CN110442026B - Extended state observer based on error correction, anti-interference control system and design method - Google Patents

Extended state observer based on error correction, anti-interference control system and design method Download PDF

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CN110442026B
CN110442026B CN201910758643.5A CN201910758643A CN110442026B CN 110442026 B CN110442026 B CN 110442026B CN 201910758643 A CN201910758643 A CN 201910758643A CN 110442026 B CN110442026 B CN 110442026B
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魏伟
夏鹏飞
左敏
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Beijing Technology and Business University
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Abstract

The invention discloses an anti-interference control system, which comprises an extended state observer based on error correction, a controller and a controlled object, wherein the extended state observer comprises: the extended state observer is used for generating an estimated value of the system state and the total disturbance of the controlled object according to the control signal and the output signal of the controlled object; and the controller is used for generating a control signal u containing interference compensation according to the system state estimated value, the total disturbance estimated value and the set value r, and inputting the control signal u to the controlled object.

Description

Extended state observer based on error correction, anti-interference control system and design method
Technical Field
The invention relates to the technical field of control, in particular to an extended state observer design method based on error correction and an anti-interference control system, which belong to the anti-interference control technology in the advanced control technology.
Background
The active disturbance rejection control has been widely paid attention to and applied due to its advantages of strong disturbance rejection capability, small dependence on model information, etc. However, the problem of limited estimation capability of the extended state observer, which is the core of the active disturbance rejection control, is the key to limit the performance improvement of the active disturbance rejection control.
Therefore, the invention provides an extended state observer design method based on error correction and an anti-interference control system, which can improve the estimation capability of the observer, reduce the estimation error and enhance the capability of the observer for estimating time-varying disturbance on the premise of not introducing nonlinearity.
Disclosure of Invention
In order to realize the purpose of the invention, the following technical scheme is adopted for realizing the purpose:
an anti-jamming control system comprising an extended state observer based on error correction, a controller and a controlled object, wherein: the extended state observer is used for generating a system state estimation value z of the controlled object according to the control signal and the output signal of the controlled object1And z2And the total disturbance estimate z3(ii) a The controller is used for generating a control signal u containing interference compensation according to the system state estimation value, the total disturbance estimation value and the set value r and inputting the control signal u to a controlled object; based on error correctionThe dilated state observer is represented by equation 7 as follows:
Figure RE-GDA0002198058400000021
z1,z2,z3for the estimation of the output, the rate of change of the output and the total disturbance of the extended state observer, β ═ β1,β2,β3]TTo observer gain, p1＝y，
Figure RE-GDA0002198058400000022
b0The gain is controlled adjustably.
The anti-jamming control system, wherein: the controller is represented by equations (8), (9) as follows:
Figure RE-GDA0002198058400000023
in the formula, kp,kdProportional and differential gains, respectively, of the controller, r being a set value, u0Is a control signal without interference compensation;
the control law of the controller considering the total disturbance estimated value is designed as
Figure RE-GDA0002198058400000024
Wherein the controller parameter is
Figure RE-GDA0002198058400000025
kd＝2ωc＝2。
A design method of an anti-interference control system comprises the following steps:
step 1: designing a generic linear extended state observer
Designing a second-order single-input single-output nonlinear system:
Figure RE-GDA0002198058400000026
in formula (1): y is the output of the system and is,
Figure RE-GDA0002198058400000027
