CN114625006B - Output feedback control method of high-speed large-inertia electrohydraulic position servo system - Google Patents
Output feedback control method of high-speed large-inertia electrohydraulic position servo system Download PDFInfo
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Abstract
The invention discloses an output feedback control method of a high-speed large-inertia electrohydraulic position servo system, which belongs to the technical field of electrohydraulic position servo control and comprises the following steps: step 1, establishing a mathematical model of a large inertia valve control asymmetric cylinder system; step 2: designing a corrected extended state observer and performing convergence analysis of the observer; step 3: designing a self-adaptive nonlinear backstepping controller and analyzing the stability of a closed-loop system; step 4: and adjusting the observer parameters and the controller parameters to enable the performance index of the control system to meet the actual requirements. According to the invention, prior information of a system model is not needed, the problems that the large-inertia electrohydraulic position servo system is easy to overshoot and steady-state oscillation during rapid movement are solved under the condition that only a displacement sensor is needed, and rapid and high-precision control of the large-inertia electrohydraulic position servo system is realized.
Description
Technical Field
The invention relates to the technical field of electrohydraulic position servo control, in particular to an output feedback control method of a high-speed large-inertia electrohydraulic position servo system.
Background
The electrohydraulic servo system has the characteristics of high response speed, large output force/moment, small size power ratio and the like, and is widely applied to the occasions requiring heavy load such as industry, military and the like. The research of the small inertia electrohydraulic servo system mainly expands around the directions of uncertainty of system parameters, external unknown disturbance and the like. Numerous algorithms such as active disturbance rejection control, feedback linearization, backstepping, adaptive control, etc. have been successfully applied to control of small inertia hydraulic servo systems and have achieved good control results. However, for the large-inertia electrohydraulic position servo system, the transient state and steady state response of the traditional design method cannot be well considered, so that the practical application of the large-inertia electrohydraulic position servo system is restricted.
Compared with a small-inertia electrohydraulic position servo system, the large-inertia electrohydraulic position servo system has two main characteristics. One is that in the transient response phase of the system, the contradiction between response rapidity and overshoot is particularly obvious. And secondly, in a steady state response stage of the system, the system can generate continuous and large oscillation in the steady state stage due to the influence of large load inertia and the closed loop action of the system. Reducing the control gain can reduce the amplitude of the oscillation and even eliminate the oscillation, but the response rapidity of the system is reduced, and the design target of stability, accuracy and rapidness in the design of the control system is not achieved. In addition, negative feedback of speed information and acceleration information helps to increase system damping and improve system performance. The actual speed sensor and the acceleration sensor are often easily affected by environmental noise, which is unfavorable for the realization of a control algorithm based on state feedback. Based on the above analysis and consideration, it is necessary to design an output feedback control method of a high-speed large-inertia electrohydraulic position servo system to meet the actual industrial requirements.
Disclosure of Invention
The invention aims to provide an output feedback control method of a high-speed large-inertia electrohydraulic position servo system, which aims to solve the problems that the large-inertia electrohydraulic position servo system is easy to generate large overshoot and steady-state oscillation during fast operation, thereby realizing fast and high-precision position tracking of the large-inertia electrohydraulic servo system.
In order to achieve the above purpose, the technical scheme provided by the invention is as follows:
An output feedback control method of a high-speed large-inertia electrohydraulic position servo system comprises the following steps:
Step 1: establishing a mathematical model of a large inertia valve control asymmetric cylinder system;
step 2: designing a correction-based extended state observer and performing convergence analysis of the observer;
Step 3: designing a self-adaptive nonlinear backstepping controller and analyzing the stability of a closed-loop system;
step 4: and adjusting the observer parameters and the controller parameters to enable the performance index of the control system to meet the actual requirements.
