CN107065522B - Fuzzy slow state feedback H-infinity control method for nonlinear switching double-time scale system - Google Patents

Fuzzy slow state feedback H-infinity control method for nonlinear switching double-time scale system Download PDF

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CN107065522B
CN107065522B CN201611249982.3A CN201611249982A CN107065522B CN 107065522 B CN107065522 B CN 107065522B CN 201611249982 A CN201611249982 A CN 201611249982A CN 107065522 B CN107065522 B CN 107065522B
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陈金香
陈璇
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Automation Research and Design Institute of Metallurgical Industry
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Abstract

Nonlinear switching double-time scale system fuzzy slow state feedback H∞A control method belongs to the field of complex system control. Describing dynamic characteristics of a controlled system by adopting a discrete fuzzy singular perturbation switching model, and fusing fuzzy logic and singular perturbation technology H∞Control and switching control theory, design fuzzy slow state feedback H∞The controller provides sufficient conditions for solving the gain of the controller, and provides a high-precision control scheme for the nonlinear switching double-time-scale system. The controller has the advantages of high response speed, strong anti-interference capability and small steady-state error; the method has a good interference suppression effect, and is suitable for high-precision control of a system with characteristics of nonlinearity, switching and double time scales.

Description

Fuzzy slow state feedback H of nonlinear switching double-time scale system∞Control method
Technical Field
The invention belongs to the technical field of complex system control, and particularly provides a nonlinear switching double-time scale system fuzzy slow state feedback H∞The control method is suitable for modeling and high-precision control of complex systems such as plate strip rolling, flexible arms of robots, medical mechanical arms and complex circuits.
Background
The nonlinear switching double-time scale system is a complex system integrating multiple characteristics such as nonlinearity, switching, double-time scale and the like, and typically comprises a strip rolling control system, a flexible arm position control system, an aircraft attitude control system and the like, and due to the complexity of the dynamics characteristics of the system, the external disturbance suppression problem of the system is more complex than that of a conventional system. In the field of actual engineering application, a reduced-order modeling and control method which ignores a switching characteristic and a time scale characteristic is generally adopted, so that high control performance is difficult to obtain. In the field of basic theory research, research on a nonlinear system, a switching system and a double-time scale system is greatly advanced, but the design difficulty of a controller is greatly increased by simultaneously considering the nonlinear characteristic, the switching characteristic and the double-time scale characteristic, the research on the modeling and control problems of the system with the three characteristics is still in an initial stage, the problem of stability analysis and stability control is concentrated, the problem of interference suppression is rarely involved, and the proposal of a new theory and method is urgently needed.
H∞The control method is an effective external disturbance suppression method, is widely researched in the last 10 years, mainly focuses on the conventional system, and is less related to H for switching a dual-time-scale system∞Control problems because this would greatly increase the controller gain solving difficulty. But over the years of non-linear dual time scale system H∞On the basis of control theory research, the switching system theory is fused, and modeling and H of a nonlinear switching double-time-scale system are researched∞The control problem has high feasibility, and the control method has high contribution to external disturbance suppression control of the system, and has important theoretical significance and practical application value.
The invention provides discrete fuzzy perturbation modeling and fuzzy state feedback H of a nonlinear switching double-time scale system under the subsidization of an item (51374082) on the national science foundation∞A control method.
Disclosure of Invention
The invention aims to provide a fuzzy state feedback H of a nonlinear switching double-time scale system∞The control method mainly solves the problem of external disturbance suppression of the nonlinear switching double-time-scale system.
The technical scheme of the invention is as follows: discrete fuzzy singular perturbation switching model construction and fuzzy state feedback H∞The control method comprises the steps of establishing a discrete singular perturbation switching model, describing the dynamic characteristics of a controlled switching double-time scale system, and designing fuzzy state feedback H∞And a controller. When the method is applied to an actual system, the adopted overall hardware structure is the same as that of a conventional control system method, and the method mainly comprises the following steps: controlled object, sensor, controller, communication part and executor.
Step 1, describing the dynamics of a controlled object as a standard discrete fuzzy singular perturbation switching model.
Regarding the state variable related to or changing quickly with small parameters of the controlled system as a fast variable, regarding the state variable changing slowly or measurable as a slow variable, and establishing a standard discrete fuzzy singular perturbation switching model set with a plurality of subsystems.
Rule i if ξ1(k) Is phii1，…，ξg(k) Is phiigThen, then
Figure BDA0001197849060000021
Wherein the content of the first and second substances,
Figure BDA0001197849060000022
