CN111577711A - Active-disturbance-rejection robust control method for double-rod hydraulic cylinder position servo system - Google Patents

Active-disturbance-rejection robust control method for double-rod hydraulic cylinder position servo system Download PDF

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CN111577711A
CN111577711A CN202010366842.4A CN202010366842A CN111577711A CN 111577711 A CN111577711 A CN 111577711A CN 202010366842 A CN202010366842 A CN 202010366842A CN 111577711 A CN111577711 A CN 111577711A
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姚建勇
陈久辉
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Nanjing University of Science and Technology
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
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Abstract

The invention discloses a self-disturbance rejection Robust (RISEESO) control method of a double-out-rod hydraulic cylinder position servo system, which combines disturbance compensation based on an Extended State Observer (ESO) with error sign integral Robust (RISE) and proves the asymptotic stable result of the system by using Lyapunov stability theory. The proposed strategy effectively combines the interference suppression method (RISE) with the interference estimation compensation method, inherits the specific advantages of both methods and avoids the respective disadvantages. The disclosed control method has the following advantages: compared with the traditional ESO method, the use of the RISE reduces the observation load of the ESO, and further reduces the observer residual error; compared with the traditional RISE method, the use of ESO ensures that the nonlinear robust gain of RISE only needs to be related to the second derivative of the state observation error, thus weakening the original harsh condition; the tracking performance of the proposed controller is superior to both RISE and ESO under the same conditions.

Description

Active-disturbance-rejection robust control method for double-rod hydraulic cylinder position servo system
Technical Field
The invention relates to an electro-hydraulic servo control technology, in particular to an active disturbance rejection robust control method of a double-rod hydraulic cylinder position servo system.
Background
The hydraulic position servo system plays a very important role in the fields of aircrafts, heavy machinery, high-performance rotation test equipment and the like by virtue of the characteristics of high power density, large force/torque output, strong load-resistant rigidity and the like. However, the inherent non-linear characteristics of hydraulic systems and various modeling uncertainties complicate the design of their controllers. At first, a great deal of research is carried out on the design of a controller, such as a PID controller, for a hydraulic system based on a linear control theory, but the design of the linear controller is based on a linearized hydraulic system model, and the nonlinear characteristic of the linear controller cannot be reflected, so that a good control effect cannot be obtained. The feedback linearization control can compensate the nonlinear characteristic of the hydraulic system in real time in the design of the controller, but requires that the system model information is completely known and is not consistent with the practical application. The Active Disturbance Rejection Control (ADRC) is widely applied because the ADRC needs less model information and can obtain excellent control performance, and is characterized in that an Extended State Observer (ESO) is adopted to expand the integrated disturbance of the system into a new state variable, and the observed disturbance acts on the system in a feedforward compensation mode to improve the control performance. In order to simplify the implementation of the non-linear ESO, a linear ESO is proposed, which has only one parameter to adjust in the actual control, thus greatly facilitating the controller design and device commissioning process, and the theoretical proof shows that the state estimation error monotonically decreases as the observer bandwidth increases. When the unmodeled dynamics of the system is large, the bandwidth of the observer must be increased in order to improve the control accuracy, however, an excessively large bandwidth may amplify system noise and even make the system unstable. The error sign integral Robust (RISE) control method can also effectively deal with the problem of modeling uncertainty, comprises a unique error sign integral robust term and can obtain asymptotically stable tracking performance under the condition that the system interference is smooth and bounded enough. However, the value of the nonlinear robust gain in the controller designed by the control method needs to satisfy a certain condition, which is closely related to the upper bound of the first derivative and the second derivative of the modeling uncertainty of the system with respect to time.
Disclosure of Invention
The invention aims to provide an active disturbance rejection robust control method of a double-output-rod hydraulic cylinder position servo system with strong disturbance rejection and high tracking performance, which can ensure that the double-output-rod hydraulic cylinder position servo system still keeps excellent control performance when having large disturbance.
