CN113126491A - Anti-interference tracking control design method based on T-S fuzzy interference modeling - Google Patents

Anti-interference tracking control design method based on T-S fuzzy interference modeling Download PDF

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CN113126491A
CN113126491A CN202110367283.3A CN202110367283A CN113126491A CN 113126491 A CN113126491 A CN 113126491A CN 202110367283 A CN202110367283 A CN 202110367283A CN 113126491 A CN113126491 A CN 113126491A
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张晓莉
裔扬
吴旭栋
周鹏
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Abstract

The invention discloses an anti-interference tracking control design method based on T-S fuzzy interference modeling in the technical field of automatic control, which comprises the following steps: 1. establishing a system state equation containing external interference; 2. establishing an interference model through a T-S fuzzy model, designing a PI tracking controller, and constructing a composite system; 3. designing a disturbance observer; 4. designing a composite control system based on the interference observer and the PI tracking controller; 5. based on the convex optimization algorithm to solve the control gain and the observer gain, the method realizes the effective suppression of the system interference and the dynamic tracking of the system output, and has good robustness and high stability.

Description

Anti-interference tracking control design method based on T-S fuzzy interference modeling
Technical Field
The invention relates to the technical field of automatic control, in particular to an anti-interference tracking control design method.
Background
It is well known that almost all control systems present external disturbances, e.g. motion control systems, robotic handling systems, terminal sliding mode systemsThe system, a Markov jump system, a flight control system and the like, so the anti-interference control problem attracts the wide attention of the academic and engineering fields. In order to maximize the applicable range of interference rejection, the interference in nature can be generally classified into norm-bounded interference, harmonic interference, gaussian/non-gaussian interference, interference generated by a neutral stable external system, and the like. The scholars propose different control methods for different types of interference, e.g. H∞Control is due to norm-bounded interference, random control is due to gaussian/non-gaussian interference, output tuning theory is due to interference generated by a neutral stable system, DOBC methods are due to neutral stable, bounded, harmonic or other dynamic interference modeled by an external system. The basic idea of a disturbance observer based control strategy (DOBC) is to use an observer to estimate the effect of external disturbances on the system and compensate in the feed-forward channel. It is worth pointing out that in the existing DOBC-related documents, the external system modeling interference is mostly considered as harmonic interference or constant interference generated by a linear system, which undoubtedly limits the modeling interference category of the DOBC method, and when the DOBC method faces irregular or other non-linear interference, some existing DOBC methods based on linear interference models are no longer applicable, and new interference modeling methods and tools need to be found.
Disclosure of Invention
The invention aims to provide an anti-interference tracking control design method based on T-S fuzzy interference modeling, and solves the problem that the existing DOBC method based on a linear interference model is poor in applicability.
The purpose of the invention is realized as follows: an anti-interference tracking control design method based on T-S fuzzy interference modeling comprises the following steps:
step 1, establishing a system state equation containing external interference;
step 2, establishing an interference model through a T-S fuzzy model, designing a PI tracking controller, and constructing a composite system;
step 3, designing a disturbance observer;
step 4, designing a composite control system based on the interference observer and the PI tracking controller;
and 5, solving the control gain and the observer gain based on a convex optimization algorithm.
Compared with the prior art, the invention has the beneficial effects that:
(1) according to the method, the T-S fuzzy model is introduced into the external interference to serve as a tool for approximating the interference, so that complex and irregular nonlinear interference can be described, and the application range of control based on the interference observer is widened;
(2) according to the method, the observer is constructed to estimate the unknown external interference of the system, so that the unknown interference can be immediately observed and the interference characteristic can be fed back when the unknown interference occurs, then the estimation information with the observed interference characteristic is introduced into the controller, and the front feed channel is compensated, so that the interference is counteracted, and the damage to the system caused by the interference is avoided;
(3) the invention combines the estimation information given by the disturbance observer and the system feedback state to construct a PI type anti-interference controller. The convex optimization algorithm is utilized to obtain the gain of the disturbance observer and the gain of the controller, so that the designed disturbance observer and controller can effectively play a role, the system stability is improved, and good output tracking performance can be obtained.
