WO2022121507A1

WO2022121507A1 - Low-complexity control method for asymmetric servo hydraulic position tracking system

Info

Publication number: WO2022121507A1
Application number: PCT/CN2021/124567
Authority: WO
Inventors: 刘爽; 王文波
Original assignee: 燕山大学
Priority date: 2020-12-07
Filing date: 2021-10-19
Publication date: 2022-06-16
Also published as: JP2023517140A; JP7448692B2; CN112486021B; CN112486021A

Abstract

A low-complexity control method for an asymmetric servo hydraulic position tracking system. The method comprises the following steps: establishing a servo hydraulic system model of a single-outlet rod (step 1); according to the servo hydraulic system model of the single-output rod, designing a controller of a servo hydraulic system of the single-output rod by using a low-complexity control strategy (step 2); and proving the stability of the servo hydraulic system of the single-output rod according to the controller of the servo hydraulic system of the single-output rod and the servo hydraulic system model of the single-output rod (step 3). The method can solve various uncertainty problems and unknown nonlinear problems in the hydraulic system, and the design of the controller does not depend on an accurate mathematical model, and only needs a state signal that can be measured.

Description

A Low-Complexity Control Method for Asymmetric Servo-Hydraulic Position Tracking System

technical field

The invention relates to the field of servo hydraulic position control, in particular to a low-complexity control method for an asymmetric servo hydraulic position tracking system.

Background technique

Servo hydraulic systems have the advantages of excellent load efficiency, small power ratio, and fast response speed, so they are widely used in modern industries (such as active suspension systems, hydraulic excavators, manipulators). The servo hydraulic system includes two kinds of hydraulic actuators, the symmetrical hydraulic actuator with double rod and the asymmetric hydraulic actuator with single rod. In active suspension control research, the areas of the two oil chambers are often considered equal, and a symmetrical hydraulic actuator is used. For asymmetric servo hydraulic systems, due to the existence of the piston rod, the effective areas of the rod cavity and the rodless cavity of the hydraulic cylinder are not equal, and the symmetrical hydraulic actuator that can achieve the same performance is often larger than the asymmetric hydraulic actuator. It is used in a few special occasions, and widely used in industry is asymmetric hydraulic actuators. Therefore, it is unreasonable to place symmetrical hydraulic actuators in the vehicle suspension system with great space limitations.

However, the asymmetric servo hydraulic system is a complex nonlinear system, and it involves various uncertainties, such as load changes, parameter uncertainties, and unknown nonlinearities, which will lead to modeling and design control. Therefore, for asymmetric servo-hydraulic systems, high-precision position control is facing a huge challenge. Among them, the parameter uncertainties mainly include leakage coefficient, oil film viscosity and viscous friction coefficient, and the unknown nonlinearity mainly includes spool dead zone and external disturbance, which hinder the development of high-performance controllers. In order to meet the operational performance of the position tracking system, many control techniques have been developed in the past ten years, such as neural network adaptation, fuzzy logic control, robust adaptive control, and backstepping adaptive control. Among these control methods, adaptive control, due to its online learning ability, can deal with parameter uncertainties, and robust adaptive control can deal with unknown disturbances. Although the above algorithms based on adaptive control can obtain good simulation results, a key problem is that these algorithms have a heavy computational burden and take a long time to reach convergence. Adaptation of the above control algorithm is difficult. In addition, most of the existing control methods of servo hydraulic system are developed on the basis of backstep control. It is well known that in the process of backstepping design of high-order systems, the problem of computational explosion will be caused due to the repeated derivation of the virtual control function. Therefore, it is necessary to find novel control strategies with low system dependence, light computational burden, and no function approximation function.

In addition, in the servo-hydraulic position tracking system, from the practical application point of view, if there is a potential bad transient response (overshoot is too large, the convergence is slow) in the above method, it may lead to the deterioration of the tracking performance and the occurrence of Dangerous or even cause hardware damage. Bechlioulis CP, Rovithakis GA, et al. initially proposed novel controller design methods that can guarantee tracking error transient performance. In this method, the transient and steady-state performance of tracking error is specified by introducing a specified performance function, and the original system with constraints is equivalent to an unlimited system by introducing error transformation. This idea is further applied to high-order nonlinear fault-tolerant control systems, hydraulic servo systems, and suspension systems. However, in the above-mentioned literatures that use a specified performance function, the control schemes are all combined with adaptive control methods to deal with unknown dynamics in the system.

To sum up, the servo hydraulic system and its position control method have the following shortcomings: 1. The single-rod nonlinear servo hydraulic system is a typical complex nonlinear system, which involves load changes, parameter uncertainty and unknown Because of the nonlinearity and other problems, there is a huge gap between the establishment of the theoretical model and the actual system, which leads to the difficulty and complexity of designing the controller; 2. Most of the current servo-hydraulic position control methods are based on backstep control adaptive However, these control methods have a high degree of dependence on the model, and in the process of backstepping design of high-order systems, a large amount of calculation will be generated, and the online learning of uncertain parameters is not conducive to actual experiments. Third, from the practical application point of view, in the servo hydraulic position tracking system, if there is a potential bad transient response (overshoot is too large, the convergence is slow), which may lead to the deterioration of the tracking performance, Dangerous or even hardware damage.

SUMMARY OF THE INVENTION

According to the problems existing in the prior art, the present invention discloses a low-complexity control method for an asymmetric servo-hydraulic position tracking system, including the following steps: S1: establishing a servo-hydraulic system model of a single-exit rod; S2: according to the single-exit rod S3: According to the controller of the servo hydraulic system with a single rod and the model of the servo hydraulic system with a single rod, it is proved that the single output rod The stability of the servo-hydraulic system of the rod.

Further, it is also characterized in that: the process of establishing the model expression of the servo-hydraulic system of the single rod is as follows: establish the dynamic model of the hydraulic cylinder according to Newton's second law:

Where: f(t) represents various disturbances, x and m represent the position and mass of the load respectively, B is the viscous damping coefficient, K is the equivalent spring stiffness of the load, when the load is an inertial load, K=0, F= A ₁ *P ₁ -A ₂ *P ₂ is the main power output by the hydraulic actuator, where P ₁ , P ₂ are the pressures of the large and small chambers of the hydraulic cylinder, A ₁ , A ₂ are the effective areas of the pistons of the large and small chambers; using three positions Five-way servo valve, the load pressure dynamics is expressed by the following formula:

In the formula: V ₁ and V ₂ are the volumes of the rod cavity and the rodless cavity respectively, V ₁ =V ₀₁ +A ₁ x, V ₂ =V ₀₂ -A ₂ x, V ₀₁ and V ₀₂ are the initial positions of the piston, respectively When , the volume of the rodless cavity and the rod cavity, C _t is the leakage coefficient inside the hydraulic cylinder, and C _e is the external leakage coefficient of the hydraulic cylinder. β _e is the elastic modulus of oil; Q ₁ is the hydraulic oil flow rate with rod cavity, Q ₂ is the hydraulic oil flow rate without rod cavity;

in:

in,

is the flow gain, and s(Γ(x _v )) is expressed as:

P _s is the oil supply pressure of the hydraulic system, P _r is the oil return pressure of the hydraulic system, C _d is the flow coefficient of the orifice, w is the area gradient of the spool valve, ρ is the oil density, and x _v is the spool of the servo valve; The spool displacement x _v of the servo valve is controlled by the voltage or current input u to obtain the required corresponding force. The dynamic characteristics of the servo valve are as follows:

τ is the time constant of the servo valve dynamics model, and u(t) is the current input; considering the spool displacement Γ(x _v ) with an unknown dead zone, its expression is as follows:

The parameters m _r and m _l represent the left and right slopes of the dead zone characteristic curve, and the parameters _br and b _l represent the breakpoints of the input nonlinearity; the spool dead zone is expressed as Γ(x _v )=m(t)x _v +d( t), m(t), d(t) expressions are as follows:

Since m _r , _ml , _br , and b _l are bounded, d(t) is bounded, and the upper bound of d(t) is written as

The state variables are defined as follows: x ₁ =x,

x ₃ =P ₁ , x ₄ =P ₂ , x ₅ =x _v , the state equation of the tracking system with unknown spool dead zone position is obtained as follows:

in:

Further: the process of designing the controller of the servo hydraulic system with a single output rod according to the servo hydraulic system model of the single output rod using a low-complexity control strategy is as follows: define the error variable:

z ₁ =x ₁ -x _r z ₂ =x ₂ -α ₁ z ₃ =F-α ₂ z ₄ =x ₅ -α ₃ (17)

Among them, α ₁ , α ₂ , α ₃ are virtual control variables, x _1r is the command signal of the hydraulic position tracking system, F (F=A ₁ x ₃ -A ₂ x ₄ ) is the active force output by the actuator, z ₁ is the position tracking error, z _i i=2...4 is the control error, four positive smooth decreasing functions

Select the specified performance function, define the standardized error ζ _i , do error conversion for zi _i = 1...4, the converted error

as follows:

Select the virtual controller as follows:

The controller of the servo hydraulic system is:

Further, according to the controller of the servo-hydraulic system of the single-exit rod and the servo-hydraulic system model of the single-exit rod, the process of proving the stability of the servo-hydraulic system of the single-exit rod is as follows: S3-1: Utilize the initial value theorem, Prove that when the normalized error vector is in the time period t∈[0,τ _max ), there is a maximum solution in the non-empty open set Ω _ζ ; S3-2: When the normalized error vector is in the time period t∈[0,τ _max ), When there is a maximum solution in the non-empty open set _Ωζ , the virtual controller and the controller of the servo-hydraulic system can guarantee that the closed-loop signal is bounded for t∈[0,τ _max ); S3-3: Use the initial value proposal to prove that when When τ _max = +∞ in S3-2, it is still true that all closed-loop signals are bounded.

Further, using the initial value theorem, it is proved that the standardized error vector has a maximum solution in the non-empty open set Ω _ζ in the time period t∈[0,τ _max ): the non-empty open set Ω _ζ , select the performance function When ρ _i , satisfy: ρ _i (0)>min{δ _-i ,δ ^- _i }|z _i (0)|, i=1...4, we can get: |ζ _i (0)|< min{δ _−i ,δ ⁻ _i }, i=1...4, therefore, the normalized error vector _ζ (0)∈Ωζ; due to the desired tracking trajectory x _r , the performance function ρ _i (t), i= 1...4, the intermediate control signal α _i , i=1...3 and the control rate u are continuously smooth and derivable, the dynamic variables in the state equation of the spool dead zone position tracking system are continuously derivable function, so the function L(t,ζ) in the standardized error vector of the formula is piecewise continuous with respect to time t, and in the open set Ω _ζ , ζ satisfies the Lipschitz condition; the conditions in the initial value theorem are all satisfied, so In the time period [0,τ _max ), there exists a unique maximum solution ζ(t)∈Ω _ζ for the normalized error vector, for

can guarantee ζ _i (t)∈Ω _ζ , i=1...4.

Further, when the normalized error vector has a maximum solution in the non-empty open set Ω _ζ in the time period t∈[0,τ _max ), the virtual controller and the controller of the servo hydraulic system, for t∈[ 0,τ _max ) can ensure that the closed-loop signal is bounded as follows: S3-2-1: Calculate according to formula (18)

Derivative to Time

in

There is a constant r _M1 such that 0>r ₁ >r _M1 ;

Further available:

remember

△ ₁ (t)=r ₁ k ₁ (32)

considering

and

ρ ₂ is bounded, according to the extreme value theorem, we know that a ₁ is bounded; from (31) and (32) we get:

Choose the Lyapunov function

(33) and (34) conform to the form of the inequality lemma, which can be known from the inequality lemma, because a ₁ is bounded, that is

|a ₁ |<A ₁ , 0<△ ₁ (t)<r _M1 k ₁ , so

because

Bounded, we can get -δ _-1 ρ ₁ (t)<z ₁ <δ ^- ₁ ρ ₁ (t), so for

The tracking error z ₁ decays within the specified bounds; it is guaranteed that the tracking error z ₁ converges within the specified performance bounds in the time period t∈[0,τ _max ), since the command is a bounded sinusoidal signal, the state variable x ₁ is bounded of;

S3-2-2: Calculated according to formula (18)

Derivative to Time

in

There is a constant r _M2 such that 0>r ₂ >r _M2 ;

Further available:

remember:

considering

and

ρ _i , is bounded, f(t) is an unknown and bounded disturbance term, B is a bounded uncertain parameter, it can be known from the extreme value theorem that a ₂ is bounded; from (36) and (37) we get:

Choose the Lyapunov function

(38) and (39) conform to the form of inequality lemma, which can be seen from the inequality lemma, because a ₂ is bounded, i.e.

such that |a ₂ |<A ₂ ,

so

because

Bounded, we can get -δ _-2 ρ ₂ (t)<z ₂ <δ ^- ₂ ρ ₂ (t), so for

Ensure that the control error z ₂ is bounded and the state variable x ₂ is bounded;

S3-2-3: Calculated according to formula (18)

Derivative to Time

in

There is a constant r _M3 such that 0>r ₃ >r _M3 ;

Substituting equations (10), (24), (11) and (21) into (40), we get:

remember

where V ₁ =V ₀₁ +A ₁ x, V ₂ =V ₀₂ -A ₂ x;