The technical solution for realizing the purpose of the invention is as follows: an active disturbance rejection robust control method of a double-out-rod hydraulic cylinder position servo system comprises the following steps:
step 1, establishing a mathematical model of a double-rod hydraulic cylinder position servo system;
step 2, designing an auto-disturbance rejection robust controller according to the mathematical model of the double-rod hydraulic cylinder position servo system;
and 3, performing stability certification on the double-rod hydraulic cylinder position servo system by applying the Lyapunov stability theory, and obtaining a result that the system can achieve gradual stability by applying the Barbalt theorem.
Compared with the prior art, the invention has the following remarkable advantages: the interference suppression (RISE) -based method is effectively combined with the interference estimation compensation (ESO), the use of the RISE further reduces the estimation residual error of the ESO, so that the control performance is improved, meanwhile, the nonlinear robust feedback gain term of the RISE after improvement is only related to the derivative of the state estimation error and is easier to meet than the original condition, and the tracking performance of the controller is improved compared with the traditional RISE and the ESO. The simulation result verifies the effectiveness of the test paper.
Drawings
FIG. 1 is a schematic diagram of a dual-out-rod hydraulic cylinder position servo system of the present invention.
FIG. 2 is a schematic diagram of the active disturbance rejection robust control (RISEESO) method of the dual-out-rod hydraulic cylinder position servo system.
FIG. 3 is a graph of command signal over time that a dual out-of-rod hydraulic cylinder position servo system is expected to track.
FIG. 4 is a graph comparing the tracking performance of the RISEESO controller, RISE controller, ESO controller, PID controller in Case 1.
FIG. 5 is a schematic diagram of the control inputs to the system under the influence of the RISEESO controller in Case 1.
FIG. 6 is a graph comparing the tracking performance of the RISEESO controller, RISE controller, ESO controller, PID controller in Case 2.
FIG. 7 is a schematic diagram of the control inputs to the system under the influence of the RISEESO controller in Case 2.
Detailed Description
The invention is described in further detail below with reference to the figures and the embodiments.
The invention is based on the traditional RISE control method, utilizes the unmodeled dynamic state of an ESO estimation system and carries out feedforward compensation, the use of the RISE reduces the observation burden of the ESO, further reduces the estimation error of the ESO, the use of the ESO ensures that the nonlinear robust gain of the RISE does not need to be related to the upper bound of the first derivative and the second derivative of the uncertainty of the system modeling, but only needs to be related to the second derivative of the state observation error, and when the state estimation error of the ESO is ensured by adjusting the bandwidth, the condition is easier to be satisfied compared with the former condition. And finally, the stability of the double-rod hydraulic cylinder position servo system is proved by applying the Lyapunov stability theory, and the result that the system can achieve gradual stability is obtained.
With reference to fig. 1-2, the active disturbance rejection robust control method of the double-rod hydraulic cylinder position servo system comprises the following steps:
step 1, establishing a mathematical model of a double-rod hydraulic cylinder position servo system;
step 1-1, the double-rod hydraulic cylinder position servo system considered by the invention drives an inertial load through a double-rod hydraulic cylinder controlled by a servo valve;
therefore, according to newton's second law, the equation of motion for an inertial load is:
Figure BDA0002476959180000031
m in the formula (1) is an inertia load parameter; pLThe pressure difference between two cavities of the hydraulic cylinder; a is the effective cross-sectional area of the piston of the hydraulic cylinder; b is a viscous friction coefficient; f (t) is other unmodeled interference; y is the displacement of the inertial load; t is a time variable;
neglecting the external leakage of the hydraulic cylinder, the dynamic equation of the pressure difference between the two cavities of the hydraulic cylinder is as follows:
Figure BDA0002476959180000032
