PMSM non-singular fast terminal sliding mode control method based on improved zebra optimization algorithm

Abstract

The invention discloses a PMSM non-singular fast terminal sliding mode control method based on an improved zebra optimization algorithm, and belongs to the technical field of PMSM terminal sliding mode control, and the method specifically comprises the steps: building a mathematical model of a PMSM system under a d-q axis coordinate system; constructing a sliding mode controller according to the established mathematical model; an improved ZOA algorithm is adopted to optimize parameters of the sliding mode controller; the improved ZOA algorithm uses Chebyshev mapping to initialize the population. According to the method, local optimum can be avoided, a globally optimal solution is obtained, the chattering phenomenon possibly generated in the switching process of sliding mode control and the influence on the overall performance of the PMSM system are avoided, and the stability and the control precision of the PMSM system are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of PMSM control methods, and specifically relates to a non-singular fast terminal sliding mode control method for PMSM based on an improved zebra optimization algorithm. Background Art

[0002] Permanent Magnet Synchronous Motors (PMSMs) have significant advantages in terms of efficiency, structure, power density, operating characteristics, etc., and have been widely used in high-performance fields such as exoskeleton robots and aerospace. A PMSM is a typical highly nonlinear and multivariable coupling system, and parameter perturbations are likely to occur during operation. Traditional PI controllers cannot meet the requirements of high-performance servo systems. Therefore, it is necessary to introduce a high-performance fast terminal sliding mode controller to control the PMSM. However, whether the parameters of the controller match directly affects the performance of the system. To avoid repeated trial and error of parameters and shorten the development cycle, researchers apply intelligent optimization algorithms to the parameter optimization problem.

[0003] The dynamic performance of a PMSM is affected by the controller parameters, and traditional parameter adjustment methods are often cumbersome and difficult to achieve the optimal effect. Existing optimization algorithms may easily fall into local optima during the search process, resulting in the inability to find the global optimal solution, thus affecting the performance of the PMSM control system.

[0004] Chattering may occur during the switching process of sliding mode control, which will affect the stability and control accuracy of the system. The parameter selection of the controller has a great influence on the performance of the system, but traditional parameter adjustment methods often require a large number of experiments and adjustments, and the process is cumbersome and difficult to find the optimal parameters. Summary of the Invention

[0005] The technical problem to be solved by the present invention is: to provide a non-singular fast terminal sliding mode control method for PMSM based on an Improved Zebra Optimization Algorithm (IZOA), which can enhance the distribution uniformity of the initial population in the search space, thereby avoiding the algorithm from falling into local optima, improving the search diversity during the search process, helping the algorithm to jump out of local optima, and ensuring that the algorithm converges efficiently to the optimal solution, improving the sliding mode control accuracy and the stability of the PMSM system.

[0006] The technical solution adopted by the present invention is: a non-singular fast terminal sliding mode control method for PMSM based on an improved zebra optimization algorithm, and the method includes the following steps:

[0007] Step 1: Establish a mathematical model of the PMSM system in the d-q axis coordinate system;

[0008] Step 2: Construct a sliding mode controller based on the mathematical model in Step 1;

[0009] Step 3: Optimize the parameters of the sliding mode controller using an improved ZOA algorithm; the improved ZOA algorithm initializes the population using Chebyshev mapping.

[0010] Furthermore, the mathematical model of the PMSM system established in the above Step 1 is as follows:

[0011]

[0012] In the formula, in the d-q axis coordinate system, u d and i d are the voltage and current on the d axis respectively, u q and i q are the voltage and current on the q axis respectively; L d and L q are the inductances on the d axis and q axis respectively; R s represents the stator resistance; ψ d and ψ q are the magnetic fluxes on the d axis and q axis respectively; ψ f represents the permanent magnet magnetic flux; P n represents the number of pole pairs of the motor; ω e is the electrical angular velocity; ω m is the mechanical angular velocity; θ m represents the mechanical angle; J is the moment of inertia; T L is the load torque; B is the friction coefficient, and the equation of the electromagnetic torque T e is expressed as: T e = 1.5P n i q ψ f .

