A design method of fractional order PID controller based on cuckoo search algorithm

Abstract

The application relates to a fractional order PID controller design method based on a cuckoo search algorithm and belongs to the automatic control field. Five parameters k p , k i , k d , lambda and mu contained in the fractional order PID controller are optimized and set by the cuckoo search algorithm. The steps are as follows: (1) initializing algorithm parameters; (2) generating a random solution by using Levy flight; (3) evaluating a specific objective function J to obtain an optimal solution of the current iteration; (4) using a probability P to determine whether to keep or change the current solution; (5) if the maximum iteration number is reached, the optimal solution is saved as the control parameter of the fractional order PID, otherwise, returning to step (2) to continue iteration. The objective function of the cuckoo search algorithm is constructed by using the frequency domain constraint index and the time domain constraint index of a control system, the controller is designed according to the control system characteristics, the fractional order PID controller setting problem is effectively solved, and the control performance of the system is optimized.

Description

Technical Field

[0001] This invention belongs to the field of automatic control, specifically relating to a design method for a fractional-order PID controller based on the cuckoo search algorithm. Background Technology

[0002] In control systems, a stronger integral PID controller results in a smaller steady-state error but reduces the relative stability of the closed-loop system; conversely, a stronger derivative PID controller increases the stability margin of the closed-loop system but weakens noise suppression capabilities. The fractional-order PID controller proposed by Professor Podlubny... λ D μ It can serve as a good compromise controller between integer-order PID controllers and fractional-order PID controllers, which can effectively balance system accuracy and stability. λ D μ The orders λ and μ can be any real numbers; the traditional PID controller is a special case where λ = 1 and μ = 1. Compared to the traditional PID controller, the fractional-order PID controller has improved handling of parameter uncertainties and good anti-interference ability, minimizing steady-state error. While five parameters increase the flexibility of controller design, they also increase the difficulty of controller tuning. Currently, the design methods for fractional-order PID controllers are mainly divided into two categories: frequency domain methods and optimization methods. A typical frequency domain method designs the controller by given the crossover frequency and phase margin; due to limited constraints, it can generally only design PI... λ Or PD μ Controllers; optimization algorithms design controllers by constructing objective functions and constraints based on given control performance indicators. If the algorithm parameters are not set properly, they can easily get trapped in local optima, and some optimization algorithms have the problem of large computational load. Summary of the Invention

[0003] (a) Technical problems to be solved

[0004] The technical problem to be solved by this invention is to provide a design method for a fractional-order PID controller based on the cuckoo search algorithm. By constructing an objective function with given frequency domain index and time domain index constraints, the method takes into account both the frequency domain and time domain performance of the control system, effectively solves the problem of difficult tuning of fractional-order PID controller, and improves the control performance of the system.

[0005] (II) Technical Solution

[0006] To address the aforementioned technical problems, this invention provides a design method for a fractional-order PID controller based on the Cuckoo Search algorithm. The Cuckoo Search algorithm is used to optimize the five parameters k to be tuned in the fractional-order PID controller. p k i k d The optimization tuning of λ and μ includes the following steps:

[0007] Step S1: Initialize algorithm parameters;

[0008] Step S2: Generate random solutions using Levy flight;

[0009] Step S3: Evaluate the specific objective function J to obtain the optimal solution for the current iteration;

[0010] Step S4: Use probability P α To decide whether to keep or change the current solution;

[0011] Step S5: If the maximum number of iterations T is reached, save the optimal solution as the control parameters for the fractional PID controller; otherwise, return to S2 to continue iterating.

[0012] The fractional-order PID controller includes: a proportional element k p Points system and the differential element k d s μ Its transfer function is:

[0013]

[0014] In the formula, s represents the complex variable in the transfer function, and k p k is the proportional coefficient of the proportional element. i k is the proportionality coefficient of the integral element. d λ is the proportionality coefficient of the differential element, λ is the differential order of the integral element, and u is the differential order of the differential element.

