Trajectory tracking optimization control method for two-connecting-rod mechanical arm with incomplete measurable state

Abstract

The invention relates to a trajectory tracking optimization control method for a two-connecting-rod mechanical arm with an incomplete measurable state, and belongs to the technical field of control engineering. The method utilizes a preset performance function and an event triggering mechanism to implement event triggering model prediction optimization control: establishing a mathematical model of a two-connecting-rod mechanical arm system, and obtaining a state-space equation of the system according to the mathematical model; designing an extended state observer, setting a preset performance function, adjusting parameters of the preset performance function to meet transient performance requirements, and performing error conversion on an output tracking error according to the transient performance requirements; designing an event triggering model predictive controller based on a preset performance function; solving an optimization problem to obtain an optimization control input variable and a state variable; and designing a self-adaptive time-varying trigger mechanism to reduce the optimization times of predictive control. The method integrates the advantages of a preset performance control method, an event triggering mechanism and a prediction control method, has the advantages of high response speed, high positioning precision, small calculation amount and the like, and has relatively good tracking performance and control flexibility.

Description

Technical Field

[0001] The invention relates to a control method, in particular to a trajectory tracking optimization control method for a two-link mechanical arm whose state is not completely measurable, and belongs to the technical field of control engineering. Background Art

[0002] Industrial robotic arms are increasingly used in various automation fields, and high-performance motion control is the key to accurate operation of robotic arms. However, in the actual control process, the robotic arm system is not only affected by constraints, uncertainties, and external interference, but the system state is often not measurable in real time. Therefore, the control problem of constrained robotic arm systems with unmeasurable states is particularly important.

[0003] Model predictive control is an optimization control method that can predict the system state and has the characteristics of rolling optimization and feedback correction. It can handle constrained control problems of the controlled system and is suitable for various uncertain and changing control environments. It has received widespread attention in the field of robot control.

[0004] Introducing the trigger mechanism into the design of the model predictive controller can solve the optimization problem online multiple times to save computing resources. Therefore, it is of great significance to study the model predictive control method under a suitable event trigger mechanism.

[0005] At present, most of the research results focus on steady-state performance, but in practical applications, whether the transient performance is satisfied is also very important. Therefore, it is necessary to study the control method that can fully consider both the transient performance and the steady-state performance of the system.

[0006] For the robotic arm system, because its state is not easy to measure in real time, the existing event-triggered predictive control method has problems such as insufficient control accuracy and the inability to take into account both stable performance and transient performance. Summary of the invention

[0007] The main purpose of the present invention is: in view of the deficiencies and gaps in the prior art, the present invention utilizes preset performance functions and event triggering mechanisms to provide an event-triggered model predictive optimization control method that can simultaneously meet transient performance and steady-state performance, so that the robotic arm not only has a fast tracking speed and a small steady-state error, but also the real-time calculation amount is greatly reduced.

[0008] In order to achieve the above purpose, for a two-link robot arm trajectory tracking problem with constraints under an incompletely measurable state, the robot arm includes two links: a first link and a second link, and two joints: a first joint and a second joint; the present invention provides a two-link robot arm trajectory tracking optimization control method with an incompletely measurable state, comprising the following steps:

[0009] Step 1, establish the mathematical model of the two-link robotic arm:

[0010]

[0011] Where q = [q1 q2] T , and are the angular position vector, angular velocity vector and angular acceleration vector of the two connecting rods respectively, q1 is the angular position of the first connecting rod, and q2 is the angular position of the second connecting rod; is the inertia matrix of a symmetric positive rigid body, represents the Coriolis force matrix, represents the gravity vector, is the external disturbance, τ=[τ1 τ2] T is the control torque, τ1 is the control torque of the first connecting rod, and τ2 is the control torque of the second connecting rod;

[0012] Step 2, define the state variable as: x1 = q, Output variable y = [y1 y2] T =x1=q, control input variable u=τ, then equation (1) can be expressed as:

[0013]

[0014] Where, f(x1,x2)=-M -1 (x1)[C(x1,x2)x2+G(x1)]; g(x1)=-M -1 (x1);ω(x1)=M -1 (x1)ω is a disturbance variable consisting of friction and external disturbance, and satisfies: ||ω(x1)||≤ω a , its derivative is also bounded and satisfies: ω a ,ω m are all constants greater than zero;