is the system internal dynamics, d is the bounded external disturbance, u is the control input; let b be0For the estimation of the control gain b, note
Figure RE-GDA0002198058400000031
For the total disturbance of the system, define x1＝y,
Figure RE-GDA0002198058400000032
Rewriting formula (1) as
Figure RE-GDA0002198058400000033
Wherein x is [ x ]1,x2,x3]TIs a variable of the state of the system,
Figure RE-GDA0002198058400000034
a general Linear Extended State Observer (LESO) is designed as a linear extended state observer
Figure RE-GDA0002198058400000035
Wherein β ═ β1,β2,β3]TFor observer gain, where the value is appropriate, the observer output ξ ═ ξ1,ξ2,ξ3]TThe system state can be estimated in real time, namely xi → x;
set observer gain to
Figure RE-GDA0002198058400000036
Get omegaoApplying a sinusoidal perturbation to the system at 9 ═ cf ═ sin (t/2-30+ pi/2) +1, as follows
Figure RE-GDA0002198058400000037
Step 2: extended state observer for correcting designed state error
Designing a modified extended state observer (4) according to the general linear extended state observer (3) obtained in step 1:
Figure RE-GDA0002198058400000038
in the formula (4), p1＝y,
Figure RE-GDA0002198058400000039
z1,z2,z3Estimating output y and output change rate respectively
Figure RE-GDA00021980584000000310
And total disturbance
Figure RE-GDA00021980584000000311
β1,β2,β3To observer gain, b0Is an adjustable control gain, and u is a control signal; wherein, the state x1Has an observation error of e1＝z1-x1＝z1-p1(ii) a State x2Is obtained by taking the observation error
Figure RE-GDA0002198058400000041
State x3Has an observation error of e3＝z3-x3＝z3-f；
And step 3: extended state observer based on error correction
Designing total disturbance estimation (6) based on proportional-integral of different states according to the modified extended state observer obtained in the step 2, and forming an extended state observer (7) based on error correction:
in equation (4), the estimate of the total disturbance is state x1Observation error e of1＝z1-x1＝z1-p1Multiplying the gain and integrating, and obtaining the result from equation (4)
Figure RE-GDA0002198058400000042
Consider a system (2) having
Figure RE-GDA0002198058400000043
Therefore, it is
Figure RE-GDA0002198058400000044
The new total disturbance estimate can then be designed as
Figure RE-GDA0002198058400000045
The extended state observer based on error correction is designed as follows:
Figure RE-GDA0002198058400000046
wherein z is1,z2,z3For the estimation of the output, the rate of change of the output and the total disturbance of the extended state observer, β ═ β1,β2,β3]TTo observer gain, p1＝y，
Figure RE-GDA0002198058400000047
b0Is an adjustable control gain, and u is a control signal;
and 4, step 4: control law for design controller
In the controller, the set point derivative information is taken into account as a usable feed forward signal
Figure RE-GDA0002198058400000048
In the formula, kp,kdProportional and differential gains, respectively, of the controller, r being a set value, u0Is a control signal without interference compensation;
the control law considering the total disturbance estimation value is designed as
Figure RE-GDA0002198058400000051
Wherein the controller parameter is
Figure RE-GDA0002198058400000052
kd＝2ωc＝2；
And 5: form a closed-loop anti-interference control system
And (4) respectively applying the control quantity obtained in the step (4) to the observer and the controlled object to form a closed-loop anti-interference control system.
Drawings
FIG. 1 is a block diagram of the disturbance rejection control system of the present invention;
FIG. 2 is a flow chart of the design of the disturbance rejection control system of the present invention;
FIG. 3 is a comparison graph of control and estimation results of the present invention.
Detailed Description
The following detailed description of the embodiments of the present invention is provided in conjunction with the accompanying drawings of fig. 1-3.
Fig. 1 is a block diagram showing the structure of the disturbance rejection control system of the present invention. The control system comprises an extended state observer (ECESO) based on error correction, a Controller (Controller) and a controlled object (plant). The extended state observer based on the error correction is used for generating a system state estimated value of the controlled object according to the control signal and the output signal of the controlled object; the controller is used for generating a control input signal containing interference compensation according to the system state estimated value and a given signal r, and applying the control input signal to a controlled object.