v in formula (2)tTotal control volume for two chambers of hydraulic cylinder βeEffective oil elastic modulus; ctIs the internal leakage coefficient; q (t) is a modeling error caused by a complex internal leakage process, unmodeled pressure dynamics, and the like; qL＝(Q1+Q2) /2 is the load flow, and Q1And Q2The flow of an oil inlet cavity and the flow of an oil return cavity of the hydraulic cylinder are respectively; qLAnd servo valve displacement xvThe relationship of (1) is:
Figure BDA0002476959180000033
flow gain coefficient of servo valve in formula (3)
Figure BDA0002476959180000034
Sign function sign (x)v) Is defined as:
Figure BDA0002476959180000035
in the formula CdA flow coefficient; omega is the valve core area gradient; rho is the oil density; psFor supply pressure, PrIs the return oil pressure;
because the servo valve dynamic state is considered, an additional displacement sensor is required to be installed to obtain the displacement of the servo valve spool, and the tracking performance is only slightly improved. Therefore, a great deal of related research ignores the dynamic state of the servo valve, and if the servo valve with high response is adopted, the valve core displacement and the control input are approximately proportional links, namely xv＝kiu，kiIs a positive electrical constant, u is the control input voltage; therefore, the formula (3) can be written as
Figure BDA0002476959180000041
In the formula (5), kt＝kqkiRepresents the total flow gain;
step 1-2, assuming that the unmodeled dynamic term f (t) is continuously differentiable, defining a state variable:
Figure BDA0002476959180000042
then the state equation of the double-out-rod hydraulic cylinder position servo system is as follows:
Figure BDA0002476959180000043
in the formula (6)
Figure BDA0002476959180000044
In equation (6), we define a new variable U to represent the control input of the system, since the actual system is equipped with a pressure sensor, therefore
Figure BDA0002476959180000045
The term can be calculated in real time, and as long as U is determined, the actual control input U can also be calculated; therefore, the following controller design will focus on providing an auto-robust controller U to handle various disturbances of the system;
in order to simplify the display format of the state equation of the double-rod hydraulic cylinder position servo system, the system is recorded with theta ═ theta1,...,θ3]TIs a known nominal value of a system parameter, thetar＝[θ1r,...,θ3r]TIs the true value of the system parameter, wherein,
Figure BDA0002476959180000046
then equation (6) can be written as
Figure BDA0002476959180000051
In the formula (8), the reaction mixture is,
Figure BDA0002476959180000052
the design target of the double-rod hydraulic cylinder position servo system controller is as follows: given system reference signal yd(t)＝x1d(t) designing a bounded control input U such that the system output y is x1Tracking the reference signal of the system as much as possible;
for the controller design, assume the following:
assume that 1: reference instruction signal x of double-rod hydraulic cylinder position servo system1d(t) is three-order continuous and the system expects that the position command, the velocity command, the acceleration command, and the jerk command are bounded; the hydraulic system operating under normal conditions, i.e. P1And P2Are all less than the supply pressure PsAnd | PLL is also less than Ps(ii) a Thus it can be seen that
Figure BDA0002476959180000053
Always bounded, then the true control input U can be guaranteed to be bounded as long as the designed U is bounded;
assume 2: general interference of double-rod hydraulic cylinder position servo system
Figure BDA0002476959180000054
Is sufficiently smooth to make
Figure BDA0002476959180000055
Are present and bounded i.e.:
Figure BDA0002476959180000056
in the formula (9)1,2Are all unknown normal numbers, i.e.
Figure BDA0002476959180000057
Have an indeterminate upper bound;
step 2, designing an auto-disturbance rejection robust controller according to the mathematical model of the double-rod hydraulic cylinder position servo system;
step 2-1, designing interference compensation based on the extended state observer:
will integrate the perturbation
Figure BDA0002476959180000058
Expansion into an additional state variable, i.e.
Figure BDA0002476959180000059
H (t) is
Figure BDA00024769591800000510
The first derivative of (c), then equation (8) can be written as:
Figure BDA00024769591800000511
since h (t) is
Figure BDA00024769591800000512
The first derivative of (a) can be obtained from equation (9):
Figure BDA0002476959180000061
the system can observe according to the structure of the formula (10), and a linear extended state observer is designed for the system as follows:
Figure BDA0002476959180000062
in the formula
Figure BDA0002476959180000063
Represents a state estimate, and wo>0 can be considered as the bandwidth of the extended state observer.
