HMDV suspension modeling method based on bridge network

Abstract

The invention discloses an HMDV suspension modeling method based on a bridge network. The HMDV suspension modeling method comprises the following steps: step (1), establishing a four-phase 8 / 6-pole switched reluctance motor structure; step (2), establishing an HMDV suspension quarter model of the bridge network; (3) calculating the impedance of the bridge type network structure; step (4), optimizing a target function rule; step (5), performing parameter optimization on the electrical elements in the bridge network by using an optimization algorithm; and step (6), performing simulation verification on the optimized parameters by using Matlab. The beneficial effects are that higher biquartic impedance is obtained by changing the type and number of components; according to the electromechanical integrated suspension, the linear motor and an external circuit are used for simulating an impedance function, so that the space occupied by mechanical parts is reduced, and great convenience is provided for layout and maintenance of practical application; and the vibration of the unsprung mass at low frequency and high frequency can be inhibited, the dynamic load of the tire and the working space of the suspension are obviously improved, and the influence of the negative effect of the vertical vibration of the HMDV suspension is reduced.

Description

Technical Field

[0001] The invention relates to the technical field of HMDV suspension outer end circuit structure optimization, in particular to a HMDV suspension modeling method based on a bridge network. Background Art

[0002] HMDV suspension refers to the suspension system installed on the hub motor drive vehicle. The hub motor drive vehicle directly installs the drive motor in the wheel hub, eliminating the drive shaft, differential and other components of traditional vehicles. It has many advantages such as compact structure and flexible layout.

[0003] Most existing HMDV suspension modeling methods are still limited to the "spring-damper" binary framework. This structure is relatively simple when dealing with complex vibration environments. It is difficult to fully suppress the negative impact of vehicle vertical vibration. It often involves complex mechanical structures and dynamic equations, which increases the difficulty of modeling and simulation. In practical applications, complex mechanical structures not only increase manufacturing costs, but may also lead to difficulties in maintenance and debugging.

[0004] Inertial elements (such as inertial vessels) are rarely used in HMDV suspension systems, mainly because they are large in size and inconvenient to install, resulting in a lack of effective inertial elements in the suspension system, thus limiting the further improvement of suspension performance;

[0005] Studying the negative vertical vibration effect of hub motor-driven vehicles from the perspective of suspension system has a very prominent advantage. As a device for transmitting force and torque between the vehicle body and wheels, suspension is closely related to the vertical motion performance of the vehicle.

[0006] The network structure obtained by electrical network synthesis still has a large number of redundant components, so passive network synthesis has been focusing on the "simplest implementation" of network structure. As a classic problem in network synthesis research, the biquadratic transfer function has been widely studied and achieved a series of results.

[0007] Although high-order transfer functions can further improve suspension performance, the difficulty of passively implementing them through the network synthesis process is greatly increased;

[0008] In view of the above situation, it is necessary to improve the existing HMDV suspension modeling method so that it can adapt to the current needs of suspension system use. Summary of the invention

[0009] The present invention provides a HMDV suspension modeling method based on a bridge network, which can be used to optimize the design of the external circuit of the electromechanical inertia container, and use the bridge network to construct a double quartic impedance function to suppress the negative effect of the vertical vibration of the HMDV; thereby improving the ride comfort, driving safety and road friendliness of the vehicle.

[0010] To achieve the above-mentioned purpose, the technical solution of the present invention is a HMDV suspension modeling method based on a bridge network, comprising the following steps:

[0011] Step (1): Establish a four-phase 8 / 6-pole switched reluctance motor structure;

[0012] Step (2): Establish a quarter model of the HMDV suspension with a bridge network;

[0013] Step (3): Calculate the impedance of the bridge network structure;

[0014] Step (4): Optimize the objective function rule;

[0015] Step (5): Optimizing the parameters of the electrical components in the bridge network using an optimization algorithm and determining the parameter optimization results;

[0016] Step (6): Use Matlab to simulate and verify the optimization parameters.

[0017] To further supplement the technical solution, the electromagnetic force equation is established according to the four-phase 8 / 6-pole switched reluctance motor structure described in step (1):

[0018]

[0019] Among them, F z is the electromagnetic force, j is the number of motor phases, i is the phase current, θ is the motor rotor position, i j is the j-phase current; m, n are the number of stator poles of the motor in opposite directions, g m (n) = (1-ε)g m , g m (m) = (1 + ε) g m , g m is the motor air gap, and |mn|=4; N r is the number of rotor poles, L u is the inductance of the misaligned position, a n , b n are the polynomial fitting coefficients.