The disturbance rejection control system is designed as follows:
step 1: designing a generic linear extended state observer
Designing a second-order single-input single-output nonlinear system:
Figure RE-GDA0002198058400000061
in formula (1): y is the output of the system and,
Figure RE-GDA0002198058400000062
is the system internal dynamics, d is the bounded external disturbance, and u is the control input. Let b0For the estimation of the control gain b, note
Figure RE-GDA0002198058400000063
For the total disturbance of the system, define x1＝y,
Figure RE-GDA0002198058400000064
Formula (1) can be rewritten as
Figure RE-GDA0002198058400000065
Wherein x is [ x ]1,x2,x3]TIs a variable of the state of the system,
Figure RE-GDA0002198058400000066
a general Linear Extended State Observer (LESO) is designed as a linear extended state observer
Figure RE-GDA0002198058400000067
Wherein β ═ β1,β2,β3]TFor observer gain, where the value is appropriate, the observer output ξ ═ ξ1,ξ2,ξ3]TCapable of estimating the state of the system in real time, i.e.ξ→x。
Set observer gain to
Figure RE-GDA0002198058400000068
ωoFor observer bandwidth, take ωo9. In order to test the estimation effect of different types of observer designs on different types of disturbances, the disturbances are set in different time periods. Namely: initially applying no perturbation; when t is 20s, applying unit step disturbance f to the simplified system with the double integrators connected in series, wherein f is 1; applying a time-varying slope disturbance f of 0.05(t-40) +1 to the system at t-40 s; at t 40s, a sinusoidal disturbance f sin (t/2-30+ pi/2) +1 is applied to the system. Is provided with
Figure RE-GDA0002198058400000069
Step 2: modified extended state observer
Designing a state error correction-based extended state observer (4) according to the general linear extended state observer (3) obtained in the step 1:
it is noted that, in the formula (3),
Figure RE-GDA0002198058400000071
and xi2，
Figure RE-GDA0002198058400000072
And xi3+b0u all adopt the state x1Observation error e of1＝ξ1-x1＝ξ1Y. This correction is used to estimate the state x1Is reasonable but used to estimate state x2But are conserved. Therefore, reconstruction should be used to estimate state x2The correction term of (1). State x2Has an observation error of e2＝ξ2-x2But due to x2Need to be estimated, i.e. x2Is unknown, and xi1Estimate x1And is
Figure RE-GDA0002198058400000073
Then, the internal signal of the observer can be used
Figure RE-GDA0002198058400000074
Approximation x2True value of (1) for correcting
Figure RE-GDA0002198058400000075
Thus, state x2Has an observation error of
Figure RE-GDA0002198058400000076
The extended state observer using state error correction is designed as
Figure RE-GDA0002198058400000077
In the formula (4), p1＝y,
Figure RE-GDA0002198058400000078
z1,z2,z3Estimating output y and output change rate respectively
Figure RE-GDA0002198058400000079
And total disturbance
Figure RE-GDA00021980584000000710
β1,β2,β3To observer gain, b0U is a control signal for adjustable control gain; wherein, the state x1Has an observation error of e1＝z1-x1＝z1-p1(ii) a State x2Of (2) observation error taking
Figure RE-GDA00021980584000000711
State x3Has an observation error of e3＝z3-x3＝z3-f。
And step 3: extended state observer based on error correction
Designing a new total disturbance estimation (6) according to the modified extended state observer obtained in the step 2, and forming a new extended state observer (7):
in equation (4), the estimate of the total disturbance is state x1Observation error e of1＝z1-x1＝z1-p1The gain is multiplied and integrated. This method is robust to steady state disturbances, but is also conservative. From the formula (4)
Figure RE-GDA00021980584000000712
Consider equation (2) having
Figure RE-GDA00021980584000000713
Therefore, it is
Figure RE-GDA00021980584000000714
The new total disturbance estimate can then be designed as
Figure RE-GDA0002198058400000081
As can be seen, the new total disturbance estimate is state x1Integral of the observed error of (2) and state x2Is calculated as a sum of the ratios of the observation errors of (1).
