A Design Method of Fractional-Order PID Controller

Abstract

A design method for a fractional-order PID controller comprises structural updating and parameter tuning of the fractional-order PID controller, and the steps are as follows: step 1: obtaining an optimal simplified Oustaloup filter; step 2: determining an adjustable coefficient in the optimal simplified Oustaloup filter; step 3: updating the structure of the fractional-order PID controller; step 4: obtaining a nonlinear dynamic chaotic particle swarm optimization algorithm; step 5: tuning the parameters of the fractional-order PID controller; step 6: obtaining a final fractional-order PID controller; the optimal simplified Oustaloup filter proposed by the present invention effectively improves the fitting accuracy of the fractional-order operator in the fractional-order PID controller; the nonlinear dynamic chaotic particle swarm optimization algorithm has a faster convergence speed and is not easy to fall into a local optimum, which effectively improves the iteration effect and can quickly tune the optimal fractional-order PID controller parameters; the fractional-order PID controller designed by the method provided by the present invention can enable the control system to have the advantages of short adjustment time, small steady-state error and small overshoot.

Description

Technical Field

[0001] The invention belongs to the field of electrical automation, and in particular relates to a design method of a fractional-order PID controller. Background Art

[0002] As a mature and practical control method, PID controller has been widely used in industrial automation. However, it is difficult to achieve ideal control accuracy and the adjustment time is relatively slow in the process of using some complex systems. The fractional-order PID controller proposed by Professor Podlubny introduces two fractional-order parameters, λ and μ, to the differential and integral terms of the PID controller, and extends the traditional PID controller to the fractional-order field. It has higher control accuracy and more flexible design process. Applying it to actual industrial control systems can achieve better adjustment effects.

[0003] Although the performance of the fractional-order PID controller is better, the difficulty of parameter tuning is correspondingly increased because the fractional-order PID controller adds two fractional-order parameters. In addition, the digital implementation of fractional-order calculus is not ideal. Therefore, the design of the fractional-order PID controller needs to overcome these technical difficulties.

[0004] At present, the digital implementation of fractional-order calculus operators is mostly completed through Oustaloup filters, but there is a defect of poor fitting effect near the boundary, especially in the low-frequency band and high-frequency band. There is still room for improvement in the fitting performance. The parameter tuning methods of fractional-order PID controllers include analytical method, frequency method and intelligent optimization algorithm. Among them, it is difficult for the analytical method and frequency method to directly tune the ideal fractional-order PID controller parameters, while the optimization ability of the intelligent algorithm is better than the other two methods. The most representative intelligent method is the parameter tuning method of fractional-order PID controller based on particle swarm algorithm, but it also has the defects of poor convergence and easy to fall into local optimality. Therefore, it is urgent to improve it to obtain better optimization ability. Summary of the invention

[0005] The invention improves the actual fitting ability of the fractional-order operator by transforming the Oustaloup filter, completes the digital realization of the fractional-order operator and updates the structure of the fractional-order PID controller; at the same time, the invention provides a complete set of methods based on nonlinear dynamic chaotic particle swarm to complete the tuning of the parameters of the fractional-order PID controller; the fractional-order PID controller designed by the method provided by the invention can make the control system have the advantages of short adjustment time, small steady-state error and small overshoot.

[0006] A design method of a fractional-order PID controller includes structural updating and parameter tuning of the fractional-order PID controller, and the steps are as follows:

[0007] Step 1: Obtain the optimal simplified Oustaloup filter;

[0008] Step 2: Determine the adjustable coefficients in the optimal simplified Oustaloup filter;

[0009] Step 3: Update the fractional-order PID controller structure;

[0010] Step 4: Obtain nonlinear dynamic chaotic particle swarm optimization algorithm;

[0011] Step 5: Adjust the parameters of the fractional-order PID controller;

[0012] Step 6: Get the final fractional-order PID controller.

[0013] In step 1, the optimal simplified Oustaloup filter obtained has the following structure:

[0014]

[0015] in, represents the optimal simplified Oustaloup filter obtained by transformation, b, c, d are adjustable coefficients, s is the Laplace operator, α is the order of the fractional calculus operator, N is the filter order, ω u represents the upper bound of the fitting frequency band, ω l Indicates the lower bound of the fitted frequency band.

