CN110978931A - Vehicle active suspension system modeling and control method based on high semi-Markov switching - Google Patents
Vehicle active suspension system modeling and control method based on high semi-Markov switching Download PDFInfo
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- B60G17/015—Resilient suspensions having means for adjusting the spring or vibration-damper characteristics, for regulating the distance between a supporting surface and a sprung part of vehicle or for locking suspension during use to meet varying vehicular or surface conditions, e.g. due to speed or load the regulating means comprising electric or electronic elements
- B60G17/018—Resilient suspensions having means for adjusting the spring or vibration-damper characteristics, for regulating the distance between a supporting surface and a sprung part of vehicle or for locking suspension during use to meet varying vehicular or surface conditions, e.g. due to speed or load the regulating means comprising electric or electronic elements characterised by the use of a specific signal treatment or control method
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B60—VEHICLES IN GENERAL
- B60G—VEHICLE SUSPENSION ARRANGEMENTS
- B60G17/00—Resilient suspensions having means for adjusting the spring or vibration-damper characteristics, for regulating the distance between a supporting surface and a sprung part of vehicle or for locking suspension during use to meet varying vehicular or surface conditions, e.g. due to speed or load
- B60G17/015—Resilient suspensions having means for adjusting the spring or vibration-damper characteristics, for regulating the distance between a supporting surface and a sprung part of vehicle or for locking suspension during use to meet varying vehicular or surface conditions, e.g. due to speed or load the regulating means comprising electric or electronic elements
- B60G17/016—Resilient suspensions having means for adjusting the spring or vibration-damper characteristics, for regulating the distance between a supporting surface and a sprung part of vehicle or for locking suspension during use to meet varying vehicular or surface conditions, e.g. due to speed or load the regulating means comprising electric or electronic elements characterised by their responsiveness, when the vehicle is travelling, to specific motion, a specific condition, or driver input
- B60G17/0165—Resilient suspensions having means for adjusting the spring or vibration-damper characteristics, for regulating the distance between a supporting surface and a sprung part of vehicle or for locking suspension during use to meet varying vehicular or surface conditions, e.g. due to speed or load the regulating means comprising electric or electronic elements characterised by their responsiveness, when the vehicle is travelling, to specific motion, a specific condition, or driver input to an external condition, e.g. rough road surface, side wind
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Abstract
The invention discloses a vehicle active suspension system modeling and control method based on high semi-Markov switching, relates to a vehicle active suspension system modeling and control method, and aims to solve the problems that an existing active suspension needs to depend on a sensor to adapt to road conditions autonomously, the use cost is high, the failure rate is high and the like. Establishing a vehicle active suspension dynamic equation; modeling a suspension system into a heterogeneous hidden semi-Markov random switching system, wherein damping and rigidity in a vehicle active suspension dynamic equation can be randomly switched among a plurality of submodes to adapt to different road conditions; carrying out stability analysis on the vehicle active suspension system; state feedback controllers that rely on observation modalities are designed. According to the method, the active suspension system is modeled into a non-homogeneous hiddensemi-Markov random switching system according to actual conditions, so that the use cost of the suspension system is reduced, and the comfort of the suspension system is improved.
Description
Technical Field
The invention relates to a modeling and control method for a vehicle active suspension system, in particular to a modeling and control method for a vehicle active suspension non-ideal switching system with uncertain parameters, and relates to the technical field of active suspension system control.
Background
With the rapid development of industrial technology, automobiles are widely concerned by many research units and students as a high-end industrial product, and people are dedicated to improving the riding comfort and the driving controllability of vehicles. Among these, vehicle suspension systems are also constantly being optimized as technology advances. From the passive suspensions such as the early dependent suspension and the independent suspension, and the active suspension and the air suspension which are more advanced at present, various suspension systems bring comfort to people, and meanwhile, the cost of the active suspension system is higher and the failure rate is higher due to the complex working condition and the precise sensing element. How to design an active suspension system that is comfortable and low in use cost becomes a focus of attention of many researchers.
In the active suspension system, the actuator is positioned between the frame and the vehicle body, so that the suspension system meets the requirements of vibration energy dissipation, vehicle body vertical dynamic stabilization, comfort increase and the like. The rigidity and damping characteristics of the existing active suspension are required to be dynamically adaptive according to the running conditions of a vehicle (the motion state of the vehicle, the road surface condition and the like), so that a suspension system is always in the optimal damping state. However, in the existing active suspension system, the monitoring of the road state mostly depends on a plurality of expensive sensors, and the sensors have high requirements on the use environment and high maintenance cost.
In many studies, researchers have modeled a class of sensor-independent active suspension systems using a Markov stochastic process. It is noted, however, that modeling active suspensions using a markov process has some drawbacks. First, the discrete markov system requires that the residence time be distributed geometrically, however, in practice, the residence time on a certain road condition may be distributed other than geometrically when switched among different road conditions. In addition, since such active suspension systems no longer rely on sensors, the vehicle cannot directly obtain modal information of the vehicle, and the prior studies assume that the information of the vehicle on which road condition is precisely known.
The prior art under reference CN106183692A provides a vehicle active suspension system, which may include a suspension actuator and a controller, and a control method thereof. The controller may be configured to: in response to detecting an object within a predetermined range of the vehicle while the vehicle speed is greater than a threshold, classifying the object into at least one of a plurality of predefined categories. The controller may be further configured to actuate the suspension actuator according to a predefined actuation pattern. This prior document also does not suggest how the active suspension system adapts autonomously to road conditions independent of the sensor.