[0015] Assume that the output variable y=x1 satisfies the constraints: in x 1, are the lower and upper bounds of x1, respectively, and ||·|| represents the Euclidean norm; the control input variable u satisfies: u min ≤||u||≤u max , where u min 、u max are the lower and upper bounds of u respectively; the output tracking error is defined as e1 = [e 11 e 12 ] T =yy d , where e 11、e 12 are the output tracking errors of the first link and the second link, y d =[y d1 ,y d2 ] T is the expected value of the angular position vector of the two links, y d1 ,y d2 are the expected values ​​of the angular positions of the first link and the second link, respectively;

[0016] Step 3, design the extended state observer:

[0017]

[0018] Wherein, L1, L2, and L3 represent the three gains of the extended state observer respectively. denote the estimated values ​​of the state variables x1, x2 and the disturbance variable ω(x1), respectively, is the estimated value of the output variable y;

[0019] The observation error is defined as i=1,2,3, then the error state equation can be obtained from equation (2) and equation (3):

[0020]

[0021] Step 4, error conversion:

[0022] Let the preset performance function ρ(t) be:

[0023]

[0024] Wherein, ρ0 is the initial value of the preset performance function ρ(t), T0 is the time required for the preset performance function ρ(t) to converge from the initial value to the steady-state value, is the steady-state value of the predetermined performance function ρ(t), and constants a1, a2, a3 are selected to satisfy the following conditions:

[0025]

[0026] Let the output tracking error e1 be converted to:

[0027] e1=ρ(t)S(ε1) (6)

[0028] In the formula, ε1=[ε 11 ,ε 12 ] T represents the error after conversion, where ε 11 , ε 12are the errors after the conversion of the first link and the second link respectively; S(ε1)=[S(ε 11 ),S(ε 12 )] T is the conversion function, where S(ε 11 ), S(ε 12 ) are all strictly increasing and reversible smooth functions, and satisfy S(0) = 0;

[0029] make Define the augmented error variable z as: Then we can have:

[0030]

[0031] in, φ(t)=diag[φ1(t),φ2(t)]; j=1,2; Its upper bound is γ, γ>0;

[0032] make Then the system (7) is a nominal system, and we have:

[0033]

[0034] In the formula, denote the augmented error variable and control input variable of the nominal system (8) respectively.

[0035] Step 5: Design an event-triggered model predictive controller based on a preset performance function:

[0036] 51) Take the objective function J as:

[0037]

[0038] Among them, Q, R and P are positive definite matrices; T is the prediction time domain; s is the integral variable, t k for the current moment;

[0039] Solve the following optimization problem:

[0040]

[0041]

[0042] Where δ is the shrinkage factor, and 0<δ<1, α is a parameter related to the terminal set; U = {u|u min ≤||u||≤u max};

[0043] Solve the optimization problem (9) to obtain the current time t kOptimization control input variables The corresponding optimized state variable is recorded as

[0044] 52) Set up event triggering mechanism:

[0045] Design a time-varying trigger mechanism with adaptive trigger conditions:

[0046] Setting trigger conditions for:

[0047]

[0048] Among them, σ(s) satisfies

[0049]

[0050] δ0>0 indicates the state deviation value allowed in the design, c1, c2, d1, d2 are adjustable parameters, and σ(s) also satisfies σ(s)∈[σ min ,σ max ],σ max and σ min is the boundary value of the trigger mechanism;

[0051] Next optimization time t k+1 Take it as:

[0052]

[0053] Step 6: Implement trajectory tracking control. The specific method is as follows:

[0054] 61) Measure the current time t of the system k The output control variable y, and the state estimation value based on the extended state observer

[0055] 62) obtaining a converted error model (7) based on the preset performance function ρ(t) and the converted output tracking error (6);

[0056] 63) Solve for the current time t k Optimization problem (9), obtain the optimized input variables and state variables

[0057] 64) At the next moment t k+1 If the trigger condition (10) is not met, Input into the system (7) and (8) respectively; if the trigger condition (10) is met, go to step 63) and solve the optimization problem (9) again.

[0058] The beneficial effects of the present invention are:

[0059] 1) The model predictive control method proposed in the present invention is designed based on preset performance and adaptive event triggering mechanism, which can not only make the closed-loop controlled system stable while satisfying constraints, but also meet transient performance requirements by adjusting the parameters of the preset performance function.