In view of the safe operation of the servo hydraulic system actuator, a safety margin is reserved, that is, according to the physical structure, the hydraulic cylinder can fluctuate up to 12 cm up and down near the neutral position, and the amplitude of the command signal is less than or equal to 10 cm to ensure h ₁ h ₂ h ₃ is bounded, that is, there are three positive numbers respectively

make

considering

and α ₁ ,

ρ _i is bounded, m(t) is the unknown slope of the spool dead zone, m(t) is bounded (ie

), from the extreme value theorem we know that a ₃ is bounded;

From (41) and (42) we get:

Choose the Lyapunov function

(43) and (44) conform to the form of the inequality lemma, which can be known from the inequality lemma, because a ₃ is bounded, namely

such that |a ₃ |<A ₃ ,

so

because

Bounded, we can get -δ _-3 ρ ₃ (t)<z ₃ <δ ^- ₃ ρ ₃ (t), so for

Ensure that the control error _z3 is bounded and the active force u _a is bounded. From the mathematical expression of the state equation, the state variables _x3 and _x4 are bounded, and they are both less than the pressure value of the hydraulic source P _s ;

S3-2-4: Calculated according to formula (18)

Derivative to Time

in

There is a constant r _M4 such that 0>r ₄ >r _M4 ;

Further get:

remember

considering

and α _i ,

ρ _i is bounded, k ₄ is an optional parameter when designing the controller, so a ₄ is bounded; from (46) and (47):

Choose a Lyapunov function:

(48) and (49) conform to the form of the inequality lemma, which can be known from the inequality lemma, because a ₄ is bounded, namely

such that |a ₄ |<A ₄ ,

so

because

Bounded, we get -δ _-4 ρ ₄ (t)<z ₄ <δ ^- ₄ ρ ₄ (t), so for

It is guaranteed that the control error _z4 is bounded and the state variable _x5 is bounded.

Further, the inequality lemma is expressed as follows:

If a is bounded (i.e.

|a|<A),

then x must be bounded and

Further, the process of using the initial value proposal to prove that when τ _max =+∞ in S3-2, all closed-loop signals are bounded is still correct as follows:

because

so

is a non-empty set;

Assuming τ _max <+∞, the initial value proposal emphasizes the existence of a time constant t ₁ ∈ [0,τ _max ) such that

Contradictory, the resulting assumption is that τ _max = +∞, for any t∈[0,+∞), all closed-loop system signals are bounded and the tracking error is guaranteed to converge within specified bounds.

Due to the adoption of the above technical solution, the present invention provides a low-complexity control method for an asymmetric servo-hydraulic position tracking system, which can solve various uncertainties in the hydraulic system (eg, unknown friction effects, variable parameters, etc.). Determined and load changes) and unknown nonlinear problems (such as spool dead zone, external disturbances), the design of the controller does not rely on an accurate mathematical model, only the state signal that can be measured, the calculation of the control rate is consistent with the existing in Compared with the algorithm developed on the basis of backstepping adaptation, the calculation process is simple, the amount of calculation is small, it is convenient for real-time control, and it is easier to realize engineering; the invention can ensure the convergence speed and steady-state accuracy of tracking error; finally, the experimental results show that, Compared with the traditional pid control method, the position tracking effect of the present invention has higher steady-state precision and smaller tracking displacement phase lag. When the frequency of the desired track signal increases, the actual tracking displacement phase lag degree obtained by the pid algorithm control is greater, the amplitude attenuation is more severe, and the displacement tracking error is greater. The degree of control displacement hysteresis is basically unchanged, the tracking error always converges within the specified boundary, and the amplitude is basically not attenuated. When the frequency of the desired signal remains unchanged and the amplitude increases, the displacement tracking error of the pid algorithm is larger, and the control algorithm of the present invention can still ensure that the tracking error converges within the specified boundary, the degree after the phase is small, and the amplitude is basically not attenuated.

Description of drawings

In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the accompanying drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only These are some embodiments described in this application. For those of ordinary skill in the art, other drawings can also be obtained based on these drawings without any creative effort.

Fig. 1 is a kind of low-complexity control method flow chart for the asymmetric servo hydraulic position tracking system of the present invention:

Figure 2 is a model block diagram of a servo hydraulic system with a single rod;

Figure 3(a) is the structure composition diagram I of the experimental platform;

Figure 3(b) is the structural composition diagram II of the experimental platform;

Figure 3(c) is the structural composition diagram III of the experimental platform;

Figure 3(d) is the structural composition diagram IV of the experimental platform;

Fig. 4 is the error convergence simulation curve graph under the action of the controller of the present invention;

Fig. 5 is the simulation curve diagram of tracking error convergence under disturbance action;

Fig. 6 is the simulation curve diagram of error convergence under the action of unknown spool dead zone;

Fig. 7 is a simulation curve diagram of spool displacement under the action of unknown spool dead zone;

FIG. 8 is a simulation graph showing the comparison of error convergence between the control method of the present invention and the adaptive backstepping control mode (SPPFBSA) with unknown dead zone of the spool that satisfies the specified performance;

Fig. 9 is a statistical diagram of control method of the present invention and SPPFBSA simulation calculation time and error adjustment time;

Fig. 10 is the command signal is TS=3s: x _r =100sin((2*pi*t)/3)mm, the control method of the present invention and pid contrast displacement tracking experiment diagram;

Figure 11 is an experimental diagram of the tracking error of the control method of the present invention when the command signal is TS=3s: x _r =100sin((2*pi*t)/3)mm;

Figure 12 is an experimental diagram of displacement tracking compared with the control method of the present invention and pid when the command signal is TS=2s: x _r =100sin(pi*t)mm;

Figure 13 is an experimental diagram of the tracking error of the control method of the present invention when the command signal is TS=2s: x _r =100sin(pi*t)mm;

Fig. 14 is an experiment diagram of displacement tracking comparing the control method of the present invention with pid when the command signal is TS=1s: x _r =50sin(2*pi*t)mm;

Figure 15 is an experimental diagram of the tracking error of the control method of the present invention when the command signal is TS=1s: x _r =50sin(2*pi*t)mm;

Figure 16 is an experimental diagram of displacement tracking compared with the control method of the present invention and pid when the command signal is TS=2s: x _r =100sin(pi*t)mm;

Figure 17 is an experimental diagram of the tracking error of the control method of the present invention when the command signal is TS=2s: x _r =100sin(pi*t)mm;

Figure 18 is an experimental diagram of displacement tracking compared with the control method of the present invention and pid when the command signal is TS=2s: x _r =80sin(pi*t)mm;

Figure 19 is an experimental diagram of the tracking error of the control method of the present invention when the command signal is TS=2s: x _r =80sin(pi*t)mm;

Figure 20 is an experimental diagram of displacement tracking compared with the control method of the present invention and pid when the command signal is TS=2s: x _r =60sin(pi*t)mm;

FIG. 21 is an experimental diagram of the tracking error of the control method of the present invention when the command signal is TS=2s: x _r =60sin(pi*t)mm.