[0013] Furthermore, the construction method of the sliding mode controller in the above Step 2 is as follows:

[0014] Let the state variables of the PMSM system be:

[0015]

[0016] In the formula, θ * is the motor set angle; θ m is the mechanical angle; e is the error angle, and the state variables of the system are x1 and x2;

[0017] Select the following non-singular fast terminal sliding mode surface:

[0018]

[0019] In the formula, p and q are odd numbers, and p > q > 0, β1 > 0 and β2 > 0 and both are integral gains;

[0020] The reaching law of the sliding mode controller is designed as follows:

[0021]

[0022] where: tanh(s) is the hyperbolic tangent function; k1, k2, α, δ, and σ are design parameters, and k1 > 0, k2 > 0, 0 < α < 1, 0 < δ < 1, and σ > 1;

[0023] Differentiating the sliding mode surface in Equation (3) gives:

[0024]

[0025] Combining with the motor motion equation in Equation (1), the system control law is:

[0026]

[0027] Furthermore, the optimized controller parameters in Step 3 above include p, q, β1, β2, k1, k2, α, δ, and σ. The improved ZOA algorithm using the optimized controller parameters is:

[0028] The ZOA algorithm initializes the ZOA algorithm population using the Chebyshev mapping to obtain:

[0029]

[0030] where, Z i,j is the initialized population individual, lb j is the lower bound of the optimal value, ub j is the upper bound of the optimal value, X i,j represents the value of the i-th individual in the current population on the j-th decision variable;

[0031] Introduce a random perturbation mechanism. In the foraging behavior stage and the defense strategy stage, according to the comparison of the random number and the perturbation frequency, randomly perturb the current solution;

[0032] The position update in the foraging stage is expressed as:

[0033]

[0034] where, represents the new position of the zebra individual in the foraging stage, r is a random number between [0, 1], I is a random value of the set {1, 2}, F i represents the objective function value of the i-th zebra, t represents the number of iterations, T max is the maximum number of iterations, F per is the adaptive perturbation frequency threshold, and its expression is:

[0035] F per = h(1 - t / T max ) (14)

[0036] where h is a constant and is the adaptive initial value;

[0037] Similarly, in the defense strategy stage, use P s to select S1 or S2 to calculate the strategy, and use r and F per to judge whether to increase the perturbation strategy. The updated S1 and S2 for position are respectively expressed as:

[0038]

[0039] where represents the new position of the zebra individual in the attack or defense stage, R = 0.01, P s ∈ [0,1] represents the probability of choosing attack or defense when threatened, T max represents the maximum number of iterations, and AZ j represents the state of the zebra when threatened.

[0040] Advantages of the present invention: Compared with the prior art, the effects of the present invention are as follows:

[0041] (1) The present invention uses an improved zebra algorithm to optimize the parameters of the sliding mode controller, which can avoid falling into local optima, obtain the global optimal solution, avoid the chattering phenomenon that may occur during the switching process of the sliding mode control and the impact on the overall performance of the PMSM system, improve the stability and control accuracy of the PMSM system, effectively solve the problem that traditional parameter adjustment methods often require a large number of experiments and adjustments, which is cumbersome and difficult to find the optimal parameters, and also solve the problem that it is easy to fall into local optima when using the traditional zebra optimization algorithm to optimize the controller parameters, resulting in the inability to obtain the global optimal solution. The method of the present invention realizes the improvement of the motor control accuracy and efficiency, can reduce energy loss and lower the motor operation cost. For example, when the motor load changes, it can more accurately adjust the speed and torque of the motor, so that it always operates in the optimal working state, avoiding energy waste caused by control lag or overshoot;

