Assessment and Future Directions of Nonlinear Model Predictive ...

516 A. Deshpande, S.C. Patwardhan, and S. Narasimhandetection under steady-state conditions [5], we propose a version of nonlinearGLR method under dynamic operating conditions. By this approach, once theFCT confirms the occurrence of a fault at instant t, we formulate a separateEKF over a time window [t, t + N] for each hypothesized fault. For example, assumingthat actuator j has failed at instant t, the process behavior over window[t, t + N] can be described as followsx mj (i +1)=x mj (i)+(i+1)T∫iTF [ x mj (t),u mj (i),p,d ] dt (5)y mj (i) =H [ x mj (i) ] + v(k) (6)where u mj (i) is given by equation 3. The magnitude estimation problem cannow be formulated as a nonlinear optimization problem as followst+Nmin ∑(Ψ mj )= γm T b j(i)V mj (i) −1 γ mj (i) (7)mji=twhere γ mj (i) andV mj (i) are the innovations and the innovations covariance matrices,respectively, generated by the EKF constructed using equations 5 and 6with initial state ̂x(t|t). The estimates of fault magnitude can be generated foreach hypothesized fault in this manner. The fault isolation is viewed as a problemof finding the observer that best explains the output behavior observed over thewindow. Thus, the fault that corresponds to minimum value of the objective function,Ψ fj , with respect to f j ,wheref ∈ (p, d, y, u, m, s) represents the fault type,is taken as the fault that has occurred at instant t. Since the above method is computationallyexpensive, we use a simplified version of nonlinear GLR proposed byVijaybaskar, [6] for fault isolation. This method makes use of the recurrence relationshipsfor signature matrices derived under linear GLR framework [4], whichcapture the effect of faults on state estimation error and innovation sequence. Ifa fault of magnitude b fj occurs at time t, the expected values of the innovationsgenerated by the normal EKF at any subsequent time are approximated asE [γ(i)] = b fj G f (i; t)e fj + g fi∀i ≽ t (8)Here, G f (i; t) andg fj (i; t) represent fault signature matrix and fault signaturevector, respectively, which depend on type, location and time of occurrence ofa fault. For example, if j th actuator fails, then the corresponding signature matricesand the signature vectors can be computed using the following recurrencerelations for i ∈ [t, t + N]:G m (i; t) =C(i)Γ u (i) − C(i)Φ(i)J m (i − 1; t) (9)[ ]g mj(i; t) =C(i)Γ u (i) e T m jm(i) e mj − C(i)Φ(i)j mj(i − 1; t) (10)J m(i; t) =Φ(i)J m(i − 1; t)+L(i)G m (i − 1; t) − Γ u (i) (11)[ ]j mj (i; t) =Φ(i)j mj (i − 1; t)+L(i)g mj(i − 1; t) − Γ u (i) e T m jm(i) e mj (12)