı / Kontrol ve Otomasyon Mühendisliği Bilim Dalı
Derece: Doktora, 2024
Konu: Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrol = Computer Engineering and Computer Science and Control
Dizin: Doğrusal olmayan kontrol = Nonlinear control ; Doğrusal olmayan kontrol teorisi = Nonlinear control theory ; Geri beslemeli kontrol = Feedback control

Özet:
Bu tez çalışmasında eksik sürülmüş sistemlerin enerji tabanlı kontrol yöntemleri olan Kontrollü Lagrange ve IDA-PBC (Interconnection and Damping Assignment Passivity-Based Control) yöntemleri ele alınmıştır. Eksik sürülmüş sistemler, sistemin eyleyici sayısının, serbestlik derecesinden az olduğu, dolayısıyla tüm hareketlerinin doğrudan kontrol edilemediği sistemlerdir Bu tür sistemler, mühendislikte özellikle robotik, havacılık, denizcilik ve otomotiv gibi alanlarda sıkça karşılaşılır. Karmaşık dinamik yapıları nedeniyle, bu sistemler kontrolü zor olan sistemler arasında yer alır.
Tez çalışmasında, öncelikle klasik mekaniğin tarihsel gelişimi ve varyasyonel hesaplama yaklaşımı ele alınarak, bu yaklaşımların Euler-Lagrange ve Hamiltonian sistemlere uygulanışı incelenmiştir. Bu bağlamda, varyasyonel hesaplamanın temel ilkeleri, Euler-Lagrange denklemlerinin türetilmesi ve bu denklemlerin mekanik sistemlerin modellemesinde nasıl kullanıldığı açıklanmıştır. Aynı zamanda, Hamiltonian sistemlerin temel prensipleri ve bu sistemlerin kontrolü için kullanılan enerji tabanlı yöntemler de ele alınmıştır.
Kontrollü Lagrange yöntemi, EL sistemlerin kontrolü için kullanılan bir yöntem olup, sistemin enerji fonksiyonunu şekillendirerek kapalı çevrim sistemin istenen denge noktasındaki kararlılığı sağlamayı hedefler. IDA-PBC yöntemi ise, Hamiltonian sistemlerin kontrolünde kullanılan bir yöntem olup, aynı şekilde sistemin enerji fonksiyonunu şekillendirerek ve sönüm ekleyerek kapalı çevrim sistemin pasifliğini ve dolayısıyla kararlılığını sağlamayı amaçlar. Her iki yöntemde de, kapalı çevrim sistemin kararlılığı sağlanırken, EL ve Hamiltonian sistem olma özelliği korunur.
IDA-PBC ve Kontrollü Lagrangian yöntemleri, eksik sürülmüş lineer olmayan sistemlerin kontrolü için güçlü birer araç olsalar da, kontrol kuralının varlığı ancak eşleşme koşulu adı verilen bir dizi lineer olmayan ve homojen olmayan PDE'lerin çözülmesiyle mümkündür. Bu PDE'lerin her durumda çözümü bulunmamaktadır, bu da bahsedilen yöntemlerin en zorlayıcı noktasıdır. Bu tez çalışmasında ayrılabilir (seperable) Hamiltonian sistemlerde eşleşme koşullarının göreceli olarak daha kolay çözülmesi fikrinden yola çıkarak geliştirilen Gören-Sümer ve Şengör (2015)'de önerilen yöntem geliştirilerek Euler Lagrange sistemlere genişletilmiştir. 
Bu tez çalışmasında önerilen yöntem, eşleşme koşullarının yaklaşık çözümleri, kapalı çevrimli sistemin genelleştirilmiş atalet matrisinin bir dizi sabit atalet matrisinin radyal baz fonksiyonları kullanılarak doğrusal olmayan bir kombinasyonu ile yaklaşık olarak ifade edilmesine dayanmaktadır Eşleşeme koşullarının yaklaşık çözümü ile elde edilen kontrol kuralı kullanıldığında dahi, kapalı çevrim sistemin kararlılığının sağlanabildiği gösterilmiştir.
Bu çalışmanın son aşamasında, tezde önerilen yöntemlerin çeşitli mekanik sistemler üzerindeki uygulamaları incelenmiştir. Vinç sistemi, TORA, top ve çubuk, top ve tabla sistemi gibi örnekler kullanılarak, önerilen yöntemin, literatürde bulunan örnekler üzerinde uygulamasına yer verilmiştir. Bu uygulamalar, önerilen yöntemin yaygın olarak kullanılabilirliğini ortaya koymuştur.
