Multi-AGV formation keeping control method based on IPSO-fuzzy PID

Abstract

The invention discloses a multi-AGV formation keeping control method based on IPSO-fuzzy PID. The multi-AGV formation keeping control method is characterized by comprising the following steps: (1) establishing a dynamic self-adaptive multi-AGV pilot following formation control model; (2) constructing a fuzzy PID controller, setting a fuzzy domain of input and output variables, a membership function and a fuzzy control rule, and obtaining the correction of PID parameters through fuzzy reasoning and defuzzification; (3) constructing an improved particle swarm optimization algorithm with dynamic balance of global search and local optimization, improving the inertia weight and learning factor of a particle swarm, and setting a fitness function; and (4) optimizing the input quantization factor and the output scaling factor of the fuzzy PID controller by using an IPSO algorithm, assigning optimized parameters to the controller, and adjusting the linear velocity and the angular velocity of the following AGV. The formation keeping precision, the response speed and the anti-interference capability can be improved.

Description

Technical Field

[0001] This invention relates to a control technology for maintaining the formation of multiple automated guided vehicles (AGVs) during collaborative transport, and more particularly to a control method for improving formation maintenance accuracy, response speed, and anti-interference capability. Specifically, it is a multi-AGV formation maintenance control method based on IPSO-fuzzy PID. Background Technology

[0002] Automated Guided Vehicles (AGVs) are programmable unmanned transport devices widely used in intelligent manufacturing, warehousing, and collaborative handling of large components. In multi-vehicle collaborative transport operations, multiple AGVs need to operate synchronously in a pre-set formation to ensure the safety and stability of the transport process. However, in actual operation, the AGV formation often shifts due to positional or heading deviations or external interference, leading to uneven stress on the support frame, deformation of the transport structure, or even complete overturning, severely impacting operational efficiency and safety.

[0003] The core issue in formation maintenance lies in ensuring the consistency of relative position and attitude among multiple AGVs during transport. Traditional formation maintenance control methods often employ PID control based on a pilot-follower model, adjusting proportional, integral, and derivative coefficients to reduce the error between the following AGV and the desired pose. However, traditional PID control typically uses fixed parameters, making it difficult to balance the high precision and rapid response requirements of formation maintenance in complex dynamic environments or when external disturbances are present. This results in large overshoot, long response time, and insufficient anti-interference capability.

[0004] Therefore, to address the requirements of high precision, short response time, and anti-interference in formation maintenance within complex collaborative handling environments, this invention proposes a fuzzy PID formation maintenance control method based on an improved particle swarm optimization (IPSO) algorithm. This method utilizes a lead AGV to calculate and publish reference motion parameters in real time, while the following AGV dynamically tunes its PID parameters using an improved fuzzy PID controller. The quantization factor and proportional factor obtained from IPSO optimization are applied to the controller, enabling rapid formation adjustment and high-precision formation maintenance, and ensuring a stable formation structure under external disturbances. Summary of the Invention

[0005] The purpose of this invention is to address the problems of low accuracy, slow response, and weak anti-interference capability in traditional multi-AGV formation holding control methods, and to propose a multi-AGV formation holding control method based on IPSO-fuzzy PID. This method combines the adaptability of fuzzy control to nonlinear systems with the global optimization capability of IPSO, and optimizes the key parameters of the PID controller online, thereby improving the dynamic response speed, steady-state accuracy, and robustness of multi-AGV system formation holding.

[0006] The technical solution of this invention is:

[0007] A multi-AGV formation-keeping control method based on IPSO-fuzzy PID is characterized by: firstly, establishing a multi-AGV cooperative transport formation control model based on the leader-follower method, where the following AGV receives the reference speed and desired pose from the leader AGV; secondly, constructing a fuzzy PID controller, taking the pose deviation and its rate of change of the following AGV as input, and outputting the increment of the PID control parameters; then, using the improved particle swarm optimization algorithm (IPSO) to optimize the quantization factor and proportional factor of the fuzzy PID controller online, using the integral of time-in-absolute error (ITAE) as the fitness function to obtain the optimal control parameters; finally, the following AGV adjusts its linear velocity and angular velocity according to the optimal parameters to achieve precise formation maintenance.

