An Optimal Control Method for Trajectory Tracking of a Two-Link Manipulator with Partially Unmeasurable States

Abstract

The present invention relates to a trajectory tracking optimization control method for a two-link manipulator with partially unmeasurable states, belonging to the technical field of control engineering. This method utilizes a preset performance function and an event-triggered mechanism to implement event-triggered model predictive optimization control: establish a mathematical model of the two-link manipulator system, and obtain the state-space equation of the system accordingly; design an extended state observer, set a preset performance function, adjust its parameters to meet the transient performance requirements, and perform error conversion on the output tracking error accordingly; design an event-triggered model predictive controller based on the preset performance function: solve the optimization problem to obtain the optimized control input variables and state variables; design an adaptive time-varying triggering mechanism to reduce the number of optimizations in predictive control. The present invention combines the advantages of the preset performance control method, the event-triggered mechanism, and the predictive control method, and has the advantages of fast response speed, high positioning accuracy, and small computational load, and has 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 \(q = [q_1\ q_2]\) T , and are the angular position vector, angular velocity vector, and angular acceleration vector of the two linkages respectively, \(q_1\) is the angular position of the first linkage, and \(q_2\) is the angular position of the second linkage; is a symmetric positive rigid body inertia matrix, represents the Coriolis force matrix, represents the gravity vector, is the external disturbance, \(\tau = [\tau_1\ \tau_2]\) T is the control torque, \(\tau_1\) is the control torque of the first linkage, and \(\tau_2\) is the control torque of the second linkage;

[0012] Step 2, define the state variables as: \(x_1 = q\), the output variable \(y = [y_1\ y_2]\) T = \(x_1 = q\), and the control input variable \(u = \tau\). Then, Equation (1) can be expressed as:

[0013]

[0014] where \(f(x_1,x_2)= -M\) -1 (x_1)>C(x_1,x_2)x_2 + G(x_1)@; \(g(x_1)= -M\) -1 (x_1); \(\omega(x_1)= M\) -1 (x_1)\(\omega\), which is a disturbance variable composed of friction and external disturbance, and satisfies: \(\|\omega(x_1)\|\leq\omega\) a , and its derivative is also bounded and satisfies: Z a ,Z m are all positive constants;

[0015] Assume that the output variable \(y = x_1\) satisfies the constraint condition: where x 1, are the lower and upper bounds of \(x_1\) respectively, and \(\|\cdot\|\) represents the Euclidean norm; the control input variable \(u\) satisfies: \(u\) min \(\leq\|u\|\leq u\) max , where \(u\) min and \(u\) max are the lower and upper bounds of \(u\) respectively; define the output tracking error as \(e_1 = [e\) 11 e 12 T = \(y - y\) d , where \(e\) 11 ​, e 12 are the output tracking errors of the first link and the second link respectively, 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 an extended state observer:

[0017]

[0018] where L1, L2, and L3 represent the three gains of the extended state observer respectively, represent 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] Define the observation error as Then, from Equation (2) and Equation (3), the error state equation can be obtained as:

[0020]

[0021] Step 4, error conversion:

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

[0023]

[0024] where ρ0 is the initial value of the preset performance function ρ(t), T0 is the time required for the predetermined 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 by selecting constants a1, a2, a3, the following conditions are satisfied:

[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 converted error, where ε 11 , ε 12 are the converted errors 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 both strictly increasing and invertible smooth functions, and satisfy S(0) = 0;

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

[0030]

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

[0032] Let Then the system (7) at this time is the nominal system, and we have:

[0033]

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

[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] where Q, R, and P are positive definite matrices; T is the prediction horizon; s is the integration variable, and t k is the current time;

[0039] Solve the following optimization problem:

[0040]

[0041]

[0042] where δ is the contraction 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 optimal control input variable at the current time t k and the corresponding optimal state variable is denoted as ​

[0044] 52) Set up an event trigger mechanism:

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

[0046] Set the trigger condition as:

[0047]

[0048] where V(s) satisfies

[0049]

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

[0051] The next optimization moment t k+1 is taken as:

[0052]

[0053] Step 6, implement trajectory tracking control, and the specific method is:

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

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

[0056] 63) Solve the optimization problem (9) at the current moment t k to obtain the optimized input variable and the state variable

[0057] 64) At the next moment t k+1 , if the trigger condition (10) is not satisfied, then are respectively input into the systems (7) and (8); if the trigger condition (10) is satisfied, then go to step 63), and re-solve the optimization problem (9).

[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 an adaptive event-triggering mechanism. It can not only make the closed-loop controlled system stable while satisfying the constraints, but also meet the transient performance requirements by adjusting the parameters of the preset performance function.