[0042] (2) IZOA uses Chebyshev mapping initialization and random perturbation strategy to effectively avoid the local optimum problem, ensure finding the global optimal solution, and improve the overall performance of the PMSM system;

[0043] (3) The use of IZOA greatly simplifies the debugging process of the controller parameters, reduces human intervention, and improves the debugging efficiency. By comparing IZOA with other intelligent optimization algorithms, the experimental results show that IZOA performs excellently in optimizing the parameters of the PMSM controller and is applicable to a variety of industrial scenarios, such as electric vehicle drive systems, industrial robot joint control, etc.;

[0044] In summary, aiming at the problems of premature convergence and local optimum entrapment of ZOA and the problems existing in traditional methods in sliding mode control, a nonsingular fast terminal sliding mode control method for PMSM based on the Improved Zebra Optimization Algorithm (IZOA) is proposed. The improved zebra optimization algorithm is used to tune the sliding mode control parameters. Specifically, the Chebyshev mapping is used to initialize the population, enhancing the distribution uniformity of the initial population in the search space, thus avoiding the algorithm falling into local optimum. During the search process, a random perturbation mechanism is introduced to improve search diversity and help the algorithm jump out of local optimum. In addition, an adaptive perturbation frequency adjustment strategy is adopted. In the initial stage, extensive search is carried out through high-frequency perturbation, and the perturbation frequency is gradually reduced in the later stage to strengthen local search, ensuring that the algorithm converges efficiently to the optimal solution. Description of the Drawings

[0045] Figure 1 It is a comparison diagram of reaching law phase trajectories;

[0046] Figure 2 It is a comparison diagram of reaching times of the reaching law;

[0047] Figure 3 It is a comparison diagram of algorithms;

[0048] Figure 4 It is a block diagram of the vector control structure of the permanent magnet synchronous motor;

[0049] Figure 5 It is an experimental curve diagram of the step response;

[0050] Figure 6 It is an experimental curve diagram of the sine signal following;

[0051] Figure 7 It is a schematic flow diagram of the nonsingular fast terminal sliding mode control method for PMSM based on the improved zebra optimization algorithm. Detailed Implementation Manner

[0052] Example 1: As Figures 1 to 7 shown, the nonsingular fast terminal sliding mode control method for PMSM based on the improved zebra optimization algorithm includes the following steps:

[0053] Step 1: Establish the mathematical model of the PMSM system in the d-q axis coordinate system;

[0054] For simplified analysis, without considering core saturation, hysteresis, and eddy current losses, and ignoring the effects caused by rotor shaft friction and stator slot irregularities, the mathematical model of the PMSM system established in Step 1 is:

[0055]

[0056] where, in the d-q axis coordinate system, u d and i d are the voltage and current on the d axis respectively, u q and i q are the voltage and current on the q axis respectively; L d and L q are the inductances on the d axis and q axis respectively; R s represents the stator resistance; ψ d and ψ q are the magnetic fluxes on the d axis and q axis respectively; ψ f represents the permanent magnet flux linkage; P n represents the number of pole pairs of the motor; ω e is the electrical angular velocity; ω m is the mechanical angular velocity; θ m represents the mechanical angle; J is the moment of inertia; T L is the load torque; B is the friction coefficient, and the equation of the electromagnetic torque T e is expressed as: T e =1.5P n i q ψ f ;

[0057] Step 2: Construct a sliding mode controller according to the mathematical model in Step 1;

[0058] The construction method of the sliding mode controller in Step 2 is as follows:

[0059] For the convenience of controller design, let the state variables of the PMSM system be:

[0060]

[0061] where, θ * is the set angle of the motor; θ m is the mechanical angle; e is the error angle, and the state variables of the system are x1 and x2;

[0062] To improve the system convergence speed and robustness, select the following non-singular fast terminal sliding mode surface:

[0063]