[0008] Specifically, the following steps are included:

[0009] Step 1: Establish a dynamic and adaptive multi-AGV navigation and following formation control model;

[0010] Step 2: Construct a fuzzy PID controller, set the fuzzy universe of discourse, membership function and fuzzy control rules for the input and output variables, and obtain the correction amount of the PID parameters through fuzzy inference and defuzzification;

[0011] Step 3: Construct an improved particle swarm optimization algorithm that dynamically balances global search and local optimization, improves the inertia weight and learning factor of the particle swarm, and sets the fitness function;

[0012] Step 4: Use the IPSO algorithm to optimize the input quantization factor and output proportional factor of the fuzzy PID controller, assign the optimized parameters to the fuzzy PID controller, and output the desired linear velocity and angular velocity to the drive unit to achieve formation holding control.

[0013] Furthermore, in step 1: the lead AGV is responsible for sending its respective reference linear velocity v to all the following AGVs in the platoon. s and angular velocity ω s Simultaneously, it calculates the desired pose of each following AGV in the formation using its own pose information and formation constraint information, and sends it to the corresponding following AGV; after receiving the desired pose information, the following AGV calculates the deviation based on its own actual pose information; the position deviation e of the i-th following AGV is then calculated. ix ,e iy and heading angle deviation e iθ And calculate the rate of change of deviation. This serves as input information for the controller. The controller dynamically adjusts the desired linear and angular velocities of the following AGVs based on changes in error, and transmits these desired values ​​to the drive unit, thereby correcting the actual posture in real time. Under the closed-loop operation of this control strategy, all following AGVs continuously update their motion parameters according to their own state and the instructions of the lead AGV, achieving dynamic formation maintenance.

[0014] Furthermore, in step 2: traditional PID controllers preset fixed PID parameters, which cannot meet the high precision and short response time requirements of formation holding control. This invention uses a fuzzy algorithm to perform online self-tuning of the PID parameters, constructing a fuzzy PID controller, thereby improving the control effect. The fuzzy PID controller adopts a dual-input, three-output structure. The inputs are the deviation e and the rate of change of deviation ec, and the outputs are the increments (ΔK) of the three parameters of the PID controller. p ,ΔK i ,ΔK d The universes of discourse for e and ec are set to [-3, 3], and discretized into 7 levels {-3, -2, -1, 0, 1, 2, 3}. A fuzzy subset is defined for each fuzzy variable as {negative large, negative medium, negative small, zero, positive small, positive medium, positive large}, denoted as {NB, NM, NS, ZO, PS, PM, PB}. A triangular membership function is used to establish the conversion relationship between the universe values ​​of e and ec and the fuzzy language. Based on the experience of manually adjusting PID, the formula "IFe = A AND ec = B THEN ΔK" is adopted. p =C,ΔK i =D,ΔK d =E” form, respectively for K p K i and K d A fuzzy rule base is established using three parameters; based on the input values ​​'e' and 'ec', rules in the rule base are matched to calculate the membership degree of the output fuzzy values, resulting in a fuzzy set of all output linguistic values; the centroid method is used for defuzzification to obtain the output parameter increment ΔK. p ,ΔK i ,ΔK d .

[0015] Furthermore, in step 3, the input quantization factor and output proportional factor of the fuzzy PID controller largely rely on empirical settings, making it difficult to obtain the optimal parameter combination. This results in certain deficiencies in the system's dynamic response speed, steady-state accuracy, and robustness. This invention utilizes an improved particle swarm optimization algorithm to tune the parameters of the fuzzy controller, thereby further improving the control effect. Addressing the problem that traditional particle swarm optimization algorithms are prone to getting trapped in local optima and struggle to balance convergence speed and optimization accuracy, improvements are made to its inertia weight ω and learning factors c1 and c2. A cosine variation weight factor strategy is used to update the inertia weight ω, while an asynchronous variation strategy is used to adjust the learning factors c1 and c2. This maintains a large ω and c1 and a small c2 in the early stages of iteration, enhancing the global search capability of the particles. As the number of iterations increases, ω and c1 are gradually decreased while c2 is increased, focusing on the local optimum region for fine-grained searching, accelerating convergence speed, and improving optimization accuracy. ITAE is selected as the fitness function, and error, time factor and error weight are combined. The error is weighted by time, so that large errors in the early stage of system operation are given smaller weights to avoid over-adjustment and overshoot. In the later stage of operation, persistent errors are given larger weights to force the algorithm to reduce steady-state error.