[0020] To further supplement the technical solution, the Laplace transform formula is established according to the HMDV suspension quarter model of the bridge network described in step (2):

[0021]

[0022] Among them, m s is the sprung mass, m us is the mass of the hub motor stator, k s is the suspension spring stiffness, k m is the motor equivalent stiffness, kt is the thrust coefficient of the linear motor, k e is the back EMF constant of the linear motor, Z s , Z us , Z ur , Z e (s) are the Laplace transforms of the corresponding variables, s is the Laplace variable, Z s is the vertical displacement of the sprung mass, Z us is the vertical displacement of the stator and shaft, Z ur is the vertical displacement of the rotor and tire, Z e is the impedance of the external bridge circuit, F EM is the electromagnetic force generated by the linear motor, F z It is the vertical unbalanced force caused by the hub motor.

[0023] To further supplement the technical solution, according to step (3), the total impedance Z of the external electrical network is calculated in the general form of Ze(s) using a bridge network consisting of two inductance elements, two capacitance elements and one resistance element:

[0024]

[0025] Among them, A1=RL1L2C1C2; B1=C1C2(L1+L2); C1=RL1C2+RL2(C1+C2); D1=C1; E1=R; F1=L1L2C1C2; G1=RL1L2C1;

[0026] Among them, A1~J1 are impedance coefficients.

[0027] As a further supplement to the technical solution, the objective function optimization rule is determined according to the impedance of the bridge network structure in step (3):

[0028] The traditional passive suspension is selected as the reference benchmark to establish the optimization objective function:

[0029]

[0030] Where J represents the root mean square value of the tire dynamic load DTL of the bridge network suspension structure under random road input i , compared with the root mean square value of tire dynamic load DTL of traditional passive suspension pass ratio.

[0031] As a further supplement to the technical solution, the step (3) is specifically as follows: the first inductor L1 and the second capacitor C2 are connected in series and then connected in parallel with the second inductor L2, one end of the second inductor L2 is connected to the first capacitor C1, and the other end of the second inductor L2 is connected to the first resistor R.

[0032] As a further supplement to the technical solution, in step (4), in order to obtain electrical network parameters with better performance and ensure that the optimized performance is better than the traditional structure, while taking into account the rationality of the components, the constraint range of the variables to be optimized is determined:

[0033]

[0034] Limit suspension performance to DTL i <DTL pas , B.A. i <BA pas , SWS i <SWS pas During the optimization process, the three performance indicators of the traditional passive suspension are used as comparative constraint indicators. When the optimization objective function value exceeds its constraint conditions, the optimization is unsuccessful. The objective function will be added with 100 and returned for re-optimization until the optimization objective function value meets the constraint conditions.

[0035] As a further supplement to the present technical solution, in step (5), the optimization algorithm adopts a pattern search method.

[0036] As a further supplement to the technical solution, the pattern search method comprises the following steps:

[0037] Step (5.1): Set the initial iteration point x 0 ∈R n , initial step length δ0, error ε>0, acceleration factor α≥1, reduction factor β∈(0,1). Let y=x 0 , k = 0, j = 1;

[0038] Step (5.2): Axial search: If condition ① is met: f(y+δ k e j )<f(y), then let y=y+δ k e j , go to step (5.3); if condition ② is met: f(y-δ k e j )<f(y), then let y=y-δ k e j , go to step (5.3); if both conditions ① and ② are not met, leave y unchanged and go to step (5.3);

[0039] Step (5.3): If j < n, set j = j + 1 and go to step (5.2); otherwise, set x k+1 =y, compare f(x k+1 )<f(x k ), if satisfied, go to step (5.4), otherwise, go to step (5.5).

[0040] Step (5.4): Pattern search: Let δ k+1 =δ k , y = x k+1 +α(x k+1 -x k ), let k = k + 1, j = 1, and go to step (5.2);

[0041] Step (5.5): If δ k ≤ε, then we get point x k , end; otherwise, let δ k+1 =βδ k , y = x k , x k+1 =x k , then let k = k + 1, j = 1, and go to step (5.2);

[0042] The meanings of the variables are as follows: i is the i-th iteration point, R n is an n-dimensional real number set, k is the number of pattern searches, j is the number of axial searches, δ0 is the initial step size, α is the acceleration factor, ε is the error, β is the reduction factor, and n represents the n coordinate axis directions.

