CN111577711B - Active-disturbance-rejection robust control method for double-rod hydraulic cylinder position servo system - Google Patents

Abstract

The invention discloses a robust active disturbance rejection (RISEESO) control method for a position servo system of a double-rod hydraulic cylinder. The method combines disturbance compensation based on extended state observer (ESO) with robust error sign integral (RISE) control Combined, Lyapunov stability theory is used to prove that the system is asymptotically stable. The proposed strategy effectively combines the interference suppression method (RISE) with the interference estimation compensation method, inheriting the unique advantages of the two methods and avoiding their respective shortcomings. The disclosed control method has the following advantages: compared with the traditional ESO method, the use of RISE reduces the observation burden of the ESO and further reduces the observer residual; The linear robust gain only needs to be related to the first and second derivatives of the state observation error, which weakens its original strict condition; the tracking performance of the proposed controller is better than that of RISE and ESO under the same conditions.

Description

Active Disturbance Rejection Robust Control Method for Position Servo System of Double Rod Hydraulic Cylinder

technical field

The invention relates to an electro-hydraulic servo control technology, in particular to an active disturbance rejection robust control method of a position servo system of a double-rod hydraulic cylinder.

Background technique

Hydraulic position servo system plays a pivotal role in aircraft, heavy machinery, high-performance rotary test equipment and other fields due to its high power density, large force/torque output, and strong anti-load rigidity. However, the inherent nonlinear characteristics of hydraulic systems and various modeling uncertainties complicate the design of their controllers. At first, a lot of researches were done on the design of the hydraulic system controller based on the linear control theory, such as the PID controller, but the design of the linear controller was based on the linearized hydraulic system model, which could not reflect its nonlinear characteristics, so it could not obtain good results. Control effect. The feedback linearization control can compensate the nonlinear characteristics of the hydraulic system in real time in the design of the controller, but it requires the system model information to be completely known, which is inconsistent with the actual application. Active disturbance rejection control (ADRC) has been widely used because it requires less model information and can obtain excellent control performance. It is characterized by using an extended state observer (ESO) to expand the integrated disturbance of the system into a A new state variable that applies the observed disturbance to the system through feedforward compensation to improve control performance. In order to simplify the implementation of nonlinear ESO, linear ESO is proposed. In actual control, only one parameter needs to be adjusted, which greatly facilitates the process of controller design and equipment debugging. The theoretical proof shows that the state estimation error increases with the The observer bandwidth increases and decreases monotonically. When the unmodeled dynamics of the system are large, in order to improve the control accuracy, the bandwidth of the observer must be increased. However, an excessively large bandwidth will amplify the system noise and even make the system unstable. The Robust Sign Integral Error (RISE) control method can also effectively deal with modeling uncertainty. It contains a unique Sign Integral Error Robust term, which can achieve asymptotically stable system disturbances that are sufficiently smooth and bounded. Track performance. However, the value of the nonlinear robust gain in the controller designed by this control method needs to meet certain conditions, which are closely related to the upper bounds of the first and second derivatives of the modeling uncertainty of the system with respect to time. , when the system is not modeled with large dynamics, in order to optimize the control performance, a larger feedback gain must be taken, which will also cause the system to have the risk of oscillation.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide an active disturbance rejection robust control method of a dual-rod hydraulic cylinder position servo system with strong anti-disturbance and high tracking performance, which can make the dual-rod hydraulic cylinder position servo system still maintain the Excellent control performance.

The technical solution for realizing the purpose of the present invention is: a robust control method for active disturbance rejection of a position servo system of a double-rod hydraulic cylinder, comprising the following steps:

Step 1, establish the mathematical model of the position servo system of the double-rod hydraulic cylinder;

Step 2, designing an ADRC robust controller according to the mathematical model of the above-mentioned dual-rod hydraulic cylinder position servo system;

Step 3, using the Lyapunov stability theory to prove the stability of the position servo system of the double-rod hydraulic cylinder, and using Barbalat's lemma to obtain the result that the system can achieve asymptotic stability.