As a further limitation of the present invention, the system state equation containing the external interference in step 1 is specifically:
Figure BDA0003007689400000021
wherein x (t) e RnRepresents the system state, RnRepresenting an n-dimensional real vector space, u (t) e RmRepresents a control input; d (t) ε RmDenotes external interference, RmRepresenting an m-dimensional real vector space,
Figure BDA0003007689400000035
the output variable of the system is represented,
Figure BDA0003007689400000036
represents p1Dimension real vector space, A0,B0,C0Is suitable for maintenanceThe system matrix of (2); f01And F02Representing a gain matrix of nonlinear terms; f. of0i(x (t), t) (i ═ 1,2) are known or unknown nonlinear terms that satisfy the Lipschitz condition, i.e., there is a known Lipschitz parameter array Ui∈Rn×n(i ═ 1,2) so that the following inequality holds:
Figure BDA0003007689400000031
wherein x is1(t),x2(t) e { x (t) t e R } is any two states in the system state set.
The overall establishment of the system model framework is carried out in the step 1, and any actual engineering system can be applied to the framework as long as the relevant parameters of the actual system are corresponded to the system matrix of the established model framework.
As a further limitation of the present invention, the interference model established in step 2 is:
Figure BDA0003007689400000032
where ω (t) represents a state variable of the interfering system and (W)j,Vj) Is the adaptive parameter matrix for the jth sub-system, r represents the number of fuzzy rules,
Figure BDA0003007689400000033
denotes thetajIn fuzzy sets
Figure BDA0003007689400000034
Of (2) a membership function ofjRepresenting a precondition variable, n representing the number of fuzzy rules, d (t) e RmIndicating system interference. Compared with a general interference model, the interference model modeled by the T-S fuzzy method can describe complex and irregular nonlinear interference, so that the interference research is not only limited to linearity, but also has a wider range.
As a further limitation of the present invention, the PI tracking controller designed in step 2 specifically includes:
Figure BDA0003007689400000041
wherein u isPI(t) is a PI controller, KpDenotes the ratio to be found, KIRepresenting an integral control gain matrix, e (t) representing a system output tracking error;
the composite system is constructed as follows:
Figure BDA0003007689400000042
wherein,
Figure BDA0003007689400000043
Figure BDA0003007689400000044
ydrepresenting the desired output of the system and I represents the identity matrix. Compared with a common proportional controller, the designed PI controller uPI(t) can satisfy the requirement that the system output finally tracks to a desired value.
As a further limitation of the present invention, the interference observer designed in step 3 specifically is:
Figure BDA0003007689400000045
wherein,
Figure BDA0003007689400000046
is an estimate of the interference of the system,
Figure BDA0003007689400000047
is an estimate of a state variable of the interfering system,v (t) denotes an auxiliary variable, L is the gain matrix to be solved,
Figure BDA0003007689400000048
indicating a new system state. Most of interference in the system is unknown, the existing anti-interference method basically inhibits the interference after the interference occurs, an effective interference observer is designed, so that the characteristics of the interference can be observed while the interference occurs, and the information is substituted into a feedforward channel to effectively counteract the interference in the system.
As a further limitation of the present invention, in step 4, based on the disturbance observer and the PI tracking controller, the specific steps of designing the composite control system are as follows:
step 4-1, designing a composite controller based on the interference observer and PI control input
Figure BDA0003007689400000051
K＝[kP KI]
Wherein u (t) e RmFor control input, KpDenotes the ratio to be found, KIRepresenting an integral control gain matrix;
step 4-2, the composite controller acts on the composite system to obtain a closed-loop system:
Figure BDA0003007689400000052
in the formula, ew(t) is an observation error, and specifically comprises the following steps:
Figure BDA0003007689400000053
and 4-3, designing a composite control system according to the closed-loop system:
Figure BDA0003007689400000054
in the formula,
Figure BDA0003007689400000055
the designed interference observer and the PI controller are integrated into a composite system, and the system which can meet the requirements of interference cancellation and has good stability and tracking performance is obtained.
