JP7448692B2 - Low complexity control method for asymmetric servohydraulic position tracking system - Google Patents

Description

The present invention relates to the field of servohydraulic position control, and in particular to a low complexity control method for an asymmetric servohydraulic position tracking system.

Servo-hydraulic systems have advantages such as high load efficiency, low power ratio, and fast response speed, so they are widely applied in modern industry (for example, active suspension systems, hydraulic excavators, manipulators). The servo-hydraulic system includes two types of hydraulic actuator mechanisms: a dual output rod symmetric hydraulic actuator mechanism and a single output rod asymmetric hydraulic actuator mechanism. In active suspension control research, the areas of the two oil cavities are often assumed to be equal and a symmetrical hydraulic actuator mechanism is used. For asymmetric servo-hydraulic systems, due to the presence of the piston rod, the effective areas of the rod cavity and rodless cavity of the hydraulic cylinder are no longer equal, and a symmetric hydraulic actuator mechanism that can achieve the same performance will always have a smaller volume than an asymmetric hydraulic actuator mechanism. Large, generally applied in a small number of special cases, and widely used in industry are asymmetric hydraulic actuator mechanisms. Therefore, it is unreasonable to place a symmetrical hydraulic actuator mechanism in a vehicle suspension system where space is severely limited.

However, the asymmetric servo-hydraulic system is a complex nonlinear system, and at the same time, it is difficult to establish the model and design the controller regarding various uncertainties such as load changes, parameter uncertainties and unknown nonlinearities. Therefore, for asymmetric servo-hydraulic systems, high-precision position control faces great challenges. The parameter uncertainties mainly include the leakage coefficient, oil film viscosity and viscous friction coefficient, and the unknown nonlinearities mainly include the spool dead zone and disturbance, and these factors are hindering the development of high-performance controllers. . To meet the operational performance of position tracking systems, many control techniques have been developed in the past decade, such as neural network adaptation, fuzzy logic control, robust adaptive control, and backstepping adaptive control. In these control schemes, adaptive control can handle the problem of parameter uncertainty due to its online learning ability, and robust adaptive control can handle the problem of unknown disturbance. Although the algorithms based on adaptive control described above can obtain good simulation results, one important issue is that these algorithms are computationally intensive and take a long time to reach convergence. It is difficult to apply the above control algorithm based on . Additionally, most of the existing control methods for servo-hydraulic systems are developed based on backstepping control. As is well known, in the backstepping design process of high-order systems, the derivation of virtual control functions is repeated, resulting in the problem of computational explosion. Therefore, there is a need to find a new control policy that is less dependent on the design system, has a lower computational load, and does not have function approximation capabilities.

In addition, in the servo-hydraulic position tracking system, from the practical application point of view, if there is a potentially bad transient response (overshoot too large and slow convergence) in the above method, the tracking performance will be degraded and there will be danger. may occur and potentially damage the hardware. Bechlioulis CP, Rovitthakis GA, et al. first proposed a novel controller design method that can ensure transient performance of tracking errors. The method specifies the transient and steady-state performance of the tracking error by introducing a specified performance function and makes the original bounded system equivalent to an unbounded system by introducing an error transformation. . This idea has been applied to higher-order nonlinear fault-tolerant control systems, servohydraulic systems, and suspension systems. However, the control schemes in the above literature using specified performance functions are all combined with adaptive control schemes to handle unknown dynamics problems in the system.

From the above, there are the following deficiencies in the servo-hydraulic system and its position control method. The first is that the single-output rod nonlinear servo-hydraulic system is a typical complex nonlinear system, and the load change, parameter Due to issues such as uncertainty and unknown nonlinearity, there is a large gap between the establishment of a theoretical model and the actual system, making controller design difficult and complex. Most of the servo-hydraulic position control methods have been developed based on backstepping control adaptation, but these control methods are highly dependent on models, and the backstepping design process of high-order systems requires a large amount of calculation. Thirdly, from the practical application point of view, the online learning of parameters has large uncertainties, which is disadvantageous to real-time control during actual testing, and the third is that the servo-hydraulic position tracking system has a potentially bad transient response. If there is (too much overshoot, slow convergence), the tracking performance will be degraded, creating a hazard and potentially damaging the hardware.

Based on the problems existing in the prior art, the present invention provides a step S1 of establishing a model of a single output rod servo-hydraulic system and a low complexity control policy based on the model of the single output rod servo-hydraulic system. step S2 of designing a controller for a single output rod servo hydraulic system using a single output rod servo hydraulic system controller and a single output rod servo hydraulic system based on the model A low complexity control method for an asymmetric servo-hydraulic position tracking system is disclosed, comprising step S3 of verifying the stability of the hydraulic system.

（ただし、ｆ（ｔ）は様々な干渉を表し、ｘとｍとはそれぞれ負荷の位置と質量とを表し、Ｂは粘性減衰係数であり、Ｋは負荷の等価ばね剛性であり、負荷が慣性負荷の場合、Ｋ＝０であり、Ｆ＝Ａ_１＊Ｐ_１－Ａ_２＊Ｐ_２は油圧アクチュエータにより出力される主動力であり、ただし、Ｐ_１、Ｐ_２は油圧シリンダの大きなキャビティと小さなキャビティとの圧力であり、Ａ_１、Ａ_２は大きなキャビティと小さなキャビティとのピストンの有効面積である。）
３ポジション５ポートサーボバルブを用い、負荷圧力の動力学は下記式で表され、

（式中、Ｖ_１、Ｖ_２はそれぞれロッドキャビティとロッドレスキャビティとの容積であり、Ｖ_１＝Ｖ_０１＋Ａ_１ｘであり、Ｖ_２＝Ｖ_０２－Ａ_２ｘであり、Ｖ_０１とＶ_０２とはそれぞれピストンが初期位置にあるときのロッドレスキャビティとロッドキャビティとの容積であり、Ｃ_ｔは油圧シリンダの内部漏れ係数であり、Ｃ_ｅは油圧シリンダの外部漏れ係数であり、β_ｅはオイルの弾性率であり、Ｑ_１はロッドキャビティの作動油の流量であり、Ｑ_２はロッドレスキャビティの作動油の流量である。）
ただし、

（ただし、

は流量ゲインであり、ｓ（Γ（ｘ_ｖ））の式は

であり、
Ｐ_ｓは油圧システムの給油圧力であり、Ｐ_ｒは油圧システムの油戻し圧力であり、Ｃ_ｄはオリフィスの流量係数であり、ｗはスライドバルブの面積勾配であり、ρはオイルの密度であり、ｘ_ｖはサーボバルブのスプールである。）
電圧又は電流入力ｕによりサーボバルブのスプールの変位ｘ_ｖを制御して、さらに必要な対応する力を取得し、サーボバルブの動的特性は以下のように示され、

（τはサーボバルブの動力学モデルの時定数であり、ｕ（ｔ）は電流入力である。）
未知の不感帯があるスプールの変位Γ（ｘ_ｖ）を考慮すると、その式は

であり、
（パラメータｍ_ｒとｍ_ｌとは不感帯特性曲線の左右の傾きを表し、パラメータｂ_ｒとｂ_ｌとは入力非線形のブレークポイントを表す。）
スプールの不感帯はΓ（ｘ_ｖ）＝ｍ（ｔ）ｘ_ｖ＋ｄ（ｔ）で表され、ｍ（ｔ）、ｄ（ｔ）の式は以下のように示され、

（ｍ_ｒ、ｍ_ｌ、ｂ_ｒ、ｂ_ｌが有界であるため、ｄ（ｔ）が有界であり、ｄ（ｔ）の上限を

とする。）
状態変数を

のように定義し、未知のスプールの不感帯を有する以下の位置追跡システムの状態方程式を取得する、ことを特徴とする。

（ただし、

。） Furthermore, the process of establishing a model of a single output rod servo-hydraulic system is as follows:
Established a dynamic model of a hydraulic cylinder based on Newton's second law,

(where f(t) represents various interferences, 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 For the 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 large cavities of the hydraulic cylinders and the small (A ₁ and A ₂ are the effective areas of the piston in the large cavity and the small cavity.)
Using a 3-position 5-port servo valve, the dynamics of the load pressure is expressed by the following formula:

(In the formula, V ₁ and V ₂ are the volumes of the rod cavity and rodless cavity, respectively, V ₁ = V ₀₁ + A ₁ x, V ₂ = V ₀₂ - _{A 2} x, and V ₀₁ and V ₀₂ are the volumes of the rodless cavity and rod cavity when the piston is in the initial position, C _t is the internal leakage coefficient of the hydraulic cylinder, C _e is the external leakage coefficient of the hydraulic cylinder, and β _e is the elastic modulus of the oil, _Q1 is the flow rate of hydraulic oil in the rod cavity, and _Q2 is the flow rate of hydraulic oil in the rodless cavity.)
however,

(however,

is the flow rate gain, and the formula for s(Γ(x _v )) is

and
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 slope of the slide valve, and ρ is the density of the oil. , x _v is the spool of the servo valve. )
Controlling the displacement x _v of the spool of the servovalve by the voltage or current input u to further obtain the required corresponding force, the dynamic characteristics of the servovalve are shown as follows:

(τ is the time constant of the servovalve dynamic model and u(t) is the current input.)
Considering the displacement Γ(x _v ) of the spool with an unknown dead zone, the formula is

and
(The parameters m _r and m _l represent the left and right slopes of the dead zone characteristic curve, and the parameters b _r and b _l represent the break points of the input nonlinearity.)
The dead zone of the spool is expressed as Γ( _xv )=m(t) _xv +d(t), and the formulas of m(t) and d(t) are shown as below,

(Since m _r , m _l , b _r , and b _l are bounded, d(t) is bounded, and the upper limit of d(t) is

shall be. )
state variables

and obtain the following state equation of the position tracking system with an unknown spool dead zone.

