Non-Asymptotic Algebraic Estimation Method For Fractional Order System

The emergence of fractional calculus in automation science and technology has injected fresh blood into the discipline and brought many new exploration ideas for researchers and engineers.Theories related to fractional order systems(FOSs)and the design of fractional order controllers(FOCs)are being continuously enriched and perfected.More and more studies have shown that FOSs have better description capabilities for some phenomena and processes in actual production and life,and FOCs are more conducive to obtaining better control performance due to higher design flexibility.However,although the concept of fractional calculus appeared very early,it has only been paid attention to by engineers in the past 30 years.Therefore,the researches related to the theory of FOSs are still in the stage of exploration and perfection,and the estimation theory of FOSs is one of them.In system theory,the acquisition of the system state and its derivative information has always been an important issue.But generally speaking,this information is not easy to obtain,or even if it can be measured by a sensor,the measured value is usually contaminated by noise.Therefore,it is naturally an indispensable method to construct an estimator that can effectively overcome noise interference through limited measurement values.As far as the FOS is concerned,the need to construct an estimator is more urgent.On the one hand,the traditional estimation theory still has some inherent flaws,and various state observers or filters are always difficult to balance accuracy and speed;on the other hand,due to the complexity of the FOS model itself,it is necessary to obtain the system state and some fractional derivative information may be more difficult.Non-asymptotic algebraic estimation theory is a kind of estimation method with accuracy,rapidity,and robustness.Its core tools are the algebraic parametric method(APM)and modulating functions method(MFM).At present,this theory has begun to be applied to some estimation problems of integer order systems,but it has not attracted people's attention in FOSs and has not formed a mature application framework.Therefore,taking some estimation problems in the FOS as an opportunity,studying the application of non-asymptotic algebraic estimation theory is a very important and valuable topic.Driven by the design of fractional order differentiator(FOD)and state estimation,the application research of non-asymptotic algebraic estimation methods in FOSs will be systematically carried out in this thesis.Since the system model is the basis for the further development of many estimation theories,this paper first studies the identification of continuous-time FOSs.Considering that there are two types of variables to be estimated in the FOS:parameter and fractional differential order,this paper divides the identification process into two stages:parameter estimation and order update.In the first stage,given the order of the fractional differential order,the data matrix required for the construction and operation of the two filters is given,and then the system parameters are estimated through the least-squares;the second stage provides a gradient-based method to update the fractional differential order.Moreover,the problem of introducing irrational terms after obtaining the gradient of the order is solved,and finally realizes the estimation of the fractional differential order.This research has achieved a good estimate of the parameters and order of FOSs,laying a foundation for the study of model-based estimation problems.Secondly,the design of model-based FOD is studied by using APM.The design of frequency domain annihilator is an important part of APM,which can eliminate the unknown initial value in the system.This paper systematically studies the issue of the design of annihilator for FOSs,and provides guidance for the use of APMs under different description forms and different definitions.At the same time,by using the fractional Leibniz law,the proposed FOD is realized in a recursive manner.Since APM will introduce a parameter to be designed in the estimation,this paper demonstrates the relationship between the design parameter and the estimation error by analyzing the estimation error.This research completes the application framework of the APM and provides a high-precision,fast and robust solution for the design of FODs.Thirdly,the problem of model-based state estimation for FOS is studied by using MFM.The idea of MFM is simple,but in the application,the state space equation must be transformed first,which is especially inconvenient when dealing with the estimation problem of FOS.For this reason,this article cleverly uses the structural characteristics of the observable standard form,and obtains the desired algebraic equations and modulation functions in a recursive manner.This method is also applicable to the situation under APM,so it is also a perfection to the non-asymptotic estimation theory.Considering that the system states can be uniquely determined by the initial states of the system,this paper discusses in detail the nature of the fractional order modulation functions and how to design them in the initial value estimation.In addition,in order to improve the estimation performance,this article also discusses how to design more suitable parameters for the modulation function and estimation process through analyzing the estimation error.This research not only simplifies the processing process of MFM,but also makes the estimation method not restricted by the system orders,which is an important breakthrough in the non-asymptotic estimation method.Finally,by introducing the model-free idea,the application range of MFM is further extended,and the model-free FOD design and the estimation of a class of nonlinear FOS are realized.In order to deal with the situation that the system is not modeled or has unknown nonlinear parts,this paper introduces the idea of model-free estimation.The core of this idea is not really without a model,but an approximate model with unknown parameters is established,so that all the problems to be estimated are converted into parameter estimation problems of approximating the model.More importantly,this paper introduces the concept of moving windows in the algebraic integral formula,which ensures estimation robustness against noise.Meanwhile,the approximation model always keeps fitting the signal on a fixed interval,thus ensuring the estimation accuracy.This research makes up for the shortcomings of model-based work that cannot deal with un-modeled or nonlinear situations,and expands the application range of non-asymptotic estimation methods in FOSs.