[0043] The beneficial effects are that the present invention uses only four energy storage elements and one energy dissipation element to realize the double quartic impedance function; compared with realizing a series-parallel network with a high-order impedance function, only five components are needed to realize a bridge network; compared with a double cubic impedance function containing five components, the present invention obtains a higher double quartic impedance by changing the type and number of components; in addition, the mechatronic suspension uses a linear motor and an external circuit to simulate the impedance function, which reduces the space occupied by mechanical components and provides great convenience for the layout and maintenance of practical applications; and it is capable of suppressing the vibration of the unsprung mass at low and high frequencies, significantly improving the tire dynamic load and the suspension working space, and reducing the influence of the negative effects of the vertical vibration of the HMDV suspension. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a flow chart of the HMDV suspension modeling method based on the bridge network;

[0045] Figure 2 It is a schematic diagram of the structure of a four-phase 8 / 6-pole switched reluctance motor;

[0046] Figure 3 It is a schematic diagram of a typical suspension structure of HMDV;

[0047] Figure 4 It is a schematic diagram of a quarter model of the vehicle HMDV suspension;

[0048] Figure 5 It is a schematic diagram of the bridge network structure;

[0049] Figure 6 It is the overall flow chart of the pattern search method;

[0050] FIG7 is a frequency domain gain diagram of the HMDV suspension, wherein (a) is a frequency domain gain comparison diagram of vehicle body acceleration, (b) is a frequency domain gain comparison diagram of suspension dynamic travel, and (c) is a frequency domain gain comparison diagram of tire dynamic load;

[0051] FIG8 is a performance time domain index diagram of the HMDV suspension, wherein (a) is a vehicle body acceleration response diagram, (b) is a suspension dynamic travel response diagram, and (c) is a tire dynamic load response diagram;

[0052] In the figure, m s is the sprung mass, m us is the mass of the hub motor stator, m ur The mass of the motor rotor and tire, k s is the suspension spring stiffness, k m is the motor equivalent stiffness, K t is the stiffness of the tire, Z s is the vertical displacement of the sprung mass, Z us is the vertical displacement of the stator and shaft, Z ur is the vertical displacement of the rotor and tire, Z r represents road surface excitation; F EM is the electromagnetic force generated by the linear motor, V is the terminal voltage generated by the linear motor, and Z e is the impedance of the external bridge circuit. The network includes 2 inductors, 2 capacitors and 1 resistor. z It is the vertical unbalanced force caused by the hub motor, L1 is the first inductor, L2 is the second inductor, C1 is the first capacitor, C2 is the second capacitor, and R is the first resistor. DETAILED DESCRIPTION

[0053] In order to make the technical solution more clear to those skilled in the art, Figure 1 -8 Detailed description of the technical solution of the present invention:

[0054] As a unique structure, the core feature of the bridge electrical network lies in its unique connection method and impedance analysis path, which is completely different from the direct series-parallel architecture. The network needs to be converted into a more intuitive series-parallel form for calculation through the exquisite △-Y structure conversion technique, and this process shows its complexity and flexibility. For the design of the external circuit of the motor equipped with a bridge network, its significant advantages are mainly reflected in two aspects: on the one hand, the bridge network realizes the deepening and diversification of impedance expression with its special structure. Even with a limited number of components, it can achieve higher-order impedance characteristics than the traditional series-parallel network, reflecting higher design efficiency and functional integration; on the other hand, when dealing with complex unbalanced bridge scenarios, the bridge network demonstrates its excellent transformation ability. Through the equivalent transformation strategy between the △ and Y structures, it is cleverly converted into an easy-to-handle series-parallel equivalent model, thereby greatly simplifying the complex process of solving the unbalanced bridge impedance and improving the flexibility and efficiency of problem solving.

[0055] like Figure 1 As shown, a HMDV suspension modeling method based on a bridge network includes the following steps:

[0056] Step (1): Establish a four-phase 8 / 6-pole switched reluctance motor structure;

[0057] Step (2): Establish a quarter model of the HMDV suspension with a bridge network;

[0058] Step (3): Calculate the impedance of the bridge network structure;

[0059] Step (4): Optimize the objective function rule;

[0060] Step (5): Optimize the parameters of the electrical components in the bridge network using an optimization algorithm and determine the parameter optimization results.

[0061] Step (6): After the model is established, it is verified by the following steps: using Matlab to simulate and verify the optimization parameters, and using Matlab to simulate and verify the optimization parameters.

