CN111781836B - An adaptive asymptotic control method for hydraulic pressure preset performance - Google Patents

Abstract

The invention discloses a hydraulic pressure preset performance adaptive asymptotic control method in the technical field of hydraulic control systems, comprising the following steps: establishing a double-rod hydraulic cylinder servo system model; designing a hydraulic pressure preset performance adaptive asymptotic controller ; Adjust the parameters in the above steps to make the system meet the control performance index, suppress the system disturbance through the robust error sign integral function, and at the same time use the adaptive controller to approximate the system parameters, greatly weakening the influence of uncertainty on the system; In addition, considering the transient control accuracy of the system, the control accuracy of the system is constrained by the design of the preset performance function, the asymptotic tracking of the system is theoretically realized, and the output force of the double-rod hydraulic cylinder servo system is guaranteed to be accurate. The desired force command can be tracked to the ground; the invention simplifies the design of the controller and is more conducive to the application in engineering practice.

Description

An adaptive asymptotic control method for hydraulic pressure preset performance

technical field

The invention relates to the technical field of hydraulic control systems, in particular to a hydraulic pressure preset performance adaptive asymptotic control method.

Background technique

The electro-hydraulic servo system has the advantages of high output power and flexible signal processing, and is easy to realize the feedback of various parameters. It is most suitable for occasions with high power and quality. control, radar and artillery control, position control of machine tool table, plate thickness control of strip mill, electrode position control of electric furnace smelting, pressure control of material testing machine and other experimental machines, etc. However, the ubiquitous uncertainty in electro-hydraulic servo systems increases the design difficulty of the control system and may seriously deteriorate the achievable control performance, resulting in low control accuracy, limit cycle oscillations, and even instability. To improve the control performance of electro-hydraulic systems, designers have studied many advanced nonlinear controllers, such as robust adaptive control, adaptive robust control (ARC), sliding mode control, and so on. Although these controllers can ensure good steady-state performance, they do not pay attention to transient control performance. Based on this, the present invention designs an adaptive asymptotic control method for hydraulic pressure preset performance to solve the above problems.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide an adaptive asymptotic control method for hydraulic pressure preset performance, so as to solve the problems raised in the above background art.

In order to achieve the above purpose, the present invention provides the following technical solutions: a hydraulic pressure preset performance adaptive asymptotic control method, comprising the following steps:

S1: Establish the servo system model of the double-rod hydraulic cylinder;

S2: Design hydraulic pressure preset performance adaptive asymptotic controller;

S3: Adjust the parameters in step S2 so that the system meets the control performance index.

Further, the step S1 is specifically: according to Newton's second law, the dynamic model equation of the inertial load of the double-rod hydraulic cylinder is:

In the formula: y is the load displacement, T is the driving force, P _L = P ₁ -P ₂ is the load driving pressure, P ₁ and P ₂ are the pressures of the two chambers of the hydraulic cylinder, A is the effective working area of the piston rod, and b is the viscosity Friction coefficient, f is other unmodeled disturbances, including nonlinear friction, external disturbances and unmodeled dynamics, the dynamic equation of hydraulic cylinder load pressure is:

In the formula: V _t is the total effective volume of the two chambers of the hydraulic cylinder, C _t is the leakage coefficient of the hydraulic cylinder, Q _L = (Q ₁ +Q ₂ )/2 is the load flow, Q ₁ is the oil supply flow of the hydraulic cylinder into the oil chamber, Q ₂ is the return oil flow of the hydraulic cylinder oil return chamber, q(t) is the modeling error and unmodeled dynamics;

Q _L is a function of servo valve spool displacement x _v :

为流量伺服阀的增益系数，C_d为伺服阀的流量系数，w为伺服阀的面积梯度；ρ为液压油的密度,P_s为供油压力，sign(x_v)为：where:

is the gain coefficient of the flow servo valve, C _d is the flow coefficient of the servo valve, w is the area gradient of the servo valve; ρ is the density of the hydraulic oil, P _s is the oil supply pressure, and sign(x _v ) is:

Assuming that the servo valve spool displacement is proportional to the control input u, that is, x _v = _ki u, where _ki > 0 is the proportionality factor and u is the control input voltage, therefore, equation (3) can be transformed into

Where: k _t = k _q k _i represents the total flow gain.

