Adaptive boundary control using backstepping for 1D variable length string-mass system under disturbances

Abstract

In this thesis, an adaptive boundary control using delayed control methodology for a 1D wave equation is examined. The outlined problem is applied in the control of an ideal string- mass system with constant or time-varying length. The dynamics of the system, which constitutes the basis for the control problem, is first derived using the extended Hamilton‘s Principle. The resulting wave PDE is then transformed into two decoupled hyperbolic equations using the method of characteristics. The solution of the characteristic equation allows one to project the input signal at one boundary onto the dynamics describing the other boundary. Here, the input appears with an explicit delay. If the domain is characterized by a moving boundary, i.e., the length of the string is non -constant, the delay is time-varying. The problem then becomes that of control of a linear ODE with an input delay. Afterward, the transport PDE representation is used to re-express the delay in terms of a PDE‘s boundary value re sulting in an ODE- PDE cascade system. The backstepping transformation then gives the control law and transforms the system into the target system characterized by fa vorable control properties. The only feedback required for the control is the boundary measurements. Thereafter, Lyapunov‘s theory is used in the stability analysis. Any unknown in-domain or boundary disturbances, as well as uncertain boundary parame ters, are handled using the adaptive control strategies. The dynamics of the string-mass system and the performance of the derived controllers are illustrated using numerical simulations. This is followed by a case study where the deployment and control of an underwater sensor in the presence of the water waves are simulated.