[0015] The algorithm parameters in step S1 include: basic parameters of the cuckoo search algorithm and control system parameters;

[0016] The basic parameters of the cuckoo search algorithm include: number of solutions N, problem dimension D, total number of iterations T, and discovery probability P. α Algorithm search boundary X max =[x p ,x i ,x d ,x λ ,x μ ],Y min =[y p ,y i ,y d ,y λ ,y μ ];

[0017] The control system parameters are set to bandwidth B. w The bandwidth B w Bandwidth B is used to measure the speed of a control system. wScope [B] min B max ].

[0018] Wherein, the algorithm boundary X max ,Y min With the fractional-order PID controller, the parameter k to be tuned p k i k d , λ, u are related, where:

[0019] k p k i k d ∈[0,∞]

[0020] λ, μ∈[0,2].

[0021] The formula for generating random solutions during Levy flight in step S2 is as follows:

[0022]

[0023] Where α is the step size, This represents point-to-point multiplication, where t represents flight time, and the Levy(β) distribution function is:

[0024]

[0025] Where Γ(·) is the gamma function, β = 1.5, and μ and v are standard Gaussian random numbers.

[0026] The objective function J of the cuckoo optimization algorithm constructed through the penalty function in step S3 is as follows:

[0027] J = [C1 L] a ]×F1

[0028] L a = [C2 ITAE]×F2

[0029] Where C1 and C2 are penalty factors, F1 and F2 are conditional constraint functions, ITAE is the integral index of time multiplied by absolute error, and L a For the adaptive value.

[0030] The system constraints processed by the penalty function in step S3 include frequency domain constraints and time domain constraints.

[0031] The frequency domain constraint includes the magnitude margin G. m Phase margin P m and bandwidth B w When G m >1, P mWhen the value is greater than 0, the system is stable. If the system is unstable, the penalty factor is C1, and the corresponding constraint function is F1, as shown in the following formula:

[0032]

[0033] in

[0034] If the system bandwidth B w If the setting range is exceeded, the corresponding penalty factor C2 and the corresponding condition constraint function F2 are as follows:

[0035]

[0036] The time-domain constraint uses the error integral function ITAE index as an indicator to measure the time-domain control performance, as shown in the following formula:

[0037]

[0038] Where e(t) represents the deviation between the actual output and the expected output, and t is time.

[0039] In step S4, the cuckoo search algorithm employs an elimination mechanism, where the host bird is eliminated with a certain probability P. α After discovering the invasive bird, reconstruct the nest location path and generate a random number r that follows a uniform distribution within the range [1,0]. If the random number r > P α Then the solution X is eliminated. t+1 And the new solution replaces the old one, and the local walk path function of the new solution is shown in the following equation. At the same time, the objective function J is calculated:

[0040]

[0041] Where γ, ∈ [a, b] are random numbers following a uniform distribution, and Heaviside(x) is the jump function, X = 1 when x > 0 and = 0 when x < 0. i X j It is any two other bird nests.

[0042] In step S5, if the maximum number of iterations T is reached, the optimal solution is saved as the control parameters for the fractional-order PID controller; otherwise, the process returns to S2 to continue iterating.

[0043] In step S4, the probability of reconstructing the nest location path after discovering the invasive bird is set as P. α =0.25.

[0044] (III) Beneficial Effects

[0045] Compared with existing technologies, the present invention has the following advantages: Currently, most frequency domain design methods are unable to design with 5 tuning parameters k due to limited frequency domain constraints.p k i k d Fractional-order PID controllers with parameters λ and μ are often designed using frequency domain methods. However, the presence of integral or derivative terms in these controllers necessitates solving complex nonlinear equations, increasing the difficulty of frequency domain design. The method proposed in this invention can design a fractional-order PID controller with five parameters without solving complex nonlinear equations. Furthermore, compared to other optimization design methods, this invention utilizes the Cuckoo Search algorithm, effectively improving optimization accuracy and convergence speed. It also integrates frequency and time domain indices into the objective function, allowing for targeted controller design and optimized system control performance. Attached Figure Description

