CN110443003B - Control and optimal design method of active stabilizer bar system - Google Patents
Control and optimal design method of active stabilizer bar system Download PDFInfo
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- 239000003381 stabilizer Substances 0.000 title claims abstract description 72
- 238000000034 method Methods 0.000 title claims abstract description 22
- 238000005457 optimization Methods 0.000 claims abstract description 45
- 230000000694 effects Effects 0.000 claims abstract description 12
- 239000011159 matrix material Substances 0.000 claims description 15
- 230000006641 stabilisation Effects 0.000 claims description 10
- 238000011105 stabilization Methods 0.000 claims description 10
- 230000001133 acceleration Effects 0.000 claims description 8
- 239000000725 suspension Substances 0.000 claims description 7
- 238000013016 damping Methods 0.000 claims description 6
- 238000004364 calculation method Methods 0.000 claims description 4
- NAWXUBYGYWOOIX-SFHVURJKSA-N (2s)-2-[[4-[2-(2,4-diaminoquinazolin-6-yl)ethyl]benzoyl]amino]-4-methylidenepentanedioic acid Chemical compound C1=CC2=NC(N)=NC(N)=C2C=C1CCC1=CC=C(C(=O)N[C@@H](CC(=C)C(O)=O)C(O)=O)C=C1 NAWXUBYGYWOOIX-SFHVURJKSA-N 0.000 claims description 3
- 101150036841 minJ gene Proteins 0.000 claims description 3
- HOWHQWFXSLOJEF-MGZLOUMQSA-N systemin Chemical compound NCCCC[C@H](N)C(=O)N[C@@H](CCSC)C(=O)N[C@@H](CCC(N)=O)C(=O)N[C@@H]([C@@H](C)O)C(=O)N[C@@H](CC(O)=O)C(=O)OC(=O)[C@@H]1CCCN1C(=O)[C@H]1N(C(=O)[C@H](CC(O)=O)NC(=O)[C@H](CCCN=C(N)N)NC(=O)[C@H](CCCCN)NC(=O)[C@H](CO)NC(=O)[C@H]2N(CCC2)C(=O)[C@H]2N(CCC2)C(=O)[C@H](CCCCN)NC(=O)[C@H](CO)NC(=O)[C@H](CCC(N)=O)NC(=O)[C@@H](NC(=O)[C@H](C)N)C(C)C)CCC1 HOWHQWFXSLOJEF-MGZLOUMQSA-N 0.000 claims 1
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B60—VEHICLES IN GENERAL
- B60G—VEHICLE SUSPENSION ARRANGEMENTS
- B60G21/00—Interconnection systems for two or more resiliently-suspended wheels, e.g. for stabilising a vehicle body with respect to acceleration, deceleration or centrifugal forces
- B60G21/02—Interconnection systems for two or more resiliently-suspended wheels, e.g. for stabilising a vehicle body with respect to acceleration, deceleration or centrifugal forces permanently interconnected
- B60G21/04—Interconnection systems for two or more resiliently-suspended wheels, e.g. for stabilising a vehicle body with respect to acceleration, deceleration or centrifugal forces permanently interconnected mechanically
- B60G21/05—Interconnection systems for two or more resiliently-suspended wheels, e.g. for stabilising a vehicle body with respect to acceleration, deceleration or centrifugal forces permanently interconnected mechanically between wheels on the same axle but on different sides of the vehicle, i.e. the left and right wheel suspensions being interconnected
- B60G21/055—Stabiliser bars
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B60—VEHICLES IN GENERAL
- B60G—VEHICLE SUSPENSION ARRANGEMENTS
- B60G2204/00—Indexing codes related to suspensions per se or to auxiliary parts
- B60G2204/80—Interactive suspensions; arrangement affecting more than one suspension unit
- B60G2204/83—Type of interconnection
- B60G2204/8302—Mechanical
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Abstract
The invention provides a control and optimal design method of an active stabilizer bar system, which comprises the following steps: on the basis of establishing a dynamic model of an active stabilizer bar system with parameter uncertainty, describing the uncertainty by adopting a fuzzy set theory so as to construct a fuzzy dynamic model of the active stabilizer bar system; aiming at the fuzzy dynamic model, a deterministic robust controller is designed to compensate the uncertainty of the system; performing stability analysis on the constructed robust controller; providing an optimization function containing the steady-state performance and the control cost of the system, and obtaining the optimal solution of the control parameters by minimizing the optimization function; and adjusting the weight coefficient in the optimization function, and analyzing the influence of the weight coefficient on the control parameters and the control effect. The control and optimization design method of the active stabilizer bar system can effectively process the influence of parameter uncertainty and external interference of the system, and simultaneously balance the performance of the system and the control cost of the system.