Compared with the prior art, the present invention has the significant advantages of effectively combining interference suppression (RISE) and interference estimation compensation (ESO), and the use of RISE further reduces the estimation residual error of ESO, so that the control performance is improved, At the same time, the nonlinear robust feedback gain term of the improved RISE is only related to the derivative of the state estimation error, which is easier to satisfy than the original condition, and the tracking performance of the controller is improved compared with the traditional RISE and ESO. Simulation results verify its effectiveness.

Description of drawings

Fig. 1 is the principle diagram of the position servo system of the double-rod hydraulic cylinder of the present invention.

FIG. 2 is a schematic diagram of the principle of the Robust Active Disturbance Rejection Control (RISEESO) method for the position servo system of the double-rod hydraulic cylinder.

Fig. 3 is a graph showing the time-dependent change of the command signal expected to be tracked by the position servo system of the double-rod hydraulic cylinder.

Figure 4 is a comparison curve of the tracking performance of the RISEESO controller, the RISE controller, the ESO controller, and the PID controller in Case1.

Figure 5 is a schematic diagram of the control input of the system under the action of the RISEESO controller in Case1.

Figure 6 is a comparison curve of the tracking performance of the RISEESO controller, the RISE controller, the ESO controller, and the PID controller in Case2.

Figure 7 is a schematic diagram of the control input of the system under the action of the RISEESO controller in Case2.

Detailed ways

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

The invention is based on the traditional RISE control method, uses ESO to estimate the unmodeled dynamics of the system and performs feedforward compensation. The use of RISE reduces the observation burden of ESO and further reduces the estimation error of ESO. The use of ESO makes RISE nonlinear The robust gain does not need to be related to the upper bounds of the first and second derivatives of the system modeling uncertainty, but only needs to be related to the first and second derivatives of the state observation error. When the state estimation error of ESO is guaranteed by adjusting the bandwidth, This condition is easier to satisfy than the former. Finally, the Lyapunov stability theory is used to prove the stability of the position servo system of the double-rod hydraulic cylinder, and the result that the system can achieve asymptotic stability is obtained.

With reference to Figures 1 to 2, the active disturbance rejection robust control method for the position servo system of the double-rod hydraulic cylinder of the present invention includes the following steps:

Step 1, establish the mathematical model of the position servo system of the double-rod hydraulic cylinder;

Step 1-1. The dual-rod hydraulic cylinder position servo system considered in the present invention is a dual-rod hydraulic cylinder controlled by a servo valve to drive an inertial load;

Therefore, according to Newton's second law, the equation of motion for an inertial load is:

In formula (1), m is the inertial load parameter; _PL is the pressure difference between the two chambers of the hydraulic cylinder; A is the effective cross-sectional area of the hydraulic cylinder piston; B is the viscous friction coefficient; f(t) is other unmodeled disturbances; y is the inertial displacement of the load; t is the time variable;

Ignoring the external leakage of the hydraulic cylinder, the dynamic equation of the pressure difference between the two chambers of the hydraulic cylinder is:

In formula (2), V _t is the total control volume of the two chambers of the hydraulic cylinder; β _e is the effective elastic modulus of oil; C _t is the internal leakage coefficient; Q(t) is the complex internal leakage process and the unmodeled pressure dynamics _{Modeling error caused by etc ;} The relationship of _xv is:

符号函数sign(x_v)的定义为：Servo valve flow gain coefficient in formula (3)

The sign function sign(x _v ) is defined as:

where C _d flow coefficient; ω is the spool area gradient; ρ is the oil density; P _s is the oil supply pressure, and P _r is the oil return pressure;

Due to the consideration of servo valve dynamics, an additional displacement sensor needs to be installed to obtain the displacement of the servo valve spool, and there is only a slight improvement in tracking performance. Therefore, a large number of related researches ignore the dynamics of the servo valve. Assuming that a high-response servo valve is used, the spool displacement and the control input are approximated as a proportional link, that is, x _v = _ki u, _ki is a positive electrical constant, and u is the control input. input voltage; so equation (3) can be written as