[0060] 2) The proposed adaptive event-triggering mechanism is flexible and variable, reducing the number of optimizations in predictive control and the computational load. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 It is a schematic diagram of a two-link manipulator according to an embodiment of the present invention.

[0062] Figure 2 It is a structural diagram of the control system of the two-link manipulator of the present invention.

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

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

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

[0066] Figure 6 It is the angular velocity error of the two-link manipulator of the present invention and the traditional methods based on PD algorithm and conventional MPC method Comparative simulation curve diagram.

[0067] Figure 7 It is the angular velocity error of the two-link manipulator of the present invention and the traditional methods based on PD algorithm and conventional MPC method Comparative simulation curve diagram.

[0068] Figure 8 It is the number of solution times of the optimization problem under event triggering of the present invention and the number of optimization solutions of conventional MPC.

[0069] Figure 9 It is a comparative simulation curve diagram of the control torque u of the two-link manipulator of the present invention and the traditional methods based on PD algorithm and conventional MPC method.

[0070] Reference numerals in the figure: 1 - first link, 2 - second link, 3 - first joint, 4 - second joint. DETAILED DESCRIPTION OF THE INVENTION

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

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

[0073] To achieve the smooth operation of the above-mentioned robotic arm, an optimized control method for trajectory tracking of a two-link robotic arm with incomplete state measurement according to the present invention includes 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 vectors, angular velocity vectors, and angular acceleration vectors of the two linkages respectively, q1 is the angular position of the first linkage 1, and q2 is the angular position of the second linkage 2; is the symmetric positive rigid body inertia matrix, 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 linkage 1, and τ2 is the control torque of the second linkage 2; and there is:

[0077]

[0078] where C 22 = 0, G1 = (m1l c2 + m2l1)gcosq1 + m2l c2 gcos(q1 + q2), G2 = m2l c2 gcos(q1 + q2), g is the acceleration due to gravity.

[0079] For any t ≥ 0, q(t), all need to satisfy the constraint conditions:

[0080]

[0081] where \(k\) uj and \(k\) vj are constant vectors, and \(k\) uj = \([k\) uj1 , \(k\) uj2 \) T , \(k\) vj = \([k\) vj1 , \(k\) vj2 \) T , \(j = 1, 2\).

[0082] Step 2, for the convenience of subsequent controller design and analysis, define the state variables as: \(x1 = q\), the output variable \(y = [y1 y2]\) T = \(x1 = q\), and the control input \(u = W\). 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); \(\omega(x1) = M\) -1 (x1)\(\omega\), which is a bounded lumped disturbance composed of friction and external disturbances, and satisfies: \(\|\omega(x1)\| \leq \omega\) a , and its derivative is also bounded and satisfies: \(\omega\) a , \(\omega\) m are both positive constants;

[0085] Assume that the output variable \(y = x1\) satisfies the constraint condition: where x 1, are the lower and upper bounds of \(x1\) respectively, and \(\|\cdot\|\) represents the Euclidean norm; the control input \(u\) satisfies: \(u\) min \(\leq \|u\| \leq u\) max , where \(u\) min and \(u\) max are the lower and upper bounds of \(u\) respectively; define the output tracking error as \(e1 = [e\) 11 \(e\) 12 \) T = \(y - y\) d , where \(e\) 11 and \(e\) 12 are the output tracking errors of the first link 1 and the second link 2 respectively, and \(y\) 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 respectively the expected values of the angular positions of the first link 1 and the second link 2;

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

[0087]

[0088] where L1, L2, and L3 respectively represent the three gains of the extended state observer. Here, take L1 = 3k0, k0 > 0. respectively represent the estimated values of the state variables x1, x2, and the disturbance variable Z(x1), represents the estimated value of the output variable y;

[0089] Define the observation error as Then, from equations (2) and (3), the error state equation can be obtained as:

[0090]

[0091] Step 4, error conversion:

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

[0093]

[0094] In the formula, 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 selecting the constants a1, a2, 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 converted error, where ε 11 、ε 12 are respectively the converted errors of the first link 1 and the second link 2; 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 invertible smooth functions, and satisfy S(0) = 0; after transformation, the error is taken as:

[0099]

[0100] where h j and are positive parameters, and satisfy 0 < h j ≤ 1,

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

[0102]

[0103] where, φ(t) = diag[φ1(t), φ2(t)]; Its upper bound is and γ > 0.