Note the book
Figure BDA0002476959180000064
i is 1,2,3,4, which represents the state estimation error, and the derivative of the state estimation error obtained from equation (10) and equation (12) is:
Figure BDA0002476959180000065
note the book
Figure BDA0002476959180000066
i is 1,2,3, 4; equation (12) can then be written as:
Figure BDA0002476959180000067
wherein the content of the first and second substances,
Figure BDA0002476959180000068
M＝[0 0 0 1]Td is a Helverz matrix; since D is a Helveltz matrix, there is a symmetric positive definite matrix H such that D isTH+HD＝-I；
The following reasoning is given:
by
Figure BDA0002476959180000069
The following conclusions can be drawn: the estimated state is always bounded and the size of this bound after any time follows the observer bandwidth woIs increased and decreased, there is a normal number
Figure BDA00024769591800000610
The following relationships are satisfied:
Figure BDA00024769591800000611
the theory proves that:
defining the Lyapunov function:
V＝TH (16)
the derivation of which is:
Figure BDA0002476959180000071
in the formula (17), the compound represented by the formula (I),
Figure BDA0002476959180000072
λmax(H)the maximum eigenvalue of H.
From (17) can be obtained:
Figure BDA0002476959180000073
from the above formula, one can obtain:
Figure BDA0002476959180000074
in formula (19), λmin(H)The minimum eigenvalue of H.
From (19), it can be known that: when h (t) is bounded, | | | | is always bounded, and so long as let w be boundedo> 1, then | | | | will decrease with increasing time, more importantly by increasing woTo increase λ to narrow the error value of the state estimate. By
Figure BDA0002476959180000075
i is 1,2,3, 4; and equation (13) is easy to see that (15) is true, so far, it is referred to.
Step 2-2, definition of z1＝x1-x1dFor systematic tracking error, according to the first equation in equation (10)
Figure BDA0002476959180000076
Let α1For virtual control, let equation
Figure BDA0002476959180000077
Tending to a steady state α1And the true state x2Has an error of z2＝x2-α1To z is to1The derivation can be:
Figure BDA0002476959180000078
designing a virtual control law:
Figure BDA0002476959180000079
in the formula k1If > 0 is adjustable gain, then
Figure BDA0002476959180000081
Due to z1(s)＝G(s)z2(s) wherein G(s) is 1/(s + k)1) Is a stable transfer function when z2When going to 0, z1Also inevitably tends to 0; so in the next design, z will be such that2Tends to 0 as the main design goal;
step 2-3, taking into account the second equation of equation (10), choose α2Is x3Virtual control of z3Is a virtual control α2And x3Deviation z between3＝x3-α2(ii) a Then z is2Has a dynamic equation of
Figure BDA0002476959180000082
Design virtual control law α2The following were used:
Figure BDA0002476959180000083
in the formula k2For a positive feedback gain, equation (24) is substituted for equation (23):
Figure BDA0002476959180000084
due to z2(s)＝G'(s)z3(s) wherein G'(s) ═ 1/(s + k)2) Is a stable transfer function when z3When going to 0, z2Also inevitably tends to 0; so in the next design, z will be such that3Tends to 0 as the main design goal;
step 2-4, defining an auxiliary error signal r (t) for obtaining an additional degree of freedom of controller design:
Figure BDA0002476959180000085
k in formula (26)3The gain is adjustable when the gain is more than 0, and the jerk signal of the position is contained in r (t), so the jerk signal is considered to be unmeasurable in practice, namely r (t) is only used for auxiliary design and is not specifically shown in a designed controller;
considering the third equation of equation (10), the expansion of r can be obtained as follows:
Figure BDA0002476959180000091
according to equation (27), the model-based controller U can be designed to:
Figure BDA0002476959180000092