Due to the adoption of the technical scheme, compared with the prior art, the invention has the following advantages:
(1) The output feedback control method of the high-speed large-inertia electrohydraulic position servo system designed by the invention does not need prior information of any electrohydraulic position servo system model parameters when an observer and a controller are implemented, so that the practicability and portability of the invention are greatly enhanced;
(2) According to the output feedback control method of the high-speed large-inertia electrohydraulic position servo system, which is designed by the invention, by introducing a correction link to a conventional extended state observer, the gain of a high-frequency signal generated in the running process of the system is reduced, and the problem that the high-inertia electrohydraulic position servo system is easy to generate steady-state oscillation under high-speed running is solved;
(3) According to the output feedback control method of the high-speed large-inertia electrohydraulic position servo system, a nonlinear error feedback mechanism is introduced under a backstepping design framework, and a nonlinear function construction and selection principle is provided, so that the contradiction between response rapidity and overshoot existing in the operation of the large-inertia electrohydraulic position servo system is greatly relieved. Meanwhile, a self-adaptive compensation mechanism is designed aiming at the observation error and the filtering error, so that the influence of the observation error on the performance of the closed-loop system is greatly reduced;
(4) According to the invention, the state information and disturbance information in the operation of the system are observed by using the corrected extended state observer, a speed sensor and an acceleration sensor are not required, and the quick and high-precision control of the large-inertia electrohydraulic position servo system is realized under the condition of only needing a displacement sensor, so that the implementation cost is greatly reduced, and the disturbance resistance of the system is enhanced.
Drawings
FIG. 1 is a flow chart of an output feedback control method of a high-speed large-inertia electrohydraulic position servo system;
FIG. 2 is a block diagram of an output feedback control method of a high-speed large-inertia electrohydraulic position servo system;
FIG. 3 is a schematic diagram of a large inertia electro-hydraulic position servo system;
FIG. 4 is an AMESim model diagram of a large inertia electro-hydraulic position servo system;
FIG. 5 is a graph of displacement tracking error versus controller (ESO-backstepping) for a large inertia electro-hydraulic position servo system of the present invention with a conventional extended state observer and backstepping controller (CESO-ANLbackstepping);
FIG. 6 is a graph of control versus control for a controller designed for a high inertia electro-hydraulic position servo system (CESO-ANLbackstepping) in accordance with the present invention, as compared to a controller formed by a conventional extended state observer and backstepping (ESO-backstepping).
Detailed Description
The invention is further illustrated by the following examples:
Referring to fig. 1 and 2, the output feedback control method of the high-speed large-inertia electrohydraulic position servo system provided by the invention comprises the following steps:
Step 1: establishing a mathematical model of a large inertia valve control asymmetric cylinder system;
step 2: designing a correction-based extended state observer and performing convergence analysis of the observer;
Step 3: designing a self-adaptive nonlinear backstepping controller and analyzing the stability of a closed-loop system;
step 4: and adjusting the observer parameters and the controller parameters to enable the performance index of the control system to meet the actual requirements.
According to the above, in conjunction with fig. 3, the specific process of step 1 is:
step 1.1, the force equation at the piston rod of the hydraulic cylinder can be described as:
in the formula (1), m is the total mass of the load and the piston rod. x p is the actual displacement of the piston rod and it is assumed that the load displacement coincides with the piston rod displacement. P 1 and P 2 are cylinder rodless cavity pressure and cylinder rod cavity pressure, respectively. A 1 and A 2 are the effective acting area of the rodless cavity of the hydraulic cylinder and the effective acting area of the rod cavity of the hydraulic cylinder respectively. B P is the viscous friction coefficient, g represents the gravitational acceleration. F L is other unmodeled factors outside the force equation, such as friction, unknown external disturbance forces, etc. F g is the extra disturbance at the piston rod under load with large inertia.
