Closed-loop Subspace Identification Methods For Dynamic Systems

In modern industrial systems,the controller is generally designed and implemented based on the mathematical model of the controlled process.The model needs to accurately describe the steady-state and dynamic characteristics of the controlled process.In addition,the mathematical model of the controlled process can also be used to help develop other functional system modules,such as process monitoring and fault diagnosis,soft sensor,etc.For systems with explicit physical laws,mathematical operations such as calculus can be used to derive the process model through theoretical modeling.However,the established mathematical model based on the first principle is often complicated and it is necessary to simplify the model based on proper assumptions for practical application,which may lead to the model mismatch to some extent.In addition,for large-scale complex systems,theoretical modeling is rather difficult,time-consuming and even impossible.When the physical laws are unknown or the system is rather complex,system identification provides a more convenient and feasible choice for modeling,which directly uses the available data measured in the experiment to determine the appropriate process model in the selected model class.In particular,as a classic system identification technique,the subspace identification methods are suitable for the identification of multiple-input multiple-output state-space models with low computational complexity.In addition,the subspace identification methods have been successfully extended to the data-driven fault detection and predictive control.In view of this,aiming to cope with the correlation between control inputs and noise under the closed-loop conditions,a novel framework for closed-loop subspace identification methods is first proposed by integrating the prior knowledge of the controller,based on which in-depth research will be done regarding the system disturbances,cascade connection and recursive implementation.First,the thesis introduces the research background and significance of system identification,the research task,the basic procedure and summarizes the related research methods.Based on several aspects including closed-loop subspace identification,subspace identification with disturbance systems,subspace identification for interconnected systems and recursive subspace identification,the current research status of subspace identification is introduced in detail and the problems to be studied for subspace identification are listed.In view of these points to be studied,the content of this article is introduced as well.Second,a closed-loop subspace identification framework is proposed to cope with the correlation between control inputs and noise by integrating the prior knowledge of the controller.Based on the coprime factorization technique,an instrumental matrix is constructed by utilizing the the prior knowledge of the controller,There is no correlation between the constructed instrumental matrix and the system noise.The future output can be directly projected into the coordinate space formed by the past input/output and the instrumental matrix and then projected back to the coordinate space formed by the past input/output and future input through coordinate transformation,which solves the problem of the biased identification due to the correlation between control inputs and noise under closed-loop feedback and at the same time establishes a unified framework for open-loop and closed-loop subspace identification.Third,a closed-loop subspace system identification method is proposed to eliminate the the influence of unknown periodic disturbance on the closed-loop system.For periodic disturbances with known frequencies,Fourier series are used to construct the row space to describe the disturbance.The row space contains both the disturbance frequencies and the time series.For periodic disturbances with unknown frequencies,the signal space of the disturbance is approximately constructed using Bernstein polynomials and the selection of the Bernstein polynomial order is given.The signal space contains only the time series.Based on the closed-loop subspace identification framework in the previous chapter,the oblique projection technique is used to eliminate the influence of periodic disturbances.Finally,the parameter matrices of the state space model are identified.Fourth,a closed-loop subspace identification method is proposed to solve the problem that information transmission for each closed-loop subsystem under the cascade topology is unknown.The proposed method uses the available known input and output data to linearly represent the unknown states from the neighboring system for each identified subsystem.Based on the extended state space model of each subsystem,the influence of the unknown states from the neighboring system for each identified subsystem is eliminated through the oblique projection under the feedback control,and each subsystem under the cascade topology can be identified.In addition,the closed-loop subspace identification can be further extended to the system identification with the directed acyclic graph topology.Fifth,a recursive subspace identification method for closed-loop systems is proposed to meet the real-time performance for online identification.The proposed method uses the Givens transform and Hyperbolic transform to recursively update the Cholesky decomposition as well as update the parameters of the extended state space model.In addition,an improved extended observable matrix identification method,as well as a propagator based recursive identification method for the extended observability matrix with forgetting factor using the sliding window,is proposed to further reduce the online computational burden.