[0016] In step 2, an optimization objective function is designed to determine The adjustable coefficients b, c, d in the optimization objective function formula are as follows:

[0017]

[0018] Among them, J bcd represents the optimization objective function, ω represents the frequency, M r , P r For α The actual amplitude and phase frequency, M a , P a is the approximate amplitude and phase frequency, ω u represents the upper bound of the fitting frequency band, ω l It represents the lower bound of the fitting frequency band. The adjustable coefficients b, c, and d can be determined according to different optimization objective functions and search ranges.

[0019] In step 3, after obtaining the optimal compact Oustaloup filter, use the optimal compact Oustaloup filter Complete the digital implementation of the fractional-order operator and update the fractional-order PID controller structure. The updated fractional-order PID controller structure expression is as follows:

[0020]

[0021] Among them, G c (s) represents the fractional-order PID controller, s is the Laplace operator, K p is the proportionality coefficient, K i is the integration coefficient, K d is the differential coefficient, λ is the integration order, and μ is the differential order; is the optimal simplified Oustaloup filter; then They are as follows:

[0022]

[0023] In step 4, the nonlinear dynamic chaotic particle swarm optimization algorithm obtained is as follows:

[0024] v id (it+1)=w(it)v id (it)+c1(it)r1(it)[p id (it)-x id (it)]+c2(it)r2(it)[p gd (it)-x id (it)]

[0025] x id (it+1)=x id (it)+v id (it+1),d=1,2,...

[0026] Among them, it represents the current iteration number, i represents a particle in the population, d represents the dimension of the solution space, and v id (it+1) represents the moving speed of the d-dimensional i-th particle at the it+1-th iteration, x id (it+1) represents the position of the d-th dimension of the i-th particle at the it+1-th iteration, p id (it) represents the individual optimal position of the d-dimensional particle at the it-th iteration, p gd (it) represents the global optimal position of the d-dimensional population at the it-th iteration, c1(it) and c2(it) are the dual asynchronous nonlinear dynamic learning factors at the it-th iteration, r1(it) and r2(it) are the chaos scaling coefficients at the it-th iteration, and w(it) is the chaos inertia weight at the it-th iteration.

[0027] The formulas of dual asynchronous nonlinear dynamic learning factors c1(it) and c2(it) are as follows:

[0028]

[0029] Among them, it is the current iteration number, it max is the maximum number of iterations, f1 and f2 are weight adjustment factors, and f 1min 、f 1max is the minimum and maximum value of f1, f 2min 、f 2max is the minimum and maximum value of f2, c 1min 、c 1max is the minimum and maximum value of the learning factor c1, c 2min 、c 2max is the minimum and maximum value of the learning factor c2.

[0030] The formulas for chaos scaling coefficients r1(it) and r2(it) are as follows:

[0031] r1(it)=0.500+0.476mod[z1(it-1)+δ+gz2(it),1]

[0032] r2(it)=0.500+0.476cos(2πz1(it-1)+e -r z2(it-1)

[0033] Wherein, it represents the current cycle number, z1 and z2 are chaos mapping variables, mod represents the modulo operation of z1(it-1)+δ+gz2(it), cos represents the cosine operation, e is the natural logarithm, π is the pi, and r, g, δ are chaos regulation parameters.

[0034] The formula of chaotic inertia weight w(it) is as follows:

[0035] w'(it)=0.650+0.238mod[z1(it-1)+δ+gz2(it),1]

[0036] w(it)=0.650+0.238cos(2πz1(it-1)+e -r z2(it-1)

[0037] Wherein, it represents the current cycle number, z1 and z2 are chaotic mapping variables, mod represents the modulo operation of z1(it-1)+δ+gz2(it), cos represents the cosine operation, e is the natural logarithm, π is the pi, w'(it) is the chaotic mapping sequence of w(it), and r, g, δ are chaotic adjustment parameters.

[0038] A method for tuning parameters of a fractional-order PID controller is provided, which is used for tuning parameters of a fractional-order PID controller. The structure of the fractional-order PID controller is as follows:

[0039]

[0040] Among them, G c (s) represents the fractional-order PID controller, s is the Laplace operator, K p is the proportionality coefficient, K i is the integration coefficient, K d is the differential coefficient, λ is the integration order, and μ is the differential order; It is the optimal simplified Oustaloup filter;

[0041] The nonlinear dynamic chaotic particle swarm optimization algorithm is used to optimize the parameter K in the following steps: p , K i , K d , λ, μ are adjusted, including the following steps:

[0042] S1: Initialization, setting the range of parameters to be adjusted, the maximum number of iterations, the population size, the particle movement speed and K p , K i , K d , λ, μ positions and fitness functions;

