Parallel Difference Algorithms Of Fractional Order Differential System And Design Of Fractional Order PID Controller

Fractional order differential(FOD)system has rich theoretical connotation and profound physical background.The research on its fast numerical algorithm is the key link of model application,which is of important scientific significance and engineering application value.In order to meet the needs of fast solution of FOD system,this paper mainly studies the parallel difference algorithm of three kinds of classical FOD systems and the design of fractional order PID(FOPID)controller based on swarm intelligence algorithm.Aiming at the problem of long time course and large amount of calculation of FOD equations,this paper studies the new parallel algorithm for solving three kinds of classical FOD systems.The new algorithm is based on the parallelization of classical difference schemes,takes into account the stability,accuracy and parallelism of calculation at the same time.And the algorithm is easy to be used on many types of parallel computers.Aiming at the problems of FOPID controller design and algorithm derivation,we improve the defects of slow convergence speed or low calculation accuracy of the classical swarm intelligence algorithm,so that it can give full play to the advantages of global search and local mining.The improved swarm intelligence algorithm can quickly and effectively complete the design of FOPID controller.The main work of this paper is as follows:(1)The time fractional anomalous diffusion(TFAD)system is studied.Firstly,the grouping explicit(GE)difference scheme and alternating segment C-N(ASC-N)scheme are constructed for TFAD equation.Secondly,an unconditionally stable alternating band C-N(ABdC-N)scheme is constructed for the two-dimensional TFAD equation.Finally,pure alternating segment explicit-implicit and implicit-explicit(PASE-I and PASI-E)schemes are constructed for multi-term TFAD equation.Using mathematical induction and matrix eigenvalue method,the parallel scheme is numerically analyzed.Numerical experiments show that compared with the classical serial difference scheme,the computing time of the parallel algorithm in this paper can be saved by more than 80%.(2)The time fractional diffusion-wave(TFDW)system is studied.The ASC-N scheme is constructed for TFDW equation with damping and multi-term TFDW equation,respectively.By using two different energy estimation methods and mathematical induction,we can know that the ASC-N scheme is unconditionally stable and its convergence order is O(τ3-α+h2).Numerical experiments show that with the encryption of spatial grid,the parallel computing characteristics of ASC-N scheme will become more and more obvious.(3)For the time fractional telegraph(TFT)equation,the PASE-I and PASI-E parallel difference schemes are constructed.Through the analysis of matrix eigenvalues and mathematical induction,it can be seen that PASE-I and PASI-E schemes are unconditionally stable,and the convergence order is O(τ3-α+h2).When the number of spatial grid points is greater than 800,PASE-I and PASI-E schemes can achieve linear speedup.Compared with serial differential format,it can save CPU time by about 80%.Through the alternating difference method,PASE-I and PASI-E schemes inherit the advantages of parallelism of explicit scheme and absolute stability of implicit scheme at the same time.(4)The design of FOPID controller is a complex nonlinear optimization problem.Firstly,a parameter tuning algorithm of FOPID controller based on adaptive Cuckoo search(CS)algorithm is proposed.In order to improve the convergence speed and calculation accuracy of the classical CS algorithm and give full play to the advantages of global search and local mining,an adaptive step strategy based on systematic error is adopted.Simulation results show the adaptive cuckoo search algorithm can effectively complete the design of FOPID controller.Secondly,a parameter tuning method of FOPID controller based on improved bat algorithm is proposed.Simulation results show that the improved bat algorithm can effectively complete the design of FOPID controller.