CN111459031A - Learning-oriented disturbance observer design method - Google Patents

Learning-oriented disturbance observer design method Download PDF

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CN111459031A
CN111459031A CN202010314920.6A CN202010314920A CN111459031A CN 111459031 A CN111459031 A CN 111459031A CN 202010314920 A CN202010314920 A CN 202010314920A CN 111459031 A CN111459031 A CN 111459031A
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state
disturbance
matrix
learning
substances
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池荣虎
魏阳春
姚文龙
林娜
惠宇
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Qingdao University of Science and Technology
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Abstract

The invention discloses a learning-oriented disturbance observer design method, and belongs to the field of intelligent control. The invention mainly aims at the problem of total disturbance estimation of a repeatable system with non-repetitive external disturbance in a data-driven framework, and the scheme of a disturbance observer is as follows: establishing a system with non-repetitive disturbances; the system iteration is linearized into a linear data model, and the non-repetitive disturbance is converted into total disturbance; designing an iterative update algorithm to estimate a gradient matrix in the linear data model; aiming at a system with a measurable state, a learning disturbance observer based on the state is designed; and aiming at the system with an unmeasurable state, designing a learning disturbance observer based on output. The learning-oriented disturbance observer design method disclosed by the invention can estimate the total disturbance in the system under a data-driven framework, only utilizes a small amount of model information, has good adaptability, can learn from the running data of the previous batch, and can estimate the total disturbance more accurately.

Description

Learning-oriented disturbance observer design method
Technical Field
The invention belongs to the field of intelligent control, and particularly relates to a design method of a learning disturbance observer based on data driving.
Background
In industrial applications, the presence of disturbances is inevitable. The system disturbance comprises non-linear disturbance, time lag, sensor measurement noise, external disturbance and unknown input disturbance. Disturbances in the system can severely affect control performance. To account for disturbances in the system, a disturbance observer for estimating uncertainty can be designed and then incorporated into the controller to compensate for its effect on control performance.
At present, linear disturbance observers are well developed, and have many theoretical achievements and practical applications, such as frequency domain-based disturbance observers, reduced order disturbance observers, time-lag disturbance observers, generalized PI type observers and the like. On the other hand, since a nonlinear system is more common than a linear system in practical application, in recent years, many researches have been made for scholars at home and abroad on a nonlinear disturbance observer, such as a high-order disturbance observer, an extended high-gain state observer, and the like.
However, most disturbance observers, whether linear or non-linear, require known model information as a priori. In other words, these methods are model-based. However, as the actual process becomes more complex and larger, modeling the control object using physicochemical principles or recognition methods is actually too difficult. Therefore, the above-described model-based observer may encounter challenges and difficulties when applied to practical problems, and thus data-driven modeling, control, and optimization methods become more popular. On the other hand, many practical systems operate repeatedly at fixed times. Such as industrial robots, high speed trains, flow systems, multi-agent systems, etc. For such a repetitive system, the iterative learning control can use the control information in the previous trial for perfect tracking. How to estimate the non-repetitive disturbance in iterative learning control in a data-driven framework is a learning disturbance observer method based on data driving, which is a problem to be solved urgently in the field at present.
In order to estimate the non-repetitive disturbances in repeatable systems, it is necessary to design a learning disturbance observer that can estimate the repetitive disturbances independent of the system model.
Disclosure of Invention
The invention discloses a learning-oriented disturbance observer design method which aims to solve the technical problem that a designed learning disturbance observer can estimate the total disturbance of a system by using system operation data along an iteration axis under the condition that non-repeated disturbance exists in a repeatable system.
The purpose of the invention is realized by the following technical scheme:
the invention discloses a learning-oriented disturbance observer design method, which mainly aims at the problem of total disturbance estimation of a repeatable system with non-repetitive external disturbance in a data-driven framework.
