CN113467233A - Time-lag finite frequency domain output feedback control method based on fuzzy model - Google Patents

Time-lag finite frequency domain output feedback control method based on fuzzy model Download PDF

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CN113467233A
CN113467233A CN202110501561.XA CN202110501561A CN113467233A CN 113467233 A CN113467233 A CN 113467233A CN 202110501561 A CN202110501561 A CN 202110501561A CN 113467233 A CN113467233 A CN 113467233A
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CN113467233B (en
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曲杰
王超
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South China University of Technology SCUT
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Abstract

The invention discloses a time-lag finite frequency domain output feedback control method based on a fuzzy model, which comprises the following steps of: establishing a dynamic model of the automobile active suspension mechanics; determining the variation range of the sprung mass and the unsprung mass of the automobile; constructing two key physical quantities for evaluating the performance of the control method; establishing a time-lag state space model of the automobile suspension; establishing a fuzzy state space model of an automobile suspension time-lag system; obtaining an integral time-lag fuzzy control space model of the automobile suspension system and establishing a static output feedback controller which meets the requirements of a limited frequency domain and the asymptotic stability of a closed-loop system. The invention can realize the high-performance control target of the automobile suspension system, meet the requirements of high comfort and high safety in the driving process, and particularly consider the frequency range of 4-8Hz which is most sensitive to vibration of a human body and the time lag problem of a sensor and an actuator; and the robustness and stability are strong, and the real-time requirement of an automobile suspension system can be met.

Description

Time-lag finite frequency domain output feedback control method based on fuzzy model
Technical Field
The invention belongs to the field of intelligent automobile manufacturing, and particularly relates to a time-lag finite frequency domain output feedback control method based on a fuzzy model.
Background
The suspension system of the automobile consists of a connecting rod, a spring and a shock absorber, and can greatly improve the riding comfort, the operation stability and the ground holding force of the automobile. As an important constituent of the chassis of a motor vehicleIn part, automotive suspensions have attracted extensive attention. In order to improve noise, vibration and harshness (NVH) performance of automotive suspensions, a great deal of research has been conducted on automotive suspension systems, such as passive suspensions, energy regenerative suspensions, semi/slow active suspensions, etc. In current research, active suspension is an effective method to combine independent actuators and controllers to improve suspension performance. It is well known that these three main performances are always conflicting, especially in the trade-off between ride comfort and road holding capacity. To improve ride comfort and maintain suspension and tire displacements within acceptable ranges, various control methods are designed and applied, such as sliding mode control, predictive control, model predictive control, H∞And (5) controlling. Wherein H2、H∞And H2/H∞Control methods are widely discussed, particularly in the context of robustness and interference attenuation.
Time delays are very common in various engineering systems, such as long transmission lines, hydraulic systems, electronic systems, and magnetorheological systems. The presence of skew may be a source of instability and poor performance. In active suspension systems, there is always a time lag in the control channel since the digital controller takes time to calculate, the actuator takes time to build the required force. It is clear that suspension systems with time lag need to be carefully analyzed and synthesized. Thus, the time lag problem present in active suspension control is investigated herein.
It is noted that all of the above controllers are designed with the model parameters known. Thus, the controller may crash in the face of various parameter changes. Some parameter uncertainty is an unavoidable phenomenon, such as the uncertainty of the sprung and unsprung masses due to passenger count, fuel consumption, tire wear, and the like. Therefore, the active suspension system has become a complex nonlinear system. For the representation of complex nonlinear systems, the Takagi-Sugeno (T-S) fuzzy model has proven to be an efficient method and utility.
However, the fuzzy controllers described above for uncertain non-linear suspension systems all involve the entire frequency range. A key goal of the controller design is to minimize wheel-to-body vibration while maximizing passenger comfort. According to ISO-2631, the human body is more sensitive to vertical vibrations between 4-8 Hz. Furthermore, all road surface excitations occur only in a limited frequency range. Thus, the controller in a certain frequency range is less conservative and more efficient than the controller in the entire frequency range.
The document (Panhui, research on nonlinear control of an automobile active suspension system [ D ]. Harbin Industrial university, 2017.) only considers time lag and limited time, does not consider the frequency range of 4-8Hz which is most sensitive to a human body, and does not consider sprung and unsprung masses, so that the output feedback control of state quantity loss is also considered in consideration of the fact that state feedback quantity in the system cannot be measured.
Similar studies exist to date as this study: a reliable fuzzy state feedback controller is provided for an active suspension system with actuator time lag. In addition, a fuzzy sampling data controller is also provided. Dynamic output feedback dissipation control is proposed for a T-S fuzzy system with time-varying input skew and output constraints. Semi-active vehicle suspensions and magneto-rheological dampers under T-S fuzzy control and experimental verification are researched by some people. In addition, researchers also adopt a dynamic sliding mode control method to carry out fuzzy control on an uncertain vehicle active suspension system.
Disclosure of Invention
The invention aims to provide a time lag limited frequency domain output feedback control method based on a fuzzy model, which is used for solving the problems of time lag and hard constraint existing in a suspension system and considering the most sensitive frequency range of a human body, thereby realizing the high-performance control target of an automobile suspension system and meeting the comfort level and high safety in the driving process.
The invention is realized by at least one of the following technical schemes.
A time lag limited frequency domain output feedback control method based on a fuzzy model comprises the following steps:
1) aiming at a time-lag automobile active suspension system, establishing a dynamic model of the following automobile suspension system according to a mechanical principle;
2) determining the variation range of the sprung mass and the unsprung mass according to the characteristics of the mechanical structure of the vehicle and the allowable number of passengers and the variation of the effective load mass;
3) constructing two physical quantities to evaluate the performance of the control method;
4) establishing a time-lag state space model of an automobile suspension system;
5) establishing a time-lag fuzzy state space model of the automobile suspension system according to the Takagi-Sugeno fuzzy model;
6) establishing a time-lag output feedback controller;
7) solving the gain of an output feedback controller of the automobile suspension system;
8) optimizing a feedback controller;
9) and a fuzzy finite frequency domain output feedback controller is adopted to carry out online control on the automobile suspension system.