Considering the difference in effective acting areas of the rod cavity and the rodless cavity of the asymmetric cylinder, the following deformation can be performed on the formula (1):
In the formula (2), P L represents a load pressure, and its expression is: p L＝P1-P2. Taking out Then formula (2) reduces to:
Step 1.2, neglecting the influence of external leakage, the flow equations of the rodless cavity and the rod cavity of the hydraulic cylinder can be described as:
in the formula (4), Q 1 and Q 2 respectively correspond to the flow rates at the rodless cavity and the rod cavity; c t is the internal leakage coefficient of the hydraulic cylinder; beta e is the hydraulic oil elastic modulus; v 1 and V 2 represent the real-time volumes of the rodless and the rod-containing chambers, respectively, expressed as V 1＝V10+A1xp and V 2＝V20-A2xp,V10 and V 20 are the initial volumes of the rodless and the rod-containing chambers, respectively. q 1 (t) and q 2 (t) represent unknown modeling errors generated during the modeling process.
Taking the load flow Q L＝(Q1+Q2)/2, then equation (4) can be expressed as:
In the formula (5), A P＝(A1+A2)/2 represents the average effective acting area of the two cavities; v t＝V1+V2 denotes the total effective volume of the hydraulic lever; q (t) = [ q 1(t)+q2 (t) ]/2 is the sum of unknown modeling errors.
Step 1.3, considering the high-frequency response characteristic of the servo valve, the load flow equation at the servo valve can be expressed as:
QL＝kvu+q3(t) (6)
in equation (6), k v represents the flow gain at the servo valve, u is the control input signal at the servo valve, and q 3 (t) represents the modeling error.
Step 1.4, defining a state vector based on the formulas (3), (5) and (6)Taking out The state space description of the large inertia valve controlled asymmetric cylinder can be expressed as:
according to the mathematical model of the large inertia valve control asymmetric cylinder obtained in the step 1, the specific process of the step 2 is as follows:
step 2.1, designing a conventional extended state observer;
2.2, correcting disturbance information obtained by observation of a conventional extended state observer;
and 2.3, analyzing the convergence of the observer.
The specific process of the step 2.1 is as follows:
taking f= -a 1x2-a2x3+d+(b-b0) u for the mathematical model as expressed by equation (7), equation (7) can be restated as:
suppose 1: f is bounded and steerable and its derivative satisfies:
When assuming that 1 is satisfied, assuming an expanded state x 4 =f, then equation (8) can be restated as:
the extensional state observer can be designed to be:
In the formula (10), the amino acid sequence of the compound, The estimated values of x i (i=1, 2,3, 4), respectively, and β i (i=1, 2,3, 4) are error correction coefficients of the observer.
Taking the observation error of the extended state observer asThe dynamic equation for the observed error can be described as:
Taking out The pull-type transforms of u, x 1 are respectivelyU (S), X 1 (S), the input-output transfer function of the extended state observer can be expressed as:
Obtainable according to formula (8)
Taking the pull-transform of F to F (S), then equation (16) can be expressed as:
S3X1(S)＝F(S)+b0U(S) (17)
the combined type (15) and the formula (17) can be obtained:
According to the bandwidth law parameter adjustment strategy, the poles of the extended state observer are usually arranged at-w o (w o > 0), namely the gain beta i of the extended state observer meets the following conditions:
S4+β1S3+β2S2+β3S+β4＝(S+wo)4 (19)
at this time, the value of beta i should satisfy:
The specific process of the step 2.2 is as follows:
Aiming at the characteristic that the large-inertia electrohydraulic position servo system is easy to generate steady-state buffeting when the displacement is converged rapidly, the high-frequency gain of the system is reduced to improve the steady-state performance of the system. Based on the considered direction, a low-pass filter is introduced to disturbance observation value Performing series hysteresis correction, and taking the disturbance observation value of the correction hysteresis asWhich is converted intoThen there is the expression:
In the formula (20), τ is an inertia coefficient. The combined equation (18) and equation (20) yields the total disturbance transfer function after correction as:
the specific process of the step 2.3 is as follows:
the state space of the extended state observer after correction available according to equation (10) and equation (20) is expressed as:
For the conventional extended state observer described by equation (10), in the case of pole configuration using the bandwidth method, the poles thereof are all located on the negative real axis, and thus they are stable in bounded input-bounded output. From equations (20) and (22), it is known that the added correction element is in series relationship with the original extended state observer, and therefore does not affect the convergence of the conventional extended state observer. From equation (21), when the selection parameter τ > 0, the poles of φ 2 (S) are all located on the negative real axis, so the disturbance observations after correction are still converging.
