(560c) Optimization-Based Sensor/Actuator Scheduling and Control of Sampled-Data Distributed Processes

Despite the extensive body of research work on control of spatially-distributed systems, the problem of designing sampled-data feedback controllers for spatially-distributed processes has received relatively limited attention (e.g., [2], [3],[7] for some notable exceptions). This is an important problem given the prevalent use of digital sensors and controllers in industrial control systems. To date, the overwhelming majority of studies on the analysis and design of sampled-data control systems have focused primarily on lumped parameter systems modeled by ordinary differential equations (e.g., see [1],[4],[5],[6],[8],[9]).

An effort to address the sampled-data control problem for spatially-distributed systems was undertaken in a series of earlier works (e.g., [10],[11]). The main idea was to include an approximate finite-dimensional model of the infinite-dimensional system in the controller to provide estimates of the dominant slow states that compensate for the unavailability of output measurements between sampling times, and to update the model states using the actual measurements at each sampling time. A key objective of this model-based control approach was to guarantee closed-loop stability with minimal sampling frequency. This is an appealing goal in situations where technological constraints on the measurement sensing techniques make it difficult or costly to collect measurements frequently. To this end, an exact characterization of the maximum allowable sampling period required for stabilization was obtained in the earlier studies and found to depend on the spatial placement of the measurement sensors and control actuators. An implication of this finding is that one could use this characterization to identify the set of feasible sensor and actuator locations that can achieve stabilization with minimal sampling frequencies.

While this is important from a sampling cost savings standpoint, the resulting actuator and sensor placement does not consider the impact of discrete measurement sampling on closed-loop performance. It is well known, for example, that faster sampling rates are typically required to enhance the control system performance. Ultimately, there is a need to resolve the inherent conflict that arises between the tight restrictions on sampling rates which are required to maintain the control system performance on the one hand, and the usual demand for limited sampling frequency which is desired to minimize measurement costs.

Motivated by these considerations, we present in this work an optimization-based methodology for the placement and scheduling of measurement sensors and control actuators in spatially-distributed processes with low-order dynamics and discretely-sampled output measurements. Initially, a sampled-data observer-based controller, with an inter-sample model predictor, is designed based on an approximate finite-dimensional system that captures the infinite-dimensional system's dominant dynamics. An explicit characterization of the interdependence between the stabilizing locations of the sensors and actuators and the maximum allowable sampling period is obtained. Based on this characterization, a constrained finite-horizon optimization problem is formulated to obtain the sensor and actuator locations, together the corresponding sampling period, that optimally balance the trade-off between the control performance requirements on the one hand, and the demand for reduced sampling, on the other. The objective function penalizes both the control performance cost, expressed in terms of the response speed and the control effort, and the sampling cost, expressed in terms of the sampling frequency. The optimization problem is solved in a receding horizon fashion, leading to a dynamic policy that varies the sensor and actuator spatial placement, together with the sampling period, over time. The developed methodology is illustrated through an application to a simulated diffusion-reaction process example.

References:

[1] Chen, T. and Francis, B. (1995). Optimal Sampled-Data Control Systems. Springer, London; New York.

[2] Cheng, M., Radisavljevic, V., Chang, C., Lin, C., and Su, W. (2009). A sampled-data singularly perturbed boundary control for a heat conduction system with non-collocated observation. IEEE Trans. Automat. Contr., 54, 1305-1310.

[3] Fridman, E. and Blighovsky, A. (2010). Sampled-data stabilization of a class of parabolic systems. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems. Budapest, Hungary.

[4] Fujioka, H. (2009). A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices. IEEE Trans. Automat. Contr., 54, 2440-2445.

[5] Hu, L., Bai, T., Shi, P., and Wu, Z. (2007). Sampled-data control of networked linear control systems. Automatica, 43, 903-911.

[6] Hu, Y. and El-Farra, N.H. (2011). Control, monitoring and reconfiguration of sampled-data hybrid process systems with actuator faults. In Proceedings of 50th IEEE Conference on Decision and Control, 736-3741. Orlando, FL.

[7] Logemann, H., Rebarber, R., and Townley, S. (2005). Generalized sampled-data stabilization of well-posed linear infinite-dimensional systems. SIAM J. Cont. Optim., 44, 1345-1369.

[8] Naghshtabrizi, P., Hespanha, J., and Teel, A. (2008). Exponential stability of impulsive systems with application to uncertain sampled-data systems. Syst. Contr. Lett., 57, 378-385.

[9] Nesic, D. and Grune, L. (2005). Lyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation. Automatica, 41, 1143-1156.

[10] Yao, Z. and El-Farra, N.H. (2011). Robust fault detection and reconfigurable control of uncertain sampled-data distributed processes. In Proceedings of 50th IEEE Conference on Decision and Control, 4925-4930. Orlando, FL.

[11] Yao, Z. and El-Farra, N.H. (2012). A predictor-corrector approach for multi-rate sampled-data control of spatially distributed systems. In Proceedings of 51st IEEE Conference on Decision and Control, 2908-2913. Maui, HI.