Detailed ways

In order to make the technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be described clearly and completely below with reference to the accompanying drawings in the embodiments of the present invention:

The specific embodiments of the present invention will be further described below in conjunction with the accompanying drawings:

1 is a flowchart of a low-complexity control method for an asymmetrical servo hydraulic position tracking system of the present invention: a low-complexity control method for an asymmetrical servo-hydraulic positional tracking system, comprising the following steps: S1: establish a Servo hydraulic system model; Figure 2 is a block diagram of the servo hydraulic system model with a single output rod; S2: According to the servo hydraulic system model of a single output rod, the controller of the single output rod servo hydraulic system is designed with a low-complex control strategy; S3: According to the controller of the single-rod servo-hydraulic system and the model of the single-rod servo-hydraulic system, the stability of the single-rod servo-hydraulic system is proved.

The process of establishing the model expression of the servo-hydraulic system with a single rod is as follows:

First, establish the dynamic model of the hydraulic cylinder according to Newton's second law:

The additional term f(t) in the mechanics equation (1) represents various disturbances (eg, unmodeled friction effects, unmodeled dynamics, external disturbances, etc.), x and m represent the position and mass of the load, respectively, B is the viscous damping coefficient, K is the equivalent spring stiffness of the load, and the load is mainly inertial load, therefore, K=0; F=A ₁ *P ₁ -A ₂ *P ₂ is the main power output by the hydraulic actuator, where P ₁ , P ₂ are the pressures of the large and small chambers of the hydraulic cylinder, which can be measured by a pressure sensor, and A ₁ and A ₂ are the effective areas of the pistons of the large and small chambers.

Assumption 1: The interference term f(t) is bounded, i.e.

Such that |f(t)|<f _m , the viscous damping coefficient B is an uncertain bounded positive parameter, namely

make

The present invention adopts a three-position five-way valve piston actuator, and the load pressure dynamics are described as follows:

In the formula, V ₁ and V ₂ are the volumes of the rod cavity and the rod-less cavity respectively (V ₁ =V ₀₁ +A ₁ x, V ₂ =V ₀₂ -A ₂ x), and V ₀₁ and V ₀₂ are the initial positions of the piston, respectively C _t is the leakage coefficient inside the hydraulic cylinder, C _e is the external leakage coefficient of the hydraulic cylinder, β _e is the elastic modulus of the oil, Q ₁ is the supply (return) of the rodless cavity Oil flow, Q ₂ is the return (supply) oil flow of the rod cavity, and the calculation formulas of Q ₁ and Q ₂ are as follows:

in,

is the flow gain, and s(Γ(x _v )) is expressed as:

P _s is the oil supply pressure of the hydraulic system, P _r is the oil return pressure of the hydraulic system, C _d is the flow coefficient of the orifice, w is the area gradient of the spool valve, ρ is the oil density, and x _v is the spool displacement of the servo valve ;

Assumption 2: The indeterminate parameter leakage coefficient _Ct is bounded, that is

make

The uncertain parameter leakage coefficient C _e is bounded, that is,

make

The uncertain parameter oil elastic modulus β _e is bounded, namely

make

In the operation of the hydraulic actuator, the spool displacement x _v of the servo valve is actually controlled by the voltage or current input u to obtain the required corresponding force. The dynamic characteristics of the servo valve are as follows:

τ is the time constant of the servo valve dynamics model and u(t) is the current input.

In this paper, the spool displacement Γ(x _v ) with unknown dead zone is considered, and its expression is as follows:

The parameters m _r and m _l represent the left and right slopes of the dead zone characteristic curve, and the parameters _br and b _l represent the breakpoints of the input nonlinearity;

Assumption 3: The parameters m _r , _ml , _br , _bl are unknown positive numbers and m _r = _ml , in this paper, the upper bound of the slope is m _max , and the upper bound of _br is

The upper bound of b _l is

The spool dead zone can be expressed as Γ(x _v )=m(t)x _v +d(t), _m (t), _d ( _t ) expression is as follows:

As we all know, the servo valve is a key mechanical component in the electro-hydraulic actuator. The current or voltage controls the displacement of the spool of the servo valve, and then controls the hydraulic oil to be drawn in or out of the oil chamber, and finally the actuator performs the corresponding movement; obviously, in the servo valve In the hydraulic position tracking system, there must be a nonlinear problem of the dead zone of the spool. Therefore, it is necessary to consider the adverse effects of this problem to obtain better system performance; in addition, considering that it is difficult to obtain an accurate slope of the dead zone model in practical applications and interval points, so a robust and strong novel control strategy is proposed to solve this problem;

In order to deduce the hydraulic asymmetric cylinder position tracking system into the form of state space expression, the state variables are defined as follows: x ₁ =x,

x ₃ =P ₁ , x ₄ =P ₂ , x ₅ =x _v , (1) to (6) the dynamic equations can obtain the state equation with the spool dead zone as follows:

in

The process of designing the controller of the servo-hydraulic system of the single-rod according to the servo-hydraulic system model of the single-rod and adopting the low-complexity control strategy is as follows: define the error variable:

Among them, α ₁ , α ₂ , α ₃ are the virtual control quantities obtained in the subsequent proof process, x _1r is the command signal of the hydraulic position tracking system, and F (F=A ₁ x ₃ -A ₂ x ₄ ) is the actuator The main power of the output; z ₁ is the position tracking error, zi _i = 2...4 is the control error; in order to solve the transient and steady-state problems of the tracking error and smoothly select the virtual control variable, four positive smooth decreasing functions

is selected as the specified performance function; define the normalized error ζ _i , do error transformation for zi _i = 1...4, the transformed error

as follows:

The virtual control function is selected as follows:

The controller of the servo hydraulic system is:

Further, described according to the controller of the servo hydraulic system of the single output rod and the servo hydraulic system model of the single output rod, the process of proving the stability of the servo hydraulic system of the single output rod is as follows: S3-1: Utilize the initial value theorem, Prove that when the normalized error vector is in the time period t∈[0,τ _max ), there is a maximum solution in the non-empty open set Ω _ζ ; S3-2: When the normalized error vector is in the time period t∈[0,τ _max ), When there is a maximum solution in the non-empty open set _Ωζ , the virtual controller and the controller of the servo-hydraulic system can guarantee that the closed-loop signal is bounded for t∈[0,τ _max ); S3-3: Use the initial value proposal to prove that when When τ _max = +∞ in S3-2, it is still true that all closed-loop signals are bounded. In order to successfully complete the controller stability proof, a lemma is proposed, and the correctness of the lemma is proved;

The inequality lemma expression is as follows:

If a is bounded (i.e.

|a|<A),

then x must be bounded and

prove:

because

so

When |x|>A,

so

Based on the position error, the relationship between the error variable (17) and the normalized error ζ _i (18), x ₁ , x ₂ , F, x ₅ can be rewritten as: x ₁ =ζ ₁ ρ ₁ +x _r x ₂ =ζ ₂ ρ ₂ +α ₁ F=ζ ₃ ρ ₃ +α ₂ x ₅ =ζ ₄ ρ ₄ +α ₃ (24)

Direct derivation of ζ _i (i=1...4) in (18), we can get:

in:

By defining the normalized error vector ζ = [ζ ₁ , ζ ₂ , ζ ₃ , ζ ₄ ] ^T , (25)-(28) can be written as follows:

Define an open set:

Further, using the initial value theorem, it is proved that when the normalized error vector is in the time period t∈[0,τ _max ), there is a maximum solution in the non-empty open set Ω _ζ as follows: Ω _ζ is a non-empty open set, and the selection performance When the function ρ _i satisfies ρ _i (0)>min{δ _-i ,δ ^- _i }|z _i (0)|, i=1...4, it can be deduced |ζ _i (0)|<min {δ _−i ,δ ⁻ _i }, i=1...4, therefore, for Eq. (29) _ζ (0)∈Ωζ; furthermore, due to the desired tracking trajectory x _r , the performance function ρ _i (t) , i=1...4, the intermediate control signals α _i , i=1...3 and the control rate u are continuously smooth and derivable, the dynamic variables in the state equation of the spool dead zone position tracking system are continuous It is a derivable function, so the function L(t,ζ) in the normalized error vector of the formula is piecewise continuous with respect to time t, and in the open set Ω _ζ , ζ satisfies the Lipschitz condition; the conditions in the inequality lemma are all is satisfied, so in the time period [0,τ _max ), there is a unique maximum solution ζ(t)∈Ω _ζ for the normalized error vector, for

can guarantee ζ _i (t)∈Ω _ζ , i=1...4;

Further, when the normalized error vector has a maximum solution in the non-empty open set Ω _ζ in the time period t∈[0,τ _max ), the virtual controller and the controller of the servo hydraulic system, for t∈[ 0,τ _max ) can ensure that the closed-loop signal is bounded as follows:

S3-2-1: Calculated according to formula (18)

Derivative to Time

in

There is a constant r _M1 such that 0>r ₁ >r _M1 ;

Substituting equations (10), (24) and (19) into (30), we get:

remember

△ ₁ (t)=r ₁ k ₁ ; (32)

considering

and

Choose a Lyapunov function:

(33) and (34) conform to the form of the inequality lemma, which is known by the inequality lemma, because a ₁ is bounded (

|a ₁ |<A ₁ ), 0<△ ₁ (t)<r _M1 k ₁ , so

because

Bounded, from (13) and (14) we can get -δ _-1 ρ ₁ (t)<z ₁ <δ ^- ₁ ρ ₁ (t), so for

The tracking error z ₁ decays within the specified bounds; so it can be guaranteed that the tracking error z ₁ converges within the specified performance bounds in the time period t∈[0,τ _max ), since the command is a bounded sinusoidal signal, the state variable x ₁ is Bounded; S3-2-2: Calculated according to formula (18)

Derivative to Time

in

There is a constant r _M2 such that 0>r ₂ >r _M2 ; taking equations (10), (24) and (20) into (35), we get:

remember

considering

and

Choose a Lyapunov function:

(38) and (39) conform to the form of the inequality lemma, which is known by the inequality lemma, because a ₂ is bounded

such that |a ₂ |<A ₂ ,

so

because

Bounded, from (13) and (14) we can get -δ _-2 ρ ₂ (t)<z ₂ <δ ^- ₂ ρ ₂ (t), so for

It can ensure that the control error z ₂ is bounded and the state variable x ₂ is bounded;

S3-2-3: Calculated according to formula (18)

Derivative to Time

in:

There is a constant r _M3 such that 0>r ₃ >r _M3 ; put equations (10), (24), (11) and (21) into (40), we get:

Note: △ ₃ (t)=r ₃ m(t)k ₃ h ₁ (42)

in:

Among them: V ₁ =V ₀₁ + A ₁ x, V ₂ =V ₀₂ -A ₂ x; in practical problems, consider the safe operation of the actuator and leave a safety margin (that is, according to the physical structure, the hydraulic cylinder is in the middle The maximum fluctuation of up and down near the bit is 12 cm, and the given command signal generally does not exceed 10 cm.), so we can ensure that h ₁ h ₂ h ₃ is bounded, that is, there are three positive numbers respectively.

make

considering

and α ₁ ,

), from the extreme value theorem we know that a ₃ is bounded; from (41) and (42) we get:

Choose a Lyapunov function:

(43) and (44) conform to the form of the inequality lemma, which is known by the inequality lemma, because a ₃ is bounded

such that |a ₃ |<A ₃ ,

so

because

Bounded, from (13) and (14) we can get -δ _-3 ρ ₃ (t)<z ₃ <δ ^- ₃ ρ ₃ (t), so for

It can ensure that the control error z ₃ is bounded and the active force u _a is bounded. From the mathematical expressions of H ₁ and H ₂ in the mathematical model, the state variables x ₃ and x ₄ are bounded, and they are all smaller than the hydraulic source The pressure value of P _s ;

S3-2-4: Calculated according to formula (18)

Derivative to Time

in

There is a constant r _M4 such that 0>r ₄ >r _M4 ;

Substituting equations (10), (24) and (22) into (45), we get:

remember:

considering

and α _i ,

Choose a Lyapunov function:

(48) and (49) conform to the form of the inequality lemma, which is known by the inequality lemma, because a ₄ is bounded

such that |a ₄ |<A ₄ ,

so

because

Bounded, from (13) and (14) we get -δ _-4 ρ ₄ (t)<z ₄ <δ ^- ₄ ρ ₄ (t), so for

We can guarantee that the control error _z4 is bounded and the state variable _x5 is bounded.

Further, the process of using the initial value proposal to prove that when τ _max =+∞ in S3-2, all closed-loop signals are bounded is still correct as follows: From the above proof, we can see that,

is bounded, i.e.

From equation (23), it can be seen that

This is a non-empty collection; obviously

Therefore, assuming τ _max <+∞, the initial value proposal emphasizes the existence of a time constant t ₁ ∈ [0,τ _max ) such that

There is clearly a contradiction. So we arrive at the correct assumption that τ _max = +∞, so for any t∈[0,+∞) all closed-loop system signals are bounded and the tracking error is guaranteed to converge within the specified bounds.