[0046] Figure 1 This is a flowchart of the design method for a fractional-order PID controller based on the cuckoo search algorithm of the present invention;

[0047] Figure 2 This is a control block diagram of the inertial stabilization platform system of the present invention;

[0048] Figure 3 The parameters of the fractional-order PID controller in this invention are based on the Levy flight random walk graph;

[0049] Figure 4 The fitness curve of the fractional-order PID controller design method based on the cuckoo search algorithm of this invention is shown below.

[0050] Figure 5 This is the open-loop Bode plot of the inertial stabilization platform system of the present invention;

[0051] Figure 6 This is the closed-loop Bode diagram of the inertial stabilization platform system of the present invention;

[0052] Figure 7 This is a comparison diagram of the unit step response of the present invention. Detailed Implementation

[0053] To make the objectives, contents, and advantages of the present invention clearer, the specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples.

[0054] This embodiment provides a design method for a fractional-order PID controller based on the Cuckoo Search algorithm. The Cuckoo Search algorithm is used to optimize the five parameters k to be tuned in the fractional-order PID controller. p k i k d The optimization tuning of λ and μ includes the following steps:

[0055] Step S1: Initialize algorithm parameters;

[0056] Step S2: Generate random solutions using Levy flight;

[0057] Step S3: Evaluate the specific objective function J to obtain the optimal solution for the current iteration;

[0058] Step S4: Use probability P α To decide whether to keep or change the current solution;

[0059] Step S5: If the maximum number of iterations T is reached, save the optimal solution as the control parameters for the fractional PID controller; otherwise, return to S2 to continue iterating.

[0060] The fractional-order PID controller includes: a proportional element k p Points system and the differential element k d s μ Its transfer function is:

[0061]

[0062] In the formula, s represents the complex variable in the transfer function, and k p k is the proportional coefficient of the proportional element. i k is the proportionality coefficient of the integral element. d λ is the proportionality coefficient of the differential element, λ is the differential order of the integral element, and u is the differential order of the differential element.

[0063] Fractional PID parameter tuning involves finding a set of parameters {k} within the solution region. p k i k d The system is configured such that λ, μ}, λ, μ}, which allows the system to meet certain dynamic performance requirements.

[0064] The algorithm parameters in step S1 include: basic parameters of the cuckoo search algorithm and control system parameters;

[0065] The basic parameters of the cuckoo search algorithm include: population size (number of solutions) N = 5; number of eggs in a nest (five control parameters) D = 5; total number of iterations T = 1000; and discovery probability P. α =0.25; Algorithm search boundary X max =[0.1,0.2,0.1,1.1,1.1], Y min = [0,0,0,1,1]; Control system parameters: bandwidth B w The (HZ) range is set to [40, 70].

[0066] Wherein, the algorithm boundary X max ,Y min With the fractional-order PID controller, the parameter k to be tunedp k i k d , λ, u are related, where:

[0067] k p k i k d ∈[0,∞]

[0068] λ, μ∈[0,2].

[0069] The formula for generating random solutions during Levy flight in step S2 is as follows:

[0070]

[0071] Where α is the step size; the step size is increased if the distance is long and decreased if the distance is short. For this system, α = [0.001, 0.002, 0.001, 0.011, 0.011]. This represents point-to-point multiplication, where t represents flight time, and the Levy(β) distribution function is:

[0072]

[0073] Where Γ(·) is the gamma function, β = 1.5, and μ and ν are standard Gaussian distributed random numbers.

[0074] Levy flight is a quasi-stochastic process whose step size follows a Levy distribution. It is a pattern of high-frequency short-range searches interspersed with low-frequency long-range walks. The foraging trajectories of animals such as albatrosses and bees in the animal kingdom exhibit typical characteristics of Levy flight. Figure 3 The figure shows the random walk mode of the five parameters of the fractional-order PID controller in this system.