[0062] Among them, step (1) is specifically: according to Figure 2 The schematic diagram of the four-phase 8 / 6-pole switched reluctance motor structure is shown, and the electromagnetic force equation is established:

[0063]

[0064] Among them, F z is the electromagnetic force, j is the number of motor phases, i is the phase current, θ is the motor rotor position, i j is the j-phase current; m, n are the number of stator poles of the motor in opposite directions, g m (n) = (1-ε)gm , g m (m) = (1 + ε) g m , g m is the motor air gap, and |mn|=4; N r is the number of rotor poles, L u is the inductance of the misaligned position, a n , b n are the polynomial fitting coefficients.

[0065] The motor used is a four-phase 8 / 6-pole switched reluctance motor, which has a simple structure, large torque and small current at startup and low speed, a wide constant power range at high speed, high output and high efficiency in a wide speed and power range, and good fault tolerance, and has a wide range of applications.

[0066] Among them, step (2) is specifically: according to Figure 3 The typical suspension structure of HMDV shown in Figure 4 The HMDV suspension quarter model of the bridge network shown is constructed with its Laplace transform:

[0067]

[0068] Among them, m s is the sprung mass, m us is the mass of the hub motor stator, k s is the suspension spring stiffness, k m is the motor equivalent stiffness, k t is the thrust coefficient of the linear motor, k e is the back EMF constant of the linear motor, Z s , Z us , Z ur , Z e (s) are the Laplace transforms of the corresponding variables, s is the Laplace variable, Z s is the vertical displacement of the sprung mass, Z us is the vertical displacement of the stator and shaft, Z ur is the vertical displacement of the rotor and tire, Z e is the impedance of the external bridge circuit, F EM is the electromagnetic force generated by the linear motor, F z It is the vertical unbalanced force caused by the hub motor.

[0069] Among them, step (3) is specifically: according to Figure 5 The electrical elements shown are a bridge network consisting of two inductance elements, two capacitance elements and one resistance element. Specifically, the first inductance L1 and the second capacitance C2 are connected in series and then connected in parallel with the second inductance L2. One end of the second inductance L2 is connected to the first capacitance C1, and the other end of the second inductance L2 is connected to the first resistance R.

[0070] Calculate the impedance of the bridge network structure (the total impedance Z of the external electrical network is generally in the form of Ze(s)):

[0071]

[0072] Among them, A1=RL1L2C1C2; B1=C1C2(L1+L2); C1=RL1C2+RL2(C1+C2); D1=C1; E1=R; F1=L1L2C1C2; G1=RL1L2C1;

[0073] Among them, A1~J1 are impedance coefficients.

[0074] Wherein, step (4) is specifically: determining the objective function optimization rule according to the bridge network structure impedance in step (3):

[0075] Furthermore, the traditional passive suspension is selected as a reference benchmark to establish the optimization objective function:

[0076]

[0077] Where J represents the root mean square value of the tire dynamic load DTL of the bridge network suspension structure under random road input i , compared with the root mean square value of tire dynamic load DTL of traditional passive suspension pass ratio.

[0078] Furthermore, in the above step (4), in order to obtain electrical network parameters with better performance and ensure that the performance after optimization is better than the traditional structure, the constraint range of the variables to be optimized is determined taking into account the rationality of the components:

[0079]

[0080] Furthermore, the suspension performance is limited to DTL i <DTL pas , B.A. i <BA pas , SWS i <SWS pas During the optimization process, the three performance indicators of the traditional passive suspension are used as comparative constraint indicators. When the optimization objective function value exceeds its constraint conditions, the optimization is unsuccessful. The objective function will be added with 100 and returned for re-optimization until the optimization objective function value meets the constraint conditions.

[0081] In step (5), the optimization algorithm adopts a pattern search method, such as Figure 6 As shown, the pattern search method includes the following steps:

[0082] Step (5.1): Set the initial iteration point x 0 ∈R n , initial step length δ0, error ε>0, acceleration factor α≥1, reduction factor β∈(0,1). Let y=x 0 , k = 0, j = 1;

[0083] Step (5.2): Axial search: If condition ① is met: f(y+δ k e j )<f(y), then let y=y+δ k e j , go to step (5.3); if condition ② is met: f(y-δ k e j )<f(y), then let y=y-δ k e j , go to step (5.3); if both conditions ① and ② are not met, leave y unchanged and go to step (5.3);

[0084] Step (5.3): If j < n, set j = j + 1 and go to step (5.2); otherwise, set x k+1 =y, compare f(x k+1 )<f(x k ), if satisfied, go to step (5.4), otherwise, go to step (5.5).