In summary, the new extended state observer based on error correction can be designed as:
Figure RE-GDA0002198058400000082
wherein z is1,z2,z3For error correction based estimation of the extended state observer output, rate of change of output, and total disturbance, β ═ β1,β2,β3]TTo observer gain, p1＝y，
Figure RE-GDA0002198058400000083
b0To be regulated and controlledAnd u is a control signal.
And 4, step 4: control law for design controller
Designing a proportional-derivative controller according to the observer output obtained in the step 3 and by considering derivative feedforward information of a set value r:
to obtain better tracking, the set point derivative information is taken into account in the controller as a usable feed forward signal
Figure RE-GDA0002198058400000084
In the formula, kp,kdProportional and differential gains, respectively, of the controller, r being a set value, u0Is a control signal without interference compensation.
The control law considering the total disturbance estimation value is designed as
Figure RE-GDA0002198058400000085
Wherein the controller parameter is
Figure RE-GDA0002198058400000086
kd＝2ωc＝2，ωcIs the controller bandwidth.
And 5: form a closed loop
And (4) respectively applying the control quantity obtained in the step (4) to an observer and a controlled object (system) to form a closed-loop system so as to realize the functions of disturbance estimation and system closed-loop control.
The simulation result is shown in fig. 3, in which diagram (a) is the result of the simulated tracking control; graph (b) shows the effect of three observers estimating the total disturbance; graph (c) gives the local amplification effect of the estimated ramp and sinusoidal perturbations.
The root mean square of the errors of the three states of the three observer estimation systems are listed in table 1, where LESO is a linear extended state observer, POESO is a phase optimized extended state observer, and ECESO is an error correction based extended state observer (ECESO).
Comparing the data in the table, it is clear that the proposed observer has the smallest observed error for all three states of the system. That is, the extended state observer based on error correction has the best estimation effect when the same parameters are chosen.
TABLE 1 State estimation error comparison of observer
Figure RE-GDA0002198058400000091
ECESO has a good estimation effect and is briefly analyzed as follows.
For comparison, xi in formula (3) is used3Is denoted by z3LZ in the formula (7)3Is denoted by z3E。
From the formula (3) z3LA transfer function of
Figure RE-GDA0002198058400000092
And the second equation in equation (2) can be written as
Figure RE-GDA0002198058400000093
Then f is equal to s2y-b0u, is provided with
Figure RE-GDA0002198058400000094
From formula (7) may give z3NA transfer function of
Figure RE-GDA0002198058400000101
The same can be obtained
Figure RE-GDA0002198058400000102
Setting the total disturbance estimation errors of different observers as
Figure RE-GDA0002198058400000103
Is provided with
Figure RE-GDA0002198058400000104
Figure RE-GDA0002198058400000105
When the total disturbance is a step signal with an amplitude K, i.e. f ═ K/s, the steady state estimated deviations are each
Figure RE-GDA0002198058400000106
From equation (18), it can be seen that both observers can achieve no difference in steady state estimation for constant disturbances such as steps. However, when the total disturbance is a ramp signal of amplitude K, i.e. f-K/s2While, the steady state estimated deviation is respectively
Figure RE-GDA0002198058400000107
This indicates that the linear extended state observer has a deviation in the estimated slope disturbance, whereas the observer proposed by the present invention has an estimated deviation of zero.