In formula (5), k _t = k _q k _i represents the total flow gain;

则双出杆液压缸位置伺服系统的状态方程为：Step 1-2, assuming that the unmodeled dynamic term f(t) is continuous and differentiable, define the state variable:

Then the state equation of the position servo system of the double-rod hydraulic cylinder is:

In formula (6)

该项可以实时计算得出，只要U确定，那么实际的控制输入u也能得以计算；因此，以下控制器设计将聚焦于提出一种自抗扰鲁棒控制器U来掌控系统的各种干扰；In Equation (6), we define a new variable U to represent the control input of the system. Since a pressure sensor is installed in the actual system, so

This term can be calculated in real time, as long as U is determined, the actual control input u can also be calculated; therefore, the following controller design will focus on proposing an ADRC robust controller U to control various disturbances of the system ;

则式(6)可写成In order to simplify the display format of the state equation of the dual-rod hydraulic cylinder position servo system, denote θ=[θ ₁ ,...,θ ₃ ] ^T is the known nominal value of the system parameters, θ _r =[θ _1r ,... ,θ _3r ] ^T is the real value of the system parameters, where,

The formula (6) can be written as

In formula (8),

The design goal of the system controller of the position servo system of the double-rod hydraulic cylinder is: given the system reference signal y _d (t) = x _1d (t), design a bounded control input U to make the system output y = x ₁ as much as possible ground tracking system reference signal;

To facilitate controller design, the following assumptions are made:

总是有界的，那么只要所设计的U有界即可以保证真实的控制输入u有界；Assumption 1: The reference command signal x _1d (t) of the position servo system of the double-rod hydraulic cylinder is continuous in third order, and the system expects that the position command, speed command, acceleration command and jerk command are all bounded; working under working conditions, that is, both P ₁ and P ₂ are less than the oil supply pressure P _s , and |P _L | is also less than P _s ; therefore, it can be seen that

is always bounded, then as long as the designed U is bounded, the real control input u can be guaranteed to be bounded;

足够光滑，使得

均存在并有界即：Assumption 2: The total disturbance of the position servo system of the double-rod hydraulic cylinder

smooth enough that

Both exist and are bounded:

具有不确定的上界；In formula (9), δ ₁ , δ ₂ are unknown constants, namely

has an indeterminate upper bound;

Step 2, designing an ADRC robust controller according to the mathematical model of the above-mentioned dual-rod hydraulic cylinder position servo system;

Step 2-1. Disturbance compensation design based on extended state observer:

扩张为一个额外的状态变量，即

记h(t)为

的一阶导数，则式(8)可以写为：will integrate perturbation

is expanded to an additional state variable, i.e.

Let h(t) be

The first derivative of , then equation (8) can be written as:

的一阶导数，由式(9)可得：Since h(t) is

The first derivative of , can be obtained from equation (9):

From the structure of formula (10), it can be known that the system can be observed, and a linear expansion state observer is designed as follows:

表示状态估计，且w_o>0可以视作扩张状态观测器的带宽。in the formula

represents the state estimate, and w _o > 0 can be regarded as the bandwidth of the extended state observer.

i＝1,2,3,4，表示状态估计误差，由式(10)和式(12)可得状态估计误差的导数为：remember

i=1, 2, 3, 4, representing the state estimation error. The derivative of the state estimation error can be obtained from equation (10) and equation (12) as:

i＝1,2,3,4；于是式(12)可以写为：remember

i=1,2,3,4; then equation (12) can be written as:

M＝[0 0 0 1]^T，D为赫尔维茨矩阵；由于D为赫尔维茨矩阵，因此存在一个对称正定矩阵H使得D^TH+HD＝-I；in,

M=[0 0 0 1] ^T , D is the Hurwitz matrix; since D is the Hurwitz matrix, there is a symmetric positive definite matrix H such that D ^T H+HD=-I;

