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Top1. Introduction
The system with error-in-variable refers to the system whose input and output are destroyed by external interference noise, which is usually abbreviated as EIV (error-in-variable). At present, due to various reasons, only the case of output error and open-loop variable error are considered in the study of control system. However, in real life, the vast majority of systems are closed-loop variable with error system, so the study of closed-loop variable with error system is very meaningful. Due to the limitations of current technology, when a small part is combined to form a complete system, there will always be a variety of interference. Therefore, it is necessary to consider these interference factors in the equipment with high precision requirements. Especially in many high-precision instruments, such as high-precision medical and surgical equipment, aircraft, etc., even a little error will cause difficult to estimate the consequences. In this paper, a closed-loop variable with error model is proposed to study the case that the input and output are damaged by noise. After considering the input and output errors, the designed system will be more accurate. This paper presents a closed-loop variable error system model, designs and studies its controller, and applies it to the position control of a quadrotor UAV.
Professor Torsten SöDerströM's team at the Uppsala University has made a significant contribution to the identification of error models for dynamic variables. The book Errors-in-Varibles Methods in System Identification, published in 2018, summarizes the findings of Professor Torsten S öderströ in the field from years of research, the book examines “The in-depth and extensive analysis of EIV modeling for Linear dynamical system, the process of identification, model selection and (parameter) estimation, and its statistical properties”. Professor Torsten Söderström won the IFAC TC 1.1 Prize for 2021 Systems Recognition for this book, in recognition of the outstanding contributions made by Professor Torsten S öderström in the field of systematic identification by publications (journals, papers, book chapters or scientific monographs) that appeared six years before the award year. This problem has been studied from different angles for a typical system with errors-in- variable. Statistically valid maximum likelihood methods can be used, which are computationally complex, but computationally simple methods are generally inaccurate. Subspace fitting methods are more accurate when the amount of computation is large (Stoica et al.,1995; Cedervall et al.,1996), and (Mahata & Söderström, 2002) reports a significant improvement in accuracy when the amount of computation is low. The processing of unperturbed input in (Söderström & Hong, 2005) is periodic. In (Söderström et al., 2005), the local convergence characteristics of bias elimination least squares (BELS) algorithm for EIV identification are studied. A simplified BELS method is proposed in (Hong, et al., 2007). A new method called extended compensation least squares (ECLS) is proposed and further analyzed in (Ekman, 2005; Ekman, et al., 2006). In (Söderström, et al., 2006), some methods of continuous-time modeling in EIV identification are discussed. Studies (Söderström, 2010; & Söderström, 2011) show how to embed all of these methods into a joint framework (called a generalized instrumental variable estimator, GIVE), some selection of algorithm parameters can lead to different special cases previously referred to as specific methods. Xiao Deyun et al put forward an identification method which decomposes the variable error output error model by UD, and then compensates the error (Deyun, et al., 2018). Chen Hanfu studies the variable error output error model with the system input being ARMA process and proves the convergence of the proposed recursive algorithm (Chen & Yang, 2005). From the current research status, the research of variable with error system focuses on the open-loop variable with error and the identification of variable with error system. Therefore, the controller of closed-loop variable system with error is studied in this paper.