Preferably, the dynamic model of the automobile suspension system is as follows:
Figure BDA0003056628470000021
wherein m iss(t) is the sprung mass formed by the mass of the automobile body, unit: kg; m isu(t) unsprung mass, unit, of vehicle tire component mass: kg; u (t) is the control input quantity of the automobile active suspension system, and the unit is as follows: n; z is a radical ofs(t) is ms(t) vertical displacement of the sprung mass in a vertically upward direction with the horizontal ground as a starting point, in units of: m;
Figure BDA0003056628470000038
and
Figure BDA0003056628470000039
acceleration and velocity of the sprung mass; z is a radical ofu(t) is mu(t) vertical displacement of the unsprung mass in the vertical upward direction with the horizontal ground as the starting point, in units of m;
Figure BDA00030566284700000310
and
Figure BDA00030566284700000311
acceleration and velocity of the unsprung mass; z is a radical ofr(t) is the vertical displacement of the road surface and the tire contact point in the vertical upward direction with the horizontal ground as the starting point, and the unit is m;
Figure BDA00030566284700000312
the speed input for the road surface; c. CsThe damping coefficient of the automobile suspension system is expressed in the unit of N/(m/s); k is a radical ofsThe unit is N/m, and the rigidity coefficient is the rigidity coefficient of the automobile suspension system; c. CtThe damping coefficient of the automobile tire is expressed by the unit of N/(m/s); k is a radical oftThe rigidity coefficient of the automobile tire is N/m.
Preferably, the sprung mass ms(t) and unsprung mass mu(t) ranges of variation are: m iss(t)∈[msmin，msmax]And mu(t)∈[mumin,mumax]，msminAnd msmaxMinimum and maximum values of sprung mass, muminAnd mumaxThe minimum and maximum values of the unsprung mass.
Preferably, the evaluation control method is:
Figure BDA0003056628470000031
wherein g is the acceleration of gravity in units; N/Kg; and it is necessary to ensure | zs(t)-zu(t)|≤zmaxAnd kt(zu(t)-zr(t))<(ms(t)+mu(t)) g is simultaneously true, z1(t) acceleration of the sprung mass, z2(t) is the relative dynamic deflection z of the suspension21(t) and relative dynamic load z of the wheel22(t) a matrix ofmaxThe maximum displacement stroke of the automobile suspension system is represented by the following unit: and m is selected.
Preferably, the time lag state space model of the automobile suspension system is as follows:
Figure BDA0003056628470000032
wherein, x (t) ═ x is defined1(t)x2(t)x3(t)x4(t)]TIn the form of a matrix of state variables of the system,
Figure BDA00030566284700000313
as derivatives of state variables, x1(t)＝zs(t)-zu(t) dynamic deflection of the suspension, x2(t)＝zu(t)-zr(t) is the dynamic deflection of the wheel,
Figure BDA0003056628470000033
Figure BDA0003056628470000034
is the velocity of the sprung mass,
Figure BDA0003056628470000035
is the velocity of the unsprung mass,
Figure BDA0003056628470000036
the speed of the road surface input is used,
Figure BDA0003056628470000037
three state variable output matrices for the system, A (t) is a system space state variable coefficient matrix, B1(t) System State space road surface disturbance coefficient matrix, B2(t) the system state space variable controls the input coefficient matrix, C1(t) is a sprung mass acceleration output state space coefficient matrix, D1(t) is a control coefficient matrix in the sprung mass acceleration output state space, C2(t) coefficient matrix of suspension relative dynamic deflection and wheel relative dynamic load, C unit matrix, tau is time lag parameter,
Figure BDA0003056628470000041
is an initial continuous function;
Figure BDA0003056628470000042
Figure BDA0003056628470000043
Figure BDA0003056628470000044
Figure BDA0003056628470000045
preferably, in step 5), according to ms(t) and mu(t) change, two fuzzy variables are selected as zeta1(t)＝1/ms(t) and ζ2(t)＝1/mu(t), and establishing a time-lag fuzzy state space model of the automobile suspension system by using a Takagi-Sugeno fuzzy model:
rule 1, if ζ1(t) is M1(ζ1(t)) represents weight, and ζ2(t) is N1(ζ2(t)) represents "heavy", then:
Figure BDA0003056628470000046
rule 2, if ζ1(t) is M1(ζ1(t)) represents weight and ζ2(t) is N2(ζ2(t)) represents light, then
Figure BDA0003056628470000047
Rule 3, if ζ1(t) is M2(ζ1(t)) represents light and ζ2(t) is N1(ζ2(t)) represents weight, then
Figure BDA0003056628470000048
Rule 4, if ζ1(t) is M2(ζ1(t)) represents light and ζ2(t) is N2(ζ2(t)) represents light, then
Figure BDA0003056628470000051
Wherein,
Figure BDA0003056628470000052
representing the sprung mass weight function weight,
Figure BDA0003056628470000053
the weight function representing the sprung mass is light,
Figure BDA0003056628470000054
represents the unsprung mass weight function weight,
Figure BDA0003056628470000055
representing unsprung mass weight function light, M1(ζ1(t)) represents the sprung mass imbalance, M2(ζ1(t)) shows a lighter sprung mass, N1(ζ1(t)) represents the unsprung mass weight, N2(ζ1(t)) means unsprung weight is lighter, A1、B11、B21、C11、D11、C21Each represents the system state space matrix at which the sprung and unsprung masses take a minimum value, namely:
Figure BDA0003056628470000056
Figure BDA0003056628470000057
Figure BDA0003056628470000058
in the formula, A2、B12、B22、C12、D12、C22A system state space matrix representing the minimum and maximum values of the sprung mass, namely:
Figure BDA0003056628470000059
Figure BDA0003056628470000061
Figure BDA0003056628470000062
in the formula, A3、B13、B23、C13、D13、C23The system state space matrix when the maximum value of the sprung mass and the minimum value of the unsprung mass are expressed is as follows:
Figure BDA0003056628470000063
Figure BDA0003056628470000064
Figure BDA0003056628470000065
wherein A is4、B14、B24、C14、D14、C24Indicating springThe system state space matrix when the maximum value of the load mass and the unsprung mass take the maximum value is as follows:
Figure BDA0003056628470000066
Figure BDA0003056628470000067
Figure BDA0003056628470000068
according to the fuzzy modeling method, the overall fuzzy state space model of the automobile suspension system is obtained as follows:
Figure BDA0003056628470000071
wherein, in the formula, h1(ζ (t)) represents a fuzzy weight coefficient under the combination of weight and weight, h2(ζ (t)) represents the blur weight coefficient in the heavy and light combination, h3(ζ (t)) represents the fuzzy weight coefficient at the combination of light and heavy, h4(ζ (t)) represents the blur weight coefficient for a combination of light and light, such that
Figure BDA0003056628470000072
Is the sum of the blur weight coefficients,
Figure BDA0003056628470000073
i is 1,2,3,4, and
h1(ζ(t))＝M1(ζ1(t))×N1(ζ2(t))
h2(ζ(t))＝M1(ζ1(t))×N2(ζ2(t))
h3(ζ(t))＝M2(ζ1(t))×N1(ζ2(t))
h4(ζ(t))＝M2(ζ1(t))×N2(ζ2(t))
wherein, C1h、D1h、C2hAll are derived state space matrices after adding the fuzzy weight coefficients.