[0075] The objective function J of the cuckoo optimization algorithm constructed through the penalty function in step S3 is as follows:

[0076] J = [C1 L] a ]×F1

[0077] L a = [C2 ITAE]×F2

[0078] Where C1 and C2 are penalty factors, F1 and F2 are conditional constraint functions, ITAE is the integral index of time multiplied by absolute error, and L a For the adaptive value.

[0079] The system constraints processed by the penalty function in step S3 include frequency domain constraints and time domain constraints.

[0080] The frequency domain constraint includes the magnitude margin G.m Phase margin P m and bandwidth B w When G m >1, P m When the value is greater than 0, the system is stable. If the system is unstable, the penalty factor is C1, and the corresponding constraint function is F1, as shown in the following formula:

[0081]

[0082] in

[0083] If the system bandwidth B w If the setting range is exceeded, the corresponding penalty factor C2 and the corresponding condition constraint function F2 are as follows:

[0084]

[0085] The time-domain constraint uses the error integral function ITAE index as an indicator to measure the time-domain control performance, as shown in the following formula:

[0086]

[0087] Where e(t) represents the deviation between the actual output and the expected output, and t is time.

[0088] The objective function has the following meaning: First, it optimizes the controller design by incorporating the gain margin and phase margin into a penalty function, thereby reducing some computational overhead. For example, G... m ≤1 or P m When the system is at critical stability or instability, the objective function is directly a large penalty factor J = C1, without needing to calculate the subsequent bandwidth B. w And the error integral function ITAE. Secondly, in practical engineering, if the relationship between the resonant frequency and the system bandwidth is not handled well, resonance will occur; the time domain index is the most intuitive reflection of the system control performance; therefore, the controller is designed by constructing an objective function that integrates the time domain index and the frequency domain index, taking into account both the frequency domain characteristics and the time domain performance of the control system.

[0089] In step S4, the cuckoo search algorithm employs an elimination mechanism, where the host bird is eliminated with a certain probability P. α After discovering the invasive bird, reconstruct the nest location path and generate a random number r that follows a uniform distribution within the range [1,0]. If the random number r > P α Then the solution X is eliminated. t+1 And the new solution replaces the old one, and the local walk path function of the new solution is shown in the following equation. At the same time, the objective function J is calculated:

[0090]

[0091] Where γ, ∈ [a, b] are random numbers following a uniform distribution, and Heaviside(x) is the jump function, X = 1 when x > 0 and = 0 when x < 0. i X j It is any two other bird nests.

[0092] If step S5 reaches the maximum number of iterations T = 1000, the optimal solution is saved as the control parameters for the fractional PID controller; otherwise, the process returns to S2 to continue iterating.

[0093] like Figure 4 The figure shows the fitness evolution curve of the objective function after 1000 iterations of this system. The optimal objective function value is J = 0.7108, and the optimal solution is X. best = [0.0749, 0.1244, 0.0008, 1.0001, 1.0003], thus obtaining the fractional-order PID controller as follows:

[0094]

[0095] like Figure 5 The figure shows the open-loop Bode plot of the system under fractional-order PID control according to the present invention. The gain margin G of the system can be calculated from the figure. m =6, phase margin P m =49°;

[0096] like Figure 6 The figure shows the closed-loop Bode plot of the system. The system bandwidth B can be calculated from the figure. w =70Hz. The fractional-order PID controller designed in this invention is compared with the fractional-order PD obtained by existing frequency domain design methods and traditional PID controllers, such as... Figure 7 The unit step response diagram shown in the figure demonstrates that the fractional-order PID controller designed in this invention has a shorter response time and settling time, making it more suitable for the inertial stability platform system. The controller designed using the method proposed in this invention exhibits better control performance.

