CN113504731B - Sliding mode disturbance observer design method based on parallel controller - Google Patents

Sliding mode disturbance observer design method based on parallel controller Download PDF

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CN113504731B
CN113504731B CN202110919088.7A CN202110919088A CN113504731B CN 113504731 B CN113504731 B CN 113504731B CN 202110919088 A CN202110919088 A CN 202110919088A CN 113504731 B CN113504731 B CN 113504731B
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毛耀
张兵
陈兴龙
谭毅
任戈
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Abstract

The invention discloses a sliding mode disturbance observer design method based on a parallel controller, which is used for compensating system disturbance and improving disturbance inhibition capability of a system so as to meet the tracking control requirement of higher precision. Aiming at the defect that the prior sliding mode disturbance observer has large disturbance estimation error on higher frequency, the invention provides a disturbance estimation method by using two controllers with parallel structures, wherein one controller realizes rough estimation on system disturbance, which is called a disturbance rough estimation controller, and the other sliding mode controller is connected in parallel outside the disturbance rough estimation controller to realize compensation on disturbance estimation error, which is called a sliding mode compensation controller, so as to finally realize more accurate estimation on system disturbance and improve disturbance inhibition capability of the system.

Description

Sliding mode disturbance observer design method based on parallel controller
Technical Field
The invention belongs to the field of disturbance estimation and suppression, and particularly relates to a sliding mode disturbance observer design method based on a parallel controller, which is mainly used for estimating internal and/or external disturbance of a photoelectric tracking system and improving disturbance suppression capability of the system.
Background
The photoelectric tracking system is often applied to various complex motion scenes such as various motion platforms, the requirement on the stable precision of the visual axis of the system is very strict, and how to better inhibit various complex disturbance sources facing the system is always a research hot spot. The disturbance faced by the photoelectric tracking system consists of external disturbance and internal disturbance, wherein external disturbance factors include vibration of a carrier, wind resistance in motion and the like, while internal disturbance factors generally include modeling errors, friction among shafting, unbalanced moment inside a platform, motor moment fluctuation and the like, and the existence of the disturbance affects system stability and tracking precision, and even instability of closed-loop system control is possibly caused. To overcome the effects of these disturbances, a disturbance observer technique that can be used to estimate the disturbance has been widely developed, and disturbance observation-based control is considered one of the most promising disturbance suppression methods.
In order to obtain a more accurate disturbance estimation result, a sliding mode controller is introduced into the design of a disturbance observer, although the sliding mode disturbance observer of the prior art provides a certain disturbance estimation accuracy, some sliding mode disturbance observers are designed based on slow time-varying disturbance, i.e. the first-order reciprocal of disturbance is assumed to be equal to zero, such as document Adaptive sliding mode current control with sliding mode disturbance observer for PMSM drives (DengY T, wang J L, li H W, et al [ J ]. Isa transactions.2018) [ Current Sliding Mode Control with a Load Sliding Mode Observer for Permanent Magnet Synchronous Machines (Jin N, wang X, wu X. [ J ]. Journal of power electronics.2014) and the like, the disturbance estimation effect obtained by such disturbance observers is not ideal, the disturbance estimation error increases with the increase of disturbance frequency, and thus the disturbance estimation effect on higher frequency disturbance is slightly poor, and the defect of such a disturbance observer is similarly illustrated in the "dynamic servo system based on a composite control method of a sliding mode expansion state observer" of patent publication No. CN104898550A of Xiong Shaofeng et al. However, in practice, the disturbance change speed is fast, the frequency division range is wide, for example, the document (Zhang Qiandan [ D ]. University of chinese academy of sciences. 2018) describes the analysis of the disturbance frequency distribution of the actual spaceborne telescope, and the disturbance distribution is between 0 and 100Hz, and the photoelectric tracking system requires high stable precision in a wide frequency range, so that a disturbance observer with high precision in the whole frequency range of the disturbance distribution is needed.
Disclosure of Invention
In order to obtain small-error estimation of disturbance in the whole frequency range of disturbance distribution and further improve disturbance rejection capability of a photoelectric tracking system, disturbance estimation errors are reduced by designing observers with two controllers in parallel connection, one of the controllers is used for achieving rough disturbance estimation, the other sliding mode compensation controller is connected in parallel with the rough disturbance estimation controller to achieve compensation of disturbance estimation errors, the disturbance amplitude range and the disturbance change rate can be effectively reduced after rough disturbance estimation, at the moment, the sliding mode compensation controller is equivalent to estimating disturbance with small amplitude and relatively slow rate change, so that disturbance with large amplitude and fast change rate is avoided, high-precision disturbance estimation can be achieved due to the fact that the controller has better rejection capability of small-amplitude slow-time-varying disturbance, and zero-error estimation of system disturbance can be achieved theoretically.