in the formula krFor positive feedback gain, UaFor model-based compensation terms, UsIs a robust control law and in which Us1For a linear robust feedback term, Us2The nonlinear robust term is used for overcoming the influence of modeling uncertainty on the system performance; substituting formula (28) into formula (27) to obtain:
Figure BDA0002476959180000093
according to the design method of an error sign integral robust controller, an integral robust term Us2Can be designed as follows:
Figure BDA0002476959180000094
beta should satisfy the following condition:
Figure BDA0002476959180000095
the two-sided derivation of the equation of equation (29) and the use of equations (7) and (11) can be obtained:
Figure BDA0002476959180000096
step 3, the stability of the double-rod hydraulic cylinder position servo system is proved by applying the Lyapunov stability theory, and a gradual stable result of the system can be obtained by applying the Barbalt theorem, which is specifically as follows:
the following arguments are given:
defining auxiliary functions
Figure BDA0002476959180000097
If the control gain β is selected to satisfy the condition shown in equation (31):
Figure BDA0002476959180000098
then
Figure BDA0002476959180000099
Figure BDA0002476959180000101
z3(0)、
Figure BDA0002476959180000102
Respectively represents z3(t) and
Figure BDA0002476959180000103
an initial value of (1);
proof of this lemma:
integrating the two sides of equation (33) and applying equation (26) to obtain:
Figure BDA0002476959180000104
the latter two terms in equation (32) are fractionally integrated to obtain:
Figure BDA0002476959180000105
therefore, it is
Figure BDA0002476959180000106
As can be seen from equation (38), if β is selected to satisfy the condition shown in equation (31), equations (34) and (35) are satisfied, which is referred to as "yes".
Defining an auxiliary function:
Figure BDA0002476959180000107
according to the above theory of justice
Figure BDA0002476959180000108
When P (t) ≧ 0, the Lyapunov function is thus defined as follows:
Figure BDA0002476959180000109
the derivation of equation (27) and substitution of equations (22), (25), (26), (32), and (39) can be obtained:
Figure BDA00024769591800001010
defining:
Z＝[z1,z2,z3,r]T(42)
Figure BDA0002476959180000111
by adjusting the parameter k1,k2,k3,krThe symmetry matrix Λ can be made positive, then:
Figure BDA0002476959180000112
in formula (44) < lambda >min(Λ) is the minimum eigenvalue of the symmetric positive definite matrix Λ.
Integration of equation (44) yields:
Figure BDA0002476959180000113
z is represented by formula (32)1，z2，z3，r∈L2Norm, and is obtained according to equations (22), (25), (26), (32) and hypothesis 2:
Figure BDA0002476959180000114
norm, and therefore W, is consistently continuous, as can be seen by the barbalt theorem: t → ∞ time, W → 0. Therefore, t → ∞ time, z1→0。
It is therefore concluded that: the auto-disturbance rejection robust controller designed for the double-rod hydraulic cylinder position servo system (2) can lead the system to obtain an asymptotically stable result and adjust the gain k1、k2、k3、krThe tracking error of the system tends to zero under the condition that the time tends to infinity; the schematic diagram of the active disturbance rejection Robust (RISEESO) control principle of the double-out-rod hydraulic cylinder position servo system is shown in FIG. 2.
Examples
In order to assess the performance of the designed controller, parameters in the table 1 are taken in simulation to model the position servo system of the double-rod hydraulic cylinder:
TABLE 1 Dual out-of-rod Hydraulic System parameters
Figure BDA0002476959180000115
Figure BDA0002476959180000121
The expected instruction for a given system is x1d＝0.2sin(πt)[1-exp(-0.01t3)](m) as shown in FIG. 3. To verify the effectiveness of the controller proposed by the present invention, the following controller is divided into two cases.