(however,

. )

を指定された性能関数として選択し、正規化誤差ζ_ｉを定義し、ｚ_ｉ，ｉ＝１…４に対して誤差変換を行い、変換された誤差

は

であり、
以下の仮想コントローラを選択し、

サーボ油圧システムのコントローラは、

である。 Furthermore, the process of designing a controller for a single output rod servo-hydraulic system using a low complexity control policy based on the model of the single-output rod 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)
(However, α ₁ , α ₂ , α ₃ are virtual control variables, 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 output signal of the actuator. z ₁ is the position tracking error, and z _i , i=2...4 are the control errors.)
4 positive smooth decreasing functions

is selected as the specified performance function, the normalized error ζ _i is defined, the error conversion is performed for z _i , i=1...4, and the converted error is

teeth

and
Select the virtual controller below and

The controller of the servo hydraulic system is

It is.

Furthermore, based on the controller of the single output rod servo hydraulic system and the model of the single output rod servo hydraulic system, the process of proving the stability of the single output rod servo hydraulic system is as follows: S3- 1: Using the initial value theorem, prove that if the normalized error vector is in the period t∈[0, τ _max ), there is a maximum solution in the non-empty open set Ω _ζ , and S3-2: The normalized error vector is in the period t∈[0,τ _max ) and there is _a maximum solution in the non-empty open set Ω _ζ S3-3: Using the initial value proposal, if τ _max = +∞ in S3-2, it is still accurate that all closed-loop signals are bounded. prove

を満たすことにより、

を得ることができるため、正規化誤差ベクトルはζ（０）∈Ω_ζであり、所望の追跡奇跡ｘ_ｒ、性能関数ρ_ｉ（ｔ），ｉ＝１…４、中間制御信号α_ｉ，ｉ＝１…３及び制御率ｕが連続的で滑らかに導出可能であり、スプールの不感帯の位置追跡システムの状態方程式における動力学変数が連続的に導出可能な関数であるため、式の正規化誤差ベクトルの関数Ｌ（ｔ，ζ）は、時間ｔに関して区分的に連続し、開集合Ω_ζでは、ζはリプシッツ条件を満たし、初期値定理における条件がいずれも満たされるため、期間［０，τ_ｍａｘ）で、正規化誤差ベクトルには唯一の最大解ζ（ｔ）∈Ω_ζがあり、∀ｔ∈［０，τ_ｍａｘ）に対して、いずれもζ_ｉ（ｔ）∈Ω_ζ，ｉ＝１…４を確保できる。 Furthermore, using the initial value theorem, in the step of proving that if the normalized error vector is in the period t∈[0, τ _max ), there is a maximum solution in the non-empty open set Ω _ζ , Ω _ζ is when choosing the performance function ρ _i ,

By satisfying

can be obtained, the normalized error vector is ζ(0)∈Ω _ζ , the desired tracking miracle 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 can be derived continuously and smoothly, and the dynamic variables in the state equation of the position tracking system for the dead zone of the spool are functions that can be derived continuously, so the normalization error of the equation The vector function L(t, ζ) is piecewise continuous with respect to time t, and in the open set Ω _ζ , ζ satisfies the Lipschitz condition and all conditions in the initial value theorem are satisfied, so the period [0, τ _max ), the normalized error vector has a unique maximum solution ζ(t)∈Ω _ζ , and for ∀t∈[0,τ _max ), both ζ _i (t)∈Ω _ζ , i= 1...4 can be secured.

の時間に対する導関数

を求めて、

（ただし、

であり、１つの正の定数ｒ_Ｍ１があり、０＞ｒ_１＞ｒ_Ｍ１となる。）
さらに、

を得ることができ、

とし、

であり、

が有界であることを考慮すると、極値定理から、ａ_１が有界であることが分かり、
（３１）と（３２）とから、

を得て、
リアプノフ関数

を選択し、
（３３）と（３４）とは、不等式補助定理の形態を満たし、不等式補助定理から分かるように、ａ_１が有界であり、すなわち∃Ａ_１＞０であり、｜ａ_１｜＜Ａ_１、０＜Δ_１（ｔ）＜ｒ_Ｍ１ｋ_１となるため、

であり、

が有界であり、

を得ることができるため、∀ｔ∈［０，τ_ｍａｘ）に対して、追跡誤差ｚ_１が指定された境界内で減衰され、期間ｔ∈［０，τ_ｍａｘ）での追跡誤差ｚ_１が指定された性能境界内で収束することを確保し、指令が有界正弦信号であるため、状態変数ｘ_１が有界であり、
Ｓ３－２－２：式（１８）に基づいて

の時間に対する導関数

を求めて、

（ただし、

であり、１つの正の定数ｒ_Ｍ２があり、０＞ｒ_２＞ｒ_Ｍ２となる。）
さらに、

を得ることができ、

とし、

であり、

が有界であり、ｆ（ｔ）が未知の有界な外乱項であり、Ｂが有界な不確実性パラメータであることを考慮すると、極値定理から、ａ_２が有界であることが分かり、
（３６）と（３７）とから、

を得て、
リアプノフ関数

を選択し、
（３８）と（３９）とは、不等式補助定理の形態を満たし、不等式補助定理から分かるように、ａ_２が有界であり、すなわち∃Ａ_１＞０であり、｜ａ_２｜＜Ａ_２、

となるため、

であり、

が有界であり、

を得ることができるため、∀ｔ∈［０，τ_ｍａｘ）に対して、制御誤差ｚ_２が有界であり及び状態変数ｘ_２が有界であることを確保し、
Ｓ３－２－３：式（１８）に基づいて

の時間に対する導関数

を求めて、

（ただし、

であり、１つの正の定数ｒ_Ｍ３があり、０＞ｒ_３＞ｒ_Ｍ３となる。）
式（１０）、（２４）、（１１）及び（２１）を（４０）に代入して、

を得て、

とし、

、ただし、Ｖ_１＝Ｖ_０１＋Ａ_１ｘ、Ｖ_２＝Ｖ_０２－Ａ_２ｘ。）
サーボ油圧システムアクチュエータの安全な動作に鑑みて、安全マージンが残され、すなわち、物理的構造に基づいて、油圧シリンダが中央位置の付近に最大１２センチだけ上下に変動でき、指令信号の振幅が１０センチ以下であり、ｈ_１、ｈ_２、ｈ_３が有界であることを確保し、すなわち、それぞれ３つの正数

があり、

となり、

であり、

が有界であり、ｍ（ｔ）がスプールの不感帯の未知の傾きであり、ｍ（ｔ）が有界である（すなわち∃Ｍ_ｒ＞０，０＜ｍ（ｔ）＜Ｍ_ｒ）ことを考慮すると、極値定理から、ａ_３が有界であることが分かり、
（４１）と（４２）とから、

を得て、
リアプノフ関数

を選択し、
（４３）と（４４）とは、不等式補助定理の形態を満たし、不等式補助定理から分かるように、ａ_３が有界であり、すなわち∃Ａ３＞０であり、｜ａ３｜＜Ａ３、

となるため、

であり、

が有界であり、

を得ることができるため、∀ｔ∈［０，τ_ｍａｘ）に対して、制御誤差ｚ_３が有界であり及び主動力ｕ_ｓが有界であることを確保し、状態方程式の数式から見ると、状態変数ｘ_３とｘ_４とが有界であり、いずれも油圧源Ｐ_ｓの圧力値よりも小さく、
Ｓ３－２－４：式（１８）に基づいて

の時間に対する導関数

を求めて、

（ただし、

であり、１つの正の定数ｒ_Ｍ４があり、０＞ｒ_４＞ｒ_Ｍ４となる。）
さらに、

を得て、

とし、

であり、

が有界であり、ｋ_４がコントローラを設計するときのオプションパラメータであることを考慮すると、ａ_４が有界であり、
（４６）と（４７）とから、

を得て、
リアプノフ関数

を選択し、
（４８）と（４９）とは、不等式補助定理の形態を満たし、不等式補助定理から分かるように、ａ_４が有界であり、すなわち∃Ａ_４＞０であり、｜ａ_４｜＜Ａ_４、