The following lemma is given:

可以得到如下结论：估计状态总是有界的，且任意时间后该界的大小随着观测器带宽w_o的增大而减小，存在正常数

满足下列关系：Depend on

The following conclusions can be drawn: the estimated state is always bounded, and the size of the bound decreases with the increase of the observer bandwidth w _o after any time, and there is a constant

Satisfy the following relationship:

This lemma proves:

Define a Lyapunov function:

V _ε = ε ^T Hε (16)

Derive it to get:

λ_max(H)为H的最大特征值。In formula (17),

λ _max(H) is the largest eigenvalue of H.

From (17) we can get:

It can be obtained from the above formula:

In formula (19), λ _min(H) is the minimum eigenvalue of H.

i＝1,2,3,4；及式(13)易得知(15)成立，至此，引理得证。It can be known from (19) that when h(t) is bounded, ||ε|| is always bounded, and as long as w _o > 1, then ||ε|| will increase with time. It is more important to increase λ by increasing w _o to reduce the error value of state estimation. Depend on

i=1, 2, 3, 4; and formula (13) is easy to know that (15) holds, so far, the lemma is proved.

令α₁为虚拟控制，使方程

趋于稳定状态；α₁与真实状态x₂的误差为z₂＝x₂-α₁，对z₁求导可得：Step 2-2, define z ₁ =x ₁ -x _1d as the tracking error of the system, according to the first equation in equation (10)

Let α ₁ be the virtual control, so that the equation

tends to a stable state; the error between α ₁ and the real state x ₂ is z ₂ =x ₂ -α ₁ , and the derivative of z ₁ can be obtained:

Design a virtual control law:

where k ₁ > 0 is the adjustable gain, then

Since z ₁ (s)=G(s)z ₂ (s), where G(s)=1/(s+k ₁ ) is a stable transfer function, when z ₂ tends to 0, z ₁ also It must tend to 0; so in the next design, the main design goal will be to make z ₂ tend to 0;

Step 2-3, consider the second equation of formula (10), select α ₂ to be the virtual control of x ₃ , and z ₃ to be the deviation between the virtual control α ₂ and x ₃ z ₃ =x ₃ -α ₂ ; then The dynamic equation of z ₂ is

The virtual control law _α2 is designed as follows:

where k ₂ is the positive feedback gain. Substitute equation (24) into equation (23) to get:

Since z ₂ (s)=G'(s)z ₃ (s), where G'(s)=1/(s+k ₂ ) is a stable transfer function, when z ₃ tends to 0, z ₂ must also tend to 0; so in the next design, the main design goal will be to make z ₃ tend to 0;

Steps 2-4. To gain an additional controller design freedom, define an auxiliary error signal r(t):

In formula (26), k ₃ > 0 is an adjustable gain. Since r(t) contains the jerk signal of the position, it is considered unmeasurable in practice, that is, r(t) is only used for auxiliary design, not Specifically appear in the designed controller;

Considering the third equation of Eq. (10), the expansion of r can be obtained as follows:

According to equation (27), the model-based controller U can be designed as:

where k _r is the positive feedback gain, U _a is the model-based compensation term, U _s is the robust control law and where U _s1 is the linear robust feedback term, U _s2 is the nonlinear robust term used to overcome the modeling The influence of uncertainty on system performance; Substitute equation (28) into equation (27) to get:

According to the error symbol integral robust controller design method, the integral robust term U _s2 can be designed as:

β must meet the following conditions:

Differentiating both sides of equation (29) and applying equations (7) and (11), we get:

Step 3, using the Lyapunov stability theory to prove the stability of the position servo system of the double-rod hydraulic cylinder, and using Barbalat's lemma to obtain the result that the system can achieve asymptotic stability, as follows:

First give the following lemma:

define helper functions

If the selection of control gain β satisfies the condition shown in equation (31), that is:

but

分别表示z₃(t)和

的初始值；z ₃ (0),

denote z ₃ (t) and

the initial value of ;

Proof of this lemma:

Integrating both sides of Equation (33) and applying Equation (26), we get:

Integrating the last two terms in Eq. (32) by parts can be obtained:

Therefore

It can be seen from equation (38) that if the selection of β satisfies the conditions shown in equation (31), equations (34) and (35) are established, that is, the lemma is proved.