Description
Technical Field
The invention relates to the field of automobile active safety control, in particular to a control and optimal design method of an active stabilizer bar system.
Background
When the automobile turns, the automobile body can incline under lateral acceleration under the action of centrifugal force, so that accidents such as rollover are easily caused, the comfort of personnel in the automobile and the maneuverability of the automobile are reduced, and the safety factor is indirectly reduced. In order to reduce the side inclination of the automobile during turning, the passenger car is generally provided with a passive stabilizer bar at the bottom of the automobile body or the automobile frame, and two ends of the passive stabilizer bar are connected with a lower swing arm of a suspension or an upright post of a shock absorber. When the automobile rolls, the twisted stabilizer bar applies an anti-rolling moment to the automobile body, so that the possibility of rolling and overturning is reduced.
The passive stabilizer bar reduces roll to some extent, but cannot actively adjust the anti-roll moment and cannot completely eliminate roll. The active stabilizer bar system can adjust the exciter according to the signal data of lateral acceleration, roll angle, vehicle speed and the like obtained by feedback or calculation of the sensor by adopting a proper control method to output anti-roll moment with corresponding magnitude, has good anti-roll effect, and can improve the safety and the controllability of the vehicle and the riding comfort of people in the vehicle.
In order to solve the problem, researchers at home and abroad successively develop various control methods, and the control methods have certain effects, such as PID control, linear feedback control, LQR (linear quadratic regulator) control, robust control and the like. The system is easy to vibrate under nonlinear disturbance by PID control and linear feedback control, the robustness of LQR control is not strong, and overshoot and unnecessary control consumption are easy to generate by robust control.
Therefore, there is a need to provide a control scheme with high stability and robustness.
Disclosure of Invention
Aiming at the technical problems, the invention provides a control and optimization design method of an active stabilizer bar system, which can effectively process the influence of uncertainty of system parameters and external interference and simultaneously carry out optimization design on control parameters, so that the balance between the performance of the system and the control cost of the system is realized.
The technical scheme adopted by the invention is as follows:
a control and optimization design method of an active stabilizer bar system comprises the following steps:
on the basis of establishing a dynamic model of an active stabilizer bar system with parameter uncertainty, describing the uncertainty by adopting a fuzzy set theory so as to construct a fuzzy dynamic model of the active stabilizer bar system;
writing a state equation form into the fuzzy dynamics model, and providing a deterministic robust controller to solve the problem of system uncertainty;
performing stability analysis on the constructed robust controller;
an optimization function containing the steady-state performance and the control cost of the system is provided, and the optimal solution of the control parameters is obtained by minimizing the optimization function;
and adjusting the weight coefficient in the optimization function, and analyzing the influence of the weight coefficient on the control parameters and the control effect.
Optionally, the constructing a fuzzy dynamic model of the active stabilizer bar system with parameter uncertainty comprises:
constructing a dynamic model of the active stabilizer bar system as shown in equation (1) below:
wherein,
for the moment of inertia of the sprung mass about the roll axis, <' >>
Is roll damping and/or is based on>
Roll stiffness, M af For anti-roll moment of the front wheel, M ar For the rear wheel anti-roll moment, m s Is sprung mass, h s Is the vertical distance of the center of mass from the roll axis, a y Is a lateral acceleration, g is a gravitational acceleration, ->
Is the side inclination angle of the vehicle body>
For front and rear suspension damping factors>
Front and rear suspension stiffness, respectively>
Respectively, the rigidity of the front and rear stabilizer bars;
assuming the desired roll angle signal for the active stabilizer bar system is
Then the roll angle signal following error e (the difference between the actual roll angle and the desired roll angle) for the system is:
therefore, there are:
therefore, an error dynamic equation of the active stabilizer bar system can be established
Considering the sprung mass m of the system s Is an uncertainty parameter and can be decomposed into:
wherein:
Δ m being a deterministic part of the sprung mass s Is the uncertainty component of the sprung mass.