Sonuç olarak, bu doktora tez çalışması, eksik sürülmüş sistemlerin kontrolünde kullanılan enerji tabanlı yöntemler olan IDA-PBC ve Kontrollü Lagrange yöntemlerinin temel zorluğu olan eşleşme koşullarının çözülmesi konusuna bir katkı sunmaktadır. Elde edilen sonuçlar, eşleşme koşullarının çözümlerinin yaklaşık olarak bulunduğu durumda dahi, enerji tabanlı kontrol yöntemlerinin, eksik sürülmüş sistemlerin kararlılığını sağlamada etkili araçlar olduğunu göstermektedir. Bu çalışma, ilgili alanda çalışacak araştırmacılara bu yöntemlerin daha karmaşık sistemler için nasıl geliştirilebileceği ve uygulanabileceği konusunda yol gösterici olacaktır.

Abstract:
In control engineering, underactuated systems have gained significant importance due to their prevalence in various practical applications and the unique challenges they present. These systems, characterized by having fewer actuators than degrees of freedom, are commonly encountered in robotics, aerospace, marine, and automotive industries. The complexity of controlling such systems arises from the need to manage unactuated degrees of freedom while achieving precise control over the actuated ones. This complexity is further amplified by the nonlinear dynamics and potential interactions between different parts of the system. As a result, underactuated systems serve as a critical testbed for developing advanced control theories and techniques that can handle real-world constraints and ensure robust, efficient, and safe operation of engineering systems.
Cranes are a classic example of underactuated systems used in various industries for lifting and transporting heavy loads. The pendulum-like oscillations of the crane's load during movement need to be controlled to avoid accidents and ensure precise positioning. Similarly, VTOL aircraft, such as drones, are another example where the control inputs are fewer than the degrees of freedom, resulting in complex flight dynamics that require advanced control strategies to manage vertical takeoff, hovering, and landing. Mobile robots, which are used in numerous applications from manufacturing to space exploration, also often operate as underactuated systems, having to perform tasks like navigation and manipulation with limited actuators. These robots must adapt to changing environments and perform precise movements, making their control particularly challenging. Additionally, systems like robotic manipulators mounted on mobile platforms, two-wheeled balancing robots, and inverted pendulums such as the Furuta pendulum and the inertia wheel pendulum (IWP) are extensively studied as benchmarks for testing new control algorithms.
Several control methods have been developed to address the unique challenges of underactuated systems. Traditional approaches include linear and nonlinear control techniques, which are designed to stabilize and manage the dynamics of these systems. For instance, model predictive control (MPC), sliding mode control (SMC), and feedback linearization have been widely used. Advanced methods such as adaptive control and intelligent control techniques like fuzzy control and neural networks have also been explored to handle the uncertainties and dynamic variations within these systems. Additionally, specific algorithms like differential flatness and backstepping are applied to ensure trajectory tracking and stabilization. Among these, energy-based control methods stand out due to their ability to leverage the system's energy properties to achieve desired control objectives, providing a robust framework for stabilizing and controlling underactuated systems.
Energy-based control methods are particularly effective for underactuated systems as they exploit the natural energy properties of the system to achieve stabilization and control. These methods include passivity-based control, where the system's passive properties are used to design stable control laws. Two prominent energy-based control techniques are the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) and the Controlled Lagrangian method. IDA-PBC method shapes the system's energy to achieve desired dynamic behavior. By modifying the interconnection and damping structures, IDA-PBC ensures that the closed-loop system behaves in a stable and desired manner. This approach leverages the natural passivity properties of the system, making it robust to various uncertainties and disturbances. Controlled Lagrangian method, on the other hand, involves shaping both the kinetic and potential energy functions to stabilize underactuated systems. In the Controlled Lagrangian method, an underactuated Euler-Lagrange system is assumed to possess symmetry, meaning that the system's Lagrangian is invariant under the action of an Abelian Lie group. The goal is to find a closed-loop system, also in Euler-Lagrange form, that stabilizes an equilibrium point while preserving the system's symmetry. This is achieved by selecting a feedback law that transforms the original control system into the proposed closed-loop Euler-Lagrange system. The existence conditions for such a feedback law are known as matching conditions.