[0016] Furthermore, in step 4: the quantization factor C of the deviation is... e Quantification factor C of the deviation change rate ec And the proportional factor CK of the three parameters of the fuzzy PID controller p CK i CK d As the particle position vector in the particle swarm, the optimal parameter combination is found by iteratively finding the particle position that minimizes the fitness function value. Finally, the AGV uses an optimized fuzzy PID controller to calculate the desired linear velocity and angular velocity, and the drive unit quickly converges the pose error to achieve high-precision formation maintenance.

[0017] The beneficial effects of this invention are:

[0018] 1. This invention utilizes fuzzy logic reasoning to achieve online adaptive adjustment of PID parameters, overcoming the limitation of fixed parameters in traditional PID when facing nonlinear disturbances, thereby significantly enhancing the robustness and anti-interference capability of the system.

[0019] 2. This invention utilizes an improved particle swarm optimization algorithm to globally optimize the key parameters of the fuzzy PID controller, and uses ITAE as the fitness function to force the algorithm to find the parameter combination that minimizes the system settling time and steady-state error. This enables the following AGV to eliminate pose errors with the fastest response speed while achieving high-precision formation maintenance. Attached Figure Description

[0020] Figure 1This is a flowchart of the present invention.

[0021] Figure 2 This is a diagram of the multi-AGV navigation and following formation control model of the present invention.

[0022] Figure 3 This is a schematic diagram of the IPSO-fuzzy PID control principle of the present invention.

[0023] Figure 4 This is a comparison diagram showing the effects of the control method of the present invention.

[0024] Figure 5 This is a diagram showing the control results of multi-vehicle cooperative transport formation maintenance according to the present invention. Detailed Implementation

[0025] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0026] like Figures 1-5 As shown.

[0027] A multi-AGV formation-keeping control method based on IPSO-fuzzy PID is proposed. First, a multi-AGV cooperative transport formation control model based on the lead-follow method is established, where the following AGV receives the reference speed and desired pose from the lead AGV. Second, a fuzzy PID controller is constructed, taking the pose deviation and its rate of change of the following AGV as input and outputting the increment of the PID control parameters. Then, the quantization factor and proportional factor of the fuzzy PID controller are optimized online using an improved particle swarm optimization (IPSO) algorithm, with the time-integral-absolute-error (ITAE) as the fitness function to obtain the optimal control parameters. Finally, the following AGV adjusts its linear velocity and angular velocity according to the optimal parameters to achieve precise formation maintenance. Figure 1 The formation maintenance control flowchart of the present invention specifically includes the following steps:

[0028] Step 1: Establish a dynamic and adaptive multi-AGV navigation and following formation control model.

[0029] Multi-AGV navigation and following formation control model, such as Figure 2 As shown, the lead AGV is responsible for sending its reference linear velocity v to all the following AGVs in the platoon. s and angular velocity ω s Simultaneously, it calculates the desired pose of each following AGV in the formation using its own pose information and formation constraint information, and sends it to the corresponding following AGV; after receiving the desired pose information, the following AGV calculates the deviation based on its own actual pose information; the position deviation e of the i-th following AGV is then calculated. ix ,e iy and heading angle deviation e iθ And calculate the rate of change of deviation. This serves as input information for the controller. The controller dynamically adjusts the desired linear and angular velocities of the following AGVs based on changes in error, and transmits these desired values ​​to the drive unit, thereby correcting the actual posture in real time. Under the closed-loop operation of this control strategy, all following AGVs continuously update their motion parameters according to their own state and the instructions of the lead AGV, achieving dynamic formation maintenance.

[0030] Step 2: Construct a fuzzy PID controller, set the fuzzy universe of discourse, membership function and fuzzy control rules for the input and output variables, and obtain the correction amount of the PID parameters through fuzzy inference and defuzzification.