The invention adopts the following technical scheme that the design method of the sliding mode disturbance observer based on the parallel controller comprises the following steps:
(1) The system state equation is established as follows:
Figure BDA0003206720310000021
wherein ,
Figure BDA0003206720310000022
C＝[1，0]t, b are parameters related to the system. />
Figure BDA0003206720310000023
Is a state vector, θ is a position response, +.>
Figure BDA0003206720310000024
For the speed response, u is the control input, y is the system output, and d is the system lumped disturbance.
(2) The composite control rate of the design system is as follows:
Figure BDA0003206720310000025
wherein ur Is an output of the closed-loop controller,
Figure BDA0003206720310000026
for disturbance estimation, +.>
Figure BDA0003206720310000027
To perturb the rough estimate, u c Is a disturbance compensation value.
(3) The state observer of the design system is:
Figure BDA0003206720310000028
wherein z= [ z ] 1 ，z 2 ] T For the estimated value of the system state x, f= [ F 1 ，F 2 ] T Is an observer gain matrix, and the matrix (a-FC) satisfies the Hurwitz condition.
(4) Designing a disturbance rough estimation controller, and estimating an approximate value of system disturbance:
let u at this point, the lumped disturbance d=0 of the system (1) is assumed, while let u c =0, defining a state estimation error e= [ E ] 1 ，e 2 ] T =z-x, then combining (1), (2) and (3) yields:
Figure BDA0003206720310000029
/>
Figure BDA0003206720310000031
definition:
Figure BDA0003206720310000032
where K is the feedback gain matrix and G is the normal number matrix. The derivative of S can be expressed as:
Figure BDA0003206720310000033
order the
Figure BDA0003206720310000034
The rough estimate of the disturbance can be expressed as:
Figure BDA0003206720310000035
(5) Designing a sliding mode compensation controller:
lumped of the actual system (1)Disturbance d is not equal to 0, and coarse disturbance estimation error is defined
Figure BDA0003206720310000036
Then combining (1), (2) and (3) to obtain:
Figure BDA0003206720310000037
/>
to compensate for the disturbance coarse estimation error, the following equation is constructed:
Figure BDA0003206720310000038
defining a sliding die surface as follows:
s＝HE
(10)
where H is the normal number matrix. The design approach rate is as follows:
Figure BDA0003206720310000039
wherein k > 0 is the sliding mode convergence rate, epsilon > 0 is the switching gain, and the defined control rate is:
Figure BDA00032067203100000310
when the system enters a sliding mode
Figure BDA00032067203100000311
E=0, then->
Figure BDA00032067203100000312
Then:
Figure BDA00032067203100000313
/>
Figure BDA0003206720310000041
(6) In order to reduce the slip-mode buffeting caused by the sign function sgn(s) in (12), use is made of
Figure BDA0003206720310000042
Function substitution:
Figure BDA0003206720310000043
wherein α > 0 is an adjustable factor.
(7) For convenience of application, assuming a sampling period of h, the first order Euler method discrete system (3) is used:
z(n+1)＝z(n)+h[(A-FC)z(n)+Fy(n)+Bu r (n)] (15)
wherein z (n) is an estimated value of the system state at the nth time, y (n) is a position state of the system at the nth time, z (n+1) is an estimated value of the system state at the n+1 time, u r And (n) is the output value of the closed-loop controller at the nth moment.
The invention has the following advantages:
(1) The invention improves the accuracy of system disturbance estimation and can theoretically realize error-free estimation lumped disturbance.
(2) The invention can estimate small error for disturbance with higher frequency, even white noise disturbance, and has the characteristic of insensitivity to disturbance frequency change.
(3) The invention comprises two controllers with parallel structure, usually a single controller needs larger controller gain to realize smaller estimation error, while when adopting parallel controller, disturbance rough estimation reduces disturbance boundary and disturbance change rate, so that the two controllers can obtain smaller disturbance estimation error when having small gain.
(4) The invention solves the differential equation by using a first-order Euler method, has simple algorithm and strong transplanting capability, and is easy to be practically implemented.
Drawings
FIG. 1 is a block diagram of a sliding mode disturbance observer based on a parallel controller of the present invention;
FIG. 2 is a graph of the results of the square wave disturbance estimation of the present invention;
FIG. 3 is a graph of the results of the present invention for low frequency high amplitude sinusoidal disturbance estimation;
FIG. 4 is a graph of the results of the present invention for high frequency low amplitude sinusoidal disturbance estimation;
FIG. 5 is a graph of the results of white noise disturbance estimation of the present invention;
fig. 6 is a graph comparing the root mean square error of the general scheme and the scheme of the invention for estimating the sinusoidal disturbance of different frequencies.
Detailed Description
The following describes the embodiments of the present invention in detail with reference to the drawings.
Example 1
A sliding mode disturbance observer block diagram based on a parallel controller is shown in fig. 1, and comprises a system model, a state observer, a disturbance rough estimation controller and a sliding mode compensation controller, and in order to achieve the purpose of the invention, the method comprises the following steps:
(1) The system state equation is established as follows:
Figure BDA0003206720310000051
wherein ,
Figure BDA0003206720310000052
C＝[1，0]t, b are parameters related to the system. />
Figure BDA0003206720310000053
Is a state vector, θ is a position response, +.>
Figure BDA0003206720310000054
For the speed response, u is the control input, y is the system output, and d is the system lumped disturbance.
(2) The composite control rate of the design system is as follows:
Figure BDA0003206720310000055
wherein ur Is an output of the closed-loop controller,
Figure BDA0003206720310000056
for disturbance estimation, +.>