[0043] S2: Initialize the dual asynchronous nonlinear dynamic learning factor, chaos scaling coefficient and chaos inertia weight;

[0044] S3: Evaluate the fitness function value of the particle according to the initial position and determine the optimal initial individual and population;

[0045] S4: Update the dual asynchronous nonlinear dynamic learning factor according to the current number of iterations;

[0046] S5: Update the chaos scaling factor and chaos inertia weight according to the current number of iterations;

[0047] S6: Update the particle's moving speed and K p , K i , K d ,λ,μ positions;

[0048] S7: By calculating the fitness function value of the particle and comparing its value, the smallest one is taken to obtain the optimal position of the contemporary individual and the optimal position of the population;

[0049] S8: Return to step S4 and enter the next iteration until the maximum number of iterations is reached. At this time, the global population optimal position is the corresponding optimal K p , K i , K d ,λ,μ values.

[0050] In step S1, the fitness function expression is as follows:

[0051]

[0052] Among them, J FOPID represents the fitness function, T is the simulation time, t is the sampling time, e(t) is the error, e'(t) is the rate of change of the error, which can predict the error change trend in advance, k1 and k2 are the weight coefficients corresponding to the error and the error change rate;

[0053] When e(t)<0, the penalty function can effectively suppress overshoot. At this time, the fitness function in step S1 becomes:

[0054]

[0055] Among them, J FOPID represents the fitness function, T is the simulation time, t is the sampling time, e(t) is the error, e'(t) is the rate of change of the error, k1 and k2 are the weight coefficients corresponding to the error and the rate of change of the error, and k3 is the penalty factor.

[0056] In step 6, the final fractional-order PID controller can be obtained by the fractional-order PID controller structure obtained in step 3 and the optimal fractional-order PID controller parameters obtained by tuning in step 5, thus completing the design process of the fractional-order PID controller.

[0057] Compared with the prior art, the present invention has the following technical effects:

[0058] 1) The optimal simplified Oustaloup filter proposed in the present invention solves the defects of the existing Oustaloup filter, such as complex structure and poor fitting ability in low frequency band and high frequency band, and effectively improves the fitting accuracy of the fractional-order operator in the fractional-order PID controller;

[0059] 2) The present invention proposes a nonlinear dynamic chaotic particle swarm optimization algorithm with fast convergence speed and not easy to fall into local optimum to adjust the parameters of the fractional-order PID controller, which effectively improves the iteration effect and can quickly adjust the optimal fractional-order PID controller parameters;

[0060] 3) The fractional-order PID controller designed by the method provided by the present invention can make the control system have the advantages of short adjustment time, small steady-state error and small overshoot. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0062] Figure 1 A comparison diagram of amplitude-phase characteristic curves of the optimal simplified Oustaloup filter disclosed in the present invention and the existing Oustaloup filter;

[0063] Figure 2 It is the principle diagram of the fractional-order PID controller (FOPID) design method proposed in the present invention;

[0064] Figure 3 To use the standard particle swarm optimization algorithm (PSO) with fitness function J FOPID Schematic diagram of the convergence curve of the fitness function used as an optimization criterion to adjust the parameters of the fractional-order PID controller inside the automatic voltage regulation system;

[0065] Figure 4 To use the nonlinear dynamic chaotic particle swarm optimization algorithm (NDCPSO) proposed by the present invention with the fitness function J FOPID Schematic diagram of the convergence curve of the fitness function used as an optimization criterion to adjust the parameters of the fractional-order PID controller inside the automatic voltage regulation system;

[0066] Figure 5 To use the nonlinear dynamic chaotic particle swarm optimization algorithm (NDCPSO) proposed by the present invention with the fitness function J FOPID The fractional-order PID controller parameter K is set as an optimization criterion in the automatic voltage regulation system. p , K i , K d Schematic diagram of the final convergence position of particles;

[0067] Figure 6 To use the nonlinear dynamic chaotic particle swarm optimization algorithm (NDCPSO) proposed by the present invention with the fitness function J FOPID Schematic diagram of the final convergence position of particles used as optimization criteria to adjust the parameters λ and μ of the fractional-order PID controller inside the automatic voltage regulation system;

[0068] Figure 7 This is the step response curve of the automatic voltage regulation system without fractional-order PID control;

[0069] Figure 8 The figure is a comparison diagram of step response curves of an automatic voltage regulation system of a fractional-order PID controller designed based on the method of the present invention and an automatic voltage regulation system of the prior art. DETAILED DESCRIPTION

[0070] A design method for a fractional-order PID controller includes structural updating and parameter tuning of the fractional-order PID controller.

