Nonlinear Dynamics

, Volume 76, Issue 4, pp 2041–2058 | Cite as

Distributed proportional plus second-order spatial derivative control for distributed parameter systems subject to spatiotemporal uncertainties

  • Jun-Wei Wang
  • Han-Xiong Li
  • Huai-Ning Wu
Original Paper


In this paper, a robust distributed control design based on proportional plus second-order spatial derivative (P-sD\(^2\)) is proposed for exponential stabilization and minimization of spatial variation of a class of distributed parameter systems (DPSs) with spatiotemporal uncertainties, whose model is represented by parabolic partial differential equations with spatially varying coefficients. Based on the Lyapunov’s direct method, a robust distributed P-sD\(^2\) controller is developed to not only exponentially stabilize the DPS for all admissible spatiotemporal uncertainties but also minimize the spatial variation of the process. The outcome of the robust distributed P-sD\(^2\) control problem is formulated as a spatial differential bilinear matrix inequality (SDBMI) problem. A local optimization algorithm that the SDBMI is treated as a double spatial differential linear matrix inequality (SDLMI) is presented to solve this SDBMI problem. Furthermore, the SDLMI optimization problem can be approximately solved via the finite difference method and the existing convex optimization techniques. Finally, the proposed design method is successfully applied to feedback control problem of the FitzHugh–Nagumo equation.


Distributed parameter systems Robust control Exponential stability Spatiotemporal uncertainty Linear matrix inequalities (LMIs) 



This work was supported in part by the National Basic Research Program of China (973 Program) (2012CB720003), in part by the National Natural Science Foundation of China under Grants 61121003, 51175519, and 91016004, and in part by the Beijing Youth Fellowship Program (YETP0378). The authors gratefully acknowledge the helpful comments and suggestions of the anonymous reviewers, which have improved the presentation.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.School of Automation and Electrical EngineeringUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  2. 2.Department of Systems Engineering and Engineering ManagementCity University of Hong KongHong Kong SARPeople’s Republic of China
  3. 3.State Key Laboratory of High Performance Complex ManufacturingCentral South UniversityChangshaChina
  4. 4.Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical EngineeringBeihang University (Beijing University of Aeronautics and Astronautics)BeijingPeople’s Republic of China

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