Figure BDA0003206720310000057
To perturb the rough estimate, u c Is a disturbance compensation value.
(3) The state observer of the design system is:
Figure BDA0003206720310000058
wherein z= [ z ] 1 ，z 2 ] T For the estimated value of the system state x, f= [ F 1 ，F 2 ] T Is an observer gain matrix, and the matrix (a-FC) satisfies the Hurwitz condition.
(4) Designing a disturbance rough estimation controller, and estimating an approximate value of system disturbance:
let u at this point, the lumped disturbance d=0 of the system (1) is assumed, while let u c =0, defining a state estimation error e= [ E ] 1 ，e 2 ] T =z-x, then combining (1), (2) and (3) yields:
Figure BDA0003206720310000059
definition:
Figure BDA00032067203100000510
where K is the feedback gain matrix and G is the normal number matrix. The derivative of S can be expressed as:
Figure BDA00032067203100000511
Figure BDA0003206720310000061
order the
Figure BDA0003206720310000062
The rough estimate of the disturbance can be expressed as:
Figure BDA0003206720310000063
(5) Designing a sliding mode compensation controller:
lumped disturbance d.noteq.0 of the real system (1), defining coarse disturbance estimation error
Figure BDA0003206720310000064
Then combining (1), (2) and (3) to obtain: />
Figure BDA0003206720310000065
To compensate for the disturbance coarse estimation error, the following equation is constructed:
Figure BDA0003206720310000066
defining a sliding die surface as follows:
s＝HE (10)
where H is the normal number matrix. The design approach rate is as follows:
Figure BDA0003206720310000067
where k > 0 is the sliding mode convergence rate and ε > 0 is the switching gain. Defining a control rate as follows:
Figure BDA0003206720310000068
when the system enters a sliding mode
Figure BDA0003206720310000069
E=0, then->
Figure BDA00032067203100000610
Then:
Figure BDA00032067203100000611
(6) In order to reduce the slip-mode buffeting caused by the sign function sgn(s) in (12), use is made of
Figure BDA00032067203100000612
Function substitution:
Figure BDA00032067203100000613
wherein α > 0 is an adjustable factor.
(7) For convenience of application, assuming a sampling period of h, the first order Euler method discrete system (3) is used:
z(n+1)＝z(n)+h[(A-FC)z(n)+Fy(n)+Bu r (n)] (15)
wherein z (n) is an estimated value of the system state at the nth time, y (n) is the position state of the system at the nth time, z (n+1) is an estimated value of the system state at the (n+1) th time, u r And (n) is the output value of the closed-loop controller at the nth moment.
Example 2
Given a pod tracking system t=0.0398 and b= 1.0524 in the field of photoelectric tracking, the following describes the design process and effect of the present invention in detail:
(1) F, designing a matrix F: the characteristic values of the matrix A-FC are all positioned on the left half-open complex plane, namely all the characteristic values of the matrix A-FC have negative real parts, and larger characteristic values can be selected to enhance the anti-interference capability of the observer, so that the characteristic values of the matrix A-FC are precededAre all-1, and F= [ -23, 582 are obtained by inverse solution] T 。
(2) Programming is performed according to the block diagram structure and the euler discrete form shown in fig. 1, taking a sampling period of h=0.001 s.
(3) Under the condition that only the disturbance rough estimate controller exists, the initial state x= [ 00 ] of the system is taken] T Given step disturbance, a reasonable K matrix is debugged to achieve a good estimation effect, and K= [900, 30 after debugging is finished]The matrix G may be generally fixed as g= [1,1]。
(4) Adding a sliding mode compensation controller, debugging the matrix H, the parameters k, epsilon and alpha to further reduce disturbance estimation errors, and after the completion of the debugging, obtaining H= [7,1], k=5, epsilon=0.1 and alpha=10.
(5) Given square wave disturbance d=sgn (sin 20 pi t), the closed loop controller outputs ur=sin 2 pi t, and the disturbance estimation result is shown in fig. 2, where the maximum absolute value of the disturbance estimation error is 0.0023, that is, at the rising edge and the falling edge of the disturbance mutation, the disturbance estimation error is 0.1132% of the disturbance variable quantity.
(6) Given a low frequency high amplitude sinusoidal disturbance d=10sin 4 pi t, the closed loop controller outputs u r As shown in fig. 3, the maximum absolute value of the disturbance estimation error is 0.0017, i.e., the disturbance estimation errors are all less than 0.017% of the maximum disturbance amplitude.
(7) Given a high frequency low amplitude sinusoidal disturbance d=sin 200 pi t, the closed loop controller outputs u r As shown in fig. 4, the maximum absolute value of the disturbance estimation error is 0.0011, i.e., the disturbance estimation errors are all less than 0.11% of the maximum disturbance amplitude.
(8) Given any white noise disturbance as shown in fig. 5 (a), the disturbance estimation result is shown in fig. 5 (b), and it can be seen that, for any white noise disturbance, the disturbance observer designed by the invention still has a high-precision estimation effect. The root mean square of the disturbance is 0.9702, the root mean square of the disturbance estimation error is 0.0029, and the root mean square of the disturbance estimation error is 0.3% of the root mean square of the disturbance.
(9) Given sinusoidal disturbance d=sin2pi ω d t, closed loop controller output u r =sin 2 pi t, the result of fig. 6 is a disturbance estimation root mean square errorThe difference varies with the sinusoidal disturbance frequency omega d The curves were compared with the results using only the slip-mode compensation controller of fig. 1 (where k=40) and only the disturbance rough estimation controller. Compared with the scheme using only a single controller, the scheme of the invention has smaller root mean square error of disturbance estimation in the disturbance frequency range of 0-100 Hz, can obviously improve the accuracy of disturbance observer to disturbance signal estimation, and proves the effectiveness of the invention.