[0031] IPSO-fuzzy PID control principle diagram is as follows: Figure 3 As shown, the fuzzy PID controller adopts a dual-input, three-output structure. The inputs are the deviation e and the rate of change of deviation ec, and the outputs are the increments (ΔK) of the three parameters of the PID controller. p ,ΔK i ,ΔK d ); Set the universe of discourse of e and ec to [-3,3], and discretize them into 7 levels {-3,-2,-1,0,1,2,3}; Define a fuzzy subset for each fuzzy variable as {negative large, negative medium, negative small, zero, positive small, positive medium, positive large}, denoted as {NB,NM,NS,ZO,PS,PM,PB}; Use the triangular membership function to establish the conversion relationship between the universe values ​​of e and ec and the fuzzy language, as shown in Equation (1):

[0032]

[0033] In a two-input, three-output fuzzy control structure, based on experience gained from manually adjusting PID controllers, the formula "IFe = A AND ec = B THENΔK" is adopted. p =C,ΔK i =D,ΔK d =E” form, respectively for K p K i and K d A fuzzy rule base is established using three parameters. Since the number of fuzzy subsets for each of E and EC is 7, each fuzzy rule base can yield 49 fuzzy control rules. Based on the inputs e and ec, rules in the rule base are matched, and the membership degree of the output fuzzy quantity is calculated to obtain the fuzzy set of all output linguistic values. The centroid method is used for defuzzification to obtain the output parameter increment ΔK. p ,ΔK i ,ΔK d It can be expressed by formula (2), where u is the precise output ΔK after defuzzification, and x i f(i) represents the discrete values ​​of the output universe of discourse. iThe corresponding membership degree, where n is the number of discrete levels in the output universe of discourse:

[0034]

[0035] Step 3: Construct an improved particle swarm optimization algorithm that dynamically balances global search and local optimization, improves the inertia weight and learning factor of the particle swarm, and sets the fitness function.

[0036] To address the problems of traditional particle swarm optimization (PSO) algorithms easily getting trapped in local optima and the difficulty in balancing convergence speed and optimization accuracy, this paper improves the inertia weight ω and learning factors c1 and c2. A cosine variation weighting strategy is adopted to update the inertia weight ω, causing the weight to decrease slowly in the early stages of iteration, maintaining a large inertia weight value and enhancing the global search capability of the particles. As the number of iterations increases, the weight decreases at a gradually faster rate, converging quickly to a smaller value in the later stages of iteration, focusing on the local optimum region for fine-grained searching, and improving optimization accuracy. The calculation formula is shown in equation (3), where ω... max ω min These represent the maximum and minimum values ​​of the inertia weight, respectively, where k is the current iteration number, K max Maximum number of iterations:

[0037]

[0038] The learning factors c1 and c2 are adjusted using an asynchronous change strategy. In the early stage of iteration, a larger c1 and a smaller c2 are set to enhance the individual cognitive ability of the particles, encourage the particles to leave the local optimal region, expand the search range, and improve the global exploration efficiency. In the later stage of iteration, c1 is gradually reduced and c2 is increased to enhance the social cognitive ability of the particles, guide all particles to gather towards the group optimal solution, accelerate the convergence speed and improve the optimization accuracy, as shown in Equation (4):

[0039]

[0040] ITAE is selected as the fitness function, and error, time factor and error weight are combined. The error is weighted by time, so that the large error in the early stage of system operation will be given a smaller weight to avoid over-adjustment and overshoot. The persistent error in the later stage of operation will be given a larger weight to force the algorithm to reduce the steady-state error; as shown in equation (5), where e(t) represents the error of the system at time t:

[0041]

[0042] Step 4: Use the IPSO algorithm to optimize the input quantization factor and output proportional factor of the fuzzy PID controller, assign the optimized parameters to the fuzzy PID controller, and output the desired linear velocity and angular velocity to the drive unit to achieve formation holding control.

[0043] exist Figure 3 In the process, the parameters to be optimized include the quantization factor C of the deviation. e Quantification factor C of the deviation change rate ec And the proportional factor CK of the three parameters of the fuzzy PID controller p CK i CK d Using the above five parameters as the particle position vectors in the particle swarm, the optimal parameter combination is obtained by iteratively finding the particle position that minimizes the fitness function value. The particle velocity and position update formulas are shown in equation (6), where... Let r1 and r2 be the velocity of the i-th particle in the (k+1)-th iteration, where r1 and r2 are both random numbers in the interval (0, 1). For the i-th particle in the k-th iteration, gbest is the optimal position of the individual. k The globally optimal position for the k-th iteration:

[0044]

[0045] After the iteration terminates, the global optimal solution gbest is output, which is the combination of the five optimal parameters of the fuzzy PID controller. At this time, the PID controller parameters can be expressed by equation (7), where K p0 ,K i0 ,K d0 ΔK is the initial parameter for the PID controller. p ,ΔK i ,ΔK d The parameter increment for the fuzzy controller output:

[0046]

[0047] Finally, the AGV uses an optimized fuzzy PID controller to calculate the desired linear velocity and angular velocity, and the drive unit quickly converges the pose error to achieve high-precision formation maintenance. The control law is calculated using an incremental PID algorithm, as shown in equation (8), where Δu(k) is the control increment at the current moment, and e(k), e(k-1), and e(k-2) are the errors at the current moment, the previous moment, and the two moments before that, respectively.

