CN111338209B - Electro-hydraulic servo system self-adaptive control method based on extended disturbance observer - Google Patents
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Abstract
The invention provides an electro-hydraulic servo system self-adaptive control method based on an extended disturbance observer, which comprises the following steps of firstly establishing a mathematical model of the electro-hydraulic servo system: establishing a force balance equation according to a Newton second law, and establishing a pressure dynamic equation and a state space equation according to the dynamics of a hydraulic system; then designing a mapping function and an adaptive law: designing an adaptive law to enable system parameters to be learned, and designing a mapping function to ensure that the parameters are adaptively bounded; secondly, designing an extended disturbance observer: based on an extended disturbance observation theory, observers are respectively designed for matching uncertainty and mismatching uncertainty and are learned; and finally, developing controller design: and for the learning of uncertainty, feedforward compensation is carried out, and a robust controller is designed to realize the tracking effect. The method considers the problems of parameter uncertainty and unmodeled interference in the electro-hydraulic servo system, integrates the extended interference observer based on the traditional self-adaptive control method, and realizes high-performance position control.
Description
Technical Field
The invention relates to the technical field of electro-hydraulic servo control, in particular to an adaptive control method based on an extended disturbance observer.
Background
In modern industrial production, the electro-hydraulic servo system is widely applied by virtue of the characteristics of high power density, large force or torque output and the like. Many advanced mechanical devices, such as engineering machinery, mechanical arms and automobile suspensions, adopt electro-hydraulic servo systems as actuators. With the development of modern industry, the method based on the classical linear control theory cannot meet the increasing performance requirement, so the research of the nonlinear control strategy with high performance is urgent. However, the strong non-linearity and modeling uncertainty of electro-hydraulic servo systems present many challenges to the design of high performance controllers.
To solve these problems, many advanced nonlinear control methods have been proposed in succession during the last 30 years. For example, the best choice of uncertainty in the processing parameters, adaptive control, can improve tracking performance well. Based on the method, a novel continuous micro-friction model is provided, compensation of nonlinear friction is achieved, and a parameter self-adaptive law is also fused in a backstepping design to process parameter uncertainty in the friction model. However, the tracking performance of adaptive control depends on the accuracy of the modeling, and extensive and intensive model studies have negligible reduction in modeling uncertainty. Once the unmodeled interference is too large, the convergence performance of the parameter adaptive law is influenced, and the tracking performance of the proposed adaptive controller is greatly reduced. In order to improve the robustness of the adaptive controller, an Adaptive Robust Control (ARC) strategy is provided for the single-rod hydraulic actuator, and the validity of the ARC strategy is quickly checked by a plurality of practical tests. A mapping type adaptive law ensures that parameter estimation values always fall into a bounded set, and meanwhile, the design of a robust control law is assisted. The high accuracy tracking performance of the ARC must be aided by high gain feedback gain, which however is usually avoided in engineering practice in view of measurement noise and high gain feedback problems.
In order to solve the problem of model uncertainty, a control method based on a disturbance observer is also widely researched. The Active Disturbance Rejection Controller (ADRC) can handle a variety of uncertainties, including unmodeled dynamics and various external disturbances. The essential point of the Active Disturbance Rejection Controller (ADRC) design is to use an extended state observer to estimate the concentration disturbance and to feed forward compensate. However, the Extended State Observer (ESO) can only estimate the match uncertainty, which does not contribute much to the mismatch uncertainty that is common in hydraulic systems. Recently, an Extended Disturbance Observer (EDO) based on a sliding mode controller has received much attention to solve the mismatch uncertainty problem in a nonlinear system. However, the sliding mode control law is discontinuous, which causes severe buffeting of the system around the sliding mode face. On the other hand, if the modeling uncertainty of the system is mainly caused by the parameter uncertainty, the disturbance observer may instead cause a reduction in the control performance, even less than an adaptive controller.
Disclosure of Invention
The invention aims to provide an electro-hydraulic servo system control method for simultaneously processing unknown parameters and uncertain interference.