[0060] 2) The proposed adaptive event triggering mechanism is flexible and changeable, which reduces the optimization times of predictive control and reduces the amount of calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 Schematic diagram of a two-link robotic arm according to an embodiment of the present invention.

[0062] Figure 2 This is a structural diagram of the control system of the two-link robotic arm of the present invention.

[0063] Figure 3 It is a simulation curve of the angular velocity of the two-link mechanical arm of the present invention and its observed value.

[0064] Figure 4 It is a simulation curve diagram comparing the angular velocity y1(t)=q1(t) of the two-link manipulator based on the present invention and the traditional PD algorithm and conventional MPC method.

[0065] Figure 5 It is a simulation curve diagram comparing the angular velocity y2(t)=q2(t) of the two-link manipulator under the present invention and the traditional PD algorithm and conventional MPC method.

[0066] Figure 6 The angular velocity error of the two-link manipulator based on the present invention and the traditional PD algorithm and conventional MPC method is Compare the simulation curves.

[0067] Figure 7 The angular velocity error of the two-link manipulator based on the present invention and the traditional PD algorithm and conventional MPC method is Compare the simulation curves.

[0068] Figure 8 It is the number of solutions of the optimization problem triggered by the event of the present invention and the number of optimization solutions of the conventional MPC.

[0069] Fig. 9 It is a simulation curve diagram comparing the control torque u of the two-link mechanical arm under the present invention and the traditional PD algorithm and conventional MPC method.

[0070] Numbers in the figure: 1-first connecting rod, 2-second connecting rod, 3-first joint, 4-second joint. DETAILED DESCRIPTION

[0071] The present invention will be further described in detail below in conjunction with the accompanying drawings.

[0072] like Figure 1 As shown, the robot arm includes two links: the first link 1 and the second link 2, and two constrained joints: the first joint 3 and the second joint 4. The mass and length of the first link 1 are m1 and l1 respectively, and the mass and length of the second link 2 are m2 and l2 respectively; the distance from the first joint 3 to the center of gravity of the first link 1 is l c1 , the distance from the second joint 4 to the center of gravity of the second link 2 is l c2 The moments of inertia of the first link 1 and the second link 2 are J1 and J2 respectively; q1 is the angular position of the first link 1, and q2 is the angular position of the second link 2. The counterclockwise direction is defined as the positive rotation direction of the robot arm.

[0073] In order to achieve the smooth operation of the above-mentioned mechanical arm, the present invention provides a trajectory tracking optimization control method of a two-link mechanical arm whose state is not completely measurable, comprising the following steps:

[0074] Step 1: Establish the mathematical model of the two-link robotic arm:

[0075]

[0076] Where q = [q1 q2] T , and are the angular position vector, angular velocity vector and angular acceleration vector of the two connecting rods, q1 is the angular position of the first connecting rod 1, and q2 is the angular position of the second connecting rod 2; is the inertia matrix of a symmetric positive rigid body, represents the Coriolis force matrix, represents the gravity vector, is the external disturbance, τ=[τ1 τ2] T is the control torque, τ1 is the control torque of the first link 1, and τ2 is the control torque of the second link 2; and:

[0077]

[0078] in, C 22 =0, G1=(m1l c2 +m2l1)g cos q1+m2l c2 gcos(q1+q2), G2=m2l c2 g cos(q1+q2), g is the acceleration due to gravity.

[0079] For any t ≥ 0, q(t), The constraints must be met:

[0080]

[0081] where k uj , k vj is a constant vector, and k uj =[k uj1 ,k uj2 ] T , k vj =[k vj1 ,k vj2 ] T , j=1,2.

[0082] Step 2: To facilitate the design and analysis of the subsequent controller, define the state variable as: x1 = q, Output variable y = [y1 y2] T =x1=q, control input u=τ, then equation (1) can be expressed as:

[0083]

[0084] Where, f(x1,x2)=-M -1 (x1)[C(x1,x2)x2+G(x1)]; g(x1)=-M -1 (x1);ω(x1)=M -1 (x1)ω, is a bounded lumped disturbance consisting of friction and external disturbance, and satisfies: ||ω(x1)||≤ω a , its derivative is also bounded and satisfies: ω a ,ω m are all constants greater than zero;