Define a helper function:

时，P(t)≥0，因此定义李雅普诺夫函数如下：According to the above proof, it can be seen that when

When P(t)≥0, the Lyapunov function is defined as follows:

Taking the derivative of equation (27) and substituting equations (22), (25), (26), (32), and (39) into equations (22), (25), (26), (32), and (39), we get:

definition:

Z=[z ₁ ,z ₂ ,z ₃ ,r] ^T (42)

By adjusting the parameters k ₁ , k ₂ , k ₃ , k _r , the symmetric matrix Λ can be made positive definite, then:

In formula (44), λ _min (Λ) is the minimum eigenvalue of the symmetric positive definite matrix Λ.

Integrating Equation (44) can get:

范数，因此W是一致连续的，由Barbalat引理可知：t→∞时，W→0。故t→∞时，z₁→0。From equation (32), it can be known that z ₁ , z ₂ , z ₃ , r∈L ₂ norm, and according to equations (22), (25), (26), (32) and assumption 2, we can get:

norm, so W is consistent and continuous, according to Barbalat's lemma: when t→∞, W→0. Therefore, when t→∞, z ₁ →0.

Therefore, there is a conclusion: the active disturbance rejection robust controller designed for the position servo system (2) of the double-rod hydraulic cylinder can make the system obtain asymptotically stable results, and adjusting the gains k ₁ , k ₂ , k ₃ , and k _r can make the system The tracking error of the system tends to zero under the condition that the time tends to infinity; Figure 2 shows the schematic diagram of the control principle of the Robust Active Disturbance Rejection (RISEESO) of the position servo system of the double-rod hydraulic cylinder.

Example

In order to evaluate the performance of the designed controller, the parameters in Table 1 are used to model the position servo system of the double-rod hydraulic cylinder:

Table 1 Parameters of double-rod hydraulic system

The desired instruction for a given system is x _1d =0.2sin(πt)[1-exp(-0.01t ³ )](m), as shown in FIG. 3 . In order to verify the effectiveness of the controller proposed by the present invention, the following controllers are compared in two cases.

Case1: Only parameter uncertainty is considered, internal leakage and unmodeled dynamics are not considered, that is, θ≠θ _r , Q(t)=0, f(t)=0.

From Table 1, the true value of the Chu parameter can be calculated as θ _r =[8.59, 6.68×10 ⁵ , 205.75] ^T , taking its nominal value as θ = [8,7×10 ⁵ ,100] ^T .

Active disturbance rejection robust controller (RISEESO): The parameters of the RISEESO controller are taken as k ₁ =3000, k ₂ =500, k ₃ =100, _{k r} =100, β=50, wo =100.

其余与之相同，参数取值也与之相同。Error Symbol Integral Robust Controller (RISE): The RISE controller removes the U _a term in RISEESO

The rest are the same, and the parameter values are also the same.

Interference compensation controller (ESO): This controller is a RISEESO controller without the U _S2 item, and the parameter values are the same as RISEESO.

PID controller: The parameter adjustment of the PID controller is carried out by trial and error. The obtained parameters have made the PID controller reach the performance limit. If the gain value is increased, the tracking error will diverge. The selected PID controller parameters are

k _P =5000, k _I =800, k _D =50.

The tracking error comparison of each controller is shown in Figure 4. It can be seen from the figure that the control performance of the proposed RISEESO controller is far better than the other three controllers in the presence of only parameter uncertainty, which shows the effectiveness of the proposed algorithm. The control input of the system under the action of the RISEESO controller is shown in Figure 5.

Case2: Consider not only parameter uncertainty, but also leakage and unmodeled dynamics in the system, take f(x,t)=1000sint, Q(t)=1×10 ^-4 sint.