According to the mathematical model described in the formula (8), and the state information and disturbance information observed in the step 2, the specific process of the step 3 is as follows:
3.1, designing a nonlinear backstepping controller;
3.2, selecting a proper nonlinear error feedback function;
And 3.3, carrying out self-adaptive compensation on the observation error and the filtering error, and completing the stability analysis of the closed-loop control system.
The specific process of the step 3.1 is as follows:
In consideration of repeated derivation of the virtual control amount designed in the back-stepping method calculation process, a tracking differentiator is adopted to estimate the derivative of the virtual control amount. The design process of the tracking differentiator is as follows:
In the formula (23), h is a sampling period, v is an input signal of the tracking differentiator, r 1 and r 2 are output signals of the tracking differentiator, and the following are: r 1 is the transition at v is the input signal and r 2 is the derivative of r 1. Delta and h 0 are adjustable parameters of the differentiator, delta is a speed factor, the tracking speed of the differentiator is determined, h 0 is a filtering factor, and the smoothness of an output curve of the differentiator is determined. The expression of f han is as follows:
The tracking differentiator is selected because of the obvious filtering effect, and is helpful for stabilizing a system and inhibiting steady-state oscillation of a large-inertia electrohydraulic position servo system. In addition, the tracking differentiator is not only used for the estimation of the derivative of the virtual control quantity, but also the derivative of the reference trajectory x d is needed in the design process of backsteping. For a large-inertia electrohydraulic position servo system, the reference track is usually a step signal, which is not beneficial to the realization of an algorithm. Thus, another effect of the tracking differentiator is to soften the step signal and to derive the derivative of the reference trajectory.
Defining a tracking error variable:
z1＝x1-xd,z2＝x2-α1 (25)
Differentiating z 1 with respect to time includes:
In the formula (26), α 10 is a virtual control amount to be designed. Alpha 10 is taken as the input of a tracking differentiator expressed as a formula (23) to obtain alpha 1 and the derivative thereof The virtual control amount is designed as follows:
In the formula (27), f 1 is a nonlinear function to be designed, and the function is to balance the contradiction between the response rapidity and overshoot of the high-inertia electrohydraulic position servo system. The adaptive compensation term is used for compensating the filtering error, and specific expression will be given in the subsequent design. And in the subsequent design, f 2,f3 andThe function of (a) is similar to that of (b) herein, and the description will not be repeated.
Combined formula (26) and formula (27), can be obtained:
selecting Lyapunov candidate functions The differentiation with respect to time was obtained by:
defining a tracking error variable:
z3＝x3-α2 (30)
Differentiating z 2 with respect to time includes:
taking virtual control quantity alpha 20 as follows:
In the method, in the process of the invention, At this timeThe method meets the following conditions: the combined type (31) and the formula (32) are:
selecting Lyapunov candidate functions Its differentiation with respect to time is:
Taking out Then equation (34) may be re-expressed as:
differentiating z 3 with respect to time includes:
the final design control amount is as follows:
the combined type (36) and the formula (37) are:
selecting Lyapunov candidate functions Its differentiation with respect to time is:
Taking out Then equation (39) may be re-expressed as:
the specific process of the step 3.2 is as follows:
In order to reconcile the contradiction between the response rapidity and overshoot of the system and ensure the control stability, the selection or construction of the nonlinear error feedback function should have a certain principle, specifically as follows:
(1) F i about tracking error Should be a monotonically increasing function. When the tracking error is larger, f i is larger, so that the rapidity of convergence is ensured; when the tracking error is smaller, f i is smaller, so that the movement speed of a large inertia load is ensured to be smaller when the displacement is close to the expected displacement, and overshoot is reduced;
(2) The f i (x) should be an odd function, so that the symbols of x and f i (x) are the same, and the stability of the system is ensured. And when the absolute values of the tracking errors are the same, the system can obtain the same gain;
(3)、 As an even function, and Monotonically increasing;
(4)、fi(x), the definition domain is bounded, and at this time, there is a constraint according to the Lagrangian median theorem This is true.
