Integration of fault diagnosis and control by finding a trade-off between the detectability of stochastic fault and economics Yuncheng Du, Hector Budman, Thomas Duever Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 (e-mail: hbudman@uwaterloo.ca, tom.duever@uwaterloo.ca) Abstract: This paper presents a first principle model based methodology for simultaneous optimal tuning of a fault detection algorithm and a feedback controller. The key idea is to calculate the effect of stochastic input disturbances on the variability of the output variables by using a generalized polynomial chaos (gPC) expansion and a mechanic model of the process. A two-level optimization is proposed for simultaneously tuning the fault detection and controller algorithms. The goal of the outer level optimization is to find a trade-off between the efficiency for detecting faults and the closed loop performance, while the inner optimization is designed to optimally calibrate the fault detection algorithm. The proposed method is illustrated for a continuous stirred tank reactor (CSTR). The results show that the computational cost of the gPC-based method is significantly lower than a Monte Carlo (MC) simulation- based approach, thus demonstrating the potential of the gPC method for dealing with large problems. Keywords: Fault detection and control, polynomial chaos, economic impact 1. INTRODUCTION Most fault detection and diagnosis (FDD) systems are implemented at the supervisory level on top of the control system. Thus, naturally fault detection approaches are based on variables, which are also used for feedback control. While there is a large body of literature on FDD (Zufiria, 2012, Dong, et al., 2012), the issue of integration of control and diagnostic algorithms has not been addressed as much. A key challenge for integrating FDD and process control is that they often have competing objectives. For instance, when process control is very accurate, the corresponding controlled variable deviates little from the setpoint while FDD requires sufficiently large deviations for effective detection (Davoodi, et al., 2013, Meng & Yang, 2012). Methods have been proposed for optimal simultaneous tuning of FDD and control based on robust norms (Jacobson & Nett, 1991, Scott, et al., 2013), but these are often conservative since they are based on worst-case scenarios or deterministic linear model. To avoid linearization and reduce conservatism, standard statistical monitoring charts have been used, but these studies were limited to simple deterministic faults (Bin Shams, et al., 2011). Compared with Monte Carlo (MC) simulations based uncertainties analysis, a power series and polynomial chaos expansions were used to propagate uncertainties onto states and outputs, which in general saves computational time (Nagy & Braatz, 2007). The current work addresses the problem of optimal simultaneous tuning of a FDD and controller in the presence of stochastic disturbances by using gPC expansions of inputs and outputs. A significant reduction in computational effort is observed by using the gPC method as compared with MC based-approach. The topics addressed in this work are as follow: (1) The tuning parameters of the closed loop controller and/or control setpoint are optimized to achieve an optimal trade-off between FDD efficacy and closed-loop performance. (2) The generalized polynomial chaos (gPC) expansion and Galerkin projections are used to quantify the variability in controlled and manipulated variables resulting from stochastic faults entering the system in the form of input disturbances. A non-isothermal continuous stirred tank reactor (CSTR) system is used as case study. (3) Numerical tests are conducted to verify the ability of the proposed method to design a fault detection/controller combination that is both efficient for detecting faults and for maintaining good closed loop performance. This paper is organized as follows. In section 2, the theoretical background on gPC is presented. The optimization problems formulated for simultaneously tuning the fault detection algorithm and the controller are given in section 3. An endothermic continuous stirred tank reactor (CSTR) is introduced as a case study in section 4. Analysis and discussion of the results are presented in section 5 followed by conclusions in section 6. 2. QUANTIFICATION OF VARIABILITY IN OUTPUTS IN RESPONSE TO RANDOM INPUTS USING gPC The objective is to quantify the effect of stochastic inputs on different output variables (states), which are described by a system of ordinary differential equations (ODEs). The generalized polynomial chaos (gPC) method (Xiu, 2010) has been proposed for approximating the states of a system with random inputs by polynomial descriptions from the Wiener- Askey family. A variable, x , is represented as a polynomial chaos expansion: Preprints of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 24-29, 2014 Copyright © 2014 IFAC 7388