One of the primary benefits of the Euler-Lagrange (EL) and Hamiltonian formalisms is their independence from the choice of coordinate systems. Unlike the Newtonian method, which often relies on Cartesian coordinates, the EL formalism utilizes generalized coordinates that can simplify the problem, especially in systems with constraints. Similarly, the Hamiltonian approach offers a coordinate-independent formulation, beneficial in systems with symmetries and conserved quantities. The EL formalism simplifies the derivation of equations of motion through generalized coordinates, incorporating constraints directly via Lagrange multipliers. This results in more manageable equations, particularly in constrained systems. The Hamiltonian approach further simplifies the problem by transforming second-order differential equations into first-order ones, making them easier to solve and analyze. Both formalisms are inherently energy-based. The EL method, through its energy formulation, facilitates the application of energy methods for stability and control, such as Lyapunov functions or energy shaping techniques like Controlled Lagrangian. The Hamiltonian approach explicitly incorporates the system's total energy, making it ideal for energy-based control methods like IDA-PBC (Interconnection and Damping Assignment Passivity-Based Control). The EL formalism systematically handles constraints using generalized coordinates and Lagrange multipliers, providing a clear framework for incorporating both holonomic and non-holonomic constraints. Similarly, the Hamiltonian approach effectively incorporates constraints within the phase space, integrating them into the Hamiltonian function. Both formalisms leverage the connection between symmetries and conservation laws. The EL method link system symmetries with conserved quantities, simplifying analysis and control design. The Hamiltonian formalism naturally respects conservation laws through its structure, aiding in the determination and exploitation of these properties for control purposes.
Energy-based control methods, such as the Controlled Lagrangian and IDA-PBC, offer significant advantages in maintaining the intrinsic structures of Euler-Lagrange (EL) and Hamiltonian systems. These methods preserve the fundamental energy properties and dynamics of the system, which is crucial for ensuring stability and robust performance. The Controlled Lagrangian method shapes both the kinetic and potential energy functions to stabilize underactuated systems while preserving the system's Lagrangian symmetry. This approach leverages the system's inherent structure, making it easier to design control laws that achieve the desired equilibrium points without altering the original dynamics of the system. Similarly, by preserving the Hamiltonian structure, IDA-PBC facilitates the design of control laws that exploit the system's natural passivity properties, making the control strategy more robust to uncertainties and disturbances. Overall, these energy-based methods ensure that the essential characteristics of EL and Hamiltonian systems are maintained, providing a systematic and effective framework for controlling complex, underactuated systems.
Specifically, since the late 1990s, studies on the control of underactuated, nonlinear systems have begun to emerge. The first significant contribution in this field came from Jerrold E. Marsden's team at Caltech, renowned for their work in mathematical physics and mechanics, dynamic systems, and control theory. In their seminal work, Bloch et al. (1997) introduced the Controlled Lagrangian method. Marsden and his team, as stated in Marsden (1999), focused on Euler-Lagrange (EL) systems based on variational principles, which can be directly generalized to the concept of general relativity.
Following the revolutionary papers by Bloch et al. (1997, 1998), which demonstrated the feasibility of controlling mechanical systems using their symmetry properties, Hamberg (1999) derived the necessary conditions for matching two Euler-Lagrange systems, known as matching conditions. Bloch et al. (2000) then systematically presented the matching conditions and the Controlled Lagrangian method. This foundational work aimed to stabilize the unactuated variables of the system by exploiting the symmetry property of mechanical systems. Therefore, in the example of the cart-pendulum system given in the study, only the stabilization of the pendulum in the upright position was achieved, while the position of the cart could not be controlled. Immediately following this study, Bloch et al. (2001) introduced the potential energy shaping defined as symmetry-breaking and developed a control rule that stabilizes both the actuated and unactuated variables of the system.
While Jerrold Marsden and his team continued their work on underactuated Euler-Lagrange (EL) systems, Romeo Ortega, known for his work on passivity and energy-based control, together with applied mathematics professor Arjan Van der Schaft, published a significant paper in 1999. The work by Ortega and Van der Schaft (1999) is a precursor to the studies by Ortega and his team on the stability of underactuated Hamiltonian systems. Although this study did not provide a specific solution for underactuated systems, it demonstrated that passivity-based control offers a solution for port-controlled Hamiltonian systems. The 2000 IFAC Congress held in Princeton, New Jersey, was particularly intriguing for participants working on nonlinear systems. At this conference, Ortega and Spong (2000) first introduced the IDA-PBC method for underactuated Hamiltonian systems. Gomez-Estern et al. (2001) defined the class of Hamiltonian systems to which the IDA-PBC method could be applied, and shortly thereafter, Blankenstein et al. (2000) defined the matching conditions used in EL systems for the IDA-PBC method. Ortega et al. (2002-a, 2002-b) further developed the IDA-PBC method for controlling a class of underactuated port-controlled Hamiltonian systems.