となるため、

であり、

が有界であり、

を得ることができるため、∀ｔ∈［０，τ_ｍａｘ）に対して、制御誤差ｚ_４が有界であり及び状態変数ｘ_５が有界であることを確保する。 Furthermore, if the normalized error vector is in the period t∈[0,τ _max ) and there is a maximum solution in the non-empty open set Ω _ζ , then the _virtual controller and the controller of the servo-hydraulic system The process that can ensure that the closed-loop signal is bounded for is as follows:
S3-2-1: Based on formula (18)

derivative with respect to time

In search of

(however,

, and there is one positive constant r _M1 such that 0>r ₁ >r _M1 . )
moreover,

you can get

year,

and

Considering that is bounded, we know from the extreme value theorem that a ₁ is bounded,
From (31) and (32),

obtained,
Lyapunov function

Select
(33) and (34) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₁ is bounded, that is, ∃A ₁ >0, and |a ₁ |<A ₁ , 0<Δ ₁ (t)<r _M1 k ₁ , so

and

is bounded,

can be obtained, so that for ∀t∈[0,τ _max ), the tracking error z ₁ is attenuated within the specified bounds, and the tracking error z ₁ in the period t∈[0,τ _max ) is Ensures convergence within the specified performance bounds, and since the command is a bounded sinusoidal signal, the state variable x ₁ is bounded,
S3-2-2: Based on formula (18)

derivative with respect to time

In search of

(however,

, and there is one positive constant r _M2 such that 0>r ₂ >r _M2 . )
moreover,

you can get

year,

and

Considering that is bounded, f(t) is an unknown bounded disturbance term, and B is a bounded uncertainty parameter, it follows from the extreme value theorem that a ₂ is bounded. I understand,
From (36) and (37),

obtained,
Lyapunov function

Select
(38) and (39) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₂ is bounded, that is, ∃A ₁ >0, and |a ₂ |<A ₂ ,

Therefore,

and

is bounded,

Since we can obtain, for ∀t∈[0,τ _max ), we ensure that the control error z ₂ is bounded and the state variable x ₂ is bounded,
S3-2-3: Based on formula (18)

derivative with respect to time

In search of

(however,

, and there is one positive constant r _M3 , such that 0>r ₃ >r _M3 . )
Substituting equations (10), (24), (11) and (21) into (40),

obtained,

year,

, 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 actuators, a safety margin is left, i.e., based on the physical structure, the hydraulic cylinder can move up and down by a maximum of 12 cm around the center position, and the amplitude of the command signal is 10 cm. centimeter or less and ensure that h ₁ , h ₂ , h ₃ are bounded, i.e. each of 3 positive numbers

There is,

Then,

and

is bounded, m(t) is the unknown slope of the dead zone of the spool, and m(t) is bounded (i.e. ∃M _r >0,0<m(t)<M _r ). Considering this, we can see from the extreme value theorem that a ₃ is bounded,
From (41) and (42),

obtained,
Lyapunov function

Select
(43) and (44) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₃ is bounded, that is, ∃A3>0, and |a3|<A3,

Therefore,

and

is bounded,

Since _{we can get} and the state variables x ₃ and x ₄ are bounded, both are smaller than the pressure value of the hydraulic source P _s ,
S3-2-4: Based on formula (18)

derivative with respect to time

In search of

(however,

, and there is one positive constant r _M4 such that 0>r ₄ >r _M4 . )
moreover,

obtained,

year,

and

Considering that is bounded and k ₄ is an optional parameter when designing the controller, a ₄ is bounded,
From (46) and (47),

obtained,
Lyapunov function

Select
(48) and (49) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₄ is bounded, that is, ∃A ₄ >0, and |a ₄ |<A ₄ ,

Therefore,

and

is bounded,

can be obtained, thus ensuring that the control error z ₄ is bounded and the state variable x ₅ is bounded for ∀t∈[0, τ _max ).

であり、
ａが有界であり（すなわち∃Ａ＞０，｜ａ｜＜Ａ）、

である場合、ｘが必ず有界であり、

である。 Furthermore, the formula of the inequality lemma is:

and
a is bounded (i.e. ∃A>0, |a|<A),

, then x is necessarily bounded,

It is.

であるため、

は、空でない集合であり、

であり、τ_ｍａｘ＜＋∞を仮定すると、初期値の提案では１つの時定数ｔ_１∈［０，τ_ｍａｘ）があることが強調され、

となり、矛盾を引き起こし、得られた仮定がτ_ｍａｘ＝＋∞であり、いずれかのｔ∈［０，＋∞）に対して、すべての閉ループシステム信号が有界であり、追跡誤差が指定された境界内で収束することを確保できる。 Furthermore, using the initial value proposal, the process of proving that it is still accurate that all closed-loop signals are bounded when τ _max = +∞ in S3-2 is as follows:

Therefore,

is a non-empty set,

, and assuming τ _max <+∞, the initial value proposal emphasizes that there is one time constant t ₁ ∈[0, τ _max ),

, which causes a contradiction and the resulting assumption is τ _max = +∞, and for any t∈[0, +∞) all closed-loop system signals are bounded and the tracking error is specified. It is possible to ensure convergence within the given boundaries.

Because the above technical solution is used, the low complexity control method for the asymmetric servo-hydraulic position tracking system according to the present invention can solve various uncertainty problems existing in the hydraulic system, such as unknown friction effects, parameter (uncertainties and load changes) and unknown nonlinear problems (e.g. spool deadbands, disturbances), the controller design does not rely on precise mathematical models and only requires measurable state signals. Compared with existing algorithms developed based on backstepping adaptation, the calculation process is simpler, the amount of calculation is lower, the real-time control is easier, and the engineering realization is easier. Therefore, the present invention can ensure the tracking error convergence speed and steady-state accuracy.Finally, according to the experimental results, compared with the traditional PID control method, the position tracking effect of the present invention can ensure the steady-state accuracy. It has been proven that the accuracy is increased and the degree of phase delay in tracking displacement is reduced. When the frequency of the desired trajectory signal increases, the larger the phase delay of the actual tracking displacement obtained by controlling the PID algorithm, the more the amplitude is attenuated and the displacement tracking error becomes larger; however, the control of the present invention The algorithm shows that if the frequency of the desired trajectory signal increases, the degree of delay of the obtained actual control displacement basically does not change, the tracking error always converges within the specified bounds, and the amplitude basically increases. Not attenuated. If the frequency of the desired signal does not change and the amplitude increases, the displacement tracking error of the PID algorithm will increase, but the control algorithm of the present invention can still ensure that the tracking error converges within the specified bounds. , the degree of phase delay is small and the amplitude is basically not attenuated.

In order to more clearly explain the technical solutions in the embodiments of the present application or the prior art, the drawings that need to be used in the embodiments or the description of the prior art will be briefly described below, and clearly described below. The drawings shown are only some examples described in this application, and a person skilled in the art can obtain other drawings based on these drawings without any creative effort.

1 is a flowchart of a low complexity control method for an asymmetric servohydraulic position tracking system according to the present invention; 1 is a block diagram of a model of a single output rod servo-hydraulic system; FIG. FIG. 1 is a structural configuration diagram of the experimental platform. FIG. 2 is a structural configuration diagram of the experimental platform. It is a structural configuration diagram of the experimental platform III. It is a structural block diagram IV of the experimental platform. FIG. 4 is a simulation curve diagram of error convergence under the action of the controller of the present invention. FIG. 3 is a simulation curve diagram of tracking error convergence under the action of disturbance. FIG. 6 is a simulation curve diagram of error convergence under the action of an unknown spool dead zone. FIG. 6 is a simulation curve diagram of spool displacement under the action of an unknown spool dead zone; FIG. 3 is a comparative simulation curve diagram of error convergence between the control method of the present invention and an adaptive backstepping control scheme with unknown spool deadband (SPPFBSA) that satisfies specified performance. FIG. 3 is a statistical diagram of simulation calculation time and error adjustment time between the control method of the present invention and SPPFBSA. FIG. 4 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when the command signal is TS=3s:x _r =100 sin ((2*pi*t)/3) mm. FIG. 6 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 =100 sin ((2*pi*t)/3) mm. FIG. 2 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when the command signal is TS=2s:x _r =100 sin (pi*t) mm. FIG. 4 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 =100 sin (pi*t) mm. FIG. 6 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when the command signal is TS=1s:x _r =50 sin (2*pi*t) mm. FIG. 4 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 =50 sin (2*pi*t) mm. FIG. 2 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when the command signal is TS=2s:x _r =100 sin (pi*t) mm. FIG. 4 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 =100 sin (pi*t) mm. FIG. 4 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when the command signal is TS=2s:x _r =80 sin (pi*t) mm. FIG. 6 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 =80 sin (pi*t) mm. FIG. 2 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when the command signal is TS=2s:x _r =60 sin (pi*t) mm. FIG. 6 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 =60 sin (pi*t) mm.

In order to make the technical solutions and advantages of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely explained below with reference to the drawings in the embodiments of the present invention.