The invention discloses a learning-oriented disturbance observer design method, which comprises the following steps:
step 1, establishing a system with non-repetitive disturbance:
(1) consider a discrete multiple-input multiple-output linear system with perturbations as follows:
Figure BDA0002459165320000021
wherein the content of the first and second substances,
xk(t+1)∈Rnrepresenting the state of the system at the kth iteration time t + 1;
uk(t)∈Rlrepresenting the control input of the system at the time t of the kth iteration;
dk(t)∈Rnrepresenting the disturbance in the system at the time t of the kth iteration;
yk(t)∈Rmrepresenting the output of the system at the time t of the kth iteration;
A(t)∈Rn×n，B(t)∈Rn×l，C(t)∈Rm×nrepresenting a state transition matrix of the system at the time t;
the system satisfies the following conditions:
initial state x of systemk(0) Not changing, i.e. xk(0)＝x0；
Disturbance d of the systemk(t) is bounded, i.e. | | dk(t)||≤bd；
The system meets the condition of broad sense L ipschitz when △ uk(t)≠0、△dk(t)≠0、△xk(t) ≠ 0, | | △ xk(t+1)||≤b1||△uk(t)||+b2||△dk(t)||+b3||△xk(t)||；
Wherein the content of the first and second substances,
x0is a constant vector;
bd、b1、b2、b3are all positive numbers;
△uk(t)＝uk(t)-uk-1(t)，△dk(t)＝dk(t)-dk-1(t)，△xk(t)＝xk(t)-xk-1(t)；
(2) considering a discrete multiple-input multiple-output non-linear non-affine system with perturbation, the system is given as follows:
Figure BDA0002459165320000031
wherein the content of the first and second substances,
f(·)∈Rnrepresenting an unknown non-linear function;
the system satisfies the following conditions:
initial state x of systemk(0) Not changing, i.e. xk(0)＝x0；
Disturbance d of the systemk(t) is bounded, i.e. | | dk(t)||≤bd；
The system meets the condition of broad sense L ipschitz when △ uk(t)≠0、△dk(t)≠0、△xk(t) ≠ 0, | | △ xk(t+1)||≤b1||△uk(t)||+b2||△dk(t)||+b3||△xk(t)||；
Step 2, the system iteration with the non-repeated disturbance is linearized into a linear data model, and the non-repeated disturbance is converted into total disturbance in the linear data model:
Figure BDA0002459165320000041
wherein the content of the first and second substances,
Φk(t)∈Rn×m·(t+1)is an unknown gradient matrix;
Uk(t)＝[uk(0)T,uk(1)T,...,uk(t)T]T∈Rm·(t+1)；
△Uk(t)＝Uk(t)-Uk-1(t)；
k(t)∈Rnis the total perturbation due to non-repetitive perturbations;
step 3, designing an iterative update algorithm to estimate a gradient matrix in the linear data model:
(1) for a state measurable system, the iterative update algorithm is as follows:
Figure BDA0002459165320000042
wherein the content of the first and second substances,
Figure BDA0002459165320000043
is phik(t) an estimate of;
η∈ (0,2) is a step factor;
μ is a positive weighting factor;
(2) for a state-unmeasured system, the iterative update algorithm is as follows:
Figure BDA0002459165320000044
wherein the content of the first and second substances,
(MC(t+1))L+represents the left inverse of the matrix MC (t +1), satisfies (MC (t +1))L+MC(t+1)＝I；
M∈Rm×mIs a constant matrix; i represents an identity matrix;
and 4, designing a learning disturbance observer based on the state aiming at a system with a measurable state:
Figure BDA0002459165320000045
Figure BDA0002459165320000051
wherein the content of the first and second substances,
Figure BDA0002459165320000052
is thatk(t) an estimate of;
zk(t)∈Rnis an intermediate state variable;
K＝(In-)∈Rn×nparameters of a disturbance observer are obtained;
＝diag{γ1,γ2,…,γnis a diagonal matrix; | gammai|<1；i＝1,2,…,n；
And 5, designing a learning disturbance observer based on output aiming at a system with an unmeasured state:
Figure BDA0002459165320000053
Figure BDA0002459165320000054
Figure BDA0002459165320000055
Figure BDA0002459165320000056
wherein the content of the first and second substances,
C(t+1)+representing momentsPseudo-inverse of the matrix C;
Figure BDA0002459165320000057
is a compensation term;
ξk(t) is an intermediate state variable;
Figure BDA0002459165320000058
W(t+1)＝V-Q(t+1)C(t+1)；
the matrix S (t), Q (t), R (t) satisfy (I)h-R(t))(V-Q(t)C(t))-S(t)C(t)＝0。
Further, the iterative linearization of the system used in step 2 mainly includes the following steps:
step 2.1, using iterative dynamic linearization on the linear system described in step 1 to obtain:
Figure BDA0002459165320000061
the system is differentiated along an iteration axis and a differential median theorem is used, so that a linear data model with total disturbance can be obtained:
Figure BDA0002459165320000062
wherein the content of the first and second substances,