[0071] Step 1: Obtain the optimal simplified Oustaloup filter;

[0072] The original Oustaloup filter structure is as follows:

[0073]

[0074] in, represents the original Oustaloup filter, s is the Laplace operator, α is the order of the fractional calculus operator, N is the filter order, ω u represents the upper bound of the fitting frequency band, ω l represents the lower bound of the fitting frequency band,

[0075] The structure of the improved Oustaloup filter is as follows:

[0076]

[0077] in, represents the improved Oustaloup filter, b and d are adjustable coefficients, s is the Laplace operator, α is the order of the fractional calculus operator, N is the filter order, ω u represents the upper bound of the fitting frequency band, ω l Indicates the lower bound of the fitted frequency band.

[0078] right The transformation is performed to obtain the optimal simplified Oustaloup filter. The optimal simplified Oustaloup filter structure is as follows:

[0079]

[0080] in, represents the optimal simplified Oustaloup filter obtained by transformation, b, c, d are adjustable coefficients, s is the Laplace operator, α is the order of the fractional calculus operator, N is the filter order, ω u represents the upper bound of the fitting frequency band, ω l Indicates the lower bound of the fitted frequency band.

[0081] Step 2: Determine the adjustable coefficients in the optimal simplified Oustaloup filter;

[0082] Design an optimization objective function to determine The adjustable coefficients b, c, d in the optimization objective function formula are as follows:

[0083]

[0084] Among them, J bcd represents the optimization objective function, ω represents the frequency, M r , P r For α The actual amplitude and phase frequency, M a , P ais the approximate amplitude and phase frequency, ω u represents the upper bound of the fitting frequency band, ω l Indicates the lower bound of the fitted frequency band.

[0085] Take ω l =10 -3 ,ω u =10 3 , order α = 0.5, filter order N = 7, and the optimal adjustable coefficients b = 2.4376, c = 1.0126, and d = 0.0488 can be obtained by taking 1 as the initial value in the search range [0,10].

[0086] See also Figure 1 , which is a comparison diagram of the amplitude-phase characteristic curves of the optimal simplified Oustaloup filter disclosed in the present invention and the existing Oustaloup filter.

[0087] The adjustable coefficients b, c, and d can be determined according to different optimization objective functions and search ranges. bcd As the optimization objective function, the optimal simplified Oustaloup filter can be obtained in the interval b∈[2,3], c∈[1,2], d∈[0.01,1]. Achieve better fitting performance.

[0088] Step 3: Update the fractional-order PID controller structure;

[0089] After obtaining the optimal simplified Oustaloup filter, use the optimal simplified Oustaloup filter Complete the digital implementation of the fractional-order operator and update the fractional-order PID controller structure. The updated fractional-order PID controller structure expression is as follows:

[0090]

[0091] Among them, G c (s) represents the fractional-order PID controller, s is the Laplace operator, K p is the proportionality coefficient, K i is the integration coefficient, K d is the differential coefficient, λ is the integration order, and μ is the differential order; is the optimal simplified Oustaloup filter; then They are as follows:

[0092]

[0093] Step 4: Obtain nonlinear dynamic chaotic particle swarm optimization algorithm;

[0094] is the parameter K of the above fractional-order PID controller p , K i , K d , λ, μ are adjusted. The present invention also proposes a nonlinear dynamic chaotic particle swarm optimization algorithm, wherein the standard particle swarm optimization algorithm is as follows:

[0095] v jg (t+1)=wv jg (t)+c1r1[pbest jg -x jg (t)]+c2r2[gbest jg -x jg (t)]

[0096] x jg (t+1)=x jg (t)+v jg (t+1)

[0097] Among them, t represents the current evolutionary generation, x j =(x j1 ,x j2 ,…,x jg ,…,x jm ) and v j =(v j1 ,v j2 ,…,v jg ,…,v jm ) are the spatial position and velocity vector of the jth particle in the group. j =(pbest j1 ,pbest j2 ,…,pbest jg ,…,pbest jm ) is the best experience of particle j, gbest=(gbest1,gbest2,…,gbest g ,…,gbest m ) is the optimal experience of the group, w is the inertia weight of the particle, the learning factors c1 and c2 are custom constants, and r1 and r2 are random numbers.