xs(k)∈Rnis a slow variable, xf(k)∈RmFor fast variables, u (k) e RqFor control input, w (k) e RqFor external perturbations, z (k) e RlTo control the output, [ phi ]i1，...，φig( i 1, 2.., r.) are fuzzy sets, ξ1(k)，...，ξg(k) As a measurable system variable, Aiσ,BiσC,Dσ,GσFor a suitable dimension matrix, the switching signal σ is 1,2, …, N is the number of subsystems, e is the singular perturbation parameter, In×n,Im×mRespectively an n-order unit array and an m-order unit array.
Given [ x (k); u (k) ], a global fuzzy model can be obtained by using standard fuzzy reasoning as
Figure BDA0001197849060000023
Wherein the content of the first and second substances,
Figure RE-GDA0001279040150000024
a function of the degree of membership,
Figure RE-GDA0001279040150000025
Figure RE-GDA0001279040150000026
φij(ξj(k) ) is ξj(k) At phiijDegree of membership in, set as wi(ξ (k)) > 0, i-1, 2, …, r, r is a regular number, mui(ξ(k))≥0，
Figure RE-GDA0001279040150000027
Let us order μ for easy recordingi＝μi(ξ(k))。
Step 2, designing fuzzy state feedback H of switching subsystem∞Controller
Because the fast state variable of the system is not measurable, the following fuzzy slow state feedback H is constructed∞The controller, its antecedent is the same as equation (1):
uσ(k)＝Hσ(μ)x(k) (4)
Figure BDA0001197849060000031
the controller switching rate may be selected as:
s(k)＝arg min(xT(k+1)P-1x(k+1)-xT(k)P-1x(k)) (5)
wherein argmin () represents rounding the minimum value of the number in brackets, P is a symmetric positive definite matrix and
Figure BDA0001197849060000032
step 3, providing a method for solving the gain of the controller
Theorem 1: for sufficiently small perturbation parameters ε>0 and a scalar γ>0, the control rate (4) acts on the controlled system (2) according to the switching rule of the switching rate (5) to gradually stabilize the closed loop system, if and only if a symmetrical positive definite matrix exists
Figure BDA0001197849060000033
Wherein P is11∈Rn×n，P22∈Rm×mAre all symmetric positive definite matrices, matrix Wiσ＝[Viσ0q×m]Wherein V isiσ∈Rq×nThe following linear matrix inequalities are set up,
Ψii<0i＝1，2，…r (6)
Ψij+Ψji<0 i＝1，2，…，r,i<j,j＝2,3,…,r (7)
wherein the content of the first and second substances,
Figure BDA0001197849060000034
switching signal sigma is 1,2, …, N, N is subsystem number, controller gain Hiσ＝[Fiσ0q×m]，
Fiσ＝P*Wiσ(8)
And 4, describing the switching model and the control law into C language codes, and implanting a controller to realize high-precision control of the controlled system.
The invention has the advantages that:
1) the fuzzy logic, the switching system and the singular perturbation theory are fused, the construction method of the fuzzy singular perturbation switching model is provided, the problem that the existing modeling theory cannot accurately describe the nonlinearity, the double time scales and the switching coexistence characteristic of the controlled system is solved, and a new thought is provided for modeling of the nonlinear switching double time scale system.
2) And binding H∞A theoretical and fuzzy logic method, and provides a fuzzy slow state feedback H∞The control method solves the problem that the existing control technology is difficult to eliminate the steady-state error caused by the switching characteristic and the fast variable of the controlled system, and greatly improves the control performance of the nonlinear switching double-time-scale system.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a closed loop system condition response graph.
FIG. 3 is a plot of switching rate s (k).
Detailed Description
The method of the invention is applied to a switching system having two modalities, as follows, and the method of implementation will be explained.
Step 1, aiming at a switching system with two modes, establishing a discrete fuzzy singular perturbation switching model
Rule i if ξ1(t) is phii1Then, then
Figure BDA0001197849060000041
Wherein the content of the first and second substances,
Figure BDA0001197849060000042
xs(k)∈R2is a slow variable, xf(k) E R is a fast variable, u (k) e R3For control input, w (k) e R3For external disturbances, z (k) e R is the control output, phii1To blur the set, ξ1(k)＝xs(t) is a measurable system variable, Aiσ,BiσC,Dσ,GσFor a suitable dimension matrix, I is 1,2, the switching signal σ is 1,2, e is 0.02, I is a singular perturbation parameter2×2Is a 2-order unit array and is composed of a plurality of unit arrays,
Figure BDA0001197849060000043
Figure BDA0001197849060000044
Figure BDA0001197849060000045
Figure BDA0001197849060000046
given [ x (k); u (k); w (k) ], a global fuzzy model can be obtained by using standard fuzzy reasoning as
Figure BDA0001197849060000051
Wherein, a is 1,2,
Figure RE-GDA0001279040150000052
Figure BDA0001197849060000053
step 2, designing fuzzy state feedback H of switching subsystem∞Controller
Because the fast state variable of the system is not measurable, the following fuzzy slow state feedback H is constructed∞A controller, the front piece of which is the same as equation (9):
uσ(k)＝Hσ(μ)x(k) (11)
Figure BDA0001197849060000054
the controller switching rate may be selected as:
s(k)＝arg min(xT(k+1)P-1x(k+1)-xT(k)P-1x(k)) (12)
wherein argmin () represents rounding the minimum value of the number in brackets, P is a symmetric positive definite matrix and
Figure BDA0001197849060000055
and 3, solving the gain of the controller.
Applying theorem 1 to find the gain of the controller
Figure BDA0001197849060000056
Figure BDA0001197849060000057
Figure BDA0001197849060000058
The initial state is as follows: x (0) ═ 1-23]TInput of disturbance
Figure BDA0001197849060000061
The simulation result is shown in the figure, and fig. 2 is a state feedback H of the uncertain switching double-time scale system∞The state response of the control, fig. 3 is a switching signal, and the simulation result shows that the designed controller can gradually stabilize the closed-loop system and obtain better control performance.
And 4, describing the switching model and the control law into C language codes, and implanting a controller to realize high-precision control of the controlled system.