Preferably, the time lag output feedback controller is:
Figure BDA0003056628470000074
wherein KjIs a local control gain matrix, such that
Figure BDA0003056628470000075
To control the gain matrix weighted sum, hi＝hi(ζ(t))，hj＝hj(ζ (t-d (t))), ζ (t) represents ζ1(t) and ζ2(t)，hiAnd hjAll are fuzzy weight coefficients, i and j are angle indexes when the coefficients take which values, and h respectively corresponds to 1,2,3 and 41、h2、h3、h4。
Obtaining a closed-loop fuzzy wired frequency domain state space model of the active suspension system considering nonlinearity, time lag and uncertainty through the steps 6) and 7):
Figure BDA0003056628470000076
wherein
Figure BDA0003056628470000077
And at ω (t) ∈ L2[0,∞)L2Representing a two-norm, the frequency and zero initial conditions need to be satisfied:
i. the closed loop system is asymptotically stable;
norm H under condition i∞The performance satisfies:
Figure BDA0003056628470000081
Figure BDA0003056628470000082
for the transfer function of road surface input to vehicle body acceleration, gamma is H∞Norm performance optimization indexes, wherein the values of the norm performance optimization indexes are variables during iterative solution;
Figure BDA0003056628470000083
and
Figure BDA0003056628470000084
inputting a minimum and a maximum input frequency for the road surface;
under condition i, generalized H2The performance satisfies: [ z ]2(t)]q|≤1 q＝1,2；
Preferably, the output feedback controller gain of the automobile suspension system is solved:
Figure BDA0003056628470000085
wherein a general matrix LiAnd LjI, j ═ 1,2,3,4, and is obtained by satisfying the following matrix inequality condition:
Figure BDA0003056628470000086
ξij TΞijξij+Γij+Γij T＜0 (34)
Figure BDA0003056628470000087
here, ,
Figure BDA0003056628470000088
Figure BDA0003056628470000089
Figure BDA00030566284700000810
Γsij＝[0 K -I]T×[0 LjC -Fj]
Γij＝[0 K 0 -I]T×[0 LjC 0 -Fj]
wherein, P1j、S1j、R1jIs an arbitrary symmetrical positive definite matrix,
Figure BDA00030566284700000811
Pj、Qj、Zj、Lj、Fj、Xjin the form of a matrix of any dimension,
Figure BDA00030566284700000812
ξsijis xisijCoefficient matrix of (xi), xiijIs xiijXi, xisij、Γsij、ΓijXi and xiijCorresponding transformation matrix in formula derivation, wherein I is a unit matrix, rho is an arbitrary constant larger than 0, and K is used for originally controlling a gain matrix;
for step 9), the gain K is feedback controlled by solving for the fuzzy statefsfAs the value of K, KfsfThe solution is as follows:
Figure BDA00030566284700000814
wherein, KfsfSolving for a linear matrix inequality condition satisfying:
Figure BDA0003056628470000091
-Qj+τZj＜0 (38)
Figure BDA0003056628470000092
Figure BDA0003056628470000093
wherein alpha isjRho is a scalar greater than 0, q is a corner mark of the matrix, Jj、VjAre all matrix, S1j、P1j、Ai、B2i、R1j、Qj、Pj、Xj、ZjAre all matrixes, and j is a corner mark during matrix transformation.
Preferably, the optimization needs to use the following two algorithms as the optimization of the problem, and comprises the following steps:
the first algorithm is as follows:
step one, solving KfsfLet Kinitial-fsf＝KfsfAs an initial control gain matrix;
step two, solving the optimization problem two to obtain
Figure BDA0003056628470000094
Step three, taking the gain in the step two as an initial value to be brought into the optimization problem one to obtain
Figure BDA0003056628470000095
FjRepresenting a matrix;
and (3) algorithm II:
step 1, solving KfsfAnd order
Figure BDA0003056628470000096
Setting i to 0 as an initial gain matrix corner mark;
step 2, use
Figure BDA0003056628470000097
As an initial value, the initial value is brought into an optimization problem II to obtain
Figure BDA0003056628470000098
And 3, substituting the value obtained in the step 2 into the optimization problem I to obtain
Figure BDA0003056628470000099
And 4, step 4: if a satisfactory control gain is obtained, the operation is exited, otherwise, i is made to be i +1,
Figure BDA00030566284700000910
and then returns to step two.
Preferably, the optimization problem one is as follows: satisfies the minimum H∞Performance of
And (3) minimizing: gamma ray2
Satisfies the following conditions:
Figure BDA0003056628470000101
wherein P is1j、R1j、S1j、Qj、Zj、Pj、Xj、K、Lj、F j1,2, · · 4, K is a priori fixed value: k ═ Kinitial-fsf；
The second optimization problem is as follows: initial KfsfShould be and KfsofC should be close; i.e. eta | Kfsf-KfsofC | is a 2-norm, minimize: eta
Satisfies the following conditions:
Figure BDA0003056628470000102
wherein, P1j、R1j、S1j、Qj、Zj、Pj、Xj、K、Lj、FjAnd i and j are 1-4, and K is a priori value, and the fixed value is as follows: k ═ Kinitial-fsfAnd replacing C, C by identity matrix I⊥A set of null-space orthogonal bases representing C;
the optimized controller gain is used for controlling the suspension system of the automobile on line, and the closed loop system is enabled to be asymptotically stable, the output constraint is limited and H is met∞The performance index γ is minimal.
Compared with the prior art, the invention has the beneficial effects that:
existing fuzzy controllers for uncertain non-linear suspension systems involve the whole frequency range, whereas according to ISO-2631 the human body is more sensitive to vertical vibrations between 4-8 Hz. Therefore, a limited frequency controller suitable for an active suspension system is designed, and the acceleration of a vehicle body in a wired frequency domain can be effectively reduced. GYKP (Generalized Kalman-Yakubovic-Popov) is an effective method for processing Frequency Domain Inequality (FDI) with equivalent Linear Matrix Inequality (LMI). The method is applied to carry out controller analysis and synthesis on the active control system in a relevant frequency range. In an actual suspension system, since not all state vectors are measurable on-line, the controller takes into account that states are not all measurable and applies an output feedback approach to deal with time lag or inertial nonlinearities and uncertainties. The control precision and the driving comfort of the automobile suspension system are greatly improved, and the effectiveness of the controller is verified through simulation experiments.