[0064] where, p and q are odd numbers, and p > q > 0, β1 > 0 and β2 > 0 and both are integral gains;

[0065] The convergence speed of the sliding mode controller is directly affected by the reaching law. The control effects of the traditional exponential reaching law, constant speed reaching law or power reaching law are not good. To improve the convergence speed and reduce chattering, a new reaching law is proposed as follows:

[0066]

[0067] In the formula: tanh(s) is the hyperbolic tangent function; k1, k2, α, δ, and σ are design parameters, and k1 > 0, k2 > 0, 0 < α < 1, 0 < δ < 1, and σ > 1;

[0068] Deriving the sliding mode surface formula (3) gives:

[0069]

[0070] Combining with the motor motion equation in formula (1), the system control law is:

[0071]

[0072] Comparison and stability analysis of reaching laws:

[0073] Compare the proposed new reaching law with the classical constant velocity reaching law exponential reaching law power reaching law for comparative analysis, as Figure 1 shown in the phase trajectory diagram and as Figure 2 shown in the reaching time.

[0074] As Figure 1 shown, it can be directly seen from the phase trajectory diagram of the sliding mode motion that the new reaching law has a faster reaching process and reaches the sliding mode surface first. From Figure 2 the reaching time required for the reaching process, it can be known that the reaching time of the new reaching law is 7.85×10 - 4 s, which is 1.21×10 -2 s faster than the constant velocity reaching law, and the reaching time is shortened by about 93.8%; it is 7.21×10 -3 s faster than the power reaching law, shortening by 90%; it is 4.16×10 -3 s faster than the exponential reaching law, shortening by 85%.

[0075] To verify the stability of the proposed new reaching law, select the Lyapunov function: V = 0.5s 2 . According to the Lyapunov principle, when is negative semi-definite, the system is stable, and thus:

[0076]

[0077] Based on the design parameters k1, k2, and m all being greater than 0, α and δ both being greater than 0 and less than 1, σ being greater than 1, and the hyperbolic tangent function having the same sign as s, it can be known that 1 + 1 / δ|x1| - α > 0, σ -1.5|s|> 0, so the numerator expression sk1 tanh(s) of the first term > 0, and the denominator expression α+(1 + 1 / δ|x1| - α)σ -1.5|s| > 0. In summary the control system is asymptotically stable;

[0078] IZOA combined with the new reaching law effectively reduces the system jitter in sliding mode control and improves the stability of the system. The optimized controller parameters can better suppress external interference and internal noise, further improve the control accuracy, and reduce the steady-state error;

[0079] Step 3: Optimize the parameters of the sliding mode controller using the improved ZOA algorithm; the improved ZOA algorithm initializes the population using the Chebyshev mapping. The optimized controller parameters in Step 3 include p, q, β1, β2, k1, k2, α, δ, and σ;

[0080] Traditional ZOA simulates the foraging behavior of zebras and their defense against predator attacks. Due to its strong search ability and fast convergence speed, it is widely used in the field of intelligent optimization. In the foraging strategy, the best-performing individual is regarded as the leading zebra, guiding other group members to move towards its position in the exploration space. The position update in the foraging stage can be expressed as:

[0081]

[0082]

[0083] where represents the new position of the zebra individual in the foraging stage, r is a random number between [0,1], PZ j represents the optimal pioneer zebra, I is a random value from the set {1,2}, F i represents the objective function value of the i-th zebra, X i,j represents the value of the i-th individual in the current population on the j-th decision variable, X i represents the final position of the i-th zebra individual, represents the new position calculated by the i-th zebra individual in the foraging stage, F i represents the objective function value of the current position of the i-th zebra individual, F i new ,P1 represents the new position of the i-th zebra individual corresponding objective function value.