[0098] The above standard particle swarm optimization algorithm is improved to obtain the nonlinear dynamic chaotic particle swarm optimization algorithm, which is as follows:

[0099] v id (it+1)=w(it)v id (it)+c1(it)r1(it)[p id (it)-x id (it)]+c2(it)r2(it)[pgd (it)-x id (it)]

[0100] x id (it+1)=x id (it)+v id (it+1),d=1,2,...

[0101] Among them, it represents the current iteration number, i represents a particle in the population, d represents the dimension of the solution space, and v id (it+1) represents the moving speed of the d-dimensional i-th particle at the it+1-th iteration, x id (it+1) represents the position of the d-th dimension of the i-th particle at the it+1-th iteration, p id (it) represents the individual optimal position of the d-dimensional particle at the it-th iteration, p gd (it) represents the global optimal position of the d-dimensional population at the it-th iteration.

[0102] The formulas of dual asynchronous nonlinear dynamic learning factors c1(it) and c2(it) are as follows:

[0103]

[0104] Among them, it is the current iteration number, it max is the maximum number of iterations, f1 and f2 are weight adjustment factors, and f 1min 、f 1max is the minimum and maximum value of f1, f 2min 、f 2max is the minimum and maximum value of f2, c 1min 、c 1max is the minimum and maximum value of the learning factor c1, c 2min 、c 2max is the minimum and maximum value of the learning factor c2.

[0105] The formulas for chaos scaling coefficients r1(it) and r2(it) are as follows:

[0106] r1(it)=0.500+0.476mod[z1(it-1)+δ+gz2(it),1]

[0107] r2(it)=0.500+0.476cos(2πz1(it-1)+e -r z2(it-1)

[0108] Wherein, it represents the current number of cycles, z1 and z2 are chaotic mapping variables, and their initial values ​​are randomly generated in the interval [-1.0512, 1.0512]. mod represents the modulo operation of z1(it-1)+δ+gz2(it), cos represents the cosine operation, e is the natural logarithm, π is the pi, r, g, δ are chaotic adjustment parameters, and when r=3, g=12.6695, δ=400, r1(it) and r2(it) are in the interval [0,1] and show chaotic characteristics.

[0109] The formula of chaotic inertia weight w(it) is as follows:

[0110] w'(it)=0.650+0.238mod[z1(it-1)+δ+gz2(it),1]

[0111] w(it)=0.650+0.238cos(2πz1(it-1)+e -r z2(it-1)

[0112] Wherein, it represents the current number of cycles, z1 and z2 are chaotic mapping variables, and their initial values ​​are randomly generated in the interval [-1.0512, 1.0512]. mod represents the modulo operation of z1(it-1)+δ+gz2(it), cos represents the cosine operation, e is the natural logarithm, π is the pi, w'(it) is the chaotic mapping sequence of w(it), r, g, δ are chaotic adjustment parameters, and when r=3, g=12.6695, δ=400, w(it) is in the interval [0.4, 0.9] and shows chaotic characteristics.

[0113] Step 5: Adjust the parameters of the fractional-order PID controller;

[0114] The nonlinear dynamic chaotic particle swarm optimization algorithm is applied to the tuning of the parameters of the fractional-order PID controller. The structure of the fractional-order PID controller is as follows:

[0115]

[0116] Among them, G c (s) represents the controller, s is the Laplace operator, K p is the proportionality coefficient, K i is the integration coefficient, K d is the differential coefficient, λ is the integration order, μ is the differential order, It is the optimal simplified Oustaloup filter;

[0117] The following is a demonstration of the tuning process of the fractional-order PID controller parameters in conjunction with a specific automatic voltage regulation system. The automatic voltage regulation system can adopt the automatic voltage regulation system disclosed in the document "A novel technique to design cuckoo search based FOPID controller for AVR in power systems", in which the data of the amplifier, exciter, generator, and sensor are as follows:

[0118] Amplifier gain K A =10, amplifier time constant τ A =0.02s, exciter gain K E =10, exciter time constant τ E =0.5s, generator gain K G =0.7, generator time constant τ G =1.0s, sensor gain K R =1.0, sensor time constant τ R =0.001s.