For uncertainty parameter Δ m s There is a fuzzy set S m In the universe of discourse
In which the membership function is expressed as->
That is to say that the position of the first electrode,
then, based on the above pair uncertainty parameter Δ m s Equation (3) represents a fuzzy dynamic model of the active stabilizer bar system.
Optionally, the fuzzy dynamic model of the driving stabilizer bar system is written in the form of a state equation, and a deterministic robust controller is provided, which includes:
writing the fuzzy kinetic equation of the active stabilizer bar system in the form of a state equation (state variable)
Wherein:
is provided with
Will->
Is decomposed into->
Will->
Is decomposed into->
Write Δ A to +>
Δ B is written as Δ B = BE,
will->
Is decomposed into->
Then equation (6) can be converted to:
wherein:
for equation of state (7), the following robust controller is proposed:
u(t)＝-R -1 B T Px(t)-h(t)-γR -1 B T Px(t)(δ 1 ||x(t)||+δ 2 ) 2 (8)
wherein: the positive definite matrix P is Riccati equation A T P+PA-2PBR -1 BTP + Q =0 solution, R, Q is an arbitrary positive definite matrix; delta 1 、δ 2 And γ is a positive constant.
Optionally, the performing the stability analysis on the constructed robust controller comprises:
the final stable boundary of the constructed robust controller was analyzed using the lyapunov function as shown in equation (9) below:
V(x)＝x T Px (9)
calculation of equation (9) yields the following equation (10):
wherein: lambda [ alpha ] min Representing the minimum eigenvalue of the matrix;
ρ E1 to satisfy the conditions
A constant of (c);
ρ E2 To satisfy the condition ρ E2 A fuzzy number more than or equal to E, a, b is a fuzzy number satisfying the condition that v is less than or equal to a x + b, and then is/are put in the air>
To satisfy the condition>
Is constant.
Obtaining a trade-off parameter R of the final stable consistent limit of the driving stabilizer bar system based on the formula (10), as shown in the following formula (11):
the size of the final stabilization limit of the driving stabilizer bar system is obtained based on equation (9), as shown in equation (12) below:
wherein,dlower limit value, gamma, representing the size of the final uniform stability limit of the active stabilizer bar system 1 ＝λ min (P)，γ 2 ＝λ max (P)，λ min (P) represents the minimum eigenvalue, λ, of the positive definite matrix P max (P represents the maximum eigenvalue of the positive definite matrix P;
according to the Lyapunov theory of stability, the time for the active stabilizer bar system to reach the final stabilization limit can be obtained, as shown in equation (13) below:
where T denotes the time for the active stabilizer bar system to reach the final uniform stabilization limit, r denotes the initial state of the system,
is arbitrarily greater thandPositive number of (c).
Optionally, the providing an optimization function including system steady-state performance and system control cost, and obtaining an optimal solution of the control parameter by minimizing the optimization function includes:
from equations (11) and (12), we give expression (14) of the final consistent stability bound lower limit of the system:
due to the fact that
Is a fuzzy number (whose membership function passes through mu) δ (v) we defuzzify it by:
then, we propose an optimization function (16):
J(γ)＝J 1 (γ)+βJ 2 (γ) (16)
wherein,
J 1 (gamma) reflects the steady state performance index of the system, and the larger the gamma is, the better the steady state performance is; j. the design is a square 2 (γ)＝γ 2 ，J 2 (gamma) reflects the control cost of the system, and the larger gamma is, the larger the control cost is; beta is a positive weight coefficient.
Constructing an optimization problem: when γ > 0, the minimum value of J (γ) is sought, i.e.
minJ(γ)γ＞0 (17)
By pairs
When the analysis is performed, the
When J (gamma) is smaller than the predetermined value, J (gamma) takes a minimum value. Then, the control parameter γ at this time is the optimal control parameter.
Optionally, the adjusting the optimization function weight coefficient and analyzing the control effect includes:
adjusting the weight coefficient of the constructed optimization function;
and analyzing whether the error between the actual roll angle and the target roll angle meets the preset error requirement or not based on the adjusted coefficient.