Hereinafter, specific embodiments of the present invention will be further described with reference to the drawings.

FIG. 1 is a flowchart of a low-complexity control method for an asymmetric servo-hydraulic position tracking system according to the present invention, the low-complexity control method for an asymmetric servo-hydraulic position tracking system is a single output rod servo-hydraulic system. a step S2 of designing a controller for the single output rod servo-hydraulic system using a low complexity control policy based on the model of the single output rod servo-hydraulic system; step S3 of proving the stability of the single output rod servo hydraulic system based on the controller of the output rod servo hydraulic system and the model of the single output rod servo hydraulic system, FIG. FIG. 2 is a block diagram of a model of a rod servo-hydraulic system.

（力学方程式（１）の追加項ｆ（ｔ）は様々な干渉（例えば、モデル化されていない摩擦効果、モデル化されていない動力学、外乱など）を表し、ｘとｍとはそれぞれ負荷の位置と質量とを表し、Ｂは粘性減衰係数であり、Ｋは負荷の等価ばね剛性であり、負荷が主に慣性負荷であるため、Ｋ＝０であり、Ｆ＝Ａ_１＊Ｐ_１－Ａ_２＊Ｐ_２は油圧アクチュエータにより出力される主動力であり、ただし、Ｐ_１、Ｐ_２は油圧シリンダの大きなキャビティと小さなキャビティとの圧力であり、圧力センサにより測定でき、Ａ_１、Ａ_２は大きなキャビティと小さなキャビティとのピストンの有効面積である。）
仮定１：干渉項ｆ（ｔ）が有界であり、すなわち∃ｆ_ｍ＞０であり、｜ｆ（ｔ）＜ｆ_ｍ｜となり、粘性減衰係数Ｂが不確実な有界な正のパラメータであり、すなわち

であり、

となり、本発明では、３ポジション５ポートバルブのピストンアクチュエータを用い、負荷圧力の動力学は以下のように示され、

（式中、Ｖ_１、Ｖ_２はそれぞれロッドキャビティとロッドレスキャビティとの容積（Ｖ_１＝Ｖ_０１＋Ａ_１ｘ、Ｖ_２＝Ｖ_０２－Ａ_２ｘ）であり、Ｖ_０１とＶ_０２とはそれぞれピストンが初期位置にあるときのロッドレスキャビティとロッドキャビティとの容積であり、Ｃ_ｔは油圧シリンダの内部漏れ係数でありであり、Ｃ_ｅは油圧シリンダの外部漏れ係数であり、β_ｅはオイルの弾性率であり、Ｑ_１はロッドキャビティの給油（油戻し）流量であり、Ｑ_２はロッドレスキャビティの油戻し（給油）流量である。）、Ｑ_１とＱ_２との計算式は、

であり、
（ただし、

は流量ゲインであり、ｓ（Γ（ｘ_ｖ））の式は

であり、
Ｐ_ｓは油圧システムの給油圧力であり、Ｐ_ｒは油圧システムの油戻し圧力であり、Ｃ_ｄはオリフィスの流量係数であり、ｗはスライドバルブの面積勾配であり、ρはオイルの密度であり、ｘ_ｖはサーボバルブのスプールの変位である。）
仮定２：不確実性パラメータの内部漏れ係数Ｃ_ｔが有界であり、すなわち

であり、

となり、不確実性パラメータの内部漏れ係数Ｃ_ｅが有界であり、すなわち

であり、

となり、不確実性パラメータのオイルの弾性率β_ｅが有界であり、すなわち

であり、

となり、
油圧アクチュエータの操作では、実際には、電圧又は電流入力ｕによりサーボバルブのスプールの変位ｘ_ｖを制御して、必要な対応する力をさらに取得し、サーボバルブの動的特性は以下のように示され、

（τはサーボバルブの動力学モデルの時定数であり、ｕ（ｔ）は電流入力である。）
本明細書では、未知の不感帯があるスプールの変位Γ（ｘ_ｖ）を考慮すると、その式は

であり、
（パラメータｍ_ｒとｍ_ｌとは不感帯特性曲線の左右の傾きを表し、パラメータｂ_ｒとｂ_ｌとは入力非線形のブレークポイントを表す。）
仮定３：パラメータｍ_ｒ、ｍ_ｌ、ｂ_ｒ、ｂ_ｌが未知の正数であり、ｍ_ｒ＝ｍ_ｌであり、本明細書では、傾きの上限はｍ_ｍａｘであり、ｂ_ｒの上限は

であり、ｂ_ｌの上限は

であり、スプールの不感帯はΓ（ｘ_ｖ）＝ｍ（ｔ）ｘ_ｖ＋ｄ（ｔ）で表されてもよく、ｍ（ｔ）、ｄ（ｔ）の式は以下のように示され、

（ｍ_ｒ、ｍ_ｌ、ｂ_ｒ、ｂ_ｌが有界であるため、ｄ（ｔ）が有界であり、ｄ（ｔ）の上限を

とする。）
周知のように、サーボバルブは、電気油圧アクチュエータにおける重要な機械部材であり、サーボバルブのスプールの変位を電流又は電圧制御し、さらにオイルキャビティへの作動油の抽入又は抽出を制御し、最後に、アクチュエータは対応する移動を実行し、明らかに、サーボ油圧位置追跡システムでは、スプールの不感帯の非線形問題が必然的に存在し、従って、この問題の悪影響を考慮してより良好なシステム性能を取得する必要があり、また、実際の応用では、不感帯モデルの正確な傾き及び間隔ポイントを取得することは困難であることを考慮するため、この問題を解決するためのロバスト性の高い新規の制御ポリシーが提案されており、
油圧非対称シリンダの位置追跡システムを状態空間式の形態に導出するために、状態変数を

のように定義し、（１）～（６）の動力学方程式によりスプールの不感帯を有する以下の状態方程式を取得できる。

（ただし、

。） The process of establishing a model of a single output rod servo-hydraulic system is as follows:
First, we established a dynamic model of the hydraulic cylinder based on Newton's second law,

(The additional term f(t) in the dynamic equation (1) represents various interferences (e.g., unmodeled friction effects, unmodeled dynamics, disturbances, etc.), and x and m are the load factors, respectively. position and mass, B is the viscous damping coefficient, K is the equivalent spring stiffness of the load, and since the load is mainly an inertial load, K=0, and F=A ₁ *P ₁ -A ₂ *P ₂ is the main power output by the hydraulic actuator, where P ₁ and P ₂ are the pressures between the large cavity and the small cavity of the hydraulic cylinder, which can be measured by a pressure sensor, and A ₁ and A ₂ are This is the effective area of the piston with the large cavity and the small cavity.)
Assumption 1: The interference term f(t) is bounded, that is, ∃f _m > 0, |f(t) < f _m |, and the viscous damping coefficient B is an uncertain bounded positive parameter. Yes, i.e.

and

Therefore, in the present invention, a piston actuator with a 3-position 5-port valve is used, and the dynamics of the load pressure is shown as follows:

(In the formula, V ₁ and V ₂ are the volumes of the rod cavity and rodless cavity, respectively (V ₁ = V ₀₁ + A ₁ x, V ₂ = V ₀₂ - A ₂ x), and V ₀₁ and V ₀₂ are They are the volumes of the rodless cavity and rod cavity when the piston is in the initial position, respectively, C _t is the internal leakage coefficient of the hydraulic cylinder, C _e is the external leakage coefficient of the hydraulic cylinder, and β _e is It is the elastic modulus of oil, _Q1 is the oil supply (oil return) flow rate of the rod cavity, and _Q2 is the oil return (oil supply) flow rate of the rodless cavity.) The calculation formula for _Q1 and _Q2 is ,

and
(however,

is the flow rate gain, and the formula for s(Γ(x _v )) is

and
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 slope of the slide valve, and ρ is the density of the oil. , x _v is the displacement of the servo valve spool. )
Assumption 2: The internal leakage coefficient C _t of the uncertainty parameter is bounded, i.e.

and

and the internal leakage coefficient C _e of the uncertainty parameter is bounded, i.e.

and

and the uncertainty parameter oil elastic modulus β _e is bounded, i.e.

and

Then,
In the operation of a hydraulic actuator, in practice, the displacement x _v of the spool of the servovalve is controlled by the voltage or current input u to further obtain the required corresponding force, and the dynamic characteristics of the servovalve are as follows: shown,

(τ is the time constant of the servovalve dynamic model and u(t) is the current input.)
Herein, considering the displacement Γ(x _v ) of the spool with an unknown dead zone, the formula is

and
(The parameters m _r and m _l represent the left and right slopes of the dead band characteristic curve, and the parameters b _r and b _l represent the break points of the input nonlinearity.)
Assumption 3: The parameters m _r , m _l , b _r , b _l are unknown positive numbers, m _r = m _l , and here the upper limit of the slope is m _max and the upper limit of b _r is

, and the upper limit of b _l is

The dead zone of the spool may be expressed as Γ(x _v )=m(t)x _v +d(t), and the expressions of m(t) and d(t) are shown as follows,

(Since m _r , m _l , b _r , b _l are bounded, d(t) is bounded, and the upper limit of d(t) is

shall be. )
As is well known, a servovalve is an important mechanical component in an electro-hydraulic actuator, which controls the displacement of the spool of the servovalve by current or voltage, and also controls the injection or extraction of hydraulic oil into the oil cavity, and finally , the actuator performs the corresponding movement, and obviously, in the servo-hydraulic position tracking system, the non-linear problem of the spool dead zone inevitably exists, and therefore, the negative effects of this problem should be taken into account for better system performance. Considering that it is difficult to obtain the accurate slope and spacing points of the dead zone model in practical applications, we developed a novel control with high robustness to solve this problem. A policy has been proposed;
In order to derive the position tracking system of a hydraulic asymmetric cylinder into a state-space form, the state variables are

Using the dynamic equations (1) to (6), the following equation of state with a spool dead zone can be obtained.