[0085] Assume that the output variable y=x1 satisfies the constraints: in x 1, are the lower and upper bounds of x1, respectively, ||·|| represents the Euclidean norm; the control input u satisfies: u min ≤||u||≤u max , where u min 、u max are the lower and upper bounds of u respectively; the output tracking error is defined as e1 = [e 11 e 12 ] T =yy d , where e 11 、e12 are the output tracking errors of the first link 1 and the second link 2, respectively, d =[y d1 ,y d2 ] T is the expected value of the angular position vector of the first link 1 and the second link 2, y d1 ,y d2 are the expected angular positions of the first link 1 and the second link 2, respectively;

[0086] Step 3, design the extended state observer:

[0087]

[0088] Among them, L1, L2, and L3 represent the three gains of the extended state observer, and here L1=3k0, k0>0. They represent the estimated values ​​of the state variables x1, x2 and the disturbance variable ω(x1), respectively. represents the estimated value of the output variable y;

[0089] The observation error is defined as i=1,2,3, then the error state equation can be obtained from equation (2) and equation (3):

[0090]

[0091] Step 4, error conversion:

[0092] Let the preset performance function ρ(t) be:

[0093]

[0094] Where ρ0 is the initial value of ρ(t), T0 is the time required for ρ(t) to converge from the initial value to the steady-state value, is the steady-state value of ρ(t). By choosing constants a1, a2, and a3, the following conditions are satisfied:

[0095]

[0096] Let the output tracking error e1 be converted to:

[0097] e1=ρ(t)S(ε1) (6)

[0098] In the formula, ε1=[ε 11 ,ε 12 ] T represents the error after conversion, where ε 11 , ε 12 are the errors after the conversion of the first link 1 and the second link 2 respectively; S(ε1)=[S(ε11 ),S(ε 12 )] T is the conversion function, where S(ε 11 ), S(ε 12 ) are all strictly increasing and reversible smooth functions, and satisfy S(0) = 0; after conversion, the error is:

[0099]

[0100] in h j and is a positive parameter and satisfies 0< h j ≤1,

[0101] make Define the augmented error variable z as: Then we can have:

[0102]

[0103] in, φ(t)=diag[φ1(t),φ2(t)]; j=1,2; Its upper bound is And γ>0.

[0104] make Then the system (7) is a nominal system, and we have:

[0105]

[0106] In the formula, denote the augmented error variable and control input variable of the nominal system (8) respectively.

[0107] Step 5: Design an event-triggered model predictive controller based on a preset performance function:

[0108] 51) Take the objective function J as:

[0109]

[0110] Among them, Q, R and P are positive definite matrices; T is the prediction time domain; s is the integral variable, t k for the current moment;

[0111] Solve the following optimization problem:

[0112]

[0113] Where δ is the shrinkage factor, and 0<δ<1, α is a parameter related to the terminal set; U = {u|u min ≤||u||≤u max};

[0114] Solve this optimization problem (9) and get the current time t k Optimization control input variables The corresponding optimized state variable is recorded as

[0115] 52) Set up event triggering mechanism:

[0116] Design a time-varying trigger mechanism with adaptive trigger conditions:

[0117] Setting trigger conditions for:

[0118]

[0119] Among them, σ(s) satisfies

[0120]

[0121] δ0>0 indicates the state deviation value allowed in the design, c1, c2, d1, d2 are adjustable parameters, and σ(s) also satisfies σ(s)∈[σ min ,σ max ],σ max and σ min is the boundary value of the trigger mechanism;

[0122] Then the next optimization time t k+1 Take it as:

[0123]

[0124] Step 6: Implement trajectory tracking control as follows:

[0125] 61) Measure the current time t of the system k The output control variable y and the state estimate based on the extended state observer

[0126] 62) Based on the preset performance function ρ(t) and the converted output tracking error functions (6) and (12), a converted error model (7) is obtained;

[0127] 63) Solve for the current time t k Optimization problem (9), obtain the optimized input variables and state variables

[0128] 64) At the next moment t k+1 If the trigger condition (10) is not met, Input into the system (7) and (8) respectively; if the trigger condition (10) is met, go to step 63) and solve the optimization problem (9) again.

[0129] The present invention will be further described below with a preferred embodiment.

[0130] Figure 1 The parameters of the two-link robotic arm system shown in FIG. 1 are shown in Table 1.