Figure BDA0002459165320000063
is an unknown gradient matrix;
Figure BDA0002459165320000064
is the total disturbance;
||Φk(t)||≤bΦ、||k(t)||≤b；bΦ，bare all positive numbers;
step 2.2, for the nonlinear system described in step 1, iterative dynamic linearization is used, so that:
Figure BDA0002459165320000065
wherein the content of the first and second substances,
gt(. h) is a state transfer function, which is a complex function of f (·);
the system is differentiated along an iteration axis and a differential median theorem is used, so that a linear data model with total disturbance can be obtained:
Figure BDA0002459165320000071
wherein the content of the first and second substances,
Φk(t) is an unknown gradient matrix;k(t) total perturbation;
||Φk(t)||≤bΦ、||k(t)||≤b；bΦ、bare all positive numbers;
still further, in step 3, the design of the iterative update algorithm for estimating the unknown gradient matrix in the linear data model of the state-unmeasured system is mainly realized by the following method:
and 3.1, expressing the state in an output form because the state of the system which cannot be measured cannot be directly obtained:
Figure BDA0002459165320000072
wherein, (MC (t +1))L+Represents the left inverse of the matrix MC (t +1), satisfying:
(MC(t+1))L+MC(t+1)＝I；
and 3.2, converting the state in the iterative update algorithm aiming at the state measurable system in the step 3 into an output form by using the method, thereby obtaining the iterative update algorithm aiming at the state undetectable system:
Figure BDA0002459165320000073
in a still further aspect of the present invention,
the design of the learning disturbance observer based on the output aiming at the state-immeasurable system in the step 5 mainly comprises the following steps:
and 5.1, aiming at the system with the unmeasured state, representing the state in the learning disturbance observer based on the state in the step 4 in an output form:
Figure BDA0002459165320000074
Figure BDA0002459165320000081
wherein the content of the first and second substances,
C(t+1)+yk-1(t +1) is used instead of the system state xk(t)；
C(t+1)+Represents the pseudo-inverse of matrix C;
KN(t+1)xk-1(t +1) is a compensation term;
N(t+1)＝In-C(t+1)+C(t+1)；
step 5.2, the unknown quantity KN (t +1) x is addedk-1(t +1) matrix-transforming to estimate:
KN(t+1)＝H(t+1)V；
wherein, V ∈ Rh×nIs a constant matrix, H (t +1) ∈ Rn×h；h＝rank(KN(t+1))；
Thus, unknown KN (t +1) xk-1(t +1) transforms:
KN(t+1)xk-1(t+1)＝H(t+1)Vxk-1(t+1)；
unknown quantity Vxk-1(t +1) is defined as an unknown state variable βk(t)∈Rh；
Step 5.3, designing an estimator for the unknown variables βk(t) estimating;
Figure BDA0002459165320000082
Figure BDA0002459165320000083
wherein the content of the first and second substances,
Figure BDA0002459165320000084
representative βk(t) estimation;
ξk(t)∈Rhis an intermediate state variable;
Figure BDA0002459165320000085
W(t+1)＝V-Q(t+1)C(t+1)；
matrix S (t) ∈ Rh×m，Q(t)∈Rh×m，R(t)∈Rh×hSatisfies the following conditions:
(Ih-R(t))(V-Q(t)C(t))-S(t)C(t)＝0；
thus, an output-based learning disturbance observer is obtained:
Figure BDA0002459165320000091
Figure BDA0002459165320000092
Figure BDA0002459165320000093
Figure BDA0002459165320000094
has the advantages that:
1. the invention discloses a learning-oriented disturbance observer design method, which utilizes a data driving method, particularly establishes a linear data model containing total disturbance by using input, state and output data of a repetitive system, wherein the total disturbance comprises all influences of the disturbance from an initial moment to a current moment between two continuous iterations on the state and the output of the system, and the dependence of the method on the system model is reduced.
2. The invention discloses a learning-oriented disturbance observer design method, which considers a state measurable system and a state immeasurable system, provides a learning disturbance observer based on a system state and a learning disturbance observer based on system output, and both the two proposed disturbance observers are pertinently executed by using system data of previous tests along an iteration direction, so that the running data of previous batches can be effectively utilized, and the estimation of the total disturbance in a linear data model is more accurate along with the increase of iteration times.
Other features and advantages of the present invention will become more apparent from the following detailed description of the invention when taken in conjunction with the accompanying drawings.
