Continuum backstepping control algorithms in partial differential equation orientation: A review

The continuum backstepping (C-BKST) control algorithm cooperated with boundary control approaches is proposed for distributed parameter systems (DPSs) modeled in partial differential equations (PDEs). The algorithm based on the Volterra transformation is robust, inverse optimal, and potential for explicit exact control laws and exact solutions of closed-loop systems. The C-BKST control algorithm is novel and can be combined with the achievements of the observer theory and the adaptive control theory to extend its application fields. The basic principles and design procedures of the algorithm are introduced in this paper. The recent development of this algorithm is concluded, covering the aspects of parabolic PDEs, hyperbolic PDEs, complex PDEs, and nonlinear PDEs. Finally the main characteristics of the algorithm are summarized, and the development direction of the algorithm is discussed.