Drawings
FIG. 1 is a flow chart of a time-lag finite frequency domain output feedback control method based on a fuzzy model according to the present invention;
FIG. 2 is a schematic representation of the suspension system of the present invention;
FIG. 3 is a frequency domain response graph of vertical acceleration of a vehicle body relating to a fuzzy finite frequency domain in accordance with an embodiment of the present invention;
FIG. 4 is a time domain response graph of the vehicle body vertical relative displacement according to the embodiment of the invention;
FIG. 5 is a time domain response graph of the vertical relative dynamic loading of the fuzzy finite frequency domain according to the embodiment of the invention.
Detailed Description
The present invention is further described in the following examples and with reference to the accompanying drawings so that one skilled in the art can better understand the present invention and can practice it, but the examples should not be construed as limiting the present invention.
Fig. 1 shows a time lag finite frequency domain output feedback control method based on a fuzzy model, which includes the following steps:
(1) aiming at the time-lag automobile active suspension system, establishing a dynamic model of the automobile suspension system according to Newton's second theorem;
(2) m is determined by considering the characteristics of the mechanical structure of the automobile and the variation of the number of allowed passengers and the mass of the effective loads(t) and mu(t) ranges of variation are: m iss(t)∈[msmin，msmax]And mu(t)∈[mumin,mumax]；
(3) Two physical quantities are constructed to evaluate the performance of the control method in consideration of main influence factors related to riding comfort and high safety of passengers and drivers;
(4) according to the dynamic model of the automobile active suspension mechanics established in the step (1), establishing a time-lag state space model of an automobile suspension system;
(5) due to ms(t) and mu(t) is varied, so two fuzzy variables are selected as ζ1(t)＝1/ms(t) and ζ2(t)＝1/mu(t) establishing a time-lag fuzzy state space model of the automobile suspension system by using a Takagi-Sugeno fuzzy model;
(6) obtaining an integral fuzzy state space model of the automobile suspension system according to a fuzzy modeling method;
(7) aiming at the integral fuzzy state space model of the automobile suspension system in the step (6), establishing a time-lag output feedback controller;
(8) rewriting a time-lag closed-loop system state space of the nonlinear uncertain active suspension system through the step (6) and the step (7);
(9) solving the gain of an output feedback controller of the automobile suspension system;
(10) for K in step (9), feedback control gain K can be obtained by solving fuzzy statefsfInputting as a K initial value;
(11) to better meet the performance requirements, two algorithms are designed to optimize K in step 9)fsof；
(12) The fuzzy finite frequency domain controller given in the step (7) can be used for controlling the suspension system of the automobile on line.
The fuzzy finite frequency domain control method based on the automobile suspension system can be applied to various active suspension systems. Hereinafter, a two-degree-of-freedom 1/4 suspension system is taken as an example, and a practical control application is performed.
Specifically, the fuzzy finite frequency domain output feedback control method of the automobile suspension system comprises the following working processes:
1) the following dynamic model of the automobile suspension system is established according to the mechanical principle:
Figure BDA0003056628470000121
wherein m iss(t) is the sprung mass of the car, in units: kg; m isu(t) is unsprung mass of the automobile in units: kg; u (t) is the control input quantity of the automobile active suspension system. Unit: n; z is a radical ofs(t) is msVertical displacement of the sprung mass in the vertically upward direction with the horizontal ground as the starting point, in units: m; z is a radical ofu(t) is muThe vertical displacement of the unsprung mass in the vertical upward direction with the horizontal ground as the starting point, in units of m; z is a radical ofr(t) is the vertical displacement of the road surface and the tire contact point in the vertical upward direction with the horizontal ground as the starting point, and the unit is m; c. CsThe damping coefficient of the automobile suspension system is expressed in the unit of N/(m/s); k is a radical ofsThe unit is N/m, and the rigidity coefficient is the rigidity coefficient of the automobile suspension system; c. CtThe damping coefficient of the automobile tire is expressed by the unit of N/(m/s); k is a radical oftThe rigidity coefficient of the automobile tire is N/m.
The main technical performance indicators and equipment parameters of the 1/4 suspension system using two degrees of freedom in this embodiment are: m iss(t)∈[256Kg,384Kg]，mu(t)∈[35Kg,45Kg]，ks＝18000N/m，kt＝200000N/m，cs＝1000N/(m/s)，ct＝10N/(m/s)，zmax＝0.1m,umax＝2500N,w1＝4Hz,w2＝8Hz,ρ＝1,τ＝5ms。
2) M is determined by considering the characteristics of the mechanical structure of the automobile and the variation of the number of allowed passengers and the mass of the effective loads(t) and mu(t) ranges of variation are: m iss(t)∈[msmin,msmax]＝[256Kg,384Kg]And mu(t)∈[mumin,mumax]＝[35Kg,45Kg]；
3) Considering the main influence factors related to the riding comfort and high safety of passengers and drivers, the following two physical quantities are used for evaluating the performance of the control method;
Figure BDA0003056628470000122
Figure BDA0003056628470000123
wherein the maximum displacement travel z of the suspension system of the vehiclemax0.1m, unit: m; g is the gravity acceleration g is 9.8N/Kg, unit; N/Kg; and it is necessary to ensure | zs(t)-zu(t) less than or equal to 0.1m and kt(zu(t)-zr(t))<(ms(t)+mu(t)). times.9.8N/Kg) are simultaneously established, z1(t) acceleration of the sprung mass, z2(t) is the relative dynamic deflection z of the suspension21(t) and relative dynamic load z of the wheel22(t) a matrix ofmaxThe maximum displacement stroke of the automobile suspension system is represented by the following unit: and m is selected.