Drawings
FIG. 1 is a flow chart of a learning-oriented disturbance observer design method proposed by the present invention;
FIG. 2 is a tracking curve of a state-based learning disturbance observer proposed by the present invention;
FIG. 3 is an estimation error curve of a state-based learning disturbance observer according to the present invention;
FIG. 4 is a comparison curve of the estimation error of the learning disturbance observer and the reduced order disturbance observer based on the state proposed by the present invention;
FIG. 5 is a comparison curve of the estimation error of the learning disturbance observer and the reduced order disturbance observer based on the state proposed by the present invention;
FIG. 6 is a tracking curve of the learning disturbance observer based on output proposed by the present invention;
FIG. 7 is an estimated error curve of the learning disturbance observer based on output proposed by the present invention;
FIG. 8 is a comparison curve of the estimated error of the learning disturbance observer and the reduced order disturbance observer based on the output proposed by the present invention;
FIG. 9 is a comparison curve of the estimated error of the learning disturbance observer and the reduced order disturbance observer based on the output proposed by the present invention;
Detailed Description
To better illustrate the objects and advantages of the present invention, the following is a comparison with the conventional reduced order disturbance observer method, and the detailed description is further provided in conjunction with the embodiments and the accompanying drawings.
The invention mainly aims at the problem of total disturbance estimation of a repeatable system with non-repeated external disturbance, and establishes a linear data model containing the total disturbance by using a data-driven method and particularly by using input, state and output data of the repeatable system under a data-driven framework, wherein the non-repeated disturbance is expressed by the total disturbance in the linear data model. In order to estimate the total disturbance, two conditions of measurable state and non-measurable state are considered, and a learning disturbance observer based on the system state and a learning disturbance observer based on the system output are provided. Next, a detailed description will be given of a specific embodiment of the disturbance observer design method for the orientation learning.
Referring to fig. 1, a learning-oriented disturbance observer design method disclosed in this embodiment includes the following steps:
step S1: a system with non-repetitive perturbations is established.
(1) For a discrete multiple-input multiple-output linear system with perturbation, the following system is given:
Figure BDA0002459165320000101
wherein x isk(t+1)∈RnRepresenting the state of the system at the kth iteration time t + 1; u. ofk(t)∈RlRepresenting the control input of the system at the time t of the kth iteration; dk(t)∈RnRepresenting the disturbance in the system at time t of the kth iteration yk (t) ∈ RmRepresenting the output of the system at time t of the kth iteration A (t) ∈ Rn×n，B(t)∈Rn×l，C(t)∈Rm×nRepresenting the state transition matrix of the system at time t.
The system isThe system satisfies the following conditions: initial state x of systemk(0) Not changing, i.e. xk(0)＝x0(ii) a Disturbance d of the systemk(t) is bounded, i.e. | | dk(t)||≤bdThe system meets the general L ipschitz condition when △ uk(t)≠0、△dk(t)≠0、△xk(t) ≠ 0, | | △ xk(t+1)||≤b1||△uk(t)||+b2||△dk(t)||+b3||△xk(t)||。
Wherein x is0Is a constant vector; bd、b1、b2、b3Are positive numbers △ uk(t)＝uk(t)-uk-1(t)，△dk(t)＝dk(t)-dk-1(t)，△xk(t)＝xk(t)-xk-1(t)。
(2) For a discrete multiple-input multiple-output non-linear non-affine system with perturbation, the following system is given:
Figure BDA0002459165320000111
wherein, f (-) ∈ RnRepresenting an unknown non-linear function.
The system satisfies the following conditions: initial state x of systemk(0) Not changing, i.e. xk(0)＝x0(ii) a Disturbance d of the systemk(t) is bounded, i.e. | | dk(t)||≤bdThe system meets the general L ipschitz condition when △ uk(t)≠0、△dk(t)≠0、△xk(t) ≠ 0, | | △ xk(t+1)||≤b1||△uk(t)||+b2||△dk(t)||+b3||△xk(t)||。
Step S2: and (3) iteratively linearizing the system with the non-repeated disturbance into a linear data model, and converting the non-repeated disturbance into total disturbance in the linear data model.
Specifically, the method comprises the following steps:
step S21: for the linear system described in step 1, iterative dynamic linearization is used, such that:
Figure BDA0002459165320000121
the system is differentiated along an iteration axis and a differential median theorem is used, so that a linear data model with total disturbance can be obtained:
Figure BDA0002459165320000122
wherein the content of the first and second substances,
Figure BDA0002459165320000123
is an unknown gradient matrix;
Figure BDA0002459165320000124
is the total disturbance; i phik(t)||≤bΦ、||k(t)||≤b；bΦ、bAre all positive numbers.