As a further limitation of the present invention, the control gain and the observer gain in step 5 are specifically:
Figure BDA0003007689400000056
Figure BDA0003007689400000057
k denotes the control gain and L denotes the observer gain, wherein the auxiliary matrix Q1And Q2For solving the control gain and observation gain, and Q1＝P1 -1＞0、R1、P2> 0 and R2The following convex optimization problem solved:
Figure BDA0003007689400000058
Figure BDA0003007689400000059
wherein,
Figure BDA0003007689400000061
λ1＞0，λ2> 0 and mu1> 0 is a given parameter, U1And U2Is given as a constant weight matrix, I denotes an identity matrix, C denotes a system matrix,
Figure BDA0003007689400000062
through the introduction of the convex optimization technology, unknown controller gain and unknown interference observer gain can be automatically and effectively solved, and then the corresponding controller and the corresponding interference observer can be guaranteed to play an effective role, so that the system is prevented from being influenced by interference and has better stability, and meanwhile, the output can be effectively tracked.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the provided drawings without creative efforts.
FIG. 1 is a flow chart of the present invention.
Fig. 2-8 are diagrams illustrating the effects of the embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
As shown in fig. 1, a method for designing anti-interference tracking control based on T-S fuzzy interference modeling specifically includes the steps of:
step 1, establishing a system state equation containing external interference, specifically:
Figure BDA0003007689400000071
wherein x (t) e RnRepresents the system state, RnRepresenting an n-dimensional real vector space, u (t) e RmRepresents a control input; d (t) ε RmWhich is indicative of an external disturbance,
Figure BDA0003007689400000072
representing the system output variable, A0,B0,C0Is a system matrix of suitable dimensions; f01And F02Representing a gain matrix of nonlinear terms; f. of0i(x (t), t) (i ═ 1,2) are known or unknown nonlinear terms that satisfy the Lipschitz condition, i.e., there is a known Lipschitz parameter array Ui∈Rn×n(i ═ 1,2) so that the following inequality holds:
Figure BDA0003007689400000073
wherein x is1(t),x2(t) e { x (t) t e R } is any two states in the system state set.
Step 2, establishing an interference model through a T-S fuzzy model, designing a PI tracking controller, and constructing a composite system, wherein the established interference model is as follows:
Figure BDA0003007689400000074
where ω (t) represents a state variable of the interfering system and (W)j,Vj) Is the adaptive parameter matrix for the jth sub-system, r represents the number of fuzzy rules,
Figure BDA0003007689400000075
denotes thetajIn fuzzy sets
Figure BDA0003007689400000076
Of (2) a membership function ofjRepresenting a precondition variable, n representing the number of fuzzy rules, d (t) e RmIndicating system interference.
The designed PI tracking controller specifically comprises the following components:
Figure BDA0003007689400000077
wherein u isPI(t) is a PI controller, KpDenotes the ratio to be found, KIRepresents the integral control gain matrix, e (t) represents the system output tracking error.
The composite system is constructed as follows:
Figure BDA0003007689400000081
wherein,
Figure BDA0003007689400000082
Figure BDA0003007689400000083
ydrepresenting the desired output of the system and I represents the identity matrix.