[0084] In response to the threat of predators, zebras will choose defensive or offensive strategies. When under attack, zebras will take evasive actions near their current location, which can be represented by S1 in Equation (10). When zebras choose the offensive strategy, other zebras in the group will approach the attacked zebra and attempt to intimidate or confuse the predator by forming a defensive structure, which can be represented by S2 in Equation (10). The value of P s is used to select either S1 or S2 as the calculation strategy.

[0085]

[0086] In the formula, represents the new position of the individual zebra in the attack or defense phase, R = 0.01, P s ∈[0,1] represents the probability of choosing attack or defense when threatened, T max represents the maximum number of iterations, AZ j represents the state of the zebra when threatened, represents the new position calculated by the i-th zebra individual in the attack or defense phase, F i new,P2 represents the new position of the i-th zebra individual corresponding to the objective function value.

[0087] In the Improved Zebra Optimization Algorithm (IZOA), to optimize the performance of the algorithm, Chebyshev mapping is used for initialization to ensure the uniform distribution of the initial solutions, enhance the search breadth, avoid local optima, and accelerate the algorithm convergence. Using Chebyshev mapping to initialize the population gives:

[0088]

[0089] In the formula, Z i,j is the initialized population individual, lb j is the lower bound of the optimal value, ub j is the upper bound of the optimal value; X i,j represents the value of the i-th individual in the current population on the j-th decision variable;

[0090] The IZOA algorithm introduces a random perturbation mechanism. In the foraging behavior phase and the defense strategy phase, according to the comparison between the random number and the perturbation frequency, the current solution is randomly perturbed; this mechanism increases the search diversity of the algorithm and helps the algorithm jump out of the local optimal solution;

[0091] The position update in the foraging phase is expressed as:

[0092]

[0093] In the formula, represents the new position of the zebra individual in the foraging stage, r is a random number between [0, 1], I is a random value of the set {1, 2}, F i represents the objective function value of the i-th zebra, t represents the number of iterations, T max is the maximum number of iterations, F per is the adaptive perturbation frequency threshold, and its expression is:

[0094] F per = h(1 - t / T max )(14)

[0095] In the formula, h is a constant and the adaptive initial value; in the early stage of the algorithm, the frequency threshold is relatively high, which can encourage individuals to explore, and the threshold gradually decreases in the later stage to promote convergence.

[0096] Similarly, in the defense strategy stage, use P s value to select S1 or S2 to calculate the strategy, use r and F per value to judge whether to increase the perturbation strategy, and S1 and S2 for position update are respectively expressed as:

[0097]

[0098] In the formula, represents the new position of the zebra individual in the attack or defense stage, R = 0.01, P s ∈[0, 1] represents the probability of choosing attack or defense when threatened, T max represents the maximum number of iterations, AZ j represents the state of the zebra when threatened.

[0099] The specific steps of the IZOA algorithm are as follows:

[0100]

[0101] To illustrate the effect of the present invention, an IZOA performance simulation comparison experiment is carried out:

[0102] To confirm the performance of IZOA, the present invention compares it with the standard zebra optimization (ZOA), whale algorithm (WOA), grey wolf optimization (GWO) and sparrow search (SSA). Using the national standard benchmark functions, including the unimodal and multimodal functions shown in Table 1, they are tested. Each total algorithm is set to run independently 30 times, the population size is 50, and the number of iterations is 500 times. The remaining parameters adopt the default values of the original algorithm. The results are as Figure 3 shown.

[0103] Table 1 National standard benchmark test functions

[0104]

[0105] It can be seen from the figure that IZOA has obvious advantages in solving unimodal and multimodal problems, with a faster convergence speed and a lower fitness value.

[0106] The specific simulation process is as follows:

[0107] 1. Model construction

[0108] To test the practical application ability of IZOA, a surface-mounted permanent magnet synchronous motor vector control system is built in MATLAB / Simulink, and it is used to adjust the controller parameters respectively with the whale optimization algorithm (WOA), gray wolf algorithm (GWO), sparrow search algorithm (SSA) and traditional zebra optimization algorithm (ZOA).

