Parameter And Differential Order Identification Of Continuous-time Fractional-order Systems

Fractional order calculus,whose order has been extended to the real or even com-plex range,has not only unified and extended the concept of traditional calculus,but also opened up new avenues for people to explore the physical world and improve the process.It can obtain more concise and accurate results in terms of modeling accuracy and control performance,which meets the increasing requirements with the rapid de-velopment of engineering technology.Modeling is an important precondition for the analysis and control of fractional order systems,and identification is almost the on-ly effective means.Although its research has had many days and accumulated some achievements,it still faces many difficulties.In addition,the continuous-time model is able to describe system characteristics better than discrete ones in some cases,and the corresponding fractional order systems occupy most of the field.Therefore,the identi-fication framework and method for designing fractional systems based on continuous-time models can enrich the theory and benefit engineering practice.First of all,this paper estimates the parameters of fractional order systems by de-signing the fractional order update law,especially considering the existence of impulse noise.Different from the traditional gradient descent method where the parameters adopt the first order difference,the fractional order update law uses the discrete frac-tional order difference feature to reduce the update amount of each step and include the historical information of the parameters,which smoothes the convergence process of the parameters.Impulse noise causes the algorithm with the minimum square objective function to fluctuate violently or even diverge.For this reason,this chapter constructs a differentiable approximate minimum absolute error function.Rigorous mathemati-cal analysis and detailed simulation examples illustrate that the algorithm can achieve better convergence and steady state performance.Secondly,this paper proposes a fractional order update gradient method to identify systems that are polluted by outliers.Using the nuclear norm and infinite norm to transform the problem of outlier detection into a matrix decomposition problem,which is the first time to accurately measure outliers while estimating measurement noise,and to avoid filtering out noise before detection reducing the detection accuracy.In addition,in order to further tap the potential of the fractional order gradient method,the variable initial value mechanism is introduced so that the step size can be adjusted adaptively.Combined with the update law of the previous chapter,the fractional order update gradient method is proposed,which not only speeds up the convergence rate but also improves the steady state accuracy,alleviating the contradiction between the two.Finally,this paper utilizes the block pulse function method to synchronously i-dentify the differential orders and parameters of non-commensurate fractional order systems.Based on the block pulse function,the generalized operational matrix which approximates the fractional order calculus is deduced.The differential form of sys-tem transfer function is transformed into the integral form,and the system predictive output can be represented by the algebraic combination of each order-corresponding operational matrix.Then,by the idea of output error method,solve the nonlinear op-timization problem to achieve the identification goal.This method not only evades the complicated calculation of the signal differential value,but also solves the problem that the every order is known or the system is assumed commensurate in the existing research.