Define the state variable x ₁ =T, then the whole system can be written in the following state space form:

θ₄＝B，

Define the unknown parameter set θ = [θ ₁ , θ ₂ , θ ₃ , θ ₄ ] ^T , where

θ ₄ =B,

Assumption 1: The derivatives of d(x,t) and d(x,t) are bounded, i.e.

In the formula: ζ ₁ and ζ ₂ are known constants.

Further, the step S2 includes the following steps:

S2a: Build a projection adaptive law with rate limit;

S2b: Design a robust controller;

S2c: Verify the stability of the controller's own system.

表示θ的估计，

表示θ的估计误差，即

建立一个投影函数如下：Further, the step S2a is specifically as follows: let

represents the estimate of θ,

represents the estimation error of θ, i.e.

Create a projection function as follows:

和

分别表示Ω_θ的内部和边界，

表示

时的外单位法向量；In the formula: ζ∈R ² , Γ(t)∈R ^2×2 is a time-varying positive definite symmetric matrix,

and

denote the interior and boundary of Ω _θ , respectively,

express

The outer unit normal vector when ;

For the projection function (8), in the control parameter estimation process, the preset adaptive limit speed is used, therefore, a saturation function is established as follows:

是一个预先设置的限制速率，通过如下引理总结得出系统将要用到的参数估计算法的结构特性；where:

is a preset limit rate, and the following lemma summarizes the structural characteristics of the parameter estimation algorithm that the system will use;

更新估计参数

Lemma 1: Assume the following projective adaptive law and a preset adaptive limiting rate are used

Update estimated parameters

In the formula: τ is the adaptive function, Γ(t)>0 is a continuous differentiable positive symmetric adaptive rate matrix, and from this adaptive law, the following ideal characteristics can be obtained:

因而由假设1可得

P1) The parameter estimates are always within the known bounded set of Ω _θ , that is, for any t, there is always

Therefore, from Assumption 1, we can get

P2)

P3) The law of parameter change is uniformly bounded, that is,

Further, the step S2b is specifically: defining the control error e=x ₁ -x _1d of the hydraulic cylinder, assuming that it needs to meet the following performance indicators:

Where: δ _l , δ _u are parameters to be designed, which are used to assist the upper and lower limits of the control error; ρ(t) is a positive strictly increasing smooth function, as shown in the following formula:

where: ρ ₀ , ρ _∞ and k are all positive designable parameters;

In formula (11), -δ _l ρ ₀ and δ _u ρ ₀ constrain the maximum undershoot and maximum overshoot of the output force control error e(t), respectively, and the parameter k constrains the convergence speed of the error e(t), ρ _∞ constrains the steady-state boundary of the error, so Equation (11) gives a specific plan for the performance of the output force control error. By choosing appropriate parameters such as ρ ₀ , ρ _∞ , k, δ _l and δ _u , The transient and stable performance of the output force control error can be planned in advance, and the transient performance can be improved according to the actual needs of the system;

Create the following increment function:

In the formula: z ₁ (t) is the conversion error variable corresponding to the control error e(t). It is easy to know by analysis that formula (13) is equivalent to e(t)=ρ(t)S(z ₁ (t) ), and when z ₁ (t) is bounded, the preset performance characteristic formula (8) always satisfies;

The increasing function S(z ₁ ) satisfying the characteristic formula (13) can be selected as follows:

Taking the inverse function of equation (14), we can get:

Then design the controller according to the conversion error z ₁ ;

The build function is as follows:

In the formula: k ₁ is the feedback gain;

The controller is designed as follows:

In the formula: k ₂ is the feedback gain;

Substitute controller (17) into equation (16) to get:

The adaptive law is designed as follows:

where:

Then get:

The robust controller u _s2 is designed as follows:

In the formula: β ₁ is the parameter to be designed.

