A transfer function approach to fault diagnosis for linear systems: inversion and low-pass filters Richard Marquez and Addison R´ ıos-Bol´ ıvar and Eduardo Teles 1 Abstract. The purpose of this paper is to propose a simple design method for fault detection and isolation (FDI) based on model inver- sion and low-pass filters. Motivated by practical situations, a tradeoff between desired fast responses of fault indicating signals and robust- ness in the presence of disturbances is considered. We point out plain frequency design rules and basic limitations of residual generation. Computations of inverse models are easy to implement by means of a symbolic algebraic language. Several illustrative examples show the performance of the proposed FDI scheme. 1 INTRODUCTION During last 20 years, Fault Detection and Isolation (FDI), or Fault Diagnosis (FD), has been a very attractive area for both Artificial In- telligence (AI) and Control research communities, see, for example, [10, 17, 12, 13, 9, 3] and [7, 2]. AI techniques and standard control methods interact today in order to produce reliable, robust FD sys- tems supporting industrial supervision, maintenance and decision- making tasks. Two complementary FD approaches are mainly considered, knowledge-based methods and model-based strategies. Physical rela- tions, which can be represented by mathematical models, determine the dynamical behavior of the process under study. AI or knowledge- based methods are useful in those cases where it is difficult to find an- alytic process models. Usually, process measurements feed “virtual instruments”, i.e. mathematical algorithms, called residuals. Statis- tical and AI tests permit to evaluate these residuals in order to ef- fectively infer malfunction conditions and apply corrective actions at the decision-making level. This is not our intention to make an account of virtues of the com- bination of AI and analytical methods. Instead, this note analyzes a simple method based on linear models where AI methods would be most useful from a modeling and identification point of view, see e.g. [11]. Fault detection and isolation (FDI) by inverse models and filters is well known, see, for example, [2, 19, 15, 18, 16, 5]. This note proposes a simple design method for FDI based on inversion of the fault model and the use of low-pass filters. Our approach relies on the algebraic formalism to FDI proposed by Fliess and Join in [5]. This work is organized as follows. Section 2 define parity equa- tions and inverse fault models. Parity equations (PE) which describes fault signals w are obtained from the (linear) system model under study, in terms of outputs y, inputs u, noise signals η, and initial con- ditions x0. Under appropriate conditions, PE can be easily computed 1 Departamento de Sistemas de Control, Universidad de Los Andes, M´ erida 5101, Venezuela email: {marquez,ilich}@ula.ve, telesed@hotmail.com by symbolic computations. Then, an inverse fault model is defined by zeroing η and x 0 . In Section 3, we construct residuals. As we will see, the addition of a suitable finite number of poles to this inverse model (in the form of a low-pass filter) results in strictly proper residuals. We show there exists a compromise when choosing the low-pass filter. We point out plain frequency design rules and basic limitations of residual genera- tion: these are given in terms of the transfer functions relating noises and initial conditions to residuals. Stable strictly proper transfer func- tions guarantee the effect of initial conditions is effectively attenu- ated. It is interesting to note our approach makes a relation between two analytical FD methods: inverse-model strategies [19, 16, 14, 5] and factorization approaches [4, 6]. In our opinion, this work clari- fies to some extend the design approach presented in [5], precisely that residuals may only depend on adding poles to the inverse model. Section 4 presents some plain design rules and basic limitations of the proposed residual generation. Some examples illustrating the performance of the proposed FDI method are presented in Section 5. Some conclusions and remarks on AI alternatives to this work finish the paper. 2 PARITY EQUATIONS AND INVERSE MODEL We refer the reader to [5] and their references. Here we only sketch the basic ideas. Consider the linear time-invariant system ˙ x = Ax + Bu + Ew , x(0) = x0 y = Cx + Du + F w + Hη (1) where the state vector is given by x = (x 1 ,...,x n ); y = (y 1 ,...,y p ) represents the outputs, u =(u 1 ,...,u m ) are con- trol inputs. Faults, noise signals, and initial conditions are given, resp., by w =(w 1 ,...,w q ) and η =(η 1 ,...,η r ), and x 0 = (x10,...,xn0). These quantities are not available to measure. By Laplace transform we obtain sx(s) - x0 = Ax(s)+ Bu(s)+ Ew(s) y(s)= Cx(s)+ Du(s)+ F w(s)+ Hη(s) (2) where now all quantities now depend on the complex variable s. A fault w i , i ∈ [1,...,q], is said to be detectable and isolable [5] if and only if we can find its associated parity equation pw i (s)wi = Q(s)y + R(s)u + N (s)η + T (s)x0 (3) where p w i (s) ∈ R[s] is a real polynomial on s, and Q(s) ∈ R[s] 1×p , R(s) ∈ R[s] 1×m , N (s) ∈ R[s] 1×r , T (s) ∈ R[s] 1×n are matrices of appropriate dimension. 105