Impulsive adaptive observer design for a class of hybrid ODE–PDE cascade systems with uncertain parameters

In this paper, an impulsive adaptive observer is proposed for online estimates of a class of hybrid cascade systems with uncertain parameters. The hybrid cascade system is formed by an ordinary differential equation (ODE) coupled with a partial differential equation (PDE) with uncertain parameters. In the hybrid cascade system, the state of the connection point between the two subsystems cannot be directly available, which makes it difficult to obtain online estimates of the inaccessible states. To deal with the complex observation problem, an impulsive adaptive observer is designed using the backstepping approach and the extended Kalman observer. In addition, the convergence of the proposed observer is examined. The backstepping transformation plays a crucial role in the convergence proof. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.