[0119] See also Figure 2 , is a schematic diagram of the fractional order PID controller (FOPID) design method proposed in the present invention;

[0120] Parameter K p , K i , K d The setting of , λ, and μ includes the following steps:

[0121] S1: Set the search range K of the parameters to be adjusted p , K i , K d ∈[0,200], λ, μ∈[0,2], the maximum number of iterations is 50, the population size is 50, the initial particle velocity is 0, the moving speed limit is 30, K p , K i , K d The initial positions of ,λ,μ are randomly generated within the search range, and the fitness function is:

[0122]

[0123] Among them, J FOPID represents the fitness function, simulation time T = 10s, sampling time t = 0.01s, e(t) is the error, e'(t) is the rate of change of the error, error weight coefficient k1 = 0.5, error change rate weight coefficient k2 = 0.5, and penalty factor k3 = 100.

[0124] S2: Initialize the dual asynchronous nonlinear dynamic learning factor, chaos scaling coefficient and chaos inertia weight;

[0125] The formulas of dual asynchronous nonlinear dynamic learning factors c1(it) and c2(it) are as follows:

[0126]

[0127] Among them, the current number of iterations it = 1, the maximum number of iterations it max =50, f1, f2 are weight adjustment factors, f 1min =0.4, f 1max =1.0,f 2min =1.0,f 2max =2.4, c1, c2 are learning factors, c 1min =1.0, c 1max =2.0, c 2min =1.0, c 2max =2.0.

[0128] The formulas for chaos scaling coefficients r1(it) and r2(it) are as follows:

[0129] r1(it)=0.500+0.476mod[z1(it-1)+δ+gz2(it),1]

[0130] r2(it)=0.500+0.476cos(2πz1(it-1)+e -r z2(it-1)

[0131] Among them, the current cycle number it=1, z1 and z2 are chaotic mapping variables, and the initial values ​​are randomly generated in [-1.0512, 1.0512]. Mod represents the modulo operation of z1(it-1)+δ+gz2(it), cos represents the cosine operation, e is the natural logarithm, π is the pi, r, g, δ are chaos adjustment parameters, when r=3, g=12.6695, δ=400, r1(it) and r2(it) are located in the interval [0, 1] and show chaotic characteristics.

[0132] The formula of chaotic inertia weight w(it) is as follows:

[0133] w'(it)=0.650+0.238mod[z1(it-1)+δ+gz2(it),1]

[0134] w(it)=0.650+0.238cos(2πz1(it-1)+e -r z2(it-1)

[0135] Among them, the current cycle number it=1, z1 and z2 are chaotic mapping variables, and the initial values ​​are randomly generated in [-1.0512, 1.0512]. Mod represents the modulo operation of z1(it-1)+δ+gz2(it), cos represents the cosine operation, e is the natural logarithm, π is the pi, w'(it) is the chaotic mapping sequence of w(it), r, g, δ are chaotic adjustment parameters, when r=3, g=12.6695, δ=400, w(it) is in the interval [0.4, 0.9] and shows chaotic characteristics.

[0136] S3: Evaluate the fitness function value of the particle according to the initial position and determine the optimal initial individual and population, that is, convert the K generated in step S1 p , K i , K d , λ, μ initial positions are substituted into the fractional-order PID controller G c (s), thereby controlling the automatic voltage regulation system, the actual output value measured by the sensor is subtracted from the given reference input value to obtain the error e(t), and then it is determined whether e(t) is less than 0, and substituted into the fitness function J of the corresponding condition FOPID , evaluate the current fitness function value.

[0137] S4: Update the dual asynchronous nonlinear dynamic learning factor according to the current number of iterations;

[0138] For example, if the current number of iterations is 6, we can get:

[0139]

[0140]

[0141] S5: Update the chaos scaling factor and chaos inertia weight according to the current number of iterations;

[0142] For example, if the current number of iterations is 6, mod[z1(6-1)+δ+gz2(6),1]=0.9815, cos(2πz1(6-1)+e -r z2(6-1)=-0.6282, the chaos scaling coefficient can be obtained as:

[0143] r1(6)=0.500+0.476mod[z1(6-1)+δ+gz2(6),1]=0.500+0.476×0.9815=0.967194

[0144] r2(6)=0.500+0.476cos(2πz1(6-1)+e -r z2(6-1)=0.500+0.476×(-0.6282)=0.200977

[0145] The chaotic inertia weight can be obtained as:

[0146] w'(6)=0.650+0.238mod[z1(6-1)+δ+gz2(6),1]=0.650+0.238×0.9815=0.883597

[0147] w(6)=0.650+0.238cos(2πz1(6-1)+e -r z2(6-1)=0.650+0.238×(-0.6282)=0.500488

[0148] S6: Update the particle's moving speed and K p , K i , K d ,λ,μ positions;

[0149] The speed and position are updated according to the following formula:

[0150] v id (it+1)=w(it)v id (it)+c1(it)r1(it)[p id (it)-x id (it)]+c2(it)r2(it)[p gd (it)-x id (it)]

[0151] x id (it+1)=x id (it)+v id (it+1),d=1,2,…,5

[0152] Among them, it represents the current iteration number, i represents a particle in the population, d represents the dimension of the solution space, and v id (it+1) represents the moving speed of the d-dimensional i-th particle at the it+1-th iteration, x id (it+1) represents the position of the d-th dimension of the i-th particle at the it+1-th iteration, p id (it) represents the individual optimal position of the d-dimensional particle at the it-th iteration, p gd (it) represents the global optimal position of the d-dimensional population at the it-th iteration, c1(it) and c2(it) are the dual asynchronous nonlinear dynamic learning factors at the it-th iteration, r1(it) and r2(it) are the chaos scaling coefficients at the it-th iteration, w(it) is the chaos inertia weight at the it-th iteration, K p Corresponding to dimension d=1, K i Corresponding to dimension d=2, Kd corresponds to dimension d=3, λ corresponds to dimension d=4, and μ corresponds to dimension d=5.

[0153] S7: By calculating the fitness function value of the particle and comparing its value, the smallest one is taken to obtain the optimal position of the contemporary individual and the optimal position of the population;

[0154] S8: Return to step S4 and enter the next iteration until the maximum number of iterations is reached. At this point, the global population optimal position is the corresponding optimal K p , K i , K d ,λ,μ values.

[0155] After S1-S8, the fractional-order PID controller parameter K of this embodiment is obtained by tuning. p =168.6856, K i =148.328, K d =1.5302, λ=1.4193, μ=1.6303.

[0156] Step 6: Get the final fractional-order PID controller.

[0157] Finally, the fractional-order PID controller of this embodiment is obtained by the technical method of the present invention:

[0158]

[0159] in, as follows:

[0160]

[0161] See also Figure 3 , is to use the standard particle swarm algorithm (PSO) with fitness function J FOPID Schematic diagram of the fitness function convergence curve used as the optimization criterion to adjust the parameters of the fractional-order PID controller in the automatic voltage regulation system. From the figure, it can be seen that the convergence of the convergence curve is poor, the iteration speed is slow, and it converges to the optimal fitness function value after about 38 generations.

[0162] See also Figure 4 , is to use the nonlinear dynamic chaotic particle swarm optimization algorithm (NDCPSO) proposed by the present invention with the fitness function J FOPID The schematic diagram of the fitness function convergence curve for tuning the parameters of the fractional-order PID controller in the automatic voltage regulation system as an optimization criterion. From the figure, it can be seen that the algorithm has strong convergence and fast iteration speed, and it converges to the optimal fitness function value in about 10 generations.

[0163] See also Figure 5, is to use the nonlinear dynamic chaotic particle swarm optimization algorithm (NDCPSO) proposed by the present invention with the fitness function J FOPID The fractional-order PID controller parameter K is set as an optimization criterion in the automatic voltage regulation system. p , K i , K d Schematic diagram of the final convergence position of particles. It can be seen from the figure that when the maximum number of iterations is reached, all particles gather into one point and the optimal K is found. p , K i , K d The value of .

[0164] See also Figure 6 , is to use the nonlinear dynamic chaotic particle swarm optimization algorithm (NDCPSO) proposed by the present invention with the fitness function J FOPID Schematic diagram of the final convergence position of particles used as optimization criteria to adjust the parameters λ and μ of the fractional-order PID controller in the automatic voltage regulation system. It can be seen from the figure that when the maximum number of iterations is reached, all particles gather into one point and find the optimal values ​​of λ and μ.

[0165] See also Figure 7 , is a step response curve diagram of the automatic voltage regulation system of this embodiment without fractional-order PID control. It can be seen that its stability is very poor, the overshoot and adjustment time are also very large, and there is a steady-state error.

[0166] See also Figure 8 , which is a comparison diagram of the step response curves of the automatic voltage regulation system of the fractional-order PID controller designed based on the method of the present invention and the automatic voltage regulation system of the prior art. It can be seen that the fractional-order PID controller designed using the technical solution of the present invention minimizes the overshoot of the automatic voltage regulation system and the shortest adjustment time.