The technical solution for realizing the purpose of the invention is as follows: an adaptive control method based on an extended disturbance observer comprises the following steps:
step 2, designing a mapping function and an adaptive law: designing an adaptive law to enable the system parameters to be learned online while controlling execution, and designing a mapping function to ensure that the parameters are self-adaptive and bounded;
step 3, expanding the design of the disturbance observer: based on the extended disturbance observation theory, observers are respectively designed for matching uncertainty and mismatching uncertainty and are learned, so that subsequent feedforward compensation is facilitated;
and 4, developing a controller design: and (3) performing feedforward compensation on the uncertain learning based on the step 2 and the step 3, designing a robust controller, stabilizing the nominal error of the system, and realizing the tracking effect.
Compared with the prior art, the invention has the following remarkable advantages:
(1) The design of the controller does not require an accurate model because uncertainty dynamics such as unmodeled nonlinear friction and external disturbances can be observed by the extended disturbance observer, while parameter uncertainties can be estimated by the adaptive law and feed forward compensated; (2) The learning burden of the observer is reduced due to the introduction of a parameter self-adaptive law, and the problem of buffeting control can be avoided; (3) The closed-loop system stability analysis based on the Lyapunov function shows that the proposed controller can ensure accurate transient tracking performance, and the range of global tracking error can be ensured to be small arbitrarily by adjusting control parameters. The simulation results verify the superior tracking performance of the proposed control strategy.
Drawings
FIG. 1 is a schematic diagram of an electro-hydraulic servo system of the present invention;
FIG. 2 is a schematic diagram of the principle of an adaptive control method of an electro-hydraulic servo system based on an extended disturbance observer;
FIG. 3 is the tracking effect of ACEDO;
FIG. 4 is a comparison of tracking errors for AC and ACEDO;
FIG. 5 is an estimate of ACEDO for matched and uncontained interference;
FIG. 6 is a comparison of control inputs for ACEDO and AC;
fig. 7 is the effect of parameter estimation of the ACEDO.
FIG. 8 is a parameter estimation effect of AC;
Detailed Description
The invention is further described in detail below with reference to the drawings and specific embodiments.
With reference to fig. 1-2, the adaptive control method of the electro-hydraulic servo system based on the extended disturbance observer of the invention comprises the following steps:
step 1.1, establishing a force balance equation and a pressure dynamic equation
The electrohydraulic servo system considered by the invention is a system for driving an inertial load by a throttling double-rod hydraulic cylinder with a servo valve. Considering that the dynamic time constant of the valve core is much smaller than the mechanical time constant, and the response speed of the servo valve is much higher than that of the speed ring and the position ring, the dynamic state of the valve core can be approximated to a proportional link.
Therefore, according to newton's second law, the equation of motion of the direct drive motor system is:
in the formula (1), m is the load mass, A f Is the Coulomb coefficient of friction, S f Is a shape function fitting Coulomb friction, B is the viscous friction coefficient, f (t) is other unmodeled disturbances, x p Position of inertial load, P L ＝P 1 –P 2 In which P is 1 And P 2 The pressures of an oil supply cavity and an oil return cavity of the hydraulic cylinder are respectively, A is the effective collision area of the piston, and t is a time variable;
neglecting the external leakage of the cylinder, the cylinder pressure dynamics can be expressed as:
V t is the total volume of the oil supply chamber and the oil return chamber, beta e Is the effective elastic modulus of oil, C t For internal leakage coefficients, Q (t) is the model error due to factors such as complex leakage problems and unmodeled dynamics, Q L ＝(Q 1 +Q 2 ) /2 is the load flow, where Q 1 And Q 2 Respectively the flow of the oil supply cavity and the oil return cavity. Load flow rate Q L And the position of the spool is
Wherein,
is the flow gain, C d Is the flow coefficient, w is the slide valve area gradient, rho oil density, P s And P r Pressure of oil supply and pressure of oil return, x v For the displacement of the valve core, sign (#) is a sign function and can be defined as
The present invention takes into account a servo valve, so that the valve core dynamics can be approximated as a proportional link x v ＝k i u,k i Is the scaling factor, u is the control input, therefore equation (3) can be converted to
Wherein k is t ＝k q k i Is the total gain.