4) Establishing a state space model of the automobile suspension system according to the dynamic model of the automobile active suspension mechanics established in the step 1):
Figure BDA0003056628470000131
z1(t)＝C1(t)x(t)+D1(t)u(t-d(t))
z2(t)＝C2(t)x(t)
y(t)＝Cx(t)
x(t)＝φ(t),t∈[-τ,0]
wherein, x (t) ═ x is defined1(t)x2(t)x3(t)x4(t)]TIn the form of a matrix of state variables of the system,
Figure BDA00030566284700001313
as derivatives of state variables, x1(t)＝zs(t)-zu(t) dynamic deflection of the suspension, x2(t)＝zu(t)-zr(t) is the dynamic deflection of the wheel,
Figure BDA0003056628470000132
Figure BDA0003056628470000133
is the velocity of the sprung mass,
Figure BDA0003056628470000134
is the velocity of the unsprung mass,
Figure BDA0003056628470000135
the speed of the road surface input is used,
Figure BDA0003056628470000136
three state variable output matrices for the system, A (t) is a system space state variable coefficient matrix, B1(t) System State space road surface disturbance coefficient matrix, B2(t) the system state space variable controls the input coefficient matrix, C1(t) is a sprung mass acceleration output state space coefficient matrix, D1(t) is a control coefficient matrix in the sprung mass acceleration output state space, C2(t) coefficient matrix of suspension relative dynamic deflection and wheel relative dynamic load, C unit matrix, tau is time lag parameter,
Figure BDA0003056628470000137
is an initial continuous function;
Figure BDA0003056628470000138
Figure BDA0003056628470000139
Figure BDA00030566284700001310
Figure BDA00030566284700001311
Figure BDA00030566284700001312
5) due to ms(t) and mu(t) is varied, so two fuzzy variables are selected as ζ1(t)＝1/ms(t) and ζ2(t)＝1/mu(t), and establishing a time-lag fuzzy state space model of the automobile suspension system by using a Takagi-Sugeno fuzzy model:
rule 1, if ζ1(t) is M1(ζ1(t)) represents weight and ζ2(t) is N1(ζ2(t)) represents weight, then
Figure BDA0003056628470000141
z1(t)＝C11x(t)+D11u(t-d(t))
z2(t)＝C21x(t)
Rule 2, if ζ1(t) is M1(ζ1(t)) represents weight and ζ2(t) is N2(ζ2(t)) represents light, then
Figure BDA0003056628470000142
z1(t)＝C12x(t)+D12u(t-d(t))
z2(t)＝C22x(t)
Rule 3, if ζ1(t) is M2(ζ1(t)) represents light and ζ2(t) is N1(ζ2(t)) represents weight, then
Figure BDA0003056628470000143
z1(t)＝C13x(t)+D13u(t-d(t))
z2(t)＝C23x(t)
Rule 4, if ζ1(t) is M2(ζ1(t)) represents light and ζ2(t) is N2(ζ2(t)) represents light, then
Figure BDA0003056628470000144
z1(t)＝C14x(t)+D14u(t-d(t))
z2(t)＝C24x(t)
Wherein,
Figure BDA0003056628470000145
representing the sprung mass weight function weight,
Figure BDA0003056628470000146
the weight function representing the sprung mass is light,
Figure BDA0003056628470000147
representing unsprung mass weightThe function is heavy and the function is heavy,
Figure BDA0003056628470000148
representing the unsprung mass weight function light, which is a fuzzy concept without distinct boundaries, M1(ζ1(t)) represents the sprung mass imbalance, M2(ζ1(t)) shows a lighter sprung mass, N1(ζ1(t)) represents the unsprung mass weight, N2(ζ1(t)) means unsprung weight is lighter, A1、B11、B21、C11、D11、C21Each represents the system state space matrix at which the sprung and unsprung masses take a minimum value, namely:
Figure BDA0003056628470000149
Figure BDA0003056628470000151
Figure BDA0003056628470000152
Figure BDA0003056628470000153
Figure BDA0003056628470000154
Figure BDA0003056628470000155
Figure BDA0003056628470000156
Figure BDA0003056628470000157
in the formula, A3、B13、B23、C13、D13、C23The system state space matrix when the maximum value of the sprung mass and the minimum value of the unsprung mass are expressed is as follows:
Figure BDA0003056628470000158
Figure BDA0003056628470000161
Figure BDA0003056628470000162
Figure BDA0003056628470000163
wherein A is4、B14、B24、C14、D14、C24The system state space matrix representing the maximum value of the sprung mass and the maximum value of the unsprung mass is:
Figure BDA0003056628470000164
Figure BDA0003056628470000165
Figure BDA0003056628470000166
Figure BDA0003056628470000167
6) according to the fuzzy modeling method, an integral fuzzy state space model of the automobile suspension system can be obtained:
Figure BDA0003056628470000171
Figure BDA0003056628470000172
Figure BDA0003056628470000173
wherein, in the formula, h1(ζ (t)) represents a fuzzy weight coefficient under the combination of weight and weight, h2(ζ (t)) represents the blur weight coefficient in the heavy and light combination, h3(ζ (t)) represents the fuzzy weight coefficient at the combination of light and heavy, h4(ζ (t)) represents the blur weight coefficient for a combination of light and light, such that
Figure BDA0003056628470000174
Is the sum of the blur weight coefficients,
Figure BDA0003056628470000175
i is 1,2,3,4, and
h1(ζ(t))＝M1(ζ1(t))×N1(ζ2(t))
h2(ζ(t))＝M1(ζ1(t))×N2(ζ2(t))
h3(ζ(t))＝M2(ζ1(t))×N1(ζ2(t))
h4(ζ(t))＝M2(ζ1(t))×N2(ζ2(t))
wherein, C1h、D1h、C2hAre all added to fuzzy weight systemThe state space matrix derived after counting.
7) Aiming at the fuzzy finite frequency domain state space model of the integral automobile suspension system described in the step 6), a time-lag output feedback controller is established:
Figure BDA0003056628470000176
wherein KjIs a local control gain matrix, such that
Figure BDA0003056628470000177
To control the gain matrix weighted sum, hi＝hi(ζ(t))，hj＝hj(ζ (t-d (t))), ζ (t) represents ζ1(t) and ζ2(t)，hiAnd hjAll are fuzzy weight coefficients, i and j are angle indexes when the coefficients take which values, and h respectively corresponds to 1,2,3 and 41、h2、h3、h4。
8) The state space for rewriteable closed loop systems and nonlinear active suspension systems from steps 6) and 7) is as follows:
Figure BDA0003056628470000178
Figure BDA0003056628470000179
Figure BDA00030566284700001710
wherein
Figure BDA00030566284700001711
And at ω ∈ L2[0, ∞) frequency and zero initial condition:
i. the closed loop system is asymptotically stable;
under condition i, H∞The performance satisfies:
Figure BDA0003056628470000181
under condition i, generalized H2The performance satisfies:
Figure BDA0003056628470000182
9) solving the gain of the output feedback control of the automobile suspension system:
Figure BDA0003056628470000183
wherein a general matrix LjAnd Lj(i, j ═ 1,2,3,4) can be obtained by satisfying the following matrix inequality condition:
Figure BDA0003056628470000184
ξij TΞijξij+Γij+Γij T＜0
Figure BDA0003056628470000185
here, ,
Figure BDA0003056628470000186
Figure BDA0003056628470000187
Figure BDA0003056628470000188
Γsij＝[0 K -I]T×[0 LjC -Fj]
Γij＝[0 K 0 -I]T×[0 LjC 0 -Fj]
the above condition is not a linear matrix inequality due to the presence of the matrix K. However, if K is a fixed matrix a priori, the above condition becomes the antecedent matrix inequality for the remaining unknown matrix. Then, K is solved as the gain of the initial fuzzy state feedback controller, and K is brought into the fuzzy static output feedback controller to obtain the output Kfsof. In addition, the above conditions replace the measured output matrix C with the identity matrix I, which can be used to update and improve the gain of the fuzzy state feedback controller.