(however,

. )

を指定された性能関数として選択し、正規化誤差ζ_ｉを定義し、ｚ_ｉ，ｉ＝２…４に対して誤差変換を行い、変換された誤差

は

であり、
以下の仮想制御関数を選択し、

サーボ油圧システムのコントローラは、

である。 Based on the model of a single output rod servo hydraulic system, the process of designing a controller for a single output rod servo hydraulic system using a low complexity control policy 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)
(However, α ₁ , α ₂ , α ₃ are the virtual control variables 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 main power output by the actuator, z ₁ is the position tracking error, and z _i , i=2...4 are the control errors.)
To solve the problem of tracking error transient and steady state, and to smoothly select the virtual control amount, four positive smooth decreasing functions are used.

is selected as the specified performance function, the normalized error ζ _i is defined, the error transformation is performed for z _i , i=2...4, and the transformed error is

teeth

and
Select the virtual control function below,

The controller of the servo hydraulic system is

It is.

Furthermore, based on the controller of the single output rod servo hydraulic system and the model of the single output rod servo hydraulic system, the process of proving the stability of the single output rod servo hydraulic system is as follows: S3- 1: Using the initial value theorem, prove that if the normalized error vector is in the period t∈[0, τ _max ), there is a maximum solution in the non-empty open set Ω _ζ , and S3-2: The normalized error vector is in the period t∈[0,τ _max ) and there is _a maximum solution in the non-empty open set Ω _ζ S3-3: Using the initial value proposal, if τ _max = +∞ in S3-2, it is still accurate that all closed-loop signals are bounded. prove In order to smoothly complete the proof of controller stability, a lemma is proposed and the correctness of the lemma is proven.

であり、
ａが有界であり（すなわち∃Ａ＞０，｜ａ｜＜Ａ）、

である場合、ｘが必ず有界であり、

であり、

であるため、

であり、
｜ｘ｜＞Ａの場合、

であるため、

であることが証明され、
位置誤差、及び誤差変数（１７）と正規化誤差ζ_ｉ（１８）との関係に基づいて、ｘ_１、ｘ_２、Ｆ、ｘ_５を、ｘ_１＝ζ_１ρ_１＋ｘ_ｒ，ｘ_２＝ζ_２ρ_２＋α_１，Ｆ＝ζ_３ρ_３＋α_２，ｘ_５＝ζ_４ρ_４＋α_３（２４）のように書き直すことができ、
（１８）におけるζ_ｉ（ｉ＝１…４）を直接導出して、

（ただし、

）

を得ることができ、
正規化誤差ベクトルζ＝［ζ_１，ζ_２，ζ_３，ζ_４］^Ｔを定義することにより、（２５）～（２８）を以下の形態に書くことができ、

開集合

を定義する。 The formula of the inequality lemma is:

and
a is bounded (i.e. ∃A>0, |a|<A),

, then x is necessarily bounded,

and

Therefore,

and
If |x|>A,

Therefore,

It has been proven that
Based on the position error and the relationship between the error variable (17) and the normalized error ζ _i (18), x ₁ , x ₂ , F, x ₅ are calculated 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),
Directly derive ζ _i (i=1...4) in (18),

(however,

)

you can get
By defining the normalized error vector ζ = [ζ ₁ , ζ ₂ , ζ ₃ , ζ ₄ ] ^T , (25) to (28) can be written in the following form,

open set

Define.

を満たすことにより、

を導出することができるため、式（２９）に対してζ（０）∈Ω_ζであり、また、所望の追跡奇跡ｘ_ｒ、性能関数ρ_ｉ（ｔ），ｉ＝１…４、中間制御信号α_ｉ，ｉ＝１…３及び制御率ｕが連続的で滑らかに導出可能であり、スプールの不感帯の位置追跡システムの状態方程式における動力学変数が連続的に導出可能な関数であるため、式の正規化誤差ベクトルの関数Ｌ（ｔ，ζ）は、時間ｔに関して区分的に連続し、開集合Ω_ζでは、ζはリプシッツ条件を満たし、初期値定理における条件がいずれも満たされるため、期間［０，τ_ｍａｘ）で、正規化誤差ベクトルには唯一の最大解ζ（ｔ）∈Ω_ζがあり、∀ｔ∈［０，τ_ｍａｘ）に対して、いずれもζ_ｉ（ｔ）∈Ω_ζ，ｉ＝１…４を確保できる。 Furthermore, using the initial value theorem, if the normalized error vector is in period t∈, the process to prove that there is a maximum solution for a non-empty open set Ω _ζ is as follows: Ω _ζ is a non-empty open set and , when choosing the performance function ρ _i ,

By satisfying

can be derived, so ζ(0)∈Ω _ζ for equation (29), and the desired tracking miracle x _r , performance function ρ _i (t), i=1...4, intermediate control Since the signal α _i , i=1...3 and the control rate u can be derived continuously and smoothly, and the dynamic variable in the state equation of the position tracking system of the dead zone of the spool is a function that can be derived continuously, The function L(t, ζ) of the normalized error vector in the equation is piecewise continuous with respect to time t, and in the open set Ω _ζ , ζ satisfies the Lipschitz condition and all the conditions in the initial value theorem are satisfied, so In period [0, τ _max ), the normalized error vector has a unique maximum solution ζ(t)∈Ω _ζ , and for ∀t∈[0, τ _max ), any ζ _i (t)∈ Ω _ζ , i=1...4 can be ensured.

の時間に対する導関数

を求めて、

（ただし、

であり、１つの正の定数ｒ_Ｍ１があり、０＞ｒ_１＞ｒ_Ｍ１となる。）
式（１０）、（２４）及び（１９）を（３０）に代入して、

を得て、

とし、

であり、

が有界であることを考慮すると、極値定理から、ａ_１が有界であることが分かり、
（３１）と（３２）とから、

を得て、
リアプノフ関数

を選択し、
（３３）と（３４）とは、不等式補助定理の形態を満たし、不等式補助定理から分かるように、ａ_１が有界∃Ａ_１＞０であり、｜ａ_１｜＜Ａ_１、０＜Δ_１（ｔ）＜ｒ_Ｍ１ｋ_１となるため、

であり、

が有界であり、（１３）と（１４）とから

を得ることができるため、∀ｔ∈［０，τ_ｍａｘ）に対して、追跡誤差ｚ_１が指定された境界内で減衰され、従って、期間ｔ∈［０，τ_ｍａｘ）での追跡誤差ｚ_１が指定された性能境界内で収束することを確保でき、指令が有界正弦信号であるため、状態変数ｘ_１が有界であり、
Ｓ３－２－２：式（１８）に基づいて

の時間に対する導関数

を求めて、

（ただし、

であり、１つの正の定数ｒ_Ｍ２があり、０＞ｒ_２＞ｒ_Ｍ２となる。）
式（１０）、（２４）及び（２０）を（３５）に代入して、

を得て、

とし、

であり、

が有界であり、ｆ（ｔ）が未知の有界な外乱項であり、Ｂが有界な不確実性パラメータであることを考慮すると、極値定理から、ａ_２が有界であることが分かり、
（３６）と（３７）とから、

を得て、
リアプノフ関数

を選択し、
（３８）と（３９）とは、不等式補助定理の形態を満たし、不等式補助定理から分かるように、ａ_２が有界∃Ａ_２＞０であり、｜ａ_２｜＜Ａ_２、

となるため、

であり、

が有界であり、（１３）と（１４）とから

を得ることができるため、∀ｔ∈［０，τ_ｍａｘ）に対して、制御誤差ｚ_２が有界であり及び状態変数ｘ_２が有界であることを確保でき、
Ｓ３－２－３：式（１８）に基づいて