Step 1.2, rewriting the model of the electro-hydraulic servo system into a state space equation
Defining a state variable:
wherein x is 1 Is the load position, x 2 Is the speed of movement of the load, x 3 Is the force output by the hydraulic cylinder, the equations (1) to (5) are converted into a state equation:
in the formula, B m ＝B/m,A fm ＝A f /m,d 1 (t)＝f(t)/m,d 2 (t)＝4Aβ e Q(t)/m/V t ,
Given a desired command signal x 1d (t) the design goal is to derive a bounded input u such that the system outputs x 1 Tracing instruction x as much as possible 1d (t).
To facilitate the design of the adaptive law, define θ = [ θ = ] 1 θ 2 θ 3 θ 4 θ 5 ] T ,θ 1 ＝B m ,θ 2 ＝A fm ,θ 3 ＝4Aβ e k t /m/V t ,θ 4 ＝4A 2 β e /m/V t ,θ 5 ＝4β e C t /V t Then, the state equation can be expressed as:
in the formula,
step 2, designing a mapping function and an adaptive law: an adaptive law is designed to enable system parameters to be learned, and a mapping function is designed to ensure that the parameters are self-adaptively bounded. The method comprises the following specific steps:
step 2.1, setting reasonable hypothesis based on actual engineering experience
For the controller design, assume the following:
assume that 1: expected instruction x 1d (t) three-order continuous micro-pressure, the pressure of the oil supply cavity satisfies P 1 <P s The pressure intensity of the oil return cavity satisfies P 2 <P s Pressure difference satisfies | P L |＜＜P s To ensure the function g (u, x) 3 ) Away from zero.
Assume 2: the unknown parameter estimate is bounded:
wherein, theta min ＝[θ 1min ,θ 2min ,θ 3min ,θ 4min ,θ 5min ] T ,θ max ＝[θ 1max ,θ 2max ,θ 3max ,θ 4max ,θ 5max ] T Wherein, θ imin And theta imax (i =1,2 \82305; 5) are respectively the parameters theta i A maximum value and a minimum value of (c).
Assume that 3: unmodeled dynamics and their derivatives are continuous and bounded:
wherein,
for the various derivatives of the dynamics being modeled, delta 1 And delta 2 They are all normal numbers.
Step 2.2, defining a discontinuous mapping function, and ensuring the self-adaptive boundedness of parameters:
step 2.3, designing an adaptive law based on discontinuous mapping:
f is a positive diagonal matrix and determines the adaptive rate; τ is an adaptive function that is subsequently designed.
Any adaptive function τ, under the influence of the mapping function, satisfies the following theorem:
(P1)
(P2)
step 3, expanding the design of the disturbance observer: based on the extended disturbance observation theory, observers are respectively designed for matching uncertainty and mismatching uncertainty and are learned. The details are as follows
Step 3.1, mismatch uncertainty Δ 1 (t) learning
According to the formula (9), firstly, an EDO observation delta of an extended disturbance observer is designed 1 (t)
Step 3.2, matching uncertainty Δ 2 (t) learning
Design extended disturbance observer EDO Observation Delta 2 (t)
And 4, developing a controller design: and (3) performing feedforward compensation on the uncertain learning based on the step 2 and the step 3, and designing a robust controller to realize a tracking effect. The method comprises the following specific steps:
due to mismatch uncertainty in the state equationsA reverse design method is adopted. Definition of z 1 ＝x 1 -x 1d Is the tracking error of the system.
Step 4.1, speed virtual control law design
Wherein alpha is 1 Is a velocity state x 2 The virtual control law of (2). The step-back error between the velocity virtual control law and the velocity state is z 2 ＝x 2 -α 1 ,k 1 And adjustable gain is more than 0.
Step 4.2, design of force virtual control law
Error z of velocity back stepping 2 Calculating time derivative
Wherein alpha is 2 Is in a force state x 3 Virtual control law of (c), z 3 ＝x 3 -α 2 Force step error. Virtual control law alpha according to equation (22) in combination with the results of the extended disturbance observer 2 Can be designed as
Wherein k is 2 For positive feedback gain, alpha 2a Compensating terms for the adaptive model to improve the tracking performance of the system, α 2s The robust feedback term is used for stabilizing the nominal error of the hydraulic system.