[0131] Table 1 Robot arm parameters

[0132]

[0133] Based on the system parameters shown in Table 1, the optimization control method of the present invention is used for simulation, and the simulation results are compared with those of the traditional PD algorithm and the conventional model predictive control algorithm (MPC) to illustrate the superiority of the present invention.

[0134] The system simulation conditions are:

[0135] The expected output is q d =[cos(t),sin(t)] T , the initial error value is e(0)=[0.57,-1.047] T The disturbance variables are: ω1 = 1 + 0.1sin(0.8q1), ω2 = 1 + 0.3cos(2q2), and the constraints on the state and control input are [-π / 2, -π / 2] T ≤x1≤[π / 2,π / 2] T , [-20,-20] T ≤u≤[20,20] T , the sampling interval and sampling time are taken as 0.05 seconds and 15 seconds respectively.

[0136] 1. Using the event-triggered prediction optimization control method of the present invention:

[0137] Set the simulation conditions as follows:

[0138] 1) In observer (3), the parameters are L1=3k0, Where k0 = 47;

[0139] 2) In the preset performance function (5), the parameters are a1 = 0, a2 = -0.49, ρ0=1.5, T0=3;

[0140] 3) In the error conversion function (6), take j = 1, 2, where h j =0.77,

[0141] 4) In the optimization problem (9), the weight matrix is ​​Q = 10I 4×4 ,R=0.1I 2×2 , prediction duration T = 1 second, α = 0.3, δ = 0.62;

[0142] 5) Event trigger mechanism In formula (10), d1 = d2 = 1, c1 = c2 = 2, δ0 = 0.1;

[0143] According to the above simulation conditions, the system is simulated to verify the trajectory tracking capability of the system.

[0144] 2. PD control algorithm

[0145] set up Where K p =diag(50,50);K D =diag(10,10), other parameters are as shown above;

[0146] 3. Conventional MPC algorithm

[0147] The parameters in the algorithm are the same as those in the event-triggered predictive control method. In this method, the preset performance function and event trigger mechanism are not used.

[0148] Figure 3 It shows the output value of the robot arm system and the output value of its observer. Figure 3 It can be seen that in the presence of disturbance, the initial observation error is slightly larger and then gradually decreases.

[0149] Figure 4 and Figure 5 The simulation curves of the angular velocity y1(t)=q1(t) and y2(t)=q2(t) of the two-link robot arm under the control method of the present invention are compared with the simulation curves based on the PD algorithm and the conventional MPC method. Figure 4 , Figure 5 It can be seen that under the control method of the present invention, the system can not only track the desired trajectory, but also the effect is significantly better than the results using the PD controller and the conventional MPC control method.

[0150] Figure 6 and Figure 7 are respectively the angular velocity errors of the two-link mechanical arm under the control method of the present invention The simulation curves are compared with those based on the PD algorithm and the conventional MPC method. Figure 6 , Figure 7 It can be seen that under the control method of the present invention, the system error can not only converge to zero, but also always be within the constraint boundary, and its effect is better than that of conventional MPC. Under the PD control method, the effect is slightly worse than the first two methods, and the error exceeds the constraint boundary in the initial stage, but the system can eventually be stabilized.

[0151] Figure 8 The number of times the optimization problem is solved under event triggering of the present invention is compared with the number of times the optimization problem is solved by conventional MPC. Figure 8 It can be found that the number of optimization problem solving times of the event-triggered MPC control method designed by the present invention is significantly less than that of the conventional MPC method.

[0152] Fig. 9 The control torque input of the two-link mechanical arm using the control method of the present invention is The simulation curve is compared with the simulation curve under the conventional MPC method. It can be seen from the figure that the control method of the present invention significantly reduces the number of solutions to the optimization problem, so the amount of calculation is small.

[0153] The above results show that the trajectory tracking optimization control method designed in the present invention combines the advantages of preset performance control method, event trigger mechanism and predictive control method, and has the advantages of fast response speed, high positioning accuracy and small calculation amount. It has ideal tracking performance and good control flexibility.