10) For step 9) the gain K can be controlled by solving the fuzzy state feedbackfsfAs K inputs:
Figure BDA0003056628470000191
wherein KfsfThe following linear matrix inequality conditions may be solved to obtain:
Figure BDA0003056628470000192
-Qj+τZj＜0
Figure BDA0003056628470000193
Figure BDA0003056628470000194
wherein alpha isjRho is a scalar greater than 0, q is a corner mark of the matrix, Jj、VjAre all matrix, S1j、P1j、Ai、B2i、R1j、Qj、Pj、Xj、ZjAre all matrixes, and j is a corner mark during matrix transformation.
11) To better meet the performance requirements, the following two algorithms need to be used as optimization of the problem
The first algorithm is as follows:
step one, solving 10) to enable Kinitial-fsf＝Kfsf；
Step two, solving the optimization problem two to obtain
Figure BDA0003056628470000195
Step three, taking the gain in the step two as an initial value to be brought into the optimization problem one to obtain
Figure BDA0003056628470000196
And (3) algorithm II:
step 1, by solving for 10) and order
Figure BDA0003056628470000201
Setting i to 0;
step 2, use
Figure BDA0003056628470000202
As an initial value, the initial value is brought into an optimization problem II to obtain
Figure BDA0003056628470000203
And 3, substituting the value obtained in the step 2 into the optimization problem I to obtain
Figure BDA0003056628470000204
And 4, if a satisfactory control gain is obtained, exiting. Otherwise, let i be i +1,
Figure BDA0003056628470000205
and then returns to step two.
Optimizing the first problem: satisfies the minimum H∞Performance of
And (3) minimizing: gamma ray2
Satisfies the following conditions:
Figure BDA0003056628470000206
ξij TΞijξij+Γij+Γij T＜0
Figure BDA0003056628470000207
P1j＞0,R1j＞0,S1j＞0,Qj＞0,Zj＞0
wherein P is1j,R1j,S1j,Qj,Zj,Pj,Xj,K,Lj,FjI, j ═ 1,2, · · 4, K is a priori value with a fixed value of: k ═ Kinitial-fsf。
And (2) optimizing a second problem: initial KfsfShould be and KfsofC should be close; then the process of the first step is carried out,
and (3) minimizing: eta
Satisfies the following conditions:
Figure BDA0003056628470000208
ξij TΞijξij+Γij+Γij T＜0
Figure BDA0003056628470000209
Figure BDA00030566284700002010
P1j＞0,R1j＞0,S1j＞0,Qj＞0,Zj＞0
wherein P is1j,R1j,S1j,Qj,Zj,Pj,Xj,K,Lj,FjI, j ═ 1,2, · · 4, K is a priori value with a fixed value of: k ═ Kinitial-fsfAnd C is replaced by an identity matrix I. C⊥A set of null-space orthogonal bases representing C. The following steps can be obtained through the steps: kfsf＝104×[1.2759 -0.2647-0.2649 -0.0723]，Kfsof＝104×[1.2738 -0.2684 -0.0709]。
12) Using the controller gain given in said step 7) can control the suspension system of the automobile on-line and make the closed loop system asymptotically stable and satisfy H∞The performance index γ is minimal.
In this embodiment, the maximum value of the actuator actuating force is umax2500N, frequency range is set to w1＝4Hz,w2The time lag is set to τ 5ms at 8Hz, and the other value ρ is 1.γ is a suppression index value for the external disturbance in the control system, and should be as small as possible when the user satisfies the condition again.
Fig. 2 shows a simplified model of the control of the method of the invention, intended for the user to apply the method more clearly to the specific example. Fig. 3 shows the frequency domain response curve of the closed loop system, and it is obvious that the invention can significantly reduce the acceleration of the vehicle body and greatly improve the riding comfort compared with the passive suspension in the frequency range of 4-8Hz where the human body is most sensitive. Fig. 4 shows a comparison diagram of the relative displacement of the suspension of the invention, and it can be seen from the diagram that the relative displacement is greatly reduced, the limit probability of the impact suspension can be effectively reduced, and the smoothness of the automobile is improved. Fig. 5 shows a comparison of the vertical relative dynamic load of the wheels and the body of the present invention, from which it can be seen that there is a significant reduction in the relative dynamic load, reducing the probability of the wheels jumping off the ground and improving the handling stability of the vehicle.
The above embodiments are merely illustrative of the technical ideas and features of the present invention and are intended to enable those skilled in the art to better understand and implement the present invention. The scope of the present invention is not limited to the embodiments described above, and all equivalent changes and modifications made based on the principles and design ideas disclosed by the present invention are within the scope of the present invention.

Claims (10)

1. A time lag limited frequency domain output feedback control method based on a fuzzy model is characterized by comprising the following steps:
1) aiming at a time-lag automobile active suspension system, establishing a dynamic model of the following automobile suspension system according to a mechanical principle;
2) determining the variation range of the sprung mass and the unsprung mass according to the characteristics of the mechanical structure of the vehicle and the allowable number of passengers and the variation of the effective load mass;
3) constructing two physical quantities to evaluate the performance of the control method;
4) establishing a time-lag state space model of an automobile suspension system;
5) establishing a time-lag fuzzy state space model of the automobile suspension system according to the Takagi-Sugeno fuzzy model;
6) establishing a time-lag output feedback controller;
7) solving the gain of an output feedback controller of the automobile suspension system;
8) optimizing a feedback controller;
9) and a fuzzy finite frequency domain output feedback controller is adopted to carry out online control on the automobile suspension system.
2. The fuzzy model-based time-lag finite frequency domain output feedback control method according to claim 1, wherein the dynamic model of the automobile suspension system is as follows:
Figure RE-RE-FDA0003238253960000011
wherein m iss(t) is the sprung mass formed by the mass of the automobile body, unit: kg; m isu(t) unsprung mass, unit, of vehicle tire component mass: kg; u (t) is the control input quantity of the automobile active suspension system, and the unit is as follows: n; z is a radical ofs(t) is ms(t) verticality of sprung mass in vertically upward direction with horizontal ground as starting pointDisplacement, unit: m;
Figure RE-RE-FDA0003238253960000012
and
Figure RE-RE-FDA0003238253960000013
acceleration and velocity of the sprung mass; z is a radical ofu(t) is mu(t) vertical displacement of the unsprung mass in the vertical upward direction with the horizontal ground as the starting point, in units of m;
Figure RE-RE-FDA0003238253960000014
and
Figure RE-RE-FDA0003238253960000015
acceleration and velocity of the unsprung mass; z is a radical ofr(t) is the vertical displacement of the road surface and the tire contact point in the vertical upward direction with the horizontal ground as the starting point, and the unit is m;
Figure RE-RE-FDA0003238253960000016
the speed input for the road surface; c. CsThe damping coefficient of the automobile suspension system is expressed in the unit of N/(m/s); k is a radical ofsThe unit is N/m, and the rigidity coefficient is the rigidity coefficient of the automobile suspension system; c. CtThe damping coefficient of the automobile tire is expressed by the unit of N/(m/s); k is a radical oftThe rigidity coefficient of the automobile tire is N/m.