Step 4.3, final control input design
Substituting (23) into (22) to obtain
It is easy to know from equation (24) if the force is wrong in reverseDifference z 3 =0, the output will remain stable. The goal of this step is therefore to design the true control input u such that the force backstepping error z 3 As small as possible. Note that equation (9), force step back error z 3 Can be represented as
And design force virtual control law alpha 2 Similarly, the control input u can be designed as
Wherein k is 3 Is a positive feedback gain, u a For the model compensation term, high-precision tracking is realized by means of the online parameter adaptive law given in (13), u s The method is used for stabilizing the error of the hydraulic system for a robust feedback item.
And 5, verifying the stability of the designed controller. And (4) carrying out stability verification on the control effect of the hydraulic servo system by applying the Lyapunov stability theory. The method comprises the following specific steps:
step 5.1, defining the estimation error of the observer:
definition of
Are each Δ 1 And
the estimation error of (2) can be obtained from (9), (16) and (17)
Two-sided subtraction
To obtain
According to (9), (18) and (19), the compounds are obtained
The estimation error dynamics of the extended disturbance observers (16) to (19) can be expressed as
Wherein,
similarly, the equation of motion for EDO (20)
Wherein,
is referred to as Δ 2 And
the error of the estimation of (2) is,
selecting a parameter l 11 ,l 12 ,l 21 And l 22 Let D be 1 And D 2 Satisfying the Helveltz condition, there must therefore be two positive definite matrices satisfying the Lyapunov equation
Wherein Q 1 And Q 2 Is any given positive definite matrix, and the observation error will be at the time T based on the stability analysis of EDO f Converge to | | ε 1 ||≤γ 1 ,||ε 2 ||≤γ 2 ,γ 1 And gamma 2 Is a sufficiently small positive number.
Substituting (26) into (25) to obtain
Step 5.2, liapunov stability verification
For the sake of verification, the following theorem is introduced: selecting an adaptive function based on a mapping adaptation law (13)
Wherein
At the same time, select the control gain k 1 ,k 2 ,k 3 Make the matrix
The proposed controller thus allows all closed loop signals to be bounded by a positive definite matrix. In addition, a positive definite Lyapunov function can be guaranteed
Is limited by
Wherein,
and λ min And (Λ) represents the minimum eigenvalue of the matrix Λ.
Evidence: according to equations (21), (24) and (33), the time derivative of V is
Note that the matrix Λ is positive, and the upper bound of equation (37) is
As can be seen from the principle of contrast, (36) is bounded. From this, V ∈ L is known ∞ I.e. z 1 ,z 2 ,z 3 Is bounded. And because of the use of the mapping function, the parameter estimation
Is bounded, according to assumption 2, error of parameter estimation
Is also bounded, and is liable to be verified for state x 1 ,x 2 ,x 3 Are bounded. It can be seen that the final control input u is bounded.
Examples
In order to evaluate the performance of the designed controller, the following parameters are taken in simulation to model the electro-hydraulic servo system:
load m =40kg, viscous friction coefficient B =80N · s/m, cylinder piston effective area a =2 × 10 -4 m 2 Coulomb friction value A f =10N · s/m, oil elastic modulus beta e ＝2×10 8 Pa, internal leakage coefficient C t ＝9×10 -12 m 5 /(N · s), input gain
Total volume V of oil supply cavity and oil return cavity of hydraulic cylinder t ＝2×10 -3 m 3 Pressure P of oil supply s =21MPa, oil return pressure P r ＝0Pa.
Continuous shape function S f (x 2 )＝arctan(100x 2 ) The parameter truth value to be estimated θ = [2,0.25,0.08,400,3.6] T The boundary of the predicted parameter estimation is theta max ＝[5,1,0.2,500,10] T And θ min ＝[0,0,0.05,0,0] T The initial value of the parameter estimation is selected as
Satisfying the condition of equation (10), and being far from the true value, is enough to verify the validity of the parameter adaptation law. Interference of matching and mismatching is set as f (t) =200sint N and Q (t) =2 × 10 -5 sint m 2 The sample offset is set to 0.2ms.
The expected instruction for a given system is x 1d (t)＝0.2sint[1-exp(-0.01t 3 )]m.