[0109] The overall control structure of the control system is shown in Figure 4. To compare the simulation results, the current loop parameters are kept unchanged, and the above-mentioned intelligent algorithms are used to optimize the controller parameters respectively. The optimized parameters include: p, q, β1, β2, k1, k2, α, δ and σ.

[0110] The parameters of the permanent magnet synchronous motor model used in the simulation are shown in the following table.

[0111] Table 2 Specific parameters of the permanent magnet synchronous motor

[0112]

[0113] 2. Step response simulation experiment

[0114] Set the given position to 1 rad, and add a 5 N·m load at 1.5 s for the step simulation experiment. The simulation results are as Figure 5 shown. It can be seen from Figure 5 (b) that the parameters adjusted by these 5 methods can all make the PMSM quickly follow the step signal. The adjustment time of GWO is 0.87 s, and the adjustment times of WOA, SSA, and ZOA are 0.74 s, 0.62 s, and 0.61 s respectively; while the parameters adjusted by IZOA have a faster adjustment speed when approaching the target position, and its adjustment time is: 0.18 s. The adjustment time of IZOA is shortened by 70.49%, 70.97%, 75.68%, and 79.31% compared with ZOA, SSA, WOA, and GWO respectively.

[0115] When the load disturbance is increased, from Figure 5It can be seen from (c) that the parameters adjusted by SSA have limited adaptability to the load, and the position fluctuation is 0.041 rad; the position fluctuations of GWO, WOA, and ZOA are 0.032 rad, 0.017 rad, and 0.015 rad respectively; while the parameters adjusted by IZOA have strong anti-interference ability, the position fluctuation is only 0.0035 rad, and the recovery time is short.

[0116] 3. Sine Signal Following Experiment

[0117] In actual working conditions, the PMSM needs to rotate at different angles according to the command. To verify the robustness of the controller against external interference, y1 = π / 3sin(4πt) is selected as the position tracking reference curve, y2 = π / 4sin(4πt) is selected as the load change curve, and the motor starts with load. The above five methods are also used to adjust the controller parameters, and the position following curves are shown in Figure 6.

[0118] From Figure 6 (a), it can be seen that all methods can make the motor effectively follow the reference curve, but there are obvious errors and hysteresis based on the SSA method, which is caused by the premature convergence of the SSA method and easy to fall into the local optimum. In Figure 6 (b) shows the local enlarged view of the following effect, and it can be concluded that the parameters adjusted by IZOA have a better following effect, which benefits from the perturbation strategy that enables the algorithm to jump out of the local optimum. Figure 6 (c) is the following error curve. The parameters adjusted by SSA cause a large following error for the PMSM, reaching 0.160 rad; the maximum following errors of GWO, WOA, and ZOA are 0.116 rad, 0.08 rad, and 0.03 rad respectively, while the maximum following error of IZOA is only 0.015 rad, which is reduced by 90.63%, 86.21%, 81.25%, and 50.00% compared with SSA, GWO, WOA, and ZOA respectively.

[0119] Conclusion of the simulation experiment: To improve the position following performance of the PMSM and thus improve the control quality of the system, a new reaching law is established and introduced into the position-speed loop. To avoid repeated trial and error of parameters, a nonsingular fast terminal sliding mode control method based on IZOA is proposed. In the IZOA method, Chebyshev mapping is used for population initialization, and an adaptive adjustment perturbation strategy is added to jump out of the local optimum, enhancing the search and convergence ability of the algorithm, and improving the efficiency and performance of the algorithm. Taking the permanent magnet synchronous motor position control system as the optimization target, the superiority of IZOA is demonstrated by comparing with ZOA, WOA, GWO, and SSA methods.

[0120] Through the above description of the embodiments in conjunction with the accompanying drawings, those skilled in the art can understand that for the convenience and conciseness of description, only the above division of each functional module is used as an example. In practical applications, the required measurement parameters and objects can be adjusted according to needs, and the proposed invention has a certain generality.