In this case, the parameters of the two controllers are the same as those in Case1.

Figure 6 is a comparison chart of the tracking performance of each controller when parameter uncertainty and unmodeled dynamics coexist. It can be seen that although a large unmodeled dynamics is added to the system, the control performance of RISEESO is still excellent, and The tracking performance is far better than the other three controllers, which shows that the proposed strategy has strong anti-interference ability. The control input of the system under the action of the RISEESO controller is shown in Figure 7.

Claims (1)

1. the active disturbance rejection robust control method of a double-out rod hydraulic cylinder position servo system, is characterized in that, comprises the following steps: Step 1, establish the mathematical model of the position servo system of the double-rod hydraulic cylinder, as follows: Step 1-1. Consider that the position servo system of the double-rod hydraulic cylinder is a double-rod hydraulic cylinder controlled by a servo valve to drive the inertial load; Therefore, according to Newton's second law, the equation of motion for an inertial load is:

In formula (1), m is the inertial load parameter; _PL is the pressure difference between the two chambers of the hydraulic cylinder; A is the effective cross-sectional area of the hydraulic cylinder piston; B is the viscous friction coefficient; f(t) is other unmodeled disturbances; y is the inertial displacement of the load; t is the time variable; Ignoring the external leakage of the hydraulic cylinder, the dynamic equation of the pressure difference between the two chambers of the hydraulic cylinder is:

In formula (2), V _t is the total control volume of the two chambers of the hydraulic cylinder; β _e is the elastic modulus of the effective oil; C _t is the internal leakage coefficient; Q(t) is the complex internal leakage process and the unmodeled pressure dynamics Modeling error caused; QL ₌ (Q ₁ +Q ₂ )/2 is the load flow, and Q ₁ and Q ₂ are the flow rate of the oil inlet chamber and the oil return chamber of the hydraulic cylinder, respectively; _QL and servo valve displacement x The relationship of _v is:

In formula (3), the servo valve flow gain coefficient

The sign function sign(x _v ) is defined as:

where C _d flow coefficient; ω is the spool area gradient; ρ is the oil density; P _s is the oil supply pressure, and P _r is the oil return pressure; Assuming that a high-response servo valve is used, the spool displacement and the control input are approximately proportional links, that is, x _v = _ki u, _ki is a positive electrical constant, and u is the control input voltage, so formula (3) is written as

In formula (5), k _t = k _q k _i represents the total flow gain; Step 1-2, assuming that the unmodeled dynamic term f(t) is continuous and differentiable, define the state variable:

Then the state equation of the position servo system of the double-rod hydraulic cylinder is:

In formula (6)

In order to simplify the display format of the state equation of the double-rod hydraulic cylinder position servo system, denote θ=[θ ₁ ,...,θ ₃ ] ^T is the known nominal value of the system parameters, θ _r =[θ _1r ,..., θ _3r ] ^T is the real value of the system parameter, where,

The formula (6) can be written as

In formula (8),

The design goal of the dual-rod hydraulic cylinder position servo system controller is: given the system reference signal y _d (t) = x _1d (t), design a bounded control input U to make the system output y = x ₁ as much as possible the reference signal of the tracking system; To facilitate controller design, the following assumptions are made: Assumption 1: The reference command signal x _1d (t) of the position servo system of the double-rod hydraulic cylinder is continuous in third order, and the system expects that the position command, speed command, acceleration command and jerk command are all bounded; working under working conditions, that is, both P ₁ and P ₂ are less than the oil supply pressure P _s , and |P _L | is also less than P _s ; therefore, it can be seen that

is always bounded, then as long as the designed U is bounded, the real control input u can be guaranteed to be bounded; Assumption 2: The total disturbance of the position servo system of the double-rod hydraulic cylinder

smooth enough that

exist and are bounded, that is:

In formula (9), δ ₁ , δ ₂ are unknown constants, namely

has an indeterminate upper bound; Step 2, designing an ADRC robust controller according to the mathematical model of the above-mentioned dual-rod hydraulic cylinder position servo system; The steps of designing the ADRC robust controller are as follows: Step 2-1. Disturbance compensation design based on extended state observer: will integrate perturbation

is expanded to an additional state variable, i.e.