Based on the above principle, the construction f i is:
in formula (41), l 1,l2 is an adjustable parameter, wherein L 2 represents the boundary of displacement tracking errors.
It should be noted that, the expected displacement of the large inertia electrohydraulic position servo system is usually a step signal, so the boundary of displacement tracking error is easy to obtain. Define the range of l 1 toThe establishment of the condition (4) can be ensured. In addition, f 2,f3 can be chosen as a function of the same form as f 1, but in order to reduce the difficulty of adjusting the system parameters, f 2,f3 is chosen directly as a linear form. It does not satisfy the conditions (1), (3) for the nonlinear function, but does not have an influence on the system closed loop performance and theoretical analysis part.
The specific process of the step 3.3 is as follows:
Lemma 1: for any z ε R, there is a positive constant α, β such that This is true.
The design of the self-adaptive rate is to reduce the influence of observation errors and filtering errors on the performance of a closed-loop system, and based on the principle, the following steps are taken:
Selecting Lyapunov candidate functions as follows:
the differentiation of the sample with respect to time was obtained, and the following were:
considering that z 2,z3 is not practically formable, formula (44) can be changed to:
Combined (42), obtainable:
according to lemma 1, there are:
the combined type (46) and the formula (47) are:
the configurable adaptation rate at this time is:
The combined type (48), (49) can be obtained:
according to the young's inequality:
from the analysis of the observed errors in the previous section, the observed errors are bounded, and s i is bounded, where it is preferable to: The method comprises the following steps:
Since z i is the same as f i(zi), without loss of generality, there must be a sufficiently large gain k i at this time such that It holds that within a certain interval, the closed loop system is bounded stable.
By adopting the method, the specific process of the step 4 is as follows:
And adjusting the parameters according to the selection principle of the observer parameters and the controller parameters required in the analysis process and the theoretical derivation process until the required control system performance index is met.
The following examples are used to verify the benefits of the present invention:
The parameters of the large-inertia electrohydraulic position servo system model constructed based on the AMESim platform shown in fig. 4 are shown in table 1.
Table 1 parameters set by the model
| Parameter name | Numerical value |
| Oil supply pressure | 210bar |
| Control signal range | ±10mA |
| Pressure drop of servo valve | 10bar |
| Flow rate of servo valve | 550L/min |
| Coefficient of viscous friction | 500N/(m/s) |
| Frequency of servo valve | 80HZ |
| Damping ratio of servo valve | 0.8 |
| Hydraulic cylinder stroke | 2m |
| Initial displacement of hydraulic cylinder | 0.5m |
| Diameter of piston | 440mm |
| Diameter of piston rod | 420mm |
| Load mass | 100000kg |
| Internal leakage coefficient | 6e-6L/min/MPa |
Output feedback controllers (CESO-ANLbackstepping) of the high-speed high-inertia electrohydraulic position servo system are designed for the system, and are compared with controllers (ESO-backstepping) formed by a conventional extended state observer and backstepping for verifying the effectiveness of the invention. Wherein ESO-backstepping is designed to be an extended state observer as shown in formula (10), and the control amount formed in the process can be described as follows:
And selecting a desired signal as a step signal with the amplitude of 0.5m according to the comparison between the system design CESO-ANLbackstepping algorithm and the ESO-backstepping algorithm. The parameters in the ESO-backstepping algorithm are set as follows: the tracking differentiator parameter for arranging the transition is set to δ=0.2; h=0.01; h 0 = 0.09. The track-differentiator parameter for the virtual control quantity derivative estimation is set to: δ=1000; h=0.01; h 0 = 0.12. Parameters of the extended state observer are set as follows: w o＝20;b0 = 20. The feedback gain at the controller is set to: k 1＝320;k2＝0.7;k3 =3. The CESO-ANLbackstepping algorithm parameters are set as follows: the low pass filter parameter is set to τ=0.05; the parameter in the nonlinear error feedback function is set to i 1＝1.3;l2 = 0.5; the parameter in the adaptive rate is set to r 1＝1;r2＝1;r3＝10;m1＝m2＝m3 =5; α=12; beta=15. The remaining parameters were consistent with the ESO-backstepping settings.