Further, the step S2c is specifically: through the performance theorem 1: select the initial condition of the system control to satisfy -δ _l ρ(0)<e(0)<δ _u ρ(0), that is, -δ _l <λ(0 )<δ _u ; at the same time, the selection of parameter β ₁ satisfies the following inequality:

At the same time, sufficiently large parameters k ₁ and k ₂ are designed, so that the following matrix Λ is a positive definite matrix

Then it can ensure that the control error of the output force is always bounded, the output force can achieve better command tracking, and by adjusting parameters such as ρ ₀ , ρ _∞ , k, δ _l and δ _u , the control error can be guaranteed to satisfy the formula (11) The designed preset performance requirements;

The proof is as follows:

Define the following Lyapunov function:

Taking the derivative of V further and substituting into equations (16) and (20), we can get:

In the formula: Z=[z ₁ , z ₂ ] ^T , the definition of matrix Λ is as in formula (23). If the matrix Λ is a positive definite matrix through reasonable design parameters k ₁ and k ₂ , the following formula can be satisfied:

In the formula: λ _min (Λ) represents the minimum eigenvalue of the matrix Λ. By analyzing formula (26), it can be seen that the Lyapunov function is bounded, and the W integral is bounded, and then it can be known that the conversion error quantities z ₁ and z ₂ are bounded, and the combined formula ( 8), (16) and (17), it can be known that all signals in the system are bounded, so the derivative of W is bounded. According to Barbalat's lemma, when time tends to infinity, W tends to zero, that is, the conversion The amount of error z ₁ tends to zero, so combined with equation (11), it can be known that the control error e(t) is always bounded, which proves that the controller is convergent and the system is stable.

Further, the third step is to adjust the parameters k ₁ , k ₂ , ρ ₀ , ρ _∞ , k, δ _l , δ _u , β ₁ , and Γ of the control law u to make the system meet the control performance index.

Compared with the prior art, the beneficial effects of the present invention are: aiming at the characteristics of the hydraulic servo system, the present invention establishes the hydraulic servo system model and the designed hydraulic system output force preset performance adaptive asymptotic controller, through the robust error The symbolic integral function estimates the unmodeled disturbance and performs feed-forward compensation. At the same time, the adaptive controller is used to estimate the system parameters, which can effectively solve the uncertain nonlinear and parameter uncertain problems of the motor servo system. Based on the preset performance function The preset performance controller is designed, and Lyapunov finally proves the overall stability of the system. Under the existing interference conditions, the parameters converge well, and the transient and steady-state control accuracy of the system meets the performance index; at the same time, the invention simplifies the The controller is designed, and the final simulation results show that it is effective and more conducive to the application in engineering practice.

Description of drawings

In order to explain the technical solutions of the embodiments of the present invention more clearly, the following briefly introduces the accompanying drawings that are used in the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can also be obtained from these drawings without any creative effort.

Fig. 1 is the flow chart of the present invention;

Fig. 2 is the schematic diagram of the double-rod hydraulic cylinder system of the present invention;

3 is a schematic diagram of the preset performance of the control error of the present invention;

4 is a schematic diagram of the function S(z ₁ ) of the present invention;

5 is a parameter estimation curve diagram according to an embodiment of the present invention;

FIG. 6 is a comparison curve diagram of the control errors of two controllers according to an embodiment of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

1-2, the present invention provides a technical solution: a hydraulic pressure preset performance adaptive asymptotic control method, including the following steps:

S1: Establish the servo system model of the double-rod hydraulic cylinder;

S2: Design hydraulic pressure preset performance adaptive asymptotic controller;

S3: Adjust the parameters in step S2 so that the system meets the control performance index.

Wherein, step S1 is specifically: according to Newton's second law, the dynamic model equation of the inertial load of the double-rod hydraulic cylinder is:

In the formula: y is the load displacement, T is the driving force, P _L = P ₁ -P ₂ is the load driving pressure, P ₁ and P ₂ are the pressures of the two chambers of the hydraulic cylinder, A is the effective working area of the piston rod, and b is the viscosity Friction coefficient, f is other unmodeled disturbances, including nonlinear friction, external disturbances and unmodeled dynamics, the dynamic equation of hydraulic cylinder load pressure is:

In the formula: V _t is the total effective volume of the two chambers of the hydraulic cylinder, C _t is the leakage coefficient of the hydraulic cylinder, Q _L = (Q ₁ +Q ₂ )/2 is the load flow, Q ₁ is the oil supply flow of the hydraulic cylinder into the oil chamber, Q ₂ is the return oil flow of the hydraulic cylinder oil return chamber, q(t) is the modeling error and unmodeled dynamics;

Q _L is a function of servo valve spool displacement x _v :

为流量伺服阀的增益系数，C_d为伺服阀的流量系数，w为伺服阀的面积梯度；ρ为液压油的密度,P_s为供油压力，sign(x_v)为：where:

is the gain coefficient of the flow servo valve, C _d is the flow coefficient of the servo valve, w is the area gradient of the servo valve; ρ is the density of the hydraulic oil, P _s is the oil supply pressure, and sign(x _v ) is:

Assuming that the servo valve spool displacement is proportional to the control input u, that is, x _v = _ki u, where _ki > 0 is the proportionality factor and u is the control input voltage, therefore, equation (3) can be transformed into

Where: k _t = k _q k _i represents the total flow gain.

Define the state variable x ₁ =T, then the whole system can be written in the following state space form:

θ₄＝B，

Define the unknown parameter set θ = [θ ₁ , θ ₂ , θ ₃ , θ ₄ ] ^T , where

θ ₄ =B,

Assumption 1: The derivatives of d(x,t) and d(x,t) are bounded, i.e.

In the formula: ζ ₁ and ζ ₂ are known constants.

Step S2 includes the following steps:

S2a: Build a projection adaptive law with rate limit;

S2b: Design a robust controller;

S2c: Verify the stability of the controller's own system.

表示θ的估计，

表示θ的估计误差，即

建立一个投影函数如下：Step S2a is specifically as follows: let

represents the estimate of θ,

represents the estimation error of θ, i.e.

Create a projection function as follows:

和

分别表示Ω_θ的内部和边界，

表示

时的外单位法向量；In the formula: ζ∈R ² , Γ(t)∈R ^2×2 is a time-varying positive definite symmetric matrix,

and

denote the interior and boundary of Ω _θ , respectively,

express

The outer unit normal vector when ;

For the projection function (8), in the control parameter estimation process, the preset adaptive limit speed is used, therefore, a saturation function is established as follows:

是一个预先设置的限制速率，通过如下引理总结得出系统将要用到的参数估计算法的结构特性；where:

is a preset limit rate, and the following lemma summarizes the structural characteristics of the parameter estimation algorithm that the system will use;

更新估计参数

Lemma 1: Assume the following projective adaptive law and a preset adaptive limiting rate are used

Update estimated parameters

In the formula: τ is the adaptive function, Γ(t)>0 is a continuous differentiable positive symmetric adaptive rate matrix, and from this adaptive law, the following ideal characteristics can be obtained:

因而由假设1可得

P1) The parameter estimates are always within the known bounded set of Ω _θ , that is, for any t, there is always

Therefore, from Assumption 1, we can get

P2)

P3) The law of parameter change is uniformly bounded, that is,

Step S2b is specifically: defining the control error e=x ₁ -x _1d of the hydraulic cylinder, assuming that it needs to meet the following performance indicators:

Where: δ _l , δ _u are parameters to be designed, which are used to assist the upper and lower limits of the control error; ρ(t) is a positive strictly increasing smooth function, as shown in the following formula:

where: ρ ₀ , ρ _∞ and k are all positive designable parameters; the approximate curve of the performance index inequality (11) is shown in Figure 3;

In formula (11), -δ _l ρ ₀ and δ _u ρ ₀ constrain the maximum undershoot and maximum overshoot of the output force control error e(t), respectively, and the parameter k constrains the convergence speed of the error e(t), ρ _∞ constrains the steady-state boundary of the error, so Equation (11) gives a specific plan for the performance of the output force control error. By choosing appropriate parameters such as ρ ₀ , ρ _∞ , k, δ _l and δ _u , The transient and stable performance of the output force control error can be planned in advance, and the transient performance can be improved according to the actual needs of the system;

Create the following increment function:

