Performance output tracking for cascaded heat partial differential equation‐ordinary differential equation systems subject to unmatched disturbance

Abstract

In this paper, we are concerned with the performance output tracking for two cascades of one-dimensional heat partial differential equation-ordinary differential equation systems with Dirichlet and Neumann interconnection, respectively, where both the external disturbances are non-collocated to the controller, and moreover, the disturbances come from exosystem. By means of the proper trajectory planning, the non-collocated configurations can be converted into the collocated ones, so that the difficulties caused by these non-collocateds are overcome. Exponential convergences of the regulation errors are proved. Finally, the numerical simulations validate the effectiveness of this method. K E Y W O R D Scascaded systems, exponential convergence, non-collocated boundary control, output tracking INTRODUCTIONThe output tracking for distributed parameter systems has received a lot of attention in recent years as practically motivated. Internal model principle (IMP), partial results of which have been generalized to the infinite-dimensional systems in References 1-7, has been applied to the output tracking systematically for ordinary differential equations (ODEs) in References 8-10. However, from a practical point of view, the IMP has some distance to solve partial differential equations (PDEs) because it is difficult to solve some related Sylvester operator equations, which may not be easily realized. 6 Feng and Guo 11 proposed a new active disturbance rejection control to stabilize an antistable wave equation, see also References 12,13, which method can also be applied to cope with output tracking in Reference 14. The output tracking problem for a general 2 × 2 system of first-order linear hyperbolic PDEs was considered in Reference 15, which no uncertainty and disturbance were taken into consideration. Recently, output tracking problem with non-collocated configuration was considered by means of adaptive control method in References 16,17, an early effort by adaptive control can also be seen in Reference 18, which rejected the harmonic disturbance only. It is noted that the reference signal is not specified due to disturbance in the above works, and moreover, the authors are only concerned with the performance output tracking for a single control system. Although there are some results about boundary control of PDE-ODE cascades, [19][20][21][22] how to deal with the output tracking problem of cascaded control system subject to unmatched disturbance is quite challenging. System control through actuator dynamics can usually be modeled as a cascaded control system. 23 In this paper, we consider the output tracking for an ODE with actuator dynamics dominated by a heat equation. To the best