The results of the comparison between CESO-ANLbackstepping algorithm and the ESO-backstepping algorithm are given in FIGS. 5 and 6. It can be seen from fig. 5 and 6 that there is a large overshoot in the trace curve of the ESO-backstepping algorithm, and after the desired displacement is reached, a large oscillation occurs gradually under the influence of the large inertia of the load and the closed loop action of the system. The CESO-ANLbackstepping algorithm designed by the invention greatly reduces the overshoot of displacement output while ensuring the response rapidity, eliminates the steady-state oscillation and obtains excellent tracking performance.
The present invention is capable of other and further embodiments and its several details are capable of modification and variation in accordance with the present invention by those skilled in the art, without departing from the spirit and scope of the invention as defined in the appended claims.
Claims (5)
1. The output feedback control method of the high-speed large-inertia electrohydraulic position servo system is characterized by comprising the following steps of:
Step 1: establishing a mathematical model of a large inertia valve control asymmetric cylinder system;
Step 2: designing an extended state observer based on correction and performing observer convergence analysis;
Step 3: designing a self-adaptive nonlinear backstepping controller and analyzing the stability of a closed-loop system;
step 4: adjusting observer parameters and controller parameters to enable the performance index of the control system to meet actual requirements;
The specific process of the step 1 is as follows:
Step 1.1, a stress equation at a piston rod of a hydraulic cylinder is as follows:
Wherein m is the total mass of a load and a piston rod, x p is the actual displacement of the piston rod, and the load displacement is assumed to be consistent with the piston rod displacement, P 1 and P 2 are the pressure of a rodless cavity of a hydraulic cylinder and the pressure of a rod cavity of the hydraulic cylinder respectively, A 1 and A 2 are the effective acting area of the rodless cavity of the hydraulic cylinder and the effective acting area of the rod cavity of the hydraulic cylinder respectively, B P is a viscous friction coefficient, g represents the gravity acceleration, F L is the factors except the stress equation, including friction force and unknown external disturbance force, and F g is the extra disturbance generated at the piston rod under the action of large inertia of the load;
considering the difference in effective acting areas of the rod cavity and the rodless cavity of the asymmetric cylinder, the following deformation can be performed on the formula (1):
Wherein P L represents a load pressure, and the expression is: p L＝P1-P2 take Then formula (2) reduces to:
Step 1.2, neglecting the influence of external leakage, the flow equations of the rodless cavity and the rod cavity of the hydraulic cylinder can be described as:
In the formula (4), Q 1 and Q 2 respectively correspond to the flow rate of the rodless cavity and the rod cavity, c t is the internal leakage coefficient of the hydraulic cylinder, beta e is the elastic modulus of hydraulic oil, V 1 and V 2 respectively represent the real-time volumes of the rodless cavity and the rod cavity, the expressions of V 1＝V10+A1xp and V 2＝V20-A2xp,V10 and V 20 respectively represent the initial volumes of the rodless cavity and the rod cavity, and Q 1 (t) and Q 2 (t) represent unknown modeling errors generated in the modeling process;
Taking the load flow Q L＝(Q1+Q2)/2, then equation (4) can be expressed as:
In the formula (5), A P＝(A1+A2)/2 represents the average effective acting area of the two cavities, V t＝V1+V2 represents the total effective volume of the hydraulic rod, and q (t) = [ q 1(t)+q2 (t) ]/2 is the sum of unknown modeling errors;
step 1.3, considering the high-frequency response characteristic of the servo valve, the load flow equation at the servo valve can be expressed as:
QL＝kvu+q3(t) (6)
In the formula (6), k v represents the flow gain at the servo valve, u is the control input signal at the servo valve, and q 3 (t) represents the modeling error;
Step 1.4, defining a state vector based on the formulas (3), (5) and (6) The state space description of the large inertia valve controlled asymmetric cylinder can be expressed as:
in the formula (7), a 1,a2 and b represent model parameters of the large-inertia electrohydraulic position servo system, which are time-varying or not precisely available, and are specifically expressed as follows: representing unknown disturbance existing in the large-inertia electrohydraulic position servo system, which can not be directly obtained;
the specific process of the step 2 comprises the following steps:
step 2.1, designing a conventional extended state observer;
2.2, correcting disturbance information obtained by observation of a conventional extended state observer;
step 2.3, analyzing the convergence of the observer;
The specific process of the step 3 comprises the following steps:
step 3.1, designing a nonlinear backstepping controller based on an estimated value of an observer;
step 3.2, selecting a proper nonlinear error feedback function;
step 3.3, performing self-adaptive compensation on the observation error and the filtering error, and completing stability analysis of the closed-loop control system;
the specific process of the step 3.1 is as follows:
Considering the repeated derivation of the virtual control amount designed in the back-step method calculation process, the derivative of the virtual control amount is estimated by adopting a tracking differentiator, and the design process of the tracking differentiator is as follows:
In the formula (23), h is a sampling period, v is an input signal of the tracking differentiator, r 1 and r 2 are output signals of the tracking differentiator, and the following are: r 1 is a transition process under v is an input signal, r 2 is a derivative of r 1, delta and h 0 are adjustable parameters of the differentiator, delta is a speed factor, the tracking speed degree of the differentiator is determined, h 0 is a filtering factor, the smoothness degree of an output curve of the differentiator is determined, and an expression of f han is as follows:
defining a tracking error variable:
z1＝x1-xd,z2＝x2-α1 (25)
Differentiating z 1 with respect to time includes:
In the formula (26), alpha 10 is a virtual control quantity to be designed, and alpha 1 and derivatives thereof can be obtained by taking alpha 10 as an input of a tracking differentiator expressed as the formula (23) The virtual control amount is designed as follows:
in the formula (27), f 1 is a nonlinear function to be designed, which is used for balancing the contradiction between the response rapidity and overshoot of the high-inertia electrohydraulic position servo system, For the adaptive compensation term, the effect is to compensate the filtering error, the concrete expression will be given in the subsequent design, and in the subsequent design, f 2,f3 andThe function of (a) is similar to that of (b) herein, and the description is not repeated;
combined formula (26) and formula (27), can be obtained:
selecting Lyapunov candidate functions The differentiation with respect to time was obtained by:
defining a tracking error variable:
z3＝x3-α2 (30)
Differentiating z 2 with respect to time includes:
taking virtual control quantity alpha 20 as follows:
In the formula (32), the amino acid sequence of the compound, At this timeThe method meets the following conditions: the combined type (31) and the formula (32) are:
selecting Lyapunov candidate functions Its differentiation with respect to time is:
Taking out Then equation (34) may be re-expressed as:
differentiating z 3 with respect to time includes:
the final design control amount is as follows:
the combined type (36) and the formula (37) are:
selecting Lyapunov candidate functions Its differentiation with respect to time is:
Taking out Then equation (39) may be re-expressed as:
the specific process of the step 3.2 is as follows:
(1) F i about tracking error The method is a monotonically increasing function, namely when the tracking error is large, f i is large, so that the convergence rapidity is ensured; when the tracking error is smaller, f i is smaller, so that the movement speed of a large inertia load is ensured to be smaller when the displacement is close to the expected displacement, and overshoot is reduced;
(2) The f i (x) is an odd function, so that the symbols of x and f i (x) are the same, the stability of the system is ensured, and the system can obtain the same gain when the absolute values of the tracking errors are the same;
(3)、 As an even function, and Monotonically increasing;
(4)、 the definition domain is bounded, and at this time, there is a constraint according to the Lagrangian median theorem Establishment;
Based on the principle, the construction f i is as follows:
wherein l 1,l2 is an adjustable parameter, in which L 2 represents the boundary of displacement tracking error;
The specific process of the step 3.3 is as follows:
Lemma 1: for any z ε R, there is a positive constant α, β such that Establishment; taking:
Selecting Lyapunov candidate functions as follows:
the differentiation of the sample with respect to time was obtained, and the following were:
considering that z 2,z3 is not practically formable, formula (44) can be changed to:
Combined (42), obtainable:
according to lemma 1, there are:
the combined type (46) and the formula (47) are:
the configurable adaptation rate at this time is:
The combined type (48), (49) can be obtained:
according to the young's inequality:
from the analysis of the observed errors in the previous section, the observed errors are bounded, and s i is bounded, where it is preferable to: The method comprises the following steps:
Since z i and f i(zi) are given the same sign, without loss of generality, there must be a gain ki of sufficient magnitude that It holds that within a certain interval, the closed loop system is bounded stable.