The invention has the technical effects that:
the invention provides a control and optimization design method of an active stabilizer bar system, which comprises the steps of firstly, describing uncertainty by adopting a fuzzy set theory on the basis of establishing a dynamic model of the active stabilizer bar system containing parameter uncertainty, thereby establishing a fuzzy dynamic model of the active stabilizer bar system; then, aiming at the fuzzy dynamic model, a deterministic robust controller is designed to compensate the uncertainty of the system; and performing stability analysis on the constructed robust controller; finally, an optimization function containing the system steady-state performance and the system control cost is provided, and the optimal solution of the control parameters is obtained by minimizing the optimization function; and adjusting the weight coefficient in the optimization function, and analyzing the influence of the weight coefficient on the control parameters and the control effect. Therefore, the method can effectively deal with the influence of parameter uncertainty and external interference of the system, simultaneously the performance of the system and the control cost of the system are balanced, and the anti-roll characteristic and the riding comfort are better.
Drawings
FIG. 1 is a schematic flow chart of a method for controlling and optimally designing an active stabilizer bar system according to an embodiment of the present invention;
FIG. 2 is a schematic structural diagram of an anti-roll system according to an embodiment of the present invention;
fig. 3 is a schematic diagram of an overall structure of a controller according to an embodiment of the present invention;
fig. 4 is a schematic diagram illustrating a stability simulation of the anti-roll system according to the embodiment of the present invention.
Detailed Description
In order to make the technical problems, technical solutions and advantages of the present invention more apparent, the following detailed description is given with reference to the accompanying drawings and specific embodiments.
Fig. 1 is a schematic flow chart of a control and optimal design method of an active stabilizer bar system according to an embodiment of the present invention. As shown in fig. 1, the method for controlling and optimally designing an active stabilizer bar system according to an embodiment of the present invention includes the following steps:
s101, on the basis of establishing a dynamic model of an active stabilizer bar system with parameter uncertainty, describing the uncertainty by adopting a fuzzy set theory, and establishing a fuzzy dynamic model of the model;
s102, writing a state equation form into the fuzzy dynamics model, and providing a deterministic robust controller to solve the problem of system uncertainty;
s103, performing stability analysis on the constructed robust controller;
s104, providing an optimization function containing the steady-state performance and the control cost of the system, and obtaining the optimal solution of the control parameters by minimizing the optimization function;
and S105, adjusting the weight coefficient in the optimization function, and analyzing the influence of the weight coefficient on the control parameters and the control effect.
In the invention, the fuzzy dynamic model constructed in the step S101 is the basis for designing a robust controller based on the model in the step S102; step S103 is to analyze the stability of the robust controller designed in step S102 to ensure that the controller can realize the final stability of the active stabilizer bar system; step S104, performing optimization design analysis on the control parameters in the step S102 by using the final steady-state performance obtained by analysis in the step S103 and combining the control cost of the system; step S105 is to analyze the influence of the weight coefficients in the optimization function proposed in step S104 on the control parameters and the control effect.
Specifically, the step S101 of constructing a fuzzy dynamic model of the active stabilizer bar system with parameter uncertainty includes:
constructing a dynamic model of the active stabilizer bar system as shown in equation (1) below:
wherein,
in relation to the moment of inertia of the sprung mass about the roll axis>
Is roll damping and/or is based on>
Roll stiffness, M af For anti-roll moment of the front wheel, M ar For the rear wheel anti-roll moment, m s Is a sprung mass, h s Is the vertical distance of the center of mass from the roll axis, a y Is a lateral acceleration, g is a gravitational acceleration, ->
Is the side inclination angle of the vehicle body>
For front and rear suspension damping factors>
Front and rear suspension stiffness, < > respectively>
Respectively the rigidity of the front and rear stabilizer bars;
assuming the desired roll angle signal for the active stabilizer bar system is
Then the roll angle signal following error e (the difference between the actual roll angle and the desired roll angle) for the system is:
therefore, the following are provided:
therefore, an error dynamic equation of the active stabilizer bar system can be established
Considering the sprung mass m of the system s Is an uncertainty parameter and can be decomposed into:
wherein:
Δ m, a deterministic part of the sprung mass s Is the uncertainty component of the sprung mass.
