IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 7, JULY 2005 1063 Fault Detection in a Mixed Setting M. J. Khosrowjerdi, R. Nikoukhah, and N. Safari-Shad Abstract—In this note, we study the problem of fault detection in linear time-invariant (LTI) systems which are simultaneously effected by two classes of unknown inputs: Noises having fixed spectral densities and unknown finite energy disturbances. This problem is formulated as a mixed filtering problem. Necessary and sufficient conditions for local optimality are presented. Moreover, it is shown that suboptimal solutionscanbecomputedbysolvingaconvexminimizationproblemwith a set of linear matrix inequality (LMI) constrains. A numerical example is given to illustrate the advantage of the mixed approach as compared to existing techniques which are based on optimization of and criteria. Index Terms—Linear matrix inequalities (LMIs), mixed fil- tering, robust fault detection. I. INTRODUCTION Safety and reliability are of paramount importance in control sys- tems. To assure reasonable measures of safety and reliability, the need for fault detection techniques have long been recognized; see [1] and [2] for an extensive bibliography and review of the literature. Typically fault detection schemes are concern with construction of a dynamical system called a residual generator. This auxiliary system takes the known input and output of a system and generates a signal called the residual. This signal is then processed to decide whether or not a fault has occurred in the system. Since systems are often subject to unknown inputs, it is highly desirable to minimize their effect on the residual generation. To do this without any apriori assumptions on the unknown inputs requires perfect decoupling between the unknown in- puts and the residual. This has been the focus of research in [3]–[6]. Alternatively, when unknown inputs are assumed to have fixed spectral densities (bounded energy), the residual generation problem can be for- mulated as an optimal filtering problem; see, e.g., [7]–[11]. It is well known that -norm cannot guarantee any reliable degree of robustness. On the other hand, while the use of -norm in detec- tion problems can reduce noise sensitivity, it can also lead to reduction of sensitivity to faults; see [10] and [11]. In general, unknown inputs cannot be solely modeled by those which have fixed spectral densities or those which have bounded energy. In fact, in many situations, unknown inputs may include both types. This view has been considered in [12] and [13] to develop a mixed optimal filtering problem. Adopting this point of view in the context of fault detection, we formulate a mixed residual generation problem. The advantage of such an approach as will be shown in the note is threefold: 1) By adjusting a single design parameter, it becomes possible to trade off between detection performance and noise sensi- tivity; 2) when this parameter is chosen sufficiently small, an almost perfect decoupling residual generator is obtained which has been the goal in, e.g., [3]–[6]; and 3) when this parameter is allowed to approach infinity, the classical Kalman filtering is obtained. Manuscript received April 16, 2004; revised January 31, 2005 and April 4, 2005. Recommended by Associate Editor Hong Wang. M. J. Khosrowjerdi is with the Department of Electrical Engineering, Sahand University of Technology, Tabriz, Iran (e-mail: khosrowjerdi@sut.ac.ir). R. Nikoukhah is with the INRIA, Rocquencourt BP 105, 78153 Le Chesnay Cedex, France (e-mail: ramine.nikoukhah@inria.fr). N. Safari-Shad is with the University of Wisconsin-Platteville, Platteville, WI 53818 USA (e-mail: safarisn@uwplatt.edu). Digital Object Identifier 10.1109/TAC.2005.851464 After restricting our class of residual generators to full-order Kalman–Luenberger filters, a mixed minimization problem is formulated where the optimal filter gain is sought via minimization of a certain upper bound of the mixed cost. Using the Lagrange multiplier formalism, two coupled Riccati equations are derived as the necessary and sufficient conditions for filter gain local optimality. However, the complexity of these conditions, and results reported in [12] and [13], suggest considering a closely related convex minimization problem with LMI constrains. This yields suboptimal solutions which can be easily computed using powerful packages for solving such problems, [14]. II. NOTATION The notation used in this note is fairly standard. For a given matrix , and denote its transpose and trace, respectively. If and are symmetric matrices, (respectively, ) denotes positive–semidefinite(respectively,positive–definite). denotes the largest singular value of . Given real matrices , and , we say that is the stabilizing solution of the algebraic Riccati equation (ARE) if is real, symmetric, and is Hurwitz. For simplicity of nota- tion, a transfer matrix is represented in terms of state–space data by . The space of real rational stable and proper transfer matrices is denoted . The notations , and denote, respectively, the , norms, and the space of square inte- grable functions [16]. III. PROBLEM FORMULATION We consider systems whose dynamics can be described by linear time-invariant (LTI) systems (1) Here, is the state, and are the known input and outputs. The unknown input is assumed to be a fixed spectral density process/measurement noise while the unknown input is assumed to be a finite energy disturbance modeling errors due to exogenous signals, linearization or parameter uncertainties. Moreover, the unknown input is a possible fault. With set to zero, system (1) describes the normal mode, i.e., fault-free system. , ’s, , and ’s are assumed to be known constant matrices of appropriate dimensions. All the weighting functions reflecting the knowledge of and are assumed to be absorbed into the system equations (1). Finally, it is assumed that is a detectable pair and has full row rank. As mentioned in the introduction, the objective of a fault detection design is to construct the residual generator which takes and and generates the residual signal to decide whether or not a fault has occurred in the system . In this setup, the fault has occurred if some norm of is larger than a prespecified threshold or there is a sudden change in its statistical properties. Remark 3.1: Note that the problem of fault location (isolation) can also be addressed within this framework. Indeed, by labeling the spe- cific fault to be located as and combining the remaining faults in the unknown disturbance , a dedicated residual generator can be designed. 0018-9286/$20.00 © 2005 IEEE