[0104] Let Then the system (7) at this time is the nominal system, and we have:

[0105]

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

[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] where, Q, R, and P are positive definite matrices; T is the prediction time domain; s is the integration variable, and t k is the current time;

[0111] Solve the following optimization problem:

[0112]

[0113] where, δ is a contraction 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) to obtain the current time tk Optimized control input variable Then the corresponding optimized state variable is denoted as

[0115] 52) Set up an event trigger mechanism:

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

[0117] Set the trigger condition as:

[0118]

[0119] where σ(s) satisfies

[0120]

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

[0122] Then the next optimization time t k+1 is taken as:

[0123]

[0124] Step 6, implement trajectory tracking control according to the following method:

[0125] 61) Measure the output control variable y at the current time t k of the system, and the state estimation value based on the extended state observer

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

[0127] 63) Solve the optimization problem (9) at the current time t k to obtain the optimized input variable and the state variable

[0128] 64) At the next time t k+1 , if the trigger condition (10) is not satisfied, then Input into systems (7) and (8) respectively; if the trigger condition (10) is satisfied, go to step 63), and re-solve the optimization problem (9).

[0129] The following uses a preferred embodiment to further illustrate the present invention.

[0130] Figure 1 For the shown two-link manipulator system, its parameters are shown in Table 1.

[0131] Table 1 Manipulator 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 as follows:

[0135] The desired output is q d =[cos(t),sin(t)] T , and 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 of the state and control input are [-π / 2,-π / 2] T ≤x1≤[π / 2,π / 2] T , [-20,-20] T ≤u≤[20,20] T , and the sampling interval and sampling time are taken as 0.05 seconds and 15 seconds respectively.

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

[0137] Set the simulation conditions as follows:

[0138] 1) In the observer (3), the parameter is taken as L1 = 3k0, where k0 = 47;

[0139] 2) In the preset performance function (5), the parameters are taken as a1 = 0, a2 = 0.49, ρ0 = 1.5, ρ T0 = 0.03, T0 = 3;

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

[0141] 4) In the optimization problem (9), the weighting matrix is taken as Q = 10I 4×4 , R = 0.1I 2×2 , the prediction horizon T = 1 s, α = 0.3, δ = 0.62;

[0142] 5) In the event-triggering mechanism formula (10), d1 = d2 = 1, c1 = c2 = 2, δ0 = 0.1;

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

[0144] 2. PD control algorithm

[0145] Let where K p = diag(50, 50); K D = diag(10, 10), and other parameters are as shown before;

[0146] 3. Conventional MPC algorithm

[0147] The parameters in the algorithm are the same as those of the event-triggered predictive control method. Under this method, the preset performance function and the event-triggering mechanism are not adopted.

[0148] Figure 3 Shows the output value of the robotic arm system and the output value of its observer. From Figure 3 it can be seen that in the case of disturbances, the initial observation error is slightly larger and then gradually decreases.

[0149] Figure 4 and Figure 5 are respectively the simulation curves of the angular velocities y1(t) = q1(t) and y2(t) = q2(t) of the two-link robotic arm under the control method of the present invention, and are compared with the simulation curves under the PD algorithm and the conventional MPC method. From 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 under the PD controller and the conventional MPC control method.

[0150] Figure 6 and Figure 7 are respectively the simulation curves of the angular velocity error of the two-link robotic arm under the control method of the present invention, and are compared with the simulation curves under the PD algorithm and the conventional MPC method. From Figure 6 , Figure 7It can be seen that under the control method of the present invention, the system error can not only converge to zero, but also always remain within the constraint boundary, and its effect is better than that of the conventional MPC. Under the PD control method, the effect is slightly worse than the previous two methods, and the error exceeds the constraint boundary at the beginning stage, but the system can also be stabilized finally.

[0151] Figure 8 Comparison of the number of times of solving the optimization problem under the event trigger of the present invention with that of the conventional MPC, from Figure 8 it can be found that the number of times of solving the optimization problem of the MPC control method designed under the event trigger of the present invention is significantly less than that of the conventional MPC method.

[0152] Figure 9 The control torque input of the two-link manipulator under the control method of the present invention of the simulation curve, and make a comparison with the simulation curve under the conventional MPC method. It can be seen from the figure that the number of times of solving the optimization problem of the control method of the present invention is significantly reduced, so the computational complexity is small.