の時間に対する導関数

を求めて、

（ただし、

であり、１つの正の定数ｒ_Ｍ３があり、０＞ｒ_３＞ｒ_Ｍ３となる。）
式（１０）、（２４）、（１１）及び（２１）を（４０）に代入して、

を得て、

とし、
（ただし、

、ただし、Ｖ_１＝Ｖ_０１＋Ａ_１ｘ、Ｖ_２＝Ｖ_０２－Ａ_２ｘ。）、実際の問題では、アクチュエータの安全な動作を考慮し、安全マージンが残され（すなわち、物理的構造に基づいて、油圧シリンダが中央位置の付近に最大１２センチだけ上下に変動でき、与えられた指令信号が、一般的に１０センチを超えない。）、このように、ｈ_１，ｈ_２，ｈ_３が有界であることを確保でき、すなわち、それぞれ３つの正数

があり、

となり、

であり、

が有界であり、ｍ（ｔ）がスプールの不感帯の未知の傾きであり、ｍ（ｔ）が有界である（すなわち∃Ｍ_ｒ＞０，０＜ｍ（ｔ）＜Ｍ_ｒ）ことを考慮すると、極値定理から、ａ_３が有界であることが分かり、
（４１）と（４２）とから、

を得て、
リアプノフ関数

を選択し、
（４３）と（４４）とは、不等式補助定理の形態を満たし、不等式補助定理から分かるように、ａ_３が有界∃Ａ_３＞０であり、｜ａ_３｜＜Ａ_３、

となるため、

であり、

が有界であり、（１３）と（１４）とから

を得ることができるため、に対して、制御誤差ｚ_３が有界であり及び主動力ｕ_ｓが有界であることを確保でき、数学的モデルのＨ_１とＨ_２との数式から見ると、状態変数ｘ_３とｘ_４とが有界であり、いずれも油圧源Ｐ_ｓの圧力値よりも小さく、
Ｓ３－２－４：式（１８）に基づいて

の時間に対する導関数

を求めて、

（ただし、

であり、１つの正の定数ｒ_Ｍ４があり、０＞ｒ_４＞ｒ_Ｍ４となる。）
式（１０）、（２４）及び（２２）を（４５）に代入して、

を得て、

とし、

であり、

が有界であり、ｋ_４がコントローラを設計するときのオプションパラメータであることを考慮すると、ａ_４が有界であり、
（４６）と（４７）とから、

を得て、
リアプノフ関数

を選択し、
（４８）と（４９）とは、不等式補助定理の形態を満たし、不等式補助定理から分かるように、ａ_４が有界∃Ａ_４＞０であり、｜ａ_４｜＜Ａ_４、

となるため、

であり、

が有界であり、（１３）と（１４）とから

を得ることができるため、∀ｔ∈［０，τ_ｍａｘ）に対して、制御誤差ｚ_４が有界であり及び状態変数ｘ_５が有界であることを確保できる。 Furthermore, if the normalized error vector is in the period t∈[0,τ _max ) and there is a maximum solution in the non-empty open set Ω _ζ , then the _virtual controller and the controller of the servo-hydraulic system The process that can ensure that the closed-loop signal is bounded for is as follows:
S3-2-1: Based on formula (18)

derivative with respect to time

In search of

(however,

, and there is one positive constant r _M1 such that 0>r ₁ >r _M1 . )
Substituting equations (10), (24) and (19) into (30),

obtained,

year,

and

Considering that is bounded, we know from the extreme value theorem that a ₁ is bounded,
From (31) and (32),

obtained,
Lyapunov function

Select
(33) and (34) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₁ is bounded ∃A ₁ >0, and |a ₁ |<A ₁ , 0<Δ ₁ (t)<r _M1 k ₁ , so

and

is bounded, and from (13) and (14)

can be obtained, so that for ∀t∈[0,τ _max ), the tracking error z ₁ is attenuated within the specified bounds, and thus the tracking error z in the period t∈[0,τ _max ) ₁ can be ensured to converge within the specified performance bounds, and since the command is a bounded sinusoidal signal, the state variable x ₁ is bounded,
S3-2-2: Based on formula (18)

derivative with respect to time

In search of

(however,

, and there is one positive constant r _M2 such that 0>r ₂ >r _M2 . )
Substituting equations (10), (24) and (20) into (35),

obtained,

year,

and

Considering that is bounded, f(t) is an unknown bounded disturbance term, and B is a bounded uncertainty parameter, it follows from the extreme value theorem that a ₂ is bounded. I understand,
From (36) and (37),

obtained,
Lyapunov function

Select
(38) and (39) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₂ is bounded ∃A ₂ >0, and |a ₂ |<A ₂ ,

Therefore,

and

is bounded, and from (13) and (14)

Since we can obtain ∀t∈[0,τ _max ), we can ensure that the control error z ₂ is bounded and the state variable x ₂ is bounded,
S3-2-3: Based on formula (18)

derivative with respect to time

In search of

(however,

, and there is one positive constant r _M3 , such that 0>r ₃ >r _M3 . )
Substituting equations (10), (24), (11) and (21) into (40),

obtained,

year,
(however,

, where V ₁ =V ₀₁ +A ₁ x, V ₂ =V ₀₂ -A ₂ x. ), in a practical problem, a safety margin is left to consider the safe operation of the actuator (i.e., based on the physical structure, the hydraulic cylinder can move up or down by up to 12 cm around the central position, and given ), thus we can ensure that h ₁ , h ₂ , h ₃ are bounded, i.e. each has three positive numbers.

There is,

Then,

and

is bounded, m(t) is the unknown slope of the dead zone of the spool, and m(t) is bounded (i.e. ∃M _r >0,0<m(t)<M _r ). Considering this, we can see from the extreme value theorem that a ₃ is bounded,
From (41) and (42),

obtained,
Lyapunov function

Select
(43) and (44) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₃ is bounded ∃A ₃ >0, and |a ₃ |<A ₃ ,

Therefore,

and

is bounded, and from (13) and (14)

Since we can obtain , we can ensure that the control error z ₃ is bounded and the main power _us is bounded, and from the viewpoint of the formulas of H ₁ and H ₂ of the mathematical model, , state variables x ₃ and x ₄ are bounded, both are smaller than the pressure value of the hydraulic source P _s ,
S3-2-4: Based on formula (18)

derivative with respect to time

In search of

(however,

, and there is one positive constant r _M4 such that 0>r ₄ >r _M4 . )
Substituting equations (10), (24) and (22) into (45),

obtained,

year,

and

Considering that is bounded and k ₄ is an optional parameter when designing the controller, a ₄ is bounded,
From (46) and (47),

obtained,
Lyapunov function

Select
(48) and (49) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₄ is bounded ∃A ₄ >0, and |a ₄ |<A ₄ ,

Therefore,

and

is bounded, and from (13) and (14)

Since we can obtain ∀tε[0, τ _max ), we can ensure that the control error z ₄ is bounded and the state variable x ₅ is bounded.

が有界であり、すなわち

であることが分かり、式（２３）から、

が空でない集合であり、

が明らかになることが分かり、従って、τ_ｍａｘ＜＋∞を仮定すると、初期値の提案では１つの時定数ｔ_１∈［０，τ_ｍａｘ）があることが強調され、

となり、矛盾を顕著に引き起こす。従って、得られた正確な仮定がτ_ｍａｘ＝＋∞であるため、いずれかのｔ∈［０，＋∞）に対して、すべての閉ループシステム信号が有界であり、追跡誤差が指定された境界内で収束することを確保できる。 Furthermore, using the initial value proposal, the above process of proving that it is still accurate that all closed-loop signals are bounded when r _max = +∞ in S3-2 is as follows: from,

is bounded, i.e.

It turns out that, from equation (23),

is a non-empty set,

It turns out that, assuming τ _max <+∞, the initial value proposal emphasizes that there is one time constant t ₁ ∈[0, τ _max );

This causes a noticeable contradiction. Therefore, since the correct assumption obtained is that τ _max =+∞, for any t∈[0,+∞) all closed-loop system signals are bounded and the tracking error is specified Convergence within the boundaries can be ensured.

に対して、τ_ｍａｘ＜∞の場合、時定数ｔ_１∈［０、τ_ｍａｘ）があり、

となる。 Considering the initial value problem,
ζ(t)=ν(t, ζ), ζ(0)=ζ ₀ ∈Ω _ζ (50)
(However, ν:R+×Ω _ζ →R _n is a continuous function vector, and Ω _ζ ∈R _n is a non-empty open set.)
Definition 1: If one solution ζ(t) of the initial value problem (50) does not become suitably large, then the solution is maximal;
As for the initial value theorem, for the initial value problem (12), (1) If t>0, ν(t, ζ) satisfies the partial Lipschitz condition with respect to ζ, ν(t, ζ), and (2 )ζ(t)∈Ω _ζ , ν(t, ζ) satisfies piecewise continuity, and (3) ζ(t)∈Ω _ζ , ν(t, ζ) is locally integrable with respect to t. In some case, in the period t∈[0, τ _max ), the initial value problem (50) has one solution ζ(t)∈Ω _ζ , where τ _max >0. As a proposal for the initial value, it is assumed that the initial value theorem holds, and the maximum solution ζ(t) in the period [0, τ _max ) and the set

For τ _max <∞, there is a time constant t ₁ ∈[0, τ _max );

becomes.