3. The fuzzy model-based time-lag finite frequency domain output feedback control method according to claim 2, wherein the sprung mass ms(t) and unsprung mass mu(t) ranges of variation are: m iss(t)∈[msmin，msmax]And mu(t)∈[mumin,mumax]，msminAnd msmaxMinimum and maximum values of sprung mass, muminAnd mumaxThe minimum and maximum values of the unsprung mass.
4. The fuzzy model-based time-lag finite frequency domain output feedback control method according to claim 3, wherein the evaluation control method comprises:
Figure RE-RE-FDA0003238253960000021
wherein g is the acceleration of gravity in units; N/Kg; and it is necessary to ensure | zs(t)-zu(t)|≤zmaxAnd kt(zu(t)-zr(t))＜(ms(t)+mu(t)) g is simultaneously true, z1(t) acceleration of the sprung mass, z2(t) is the relative dynamic deflection z of the suspension21(t) and relative dynamic load z of the wheel22(t) a matrix ofmaxThe maximum displacement stroke of the automobile suspension system is represented by the following unit: and m is selected.
5. The fuzzy model-based time lag finite frequency domain output feedback control method according to claim 4, wherein the time lag state space model of the automobile suspension system is:
Figure RE-RE-FDA0003238253960000022
wherein, x (t) ═ x is defined1(t) x2(t) x3(t) x4(t)]TIn the form of a matrix of state variables of the system,
Figure RE-RE-FDA0003238253960000023
as derivatives of state variables, x1(t)＝zs(t)-zu(t) dynamic deflection of the suspension, x2(t)＝zu(t)-zr(t) is the dynamic deflection of the wheel,
Figure RE-RE-FDA0003238253960000024
Figure RE-RE-FDA0003238253960000025
is the velocity of the sprung mass,
Figure RE-RE-FDA0003238253960000026
is the velocity of the unsprung mass,
Figure RE-RE-FDA0003238253960000027
the speed of the road surface input is used,
Figure RE-RE-FDA0003238253960000028
three state variable output matrices for the system, A (t) is a system space state variable coefficient matrix, B1(t) System State space road surface disturbance coefficient matrix, B2(t) the system state space variable controls the input coefficient matrix, C1(t) is a sprung mass acceleration output state space coefficient matrix, D1(t) is a control coefficient matrix in the sprung mass acceleration output state space, C2(t) coefficient matrix of suspension relative dynamic deflection and wheel relative dynamic load, C unit matrix, tau is time lag parameter,
Figure RE-RE-FDA0003238253960000029
is an initial continuous function;
Figure RE-RE-FDA00032382539600000210
Figure RE-RE-FDA00032382539600000211
Figure RE-RE-FDA0003238253960000031
Figure RE-RE-FDA0003238253960000032
6. the fuzzy model-based time-lag finite frequency domain output feedback control method as claimed in claim 5, wherein in step 5), the feedback control method is based on ms(t) and mu(t) change, two fuzzy variables are selected as zeta1(t)＝1/ms(t) and ζ2(t)＝1/mu(t), and establishing a time-lag fuzzy state space model of the automobile suspension system by using a Takagi-Sugeno fuzzy model:
rule 1, if ζ1(t) is M1(ζ1(t)) represents weight, and ζ2(t) is N1(ζ2(t)) represents weight, then:
Figure RE-RE-FDA0003238253960000033
rule 2, if ζ1(t) is M1(ζ1(t)) represents weight and ζ2(t) is N2(ζ2(t)) represents light, then
Figure RE-RE-FDA0003238253960000034
Rule 3, if ζ1(t) is M2(ζ1(t)) represents light and ζ2(t) is N1(ζ2(t)) represents weight, then
Figure RE-RE-FDA0003238253960000035
Rule 4, if ζ1(t) is M2(ζ1(t)) represents light and ζ2(t) is N2(ζ2(t)) represents light, then
Figure RE-RE-FDA0003238253960000036
Wherein,
Figure RE-RE-FDA0003238253960000041
representing the sprung mass weight function weight,
Figure RE-RE-FDA0003238253960000042
the weight function representing the sprung mass is light,
Figure RE-RE-FDA0003238253960000043
represents the unsprung mass weight function weight,
Figure RE-RE-FDA0003238253960000044
representing unsprung mass weight function light, M1(ζ1(t)) represents the sprung mass imbalance, M2(ζ1(t)) shows a lighter sprung mass, N1(ζ1(t)) represents the unsprung mass weight, N2(ζ1(t)) means unsprung weight is lighter, A1、B11、B21、C11、D11、C21Each represents the system state space matrix at which the sprung and unsprung masses take a minimum value, namely:
Figure RE-RE-FDA0003238253960000045
Figure RE-RE-FDA0003238253960000046
Figure RE-RE-FDA0003238253960000047
in the formula, A2、B12、B22、C12、D12、C22A system state space matrix representing the minimum and maximum values of the sprung mass, namely:
Figure RE-RE-FDA0003238253960000048
Figure RE-RE-FDA0003238253960000051
Figure RE-RE-FDA0003238253960000052
in the formula, A3、B13、B23、C13、D13、C23The system state space matrix when the maximum value of the sprung mass and the minimum value of the unsprung mass are expressed is as follows:
Figure RE-RE-FDA0003238253960000053
Figure RE-RE-FDA0003238253960000054
Figure RE-RE-FDA0003238253960000055
wherein A is4、B14、B24、C14、D14、C24The system state space matrix representing the maximum value of the sprung mass and the maximum value of the unsprung mass is:
Figure RE-RE-FDA0003238253960000056
Figure RE-RE-FDA0003238253960000057
Figure RE-RE-FDA0003238253960000058
according to the fuzzy modeling method, the overall fuzzy state space model of the automobile suspension system is obtained as follows:
Figure RE-RE-FDA0003238253960000061
wherein, in the formula, h1(ζ (t)) represents a fuzzy weight coefficient under the combination of weight and weight, h2(ζ (t)) represents the blur weight coefficient in the heavy and light combination, h3(ζ (t)) represents the fuzzy weight coefficient at the combination of light and heavy, h4(ζ (t)) represents the blur weight coefficient for a combination of light and light, such that
Figure RE-RE-FDA0003238253960000062
Is the sum of the blur weight coefficients,
Figure RE-RE-FDA0003238253960000063
i is 1,2,3,4, and
h1(ζ(t))＝M1(ζ1(t))×N1(ζ2(t))
h2(ζ(t))＝M1(ζ1(t))×N2(ζ2(t))
h3(ζ(t))＝M2(ζ1(t))×N1(ζ2(t))
h4(ζ(t))＝M2(ζ1(t))×N2(ζ2(t))
wherein, C1h、D1h、C2hAll are derived state space matrices after adding the fuzzy weight coefficients.