In the formula: z ₁ (t) is the conversion error variable corresponding to the control error e(t). It is easy to know by analysis that formula (13) is equivalent to e(t)=ρ(t)S(z ₁ (t) ), and when z ₁ (t) is bounded, the preset performance characteristic formula (8) always satisfies;

The increasing function S(z ₁ ) satisfying the characteristic formula (13) can be selected as follows:

The curve of the increasing function S(z ₁ ) is shown in Figure 4;

Taking the inverse function of equation (14), we can get:

Then design the controller according to the conversion error z ₁ ;

The build function is as follows:

In the formula: k ₁ is the feedback gain;

The controller is designed as follows:

In the formula: k ₂ is the feedback gain;

Substitute controller (17) into equation (16) to get:

The adaptive law is designed as follows:

where:

Then get:

The robust controller u _s2 is designed as follows:

In the formula: β ₁ is the parameter to be designed.

Step S2c is specifically: through the performance theorem 1: select the initial condition of the system control to satisfy -δ _l ρ(0)<e(0)<δ _u ρ(0), namely -δ _l <λ(0)<δ _u ; At the same time, the selection of parameter β ₁ satisfies the following inequality:

At the same time, sufficiently large parameters k ₁ and k ₂ are designed, so that the following matrix Λ is a positive definite matrix

Then it can ensure that the control error of the output force is always bounded, the output force can achieve better command tracking, and by adjusting parameters such as ρ ₀ , ρ _∞ , k, δ _l and δ _u , the control error can be guaranteed to satisfy the formula (11) The designed preset performance requirements;

The proof is as follows:

Define the following Lyapunov function:

Taking the derivative of V further and substituting into equations (16) and (20), we can get:

In the formula: Z=[z ₁ , z ₂ ] ^T , the definition of matrix Λ is as in formula (23). If the matrix Λ is a positive definite matrix through reasonable design parameters k ₁ and k ₂ , the following formula can be satisfied:

In the formula: λ _min (Λ) represents the minimum eigenvalue of the matrix Λ. By analyzing formula (26), it can be seen that the Lyapunov function is bounded, and the W integral is bounded, and then it can be known that the conversion error quantities z ₁ and z ₂ are bounded, and the combined formula ( 8), (16) and (17), it can be known that all signals in the system are bounded, so the derivative of W is bounded. According to Barbalat's lemma, when time tends to infinity, W tends to zero, that is, the conversion The amount of error z ₁ tends to zero, so combined with equation (11), it can be known that the control error e(t) is always bounded, which proves that the controller is convergent and the system is stable.

The third step is to adjust the parameters k ₁ , k ₂ , ρ ₀ , ρ _∞ , k, δ _l , δ _u , β ₁ , and Γ of the control law u to make the system meet the control performance index.

Example 1:

θ_min＝[0.01,0.1,1000,20]^T,θ_max＝[10,30,500000，500]^T,

Г＝diag{6,10,390,50}，所选取的

远离于参数的真值，以考核自适应控制律的效果，力输入信号

单位KN；系统所加干扰为f＝30sin(2πt)KN。In the simulation, the following parameters are taken to model the system: m=40kg, A=2×10 ^-4 m ² , B=80N·s/m, β _e =200Mpa, V ₀₁ =1×10 ^-3 m ³ , V ₀₂ =1×10 ^-3 m ³ , C _t =9×10 ^-12 m ⁵ /Ns,

θ _min = [0.01, 0.1, 1000, 20] ^T , θ _max = [10, 30, 500000, 500] ^T ,

Г=diag{6,10,390,50}, the selected

away from the true value of the parameter to evaluate the effect of the adaptive control law, force the input signal

The unit is KN; the interference added by the system is f=30sin(2πt)KN.

The hydraulic pressure adaptive preset performance controller (APFRISE) based on the robust error sign integration proposed by the present invention, the controller design parameters The relevant parameters of the controller are selected as: k ₁ =200, k ₂ =300, β=100, ρ ₀ = 800, ρ _∞ = 500, k = 0.0001, δ _l = δ _u = 1; PID controller parameters are k _p = 500, _ki = 100, k _d = 0;

The effect of the control law is shown in Figure 5 and Figure 6, Figure 5 is the parameter estimation curve; Figure 6 is the control error comparison curve of the designed controller (APFRISE) and the PID controller.