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

Claims

1. An optimized control method for trajectory tracking of a two-link manipulator with partially unmeasurable states. The two-link manipulator includes two links: a first link and a second link, and two joints: a first joint and a second joint. It is characterized in that: It includes the following steps: Step 1, establish the mathematical model of the two-link manipulator: where \(q = [q_1\ q_2]\) T , and are the angular position vector, angular velocity vector, and angular acceleration vector of the two linkages respectively, \(q_1\) is the angular position of the first linkage, and \(q_2\) is the angular position of the second linkage; is a symmetric positive rigid body inertia matrix, represents the Coriolis force matrix, represents the gravity vector, is the external disturbance, \(\tau = [\tau_1\ \tau_2]\) T is the control torque, \(\tau_1\) is the control torque of the first linkage, and \(\tau_2\) is the control torque of the second linkage; Step 2, define the state variables as: x1 = q, The output variable y = [y1 y2] T = x1 = q, and the control input variable u = τ, then Equation (1) can be expressed as: where \(f(x_1,x_2)= -M\) -1 (x_1)[C(x_1,x_2)x_2 + G(x_1)]; \(g(x_1)= -M\) -1 (x_1); \(\omega(x_1)= M\) -1 \(\omega(x_1)\), a disturbance variable composed of frictional force and external disturbance, satisfies: \(\|\omega(x_1)\|\leq\omega\) a , its derivative is also bounded and satisfies: \(\omega\) a , \(\omega\) m are all constants greater than zero; Let the output variable \(y = x_1\) satisfy the constraint conditions: where x 1, are the lower and upper bounds of \(x_1\) respectively, and \(\|\cdot\|\) represents the Euclidean norm; the control input \(u\) satisfies: \(u\) min \(\leq\|\ u\|\leq u\) max , where \(u\) min 、\(u\) max are the lower and upper bounds of \(u\) respectively; define the output tracking error as \(e_1=[e\) 11 \(e\) 12 T \(=y - y\) d , where \(e\) 11 、\(e\) 12 are the output tracking errors of the first link and the second link respectively, \(y\) 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 an extended state observer: Among them, L1, L2, and L3 respectively represent the three gains of the expansion state observer, respectively represent the estimated values of the state variables x1, x2, and the disturbance variable ω(x1), is the estimated value of the output variable y; Define the observation error as For \(i = 1, 2, 3\), the error state equation can be obtained from equations (2) and (3) as follows: Step 4, error transformation: Let the preset performance function ρ(t) be: where 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 preset performance function ρ(t), and by selecting constants a1, a2, a3 to satisfy the following conditions: ρ(0) = ρ0, Let the output tracking error e1 be transformed into: e1 = ρ(t)S(ε1) (6) where ε1 = [ε 11 , ε 12 T represents the error after conversion, where ε 11 , ε 12 are respectively the errors after conversion of the first link and the second link; S(ε1) = [S(ε 11 ), S(ε 12 )] T is a conversion function, where S(ε 11 ), S(ε 12 ) are both strictly increasing and invertible smooth functions, and satisfy S(0) = 0;​ Let Define the augmented error variable z as: Then we have: Among them, φ(t) = diag[φ1(t), φ2(t)]; j = 1, 2; Its upper bound is γ, γ > 0; Let Then the system (7) at this time is the nominal system, and we have: wherein, respectively represent the augmented error variable and the control input variable of the nominal system (8); Step 5, design an event-triggered model predictive controller based on the preset performance function: 51) Take the objective function J as: where Q, R, and P are positive definite matrices; T is the prediction horizon; s is the integration variable, and t k is the current time; Solve the following optimization problem: where δ is a contraction factor with 0 < δ < 1, and α is a parameter related to the set of terminals; U = {u|u min ≤ ||u|| ≤ u max}; Solve the optimization problem (9) to obtain the optimal control input variable at the current time t k The corresponding optimal state variable is denoted as Then, the corresponding optimal state variable is denoted as 52) Set the event-triggering mechanism: Design a time-varying triggering mechanism with an adaptive triggering condition: Set trigger conditions as follows: where σ(s) satisfies δ0 > 0 represents the allowable state deviation value in the design, c1, c2, d1, d2 are adjustable parameters, and in addition, σ(s) also satisfies σ(s) ∈ [σ min , σ max , σ max and σ min are the boundary values of the triggering mechanism; Next optimization time t k+1 is taken as: Step 6, implement trajectory tracking control, and the specific method is: 61) Measure the output control variable y of the system at the current moment t k and the state estimation value based on the extended state observer 62) Based on the preset performance function ρ(t) and the transformed output tracking error (6), obtain the transformed error model (7); 63) Solve the optimization problem (9) at the current moment t k to obtain the optimized input variable and the state variable 64) At the next moment t k+1 , if the trigger condition (10) is not satisfied, then are respectively input into systems (7) and (8); if the trigger condition (10) is satisfied, then go to step 63), and re-solve the optimization problem (9).

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