Consider f ═ s2y-b0u, formulas (16), (17) can be written as
Figure RE-GDA0002198058400000108
Figure RE-GDA0002198058400000111
The phase of the total disturbance estimation error is obtained by equations (20), (21) using j ω instead of s
Figure RE-GDA0002198058400000112
A phase difference of
Figure RE-GDA0002198058400000113
Due to the fact that
Figure RE-GDA0002198058400000114
Can obtain the product
Figure RE-GDA0002198058400000115
I.e. the observer's estimate z of the total disturbance3NPhase lead from z3L。
For a continuous periodic uncertain disturbance f (t) ═ Ksin (ω t), K and ω are the amplitude and frequency, respectively, of the sinusoidal signal. To quantitatively obtain the estimated deviation of the observer to the sinusoidal disturbance, the observed deviation e is defined as z-x, and the system error dynamics obtained from (2) and (3) are
Figure RE-GDA0002198058400000116
Can be rewritten as
Figure RE-GDA0002198058400000117
In the formula
Figure RE-GDA0002198058400000118
Solve (25) by
Figure RE-GDA0002198058400000119
Since the matrix A has negative eigenvalues, in the case of a bounded initial error e (0), there is
Figure RE-GDA0002198058400000121
Then, the second part of the analytical formula (26) is emphasized
Figure RE-GDA0002198058400000122
In case of sinusoidal interference, h (t) ═ K ω cos (ω t), there are
Figure RE-GDA0002198058400000123
Division and integration (29) to obtain
Figure RE-GDA0002198058400000124
Due to the matrix
Figure RE-GDA0002198058400000125
Is positively and reversibly fixed, so
Figure RE-GDA0002198058400000126
Then
Figure RE-GDA0002198058400000127
Equation (32) shows that the upper bound of the estimation error is related to the sine perturbationThe frequency ω and amplitude K of the motion are related to the matrix a. Since the frequency and amplitude ω, K of the disturbance is determined by the disturbance itself, the matrix a is determined by the construction of the observer. Then, when different observer designs are quantitatively analyzed, the limit of the sinusoidal disturbance observation error is estimated. For the distinction, let the matrix of the linear extended observer be AL(ii) a The newly proposed observer matrix is AE. Is provided with
Figure RE-GDA0002198058400000128
By
Figure RE-GDA0002198058400000129
Upper bound of estimation error of linear extended state observer is obtained by equation (32)
Figure RE-GDA00021980584000001210
Is composed of
Figure RE-GDA0002198058400000131
Obtained by the formulae (2), (7)
Figure RE-GDA0002198058400000132
Similarly, the upper bound of the estimated error of the observer is obtained from equation (32)
Figure RE-GDA0002198058400000133
Is composed of
Figure RE-GDA0002198058400000134
Due to the fact that
Figure RE-GDA0002198058400000135
So the first and second terms of the proposed observer estimation error are smaller than the linear extended state observerThe comparison result is consistent with the data in table 1.
The estimation value of the estimated state is corrected by using the error of the estimated state, so that the state estimation is more accurate, and therefore, the method has the advantages that the estimation value of the system state and the total disturbance estimation value can be obtained more quickly and accurately, and the control effect of the whole system is improved.

Claims (2)

1. An anti-interference control system comprises an extended state observer based on error correction, a controller and a controlled object, and is characterized in that: the extended state observer is used for generating a system state estimated value z of the controlled object according to the control signal and the output signal of the controlled object1、z2And the total disturbance estimate z3(ii) a The controller is used for generating a control signal u containing interference compensation according to the system state estimation value, the total disturbance estimation value and the set value r and inputting the control signal u to a controlled object; wherein the dilated state observer based on error correction is represented by equation 7 as follows:
Figure FDA0003544932890000011
z1,z2,z3for the estimation of the output, the rate of change of the output and the total disturbance of the extended state observer, β ═ β1,β2,β3]TFor observer gain, β 1, β 2, β 3 are components of the observer gain vector, p1＝y，
Figure FDA0003544932890000013
b0For adjustable control gain, y is the system output.
2. A design method of an anti-interference control system is characterized by comprising the following steps:
step 1: designing a generic linear extended state observer
Step 2: extended state observer for correcting designed state error
And step 3: extended state observer based on error correction
The extended state observer based on error correction is designed as follows:
Figure FDA0003544932890000012
wherein z is1,z2,z3For the estimation of the output, the rate of change of the output and the total disturbance of the extended state observer, β ═ β1,β2,β3]TFor observer gain, β 1, β 2, β 3 are components of the observer gain vector, p1＝y，
Figure FDA0003544932890000021
b0For adjustable control gain, u is the control signal and y is the system output;
and 4, step 4: control law for design controller
And 5: forming a closed loop immunity control system.
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