Step S22: for the nonlinear system described in step 1, iterative dynamic linearization is used such that:
Figure BDA0002459165320000125
wherein, gt(. cndot.) is a state transfer function that is a complex function of f (. cndot.).
The system is differentiated along an iteration axis and a differential median theorem is used, so that a linear data model with total disturbance can be obtained:
Figure BDA0002459165320000126
wherein phik(t) is an unknown gradient matrix;k(t) total perturbation; i phik(t)||≤bΦ、||k(t)||≤b；bΦ、bAre all positive numbers.
Step S3: an iterative update algorithm is designed to estimate the gradient matrix in the linear data model.
Specifically, the method comprises the following steps:
s31: for a state measurable system, the iterative update algorithm is as follows:
Figure BDA0002459165320000131
wherein the content of the first and second substances,
Figure BDA0002459165320000132
is phik(t) estimate, η∈ (0,2) is a step factor, and μ is a positive weighting factor.
S32: for a state-undetectable system, since the state of the state-undetectable system is not directly available, the state is represented in output form:
Figure BDA0002459165320000133
wherein, (MC (t +1))L+Represents the left inverse of the matrix MC (t +1), satisfies (MC (t +1))L+MC (t +1) ═ I; i denotes an identity matrix.
S33: transforming states in an iterative update algorithm for a state-measurable system to an output form, thereby resulting in an iterative update algorithm for a state-untestable system:
Figure BDA0002459165320000134
step S4: aiming at a system with a measurable state, a learning disturbance observer based on the state is designed to estimate the total disturbance.
Figure BDA0002459165320000135
Figure BDA0002459165320000136
Wherein the content of the first and second substances,
Figure BDA0002459165320000137
is thatk(t) an estimate of; z is a radical ofk(t)∈RnIs an intermediate state variable; k ═ I (I)n-)∈Rn×nParameters of a disturbance observer are obtained; biag { γ ═1,γ2,...,γnIs a diagonal matrix; | gammai|<1；i＝1,2,...,n。
Step S5: and aiming at the system with an unmeasurable state, designing a learning disturbance observer based on output.
Specifically, the method comprises the following steps:
step S51: for a system whose state is not measurable, the state in the state-based learning disturbance observer in step 4 is represented in output form:
Figure BDA0002459165320000141
Figure BDA0002459165320000142
wherein, C (t +1)+yk-1(t +1) is used instead of the system state xk(t)；C(t+1)+Represents the pseudo-inverse of matrix C; KN (t +1) xk-1(t +1) is a compensation term; n (t +1) ═ In-C(t+1)+C(t+1)。
Step S52: mixing the unknown quantity KN (t +1) xk-1(t +1) performing matrix transformation:
KN(t+1)＝H(t+1)V (14)
wherein, V ∈ Rh×nIs a constant matrix, H (t +1) ∈ Rn×h；h＝rank(KN(t+1))。
Thus, unknown KN (t +1) xk-1(t +1) transforms:
KN(t+1)xk-1(t+1)＝H(t+1)Vxk-1(t+1) (15)
unknown quantity Vxk-1(t +1) is defined as an unknown state variable βk(t)∈Rh。
Step S53, designing an estimator for unknown variables βk(t) estimating:
Figure BDA0002459165320000143
Figure BDA0002459165320000144
wherein the content of the first and second substances,
Figure BDA0002459165320000145
representative βk(t) estimation ξk(t)∈RhIs an intermediate state variable;
Figure BDA0002459165320000146
W(t +1) ═ V-Q (t +1) C (t +1), matrix S (t) ∈ Rh×m，Q(t)∈Rh×m，R(t)∈Rh×hSatisfies the following conditions:
(Ih-R(t))(V-Q(t)C(t))-S(t)C(t)＝0 (18)
thus, an output-based learning disturbance observer is obtained:
Figure BDA0002459165320000151
Figure BDA0002459165320000152
Figure BDA0002459165320000153
Figure BDA0002459165320000154
the system with perturbations employed in the examples is as follows:
Figure BDA0002459165320000155
where, t ∈ {0., N }, N ═ 80, the state transition matrix of the system is:
Figure BDA0002459165320000156
Figure BDA0002459165320000157
the system disturbance is set as:
Figure BDA0002459165320000158
since the total disturbance of the system is difficult to obtain, the system state with the estimated disturbance is used, the estimated value of the output is compared with the actual value for verification, and a linear data model with the estimated gradient parameter vector and the estimated total disturbance is used as an estimation model of the system state and the output, as follows:
Figure BDA0002459165320000159
wherein the content of the first and second substances,
Figure BDA00024591653200001510
is to xk(t) estimating the value of the average of the measured values,
Figure BDA00024591653200001511
is as a pair yk(t) estimation.