Case 1: considering only parameter uncertainty, not considering inner leakage and unmodeled dynamics, i.e. theta ≠ thetar，Q(t)＝0，f(t)＝0。
From Table 1, the true value of the clear parameter is calculated as θr＝[8.59,6.68×105,205.75]TThe nominal value is taken as [8,7 × 10 ]5,100]T。
Auto-robust controller (RISEESO): the RISEESO controller parameter is taken as k1＝3000,k2＝500,k3＝100,kr＝100,β＝50,wo＝100。
Error sign integral robust controller (RISE): the RISE controller is U in RISEESOaItem removal
Figure BDA0002476959180000122
The rest are the same, and the parameter values are also the same.
Interference compensation controller (ESO): the controller is not provided with a US2The parameter value of the RISEESO controller is the same as that of the RISEESO controller.
A PID controller: the PID controller parameter adjustment is performed by a trial and error method, and the obtained parameters enable the PID controller to reach the performance limit, if the gain value is increased, the tracking error will diverge. The selected PID controller parameter is
kP＝5000,kI＝800,kD＝50。
The tracking error ratio for each controller is shown in fig. 4. As can be seen from the figure, the control performance of the RISEESO controller is far better than that of the other three controllers under the condition of only parameter uncertainty, which illustrates the effectiveness of the algorithm. The control inputs to the system by the RISEESO controller are shown in FIG. 5.
Case2, not only considering parameter uncertainty, but also considering leakage and unmodeled dynamics in the system, and taking f (x, t) 1000sint, and Q (t) 1 × 10-4sint。
In this Case the parameters of both controllers are the same as the corresponding parameters in Case 1.
Fig. 6 is a graph comparing the tracking performance of each controller when the uncertainty of the parameter and the unmodeled dynamics coexist, and it can be seen that although the system adds a large unmodeled dynamics, the control performance of the RISEESO is still excellent and far better than the tracking performance of the other three controllers, which indicates that the proposed strategy has a strong anti-interference capability. The control inputs to the system by the RISEESO controller are shown in FIG. 7.

Claims (4)

1. An active disturbance rejection robust control method of a double-out-rod hydraulic cylinder position servo system is characterized by comprising the following steps:
step 1, establishing a mathematical model of a double-rod hydraulic cylinder position servo system;
step 2, designing an auto-disturbance rejection robust controller according to the mathematical model of the double-rod hydraulic cylinder position servo system;
and 3, performing stability certification on the double-rod hydraulic cylinder position servo system by applying the Lyapunov stability theory, and obtaining a result that the system can achieve gradual stability by applying the Barbalt theorem.
2. The active disturbance rejection robust control method of the double-out-rod hydraulic cylinder position servo system according to claim 1, wherein a mathematical model of the double-out-rod hydraulic cylinder position servo system is established in step 1, specifically as follows:
step 1-1, considering that a double-rod hydraulic cylinder position servo system drives an inertial load through a double-rod hydraulic cylinder controlled by a servo valve;
therefore, according to newton's second law, the equation of motion for an inertial load is:
Figure FDA0002476959170000011
m in the formula (1) is an inertia load parameter; pLThe pressure difference between two cavities of the hydraulic cylinder; a is the effective cross-sectional area of the piston of the hydraulic cylinder; b isCoefficient of viscous friction; f (t) is other unmodeled interference; y is the displacement of the inertial load; t is a time variable;