Step 3, designing a disturbance observer, which specifically comprises the following steps:
Figure BDA0003007689400000084
wherein,
Figure BDA0003007689400000085
is an estimate of the interference of the system,
Figure BDA0003007689400000086
v (t) represents an auxiliary variable, L is the gain matrix to be solved,
Figure BDA0003007689400000087
indicating a new system state. The expression of the auxiliary variable v (t) is:
Figure BDA0003007689400000088
and 4, designing a composite control system based on the interference observer and the PI tracking controller, and specifically comprising the following steps:
step 4-1, designing a composite controller based on the interference observer and PI control input:
Figure BDA0003007689400000089
K＝[kP KI]
wherein u (t) e RmFor control input, KpDenotes the ratio to be found, KIRepresenting an integral control gain matrix;
step 4-2, the composite controller acts on the composite system to obtain a closed-loop system:
Figure BDA00030076894000000810
and 4-3, designing a composite control system according to the closed-loop system:
Figure BDA0003007689400000091
in the formula,
Figure BDA0003007689400000092
step 5, solving control gain and observer gain based on a convex optimization algorithm, wherein the control gain and the observer gain are specifically as follows:
Figure BDA0003007689400000093
Figure BDA0003007689400000094
k denotes a control gain and L denotes an observer gain, wherein Q1＝P1 -1＞0、R1、P2> 0 and R2The following convex optimization problem solved:
Figure BDA0003007689400000095
Figure BDA0003007689400000096
wherein,
Figure BDA0003007689400000097
λ1＞0，λ2> 0 and mu1> 0 is a given parameter, U1And U2Is a given constant weight matrix, I denotes an identity matrix, and C denotes a system matrix.
The principle of the invention is as follows: firstly, a T-S fuzzy model can be used as a tool for approximating a nonlinear system to model external interference, and then the interference with known characteristics can be counteracted based on a method of an interference observer; constructing a composite control system based on a PI type control input and an interference dynamic observer, and finally converting the design problem of the composite control system into a convex optimization problem based on a Linear Matrix Inequality (LMI); the convex optimization problem is solved, and control gain and observation gain are solved from the feasible solution of the convex optimization problem through corresponding algebraic transformation, so that effective suppression on system interference and dynamic tracking on system output are realized, and the requirements on robustness and stability of the system are met.
The present invention is further illustrated by the following examples.
Example 1
The embodiment takes a hypersonic aircraft as an example to illustrate the specific implementation of the method:
1. establishing a system motion equation and a state equation of the hypersonic aircraft:
the hypersonic aircraft is a complex nonlinear system, adopts a cone (gained-cone) hypersonic concept aircraft disclosed by Lanli laboratories, and has an all-state nonlinear motion equation as follows:
Figure BDA0003007689400000101
wherein x, y, z, V respectively represent coordinates relative to the ground and flight speed, m represents mass of the aircraft and represents track pitch angle, α represents flight angle of attack, p, q, r respectively represent roll rate, pitch angle rate and yaw rate, Tx,Ty,TzRepresenting thrust in the x, y, z directions, respectively. g0Is standard gravitational acceleration, r is the radius of the earth, and r is equal to r0+ h, μ is gravitational constant, L is lift, and drag is D, IyyMoment of pitch M for moment of inertia about the y-axisyyEtc. are complex algebraic functions of system state and input, so a simplified expression can be obtained by means of cubic spline interpolation, specifically:
Figure BDA0003007689400000111
wherein, C*The expression (force and moment constants) is as follows:
Figure BDA0003007689400000112
wherein,
Figure BDA0003007689400000113
s, rho are the average aerodynamic chord length, the reference area and the air density of the hypersonic aircraft respectively, and the air density is related to the flight altitude; through testing the taped auxioin deviation deltaeThe effect on lift and drag was found to be negligible by the weak coupling caused by itself, i.e.,
Figure BDA0003007689400000114
approximately zero; the magnitude of the thrust is mainly determined by the throttleSettings, angle of attack, airspeed, and altitude.
Expressing the hypersonic aircraft motion equation into a state space form:
Figure BDA0003007689400000115
wherein x (t) ([ V γ α h q)]T∈R5,u(t)＝[β δe]T∈R2，
Figure BDA0003007689400000116
Figure BDA0003007689400000121
Figure BDA0003007689400000122
Figure BDA0003007689400000123
2. Establishing a T-S fuzzy interference model and designing a PI tracking controller:
the established interference model is as follows:
Figure BDA0003007689400000124
where ω (t) represents a state variable of the interfering system and (W)j,Vj) Is the adaptive parameter matrix for the jth sub-system, r represents the number of fuzzy rules,
Figure BDA0003007689400000125
denotes thetajIn fuzzy sets
Figure BDA0003007689400000126
Of (2) a membership function ofjRepresenting a precondition variable, n representing the number of fuzzy rules, d (t) e RmIndicating system interference.