In the described embodiment, the control input is set to uk(t)＝0.5sin(t/k)。
When the system state is available, the parameters are set to η ═ 1, μ ═ 1, ═ diag {0.1,0.3}, K ═ diag {0.9,0.7}, and M ═ InThe proposed state-based learning disturbance observer (fig. 2 and 3) is used. For comparison, the same parameters were used with a state-based learning disturbance observer and a state-based reduced order disturbance observer (fig. 4 and 5).
Wherein, the state-based reduced order disturbance observer is:
Figure BDA0002459165320000161
Figure BDA0002459165320000162
when the system state is not available, the parameters are set to η ═ 1, μ ═ 1, diag {0.1,0.3}, K ═ diag {0.9,0.7}, and M ═ In，V＝[1 -1]The proposed output-based learning disturbance observer (fig. 6 and 7) is used. For comparison, the same parameters were used with the output-based learning disturbance observer and the state-based reduced order disturbance observer (fig. 8 and 9).
Wherein the output-based reduced order disturbance observer is:
Figure BDA0002459165320000163
Figure BDA0002459165320000164
Figure BDA0002459165320000165
Figure BDA0002459165320000166
H1(t)V＝KN(t)，H2(t)V＝K(A-In)N(t)，Wu(t)＝(V-Q(t)C(t))B(t)，Wd(t) V-q (t) c (t), q (t), r (t), s (t) satisfy:
(V-Q(t)C(t))B(t)-R(t)(V-Q(t)C(t))-S(t)C(t)＝0 (31)
as can be seen from fig. 2 and 3: when the iteration times are small, the state-based learning disturbance observer has a poor estimation effect on the total disturbance and a large estimation error, but as the iteration times are increased, the state estimation error of the state-based learning disturbance observer is smaller and smaller, and the performance in estimating the total disturbance is good.
As can be seen from fig. 4 and 5: when the iteration times are smaller, the estimation effect of the state-based reduced order disturbance observer is better than that of the state-based learning disturbance observer, however, the estimation effect of the state-based learning disturbance observer can be improved along with the increase of the iteration times, and after a certain iteration times is reached, the estimation effect of the state-based learning disturbance observer is better than that of the reduced order disturbance observer.
As can be seen from fig. 6 and 7: when the iteration times are small, the estimation effect of the learning disturbance observer based on the output is poor, the maximum estimation error of each iteration is large, however, with the increase of the iteration times, the estimation error of the learning disturbance observer based on the output is smaller and smaller, and the total disturbance can be effectively estimated.
As can be seen from fig. 8 and 9: when the iteration times are smaller, the estimation effect of the reduced order disturbance observer is better than that of the learning disturbance observer based on the output, however, the learning disturbance observer based on the output can utilize the running data of the previous batch, so that after a certain iteration times is reached, the estimation effect of the learning disturbance observer based on the output is better than that of the reduced order disturbance observer.
In the embodiment, the system state of the estimated disturbance, the output estimated value and the estimation error are mainly used as performance parameters, and fig. 2 shows the estimation effect curves of the learning disturbance observer based on the state in the iterations 4, 10, 30 and 80. FIG. 3 shows a maximum estimation error curve for each iteration using a state-based learning disturbance observer. Fig. 4 and 5 show maximum estimation error contrast curves using a state-based learning disturbance observer and a reduced order disturbance observer. Fig. 6 shows the estimated effect curves at iterations 4, 10, 30, 80 using an output-based learning perturbation observer. FIG. 7 presents a maximum estimation error curve using an output-based learning disturbance observer. Fig. 8 and 9 present maximum estimation error contrast curves using an output-based learning disturbance observer and a reduced order disturbance observer. The figure shows that the learning-oriented disturbance observer method provided by the invention can well estimate the total disturbance in a data-driven framework, and compared with a reduced-order disturbance observer method, the learning disturbance observer provided by the invention can reduce the estimation error along with the increase of iteration times, is more suitable for a repeatable system, only utilizes a small amount of model information, and has better adaptability.