Let h(t) be

The first derivative of , then equation (8) can be written as:

Since h(t) is

The first derivative of , is obtained from equation (9):

From the structure of formula (10), it can be known that the system can be observed, and a linear expansion state observer is designed as follows:

in the formula

represents the state estimation, and w _o >0 is regarded as the bandwidth of the extended state observer; remember

represents the state estimation error, and the derivative of the state estimation error obtained from equations (10) and (12) is:

remember

So, equation (12) can be written as:

in,

M=[0 0 0 1] ^T , D is the Hurwitz matrix; It has been proved that by |h(t)|≤δ ₁ ,

The following conclusion is obtained: the estimated state is always bounded, and the size of the bound decreases with the increase of the observer bandwidth w _o after any time, and there is a constant

The following relationship is satisfied:

Step 2-2, define z ₁ =x ₁ -x _1d as the tracking error of the system, according to the first equation in equation (10)

Let α ₁ be the virtual control, so that the equation

tends to a stable state; the error between α ₁ and the real state x ₂ is z ₂ =x ₂ -α ₁ , and the derivative of z ₁ can be obtained:

Design a virtual control law:

where k ₁ > 0 is the adjustable gain, then

Since z ₁ (s)=G(s)z ₂ (s), where G(s)=1/(s+k ₁ ) is a stable transfer function, when z ₂ tends to 0, z ₁ also It must tend to 0, so in the next design, the main design goal will be to make z ₂ tend to 0; Step 2-3, consider the second equation of formula (10), select α ₂ as the virtual control of x ₃ , and z ₃ as the deviation between the virtual control α ₂ and x ₃ z ₃ =x ₃ -α ₂ ; then The dynamic equation of z ₂ is

The virtual control law _α2 is designed as follows:

where k ₂ is the positive feedback gain. Substitute equation (20) into equation (19) to obtain:

Since z ₂ (s)=G'(s)z ₃ (s), where G'(s)=1/(s+k ₂ ) is a stable transfer function, when z ₃ tends to 0, z ₂ must also tend to 0; so in the next design, the main design goal will be to make z ₃ tend to 0; Steps 2-4. To gain an additional controller design freedom, define an auxiliary error signal r(t):

In formula (22), k ₃ > 0 is an adjustable gain. Since r(t) contains the jerk signal of the position, it is considered unmeasurable in practice, that is, r(t) is only used for auxiliary design, not Specifically appear in the designed controller; Considering the third equation of Eq. (10), the expansion of r is obtained as follows:

According to equation (23), the model-based controller U can be designed as:

where k _r is the positive feedback gain, U _a is the model-based compensation term, U _s is the robust control law and U _s1 is the linear robust feedback term, and U _s2 is the nonlinear robust term used to overcome the modeling The influence of uncertainty on system performance; Substitute equation (24) into equation (23) to get:

According to the error symbol integral robust controller design method, the integral robust term U _s2 can be designed as:

The variable β must satisfy the following conditions:

Differentiating both sides of equation (25) and applying equations (7) and (11), we get:

Step 3: Use Lyapunov stability theory to prove the stability of the position servo system of the double-rod hydraulic cylinder, and use Barbalat's lemma to obtain the result that the system can achieve asymptotic stability, as follows: define helper functions

in:

z ₃ (0),

denote z ₃ and

the initial value of ; proven when

When P(t)≥0, the Lyapunov function is defined as follows:

Use Lyapunov stability theory to prove the stability, and use Barbalat's lemma to get the result of system asymptotic stability. Therefore, adjust the gains k ₁ , k ₂ , k ₃ , k _r and make the position servo system of the double-rod hydraulic cylinder. The tracking error tends to zero under the condition that time tends to infinity.

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