For uncertainty parameter Δ m s There is a fuzzy set S m In the universe of discourse
In which the membership function is expressed as->
That is to say that the position of the first electrode,
then, based on the above pair uncertainty parameter Δ m s Equation (3) represents a fuzzy dynamic model of the active stabilizer bar system.
Further, the fuzzy dynamic model of the driving stabilizer bar system in step S102 is written in the form of a state equation, and a deterministic robust controller is provided, which includes:
writing the fuzzy kinetic equation of the active stabilizer bar system in the form of a state equation (state variable)
Wherein:
is provided with
Will->
Is decomposed into->
Will->
Is decomposed into->
Write Δ A to +>
Write Δ B to
Will be/are>
Is decomposed into->
Then equation (6) can be converted to:
wherein:
for equation of state (7), the following robust controller is proposed:
u(t)＝-R -1 B T Px(t)-h(t)-γR -1 B T Px(t)(δ 1 ||x(t)||+δ 2 ) 2 (8)
wherein: the positive definite matrix P is Riccati equation A T P+PA-2PBR -1 BTP + Q =0 solution, R, Q is an arbitrary positive definite matrix; delta 1 、δ 2 And γ is a positive constant.
Further, the stability analysis of the constructed robust controller in step S103 includes:
the final consistent stable bound of the constructed robust controller was analyzed using the lyapunov function as shown in equation (9) below:
V(x)＝x T Px (9)
calculation of equation (9) yields the following formula (10):
wherein: lambda [ alpha ] min Representing the minimum eigenvalue of the matrix;
ρ E1 to satisfy the conditions
A constant of (d);
ρ E2 To satisfy the condition ρ E2 A fuzzy number more than or equal to E, a, b is a fuzzy number satisfying the condition that v is less than or equal to a x + b, and then is/are put in the air>
To satisfy the condition>
Is constant.
A trade-off parameter R of the final stabilization limit of the driving stabilizer bar system is obtained based on equation (10), as shown in equation (11) below:
the size of the final stabilization limit of the driving stabilizer bar system is obtained based on equation (9), as shown in equation (12) below:
where d represents the lower limit value of the size of the final uniform stability limit of the active stabilizer bar system, γ 1 ＝λ min (P)，γ 2 ＝λ max (P)，λ min (P) represents the minimum eigenvalue, λ, of the positive definite matrix P max (P) represents the maximum eigenvalue of the positive definite matrix P;
according to the Lyapunov theory of stability, the time for the active stabilizer bar system to reach the final stabilization limit can be obtained, as shown in equation (13) below:
where T denotes the time for the active stabilizer bar system to reach the final uniform stabilization limit, r denotes the initial state of the system,
is arbitrarily greater thandPositive number of (c).
Further, step S104 provides an optimization function including the steady-state performance of the system and the control cost of the system, and the step of minimizing the optimization function to obtain the optimal solution of the control parameter includes:
from equations (11) and (12), we give the expression (14) for the lower limit of the final consistent stability limit of the system:
due to the fact that
Is a fuzzy number (whose membership function is represented by μ δ (v)), which we defuzzify by:
then, we propose an optimization function (16):
J(γ)＝J 1 (γ)+βJ 2 (γ) (16)
wherein,
J 1 (gamma) reflects the steady state performance index of the system, and the larger the gamma is, the better the steady state performance is; j. the design is a square 2 (γ)＝γ 2 ，J 2 (gamma) reflects the control cost of the system, and the larger gamma is, the larger the control cost is; beta is a positive weight coefficient.
Constructing an optimization problem: when γ > 0, the minimum value of J (γ) is sought, i.e.
minJ(γ)γ＞0 (17)
By making a pair
When the analysis is performed, the
When J (gamma) is smaller than the predetermined value, J (gamma) takes a minimum value. Then, the control parameter γ at this time is the optimal control parameter.
Further, the step S105 of adjusting the weight coefficient of the optimization function, and analyzing the control effect includes:
adjusting the weight coefficient beta of the constructed optimization function; and analyzing whether the error between the actual roll angle and the target roll angle meets the preset error requirement or not based on the adjusted coefficient. Specifically, matlab software can be used for carrying out performance simulation on a control system, whether the error between the actual roll angle and the target roll angle meets the preset error requirement or not is analyzed, if the preset error requirement is met, the process is ended, and if the preset error requirement is not met, the control parameters are continuously adjusted until the preset error requirement is met.