[0085] Step (5.4): Pattern search: Let δ k+1 =δ k , y = x k+1 +α(x k+1 -x k ), let k = k + 1, j = 1, and go to step (5.2);

[0086] Step (5.5): If δ k ≤ε, then we get point x k , end; otherwise, let δ k+1 =βδ k , y = x k , x k+1 =x k , then let k = k + 1, j = 1, and go to step (5.2);

[0087] The meanings of the variables are as follows: i is the i-th iteration point, R n is an n-dimensional real number set, k is the number of pattern searches, j is the number of axial searches, δ0 is the initial step size, α is the acceleration factor, ε is the error, β is the reduction factor, and n represents the n coordinate axis directions.

[0088] Table 1 Bridge network parameter optimization results:

[0089] Parameters to be optimized Optimization results <![CDATA[Capacitance 1C1 / (mF)]]> 193.1 <![CDATA[Capacitance 2C2 / (mF)]]> 644.7 <![CDATA[Resistance 1R1 / (Ω)]]> 615 <![CDATA[Inductance 1L1 / (mH)]]> 1 <![CDATA[Inductance 2L2 / (mH)]]> 1

[0090] After the model is established, it is verified through the following steps: Matlab is used to simulate and verify the optimization parameters.

[0091] Through Matlab / Simulink, a quarter model of the traditional passive suspension and the bridge network HMDV suspension was built, and the two types of suspension were simulated and analyzed, with the body acceleration, suspension dynamic travel and tire dynamic load as the evaluation indicators of the suspension performance. Under the input of random road surface, it is assumed that the vehicle travels at a speed of 20m / s.

[0092] The simulation can obtain the frequency domain gain comparison diagram of the vehicle electromechanical inertial suspension based on the bridge network, as shown in Figure 7, where (a) is the vehicle body acceleration frequency domain gain comparison diagram, (b) is the suspension dynamic stroke frequency domain gain comparison diagram, and (c) is the tire dynamic load frequency domain gain comparison diagram; the performance time domain index diagram of the suspension can also be obtained, as shown in Figure 8, where (a) is the vehicle body acceleration response diagram, (b) is the suspension dynamic stroke response diagram, and (c) is the tire dynamic load response diagram; and the RMS value improvement effect of the suspension is obtained according to the simulation results, as shown in Table 2.

[0093] Table 2 shows the improvement effect of the RMS value at a speed of 20 m / s:

[0094] Traditional passive suspension Bridge Network promote Body acceleration root mean square value / (m / s2) 1.3178 1.2986 1.45％ Suspension dynamic travel RMS value / (m) 0.0139 0.0114 18.10％ Tire dynamic load root mean square value / (N) 1228.16 1096.06 10.76％

[0095] The above simulation results show that the double quartic impedance function of the bridge network structure adopted in the HMDV suspension based on the bridge network of the present invention can be equivalently realized as mechanical impedance, and suppress the vibration of the unsprung mass at low and high frequencies, significantly improve the tire dynamic load and suspension working space, and reduce the influence of the negative effect of the vertical vibration of the HMDV.

[0096] The above technical solutions only reflect the preferred technical solutions of the technical solutions of the present invention. Some changes that may be made to certain parts thereof by technicians in this technical field all reflect the principles of the present invention and fall within the protection scope of the present invention.

Claims

1. A HMDV suspension modeling method based on a bridge network, characterized in that: The following steps are involved: Step (1): Establish a four-phase 8 / 6-pole switched reluctance motor structure; Step (2): Establish a quarter model of the HMDV suspension with a bridge network; Step (3): Calculate the impedance of the bridge network structure; Step (4): Optimize the objective function rule; Step (5): Optimizing the parameters of the electrical components in the bridge network using an optimization algorithm and determining the parameter optimization results; Step (6): Use Matlab to simulate and verify the optimization parameters.

2. The HMDV suspension modeling method based on a bridge network according to claim 1, characterized in that: The electromagnetic force equation is established according to the four-phase 8 / 6-pole switched reluctance motor structure described in step (1): Among them, F z is the electromagnetic force, j is the number of motor phases, i is the phase current, θ is the motor rotor position, i j is the j-phase current; m, n are the number of stator poles of the motor in opposite directions, g m (n) = (1-ε)g m , g m (m) = (1 + ε) g m , g m is the motor air gap, and |mn|=4; N r is the number of rotor poles, L u is the inductance of the misaligned position, a n , b n are the polynomial fitting coefficients.

