阅读说明：本技术 一种多输入多输出机械系统自适应容错预设性能控制方法 (Self-adaptive fault-tolerant preset performance control method for multi-input multi-output mechanical system ) 是由 张刚 刘志坚 侯文宝 沈永跃 吴玮 李德路 于 2020-04-27 设计创作，主要内容包括：本发明公开了一种多输入多输出机械系统自适应容错预设性能控制方法。其步骤包括：步骤一,建立存在执行器故障的多输入多输出机械系统模型；步骤二,基于障碍李雅普诺夫函数设计虚拟控制器；步骤三,进行自适应容错控制器设计与自适应律的设计；步骤四,基于微分跟踪器进行虚拟控制器微分量估计。本控制方法使容错控制器可以自适应地对执行器故障造成的影响进行主动辨识和补偿,保证了系统在故障发生时的稳定；并利用障碍李雅普诺夫函数的有界性对系统跟踪误差进行主动约束,保证了系统跟踪误差在存在故障的情况下,预设的性能指标安全、准确、高效地完成。(The invention discloses a self-adaptive fault-tolerant preset performance control method for a multi-input multi-output mechanical system. The method comprises the following steps: establishing a multi-input multi-output mechanical system model with actuator faults; designing a virtual controller based on the barrier Lyapunov function; step three, designing a self-adaptive fault-tolerant controller and a self-adaptive law; and step four, carrying out virtual controller differential quantity estimation based on the differential tracker. The control method enables the fault-tolerant controller to adaptively carry out active identification and compensation on the influence caused by the fault of the actuator, thereby ensuring the stability of the system when the fault occurs; and the boundedness of the barrier Lyapunov function is utilized to actively constrain the system tracking error, so that the preset performance index is ensured to be safely, accurately and efficiently completed under the condition that the system tracking error has a fault.)

1. A self-adaptive fault-tolerant preset performance control method for a multi-input multi-output mechanical system is characterized by comprising the following steps:

step one, establishing a multi-input multi-output mechanical system model with actuator faults:

1) modeling a multi-input multi-output nonlinear mechanical system according to an Euler-Lagrange type nonlinear formula, wherein the formula is as follows:

wherein the content of the first and second substances,is the n-dimensional state quantity of the nonlinear mechanical system, the generalized inertia matrix, the Coriolis force and centrifugal force matrix and the generalized gravity vector of the nonlinear mechanical system are respectively, the specific parameters of the generalized inertia matrix, the Coriolis force and centrifugal force matrix and the generalized gravity vector of the nonlinear mechanical system are unknown to the control system,is a control input variable of the nonlinear mechanical system;

2) in non-linear mechanical systems in engineering, commonly encounteredTwo typical actuator faults are multiplicative and additive, respectively, which result in the control input actually applied by the actuatorThe following mapping relationship exists with the design control amount u:

u_actual(t)＝B(t)u(t)+u_b(t) (2)，

wherein the content of the first and second substances,for multiplicative failure of the actuator, B_i(t) (i ═ 1, …, n) satisfies 0 < B ≦ B_i(t). ltoreq.1, where b is the lower bound for known execution efficiency;the method comprises the following steps of (1) performing additive fault on an actuator, wherein the norm of the additive fault meets the bounded property;

3) consider multiplicative B (t) and additive u faults_bThe Euler-Lagrange type nonlinear mechanical system after (t) can be expressed in the form:

defining a state quantity p of a non-linear mechanical system to its desired trajectory p_dHas a tracking error of

For a typical second order mechanical system, i.e. a two-link arm system, the mass and length of the two links are each m₁,m₂,l₁And l₂Assuming that the local gravitational acceleration is g, thenThe specific forms of (b) can be written as:

step two, designing a virtual controller based on the barrier Lyapunov function:

1) based on the idea of preset performance control, let the system track error p_eThe ith-dimension vector p of_e,iThe upper and lower bound performance constraints shown by the following formula are satisfied:

-α_i(t)＜p_e,i(t)＜α_i(t) (5)，

wherein, α_i(t) > 0 is a predetermined performance function and is defined as:

