Identification of Stochastic Systems Under Multiple Operating Conditions: The Vector Dependent FP–ARX Parametrization Fotis P. Kopsaftopoulos and Spilios D. Fassois † Abstract— The problem of identifying stochastic systems under multiple operating conditions, by using excitation – re- sponse signals obtained from each condition, is addressed. Each operating condition is characterized by several measurable variables forming a vector operating parameter. The problem is tackled within a novel framework consisting of postulated Vector dependent Functionally Pooled ARX (VFP–ARX) mod- els, proper data pooling techniques, and statistical parameter estimation. Least Squares (LS) and Maximum Likelihood (ML) estimation methods are developed. Their strong consistency is established and their performance characteristics are assessed via a Monte Carlo study. I. INTRODUCTION In conventional system identification a mathematical model representing a system at a specific operating condition is identified based upon a single data record of excitation – response signals. Yet, in many applications, a system may operate under different operating conditions in differ- ent intervals of time, maintaining one such condition in each interval. These operating conditions affect the system characteristics, and thus its dynamics. Typical examples include physiological systems under different environmental conditions, mechanical systems under different load or lu- brication conditions, systems under different configurations, hydraulic systems operating under different temperatures or fluid pressures, material and structures (civil–mechanical– aerospace) under different environmental conditions (such as temperature and humidity), and so on. In such cases it is of interest to identify a “global” model describing the system under any operating condition, based upon excitation – response data records available from each condition. 14th Mediterranean Conference on Control and Automation (MED 2006) Ancona, Italy, June 2006 It could be, perhaps, argued that this may be handled by using conventional mathematical models and customary identification techniques that could artificially split the prob- lem into a number of seemingly unrelated subproblems and derive a model based upon a single data record at a time. Nevertheless, such a solution would be both awkward and statistically suboptimal. Awkwardness has to do with the fact that a potentially large number of seemingly unrelated models (one per operating condition) would be obtained. Statistical suboptimality has to do with the fact that the set of F.P. Kopsaftopoulos and S.D. Fassois are with the Stochastic Mechanical Systems & Automation (SMSA) Group, Department of Mechanical & Aeronautical Engineering, University of Patras, GR 26500, Patras, Greece. Tel/fax: +30 2610 997 405. E-mail: {fkopsaf,fassois}@mech.upatras.gr, Internet: http://www.mech.upatras.gr/∼sms †Corresponding author. identified models would be of suboptimal accuracy. This is due to two reasons. The first is the violation of the principle of statistical parsimony (model economy) as a large number of models would be used for representing the system. This would result in a large number of estimated parameters, and thus reduced accuracy. The second is the ineffective use of the information available in the totality of the data records. Indeed, not all available information would be extracted, as the interrelations among the different records would be ignored as a result of separating the problem into seemingly unrelated subproblems. This work aims at the postulation of a proper framework and methods for effectively tackling the problem of identify- ing stochastic systems under multiple operating conditions. This is to be based upon three important entities: (a) A novel, Functionally Pooled (FP), stochastic model structure that explicitly allows for system modelling under multiple operating conditions via a single mathematical representation. This repre- sentation uses parameters that functionally depend upon the operating condition. It also uses a stochas- tic structure that accounts for the statistical depen- dencies among the different data records. (b) Data pooling techniques (see [1]) for combining and optimally treating (as one entity) the data obtained from the various experiments. (c) Statistical techniques for model estimation. The resulting framework is referred to as a statistical Func- tional Pooling framework, and the corresponding models as stochastic Functionally Pooled (FP) models. A schematic representation is provided in Fig. 1. The only essential practical condition for using this frame- work and identifying “global” system models is that each operating condition corresponds to a specific value of a measurable variable, henceforth referred to as the operating parameter. The case of a scalar operating parameter (for instance operating temperature) is treated in a companion paper [2]. The present paper focuses on the case of a vector operating parameter (consisting of two or more scalars, for instance operating temperature and humidity). It should be also noted that early versions of the Functional Pooling framework, including certain simple models and estimation methods, have been already applied to practical fault diagnosis problems with very promising results. The interested reader is referred to [3], [4] for details.