Identification of Interconnected Systems by Instrumental Variables Method (Regular Paper) Grzegorz Mzyk Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-372 Wroclaw, Poland Phone: +48 71 320 32 77, E-mail: grzegorz.mzyk@pwr.wroc.pl Abstract—The paper addresses the problem of parameter esti- mation of elements in complex, interconnected systems. Similarity between causes of biases in the least squares estimates for a simple SISO linear dynamic object, and for a MIMO linear static system with composite structure, was noticed in the paper. For linear complex static system, the instrumental variable estimate was proposed and compared with the least squares approach. The strong consistency of the presented parameter estimate was proved. Also the optimal values of instrumental variables were established, and the method of their suboptimal generation was presented. The conclusions were verified in numerical experi- ments. CONTENTS I Introduction 1 II Statement of the Problem 1 III Least Squares Approach 2 IV Instrumental Variables Approach 3 V Simulation Example 4 VI Summary 4 VII Appendix 4 References 4 I. I NTRODUCTION We consider the problem of parameter estimation in com- plex, interconnected systems with the presence of random noises. In a lot of commonly met hierarchical control prob- lems, the accurate mathematical models of the particular system components are needed. Under the term ’complex’ we understand the fact that the system is built of a number of interconnected components (subsystems), e.g., in the typical production system each element is excited by the outputs of other blocks (see [1]). In consequence of mutual intercon- nections, the components are dependent and their separation may be impossible or too expensive. In general, excitations of particular element cannot be freely generated in the ex- periment. It leads to the problem of structural identifiability (i.e. identifiability of separate elements does not imply identi- fiability of the whole interconnected system [3]) and usually badly conditioned numerical tasks. Moreover, some interaction signals are hidden, and cannot be directly measured. For these reasons, the algorithms dedicated for single element cannot be directly applied in complex system analysis. Identifiability of the element, which operates in complex system, depends additionally on the system structure and the values of parameters of other elements. Particularly, the com- ponents preceding identified object must guarantee persistency of the input excitation. In the paper we apply and compare two methods – least squares (l.s.) and instrumental variables (i.v.). It is commonly known from the linear system theory, that the least squares approach applied for the simple SISO linear dynamic object leads to biased estimate. The reason of the bias results from the property of autoregression, i.e. the correlation between the noise and the values of previous outputs of the identified object (see the Appendix). Analogously, for the complex, interconnected systems with random noises, the least squares estimate has the non-zero systematic error even if the number of measurement data tends to infinity. The reason of the bias is that the output noises are transferred to the inputs through the structural feedback. In the paper, the formal similarity of these problems is shown and the instrumental variables technique, used so far for the linear dynamics identification, was successfully gener- alized for the systems with complex structure. It is shown that the proposed i.v. estimate is strongly consistent independently of the system structure and the color of the noise. Moreover, the computational complexity of the method is comparable with the l.s.algorithm. In Section II the identification problem and the purpose is formulated in detail. Next, in Section 3, the properties of the least squares based algorithm proposed in [3] are reminded. In particular, the reason of its bias is shown in detail, and finally, in Section IV the new i.v. estimate is introduced and analyzed. Finally, in Section V, the performance of the method is demonstrated by the simulation example. II. STATEMENT OF THE PROBLEM Consider the system shown in Fig. 1. It consists of n linear elements described as follows y i = a i x i + b i u i +  i (i =1; 2; :::; n),