In the embodiment of the invention, in order to balance the performance of the active stabilizer bar system and the control cost, the parameter gamma needs to be optimally designed. Based on this idea, we propose an optimization function J (γ) in which a weight coefficient β exists. When the beta value is smaller, the value of the optimal control parameter gamma is larger, the first item of the optimization function is dominant, and the steady-state performance of the system is better; when the value of beta is larger, the value of the optimal control parameter gamma is smaller, the second term of the optimization function is dominant, and the steady-state performance of the system is poorer.
The active anti-roll system shown in fig. 2 is the subject of control of the present invention. As shown in fig. 2, the system includes two stabilizer bars, a front stabilizer bar and a rear stabilizer bar, and a driving actuator, wherein the front driving actuator is integrated in the middle of the front stabilizer bar, and the rear driving actuator is integrated in the middle of the rear stabilizer bar. When the automobile turns and rolls, the active actuator outputs the rotating torque in the opposite direction, and the anti-rolling control target is realized.
Fig. 3 shows the overall structure of the controller design and optimization of the present invention. As shown in fig. 3, firstly, a dynamic equation based on a state equation is established according to a difference value between an actual roll angle and a target roll angle of the active stabilizer bar system, then, a corresponding robust controller is designed, an optimization objective function is proposed based on the designed controller, an optimal control parameter is obtained by minimizing the optimal control function, and finally, the control is applied to the active stabilizer bar system.
FIG. 4 is a schematic diagram of a simulation structure of a difference between an actual roll angle and a target roll angle of a vehicle body under different control parameter selections. The figure shows that the roll angle of the vehicle body can accurately, quickly and stably follow the expected track through the torque output by the controller under the condition that the system has uncertainty. Meanwhile, the method disclosed by the invention can be found to be more accurate and smooth by comparing with the control mode of the LQG in simulation, and the effectiveness and superiority of the design method disclosed by the invention are proved.
The above-mentioned embodiments are only specific embodiments of the present invention, which are used for illustrating the technical solutions of the present invention and not for limiting the same, and the protection scope of the present invention is not limited thereto, although the present invention is described in detail with reference to the foregoing embodiments, those skilled in the art should understand that: those skilled in the art can still make modifications or changes to the embodiments described in the foregoing embodiments, or make equivalent substitutions for some features, within the scope of the disclosure; such modifications, changes or substitutions do not depart from the spirit and scope of the embodiments of the present invention, and they should be construed as being included therein. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
Claims (1)
1. A control and optimization design method of an active stabilizer bar system is characterized by comprising the following steps:
on the basis of establishing a dynamic model of an active stabilizer bar system with parameter uncertainty, describing the uncertainty by adopting a fuzzy set theory so as to construct a fuzzy dynamic model of the active stabilizer bar system;
writing a state equation form into the fuzzy dynamics model, and providing a deterministic robust controller to solve the problem of system uncertainty;
performing stability analysis on the constructed robust controller;
providing an optimization function containing the steady-state performance and the control cost of the system, and obtaining the optimal solution of the control parameters by minimizing the optimization function;
adjusting the weight coefficient in the optimization function, and analyzing the influence of the weight coefficient on the control parameters and the control effect;
constructing the fuzzy dynamics model comprises:
constructing a dynamic model of the active stabilizer bar system as shown in equation (1) below:
wherein,
for the moment of inertia of the sprung mass about the roll axis, <' >>
Is roll damping and/or is based on>
Roll stiffness, M af For anti-roll moment of the front wheel, M ar For the rear wheel anti-roll moment, m s Is sprung mass, h s Is the vertical distance of the center of mass from the roll axis, a y Is a lateral acceleration, g is a gravitational acceleration, ->
Is the side inclination angle of the vehicle body>
For front and rear suspension damping factors>
Front and rear suspension stiffness, < > respectively>