[0097] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the technical principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A design method for a fractional-order PID controller based on the cuckoo search algorithm, characterized in that, The five parameters to be tuned in the fractional-order PID controller were determined using the Cuckoo algorithm. , , The optimization tuning process includes the following steps: Step S1: Initialize algorithm parameters; Step S2: Generate random solutions using Levy flight; Step S3: Evaluate the specific objective function This yields the optimal solution for the current iteration. Step S4: Use probability To decide whether to keep or change the current solution; Step S5: If the maximum number of iterations T is reached, save the optimal solution as the control parameters for the fractional PID controller; otherwise, return to S2 to continue iterating. The formula for generating random solutions in step S2 using Levy flight is as follows: in, Step size, This represents dot-matrix. Represents flight time. The distribution function is: in, For the gamma function, take , and These are random numbers distributed according to a standard Gaussian distribution. The objective function of the cuckoo search algorithm constructed in step S3 using the penalty function is... as follows: in, , As a penalty factor, , For condition constraint functions, The integral index is time multiplied by the absolute error. For the adaptive value.

2. The design method of the fractional-order PID controller based on the cuckoo search algorithm as described in claim 1, wherein the fractional-order PID controller comprises: Proportional Link Points system and differential elements Its transfer function is: = In the formula, Represents the complex variable in the transfer function. This is the proportional coefficient for the proportional element. This is the proportionality coefficient for the integral stage. The proportionality coefficient of the differential element. For the differential order of the integration process, This is the order of the differential element.

3. The design method of the fractional-order PID controller based on the cuckoo search algorithm as described in claim 1, characterized in that, The algorithm parameters in step S1 include: basic parameters of the cuckoo search algorithm and control system parameters; The basic parameters of the cuckoo search algorithm include: number of solutions N, problem dimension D, total number of iterations T, and discovery probability. Algorithm search boundary =[ ], ; The control system parameters are set to bandwidth. B w The bandwidth Used to measure the speed and bandwidth of a control system .

4. The design method of the fractional-order PID controller based on the cuckoo search algorithm as described in claim 3, characterized in that, The algorithm's search boundary , Parameters to be tuned for fractional-order PID controller , , Related, among which: 、 、 。 5. The fractional-order PID controller design method for the cuckoo search algorithm as described in claim 1, characterized in that, The system constraints processed by the penalty function in step S3 include frequency domain constraints and time domain constraints. The frequency domain constraints include magnitude margin. Phase margin and bandwidth ,when , The system is stable at the time of occurrence; if the system is unstable, the penalty factor is... Corresponding condition constraint function The formula is as follows: in , ; If system bandwidth Penalty factor for exceeding the set range Corresponding condition constraint function The formula is as follows: The time-domain constraint uses the error integral function ITAE index as an indicator to measure the time-domain control performance, as shown in the following formula: ITAE= in This represents the deviation between the actual output and the expected output. For time.

6. The fractional-order PID controller design method for the cuckoo search algorithm as described in claim 1, characterized in that, In step S4, the cuckoo search algorithm employs an elimination mechanism, where the host bird is eliminated with a certain probability. After discovering the invasive bird, reconstruct the nest location path and generate a random number within the range [1,0] that follows a uniform distribution. If random number Then the solution is eliminated. And it is replaced by the new solution. The local walk path function of the new solution is shown in the following equation. At the same time, the objective function is calculated. : in, , Random numbers that follow a uniform distribution For jump functions, when , , It is any two other bird nests.

7. The fractional-order PID controller design method for the cuckoo search algorithm as described in claim 1, characterized in that, If step S5 reaches the maximum number of iterations T, the optimal solution is saved as the control parameters for the fractional-order PID controller; otherwise, the process returns to S2 to continue iterating.

8. The fractional-order PID controller design method for the cuckoo search algorithm as described in claim 6, characterized in that, In step S4, the probability of reconstructing the nest location path after discovering the invasive bird is set as follows: .

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