[0048]

[0049] from Figure 4 It can be seen that the IPSO-fuzzy PID controller designed in this invention exhibits significant advantages in both dynamic response speed and overshoot suppression. Simulation results show that, under the regulation of this controller, the following AGV can eliminate deviations more quickly and achieve precise maintenance of the target formation. Figure 5Simulation results of formation-keeping control of multiple AGVs on straight and circular trajectories are presented respectively, by introducing a time-varying function d at random times. y =sin(πt / 10) to simulate external disturbances. It can be seen that the multi-AGV system exhibits strong robustness after being subjected to external disturbances, and can suppress oscillations and restore the target formation in a short time.

[0050] The parts not covered in this invention are the same as or can be implemented using existing technologies.

Claims

1. A multi-AGV formation-keeping control method based on IPSO-fuzzy PID, characterized in that: First, a multi-AGV cooperative transport formation control model based on the pilot-follower method is established, in which the following AGV receives the reference speed and desired pose issued by the pilot AGV; second, a fuzzy PID controller is constructed, which takes the pose deviation and its rate of change of the following AGV as input and outputs the increment of the PID control parameters. Then, the improved particle swarm optimization algorithm (IPSO) is used to optimize the quantization factor and proportional factor of the fuzzy PID controller online, and the time-integral-absolute-error (ITAE) is used as the fitness function to obtain the optimal control parameters. Finally, the AGV adjusts its linear velocity and angular velocity according to the optimal parameters to achieve precise formation maintenance.

2. The method according to claim 1, characterized in that, Includes the following steps: Step 1: Establish a dynamic and adaptive multi-AGV navigation and following formation control model; Step 2: Construct a fuzzy PID controller, set the fuzzy universe of discourse, membership function and fuzzy control rules for the input and output variables, and obtain the correction amount of the PID parameters through fuzzy inference and defuzzification; Step 3: Construct an improved particle swarm optimization algorithm that dynamically balances global search and local optimization, improves the inertia weight and learning factor of the particle swarm, and sets the fitness function; Step 4: Use the IPSO algorithm to optimize the input quantization factor and output proportional factor of the fuzzy PID controller, assign the optimized parameters to the fuzzy PID controller, and output the desired linear velocity and angular velocity to the drive unit to achieve formation holding control.

3. The method according to claim 2, characterized in that, In step 1: The lead AGV is responsible for sending its reference linear velocity v to all the following AGVs in the platoon. s and angular velocity ω s Simultaneously, it calculates the desired pose of each following AGV in the formation using its own pose information and formation constraint information, and sends it to the corresponding following AGV; after receiving the desired pose information, the following AGV calculates the deviation based on its own actual pose information; the position deviation e of the i-th following AGV is then calculated. ix ,e iy and heading angle deviation e iθ And calculate the rate of change of deviation. As input information to the controller, the controller can dynamically adjust the expected linear velocity and angular velocity values ​​of the following AGVs according to the changes in error, and transmit the expected values ​​to the drive unit to correct the actual posture in real time. Under the closed-loop action of this control strategy, all following AGVs continuously update their motion parameters according to their own status and the instructions of the lead AGV, so as to achieve dynamic maintenance of formation.