α_i(t)＝(α_i,0-α_i,∞)exp(-γ_it)+α_i,∞(6)，

wherein the content of the first and second substances,is α_i(ii) an initial value of (t),is α_iEnd value of (t), γ_iExpressing the speed of exponential convergence, Performance function α_iInitial value of (t) α_i,0Is selected so that α_i,0＞|p_e,i(0)|；

If the constraint in equation (5) is always true, then p_e,i(i-1, …, n) will be at least at an exponential rate exp (- γ)_it) converges to a stable region (- α)_i,∞,α_i,∞) Due to α_i(t) > 0 is always true, so equation (5) can be converted to the following form:

-1＜p_e,i(t)/α_i(t)＜1 (7)；

defining quantities of stateSo that its i-th dimension component x_1,iIs defined as:

x_1,i(t)＝p_e,i(t)/α_i(t) (8)；

then only need toThe controller is designed to ensure that for any i 1, …, n, the state quantity x_1,iAlways within the interval (-1,1), the following barrier, lyapunov function, is constructed:

2) the derivation of equation (9) can result in:

the derivation of equation (8) is substituted into (10):

due to tracking error p_e,i＝p_i-p_d,iSubstituting it into equation (11) can yield:

continue to define state variablesComprises the following steps:

wherein the content of the first and second substances,is a virtual controller;

substituting equation (13) into (12) can result in:

scaling with the Young inequality yields:

3) combining the formulas (14) and (15), obtaining a virtual controller upsilon_i(t) the design form is as follows:

wherein the content of the first and second substances,a positive definite control gain matrix;

step three, designing a self-adaptive fault-tolerant controller and a self-adaptive law:

1) substituting equations (16) and (15) into equation (14) respectively is:

based on Lyapunov function V₁In consideration of the state quantity x, the formula (9)₂Continue to define the Lyapunov function V₂The following were used:

to lyapunov function V₂Derivative and substitute x₂The definition of (8) can be given as:

substituting equation (3) for the dynamics of a nonlinear mechanical system into equation (19) may further yield:

continue to combine x₂By substituting (20) into (8):

2) defining scalar statesThe adaptive fault tolerant predictive performance controller is as follows:

wherein the content of the first and second substances,for positively determining the control gain matrix, σ₀The value of > 0 is small and the value of,andis an adaptive parameter, andandthe adaptation law of (1) is as follows:

step four, estimating the differential quantity of the virtual controller based on a differential tracker:

computing scalar statesIt is necessary to obtain a derivative value of the virtual controller vReal-time acquisition using a linear tracking differentiatorThe linear tracking differentiator is defined as follows:

wherein the content of the first and second substances,andto track the state variables of the differentiator, c₀,c₁And c₂In order to have a positive value for the adjustable parameter,component of the ith dimension ofFrom state z_2,iAnd obtaining the estimation.

2. The adaptive fault-tolerant default performance control method for a multiple-input multiple-output mechanical system according to claim 1, wherein the performance indicators of the multiple-input multiple-output mechanical system are actively designed by a technical user, including convergence rate, overshoot, and steady-state error.

Technical Field

The invention relates to a self-adaptive fault-tolerant preset performance control method for a multi-input multi-output mechanical system, and belongs to the technical field of self-adaptive control of mechanical systems.

Background

Most of the multiple-input multiple-output mechanical systems in engineering can be modeled in the form of Euler-Lagrange type nonlinear systems, such as a robot system, a spacecraft attitude control system, an energy transmission system, a helicopter control system and the like. Therefore, many scholars perform dynamics analysis and robust controller design for Euler-Lagrange type nonlinear mechanical systems. For a multi-input multi-output Euler-Lagrange type nonlinear mechanical system, some effective control methods are proposed at present, such as a sliding mode control method, an optimal control method, a control method and the like.

Although the stability of a nonlinear mechanical system can be guaranteed through simulation analysis or experimental verification, the existing control method does not consider the state constraint problem in the system stabilization process. In practical conditions, state constraints of the mechanical system are widely existed, such as joint angle constraints in a mechanical arm system, range constraints of a motion system, obstacle constraints in a running space, and the like. If the nonlinear mechanical system violates the constraint in work, serious results such as collision, instability and even task failure can occur; in addition to the state constraint problem, nonlinear mechanical systems often suffer from actuator output inaccuracy, efficiency degradation, and even partial actuator failure.