Claims

1. A design method for a fractional-order PID controller, characterized in that: Including the structural update and parameter tuning of the fractional-order PID controller, the steps are as follows: Step 1: Obtain the optimal simplified Oustaloup filter; Step 2: Determine the adjustable coefficients in the optimal simplified Oustaloup filter; Step 3: Update the fractional-order PID controller structure; Step 4: Obtain nonlinear dynamic chaotic particle swarm optimization algorithm; Step 5: Adjust the parameters of the fractional-order PID controller; Step 6: Get the final fractional-order PID controller; In step 1, the optimal simplified Oustaloup filter obtained has the following structure: in, represents the optimal simplified Oustaloup filter obtained by transformation, b, c, d are adjustable coefficients, s is the Laplace operator, α is the order of the fractional calculus operator, N is the filter order, ω u represents the upper bound of the fitting frequency band, ω l Indicates the lower bound of the fitting frequency band; In step 2, an optimization objective function is designed to determine The adjustable coefficients b, c, d in the optimization objective function formula are as follows: Among them, J bcd represents the optimization objective function, ω represents the frequency, M r , P r For α The actual amplitude and phase frequency, M a , P a is the approximate amplitude and phase frequency, ω u represents the upper bound of the fitting frequency band, ω l It represents the lower bound of the fitting frequency band. The adjustable coefficients b, c, and d can be determined according to different optimization objective functions and search ranges. In step 3, after obtaining the optimal compact Oustaloup filter, use the optimal compact Oustaloup filter Complete the digital implementation of the fractional-order operator and update the fractional-order PID controller structure. The updated fractional-order PID controller structure expression is as follows: Among them, G c (s) represents the fractional-order PID controller, s is the Laplace operator, K p is the proportionality coefficient, K i is the integration coefficient, K d is the differential coefficient, λ is the integration order, and μ is the differential order; is the optimal simplified Oustaloup filter; then They are as follows:

2. The method according to claim 1, characterized in that In step 4, the nonlinear dynamic chaotic particle swarm optimization algorithm obtained is as follows: v id (it+1)=w(it)v id (it)+c1(it)r1(it)[p id (it)-x id (it)]+c2(it)r2(it)[p gd (it)-x id (it)] x id (it+1)=x id (it)+v id (it+1),d=1,2,... Among them, it represents the current iteration number, i represents a particle in the population, d represents the dimension of the solution space, and v id (it+1) represents the moving speed of the d-dimensional i-th particle at the it+1-th iteration, x id (it+1) represents the position of the d-th dimension of the i-th particle at the it+1-th iteration, p id (it) represents the individual optimal position of the d-dimensional particle at the it-th iteration, p gd (it) represents the global optimal position of the d-dimensional population at the it-th iteration, c1(it) and c2(it) are the dual asynchronous nonlinear dynamic learning factors at the it-th iteration, r1(it) and r2(it) are the chaos scaling coefficients at the it-th iteration, and w(it) is the chaos inertia weight at the it-th iteration.

3. The method according to claim 2, characterized in that The formulas of dual asynchronous nonlinear dynamic learning factors c1(it) and c2(it) are as follows: Among them, it is the current iteration number, it max is the maximum number of iterations, f1 and f2 are weight adjustment factors, and f 1min 、f 1max is the minimum and maximum value of f1, f 2min 、f 2max is the minimum and maximum value of f2, c 1min 、c 1max is the minimum and maximum value of the learning factor c1, c 2min 、c 2max is the minimum and maximum value of the learning factor c2.

4. The method according to claim 2, characterized in that: The formulas for chaos scaling coefficients r1(it) and r2(it) are as follows: r1(it)=0.500+0.476mod[z1(it-1)+δ+gz2(it),1] r2(it)=0.500+0.476cos(2πz1(it-1)+e -r z2(it-1) Wherein, it represents the current cycle number, z1 and z2 are chaos mapping variables, mod represents the modulo operation of z1(it-1)+δ+gz2(it), cos represents the cosine operation, e is the natural logarithm, π is the pi, and r, g, δ are chaos regulation parameters.

5. The method according to claim 2, characterized in that: The formula of chaotic inertia weight w(it) is as follows: w'(it)=0.650+0.238mod[z1(it-1)+δ+gz2(it),1] w(it)=0.650+0.238cos(2πz1(it-1)+e -r z2(it-1) Wherein, it represents the current cycle number, z1 and z2 are chaotic mapping variables, mod represents the modulo operation of z1(it-1)+δ+gz2(it), cos represents the cosine operation, e is the natural logarithm, π is the pi, w'(it) is the chaotic mapping sequence of w(it), and r, g, δ are chaotic adjustment parameters.

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