The above detailed description further illustrates the objects, technical solutions and advantages of the present invention, and it should be understood that the embodiments are only used for explaining the present invention and not for limiting the scope of the present invention, and modifications, equivalent substitutions, improvements and the like under the same principle and concept of the present invention should be included in the scope of the present invention.

Claims (4)

1. A learning-oriented disturbance observer design method is characterized by comprising the following steps: the method comprises the following steps:
step 1, establishing a system with non-repetitive disturbance:
(1) consider a discrete multiple-input multiple-output linear time-varying system with perturbations as follows:
Figure FDA0002459165310000011
wherein the content of the first and second substances,
xk(t+1)∈Rnrepresenting the state of the system at the kth iteration time t + 1;
uk(t)∈Rlrepresenting the control input of the system at the time t of the kth iteration;
dk(t)∈Rnrepresenting the disturbance in the system at the time t of the kth iteration;
yk(t)∈Rmrepresenting the output of the system at the time t of the kth iteration;
A(t)∈Rn×n，B(t)∈Rn×l，C(t)∈Rm×nrepresenting a state transition matrix of the system at the time t;
the system satisfies the following conditions:
initial state x of systemk(0) Not changing, i.e. xk(0)＝x0；
Disturbance d of the systemk(t) is bounded, i.e. | | dk(t)||≤bd；
The system meets the condition of broad sense L ipschitz when △ uk(t)≠0、△dk(t)≠0、△xk(t) ≠ 0, | | △ xk(t+1)||≤b1||△uk(t)||+b2||△dk(t)||+b3||△xk(t)||；
Wherein the content of the first and second substances,
x0is a constant vector;
bd、b1、b2、b3are all positive numbers;
△uk(t)＝uk(t)-uk-1(t)，△dk(t)＝dk(t)-dk-1(t)，△xk(t)＝xk(t)-xk-1(t)；
(2) consider a discrete multiple-input multiple-output non-linear non-affine system with perturbation as follows:
Figure FDA0002459165310000012
wherein the content of the first and second substances,
f(·)∈Rnrepresenting an unknown non-linear function;
the system satisfies the following conditions:
initial state x of systemk(0) Not changing, i.e. xk(0)＝x0；
Disturbance d of the systemk(t) is bounded, i.e. | | dk(t)||≤bd；
The system meets the condition of broad sense L ipschitz when △ uk(t)≠0、△dk(t)≠0、△xk(t) ≠ 0, | | △ xk(t+1)||≤b1||△uk(t)||+b2||△dk(t)||+b3||△xk(t)||；
Step 2, the system iteration with the non-repeated disturbance is linearized into a linear data model, and the non-repeated disturbance is converted into total disturbance in the linear data model:
Figure FDA0002459165310000021
wherein the content of the first and second substances,
Φk(t)∈Rn×m·(t+1)is an unknown gradient matrix;
Uk(t)＝[uk(0)T,uk(1)T,...,uk(t)T]T∈Rm·(t+1)；
△Uk(t)＝Uk(t)-Uk-1(t)；
k(t)∈Rnis the total perturbation due to non-repetitive perturbations;
step 3, designing an iterative update algorithm to estimate a gradient matrix in the linear data model:
(1) for a state measurable system, the iterative update algorithm is as follows:
Figure FDA0002459165310000022
wherein the content of the first and second substances,
Figure FDA0002459165310000023
is phik(t) an estimate of;
η∈ (0,2) is a step factor;
μ is a positive weighting factor;
(2) for a state-unmeasured system, the iterative update algorithm is as follows:
Figure FDA0002459165310000031
wherein the content of the first and second substances,
M∈Rm×mis a constant matrix;
(MC(t+1))L+represents the left inverse of the matrix MC (t +1), satisfies (MC (t +1))L+MC(t+1)＝I；
I represents an identity matrix;
and 4, designing a learning disturbance observer based on the state aiming at a system with a measurable state:
Figure FDA0002459165310000032
Figure FDA0002459165310000033
wherein the content of the first and second substances,
Figure FDA0002459165310000034
is thatk(t) an estimate of;
zk(t)∈Rnis an intermediate state variable;
K＝(In-)∈Rn×nparameters of a disturbance observer are obtained;
＝diag{γ1,γ2,…,γnis a diagonal matrix; | gammai|<1；i＝1,2,…,n；
And 5, designing a learning disturbance observer based on output aiming at a system with an unmeasured state:
Figure FDA0002459165310000035