Disclosure of Invention

Aiming at the problems in the prior art, the invention provides a self-adaptive fault-tolerant preset performance control method for a multi-input multi-output mechanical system, which combines preset performance control and fault-tolerant control, so that an actuator can keep state constraint, can inhibit the fault influence of the actuator, and ensure that preset performance indexes are completed safely, accurately and efficiently.

In order to achieve the above object, the present invention provides a method for controlling adaptive fault-tolerant preset performance of a multiple-input multiple-output mechanical system, comprising the following steps:

step one, establishing a multi-input multi-output mechanical system model with actuator faults:

1) modeling a multi-input multi-output nonlinear mechanical system according to an Euler-Lagrange type nonlinear formula, wherein the formula is as follows:

wherein the content of the first and second substances,is the n-dimensional state quantity of the nonlinear mechanical system, the generalized inertia matrix, the Coriolis force and centrifugal force matrix and the generalized gravity vector of the nonlinear mechanical system are respectively, the specific parameters of the generalized inertia matrix, the Coriolis force and centrifugal force matrix and the generalized gravity vector of the nonlinear mechanical system are unknown to the control system,is a control input variable of the nonlinear mechanical system;

2) two typical actuator faults commonly encountered in non-linear mechanical systems in engineering are multiplicative faults and additive faults, which result in control inputs actually applied by the actuatorsThe following mapping relationship exists with the design control amount u:

u_actual(t)＝B(t)u(t)+u_b(t) (2)，

wherein the content of the first and second substances,for multiplicative failure of the actuator, B_i(t) (i ═ 1, …, n) satisfies 0 <b≤B_i(t) is less than or equal to 1, whereinbA lower bound for known execution efficiency;the method comprises the following steps of (1) performing additive fault on an actuator, wherein the norm of the additive fault meets the bounded property;

3) consider multiplicative B (t) and additive u faults_bThe Euler-Lagrange type nonlinear mechanical system after (t) can be expressed in the form:

defining a state quantity p of a non-linear mechanical system to its desired trajectory p_dHas a tracking error of

For a typical second order mechanical system, i.e. a two-link arm system, the mass and length of the two links are each m₁,m₂,l₁And l₂Assuming that the local gravitational acceleration is g, thenThe specific forms of (b) can be written as:

step two, designing a virtual controller based on the barrier Lyapunov function:

1) based on the idea of preset performance control, let the system track error p_eThe ith-dimension vector p of_e,iThe upper and lower bound performance constraints shown by the following formula are satisfied:

-α_i(t)＜p_e,i(t)＜α_i(t) (5)，

wherein, α_i(t) > 0 is a predetermined performance function and is defined as:

α_i(t)＝(α_i,0-α_i,∞)exp(-γ_it)+α_i,∞(6)，

wherein the content of the first and second substances,is α_i(ii) an initial value of (t),is α_iEnd value of (t), γ_iExpressing the speed of exponential convergence, Performance function α_iInitial value of (t) α_i,0Is selected so that α_i,0＞|p_e,i(0)|；

If the constraint in equation (5) is always true, then p_e,i(i-1, …, n) will be at least at an exponential rate exp (- γ)_it) converges to a stable region (- α)_i,∞,α_i,∞) Due to α_i(t) > 0 is always true, so equation (5) can be converted to the following form:

-1＜p_e,i(t)/α_i(t)＜1 (7)；

defining quantities of stateSo that its i-th dimension component x_1,iIs defined as:

x_1,i(t)＝p_e,i(t)/α_i(t) (8)；

it is only necessary to design the controller to ensure that for any i 1, …, n, the state quantity x_1,iAlways within the interval (-1,1), the following barrier, lyapunov function, is constructed:

2) the derivation of equation (9) can result in:

the derivation of equation (8) is substituted into (10):

due to tracking error p_e,i＝p_i-p_d,iSubstituting it into equation (11) can yield:

continue to define state variablesComprises the following steps:

wherein the content of the first and second substances,is a virtual controller;

substituting equation (13) into (12) can result in:

scaling with the Young inequality yields:

3) combining the formulas (14) and (15), obtaining a virtual controller upsilon_i(t) the design form is as follows:

wherein the content of the first and second substances,a positive definite control gain matrix;

step three, designing a self-adaptive fault-tolerant controller and a self-adaptive law:

1) substituting equations (16) and (15) into equation (14) respectively is:

based on Lyapunov function V₁In consideration of the state quantity x, the formula (9)₂Continue to define the Lyapunov function V₂The following were used:

to lyapunov function V₂Derivative and substitute x₂The definition of (8) can be given as:

substituting equation (3) for the dynamics of a nonlinear mechanical system into equation (19) may further yield:

continue to combine x₂By substituting (20) into (8):

2) defining scalar statesThe adaptive fault tolerant predictive performance controller is as follows:

wherein the content of the first and second substances,for positively determining the control gain matrix, σ₀The value of > 0 is small and the value of,andis an adaptive parameter, andandthe adaptation law of (1) is as follows:

step four, estimating the differential quantity of the virtual controller based on a differential tracker:

computing scalar statesIt is necessary to obtain a derivative value of the virtual controller vReal-time acquisition using a linear tracking differentiatorThe linear tracking differentiator is defined as follows:

wherein the content of the first and second substances,andto track the state variables of the differentiator, c₀,c₁And c₂In order to have a positive value for the adjustable parameter,component of the ith dimension ofFrom state z_2,iAnd obtaining the estimation.

Furthermore, the performance indexes of the multi-input multi-output mechanical system are actively designed by technical users, and comprise convergence speed, overshoot and steady-state error.

The control method comprises the steps of modeling a multi-input multi-output mechanical system model with an actuator fault, further designing a virtual controller, a self-adaptive fault-tolerant controller and a self-adaptive law on the basis of the modeling, and obtaining the differential estimation of the virtual controller, so that the fault-tolerant controller can adaptively carry out active identification and compensation on the influence caused by the actuator fault, and the stability of the system when the fault occurs is ensured; and the boundedness of the barrier Lyapunov function is utilized to actively constrain the system tracking error, so that the preset performance index can still be safely, accurately and efficiently completed under the condition that the system tracking error has a fault.

Drawings

FIG. 1 is a diagram of the change of joint angle error under the constraint of performance function in the stable control of the invention;

FIG. 2 is a graph showing the variation of tracking error of angular velocity of a joint in steady control;

FIG. 3 is a graph showing a change in control input in the steady control;

FIG. 4 is a graph showing adaptive parameter changes in steady control;

FIG. 5 is a graph of the change of joint angle error under the constraint of a performance function during tracking control;

FIG. 6 is a diagram showing a change in a joint angle tracking desired trajectory in tracking control;

fig. 7 is a control input variation diagram in tracking control;

fig. 8 is a diagram of adaptive parameter changes during tracking control.

Detailed Description

The invention will be further explained with reference to the drawings.

A self-adaptive fault-tolerant preset performance control method for a multi-input multi-output mechanical system comprises the following steps:

step one, establishing a multi-input multi-output mechanical system model with actuator faults:

1) modeling a multi-input multi-output nonlinear mechanical system according to an Euler-Lagrange type nonlinear formula, wherein the formula is as follows:

wherein the content of the first and second substances,is the n-dimensional state quantity of the nonlinear mechanical system, the generalized inertia matrix, the Coriolis force and centrifugal force matrix and the generalized gravity vector of the nonlinear mechanical system are respectively, the specific parameters of the generalized inertia matrix, the Coriolis force and centrifugal force matrix and the generalized gravity vector of the nonlinear mechanical system are unknown to the control system,is a control input variable of the nonlinear mechanical system;

2) two typical actuator faults commonly encountered in non-linear mechanical systems in engineering are multiplicative faults and additive faults, which result in control inputs actually applied by the actuatorsThe following mapping relationship exists with the design control amount u:

u_actual(t)＝B(t)u(t)+u_b(t) (2)，

wherein the content of the first and second substances,for multiplicative failure of the actuator, B_i(t) (i ═ 1, …, n) satisfies 0 <b≤B_i(t) is less than or equal to 1, whereinbA lower bound for known execution efficiency;the method comprises the following steps of (1) performing additive fault on an actuator, wherein the norm of the additive fault meets the bounded property;