Claims (4)

1. A sliding mode disturbance observer design method based on a parallel controller is characterized in that: the method comprises the following implementation steps:
step (1): establishing a system state equation; the system state equation is:
Figure FDA0003206720300000011
wherein ,
Figure FDA0003206720300000012
C＝[1，0]t, b is a system-related parameter, +.>
Figure FDA0003206720300000013
Is a state vector, θ is a position response, +.>
Figure FDA0003206720300000014
The speed response is realized, u is a control input, y is a system output, and d is a system lumped disturbance;
step (2): designing a system state observer; the composite control rate of the state observer is as follows:
Figure FDA0003206720300000015
wherein ur Is an output of the closed-loop controller,
Figure FDA0003206720300000016
for disturbance estimation, +.>
Figure FDA0003206720300000017
To perturb the rough estimate, u c Is a disturbance compensation value;
the system state observer is:
Figure FDA0003206720300000018
wherein z= [ z ] 1 ，z 2 ] T For the estimated value of the system state x, f= [ F 1 ，F 2 ] T Is an observer gain matrix, and the matrix (A-FC) satisfies the Hurwitz condition;
step (3): designing a disturbance rough estimation controller, and estimating a disturbance approximation value of a system; the disturbance coarse estimation controller is as follows:
Figure FDA0003206720300000019
wherein K is a feedback gain matrix, G is a normal number matrix, e= [ E ] 1 ，e 2 ] T =z-x is the state estimation error;
step (4): designing a sliding mode compensation controller to compensate disturbance coarse estimation errors; the sliding mode compensation controller is as follows:
the sliding die surface is as follows:
s＝HE
the control rate is as follows:
u c ＝-ks-εsgn(s)
wherein H is a normal number matrix, k is more than 0 and is the sliding mode convergence rate, epsilon is more than 0 and is the switching gain, sgn is a sign function;
step (5): for ease of application, the discrete observer system differential equation.
2. A parallel controller-based sliding mode disturbance observer as claimed in claim 1The design method is characterized in that: in step (4), to avoid slip form buffeting, the following is used
Figure FDA00032067203000000110
The function replaces the sign function sgn(s):
Figure FDA00032067203000000111
wherein α > 0 is an adjustable factor.
3. The sliding mode disturbance observer design method based on the parallel controller according to claim 1, wherein the method comprises the following steps: and (5) solving a differential equation by using a first-order Euler method for practical application convenience.
4. A sliding mode disturbance observer based on a parallel controller is characterized in that: the sliding mode disturbance observer is designed according to the sliding mode disturbance observer design method based on the parallel controller, which can realize small error estimation on system disturbance and improve disturbance inhibition capability of the system.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2004038697A (en) * 2002-07-05 2004-02-05 Fujitsu Ltd Positioning controller
CN103166565A (en) * 2011-12-13 2013-06-19 上海电气集团股份有限公司 Sliding mold controller and bimodal control system
CN105305425A (en) * 2015-10-20 2016-02-03 国家电网公司 UPFC buffeting control method of additional interference observer
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