Research On Fractional Order System Identification And Control Based On Hybrid Functions

Fractional calculus differs from integer-order calculus in that its integration or differ-entiation order is arbitrary real number.Fractional calculus operators exhibit global and infinite-dimensional characteristics,enabling Fractional models to more accurately describe physical systems with historical memory and distributed behavior.Due to the historical memory of Fractional calculations,system identification and controller design for Fractional systems face challenges such as large computational burden and difficulty in tuning con-troller parameters.To address these issues,this study proposes a method based on Fractional operation matrices of hybrid functions consisting of Block-pulse functions and Bernstein polynomials,transforming Fractional operations into algebraic operations and simplifying the identification and controller design process for Fractional systems.The main research work includes:(1)In solving complex dynamic systems,fractional calculus faces significant chal-lenges due to its high computational complexity and the lack of general solution methods.This paper proposes a method based on the basic concepts and properties of fractional cal-culus,which combines Block-pulse functions and Bernstein polynomials into a hybrid func-tions.By constructing the fractional integral operation matrix of the hybrid functions based on Block-pulse functions transformation,the fractional differential equation is converted into an algebraic equation for solving.The proposed method is demonstrated to be a reli-able tool for dealing with complex fractional-order problems through examples.(2)To address the issues of low smoothness of Block-pulse functions and low accuracy when using a finite number of Block-pulse functions to represent arbitrary square-integrable functions,this paper mixes Block-pulse functions with Bernstein polynomials and derives the fractional integral operation matrix of the hybrid functions based on Laplace transform.Based on this,a method for identifying the parameters and orders of fractional-order systems is proposed.This method transforms the fractional-order system into an algebraic system,and identifies the system’s parameters and orders by minimizing the mean square error be-tween the real system output and the estimated system output.The simulation results validate the effectiveness of this method.(3)Optimal control of fractional-order systems is challenging to obtain analytical opti-mality conditions.To address this issue,this paper proposes a method for solving fractional-order optimal control based on the integral operation matrix of hybrid functions composed of Block-pulse and Bernstein polynomials.The method simplifies the performance index and fractional-order dynamic constraints of optimal control problems through the hybrid func-tions operation matrix,converts the optimization problem with fractional-order dynamic constraints into an unconstrained optimization problem,and provides a set of nonlinear equa-tions for the first-order optimal conditions.The Newton method is then used to solve the nonlinear equations to obtain the optimal control input.The simulation results validate the applicability of this method.(4)The fractional-order optimal PI~λD~μcontroller has difficulties in determining param-eters and requires large computational effort.To solve this problem,a two-step tuning design method is proposed.First,the optimal control input is calculated based on the system state equation and performance constraint index,which is then applied to the controlled system to compensate for the shortcomings of the PI~λD~μcontroller’s performance.Next,the initial values of controller parameters are determined using fractional-order system identification methods,and particle swarm optimization with ITAE performance index is employed for parameter optimization.Finally,the superiority of the proposed method is verified through simulation.(5)To verify the practicality of the designed fractional-order optimal PI~λD~μcontroller in engineering systems,this paper takes the Stewart parallel stabilization platform as the research object and proposes two fractional-order PI~λD~μcontrollers based on joint space motion control and workspace motion control.The controller based on joint space motion uses the two-step tuning method for controlling each driving link,validating its feasibility.Moreover,the controller based on workspace motion considers the entire platform’s dynamic model,proposing a method based on ADAMS dynamic modeling and Simulink simulation model,and applying the particle swarm algorithm for controller parameter tuning calcula-tion.Experimental results confirm the effectiveness and superiority of the fractional-order PI~λD~μcontroller.