2. The output feedback control method of the high-speed large-inertia electrohydraulic position servo system of claim 1, wherein the specific process of step 2.1 is as follows:
taking f= -a 1x2-a2x3+d+(b-b0) u for the mathematical model as expressed by equation (7), equation (7) can be restated as:
suppose 1: f is bounded and steerable and its derivative satisfies:
When assuming that 1 is satisfied, assuming an expanded state x 4 =f, then equation (8) can be restated as:
the extensional state observer can be designed to be:
In the formula (10), the amino acid sequence of the compound, Estimated values of x i (i=1, 2,3, 4), respectively, and β i (i=1, 2,3, 4) is an error correction coefficient of the observer;
taking the observation error of the extended state observer as The dynamic equation for the observed error can be described as:
Taking out The pull-type conversion of (a) is respectivelyThe input-output transfer function of the extended state observer can be expressed as:
Obtainable according to formula (8)
Taking the pull-transform of F to F (S), then equation (16) can be expressed as:
S3X1(S)＝F(S)+b0U(S) (17)
the combined type (15) and the formula (17) can be obtained:
According to the bandwidth law parameter adjustment strategy, the poles of the extended state observer are usually arranged at-w o (w o > 0), namely the gain beta i of the extended state observer meets the following conditions:
S4+β1S3+β2S2+β3S+β4＝(S+wo)4 (19)
at this time, the value of beta i should satisfy:
3. the output feedback control method of the high-speed large-inertia electrohydraulic position servo system of claim 2, wherein the specific process of step 2.2 is as follows:
Aiming at the characteristic that steady-state buffeting is easy to occur when displacement of a large-inertia electrohydraulic position servo system is rapidly converged, the high-frequency gain of the system is reduced to improve the steady-state performance of the system, and a low-pass filter is introduced to disturbance observation values based on the considered direction Performing series hysteresis correction, and taking the disturbance observation value of the correction hysteresis asWhich is converted intoThen there is the expression:
in the formula (20), τ is an inertia coefficient, and the total disturbance transfer function after correction is obtained by combining the formula (18) and the formula (20):
4. The output feedback control method of the high-speed large-inertia electrohydraulic position servo system of claim 3, wherein the specific process of step 2.3 is as follows:
the state space of the extended state observer after correction available according to equation (10) and equation (20) is expressed as:
for the conventional extended state observer described by the formula (10), in the case of pole configuration using the bandwidth method, the poles are all located on the negative real axis, so that the finite input-finite output is stable, the added correction link is in series relation with the original extended state observer as shown by the formula (20) and the formula (22), so that the convergence of the conventional extended state observer is not affected, and the pole of phi 2 (S) is located on the negative real axis when the selection parameter τ > 0 as shown by the formula (21), so that the disturbance observation value after correction is still converged.
5. The output feedback control method of the high-speed large-inertia electrohydraulic position servo system of claim 2, wherein the specific process of step 4 includes:
and (3) adjusting the parameters according to the analysis process of the steps 1-3 and the selection principle of the observer parameters and the controller parameters required in the theoretical derivation process until the required control system performance index is met.
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