7. The fuzzy model-based time-lag limited frequency domain output feedback control method of claim 6, wherein the time-lag output feedback controller is:
Figure RE-RE-FDA0003238253960000064
wherein KjIs a local control gain matrix, such that
Figure RE-RE-FDA0003238253960000065
To control the gain matrix weighted sum, hi＝hi(ζ(t))，hj＝hj(ζ (t-d (t))), ζ (t) represents ζ1(t) and ζ2(t)，hiAnd hjAll are fuzzy weight coefficients, i and j are angle indexes when the coefficients take which values, and h respectively corresponds to 1,2,3 and 41、h2、h3、h4；
Obtaining a closed-loop fuzzy wired frequency domain state space model of the active suspension system considering nonlinearity, time lag and uncertainty through the steps 6) and 7):
Figure RE-RE-FDA0003238253960000066
wherein
Figure RE-RE-FDA0003238253960000067
And at ω (t) ∈ L2[0,∞)L2Representing a two-norm, the frequency and zero initial conditions need to be satisfied:
i. the closed loop system is asymptotically stable;
norm H under condition i∞The performance satisfies:
Figure RE-RE-FDA0003238253960000071
Figure RE-RE-FDA0003238253960000072
for the transfer function of road surface input to vehicle body acceleration, gamma is H∞Norm performance optimization indexes, wherein the values of the norm performance optimization indexes are variables during iterative solution;
Figure RE-RE-FDA0003238253960000073
and
Figure RE-RE-FDA0003238253960000074
inputting a minimum and a maximum input frequency for the road surface;
under condition i, generalized H2The performance satisfies: [ z ]2(t)]q|≤1 q＝1,2。
8. The fuzzy model-based time-lag finite frequency domain output feedback control method of claim 7, wherein the output feedback controller gain of the automotive suspension system is solved:
Figure RE-RE-FDA0003238253960000075
wherein a general matrix LiAnd LjI, j ═ 1,2,3,4, and is obtained by satisfying the following matrix inequality condition:
Figure RE-RE-FDA0003238253960000076
Figure RE-RE-FDA0003238253960000077
Figure RE-RE-FDA0003238253960000078
here, ,
Figure RE-RE-FDA0003238253960000079
Figure RE-RE-FDA00032382539600000710
Figure RE-RE-FDA00032382539600000711
Γsij＝[0 K -I]T×[0 LjC -Fj]
Γij＝[0 K 0 -I]T×[0 LjC 0 -Fj]
wherein, P1j、S1j、R1jIs an arbitrary symmetrical positive definite matrix,
Figure RE-RE-FDA00032382539600000712
Pj、Qj、Zj、Lj、Fj、Xjin the form of a matrix of any dimension,
Figure RE-RE-FDA00032382539600000713
ξsijis xisijCoefficient matrix of (xi), xiijIs xiijXi, xisij、Γsij、ΓijXi and xiijCorresponding transformation matrix in formula derivation, wherein I is a unit matrix, rho is an arbitrary constant larger than 0, and K is used for originally controlling a gain matrix;
for step 9), the gain K is feedback controlled by solving for the fuzzy statefsfAs the value of K, KfsfThe solution is as follows:
Figure RE-RE-FDA0003238253960000081
wherein, KfsfSolving for a linear matrix inequality condition satisfying:
Figure RE-RE-FDA0003238253960000082
-Qj+τZj＜0 (17)
Figure RE-RE-FDA0003238253960000083
Figure RE-RE-FDA0003238253960000084
wherein alpha isjRho is a scalar greater than 0, q is a corner mark of the matrix, Jj、VjAre all matrix, S1j、P1j、Ai、B2i、R1j、Qj、Pj、Xj、ZjAre all matrixes, and j is a corner mark during matrix transformation.
9. The fuzzy model-based time-lag finite frequency domain output feedback control method of claim 8, wherein the optimization using the following two algorithms as the optimization of the problem comprises the following steps:
the first algorithm is as follows:
step one, solving KfsfLet Kinitial-fsf＝KfsfAs an initial control gain matrix;
step two, solving the optimization problem two to obtain
Figure RE-RE-FDA0003238253960000085
Step three, taking the gain in the step two as an initial value to be brought into the optimization problem one to obtain
Figure RE-RE-FDA0003238253960000086
FjRepresenting a matrix;
and (3) algorithm II:
step 1, solving KfsfAnd order
Figure RE-RE-FDA0003238253960000087
Setting i to 0 as an initial gain matrix corner mark;
step 2, use
Figure RE-RE-FDA0003238253960000088
As an initial value, the initial value is brought into an optimization problem II to obtain
Figure RE-RE-FDA0003238253960000089
And 3, substituting the value obtained in the step 2 into the optimization problem I to obtain
Figure RE-RE-FDA00032382539600000810
And 4, step 4: if a satisfactory control gain is obtained, the operation is exited, otherwise, i is made to be i +1,
Figure RE-RE-FDA00032382539600000811
and then returns to step two.
10. The method according to claim 9, wherein the first optimization problem is that: satisfies the minimum H∞Performance of
And (3) minimizing: gamma ray2
Satisfies the following conditions:
Figure RE-RE-FDA0003238253960000091
wherein P is1j、R1j、S1j、Qj、Zj、Pj、Xj、K、Lj、Fj1,2, · · 4, K is a priori fixed value:
K＝Kinitial-fsf；
the second optimization problem is as follows: initial KfsfShould be and KfsofC should be close; i.e. eta | Kfsf-KfsofC | is a 2-norm, minimize: eta
Satisfies the following conditions:
Figure RE-RE-FDA0003238253960000092
wherein, P1j、R1j、S1j、Qj、Zj、Pj、Xj、K、Lj、FjAnd i and j are 1-4, and K is a priori value, and the fixed value is as follows: k ═ Kinitial-fsfAnd replacing C, C by identity matrix I⊥A set of null-space orthogonal bases representing C;
the optimized controller gain is used for controlling the suspension system of the automobile on line, and the closed loop system is enabled to be asymptotically stable, the output constraint is limited and H is met∞The performance index γ is minimal.
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