3. The HMDV suspension modeling method based on bridge network according to claim 2 is characterized in that: The Laplace transform equation is established based on the HMDV suspension quarter model of the bridge network described in step (2): Among them, m s is the sprung mass, m us is the mass of the hub motor stator, k s is the suspension spring stiffness, k m is the motor equivalent stiffness, k t is the thrust coefficient of the linear motor, k e is the back EMF constant of the linear motor, Z s , Z us , Z ur , Z e (s) are the Laplace transforms of the corresponding variables, s is the Laplace variable, Z s is the vertical displacement of the sprung mass, Z us is the vertical displacement of the stator and shaft, Z ur is the vertical displacement of the rotor and tire, Z e is the impedance of the external bridge circuit, F EM is the electromagnetic force generated by the linear motor, F z It is the vertical unbalanced force caused by the hub motor.

4. The HMDV suspension modeling method based on a bridge network according to claim 1, characterized in that: According to step (3), the total impedance Z of the external electrical network is calculated in the general form of Ze(s) using a bridge network consisting of two inductance elements, two capacitance elements and one resistance element: Among them, A1=RL1L2C1C2; B1=C1C2(L1+L2); C1=RL1C2+RL2(C1+C2); D1=C1; E1=R; F1=L1L2C1C2; G1=RL1L2C1; Among them, A1~J1 are impedance coefficients.

5. The HMDV suspension modeling method based on bridge network according to claim 4 is characterized in that: According to the bridge network structure impedance in step (3), the objective function optimization rule is determined: The traditional passive suspension is selected as the reference benchmark to establish the optimization objective function: Where J represents the root mean square value of the tire dynamic load DTL of the bridge network suspension structure under random road input i , compared with the root mean square value of tire dynamic load DTL of traditional passive suspension pass ratio.

6. The HMDV suspension modeling method based on bridge network according to claim 5, characterized in that: The step (3) specifically includes: the first inductor L1 and the second capacitor C2 are connected in series and then connected in parallel with the second inductor L2; one end of the second inductor L2 is connected to the first capacitor C1; and the other end of the second inductor L2 is connected to the first resistor R.

7. The HMDV suspension modeling method based on bridge network according to claim 1 is characterized in that: In step (4), in order to obtain electrical network parameters with better performance and ensure that the optimized performance is better than the traditional structure, while taking into account the rationality of the components, the constraint range of the variables to be optimized is determined: Limit suspension performance to DTL i <DTL pas , B.A. i <BA pas , SWS i <SWS pas During the optimization process, the three performance indicators of the traditional passive suspension are used as comparative constraint indicators. When the optimization objective function value exceeds its constraint conditions, the optimization is unsuccessful. The objective function will be added with 100 and returned for re-optimization until the optimization objective function value meets the constraint conditions.

8. The HMDV suspension modeling method based on bridge network according to claim 1, characterized in that: In the step (5), the optimization algorithm adopts a pattern search method.

9. The HMDV suspension modeling method based on bridge network according to claim 8, characterized in that: The pattern search method comprises the following steps: Step (5.1): Set the initial iteration point x 0 ∈R n , initial step length δ0, error ε>0, acceleration factor α≥1, reduction factor β∈(0,1). Let y=x 0 , k = 0, j = 1; Step (5.2): Axial search: If condition ① is met: f(y+δ k e j )<f(y), then let y=y+δ k e j , go to step (5.3); if condition ② is met: f(y-δ k e j )<f(y), then let y=y-δ k e j , go to step (5.3); if both conditions ① and ② are not met, leave y unchanged and go to step (5.3); Step (5.3): If j < n, set j = j + 1 and go to step (5.2); otherwise, set x k+1 =y, compare f(x k+1 )<f(x k ), if satisfied, go to step (5.4), otherwise, go to step (5.5). Step (5.4): Pattern search: Let δ k+1 =δ k , y = x k+1 +α(x k+1 -x k ), let k = k + 1, j = 1, and go to step (5.2); Step (5.5): If δ k ≤ε, then we get point x k , end; otherwise, let δ k+1 =βδ k , y = x k , x k+1 =x k , then let k = k + 1, j = 1, and go to step (5.2); The meanings of the variables are as follows: i is the i-th iteration point, R n is an n-dimensional real number set, k is the number of pattern searches, j is the number of axial searches, δ0 is the initial step size, α is the acceleration factor, ε is the error, β is the reduction factor, and n represents the n coordinate axis directions.

Images