Figure FDA0002459165310000037
Figure FDA0002459165310000038
Figure FDA0002459165310000036
wherein the content of the first and second substances,
C(t+1)+represents the pseudo-inverse of matrix C;
Figure FDA0002459165310000041
is a compensation term;
ξk(t) is an intermediate state variable;
Figure FDA0002459165310000042
W(t+1)＝V-Q(t+1)C(t+1)；
the matrix S (t), Q (t), R (t) satisfy (I)h-R(t))(V-Q(t)C(t))-S(t)C(t)＝0。
2. The method of claim 1, further comprising: the iterative linearization process of the step 2 mainly comprises the following steps:
step 2.1, for the linear system described in step 1, iterative dynamic linearization is used, so that:
Figure FDA0002459165310000043
the system is differentiated along an iteration axis and a linear data model can be obtained by using a differential median theorem:
Figure FDA0002459165310000044
wherein the content of the first and second substances,
Figure FDA0002459165310000045
is an unknown gradient matrix;
Figure FDA0002459165310000046
is the total disturbance;
||Φk(t)||≤bΦ、||k(t)||≤b；bΦ、bare all positive numbers;
step 2.2, for the nonlinear system described in step 1, iterative dynamic linearization is used, so that:
Figure FDA0002459165310000051
wherein the content of the first and second substances,
gt(. h) is a state transfer function, which is a complex function of f (·);
the system is differentiated along an iteration axis and a linear data model can be obtained by using a differential median theorem:
Figure FDA0002459165310000052
wherein the content of the first and second substances,
Φk(t) is an unknown gradient matrix;k(t) total perturbation;
||Φk(t)||≤bΦ、||k(t)||≤b；bΦ，bare all positive numbers.
3. The method of claim 1, further comprising: in step 3, for the state-undetectable system, the design of the iterative update algorithm for estimating the gradient matrix in the linear data model mainly comprises the following steps:
and 3.1, expressing the state in an output form because the state of the system which cannot be measured cannot be directly obtained:
Figure FDA0002459165310000053
wherein the content of the first and second substances,
(MC(t+1))L+represents the left inverse of the matrix MC (t +1), satisfies (MC (t +1))L+MC(t+1)＝I；
I represents an identity matrix;
and 3.2, converting the state in the iterative update algorithm aiming at the state measurable system in the step 3 into an output form, thereby obtaining the iterative update algorithm aiming at the state undetectable system:
Figure 1
。
4. the method of claim 1, further comprising: for the state-undetectable system in step 5, the design of the learning disturbance observer based on output mainly comprises the following steps:
and 5.1, aiming at the system with the unmeasured state, representing the state in the learning disturbance observer based on the state in the step 4 in an output form:
Figure FDA0002459165310000062
Figure FDA0002459165310000063
wherein the content of the first and second substances,
C(t+1)+yk-1(t +1) is used instead of the system state xk(t)；
C(t+1)+Represents the pseudo-inverse of matrix C;
KN(t+1)xk-1(t +1) is a compensation term;
N(t+1)＝In-C(t+1)+C(t+1)；
step 5.2, the unknown quantity KN (t +1) x is addedk-1(t +1) performing matrix transformation:
KN(t+1)＝H(t+1)V；
wherein, V ∈ Rh×nIs a constant matrix, H (t +1) ∈ Rn×h；h＝rank(KN(t+1))；
Thus, unknown KN (t +1) xk-1(t +1) transforms:
KN(t+1)xk-1(t+1)＝H(t+1)Vxk-1(t+1)；
unknown quantity Vxk-1(t +1) is defined as an unknown state variable βk(t)∈Rh；
Step 5.3, designing an estimator for the unknown variables βk(t) estimating;
Figure FDA0002459165310000064
Figure FDA0002459165310000065
wherein the content of the first and second substances,
Figure FDA0002459165310000071
representative βk(t) estimation;
ξk(t)∈Rhis an intermediate state variable;
Figure FDA0002459165310000072
W(t+1)＝V-Q(t+1)C(t+1)；
matrix S (t) ∈ Rh×m，Q(t)∈Rh×m，R(t)∈Rh×hSatisfies the following conditions:
(Ih-R(t))(V-Q(t)C(t))-S(t)C(t)＝0；
thus, an output-based learning disturbance observer is obtained:
Figure FDA0002459165310000073
Figure FDA0002459165310000074
Figure FDA0002459165310000075
Figure FDA0002459165310000076
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