It can be seen from the above figure that the algorithm proposed by the present invention can accurately estimate the system parameters in the simulation environment. Compared with the PID controller, the controller designed by the present invention can achieve good control accuracy, and can ensure the preset transient and stable conditions of the system. state control accuracy requirements.

In the description of this specification, description with reference to the terms "one embodiment," "example," "specific example," etc. means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one aspect of the present invention. in one embodiment or example. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

The above-disclosed preferred embodiments of the present invention are provided only to help illustrate the present invention. The preferred embodiments do not exhaust all the details, nor do they limit the invention to only the described embodiments. Obviously, many modifications and variations are possible in light of the contents of this specification. These embodiments are selected and described in this specification in order to better explain the principles and practical applications of the present invention, so that those skilled in the art can well understand and utilize the present invention. The present invention is to be limited only by the claims and their full scope and equivalents.

Claims (3)

1. a hydraulic pressure preset performance adaptive asymptotic control method, is characterized in that, comprises the steps: S1: Establish the servo system model of the double-rod hydraulic cylinder; S2: Design hydraulic pressure preset performance adaptive asymptotic controller; S3: adjust the parameters in step S2 to make the system meet the control performance index; The step S2 includes the following steps: S2a: Build a projection adaptive law with rate limit; S2b: Design a robust controller; S2c: Verify the stability of the controller's own system; The step S2a is specifically: let

represents the estimate of θ,

represents the estimation error of θ, i.e.

Create a projection function as follows:

In the formula: ζ∈R ² , Γ(t)∈R ^2×2 is a time-varying positive definite symmetric matrix,

and

denote the interior and boundary of Ω _θ , respectively,

express

The outer unit normal vector when ; For the projection function (8), in the control parameter estimation process, the preset adaptive limit speed is used, therefore, a saturation function is established as follows:

where:

is a preset limit rate, and the following lemma summarizes the structural characteristics of the parameter estimation algorithm that the system will use; Lemma 1: Assume the following projective adaptive law and a preset adaptive limiting rate are used

Update estimated parameters

In the formula: τ is the adaptive function, Γ(t)>0 is a continuous differentiable positive symmetric adaptive rate matrix, and from this adaptive law, the following ideal characteristics can be obtained: P1) The parameter estimates are always within the known bounded set of Ω _θ , that is, for any t, there is always

Therefore, from Assumption 1, we can get

P2)

P3) The law of parameter change is uniformly bounded, that is,

The step S2b is specifically: defining the control error e=x ₁ -x _1d of the hydraulic cylinder, assuming that it needs to meet the following performance indicators:

Where: δ _l , δ _u are parameters to be designed, which are used to assist the upper and lower limits of the control error; ρ(t) is a positive strictly increasing smooth function, as shown in the following formula:

where: ρ ₀ , ρ _∞ and k are all positive designable parameters; In formula (11), -δ _l ρ ₀ and δ _u ρ ₀ constrain the maximum undershoot and maximum overshoot of the output force control error e(t), respectively, and the parameter k constrains the convergence speed of the error e(t), ρ _∞ constrains the steady-state boundary of the error, so Equation (11) gives a specific plan for the performance of the output force control error. By choosing appropriate parameters such as ρ ₀ , ρ _∞ , k, δ _l and δ _u , The transient and stable performance of the output force control error can be planned in advance, and the transient performance can be improved according to the actual needs of the system; Create the following increment function:

In the formula: z ₁ (t) is the conversion error variable corresponding to the control error e(t). It is easy to know by analysis that formula (13) is equivalent to e(t)=ρ(t)S(z ₁ (t) ), and when z ₁ (t) is bounded, the preset performance characteristic formula (8) always satisfies; The increasing function S(z ₁ ) satisfying the characteristic formula (13) can be selected as follows:

Taking the inverse function of equation (14), we can get:

Then design the controller according to the conversion error z ₁ ; The build function is as follows:

In the formula: k ₁ is the feedback gain; The controller is designed as follows:

In the formula: k ₂ is the feedback gain; Substitute controller (17) into equation (16) to get:

The adaptive law is designed as follows:

where:

Then get:

The robust controller u _s2 is designed as follows:

In the formula: β ₁ is the parameter to be designed; The step S2c is specifically: through the performance theorem 1: select the initial condition of the system control to satisfy -δ _l ρ(0)<e(0)<δ _u ρ(0), that is, -δ _l <λ(0)<δ _u ; at the same time, the selection of parameter β ₁ satisfies the following inequality:

At the same time, sufficiently large parameters k ₁ and k ₂ are designed, so that the following matrix Λ is a positive definite matrix

Then it can ensure that the control error of the output force is always bounded, the output force can achieve better command tracking, and by adjusting parameters such as ρ ₀ , ρ _∞ , k, δ _l and δ _u , the control error can be guaranteed to satisfy the formula (11) The designed preset performance requirements; The proof is as follows: Define the following Lyapunov function:

Taking the derivative of V further and substituting into equations (16) and (20), we can get:

In the formula: Z=[z ₁ , z ₂ ] ^T , the definition of matrix Λ is as in formula (23). If the matrix Λ is a positive definite matrix through reasonable design parameters k ₁ and k ₂ , the following formula can be satisfied:

In the formula: λ _min (Λ) represents the minimum eigenvalue of the matrix Λ. By analyzing formula (26), it can be seen that the Lyapunov function is bounded, and the W integral is bounded, and it can be known that the conversion error quantities z ₁ and z ₂ are both bounded, and the combined formula ( 8), (16) and (17), it can be known that all signals in the system are bounded, so the derivative of W is bounded. According to Barbalat's lemma, when time tends to infinity, W tends to zero, that is, the conversion The amount of error z ₁ tends to zero, so combined with equation (11), it can be known that the control error e(t) is always bounded, which proves that the controller is convergent and the system is stable. 2. The method for adaptive asymptotic control of hydraulic pressure preset performance according to claim 1, wherein the step S1 is specifically: according to Newton's second law, the dynamics of the inertial load of the double-rod hydraulic cylinder The model equation is:

In the formula: y is the load displacement, T is the driving force, P _L = P ₁ -P ₂ is the load driving pressure, P ₁ and P ₂ are the pressures of the two chambers of the hydraulic cylinder, A is the effective working area of the piston rod, and b is the viscosity Friction coefficient, f is other unmodeled disturbances, including nonlinear friction, external disturbances and unmodeled dynamics, the dynamic equation of hydraulic cylinder load pressure is:

In the formula: V _t is the total effective volume of the two chambers of the hydraulic cylinder, C _t is the leakage coefficient of the hydraulic cylinder, Q _L = (Q ₁ +Q ₂ )/2 is the load flow, Q ₁ is the oil supply flow of the hydraulic cylinder into the oil chamber, Q ₂ is the return oil flow of the hydraulic cylinder oil return chamber, q(t) is the modeling error and unmodeled dynamics; Q _L is a function of servo valve spool displacement x _v :

where:

is the gain coefficient of the flow servo valve, C _d is the flow coefficient of the servo valve, w is the area gradient of the servo valve; ρ is the density of the hydraulic oil, P _s is the oil supply pressure, and sign(x _v ) is:

Assuming that the servo valve spool displacement is proportional to the control input u, that is, x _v = _ki u, where _ki > 0 is the proportionality factor and u is the control input voltage, therefore, equation (3) can be transformed into

Where: k _t = k _q k _i represents the total flow gain. Define the state variable x ₁ =T, then the whole system can be written in the following state space form:

Define the unknown parameter set θ = [θ ₁ , θ ₂ , θ ₃ , θ ₄ ] ^T , where

θ ₄ =B,

Assumption 1: The derivatives of d(x,t) and d(x,t) are bounded, i.e.

In the formula: ζ ₁ and ζ ₂ are known constants. 3 . The method for adaptive asymptotic control of hydraulic pressure preset performance according to claim 1 , wherein the step 3 is to adjust the parameters k ₁ , k ₂ , ρ ₀ , ρ _∞ of the control law u. 4 . , k, δ _l , δ _u , β ₁ , Γ make the system meet the control performance index.

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