IEEE Conference on Intelligent Control, Limassol, Cyprus, June 2005
Residual Generation from Principal Component Models for Fault Diagnosis in Linear Systems Part I: Review of static systems Janos Gertler, Fellow, IEEE1 Abstract. First some fundamental concepts of principal component analysis are summarized. This is followed by the review of our recently developed and published results on the generation of enhanced (structured) residuals from PC models, both by algebraic transformation and by direct structured modeling. Recent results on residual optimization and extensions to dynamic systems are discussed in Part II of the paper. 1. INTRODUCTION In model-based fault detection and diagnosis, the observed plant variables are checked against a mathematical model of the system. This concept is referred to as analytical redundancy. Any discrepancy is expressed as residuals; the analysis of these residuals leads then to decisions concerning the presence and location of faults. The mathematical model may arise from a first principle description of the system, or from systems identification using empirical data. The two main approaches within the analytical redundancy framework are consistency (parity) relations [1,2] and diagnostic observers [3]. Consistency relations are suitably rearranged equations of the plant’s inputoutput model. These are subjected to transformations to yield residuals enhanced for fault isolation; such residuals possess structured or directional properties with respect to the various faults [2]. In the parity-space formulation of the same fundamental idea, residuals are obtained as projections of the observation vector onto “parity vectors”, orthogonal to the space of fault-free variables [4]. While the space spanned by the parity vectors (the parity space) is defined by the system, with the appropriate choice within this space the desired residual properties may be achieved. Principal component analysis (PCA) [5] has been applied successfully in the supervision of complex chemical process systems. By revealing linear relations among the process variables, PCA allows the representation of large systems with models of significantly reduced dimensionality. PCA has also been suggested as a tool for the identification of faulty sensors, by reconstruction via iterative substitution and optimization [6]. ….. 1 Janos Gertler is with the School of Information Technology, George Mason University, Fairfax, VA 22030, USA; (703) 993-1604;
[email protected]
As it has been shown recently [7,8], principal component models can also serve as a basis for analytical redundancy type fault detection and diagnosis. The PC directions (eigenvectors) associated with the last principal components are, in fact, parity vectors, and the space they span (the residual space of the PC model) is the parity space. Thus the residual enhancement techniques developed in the analytical redundancy framework may be applied to PC models as well. Structured residuals are powerful tools in fault isolation [2]. In structured design, each residual responds to a specific subset of faults, and thus a fault-specific subset of the residuals responds to each fault, resulting in a unique fault code. Structured residuals may be obtained from consistency relations, or from a full PC model, by algebraic transformation. But they can also be generated by first specifying residual structures and then obtaining subsystem models, each corresponding to a residual, by systems identification or by PC modeling [7]. In this paper, we first summarize some fundamental concepts of principal component analysis. This is followed by the review of our recently developed and published results on the generation of enhanced (structured) residuals from PC models, both by algebraic transformation and by direct structured modeling. Recent results on residual optimization and extensions to dynamic systems are discussed in the second part of the paper. 2. PRINCIPAL COMPONENT MODELS 2.1 Full PC models Consider a linear static (steady-state) system with n observed (measured or manipulated) variables x°=[x°1 ... x°n]’ and k unobserved disturbances z=[z1 ... zk]’. The variables and the disturbances are centered (their mean value is deducted) and normalized (divided with their standard deviation). The superscript ° signifies that the observations are free from faults. Assume that there are m linear relations among the variables, so that B x°(t) + D z(t) + v(t) = 0
(1)
where B and D are m.n and m.k matrices and v=[v1 ... vm]’ represents the effect of noise in the equations. In
many cases, it is meaningful to decompose x° into outputs y°=[y°1 ... y°m]’=[x°1 ... x°m]’ and inputs u°=[u°1 ... u°n-m]’ =[x°m+1 ... x°n]’; then (1) may be written as y°(t) = A u°(t) + D z(t) + v(t)
(2)
where A is an m(n-m) matrix and B = [-I A]. If k
