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Open Access Publications from the University of California

Boundary Controllers and Observers for Korteweg-de Vries, Hyperbolic, and Parabolic PDEs

  • Author(s): Tang, Shuxia
  • Advisor(s): Krstic, Miroslav
  • et al.

This dissertation presents the study of some advancements in the control theory and application of the systems described by partial differential equations. Three classes of partial differential equations are discussed, mainly concerning three topics: stability analysis, boundary controller design and disturbance rejection.

Stability analysis is a topic of great interest and importance in the control area, for which the Lyapunov method and the operator semigroup theory are used in this dissertation. Stability analysis can serve as a guidance of controller design with the objective of stabilization, and the most frequently employed approach for system controller designs in this work is the backstepping methodology. We design state feedback boundary controllers based on backstepping, and further construct output feedback controllers in assistance with the state observers designed to re- cover the full system states. Moreover, the sliding mode control and the active disturbance rejection control techniques are applied for the purpose of disturbance rejection.

In detail, the following three classes of systems are investigated. The first class consists of the Korteweg-de Vries systems. For a (nonlinear) Korteweg-de Vries equation, the asymptotic stability analysis is conducted. Two methods, one relying on normal forms and the other relying on a Lyapunov approach, are em- ployed based on the center manifold theory. Both prove that the equation is (lo- cally) asymptotically stable around the origin. A class of linear Korteweg-de Vries systems with possible anti-diffusion is also discussed, focusing on the backstepping controller designs. The resulting closed-loop control systems exhibit exponential decay with respect to the corresponding state norms.

The second is a general class of coupled bidirectional hyperbolic systems with spatially varying coefficients, concerning stability analysis, controller design and disturbance rejection problems. This study borrows the controller design idea from an existing result for the systems with constant coefficients, and extends it to deal with time-varying coefficients. As a result, finite-time stability is achieved. With regards to the systems running into control matched uncertainties and disturbances, disturbance rejection and disturbance attenuation are achieved for the closed-loop feedback systems with sliding mode control and active disturbance rejection control, respectively.

Furthermore, the application of estimation techniques to a real-world issue is studied, i.e., the state-of-charge estimation problem in lithium-ion batteries. A thermal-compensated electrochemical model of lithium-ion batteries is proposed. Adding thermal dynamics serves a two-fold purpose: improving the accuracy of state-of-charge estimation and keeping track of the average temperature which is critical for battery safety management. This model can be included by the third class, i.e., the parabolic systems with time-varying coefficients. Note that in these systems, the time dependency of the system coefficients makes the observer design problem nontrivial. This dissertation can be conducive to future related endeavors.

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