The designed PI tracking controller comprises the following components:
Figure BDA0003007689400000131
wherein u isPI(t) is a PI controller, Kp，KIRespectively representing the proportional and integral control gain matrix to be solved, ydRepresenting the desired output of the system, e (t) ═ y (t) — ydRepresenting the tracking error of the system output, constructing a composite system:
Figure BDA0003007689400000132
wherein,
Figure BDA0003007689400000133
3. constructing a disturbance observer:
Figure BDA0003007689400000134
wherein,
Figure BDA0003007689400000135
d (t), ω (t), respectively. v (t) denotes an auxiliary variable, L is the gain matrix to be solved, x (t) e RnRepresenting a system state; the expression of the auxiliary variable v (t) is:
Figure BDA0003007689400000136
4. designing a composite control system based on the control input of a disturbance observer and a PI (proportional integral) controller:
Figure BDA0003007689400000137
K＝[kP KI]
wherein u (t) e RmFor control input, Kp，KIRespectively representing a proportional control gain matrix and an integral control gain matrix to be solved,
Figure BDA0003007689400000138
representing a new state variable, and enabling the composite controller to act on the composite system to obtain a closed-loop system:
Figure BDA0003007689400000139
designing a composite control system according to a closed loop system:
Figure BDA0003007689400000141
in the formula,
Figure BDA0003007689400000142
5. solving control gain based on convex optimization algorithm
Figure BDA0003007689400000143
And observer gain and
Figure BDA0003007689400000144
wherein Q1＝P1 -1＞0、R1、P2> 0 and R2The following convex optimization problem solved:
Figure BDA0003007689400000145
Figure BDA0003007689400000146
wherein,
Figure BDA0003007689400000147
λ1＞0，λ2> 0 and mu1> 0 is a given parameter, U1And U2Is given as a constant weight matrix, I denotes an identity matrix, C denotes a system matrix,
Figure BDA0003007689400000148
6. the simulation results are shown in fig. 2-8. Wherein, fig. 2 and fig. 3 respectively show velocity and altitude tracking effect graphs of the hypersonic flight vehicle, and fig. 4, fig. 5 and fig. 6 respectively show flight track angle, attack angle and pitch angle change rate graphs of the hypersonic flight vehicle; in order to verify the influence of the disturbance observer on the speed, fig. 7 shows a speed variation curve chart under the condition that the disturbance observer exists, and it can be seen that when the disturbance observer is added, the speed tends to designate a reference value in a short time, so that the disturbance observer has a good anti-jamming effect; FIG. 8 is a trace diagram of nonlinear interference and its estimated value, from which it can be seen that the algorithm of this chapter can effectively estimate irregular interference; in general, the hypersonic aircraft closed-loop model can be proved to meet the control requirement of multiple targets, namely, the hypersonic aircraft closed-loop model has good stability, dynamic tracking performance and robust performance.
The above description of the embodiments is only intended to facilitate the understanding of the method of the invention and its core idea. It should be noted that, for those skilled in the art, it is possible to make various improvements and modifications to the present invention without departing from the principle of the present invention, and those improvements and modifications also fall within the scope of the claims of the present invention.

Claims (7)

1. An anti-interference tracking control design method based on T-S fuzzy interference modeling is characterized by comprising the following steps:
step 1, establishing a system state equation containing external interference;
step 2, establishing an interference model through a T-S fuzzy model, designing a PI tracking controller, and constructing a composite system;
step 3, designing a disturbance observer;
step 4, designing a composite control system based on the interference observer and the PI tracking controller;
and 5, solving the control gain and the observer gain based on a convex optimization algorithm.
