Linear matrix inequalities (LMIs) observer and controller design synthesis for parabolic PDE
Introduction
The transport-reaction processes with diffusive and dispersive mechanisms, modeled by the parabolic partial differential equations (PDE) are ubiquitous in many chemical, petrochemical and pharmaceutical industry plants. Different control strategies, such as linear [18], [5], robust control [4], backstepping [20], theory [10], and model predictive controller (MPC) [6], have been proposed and employed to stabilize such processes and satisfy various performance requirements.
In a large number of aforementioned studies a salient feature of a parabolic PDE, whose eigenspectrum of the spatial operator can be divided into finite-dimensional slow and infinite-dimensional stable fast modal spaces [4], has been realized as a theoretical foundation for the order reduction of the infinite-dimensional system to the finite one through Galerkin׳s method. This order reduction technique adequate for parabolic PDEs scheme converts the original PDE state into the modal state space representation and only a finite number of slow modal states are considered in the controller synthesis. Thereafter, those mature control techniques suitable for the finite-dimensional ODE system, including the Lyapunov based controller [7] and MPC [6], can be applied in the modal state space for stabilization or optimization purposes. Once a finite number of dominant modal states converge to zero, it is not difficult to show that the original process states are also stabilized.
A successful state feedback based control realization, such as MPC, usually requires the knowledge and information of the process states in real time. However, the nature of most physically relevant cases is such that it is not possible to have a full and reliable information of the system׳s state due to the fact that either all process variables cannot be measured or the real time calibration is not available. Furthermore, for the distributed parameter system (DPS), the measurement for the entire spatial profile, which is crucial to determine the process state status, cannot be acquired in most of cases. In such a situation, there is a need to design a state estimator (observer) to provide the feedback information to the controller and guide the output by an appropriate actuation signal. A basic and celebrated state estimator for linear lumped parameter systems is given by Luenberger [14], which has attracted considerable attention and become the basis for many state estimation methodologies for both linear and nonlinear systems, see [16] for nonlinear observers and [21] for sliding-mode observers. Such design procedure guarantees the existence of the observer gain to diminish the state estimation error provided that the system is observable. In addition, there are also some Luenberger style observers developed for DPS, involving early lumping [23] or late lumping techniques [18]. In particular, the observer design based on an early lumping approach leads to finite-dimensional observers in which models are reduced by using the Galerkin type of approximation and design may account for the influence of sensor locations, see [17], [11]. On contrary to the early lumping, the observer designs based on the late lumping approach are more complex for realization since the underlying operators and the associated functional spaces based description introduce technical difficulties in its realization [1].
The observer based controller synthesis has been studied for a while and achieved great success for the lumped parameter system in which state dynamics is described by a set of ordinary differential equations (ODE). The counterpart for the DPS, governed by the PDE, has not been investigated extensively and very well established due to the complexity of the infinite-dimensional system representation. The Galerkin model reduction method provides foundation to represent the original abstract state as separable partitions of the slow and fast modal sub-space. Since slow modal states dominate the process dynamics, most of research efforts only focus on the finite slow modal states estimation and controller synthesis, and ignore effects of fast modal states. As a matter of fact, the long term dynamics of fast modal states can indeed be neglected if the input keeps zero along the entire evolution [4]. However, the zero input rarely happens in practice especially when the plant is inherently unstable. Hence, those fast modal states are always excited by the external actuation arising from the inputs and show control spillovers if the regulator does not take them into account. Although only the control spillover will not lead to a closed-loop instability owing to the naturally stable features of the fast modal states, the observer spillover definitely introduces more hazards by omitting fast modal states in the observer design procedure as this spillover, similar to measurement disturbances, are amplified by the observer gain to affect the reconstruction of all states. This phenomenon has been realized and discussed [15], [2], [8], [9], in which various techniques, such as the residual observer, control Lyapunov function (CLF) and observer Lyapunov function (OLF), fictitious output observer, have been proposed to achieve the stabilization and observation of entire infinite-dimensional modal states.
In this paper, we present a linear matrix inequality (LMI) based approach to regulate the second order parabolic PDE in the presence of the input constraint. The spillover is suppressed by considering the bound of the fast modal evolution. It is well known that the LMI is an efficient method to deal with the disturbance rejection and model uncertainties [3], [12]. Hence, we adopt the LMI to search proper control and observer gain when bound of the fast modal output is represented by the linear difference inclusion (LDI). As a result, if the proposed LMI is feasible, the augmented state, consisting of the unstable modal state, estimation error and fast modal output bound, will converge to zero by using a proper input signal which also respects constraints. The remainder of the paper is organized as follows: the preliminaries section provides the information about model setting; an infinite-dimensional state space is developed based on the boundary transformation and its formulation is presented in Section 3; in Section 4, the LMI based formulation for the controller, observer and invariant set design is demonstrated and discussed. Finally, a simulation study for the second order parabolic PDE in Section 5 shows the performance of the proposed approach which is followed by the conclusion in Section 6.
Section snippets
Problem statement
We consider the model of an axial dispersion reactor given by the second order linear parabolic PDE within spatial domain [18]:with following boundary conditions:The output iswhere impulse function represents the point-wise measurement at location zc; and d are known system parameters. The initial condition of the process state for the entire profile is unknown, which requires to
Sturm–Liouville eigenvalue problem
We seek to formulate a parabolic PDE given by Eqs. (1), (2), (3) by representation in a well defined Hilbert space setting [5]. Namely, the second order PDE can be converted into the Sturm–Liouville (SL) form:One can define the system operator with its domain are absolutely continuous, and boundary operators:
Main result
The main contribution of this work is to design a state feedback controller in conjunction with the Luenberger modal observer to stabilize the process with input constraint and single measurement. Although the same topic has been well studied for the linear ODE system, the counterpart research for the infinite-dimensional system is non-trivial. The first and also the major challenge is due to the spillover of observations which introduces more disturbances to the state reconstruction, thereby
Simulation study
Let us consider the following parameters associated with the second order parabolic PDE given by Eqs. (1), (2), (3), , zc=0.5, and . We easily verify that the system is approximately observable and its dominant 10 eigenvalues and eigenvectors are plotted in Fig. 1, Fig. 2. According to Fig. 1, there are two unstable modal states, which should be stabilized by the proposed controller and the observer. As mentioned before, we specify the ,
Conclusion
In this work, a boundary observer based state feedback control scheme is proposed to regulate the process, modeled by the second order parabolic PDE. The original PDE system is converted by the exact state transformation into the infinite state space model and the reduction technique is employed to obtain a finite-dimensional model where the unstable modal state is captured for the aim of controller synthesis and observer design. The bound of the fast modal output is modeled by the LDI and
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