In the past decade, the infinite-dimensional backstepping method has been verified as a powerful tool for designing boundary controllers for partial differential equations (PDEs) coupled with ordinary differential equations (ODEs) systems. With the backstepping approach, the control gains are explicitly parametrized which allows us to make a major leap forward methodologically and to design adaptive controllers. The literature considers systems with a strict-feedback structure, i.e., ODE and PDE systems are cascaded at the boundary, or non-strict-feedback structure. Moreover, the works consider ODE to be stabilized through compensating PDEs on a fixed domain, or a time-varying domain in the actuating path. This thesis investigates adaptive schemes for the boundary stabilization of the systems characterized by the wave PDE-ODE model in two applications. In the first application, we propose an adaptive scheme for the boundary stabilization of thermoacoustic instability in the Rijke tube system using the backstepping method. The mathematical model of the system is characterized by a linear first-order 2*2 partial differential equation that behaves like a wave equation coupled with a first-order ordinary differential equation in a non-strict-feedback form. Recently, a full state feedback controller has been developed to stabilize the system, assuming that the parameters of the model are known. We take into account the most common uncertain parameters which result in the coefficients of the first-order ODE system being unknown parameters. The technique of adaptive identifier is then used along with the normalized gradient algorithm to achieve the parameter update laws. The adaptive control law is obtained by replacing the output of the identifier and estimated parameters in the non-adaptive state feedback control law. The adaptive control law is then manipulated such that it uses a few measurements of the PDE states. According to the stability analysis of the system, the proposed controller guarantees that all closed-loop system states are bounded, while the ODE-PDE system states are convergent to zero. The performance of the proposed scheme is eva‎luated by the simulation examples. In the second application, we propose an adaptive output-feedback boundary control approach based on an observer to stabilize the vibration of a dual-cable mining elevator that is used to transport heavy loads to considerable depths. The elevator's vibration dynamics are described by two pairs of 2*2 heterodirectional coupled hyperbolic partial differential equations (PDEs) on a time-varying domain, which are coupled with a first-order ordinary differential equation (ODE) in uncontrolled boundaries. To develop the control design, we take into account the fact that the system is subjected to sinusoidal disturbances with unknown amplitudes appearing at the ODE, and the damping coefficients of the system are also unknown, resulting in the existence of unknown parameters in the state matrix of the ODE. Using the infinite-dimensional backstepping method and performing a Lyapunov analysis, the control input and update laws are derived to ensure the asymptotic convergence to zero of the ODE state and the boundedness of the PDE states. A numerical simulation is presented to support the theoretical results.

محسن مجيري فروشاني

محمدعلي قديري مدرس

فريد شيخ الاسلام , مرضيه كمالي