Feedback control of nonlinear differential difference equation systems

Abstract

This paper proposes a methodology for the synthesis of nonlinear output feedback controllers for single-input single-output nonlinear differential difference equation (DDE) systems which include time delays in the states, the control actuator and the measurement sensor. Initially, DDE systems which only include state delays are considered and a novel combination of geometric and Lyapunov-based techniques is employed for the synthesis of nonlinear state feedback controllers that guarantee stability and enforce output tracking in the closed-loop system, independently of the size of the state delays. Then, the problem of designing nonlinear distributed state observers, which reconstruct the state of the DDE system while guaranteeing that the discrepancy between the actual and the estimated state tends exponentially to zero, is addressed and solved by using spectral decomposition techniques for DDE systems. The state feedback controllers and the distributed state observers are combined to yield distributed output feedback controllers that enforce stability and output tracking in the closed-loop system, independently of the size of the state delays. For DDE systems with state, control actuator and measurement delays, distributed output feedback controllers are synthesized on the basis of an auxiliary output constructed within a Smith-predictor framework. The proposed control method is successfully applied to a reactor-separator process with recycle and a fluidized catalytic cracker and is shown to outperform nonlinear controller designs that do not account for the presence of dead time associated with the recycle loop and the pipes transferring material from the reactor to the regenerator and vice versa, respectively.