Claims

1. A trajectory tracking optimization control method for a two-link robot arm whose state is not completely measurable, wherein the two-link robot arm comprises two links: a first link and a second link, and two joints: a first joint and a second joint; characterized in that: The following steps are involved: Step 1, establish the mathematical model of the two-link robotic arm: Where q = [q1 q2] T , and are the angular position vector, angular velocity vector and angular acceleration vector of the two connecting rods respectively, q1 is the angular position of the first connecting rod, and q2 is the angular position of the second connecting rod; is the inertia matrix of a symmetric positive rigid body, represents the Coriolis force matrix, represents the gravity vector, is the external disturbance, τ=[τ1 τ2] T is the control torque, τ1 is the control torque of the first connecting rod, and τ2 is the control torque of the second connecting rod; Step 2, define the state variable as: x1 = q, Output variable y = [y1y2] T =x1=q, control input variable u=τ, then equation (1) can be expressed as: Where, f(x1,x2)=-M -1 (x1)[C(x1,x2)x2+G(x1)]; g(x1)=-M -1 (x1);ω(x1)=M -1 (x1)ω is a disturbance variable consisting of friction and external disturbance, and satisfies: ||ω(x1)||≤ω a , its derivative is also bounded and satisfies: ω a ,ω m are all constants greater than zero; Assume that the output variable y=x1 satisfies the constraints: in x 1, are the lower and upper bounds of x1, respectively, ||·|| represents the Euclidean norm; the control input u satisfies: u min ≤||u||≤u max , where u min 、u max are the lower and upper bounds of u respectively; the output tracking error is defined as e1 = [e 11 e 12 ] T =yy d , where e 11 、e 12 are the output tracking errors of the first link and the second link, y d =[y d1 ,y d2 ] T is the expected value of the angular position vector of the two links, y d1 ,y d2 are the expected values ​​of the angular positions of the first link and the second link, respectively; Step 3, design the extended state observer: Wherein, L1, L2, and L3 represent the three gains of the extended state observer respectively. denote the estimated values ​​of the state variables x1, x2 and the disturbance variable ω(x1), respectively, is the estimated value of the output variable y; The observation error is defined as i=1,2,3, then the error state equation can be obtained from equation (2) and equation (3): Step 4, error conversion: Let the preset performance function ρ(t) be: Wherein, ρ0 is the initial value of the preset performance function ρ(t), T0 is the time required for the preset performance function ρ(t) to converge from the initial value to the steady-state value, is the steady-state value of the predetermined performance function ρ(t), and constants a1, a2, a3 are selected to satisfy the following conditions: Let the output tracking error e1 be converted to: e1=ρ(t)S(ε1) (6) In the formula, ε1=[ε 11 ,ε 12 ] T represents the error after conversion, where ε 11 , ε 12 are the errors after the conversion of the first link and the second link respectively; S(ε1)=[S(ε 11 ),S(ε 12 )] T is the conversion function, where S(ε 11 ), S(ε 12 ) are all strictly increasing and reversible smooth functions, and satisfy S(0) = 0; make Define the augmented error variable z as: Then we can have: in, φ(t)=diag[φ1(t),φ2(t)]; Its upper bound is γ, γ>0; make Then the system (7) is a nominal system, and we have: In the formula, denote the augmented error variable and control input variable of the nominal system (8) respectively; Step 5: Design an event-triggered model predictive controller based on a preset performance function: 51) Take the objective function J as: Among them, Q, R and P are positive definite matrices; T is the prediction time domain; s is the integral variable, t k for the current moment; Solve the following optimization problem: Where δ is the shrinkage factor, and 0<δ<1, α is a parameter related to the terminal set; U = {u|u min ≤||u||≤u max }; Solve the optimization problem (9) to obtain the current time t k Optimization control input variables The corresponding optimized state variable is recorded as 52) Set up event triggering mechanism: Design a time-varying trigger mechanism with adaptive trigger conditions: Setting trigger conditions for: Among them, σ(s) satisfies δ0>0 indicates the state deviation value allowed in the design, c1, c2, d1, d2 are adjustable parameters, and σ(s) also satisfies σ(s)∈[σ min ,σ max ],σ max and σ min is the boundary value of the trigger mechanism; Next optimization time t k+1 Take it as: Step 6: Implement trajectory tracking control. The specific method is as follows: 61) Measure the current time t of the system k The output control variable y, and the state estimation value based on the extended state observer 62) Based on the preset performance function ρ(t) and the converted output tracking error (6), obtaining a converted error model (7); 63) Solve for the current time t k Optimization problem (9), obtain the optimized input variables and state variables 64) At the next moment t k+1 If the trigger condition (10) is not met, Input into the system (7) and (8) respectively; if the trigger condition (10) is met, go to step 63) and solve the optimization problem (9) again.

Images