Backstepping output-feedback control of moving boundary parabolic PDEs
Introduction
In the application of many control strategies, the knowledge of the state of the system is essential, yet, in most cases this information is not fully available due to reasons such as variables may not be measurable and using sensors may not be physically possible or economically justified. For these systems, a part of the design problem is the synthesis of state estimators (observers) that generate estimations of system states. Standard systematic techniques are available to design observers for linear and nonlinear finite-dimensional systems. However, for infinite-dimensional (or distributed-parameter) systems governed by partial differential equations (PDEs) the state variables do not depend only on time, but they also depend on spatial coordinates. In addition, changes in the shape and material properties characterized by phenomena such as material deformation, phase change, and mass transfer, may result in system models as moving boundary parabolic PDEs. Both infinite dimensionality and changes in the domain of these models impose further complexities and limitations to analysis and design.
One can address the observer (or controller) design problem for distributed parameter systems (DPSs) by using one of the two approaches: in the early-lumping method, at first the DPS is discretized by the use of suitable approximation techniques such as modal decomposition or Galerkin׳s method, to yield an approximate finite-dimensional model based on which state estimator is designed, see, e.g. [24], [15]. However, this approximation may change the system properties such as observability and/or possibly induce the physical problem model failure. Moreover, stability of a closed-loop system cannot be established in general, and neglected dynamics may result in destabilization due to the observer spillover [3]. In the late-lumping approach, on the other hand, the designer takes full advantage of the physical nature of the process and uses available DPSs theory to design an observer for the PDE model, then the resulting observer is lumped for implementation. The works of [13], [8], [25], [22], [4], [5], [23] are notable in the generalization of the finite-dimensional systems observer theory to infinite-dimensional systems, more specifically the study of observability and detectability properties of DPSs. Along this line, several observer design methods are suggested in the literature including an extension of the Luenberger-type observer to PDE systems [7] and Lyapunov-based methods [20].
Another state estimation approach introduced by Smyshlyaev and Krstic [26] is the use of backstepping concept for boundary observation of PDE systems. In this methodology, an invertible Volterra integral transformation is used to transform the estimation error dynamics into a suitably selected stable distributed-parameter target system. The kernel of this transformation is defined by the solution of the so-called kernel PDE that is of a higher-order in space in the form of Klein–Gordon equation [19]. Having the solution of the kernel PDE, the observer gains can be found to be realized in the state estimator. From theoretical point of view, the technique of the PDE backstepping observer design is extended to the state estimation of unstable hyperbolic equations [18], compensation of sensor dynamics and/or PDE-ODE cascades [17], [16], [2], state estimation and output-feedback control for coupled PDE-ODE systems [27], linear parabolic PDEs with spatially- and time-varying reaction parameters [11], parabolic PDEs with nonlinear reactive-convective terms [12] and ultimately, observer design to semilinear parabolic PDEs [21].
In the case of a moving boundary parabolic PDE, even if the process parameters are time-invariant, the system is inherently nonautonomous [14]. Considering such problems as infinite-dimensional systems, direct state estimation is not possible in a large number of cases, because analytic expressions for the two-parameter semigroups describing the nonautonomous behavior of the system cannot be found. As an early-lumping approach, Galerkin׳s method is used for an eigenfunctions-based observer design in [1] for the boundary control of a 2D heat equation with time-dependent spatial domain.
Our recent work on the state-feedback boundary control of parabolic PDEs with time-varying domain using backstepping approach [9] motivates the observer design for the parabolic PDEs with time-varying domain. In this paper, the formulation to the PDE backstepping observer design for an unstable 1D heat equation described on a domain with moving boundaries is presented. The PDE system that governs the observation error dynamics is transformed to an exponentially stable target system through the Volterra-type integral transformation to obtain the kernel PDE, which has time-dependent parameters and is defined on the 2D time-varying domain. Then, the separation principle is validated, which provides exponential stability of the observer-based output-feedback controller setup. Finally, numerical solutions to the kernel PDEs and various simulations are given to demonstrate successful stabilization of an unstable system.
Section snippets
Problem statement
Consider the linear 1D parabolic PDE system of the form:where is the state variable, is the time-varying domain of the definition of PDE, is the spatial coordinate, is the time-dependent length of the domain and is its time derivative, and is the time. α and λ0 are process parameters and the advection term appearing in (1) is due to the moving boundaries of the PDE domain [6], [10]. The
Backstepping state-feedback controller
The backstepping state-feedback results in [9] are frequently used in this work and they are briefly given in this section for easy reference.
The following state-feedback control law stabilizes the PDE system (5), (6):where the control gain function is the kernel of the Volterra transformation:that transforms (5), (6) to the following exponentially stable (in
Observer design
In the following, a distributed-parameter Luenberger-type observer is designed by modifying the PDE system (5), (6) with corrector terms aswhere and are observer gain functions to be designed. The observer error satisfies the following PDE:
Output-feedback
The development of the exponentially convergent state estimator in previous section is independent of the control input. In this section, the observer is combined with the backstepping control to explore the output-feedback controller and establish separation principle, i.e., the incorporation of a separately designed state-feedback controller and observer results in a stabilizing output-feedback controller for the possibly unstable PDE with time-varying domain.
Consider the PDE plant (5), (6)
Numerical solution to the kernel PDEs
To apply the control (26), one should know control and observer gains given in terms of kernels and that are described by associated PDEs (13), (14) and (22), (23), respectively. However, finding the solution to these equations on the time-dependent domains and is not straightforward. In this section, these PDEs are transformed to fixed spatial domains and then they are solved numerically.
Consider the space given in Fig. 1c. The
Simulation results
The given approach to output-feedback control of the PDE system (5), (6) is simulated and some results are shown in this section. Particularly, system parameters are and and the change in the domain h(t) is depicted in Fig. 3. The numerical discretization for the following simulations is obtained for N=80. The set of ODEs (41) is numerically realized using a first-order implicit integration scheme with .
Fig. 4 shows the numerical approximation of the time-varying observer
Conclusions
The observer design of a 1D unstable heat equation on the time-varying domain is formulated in this work, where the observer gains are determined by the use of backstepping methodology. This includes a Volterra integral transformation to transform the estimation error PDE to a prescribed exponentially stable target system. The kernel function of this transformation is described by the 2D time-varying PDE on the moving boundary domain. Then, the designed observer is incorporated with the
Acknowledgments
Financial support by Natural Science and Engineering Research Council of Canada (NSERC) Discovery Grant 386508-2011 is gratefully acknowledged.
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