であり、
ａが有界であり（すなわち∃Ａ４＞０，｜ａ｜＜Ａ）、

である場合、ｘが必ず有界であり、

である。 Furthermore, the formula of the inequality lemma is:

and
a is bounded (i.e. ∃A4>0, |a|<A),

, then x is necessarily bounded,

It is.

In order to verify the accuracy of the design of the controller of the present invention, an experimental platform was built to debug the experimental software program, and the construction of the experimental platform is mainly divided into two parts: software design and hardware circuit connection. 3(a) is a structural diagram I of the experimental platform, FIG. 3(b) is a structural diagram II of the experimental platform, and FIG. 3(c) is a structural diagram III of the experimental platform. Figure 3(d) is the structural configuration diagram IV of the experimental platform.The hardware of the experimental platform is mainly divided into three parts.The first part is the actuator mechanism, which mainly includes hydraulics such as accumulators and hydraulic pumps. A three-position, five-port servo valve with model number FD234-01K004VSX2A is used in the experimental platform of the present invention. The second part is a signal acquisition mechanism, the hardware of the signal acquisition mechanism is mainly a signal conversion board and an A/D board card, the function of the signal conversion board is to convert voltage and current signals into each other, Specifically, a current signal of 4 to 20 ma, which is a sensor signal, is converted to a voltage signal of 1 to 5 v, and a voltage signal of ±5 v is converted to a current signal of ±10 ma. The /D board card is ADT882, and its function is to realize the conversion of analog quantity continuous signal and digital quantity dispersion signal into each other. The third part is a control mechanism, and the core of the control mechanism is an industrial control computer.The experimental platform of the present invention uses an industrial computer PC104, and its functions are to realize the construction of a software platform and realize the control algorithm. It is to be. The hardware circuit connection mainly refers to the connection between each sensor signal line and the control current signal line of the servo valve and the signal conversion board, and the connection between the signal conversion board and the ADT882 board card. The software design environment is VC++ 6.0, window operating system, and the software program mainly consists of editing interface functions, arranging A/D board cards, setting interrupt programs to complete signal acquisition, and controlling control variables. Includes calculations and output.

Furthermore, we debug the experiment and optimize the parameters until the experimental results reach the expected control effect, step 1, the boundary initial value ρ _i0 of the specified boundary of our algorithm, the upper limit of the error convergence rate h _i and the upper limit value ρ _i∞ of the steady-state residual set of the convergence error, where i = 1...4, the boundary initial value ρ _i0 is made as large as possible, and the upper limit h _i of the error convergence rate is set as possible. and the upper limit value ρ _i∞ of the steady-state residual set of convergence errors is made as large as possible, and in step 2, the virtual control rate gain k ₁ , until the control rate gain k _i is adjusted to an appropriate value. Adjust k ₂ , k ₃ , k ₄ , and step 3: slowly and appropriately reduce the boundary initial value ρ _i0 until the expected control effect is reached, increase the upper limit h _i of the error convergence rate, and reduce the convergence error. The upper limit value ρ _i∞ of the steady-state residual set of is decreased.

FIG. 4 is a simulation curve diagram of error convergence under the action of the controller of the present invention. From FIG. 4, it can be seen that the controller of the present invention can ensure the convergence speed and control accuracy of the displacement tracking error, and FIG. FIG. 5 is a simulation curve diagram of tracking error convergence under the action of a disturbance, and FIG. 5 shows displacement tracking of the controller of the present invention and the backstepping controller under the action of a strong disturbance f(t) = 9000 sin (10t) of the system model. A comparison curve of the error is shown, and from the comparison of Fig. 4 and Fig. 5, it can be seen that the presence of strong disturbance is greatly influenced by the control effect of the backstepping controller, which can make the displacement tracking error larger and the fluctuation frequency higher. I understand. However, from FIGS. 4 and 5, it can be seen that the controller of the present invention has a strong suppressing effect against strong disturbances, and can still ensure performance in transient and steady state convergence errors.

6 is a simulation curve diagram of error convergence under the action of an unknown spool dead zone, and FIG. 7 is a simulation curve diagram of spool displacement under the action of an unknown spool dead zone. , it is verified that the low-complexity control scheme proposed herein is highly robust when handling the unknown spool deadband. The unknown nonlinearity of the deadband of the spool is added to the system model, and in order to better embody the unknown nonlinearity of the deadband of the spool, the slope of the time-varying deadband model m(t)=1+0. 3sin(2t), the discontinuous point bl=br=0.3|sin(2t)| of the time-varying dead zone model is used, and from FIG. It can be seen from FIG. 6 that although there is an unknown spool dead zone problem in the system, the controller of the present invention can still ensure a good tracking error convergence speed and steady-state accuracy.

FIG. 8 is a comparison simulation curve diagram of error convergence between the control method of the present invention and an adaptive backstepping control method (SPPFBSA) with an unknown dead zone of the spool that satisfies the specified performance. Fig. 8 is a statistical diagram of the simulation calculation time and error adjustment time between the control method and SPPFBSA, and it can be seen from Fig. 8 that both of the two types of controllers can ensure the convergence speed of the displacement tracking error and the accuracy of the steady state. From FIG. 9, it can be seen that the controller of the present invention reduces the simulation execution time by 91.7% and the error convergence adjustment time by 95.5% compared to the SPPFBSA controller. The controller design of the present invention has low model dependence, requires less computation, and does not require online learning, making real-time control easier compared to algorithms developed based on backstepping adaptation. Yes, it is easy to realize in engineering.

FIG. 10 is a comparison experimental diagram of displacement tracking between the control method of the present invention and PID when the command signal is TS=3s:x _r =100sin((2*pi*t)/3)mm. 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, and FIG. FIG. 13 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when TS=2s:x _r =100 _sin (pi * t) mm, and FIG. 14 is an experimental diagram of the tracking error of the control method of the present invention when the command signal is 100 sin (pi * t) mm, and _FIG . 14 is a diagram when the command signal is TS = 1 s: FIG. 15 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when the command signal is TS=1s:x _r =50 sin (2*pi*t) mm. Fig. 16 is an experimental diagram of the tracking error of the control method of the present invention, and Fig. 16 is a comparison of displacement tracking between the control method of the present invention and PID when the command signal is TS = 2s:x _r = 100 sin (pi * t) mm. FIG. 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 =100 sin (pi*t) mm, and FIG. 18 is an experimental diagram of the tracking error when the command signal is 19 is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID when TS=2s:x _r =80 _sin (pi * t) mm, and FIG. 20 is an experimental diagram of the tracking error of the control method of the present invention when the command signal is 80 sin (pi * t) mm, and _FIG . 20 is an experimental diagram of the tracking error when the command signal is TS = 2s: , is a comparative experimental diagram of displacement tracking between the control method of the present invention and PID, and FIG _. 21 shows the comparison experiment of the control method of the present invention when the command signal is TS = 2s: 11, FIG. 13, FIG. 15, FIG. 17, FIG. 19, and FIG. 21, regardless of how the frequency and amplitude of the command signal change, The controller can ensure that the tracking error converges within specified performance boundaries, and thus the controller of the present invention can ensure the speed of tracking error convergence and steady state accuracy. As can be seen from any of FIGS. 10, 12, 14, 16, 18 and 20, the displacement tracking curve obtained by the controller of the present invention has almost a small phase lag in the cylinder rising process; When the cylinder descends compared to the cylinder's ascending process, the phase delay of the displacement tracking curve is large; however, regarding the control effect of the controller of the present invention, in the entire displacement tracking process, the displacement tracking curve is The degree of phase delay is smaller than that of the displacement tracking curve.

Combining FIGS. 10 to 13, the higher the frequency of the command signal, the higher the phase lag of the PID tracking curve, and the greater the amplitude attenuation. In the steady state, the control effect of the controller of the present invention is that in the cylinder rising process, the phase lag degree basically does not change, and in the cylinder descending process, the phase lag degree becomes smaller and the amplitude basically attenuates. Not done. Although the higher the frequency of the command signal, the greater the PID tracking error, the tracking error of the controller of the present invention remains essentially unchanged and converges within specified performance boundaries.

Combining FIGS. 17, 19, and 21, it can be seen that the higher the frequency of the command signal, the larger the PID tracking error, but the tracking error of the controller of the present invention basically remains unchanged and remains within the specified performance boundaries. Converge.

Although the above are only preferred specific embodiments of the present invention, the protection scope of the present invention is not limited thereto. Any equivalent substitutions or changes made based on the solution and its inventive concept should be included within the protection scope of the present invention.