Consider the initial value problem: ζ(t) = ν(t, ζ), ζ(0) = ζ ₀ ∈Ω _ζ (50)

where ν:R+×Ω _ζ →R _n is a continuous function vector, and Ω _ζ ∈R _n is a non-empty open set.

Definition 1: A solution ζ(t) of the initial value problem (50), without proper right extension, the solution is the largest;

The initial value theorem is as follows: For the initial value problem (12), if ν(t, ζ) satisfies: (1) when t>0, ν(t, ζ) satisfies the local Lipschitz condition for ζ; (2) for ζ (t)∈Ω _ζ , ν(t,ζ) is piecewise continuous; (3) For ζ(t)∈Ω _ζ , ν(t,ζ) is locally integrable with respect to t; then in the time period t∈[ 0, τ _max ), there is a solution to the initial value problem (50) ζ(t)∈Ω _ζ , where τ _max >0. The initial value proposal is as follows: Assuming the initial value theorem holds; for the maximum solution ζ(t) on the time period [0, τ _max ) and the set

When τ _max <∞, there exists a time constant t ₁ ∈ [0, τ _max ) such that

Further, the inequality lemma is expressed as follows:

If a is bounded (i.e.

|a|<A),

then x must be bounded and

In order to verify the correctness of the controller design of the present invention, an experimental platform is built, and the experimental software program is debugged. The experimental platform is mainly divided into two parts: software design and hardware circuit connection; Fig. 3 (a) is the structural composition diagram I of the experimental platform; Fig. 3(b) is the structural composition of the experimental platform II; Fig. 3(c) is the structural composition of the experimental platform III; Fig. 3(d) is the structural composition of the experimental platform IV; the hardware of the experimental platform is mainly divided into three parts, the first part is The actuator mainly includes a hydraulic source system (such as accumulator, hydraulic pump, etc.), a servo valve and a single-rod hydraulic cylinder actuator. The experimental platform of the present invention adopts a three-position five-way servo valve, and its model is FD234-01K004VSX2A . The second part is the signal acquisition mechanism. The hardware of the signal acquisition mechanism is mainly the signal conversion board and the A/D board. The function of the signal conversion board is to convert the voltage and current signals to each other. Specifically, the sensor signal 4-20ma current signal is converted into 1-5v voltage signal, convert the voltage signal of ±5v to the current signal of ±10ma, the A/D board used in the experiment platform of the present invention is ADT882, its function is to realize the mutual conversion of analog continuous signal and digital discrete signal . The third part is the control mechanism. The core of the control mechanism is the industrial control computer. The experimental platform of the present invention adopts the industrial computer pc104, and its function realizes the construction of the software platform and the realization of the control algorithm. The hardware line connection mainly refers to the connection between each sensor signal line and the servo valve control current signal line and the signal conversion board, and the connection between the signal conversion board and the ADT882 board. The software design environment is VC++6.0, window operating system, the software program mainly includes writing interface functions, configuring A/D board, setting interrupt program to complete signal acquisition, control quantity calculation and output;

Further, debug the experiment and optimize the parameters until the experimental results reach the expected control effect; the first step: adjust the initial value ρ _i0 of the boundary specified in the algorithm of the present invention, the upper bound _hi of the error convergence rate and the steady-state residual of the convergence error. Difference upper bound value ρ _i∞ , where i=1...4. The initial value of the boundary ρ _i0 should be as large as possible, the upper bound of the error convergence rate _hi should be as small as possible, and the upper bound of the steady-state residual set of the convergence error should be as large _as possible; the second step: adjust the virtual control rate gains k ₁ , k ₂ , k ₃ , k ₄ , when the control rate gain _ki is adjusted to an appropriate value; the third step: slowly and appropriately reduce the initial value of the boundary ρ _i0 , increase the upper bound _hi of the error convergence rate, and reduce the steady-state residual error of the convergence error Set the upper bound value ρ _i∞ until the desired control effect is achieved.

Fig. 4 is the simulation curve diagram of error convergence under the action of the controller of the present invention, as can be seen from Fig. 4, the controller of the present invention can ensure the convergence speed and control accuracy of the displacement tracking error; Fig. 5 is the tracking error convergence under the disturbance action Simulation curve; the comparison curve of the displacement tracking error of the controller of the present invention and the backstepping controller under the action of the strong disturbance f(t)=9000sin(10t) of the system model shown in Fig. 5 can be seen from the comparison of Fig. 4 and Fig. 5 , the existence of strong disturbance has a great influence on the control effect of the backstep controller, which increases the displacement tracking error and the fluctuation frequency. However, it can also be seen from FIG. 4 and FIG. 5 that the controller of the present invention has a strong inhibitory effect on strong disturbances, and can still ensure the transient and steady-state performance of the convergence error.

Figure 6 is the simulation curve of error convergence under the action of unknown spool dead zone; Figure 7 is the simulation curve of spool displacement under the action of unknown spool dead zone; Figures 6 and 7 verify the low-complexity control scheme proposed in this paper It has better robustness to deal with unknown spool dead zone problem. The unknown dead zone nonlinearity of the valve core is added to the system model. In order to better reflect the unknown nonlinear characteristics of the valve core dead zone, this paper adopts the time-varying dead zone model slope m(t)=1+0.3sin(2t) , the discontinuous point bl=br=0.3|sin(2t)| of the time-varying dead zone model, it can be seen from Fig. 7 that the controller of the present invention can effectively compensate the adverse effects of the unknown dead zone of the spool. From Fig. 6 It can be seen that even if there is an unknown dead zone problem of the spool in the system, the controller of the present invention can still well guarantee the convergence speed and steady-state accuracy of the tracking error.

Fig. 8 is a comparative simulation graph of error convergence between the control method of the present invention and the adaptive backstepping control method (SPPFBSA) with unknown dead zone of the spool that meets the specified performance; Fig. 9 is the simulation calculation time and error of the control method of the present invention and SPPFBSA Adjustment time statistics chart; it can be seen from Figure 8 that both controllers can ensure the convergence speed and steady-state accuracy of the displacement tracking error, but as can be seen from Figure 9, in the simulation running time, the controller of the present invention and Compared with the SPPFBSA controller, it is reduced by 91.7%, and the error convergence adjustment time is reduced by 95.5%. Because the controller design of the invention has low dependence on the model, compared with the algorithm developed on the basis of backstepping self-adaptation, the calculation amount is small, and online learning is not required, so it is convenient for real-time control and engineering realization.