neglecting the external leakage of the hydraulic cylinder, the dynamic equation of the pressure difference between the two cavities of the hydraulic cylinder is as follows:
Figure FDA0002476959170000012
v in formula (2)tTotal control volume for two chambers of hydraulic cylinder βeEffective oil elastic modulus; ctIs the internal leakage coefficient; q (t) is a modeling error caused by a complex internal leakage process, unmodeled pressure dynamics; qL＝(Q1+Q2) /2 is the load flow, and Q1And Q2The flow of an oil inlet cavity and the flow of an oil return cavity of the hydraulic cylinder are respectively; qLAnd servo valve displacement xvThe relationship of (1) is:
Figure FDA0002476959170000013
in the formula (3), the flow gain coefficient of the servo valve
Figure FDA0002476959170000014
Sign function sign (x)v) Is defined as:
Figure FDA0002476959170000021
in the formula CdA flow coefficient; omega is the valve core area gradient; rho is the oil density; psFor supply pressure, PrIs the return oil pressure;
assuming a high-response servo valve is adopted, the displacement of the valve core and the control input are approximately proportional links, namely xv＝kiu，kiIs a positive electrical constant, u is the control input voltage, so equation (3) is written as
Figure FDA0002476959170000022
In the formula (5), kt＝kqkiRepresents the total flow gain;
step 1-2, assuming that the unmodeled dynamic term f (t) is continuously differentiable, defining a state variable:
Figure FDA0002476959170000023
then the state equation of the double-out-rod hydraulic cylinder position servo system is as follows:
Figure FDA0002476959170000024
in the formula (6)
Figure FDA0002476959170000025
In order to simplify the state equation display format of the double-rod hydraulic cylinder position servo system, the value theta is recorded as [ theta ]1,...,θ3]TIs a known nominal value of a system parameter, thetar＝[θ1r,...,θ3r]TIs the true value of the system parameter, wherein,
Figure FDA0002476959170000026
then the formula (6) is written as
Figure FDA0002476959170000027
In the formula (8), the reaction mixture is,
Figure FDA0002476959170000028
the design target of the double-rod hydraulic cylinder position servo system controller is as follows: given system reference signal yd(t)＝x1d(t) designing a bounded control input U such that the system output y is x1Tracking the reference signal of the system as much as possible;
for the controller design, assume the following:
assume that 1: reference instruction signal x of double-rod hydraulic cylinder position servo system1d(t) is three-order continuous and the system expects that the position command, the velocity command, the acceleration command, and the jerk command are bounded; the hydraulic system operating under normal conditions, i.e. P1And P2Are all less than the supply pressure PsAnd | PLL is also less than Ps(ii) a Thus it can be seen that
Figure FDA0002476959170000031
Always bounded, then the true control input U can be guaranteed to be bounded as long as the designed U is bounded;
assume 2: general interference of double-rod hydraulic cylinder position servo system
Figure FDA0002476959170000032
Is sufficiently smooth to make
Figure FDA0002476959170000033
Are present and bounded, i.e.:
Figure FDA0002476959170000034
in the formula (9)1,2Are all unknown normal numbers, i.e.
Figure FDA0002476959170000035
With an uncertain upper bound.
3. The robust control method of auto-disturbance rejection of the double-out-rod hydraulic cylinder position servo system according to claim 1, wherein the step 2 of designing the robust controller of auto-disturbance rejection comprises the following steps:
step 2-1, designing interference compensation based on the extended state observer:
will integrate the perturbation
Figure FDA0002476959170000036
Expansion into an additional state variable, i.e.