Claims (7)

A low complexity control method for an asymmetric servohydraulic position tracking system, the method comprising:
Step S1 of establishing a model of a single output rod servo-hydraulic system;
S2: designing a controller for the single output rod servo hydraulic system using a low complexity control policy based on the model of the single output rod servo hydraulic system;
certifying the stability of the single output rod servo hydraulic system based on the controller of the single output rod servo hydraulic system and the model of the single output rod servo hydraulic system ;
Said step S1 of establishing a model of a single output rod servo-hydraulic system is as follows:
Established a dynamic model of a hydraulic cylinder based on Newton's second law,
(where f(t) represents various interferences, 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 For the 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 large cavities of the hydraulic cylinders and the small (A ₁ and A ₂ are the effective areas of the piston in the large cavity and the small cavity.)
Using a 3-position 5-port servo valve, the dynamics of the load pressure is expressed by the following formula:
(In the formula, V ₁ and V ₂ are the volumes of the rod cavity and rodless cavity, respectively, V ₁ = V ₀₁ + A ₁ x, V ₂ = V ₀₂ - _{A 2} x, and V ₀₁ and V ₀₂ are the volumes of the rodless cavity and rod cavity when the piston is in the initial position, C _t is the internal leakage coefficient of the hydraulic cylinder, C _e is the external leakage coefficient of the hydraulic cylinder, and β _e is the elastic modulus of the oil, Q1 _is the flow rate of hydraulic oil in the rod cavity, and Q2 _is the flow rate of hydraulic oil in the rodless cavity.)
however,
(however,
is the flow rate gain, and the formula for s(Γ(x _v )) is
and
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 slope of the slide valve, and ρ is the density of the oil. , x _v is the spool of the servo valve. )
Controlling the displacement x _v of the spool of the servovalve by voltage or current input u to further obtain the required corresponding force, the dynamic characteristics of the servovalve are given as follows:
(τ is the time constant of the servovalve dynamic model and u(t) is the current input.)
Considering the displacement Γ(x _v ) of the spool with an unknown dead zone , the formula is
and
(The parameters m _r and m _l represent the left and right slopes of the dead zone characteristic curve, and the parameters b _r and b _l represent the break points of the input nonlinearity.)
The dead zone of the spool is expressed as Γ(xv ₎ =m(t)xv ₊ d(t), and the formulas of m(t) and d(t) are shown as below,
(Since m _r , m _l , b _r , and b _l are bounded, d(t) is bounded, and the upper limit of d(t) is
shall be. )
state variables
Obtain the equation of state for the position tracking system below with an unknown spool deadband defined as
(however,
. ), a low complexity control method for an asymmetric servohydraulic position tracking system. The step S2 of designing a controller for a single output rod servo hydraulic system using a low complexity control policy based on the model of the single output rod 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)
(However, α ₁ , α ₂ , α ₃ are virtual control variables, 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 output signal of the actuator. z ₁ is the position tracking error, and z _i , i=2...4 are the control errors.)
4 positive smooth decreasing functions
is selected as the specified performance function, the normalized error ζ _i is defined, the error conversion is performed for z _i , i=1...4, and the converted error is
teeth
and
Select the virtual controller below and
The controller of the servo hydraulic system is
A low complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 1. Said step S3 of proving the stability of the single output rod servo hydraulic system based on the controller of the single output rod servo hydraulic system and the model of the single output rod servo hydraulic system,
Step S3-1, using the initial value theorem, to prove that if the normalized error vector is in the period t∈[0, τ _max ), there is a maximum solution in the non-empty open set Ω _ζ ;
If the normalized error vector is in the period t∈[0,τ _max ) and there is a maximum solution in the non-empty open set Ω _ζ , then the virtual controller and the controller of the servo-hydraulic system _are On the other hand , step S3-2 can ensure that the closed loop signal is bounded;
using the initial value proposal to prove that it is still accurate that all closed-loop signals are bounded if τ _max =+∞ in S3-2 ; A low complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 1. In the step of using the initial value theorem to prove that if the normalized error vector is in the period t∈[0, τ _max ), there is a maximum solution in the non-empty open set Ω _ζ ,
A non-empty open set Ω _ζ is, when choosing a performance function ρ _i ,
By satisfying
can be obtained, the normalized error vector is ζ(0)∈Ω _ζ , the desired tracking miracle 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 can be derived continuously and smoothly, and the dynamic variables in the state equation of the position tracking system for the dead zone of the spool are functions that can be derived continuously, so the normalization error of the equation The vector function L(t, ζ) is piecewise continuous with respect to time t, and in the open set Ω _ζ , ζ satisfies the Lipschitz condition and all conditions in the initial value theorem are satisfied, so the period [0, τ _max ), the normalized error vector has a unique maximum solution ζ(t)∈Ω _ζ , and for ∀t∈[0,τ _max ), both ζ _i (t)∈Ω _ζ , i= 4. The low complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 3, characterized in that it is possible to ensure 1...4. If the normalized error vector is in the period t∈[0,τ _max ) and there is _a maximum solution in the non-empty open set Ω _ζ , then the virtual controller and the controller of the servo-hydraulic system will The step S3-2 that can ensure that the closed loop signal is bounded is as follows:
S3-2-1: Based on formula (18)
derivative with respect to time
In search of
(however,
, and there is one positive constant r _M1 such that 0>r ₁ >r _M1 . )
moreover,
you can get
year,
and
Considering that is bounded, we know from the extreme value theorem that a ₁ is bounded,
From (31) and (32),
obtained,
Lyapunov function
Select
(33) and (34) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₁ is bounded, that is, ∃A ₁ >0, and |a ₁ |<A ₁ , 0<Δ ₁ (t)<r _M1 k ₁ , so
and
is bounded,
can be obtained, so that for ∀t∈[0,τ _max ), the tracking error z ₁ is attenuated within the specified bounds, and the tracking error z ₁ in the period t∈[0,τ _max ) is Ensures convergence within the specified performance bounds, and since the command is a bounded sinusoidal signal, the state variable x ₁ is bounded,
S3-2-2: Based on formula (18)
derivative with respect to time
In search of
(however,
, and there is one positive constant r _M2 such that 0>r ₂ >r _M2 . )
moreover,
you can get
year,
and
Considering that is bounded, f(t) is an unknown bounded disturbance term, and B is a bounded uncertainty parameter, it follows from the extreme value theorem that a ₂ is bounded. I understand,
From (36) and (37),
obtained,
Lyapunov function
Select
(38) and (39) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₂ is bounded, that is, ∃A ₂ >0, and |a ₂ |<A ₂ ,
Therefore,
and
is bounded,
Since we can obtain, for ∀t∈[0,τ _max ), we ensure that the control error z ₂ is bounded and the state variable x ₂ is bounded,
S3-2-3: Based on formula (18)
derivative with respect to time
In search of
(however,
, and there is one positive constant r _M3 , such that 0>r ₃ >r _M3 . )
Substituting equations (10), (24), (11) and (21) into (40),
obtained,
year,
(
, 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 actuators, a safety margin is left, i.e., based on the physical structure, the hydraulic cylinder can move up and down by a maximum of 12 cm around the center position, and the amplitude of the command signal is 10 cm. centimeter or less and ensure that h ₁ , h ₂ , h ₃ are bounded, i.e. each of three positive numbers
There is,
Then,
and
is bounded, m(t) is the unknown slope of the dead zone of the spool, and m(t) is bounded (i.e. ∃M _r >0,0<m(t)<M _r ). Considering this, we can see from the extreme value theorem that a ₃ is bounded,
From (41) and (42),
obtained,
Lyapunov function
Select
(43) and (44) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₃ is bounded, that is, ∃A ₃ >0, and |a ₃ |<A ₃ ,
Therefore,
and
is bounded,
Since _{we can get} and the state variables x ₃ and x ₄ are bounded, both are smaller than the pressure value of the hydraulic source P _s ,
S3-2-4: Based on formula (18)
derivative with respect to time
In search of
(however,
, and there is one positive constant r _M4 such that 0>r ₄ >r _M4 . )
moreover,
obtained,
year,
and
Considering that is bounded and k ₄ is an optional parameter when designing the controller, a ₄ is bounded,
From (46) and (47),
obtained,
Lyapunov function
Select
(48) and (49) satisfy the form of the inequality lemma, and as can be seen from the inequality lemma, a ₄ is bounded, that is, ∃A ₄ >0, and |a ₄ |<A ₄ ,
Therefore,
and
is bounded,
Asymmetric servo-hydraulic position according to claim 3 , characterized in that it ensures that the control error _z4 is bounded and the state variable _x5 is bounded, for A low complexity control method for tracking systems. The formula of the inequality lemma is:
and
a is bounded (i.e. ∃A>0, |a|<A),
, then x is necessarily bounded,
A low complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 5 . Using the initial value proposal to prove that it is still accurate that all closed-loop signals are bounded if τ _max =+∞ in S3-2, the step S3-3 is as follows:
Therefore,
is a non-empty set,
and
Assuming τ _max <+∞, the initial value proposal emphasizes that there is one time constant t ₁ ∈[0, τ _max );
, which causes a contradiction and the resulting assumption is τ _max = +∞, and for any t∈[0, +∞) all closed-loop system signals are bounded and the tracking error is specified. 4. A low complexity control method for an asymmetric servo-hydraulic position tracking system according to claim 3 , characterized in that convergence within certain boundaries can be ensured.

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