2. The anti-interference tracking control design method based on T-S fuzzy interference modeling according to claim 1, characterized in that the system state equation containing external interference in step 1 is specifically:
Figure FDA0003007689390000011
wherein x (t) e RnRepresents the system state, RnRepresenting an n-dimensional real vector space, u (t) e RmRepresents a control input; d (t) ε RmDenotes external interference, RmRepresenting an m-dimensional real vector space,
Figure FDA0003007689390000012
the output variable of the system is represented,
Figure FDA0003007689390000013
represents p1Dimension real vector space, A0,B0,C0Is a system matrix of suitable dimensions; f01And F02Representing a gain matrix of nonlinear terms; f. of0i(x (t), t) (i ═ 1,2) are known or unknown nonlinear terms that satisfy the Lipschitz condition, i.e., there is a known Lipschitz parameter array Ui∈Rn×n(i ═ 1,2) so that the following inequality holds:
Figure FDA0003007689390000014
wherein x is1(t),x2(t) e { x (t) t e R } is any two states in the system state set.
3. The anti-interference tracking control design method based on T-S fuzzy interference modeling according to claim 2, characterized in that the interference model established in step 2 is:
Figure FDA0003007689390000021
where ω (t) represents a state variable of the interfering system and (W)j,Vj) Is the adaptive parameter matrix for the jth sub-system, r represents the number of fuzzy rules,
Figure FDA0003007689390000022
Figure FDA0003007689390000023
denotes thetajIn fuzzy set Ai jOf (2) a membership function ofjRepresenting a precondition variable, n representing the number of fuzzy rules, d (t) e RmIndicating system interference.
4. The method for designing anti-interference tracking control based on T-S fuzzy interference modeling according to claim 3, wherein the PI tracking controller designed in the step 2 is specifically:
Figure FDA0003007689390000024
wherein u isPI(t) is a PI controller, KpDenotes the ratio to be found, KIRepresenting an integral control gain matrix, e (t) representing a system output tracking error;
the composite system is constructed as follows:
Figure FDA0003007689390000025
wherein,
Figure FDA0003007689390000026
Figure FDA0003007689390000027
ydrepresenting the desired output of the system and I represents the identity matrix.
5. The anti-interference tracking control design method based on T-S fuzzy interference modeling according to claim 4, characterized in that the interference observer designed in step 3 is specifically:
Figure FDA0003007689390000028
wherein,
Figure FDA0003007689390000031
is an estimate of the interference of the system,
Figure FDA0003007689390000032
v (t) represents an auxiliary variable, L is the gain matrix to be solved,
Figure FDA0003007689390000033
indicating a new system state.
6. The anti-interference tracking control design method based on T-S fuzzy interference modeling according to claim 5, characterized in that, based on the interference observer and the PI tracking controller in step 4, the specific steps of designing the composite control system are as follows:
step 4-1, designing a composite controller based on the interference observer and PI control input
Figure FDA0003007689390000034
K＝[kP KI]
Wherein u (t) e RmFor control input, KpDenotes the ratio to be found, KIRepresenting an integral control gain matrix;
step 4-2, the composite controller acts on the composite system to obtain a closed-loop system:
Figure FDA0003007689390000035
in the formula, ew(t) is an observation error, and specifically comprises the following steps:
Figure FDA0003007689390000036
and 4-3, designing a composite control system according to the closed-loop system:
Figure FDA0003007689390000037
in the formula,
Figure FDA0003007689390000038
7. the anti-interference tracking control design method based on T-S fuzzy interference modeling according to claim 1, wherein the control gain and the observer gain in step 5 are specifically:
Figure FDA0003007689390000039
Figure FDA00030076893900000310
k denotes the control gain and L denotes the observer gain, wherein the auxiliary matrix Q1And Q2For solving the control gain and observation gain, and
Figure FDA0003007689390000041
R1、P2> 0 and R2The following convex optimization problem solved:
Figure FDA0003007689390000042
Figure FDA0003007689390000043
wherein,
Figure FDA0003007689390000044
λ1＞0，λ2> 0 and mu1> 0 is a given parameter, U1And U2Is given as a constant weight matrix, I denotes an identity matrix, C denotes a system matrix,
Figure FDA0003007689390000045
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Application publication date: 20210716