(1) ACEDO: the method disclosed by the invention takes the control gain as k 1 ＝100,k 2 ＝10,k 3 =50. The extended disturbance observer parameter is l 11 ＝200,l 12 ＝50,l 21 ＝50,l 22 =2, the adaptive gain matrix is f = diag {30,1,2.2 × 10 -5 ,1300,20}；
(2) AC: and (3) a classical self-adaptive control method. The difference is that there is no observation or compensation for interference. Thus, it can be used to verify the validity of the observer. To ensure fair verification, the control parameters are the same as ACEDO.
Fig. 3 is the tracking effect of the ACEDO. The comparison of the tracking error under the action of two different controllers is shown in fig. 4, from which it can be seen that the steady-state tracking error of AC is greater than 3.3 × 10 -4 m, while the steady state error of ACEDO does not exceed 3.9 x 10 -5 m, is significantly better than AC, and furthermore, as can be seen from the figure, the maximum transient error for AC is about 1.25 x 10 -3 m, and the maximum error of ACEDO is 2X 10 -4 Still better than ac fig. 5 is the effect of interference estimation, it is clear that EDO effectively estimates unmodeled interference. Hence parameter estimation error
Is also small, Δ 1 (t) approaching d 1 (t) = f (t)/m however, the third channel parameter estimate has a large impact on interference estimation, which is easy to see, Δ 2 The estimated value of (t) gradually decays. Fig. 6 shows the control inputs of two controllers. It shows that the input of the ACEDO disclosed in the present invention is similar to AC, but more effect is obtained. Fig. 7 and 8 show the effect of the ACEDO and AC parameter estimation. Since EDO estimates unmodeled interference, the parameter estimation of ACEDO converges better. And the AC is influenced by unmodeled disturbance, and the convergence performance of parameter estimation is poor. Thus, the above results demonstrate that the ACEDO controller proposed by the present invention is due to AC in terms of both process parameter uncertainty and unmodeled disturbances.
Claims (3)
1. An electro-hydraulic servo system self-adaptive control method based on an extended disturbance observer is characterized by comprising the following steps:
step 1, establishing a mathematical model of the electro-hydraulic servo system: establishing a force balance equation according to a Newton second law, and establishing a pressure dynamic equation according to the dynamics of a hydraulic system; converting the force balance equation and the pressure dynamic equation into a state space equation;
step 1.1, establishing a force balance equation and a pressure dynamic equation:
the motion equation of the direct drive motor system is as follows:
where m is the loading mass, A f Is the Coulomb coefficient of friction, S f (. Cndot.) is a shape function of the fitted coulombic friction for, B is the viscous coefficient of friction, f (t) is the other unmodeled disturbance, x p Position of inertial load, P L The pressure difference of an oil supply cavity and an oil return cavity of the hydraulic cylinder is obtained; a is the effective collision area of the piston, and t is a time variable;
the cylinder pressure dynamics are represented as:
V t beta e is the total volume of the oil supply cavity and the oil return cavity, and beta e is the effective elastic modulus of oil, C t For internal leakage coefficients, Q (t) is the model error due to complex leakage problems and unmodeled dynamic factors, Q L Is the load flow;
step 1.2, rewriting a model of the electro-hydraulic servo system into a state space equation:
defining the state variables:
wherein x is 1 Is the load position, x 2 Is the speed of movement of the load, x 3 Is the force output by the hydraulic cylinder, and defines theta = [ theta ]) 1 θ 2 θ 3 θ 4 θ 5 ] T ,θ 1 ＝B m ,θ 2 ＝A fm ,θ 3 ＝4Aβ e k t /m/V t ,θ 4 ＝4A 2 β e /m/V t ,θ 5 ＝4β e C t /V t ，k t Is the total gain; intermediate variable B m ＝B/m,A fm ＝A f /m,d 1 (t)＝f(t)/m,d 2 (t)＝4Aβ e Q(t)/m/V t ,
rewritten as the following equation of state:
in the formula,
d 1 (t)＝f(t)/m,d 2 (t)＝4Aβ e Q(t)/m/V t ,
P s sign (u) is a sign function with respect to the control input u for the supply pressure;
step 2, designing a mapping function and an adaptive law: designing an adaptive law to enable the system parameters to be learned online while controlling execution, and designing a mapping function to ensure that the parameters are self-adaptive and bounded;