4. The method according to claim 2, characterized in that, In step 2: The fuzzy PID controller adopts a dual-input, three-output structure. The inputs are the deviation e and the rate of change of deviation ec, and the outputs are the increments (ΔK) of the three parameters of the PID controller. p ,ΔK i ,ΔK d ); Set the universe of discourse of e and ec to [-3,3], and discretize them into 7 levels {-3,-2,-1,0,1,2,3}; Define a fuzzy subset for each fuzzy variable as {negative large, negative medium, negative small, zero, positive small, positive medium, positive large}, denoted as {NB,NM,NS,ZO,PS,PM,PB}; Use the triangular membership function to establish the conversion relationship between the universe values ​​of e and ec and the fuzzy language, as shown in Equation (1): In a two-input, three-output fuzzy control structure, based on experience gained from manually adjusting PID controllers, the formula "IFe=A ANDec=B THENΔK" is adopted. p =C,ΔK i =D,ΔK d =E” form, respectively for K p K i and K d A fuzzy rule base is established using three parameters. Since the number of fuzzy subsets of E and EC is 7, each fuzzy rule base can obtain 49 fuzzy control rules. Based on the inputs e and ec, the rules in the rule base are matched, the membership degree of the output fuzzy quantity is calculated, and a fuzzy set of all output linguistic values ​​is obtained. The centroid method is used for defuzzification to obtain the output parameter increment ΔK. p ,ΔK i ,ΔK d It is expressed by formula (2), where u is the precise output ΔK after defuzzification, and x is the value of x. i f(i) represents the discrete values ​​of the output universe of discourse. i The corresponding membership degree, where n is the number of discrete levels in the output universe of discourse:

5. The method according to claim 2, characterized in that, In step 3: the improvement of the inertia weight ω is to use a cosine variation weight factor strategy to update the inertia weight ω, so that the weight decreases slowly in the early stage of the iteration, maintains a large inertia weight value, strengthens the global search capability of the particles, and gradually accelerates the weight decrease rate as the number of iterations increases, and quickly converges to a small value in the later stage of the iteration, focusing on the local optimal solution region for fine search and improving the optimization accuracy. The calculation formula is shown in equation (3), where ω max ω min These represent the maximum and minimum values ​​of the inertia weight, respectively, where k is the current iteration number, K max Maximum number of iterations: The learning factors c1 and c2 are improved by using an asynchronous change strategy. In the early stage of iteration, a larger c1 and a smaller c2 are set to enhance the individual cognitive ability of the particles, encourage the particles to leave the local optimal region, expand the search range, and improve the global exploration efficiency. In the later stage of iteration, c1 is gradually reduced and c2 is increased to enhance the social cognitive ability of the particles, guide all particles to gather towards the group optimal solution, accelerate the convergence speed and improve the optimization accuracy, as shown in Equation (4): ITAE is selected as the fitness function, and error, time factor and error weight are combined. The error is weighted by time, so that the large error in the early stage of system operation will be given a smaller weight to avoid over-adjustment and overshoot. The persistent error in the later stage of operation will be given a larger weight to force the algorithm to reduce the steady-state error; as shown in equation (5), where e(t) represents the error of the system at time t:

6. The method according to claim 2, characterized in that, In step 4: The parameters to be optimized include the quantization factor C of the deviation. e Quantification factor C of the deviation change rate ec And the proportional factor CK of the three parameters of the fuzzy PID controller p CK i CK d Using the above five parameters as the particle position vectors in the particle swarm, the optimal parameter combination is obtained by iteratively finding the particle position that minimizes the fitness function value. The particle velocity and position update formulas are shown in equation (6), where... Let r1 and r2 be the velocity of the i-th particle in the (k+1)-th iteration, where r1 and r2 are both random numbers in the interval (0, 1). For the i-th particle in the k-th iteration, gbest is the optimal position of the individual. k The globally optimal position for the k-th iteration: After the iteration terminates, the global optimal solution gbest is output, which is the combination of the five optimal parameters of the fuzzy PID controller. At this time, the PID controller parameters can be expressed by equation (7), where K p0 ,K i0 ,K d0 ΔK is the initial parameter for the PID controller. p ,ΔK i ,ΔK d The parameter increment for the fuzzy controller output: Finally, the AGV uses an optimized fuzzy PID controller to calculate the desired linear velocity and angular velocity, and the drive unit quickly converges the pose error to achieve high-precision formation maintenance. The control law is calculated using an incremental PID algorithm, as shown in equation (8), where Δu(k) is the control increment at the current moment, and e(k), e(k-1), and e(k-2) are the errors at the current moment, the previous moment, and the two moments before that, respectively. Δu(k)=K p [e(k)-e(k-1)]+K i e(k) (8) +K d [e(k)-2e(k-1)+e(k-2)]。

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