Respectively the rigidity of the front and rear stabilizer bars;
assuming the desired roll angle signal for the active stabilizer bar system is
Then the roll angle signal of the system follows the error e, i.e. the difference between the actual roll angle and the desired roll angle is:
therefore, the following are provided:
therefore, an error dynamic equation of the active stabilizer bar system can be established
Considering the sprung mass m of the system s Is an uncertainty parameter and can be decomposed into:
wherein:
Δ m being a deterministic part of the sprung mass s Is the uncertainty component of the sprung mass;
for uncertainty parameter Δ m s There is a fuzzy set S m In the universe of discourse
Wherein the membership function is represented by
That is to say that the position of the first electrode,
then, based on the above pair uncertainty parameter Δ m s The equation (3) represents a fuzzy dynamic model of the active stabilizer bar system;
the fuzzy dynamic model is written in a form of a state equation, and a deterministic robust controller is provided, which comprises:
writing the fuzzy kinetic equation of the active stabilizer bar system in the form of a state equation (state variable)
Wherein:
is provided with
Will->
Is decomposed into->
Will be/are>
Is decomposed into->
Write Δ A to +>
Δ B is written as Δ B = BE,
will->
Is decomposed into->
Then equation (6) can be converted to:
wherein:
for equation of state (7), the following robust controller is proposed:
u(t)＝-R -1 B T Px(t)-h(t)-γR -1 B T Px(t)(δ 1 ||x(t)||+δ 2 ) 2 (8)
wherein: the positive definite matrix P is Riccati equation A T P+PA-2PBR -1 B T Solution of P + Q =0, R, Q is an arbitrary positive definite matrix; delta 1 、δ 2 And γ is a positive constant;
the stability analysis of the constructed robust controller comprises:
the final consistent stable bound of the constructed robust controller was analyzed using the lyapunov function as shown in equation (9) below:
V(x)＝x T Px (9)
calculation of equation (9) yields the following formula (10):
wherein: lambda min Representing the minimum eigenvalue of the matrix;
ρ E1 to satisfy the condition
A constant of (c);
ρ E2 To satisfy the condition ρ E2 A fuzzy number more than or equal to E, a, b is a fuzzy number satisfying the condition that v is less than or equal to a x + b, and then is/are put in the air>
To satisfy the condition>
A constant of (d);
a trade-off parameter R of the final stabilization limit of the driving stabilizer bar system is obtained based on equation (10), as shown in equation (11) below:
and (3) obtaining the size of the final stable limit of the driving stabilizer bar system based on the formula (9), as shown in the following formula (12):
wherein,dlower limit value, gamma, representing the size of the final uniform stability limit of the active stabilizer bar system 1 ＝λ min (P)，γ 2 ＝λ max (P)，λ min (P) represents the minimum eigenvalue, λ, of the positive definite matrix P max (P) represents the maximum eigenvalue of the positive definite matrix P;
according to the Lyapunov theory of stability, the time for the active stabilizer bar system to reach the final stabilization limit can be obtained, as shown in equation (13) below:
wherein T represents the time for the active stabilizer bar system to reach the final uniform stabilization limit, and r represents the systemIn the initial state of the mobile terminal,
is arbitrarily greater thandA positive number of;
the method for solving the optimal solution of the control parameter by minimizing the optimization function containing the steady-state performance and the control cost of the system comprises the following steps:
from equations (11) and (12), we give the expression (14) for the lower limit of the final consistent stability limit of the system:
due to the fact that
Is a fuzzy number (whose membership function passes through mu) δ (v) To express), we defuzzify it by: />
Then, we propose an optimization function (16):
J(γ)＝J 1 (γ)+βJ 2 (γ) (16)
wherein,
J 1 (gamma) reflects the steady state performance index of the system, and the larger the gamma is, the better the steady state performance is; j. the design is a square 2 (γ)＝γ 2 ，J 2 (gamma) reflects the control cost of the system, and the larger gamma is, the larger the control cost is; beta is a positive weight coefficient;
constructing an optimization problem: when γ > 0, the minimum value of J (γ) is sought, i.e.
minJ(γ) γ＞0 (17)
By pairs
When the analysis is carried out, the
When J (gamma) is the minimum value; then, the control parameter γ at this time is the optimal control parameter;
the adjusting the weight coefficient of the optimization function and analyzing the control effect comprises:
adjusting the weight coefficient of the constructed optimization function;
and analyzing whether the error between the actual roll angle and the target roll angle meets the preset error requirement or not based on the adjusted coefficient.
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