Claims

1. A PMSM non-singular fast terminal sliding mode control method based on an improved zebra optimization algorithm is characterized by: The method comprises the following steps: Step 1: Establish a mathematical model of the PMSM system in the dq axis coordinate system; Step 2: construct a sliding mode controller based on the mathematical model of step 1; Step 3: Use the improved ZOA algorithm to optimize the sliding mode controller parameters; the improved ZOA algorithm uses Chebyshev mapping to initialize the population.

2. The PMSM non-singular fast terminal sliding mode control method based on the improved zebra optimization algorithm according to claim 1 is characterized in that: The mathematical model of the PMSM system established in step 1 is: In the formula, in the dq axis coordinate system, u d and i d are the voltage and current on the d-axis, u q and i q are the voltage and current on the q axis respectively; L d and L q are the inductances on the d-axis and q-axis respectively; R s represents the stator resistance; ψ d and ψ q are the magnetic flux on the d-axis and q-axis respectively; ψ f represents the permanent magnet flux; P n Indicates the number of motor poles; ω e is the electrical angular velocity; ω m is the mechanical angular velocity; θ m represents the mechanical angle; J is the moment of inertia; T L is the load torque; B is the friction coefficient, and the electromagnetic torque T e The equation is expressed as: e =1.5P n i q ψ f .

3. The PMSM non-singular fast terminal sliding mode control method based on the improved zebra optimization algorithm according to claim 2 is characterized in that: The construction method of the sliding mode controller in step 2 is: Assume the state variables of the PMSM system are: In the formula, θ * Set the angle for the motor; θ m is the mechanical angle; e is the error angle, and the state variables of the system are x1 and x2; Select the following non-singular fast terminal sliding surface: Where p and q are odd numbers, and p>q>0, β1>0 and β2>0 and are both integral gains; The reaching law of the sliding mode controller is as follows: Wherein: tanh(s) is the hyperbolic tangent function; k1, k2, α, δ, and σ are design parameters, and k1>0, k2>0, 0<α<1, 0<δ<1, and σ>1; Derivative of the sliding surface formula (3) yields: Combining the motor motion equation in formula (1), the system control law is:

4. The PMSM non-singular fast terminal sliding mode control method based on the improved zebra optimization algorithm according to claim 3 is characterized in that: The controller parameters optimized in step 3 include p, q, β1, β2, k1, k2, α, δ and σ. The improved ZOA algorithm for optimizing the controller parameters is: The ZOA algorithm uses Chebyshev mapping to initialize the ZOA algorithm population: In the formula, Z i,j is the initialized population individual, lb j is the lower bound of the optimal value, ub j is the upper bound of the optimal value, X i,j Represents the value of the i-th individual in the current population on the j-th decision variable; A random perturbation mechanism is introduced. In the foraging behavior stage and the defense strategy stage, the current solution is randomly perturbed according to the comparison between the random number and the perturbation frequency. The position update during the foraging phase is expressed as: In the formula, represents the new position of the zebra individual during the foraging phase, r is a random number between [0,1], I is a random value in the set {1,2}, and F i represents the objective function value of the i-th zebra, t represents the number of iterations, T max is the maximum number of iterations, F per is the adaptive disturbance frequency threshold, and its expression is: F per =h(1-t / T max ) (14) In the formula, h is a constant and is the adaptive initial value; Similarly, in the defensive strategy phase, use P s The value of selects S1 or S2 calculation strategy, using r and F per The value of determines whether to add a disturbance strategy. The S1 and S2 of the position update are expressed as: In the formula, represents the new position of the zebra in the attack or defense phase, R = 0.01, P s ∈[0,1] represents the probability of attacking or defending when threatened, T max Indicates the maximum number of iterations, AZ j Indicates the state of a zebra when threatened.

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