(613a) Output Feedback Control of Transport-Reaction Processes Based On Adaptive Model Reduction in the Presence of Sensor Communication Constraints

In recent years research in the process control area has focused on the development of control and state estimator structures for complex transport-reaction processes. The control and observer problems for these processes are nontrivial due to the spatially distributed nature of the associated system dynamics. Such distributed systems can be mathematically described by parabolic partial differential equations (PDEs). A wellestablished approach to address the control and state estimator problem is via model reduction [4]. For industrial processes of current interest the issue of nonlinear spatial dynamics and complex domains limits the applicability of analytical model reduction methods. Statistical techniques circumvent this limitation via off-line construction of the required basis functions [1].

A common approach to compute the reduced order model (ROM) is via a combination of a variant of proper orthogonal decomposition (POD) with the method of weighted residuals. The performance of the controller structure hinges on the local validity of the ROM. To have a valid ROM it is necessary the basis functions are correct. There are two critical issues when using POD for control. First, the POD approach assumes an a priori availability of a sufficiently large ensemble of PDE solution data in which the most important spatial modes have been excited to produce valid basis functions. Second, as process evolves the previously identified basis functions may become unimportant.

A possible solution to circumvent this limitation is to continue augmenting the ensemble of snapshots, and then to recompute the eigenfunctions as more information regarding the process becomes available. However, this would require the solution of an eigenvalue-eigenvector problem every time, which may become computationally expensive, and hence, unsuitable for online computations as the process evolves. Our research efforts have focused on the recursive computation of empirical eigenfunctions as additional data from the process becomes available. The resulting recursive computation of eigenfunctions method, known as adaptive proper orthogonal decomposition (APOD) is used as additional data from the process becomes available during process operation. APOD was used for dynamic observer design and output feedback control of distributed parameter systems [2, 3, 5].

One of the prerequisites of APOD is obtaining snapshots frequently enough. In this paper, we will address the question how infrequent can this measurements be. To achieve our goals we will use a similar approach to the networked Lyapunov-based feedback controller designs for distributed parameter systems presented in [6]. The controller was synthesized for distributed parameter systems the infinite dimensional representation of which can be decomposed to finite dimensional slow and infinitedimensional fast subsystems. The slow subsystem model is included in the controller structure to reduce the frequency of sensor measurements over the network when communication is suspended. To determine when communication must be reestablished, the Lyapunov function evolution is monitored.

The main objective is to identify a criteria for minimizing communication bandwidth (snapshots transfer rate) from the periodic measurement sensors to the controller considering closed-loop stability. To determine the smallest frequency at which the ROM must be updated, the Lyapunov function evolution is monitored. When the value of Lyapunov function begins to violate stability threshold, the ROM will be updated using the snapshots from the periodic distributed measurement sensors and the updating procedure is then continued for as long as the value of Lyapunov function satisfies the stability criteria. The proposed approach is successfully used to control the temperature and concentration of a tubular chemical reactor.

[1] A. Armaou and P. D. Christofides. Finite-dimensional control of nonlinear parabolic PDE systems with time-dependent spatial domains using empirical eigenfunctions. Int. J. Appl. Math. & Comp. Sci., 11:287–317, 2001.

[2] D. Babaei Pourkargar and A. Armaou. Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients. AIChE J., accepted, 2013.

[3] D. Babaei Pourkargar and A. Armaou. Output feedback control of distributed parameter systems using APOD based dynamic observer designs. Automatica, submitted, 2013.

[4] P. D. Christofides. Nonlinear and robust control of PDE systems. Birkh ¨ auser, New York, 2000.

[5] S. Pitchaiah and A. Armaou. Output feedback control of dissipative PDE systems with partial sensor information based on adaptive model reduction. AICHE J., 59(3):747–760, 2013.

[6] Z. Yao and N. H. El-Farra. Networked control of specially distributed processes using an adaptive communication policy. In Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, GA, 2010.