Optimal unknown input distribution matrix selection in robust fault diagnosis

Uncertainties in dynamic systems are an inevitable consequence of non-linearity and complexity, and obscure the performance of fault diagnosis. In order to achieve robust and reliable fault diagnosis, the unknown input (disturbance) de-coupling principle has been employed in recent research. In this paper, a method of computing the unknown input distribution matrix is proposed as a powerful alternative method to either re-identification of plant parameters arising from different operating points or to the use of non-linear residual generation. The determination of a suitable unknown input distribution matrix to achieve disturbance de-coupling is described as an optimization problem which is solved here via a Singular Value Decomposition. An example of robust fault detection applied to a jet engine system is included as an illustration. INTRODUCTION Safety is an increasingly important issue in modern process plant and aerospace systems. To ensure reliable operation, a fault monitoring and diagnosis scheme needs to be available. The fault diagnosis (mainly including fault detection & isolation (FDI) tasks) approaches of today are either based on hardware redundancy, realized by additional physical equipment or on analytical redundancy (AR), mostly implemented as computer software on process supervision computers. The AR approach has gained increasing consideration world-wide (Willsky, 1976; Patton, Frank & Clark, 1989; Frank, 1990; Patton & Chen, 1991a). Research is still under way into the development of more reliable and robust methods for achieving effective FDI. Certainly, the reliability of a FDI scheme must be higher than the monitored system and, for some cases (eg. for uncertain systems) this is difficult to achieve. The main problem obstructing the progress and improvement in reliability of FDI schemes is the robustness with respect to uncertainty which arises for example, due to process noise, turbulence, parameter variations and modelling error. All AR approaches to FDI employ a model of the monitored system. If the model is accurate and the characteristics of all the disturbances are known, FDI is very straightforward and robust solutions are trivial. Uncertainties are inevitably present and may interfere seriously with the FDI performance. AR will be a practically viable alternative to the use of hardware & software replication in safety-critical systems if the robustness with respect to uncertainty can be demonstrated in a simple & certifiable manner. To design robust FDI schemes, we need the description of uncertainties acting upon the system during typical process operations. Furthermore, it is necessary to find a description of these uncertainties which can be handled in a straightforward and systematic manner. A typical uncertainty description makes use of the concept of “unknown inputs” acting upon a nominal linear model of the system. All uncertainties of the system are summarised as unknown inputs (disturbances) acting on the system and their effect can be considered as bounded or unbounded and structured or unstructured. Based on this description, one can use the “unknown input observer (UIO)” to estimate the state (e.g. Yang & Richard, 1988). Watanabe & Himmelblau(1982) and Frank & Wünnenberg (1989) used the UIO approach to robust FDI. In this scheme, the unknown input does not affect the residual so that robust FDI is achievable. Patton et al (1987, 1989, 1991b) have shown that an approach to solving this problem using assignment of suitable eigenvectors and eigenvalues (eigenstructure) as a way of providing robustness through disturbance de-coupling. By assigning the suitable eigenstructure to an observer, the residual signal can be completely de-coupled from the disturbance. In this way, robust FDI is achievable. Gertler (1991) has shown that the two approaches can be used to give identical designs. The most important contributions to robust FDI make direct use of the disturbance de-coupling principle. An observer can be designed to be robust in the sense of disturbance de-coupling; the robustness will ensure that the residual is insensitive to disturbances and modelling errors. For all disturbance de-coupling methods, a necessary assumption is that the unknown input distribution matrix must be known, but within the framework of international research on this subject, the approaches to obtain this distribution matrix have been lacking. This shortage obstructs the application of the disturbance de-coupling approach for robust FDI in real engineering systems. Here, a method of computing this matrix is proposed. In order to achieve the de-coupling condition, an optimal approximation to the distribution matrix is used. For complex non-linear systems, the operating point changes according to working conditions, and different operating points correspond to different unknown input distribution matrices. The paper gives a method that uses an optimal matrix to represent the changing structure of the disturbances over the practical plant operation. A 17th order jet engine system model is used to illustrate the method proposed in this paper. PROBLEM STATEMENT Consider a real system subject to parameter variations, disturbances, etc: ẋt  ẋ0 ẋu  A0  ΔA Ac0 Au1 Au2 x0t xut  B0  ΔB Bu ut  G0 Gu t yt  C0 0 x0t xut  D0ut  fst #