Least Squares Learning Identification Method For Time-varying Systems

System modeling is the basis for the analysis and design of many control systems.Recently,there is a relatively complete theoretical basis for the identification of time-invariant systems.However,time-varying parameters widely exists in practical control systems.Therefore,it is of great significance to study the identification of systems with time-varying parameters.Compared with other identification algorithms,the recursive least squares identification algorithm has a fast convergence speed and has been widely used in practice.However,the identification theory based on the recursive least squares algorithm is often used in the scenario of time-invariant systems.It has been found that in dealing with the problem of time-varying systems,especially fast-varying parameters,the recursive least squares algorithm has little ability to track time-varying parameters.Based on the existing work,this paper studies the parameter identification problem of a system with fast time-varying parameters based on repeated operations over a finite interval.The main contents of the studyare as follows:(1)For the ARX model,firstly,the derivation process of the least squares batch identification algorithm,the least squares recursive identification algorithm,the recursive least squares identification algorithm with forgetting factor,the iterative learning random gradient identification algorithm,and the iterative learning least square identification are briefly introduced.Through MATLAB simulations,the effects of the latter three algorithms in identifying systems with time-varying parameters are discussed.The results illustrate the limitations of the least squares algorithm with forgetting factor in identifying fast time-varying systems.Under repeated excitation conditions,learning identification can enables consistent estimation of time-varying parameters.(2)Derived three basic models of the equation error class,the equation error moving average model(CARMA),the equation error autoregressive model(CARAR),and the equation error autoregressive moving average model(CARARMA)iterative learning least squares algorithm.Through MATLAB simulation,the results show that under repeated excitation conditions,learning identification can achieve consistentestimation of three time-varying models.(3)For the output error model(OE),using the idea of the auxiliary model,the iterative learning stochastic gradient algorithm and the iterative learning least squares algorithm for the single-input single-output(SISO)output error model are derived and generalized to multiple from the input single output(MISO)output error model,an iterative learning random gradient algorithm and an iterative learning least square algorithm based on the MISO-OE model are derived.Through numerical simulations,the limitation of the least squares algorithm with forgetting factor in identifying fast time-varying systems is verified again.It is shown that the iterative learning least squares algorithm has faster congverence speed under the same external conditions,and the number of iterations needed to complete the identification is much less than the random gradient algorithm.(4)Applying learning identification to characteristic modeling,two adaptive iterative learning control algorithms based on second-order characteristic models are derived.The coefficients of the characteristic model are set to constant values,the input coefficients are controlled to be time-varying or even fast time-varying values,the recursive least squares algorithm with forgetting factor and iterative learning least squares algorithm are used for the parameters identification of the characteristic model.Numerical simulation results show that the adaptive iterative learning control algorithm based on the second-order characteristic model can completely track the given expected trajectory.