Fig. 10 is the command signal TS=3s: x _r =100sin((2*pi*t)/3) mm, the control method of the present invention is compared with the pid displacement tracking experiment diagram; Fig. 11 is the command signal TS=3s: x _r =100sin((2*pi*t)/3)mm, the tracking error experimental diagram of the control method of the present invention; Fig. 12 is the command signal TS=2s: x _r =100sin(pi*t)mm, the present invention controls Method and pid contrast displacement tracking experimental diagram; Figure 13 is the command signal is TS=2s: x _r =100sin(pi*t)mm, the tracking error experimental diagram of the control method of the present invention; Figure 14 is the command signal is TS=1s: x _r =50sin(2*pi*t)mm, the control method of the present invention is compared with the pid displacement tracking experimental diagram; Fig. 15 is the command signal is TS=1s: x _r =50sin(2*pi*t)mm, the present invention The experimental diagram of the tracking error of the control method; Figure 16 is the command signal TS=2s: x _r =100sin(pi*t)mm, the control method of the present invention is compared with the pid displacement tracking experimental diagram; Figure 17 is the command signal TS=2s : x _r =100sin(pi*t)mm, the tracking error experimental diagram of the control method of the present invention; Fig. 18 is the command signal TS=2s: x _r =80sin(pi*t)mm, the control method of the present invention is compared with pid Displacement tracking experiment diagram; Fig. 19 is the command signal TS=2s: x _r =80sin(pi*t)mm, the tracking error experiment diagram of the control method of the present invention; Fig. 20 is the command signal TS=2s: x _r =60sin (pi*t)mm, the control method of the present invention and pid contrast displacement tracking experimental diagram; Figure 21 is the command signal is TS=2s: x _r =60sin (pi*t)mm, the tracking error experimental diagram of the control method of the present invention; It can be seen from Figure 11, Figure 13, Figure 15, Figure 17, Figure 19 and Figure 21 that no matter how the frequency and amplitude of the command signal change, the controller of the present invention can ensure that the tracking error converges within the specified performance boundary, Therefore, the controller of the present invention can ensure the convergence speed and steady-state accuracy of the tracking error. It can be seen from any one of Fig. 10, Fig. 12, Fig. 14, Fig. 16, Fig. 18 and Fig. 20 that the displacement tracking curve obtained by the controller of the present invention has almost only a small phase lag during the upward process of the cylinder, Compared with the ascending process of the cylinder block, when the cylinder block descends, the phase lag degree of the displacement tracking curve is larger, but the control effect of the controller of the present invention is in the whole process of the displacement tracking, the displacement tracking curve is higher than the pid displacement tracking curve. The degree of phase lag is small.

10-13, when the frequency of the command signal is higher, the phase lag degree of the pid tracking curve is higher, and the amplitude attenuation is larger. In the steady state, the control effect of the controller of the present invention is that the phase lag degree is basically unchanged during the cylinder body ascending process, and the phase lag degree is getting smaller and smaller during the cylinder body descending process, and the amplitude is basically not attenuated. When the frequency of the command signal is higher, the pid tracking error becomes larger and larger, while the tracking error of the controller of the present invention is basically unchanged and converges within the specified performance boundary.

17, 19 and 21, when the command signal frequency is higher, the pid tracking error becomes larger and larger, while the tracking error of the controller of the present invention is basically unchanged and converges within the specified performance boundary.

The above description is only a preferred embodiment of the present invention, but the protection scope of the present invention is not limited to this. The equivalent replacement or change of the inventive concept thereof shall be included within the protection scope of the present invention.

Claims

A low-complexity control method for an asymmetric servo-hydraulic position tracking system, characterized in that it comprises the following steps:

S1: Establish a servo hydraulic system model of a single rod;

S2: According to the model of the servo-hydraulic system of the single-rod, the controller of the servo-hydraulic system of the single-rod is designed with a low-complex control strategy;

S3: According to the controller of the single-rod servo-hydraulic system and the single-rod servo-hydraulic system model, the stability of the single-rod servo-hydraulic system is proved.
A low-complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 1, further characterized in that: the process of establishing the model expression of the servo-hydraulic system with a single output rod is as follows:

The dynamic model of the hydraulic cylinder is established according to Newton's second law:

Where: f(t) represents various disturbances, x and m represent the position and mass of the load respectively, B is the viscous damping coefficient, K is the equivalent spring stiffness of the load, when the load is an inertial load, K=0, F= A 1 *P 1 -A 2 * P 2 is the main power output by the hydraulic actuator, wherein P 1 and P 2 are the pressures of the large and small chambers of the hydraulic cylinder, and A 1 and A 2 are the effective areas of the pistons of the large and small chambers;

Using a 5/3-way servo valve, the load pressure dynamics is expressed by the following formula:

In the formula: V 1 and V 2 are the volumes of the rod cavity and the rodless cavity respectively, V 1 =V 01 +A 1 x, V 2 =V 02 -A 2 x, V 01 and V 02 are the initial positions of the piston, respectively When , the volume of the rodless cavity and the rod cavity, C t is the leakage coefficient inside the hydraulic cylinder, C e is the external leakage coefficient of the hydraulic cylinder, β e is the elastic modulus of the oil; Q 1 is the hydraulic oil flow rate in the rod cavity, Q 2 is the hydraulic oil flow in the rodless cavity;

in:

in,
is the flow gain, and s(Γ(x v )) is expressed as:

P s is the oil supply pressure of the hydraulic system, P r is the oil return pressure of the hydraulic system, C d is the flow coefficient of the orifice, w is the area gradient of the spool valve, ρ is the oil density, and x v is the spool of the servo valve;

The spool displacement x v of the servo valve is controlled by the voltage or current input u to obtain the required corresponding force. The dynamic characteristics of the servo valve are as follows:

τ is the time constant of the servo valve dynamics model, u(t) is the current input;

Considering the spool displacement Γ(x v ) with unknown dead zone, its expression is as follows:

The parameters m r and m l represent the left and right slopes of the dead zone characteristic curve, and the parameters br and b l represent the breakpoints of the input nonlinearity;

The spool dead zone is expressed as Γ(x v )=m(t)x v +d(t), m(t), d(t) expression is as follows:

Since m r , ml , br , and b l are bounded, d(t) is bounded, and the upper bound of d(t) is written as

The state variables are defined as follows: x 1 =x,
x 3 =P 1 , x 4 =P 2 , x 5 =x v , the state equation of the tracking system with unknown spool dead zone position is obtained as follows:

in:
The low-complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 1, further characterized in that: the single-outlet rod is designed with a low-complexity control strategy according to the servo-hydraulic system model of the single-outlet rod The process of the controller of the servo hydraulic system is as follows:

Define the error variable:

z 1 =x 1 -x r z 2 =x 2 -α 1 z 3 =F-α 2 z 4 =x 5 -α 3 (17)

Among them, α 1 , α 2 , α 3 are virtual control variables, x 1r is the command signal of the hydraulic position tracking system, F (F=A 1 x 3 -A 2 x 4 ) is the active force output by the actuator, z 1 is the position tracking error, z i i=2...4 is the control error, four positive smooth decreasing functions
Select the specified performance function, define the standardized error ζ i , do error conversion for zi i = 1...4, the converted error
as follows:

Select the virtual controller as follows:

The controller of the servo hydraulic system is
The low-complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 1, further characterized in that: the controller of the servo-hydraulic system based on the single-rod and the servo-hydraulic system model of the single-rod , the process of proving the stability of the single-rod servo-hydraulic system is as follows:

S3-1: Using the initial value theorem, prove that the standardized error vector has a maximum solution in the non-empty open set Ω ζ in the time period t∈[0,τ max );

S3-2: When the normalized error vector has a maximum solution in the non-empty open set Ω ζ over the time period t∈[0,τ max ), the virtual controller and the controller of the servo hydraulic system, for t∈[0 ,τ max ) can ensure that the closed-loop signal is bounded;

S3-3: Prove that when τ max =+∞ in S3-2, it is still correct that all closed-loop signals are bounded by using the initial value proposal.
A low-complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 4, further characterized in that: the initial value theorem is used to prove that the normalized error vector is in the time period t∈[0,τ max ) When , there is a maximum solution in the non-empty open set Ω ζ :

For the non-empty open set Ω ζ , when the performance function ρ i is selected, it satisfies: ρ i (0)>min{δ -i ,δ - i }|z i (0)|, i=1...4 thus It can be obtained: |ζ i (0)|<min{δ -i ,δ - i }, i=1...4, therefore, the normalized error vector ζ(0)∈Ω ζ ; due to the expected tracking trajectory x r , the performance function ρ i (t), i=1...4, the intermediate control signal α i , i=1...3 and the control rate u are continuously smooth and steerable, the spool dead zone position tracks the state of the system The dynamic variables in the equation are continuously derivable functions, so the function L(t,ζ) in the standardized error vector is piecewise continuous with respect to time t, and in the open set Ω ζ , ζ satisfies the Lipschitz condition ; the conditions in the initial value theorem are all satisfied, so in the time period [0,τ max ), the standardized error vector has a unique maximum solution ζ(t)∈Ω ζ , for
can guarantee ζ i (t)∈Ω ζ , i=1...4.
The low-complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 4, further characterized in that: when the normalized error vector is in the time period t∈[0,τ max ), in a non-null When there is a maximum solution in the open set Ω ζ , the virtual controller and the controller of the servo hydraulic system can guarantee the bounded process of the closed-loop signal for t∈[0,τ max ) as follows:

S3-2-1: Calculated according to formula (18)
Derivative to Time

in
There is a constant r M1 such that 0>r 1 >r M1 ;

Further available:

remember

△ 1 (t)=r 1 k 1 (32)

considering
and
ρ 2 is bounded, and a 1 is bounded by the extreme value theorem;

From (31) and (32) we get:

Choose the Lyapunov function

(33) and (34) conform to the form of the inequality lemma, which can be known from the inequality lemma, because a 1 is bounded, that is
|a 1 |<A 1 , 0<△ 1 (t)<r M1 k 1 , so
because
Bounded, we can get -δ -1 ρ 1 (t)<z 1 <δ - 1 ρ 1 (t), so for
The tracking error z 1 decays within the specified bounds; it is guaranteed that the tracking error z 1 converges within the specified performance bounds in the time period t∈[0,τ max ), since the command is a bounded sinusoidal signal, the state variable x 1 is bounded of;

S3-2-2: Calculated according to formula (18)
Derivative to Time

in
There is a constant r M2 such that 0>r 2 >r M2 ;

Further available:

remember:

considering
and
ρ i is bounded, f(t) is an unknown bounded disturbance term, B is a bounded uncertain parameter, and it can be known from the extreme value theorem that a 2 is bounded;

From (36) and (37) we get:

Choose the Lyapunov function

(38) and (39) conform to the form of inequality lemma, which can be seen from the inequality lemma, because a 2 is bounded, i.e.
such that |a 2 |<A 2 ,
so
because
Bounded, we can get -δ -2 ρ 2 (t)<z 2 <δ - 2 ρ 2 (t), so for
Ensure that the control error z 2 is bounded and the state variable x 2 is bounded;

S3-2-3: Calculated according to formula (18)
Derivative to Time

in
There is a constant r M3 such that 0>r 3 >r M3 ;

Substituting equations (10), (24), (11) and (21) into (40), we get:

remember

where V 1 =V 01 +A 1 x, V 2 =V 02 -A 2 x;

In view of the safe operation of the servo hydraulic system actuator, a safety margin is reserved, that is, according to the physical structure, the hydraulic cylinder can fluctuate up to 12 cm up and down near the neutral position, and the amplitude of the command signal is less than or equal to 10 cm to ensure h 1 h 2 h 3 is bounded, that is, there are three positive numbers respectively
make

considering
and α 1 ,
ρ i is bounded, m(t) is the unknown slope of the spool dead zone, m(t) is bounded (ie
), from the extreme value theorem we know that a 3 is bounded;

From (41) and (42) we get:

Choose the Lyapunov function

(43) and (44) conform to the form of the inequality lemma, which can be known from the inequality lemma, because a 3 is bounded, namely
such that |a 3 |<A 3 ,
so
because
Bounded, we get - δ -3 ρ 3 (t)<z 3 <δ - 3 ρ 3 (t), so for
Ensure that the control error z 3 is bounded and the active force u a is bounded. From the mathematical expression of the state equation, the state variables x 3 and x 4 are bounded, and they are both less than the pressure value of the hydraulic source P s ;

S3-2-4: Calculated according to formula (18)
Derivative to Time

in
There is a constant r M4 such that 0>r 4 >r M4 ;

Further get:

remember

considering
and α i ,
ρ i is bounded, k 4 is an optional parameter when designing the controller, so a 4 is bounded;

From (46) and (47) we get:

Choose a Lyapunov function:

(48) and (49) conform to the form of the inequality lemma, which can be known from the inequality lemma, because a 4 is bounded, namely
such that |a 4 |<A 4 ,
so
because
Bounded, we get -δ -4 ρ 4 (t)<z 4 <δ - 4 ρ 4 (t), so for
It is guaranteed that the control error z4 is bounded and the state variable x5 is bounded.
A low-complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 6, further characterized in that:

The inequality lemma expression is as follows:

If a is bounded (i.e.
),
then x must be bounded and
The low-complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 4, further characterized in that: the initial value proposal is used to prove that when τ max =+∞ in S3-2, all closed loops The signal is still bounded and the correct procedure is as follows:

because

so
is a non-empty set;

Assuming τ max <+∞, the initial value proposal emphasizes the existence of a time constant t 1 ∈ [0,τ max ) such that
Contradicting, the resulting assumption is that τ max = +∞, for any t∈[0,+∞), all closed-loop system signals are bounded and the tracking error is guaranteed to converge within specified bounds.