IDA-PBC for Hamiltonian systems and Controlled Lagrangian method for Euler-Lagrange (EL) systems were established in the literature by 2002 as valid methods for stabilizing closed-loop systems. As Marsden (1999) points out, Hamiltonian mechanics, which is directly based on the concept of energy and is closer to quantum mechanics, and Lagrangian mechanics were equivalent in many cases. This equivalence should also apply to the IDA-PBC and Controlled Lagrangian methods. In his PhD thesis, Chang (2002) of the Marsden team showed that the Controlled Lagrangian method can also be applied to Hamiltonian systems. In the same year, Chang et al. (2002) showed the equivalence of the Controlled Lagrangian method with IDA-PBC, emphasizing that the Controlled Lagrangian method is a more general solution than IDA-PBC. Blankenstein (2006), by applying the IDA-PBC method on Euler-Lagrange systems, showed that the matching conditions in the Controlled Lagrangian method are a special case of IDA-PBC and emphasized that IDA-PBC is a more general solution.
While the equivalence discussions between the two methods continued, efforts to solve the matching conditions—a set of nonlinear partial differential equations (PDEs)—also continued. Chang (2006) generalized the λ-method, presented by Aucky and Kapitanski (2003) for solving the matching conditions of a class of EL systems, by including gyroscopic forces. Similarly, Viola et al. (2007) proposed a method for solving the matching conditions for a class of Hamiltonian systems using coordinate transformations. Crasta and Ortega (2015), while introducing a class of Hamiltonian systems for which the kinetic energy matching condition can be solved using gyroscopic forces, highlighted the importance of the solvability of the kinetic and potential energy PDEs for the success of IDA-PBC. Post-2010 studies have largely focused on solving these PDEs. Similarly, Donaire et al. (2016) demonstrated that under certain restrictive conditions, ordinary differential equations (ODEs) could be used instead of PDEs to solve the matching conditions. To address the same problem, Gören Sümer and Yalçın (2011) tackled the discrete-time counterpart of the problem and obtained matching conditions in the discrete-time setting to provide a relatively easy technique for stabilizing the unstable equilibrium points of underactuated Hamiltonian systems. For the same problem, Sarras et al. (2013) proposed an approach using the "immersion and invariance" technique to bypass the solution of PDEs for a class of port-controlled Hamiltonian systems.
In this thesis, a method is proposed for obtaining approximate solutions to the partial differential equations (PDEs) known as matching conditions. The inspiration for the proposed method is the fact that the matching conditions for underactuated Euler-Lagrange (EL) systems with a constant generalized inertia matrix are linear PDEs, meaning their analytical solutions can be easily found. The main idea of the proposed method relies on expressing both the controlled and the targeted closed-loop system's generalized inertia matrices approximately as a nonlinear combination of a set of constant inertia matrices using radial basis functions. Consequently, the system can be approximately modeled in terms of a set of linear EL systems, and a common potential energy function that provides the solution to the matching conditions, now a series of linear PDEs, can be parametrically constructed for all these systems. As a result, a potential energy function V_c (q) that approximately satisfies the potential energy matching condition and the generalized inertia matrix M ̂_c (q) of the closed-loop EL system can be obtained by selecting the parameters of a set of constant matrices and radial basis functions.
In conclusion, this thesis demonstrates that the stability of energy-based underactuated nonlinear systems can be ensured by using an approximate solution to the matching conditions, which is the challenging aspect of the IDA-PBC and Controlled Lagrangian methods. The method proposed has been applied to various systems in the literature, showcasing its versatility and effectiveness. Further research could focus on refining these approximate solutions and extending them to more complex and varied underactuated systems, potentially improving the robustness and performance of control strategies in practical applications.
This PhD research consists of six chapters including the Introduction. In the second part of the dissertation, variational methods for EL equations and Hamiltonian systems will be explained and then both IDA-PBC and controlled Lagrangian methods will be introduced in detail. 
In the third part of the thesis, a method that ensures the stability of a class of underactuated Euler Lagrangian systems and allows to obtain an approximate solution of the matching conditions will be introduced. An analysis will be made explaining how the control rule obtained using the approximate solution of the matching conditions guarantees the stability of the system. 
In the fourth part of the thesis, a control method using approximate solutions of the matching conditions in IDA-PBC which is developed for port-controlled Hamiltonian systems similar to EL systems will be presented.
The fifth chapter of the thesis includes applications of a set of underactuated EL systems existing in the literature which are controlled with the method proposed in the thesis.
The sixth and final chapter presents the conclusions of this thesis and the recommendations related to the study. It also discusses the conclusions and comparisons made in the thesis and provides information about future studies.