3) consider multiplicative B (t) and additive u faults_bThe Euler-Lagrange type nonlinear mechanical system after (t) can be expressed in the form:

defining a state quantity p of a non-linear mechanical system to its desired trajectory p_dHas a tracking error of

For a typical second order mechanical system, i.e. a two-link arm system, the mass and length of the two links are each m₁,m₂,l₁And l₂Assuming that the local gravitational acceleration is g, thenThe specific forms of (b) can be written as:

step two, designing a virtual controller based on the barrier Lyapunov function:

1) based on the idea of preset performance control, let the system track error p_eThe ith-dimension vector p of_e,iThe upper and lower bound performance constraints shown by the following formula are satisfied:

-α_i(t)＜p_e,i(t)＜α_i(t) (5)，

wherein, α_i(t) > 0 is a predetermined performance function and is defined as:

α_i(t)＝(α_i,0-α_i,∞)exp(-γ_it)+α_i,∞(6)，

wherein the content of the first and second substances,is α_i(ii) an initial value of (t),is α_iEnd value of (t), γ_iExpressing the speed of exponential convergence, Performance function α_iInitial value of (t) α_i,0Is selected so that α_i,0＞|p_e,i(0)|；

If the constraint in equation (5) is always true, then p_e,i(i-1, …, n) will be at least at an exponential rate exp (- γ)_it) converges to a stable region (- α)_i,∞,α_i,∞) Due to α_i(t) > 0 is always true, so equation (5) can be converted to the following form:

-1＜p_e,i(t)/α_i(t)＜1 (7)；

defining quantities of stateSo that its i-th dimension component x_1,iIs defined as:

x_1,i(t)＝p_e,i(t)/α_i(t) (8)；

it is only necessary to design the controller to ensure that for any i 1, …, n, the state quantity x_1,iAlways within the interval (-1,1), the following barrier, lyapunov function, is constructed:

2) the derivation of equation (9) can result in:

the derivation of equation (8) is substituted into (10):

due to tracking error p_e,i＝p_i-p_d,iSubstituting it into equation (11) can yield:

continue to define state variablesComprises the following steps:

wherein the content of the first and second substances,is a virtual controller;

substituting equation (13) into (12) can result in:

scaling with the Young inequality yields:

3) combining the formulas (14) and (15), obtaining a virtual controller upsilon_i(t) the design form is as follows:

wherein the content of the first and second substances,a positive definite control gain matrix;

step three, designing a self-adaptive fault-tolerant controller and a self-adaptive law:

1) substituting equations (16) and (15) into equation (14) respectively is:

based on Lyapunov function V₁In consideration of the state quantity x, the formula (9)₂Continue to define the Lyapunov function V₂The following were used:

to lyapunov function V₂Derivative and substitute x₂The definition of (8) can be given as:

substituting equation (3) for the dynamics of a nonlinear mechanical system into equation (19) may further yield:

continue to combine x₂By substituting (20) into (8):

2) defining scalar statesThe adaptive fault tolerant predictive performance controller is as follows:

wherein the content of the first and second substances,for positively determining the control gain matrix, σ₀The value of > 0 is small and the value of,andis an adaptive parameter, andandthe adaptation law of (1) is as follows:

step four, estimating the differential quantity of the virtual controller based on a differential tracker:

computing scalar statesIt is necessary to obtain a derivative value of the virtual controller vReal-time acquisition using a linear tracking differentiatorThe linear tracking differentiator is defined as follows:

wherein the content of the first and second substances,andto track the state variables of the differentiator, c₀,c₁And c₂In order to have a positive value for the adjustable parameter,component of the ith dimension ofFrom state z_2,iAnd obtaining the estimation.

Specifically, the performance index of the multiple-input multiple-output mechanical system can be actively designed by a technical user, and mainly comprises a convergence rate, an overshoot and a steady-state error.