Figure FDA0002476959170000037
H (t) is
Figure FDA0002476959170000038
The first derivative of (c), then equation (8) is written as:
Figure FDA0002476959170000039
since h (t) is
Figure FDA00024769591700000310
Is derived from equation (9):
Figure FDA00024769591700000311
the system can observe according to the structure of the formula (10), and a linear extended state observer is designed for the system as follows:
Figure FDA0002476959170000041
in the formula
Figure FDA0002476959170000042
Represents a state estimate, and wo>0 is taken as the bandwidth of the extended state observer;
note the book
Figure FDA0002476959170000043
Representing the state estimation error, the derivative of the state estimation error from equations (10) and (12) is:
Figure FDA0002476959170000044
note the book
Figure FDA0002476959170000045
Then, equation (12) is written as:
Figure FDA0002476959170000046
wherein the content of the first and second substances,
Figure FDA0002476959170000047
M＝[0 0 0 1]Td is a Helverz matrix;
is proved to be composed of
Figure FDA0002476959170000048
The following conclusions were made: the estimated state is always bounded and the size of this bound after any time follows the observer bandwidth woIs increased and decreased, there is a normal number
Figure FDA0002476959170000049
The following relationships are satisfied:
Figure FDA00024769591700000410
step 2-2, definition of z1＝x1-x1dFor systematic tracking error, according to the first equation in equation (10)
Figure FDA00024769591700000411
Let α1For virtual control, let equation
Figure FDA00024769591700000412
Tending to a steady state α1And the true state x2Has an error of z2＝x2-α1To z is to1The derivation can be:
Figure FDA00024769591700000413
designing a virtual control law:
Figure FDA0002476959170000051
in the formula k1If > 0 is adjustable gain, then
Figure FDA0002476959170000052
Due to z1(s)＝G(s)z2(s) wherein G(s) is 1/(s + k)1) Is a stable transfer function when z2When going to 0, z1And necessarily tends to 0, so in the next design, z will be made to2Tends to 0 as the main design goal;
step 2-3, taking into account the second equation of equation (10), choose α2Is x3Virtual control of z3Is a virtual control α2And x3Deviation z between3＝x3-α2(ii) a Then z is2Has a dynamic equation of
Figure FDA0002476959170000053
Design virtual control law α2The following were used:
Figure FDA0002476959170000054
in the formula k2For positive feedback gain, equation (20) is substituted for equation (19):
Figure FDA0002476959170000055
due to z2(s)＝G'(s)z3(s) wherein G'(s) ═ 1/(s + k)2) Is a stable transfer function when z3When the concentration of the carbon dioxide tends to be 0,z2also inevitably tends to 0; so in the next design, z will be such that3Tends to 0 as the main design goal;
step 2-4, defining an auxiliary error signal r (t) for obtaining an additional degree of freedom of controller design:
Figure FDA0002476959170000056
k in formula (22)3The gain is adjustable when the gain is more than 0, and the jerk signal of the position is contained in r (t), so the jerk signal is considered to be unmeasurable in practice, namely r (t) is only used for auxiliary design and is not specifically shown in a designed controller;
considering the third equation of equation (10), the expansion of r is obtained as follows:
Figure FDA0002476959170000061
according to equation (23), the model-based controller U can be designed to:
Figure FDA0002476959170000062
in the formula krFor positive feedback gain, UaFor model-based compensation terms, UsIs a robust control law and in which Us1For a linear robust feedback term, Us2The nonlinear robust term is used for overcoming the influence of modeling uncertainty on the system performance; substituting formula (24) into formula (23):
Figure FDA0002476959170000063
according to the design method of an error sign integral robust controller, an integral robust term Us2Can be designed as follows:
Figure FDA0002476959170000064
the variable β must satisfy the following condition:
Figure FDA0002476959170000065
the two-sided derivation of the equation of equation (25) and the use of equations (7) and (11) can be obtained:
Figure FDA0002476959170000066
4. the robust auto-disturbance-rejection control method for the position servo system of the double-out-rod hydraulic cylinder according to claim 1, wherein the stability of the position servo system of the double-out-rod hydraulic cylinder is proved by applying Lyapunov stability theory in step 3, and the gradual stabilization result of the system is obtained by applying barbalt theorem, which is as follows:
defining auxiliary functions
Figure FDA0002476959170000067
Figure FDA0002476959170000068
Wherein:
Figure FDA0002476959170000069
z3(0)、
Figure FDA0002476959170000071
respectively represents z3And
Figure FDA0002476959170000072
an initial value of (1);
is proved to be when
Figure FDA0002476959170000073
When P (t) ≧ 0, the Lyapunov function is thus defined as follows:
Figure FDA0002476959170000074
the Lyapunov stability theory is used for stability verification, and the Barbalt theorem is used for obtaining the result of asymptotic stability of the system, so that the gain k is adjusted1、k2、k3、krAnd the tracking error of the double-rod hydraulic cylinder position servo system tends to zero under the condition that the time tends to be infinite.
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