Claims (1)

1. Nonlinear switching double-time scale system fuzzy slow state feedback H∞The control method is characterized by comprising the following steps:
step 1, describing the dynamics of a controlled object as a standard discrete fuzzy singular perturbation switching model
Regarding the state variable related to or changing quickly by small parameters of a controlled system as a fast variable, regarding the state variable changing slowly or measurable as a slow variable, and establishing a standard discrete fuzzy singular perturbation switching model set with a plurality of subsystems:
rule i if ξl(k) Is phii1，…，ξg(k) Is phiigThen, then
Figure FDA0001197849050000011
Wherein the content of the first and second substances,
Figure FDA0001197849050000012
xs(k)∈Rnis a slow variable, xf(k)∈RmFor fast variables, u (k) e RqFor control input, w (k) e RqFor external perturbations, z (k) e RlTo control the output, [ phi ]i1，...，φig(i 1, 2.., r.) are fuzzy sets, ξ1(k)，...，ξg(k) As a measurable system variable, Aiσ,BiσC,Dσ,GσFor a suitable dimension matrix, the switching signal σ is 1,2, …, N is the number of subsystems, e is the singular perturbation parameter, In×n,Im×mRespectively an n-order unit array and an m-order unit array;
given [ x (k); u (k) ], a global fuzzy model can be obtained by using standard fuzzy reasoning as
Figure FDA0001197849050000013
Wherein the content of the first and second substances,
Figure FDA0001197849050000014
function of degree of membership
Figure FDA0001197849050000015
φij(ξj(k) ) is ξj(k) At phiijDegree of membership in, set as wi(ξ (k)) > 0, i-1, 2, …, r, r is a regular number, mui(ξ(k))≥0，
Figure FDA0001197849050000016
Let us order μ for easy recordingi＝μi(ξ(k))；
Step 2, designing fuzzy state feedback H of switching subsystem∞Controller
The fuzzy slow state feedback H is constructed as follows∞The controller, its antecedent is the same as equation (1):
uσ(k)＝Hσ(μ)x(k) (4)
Hσ(μ)＝[Fσ(μ) 0q×m]，
Figure FDA0001197849050000021
Fiσ∈Rq×n，
the switching rate of the controller is selected as follows:
s(k)＝arg min(xT(k+1)P-1x(k+1)-xT(k)P-1x(k)) (5)
wherein argmin () represents rounding the minimum value of the number in brackets, P is a symmetric positive definite matrix and
Figure FDA0001197849050000022
P11∈Rn×n,P22∈Rm×m；
step 3, providing a method for solving the gain of the controller
Theorem 1: for sufficiently small perturbation parameters ε>0 and a scalar γ>0, the control rate (4) acts on the controlled system (2) according to the switching rule of the switching rate (5) to gradually stabilize the closed loop system, if and only if a symmetrical positive definite matrix exists
Figure FDA0001197849050000023
Wherein P is11∈Rn×n，P22∈Rm×mAre all symmetric positive definite matrices, matrix Wiσ＝[Viσ0q×m]Wherein V isiσ∈Rq×nThe following linear matrix inequalities are set up,
Ψii<0 i＝1，2，…r (6)
Ψij+Ψji<0i＝1，2，…，r,i<j,j＝2,3,…,r (7)
wherein the content of the first and second substances,
Figure FDA0001197849050000024
switching signal sigma is 1,2, …, N, N is subsystem number, controller gain Hiσ＝[Fiσ0q×m]，
Fiσ＝P*Wiσ(8)
And 4, describing the switching model and the control law into C language codes, and implanting a controller to realize high-precision control of the controlled system.
CN201611249982.3A 2016-12-29 2016-12-29 Fuzzy slow state feedback H-infinity control method for nonlinear switching double-time scale system Expired - Fee Related CN107065522B (en)

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