comprises the following steps
Step 2.1, setting hypothesis: assume that 1: expected instruction x 1d (t) three-order continuous micro-pressure, the pressure of the oil supply cavity satisfies P 1 <P s The pressure intensity of the oil return cavity meets P 2 <P s Pressure difference satisfies | P L |＜＜P s ；
Assume 2: the estimation value of the unknown parameter is bounded;
step 2.2, defining a discontinuous mapping function, and ensuring the self-adaptive boundedness of parameters:
step 2.3, designing an adaptive law based on discontinuous mapping:
r is a positive determined diagonal matrix; τ is an adaptation function;
step 3, expanding the design of the disturbance observer: based on the extended disturbance observation theory, observers are respectively designed for matching uncertainty and mismatching uncertainty and are learned, so that subsequent feedforward compensation is facilitated; the method specifically comprises the following steps:
step 3.1, not matchingUncertainty of distribution Δ 1 In the study of
Firstly, designing EDO observation delta of an extended disturbance observer 1
step 3.2, matching uncertainty Δ 2 In the study of
Design extended disturbance observer EDO Observation Delta 2
and 4, developing a controller design: performing feedforward compensation on the uncertain learning based on the step 2 and the step 3, designing a robust controller, stabilizing the nominal error of the system and realizing the tracking effect; developing a controller design specifically includes the steps of:
definition of z 1 ＝x 1 -x 1d Is the tracking error of the system; x is the number of 1d Is a desired instruction;
step 4.1, speed virtual control law design
Wherein alpha is 1 Is the load motion speed state x 2 The virtual control law of (a); the step-back error between the velocity virtual control law and the velocity state is z 2 ＝x 2 -α 1 ,k 1 Is an adjustable gain;
step 4.2, designing a force virtual control law: pseudo control law alpha 2 Is designed as
Wherein k is 2 For positive feedback gain, alpha 2a Is an adaptive model compensation term to improve the tracking performance of the system, alpha 2s Is a robust feedback term;
step 4.3, final control input design, wherein the control input u can be designed as
Wherein k is 3 In order to have a positive feedback gain, the feedback gain,u a as a model compensation term, u s For robust feedback terms to stabilize the error of the hydraulic system, z 3 Force step error.
2. The adaptive control method according to claim 1, further comprising step 5 of verifying controller stability: and (3) carrying out stability verification on the control effect of the hydraulic servo system by applying the Lyapunov stability theory.
3. The adaptive control method according to claim 2, wherein the step 5 of verifying the stability of the controller comprises the following steps:
step 5.1, defining the estimation error of the observer:
the estimation error dynamics of the extended disturbance observer can be expressed as
Wherein,
E 1 ＝[0,1] T ；
dynamic equation of extended disturbance observer EDO
Wherein,
is referred to as Δ 2 Derivative sum
The error of the estimation of (2) is,
E 2 ＝[0,1] T ；
let D be 1 And D 2 Satisfy the Hervitz condition to obtain
Step 5.2, the stability of the Lyapunov is verified:
ensuring positive definite Lyapunov function
Is limited by
Wherein λ =2 λ min (Λ),
γ 1 And gamma 2 Is a positive number, λ min (Λ) represents the minimum eigenvalue of matrix Λ;
the time derivative of V is
Note that the matrix Λ is positive, and the upper bound of equation (37) is
k 1 ,k 2 ,k 3 Representing control gain, parameter estimate
Is bounded, error of parameter estimation
Is also bounded, and is easily verified for state x 1 ,x 2 ,x 3 Are bounded and the final control input u is bounded.
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| CN116300420B (en) * | 2022-12-13 | 2024-03-15 | 南方海洋科学与工程广东省实验室(湛江) | Servo control method, system, device, terminal equipment and storage medium |
| CN115903748B (en) * | 2022-12-14 | 2024-07-26 | 燕山大学 | Electrohydraulic servo displacement tracking test device based on reference model and control method |
| CN117784622B (en) * | 2024-02-27 | 2024-05-03 | 中国矿业大学 | Second-order observer-based electrohydraulic servo system global sliding mode control method |
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