It is well known that in practical situations the observed input-output data of an identified plant are usually corrupted by measurement noise. In this case the ordinary least-squares estimator of the system parameters is biased. In order to obtain a consistent estimator, a new type of modified least-squares estimation method, which is called the bias-eliminated least-squares (BELS) method, is presented in this paper. It is shown that the estimation bias can be determined if the variance of the white measurement noise can be obtained accurately. A designed first-order prefilter is connected in parallel to the input of the identified system. Based on asymptotic analysis, the noise variance can be estimated correctly by using the processed sampled data. Both a batch algorithm and a recursive algorithm are presented. It is shown that the presented BELS method gives a consistent estimate without a priori knowledge of the variance of the white input and output noise. Simulation results are presented to support the theoretical discussions.
KEY WORDS Identification Parameter estimation Least-squares method Estimation techniques
Errors in variables
During. recent years, many different identification methods have been proposed (see e.g. References 1-3). However, most methods are based on the assumption that the inputs of the system are noise-free. This statement is not always true because of measurement errors in most practical situations. As pointed out by S o d e r ~t r o m , ~ it can be shown that neglecting input noise, if present, leads to error-prone results. This is especially true if the intention is t o identify the physics of the process, which describes the behaviour of the system with respect to the true input of the system. Moreover, it has also been proved that in the presence of input and output noise, many currently used identification methods fail to compute asymptotic unbiased or consistent parameter estimators (see e.g. References 5 and 6). That is why estimation of system parameters by using noise-corrupted input-output data is of fundamental importance from both theoretical and practical points of view.
However, only a limited number of identification methods for systems with noisy input and output measurements have been developed. Among them, two methods, namely the jointoutput (JO) method4 and the Koopmans-Levin (KL) method, ' have received much more attention recently. The JO method can be expected to give good accuracy of the parameter
This paper was recommended for publication by editor A . Benveniste