The existing suspension system for monitoring road conditions based on sensors to adjust the damping and rigidity of the suspension has high manufacturing cost and high failure rate; while another method for modeling a suspension system by adopting a Markov random process has ideal assumptions such as completely known system modes, completely known residence time information, unchanged transition probability and the like, and does not accord with the practical application situation. By combining the reasons, an active suspension system which can be automatically adapted to road conditions without depending on a sensor is urgently needed to be designed, so that the use cost of the active suspension is reduced while the riding comfort is met.
Disclosure of Invention
The technical problem to be solved by the invention is as follows:
the invention provides a vehicle active suspension system modeling and control method based on high sensitivity semi-Markov switching, aiming at the problems that the existing active suspension needs to be self-adapted to the road condition by depending on a sensor, the existing vehicle active suspension system is high in use cost, high in failure rate and the like.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a method of vehicle active suspension system modeling and control based on high semi-Markov switching, the method comprising:
the method comprises the following steps: establishing a vehicle active suspension dynamic equation:
wherein x1And x3Respectively representing the displacement of the tire from the balance point and the displacement of the suspension from the balance point; x is the number of2And x4Respectively representing unsprung mass velocity and sprung mass velocity; u is the actuator output force of the suspension system; m issAnd mtRespectively representing the sprung and unsprung masses; k is a radical oftIs the coefficient of compression of the vehicle tire; c. CsAnd ksRespectively representing the damping and the rigidity of the suspension system;
step two: converting a suspension dynamics equation into a nonhomogeneous high semi-Markov random switching system equation and damping c in a vehicle active suspension dynamics equationsAnd stiffness ksCan be randomly switched among a plurality of submodes to adapt to different road conditions:
let x be [ x ]1x2x3x4]TRepresenting a state vector of a vehicle active suspension dynamic equation (1), setting a sampling period as T, and obtaining the following discrete time vehicle suspension state space expression by a first-order Euler approximation method
x(k+1)＝Aix(k)+Biu(k) (2)
Equation (2) is an inhomogeneous high semi-Markov random switching vehicle suspension system equation, wherein i represents the sub-mode state of the system, i belongs to N, the mode switching of the system is governed by a high semi-Markov chain, and
next, a state feedback controller for observing mode dependence and residence time dependence is designed to eliminate the influence of the system mode that cannot directly obtain the stabilizing effect of the system, and the form is as follows:
wherein Kl(t)For the controller gain required, and l (t) e M, further, combining (2) with (3) will result in a closed loop heterogeneous high semi-Markov vehicle suspension system of the form:
wherein
l (t) is the observation modality at the dwell time t within the current modality i, and
l(t)∈M，πil(t)＝Pr{rk *＝l(t)|rki represents a conditional probability;
time-varying half-horse core in time-non-uniform high semi-Markov vehicle suspension system
Transition probability of inhomogeneous Markov update chains
And a time-varying dwell time probability density function
Upper and lower bounds of (c):
hypothesis Embedded heterogeneous Markov chains
Can be completely known, and simultaneously takes the probability density function of the residence time of the system into consideration
Partially unknown, the fact that N ═ N is definedi a∪Ni uAnd is
Wherein N isi a＝{na(1),na(2),...,na(Na) τ is the dwell time within a certain modality;
definition of
The transition probabilities for the Markov update chains of the nonhomogeneous hidden semi-Markov system are mathematically expressed as:
defining the residence time probability density function in the current mode to depend on the next target mode and the last switching time, which is defined as
Wherein SnRepresenting the time of residence within a modality, RnDenotes that the nth switching occurs in the mode of the system, k denotes the sampling time, knRepresenting the system switching time, i representing the current mode of the system, and j representing the next mode to be processed by the system; the concept of the half horse nucleus in the nonhomogeneous high semi-Markov random switching system is obtained by the two definitions, and the half horse nucleus is an element
Formed matrix of elements therein
The method comprises the steps of obtaining system modal jump information and system residence time information;
step three: and (3) carrying out stability analysis on the vehicle active suspension system: constructing a Lyapunov function which depends on the system mode and the stay time in the mode; the stability criterion of the vehicle active suspension system is pushed down on the premise of ensuring the energy reduction of the suspension system Lyapunov function switching point, so that the stability analysis of the vehicle active suspension system is realized;
step four: designing a state feedback controller dependent on the observation modality: according to the deduced stability criterion of the vehicle suspension system, the existence condition of the state feedback controller depending on the observation mode of the system is deduced through Schur supplementary theory, so that the controller gain capable of stabilizing the suspension system is obtained, and finally modeling and control of the vehicle active suspension system are completed.
Further, in step three, the process of performing stability analysis on the vehicle active suspension system is as follows:
the stability condition of the vehicle active suspension system is given as follows:
in the formula:
and carrying out stability analysis on the vehicle active suspension system according to the conditions.
Further, in step four, the expression of the observation-modality-dependent state feedback controller is designed as follows:
given a finite constant Δi> 0 and maximum residence time in i-mode
If there is a set of intermediate variable matrices Fi(t,υ)，Fi(τ,m)，
And
wherein, the system mode i belongs to N, and the system mode
The residence time t is satisfied
The residence time of the vehicle suspension system in the i mode is satisfied
The staying time in a certain mode of the suspension system satisfies upsilon ∈ [0, t ∈ >]The self-defined variable m satisfies m E [0, tau]And a set of intermediate variable matrices Ul(υ+1)Y, V, such that for any suspension system mode i ∈ N, and the intra-mode dwell time is satisfied
The following inequalities hold:
wherein the content of the first and second substances,
and is
Wherein V and Y are intermediate variable matrices to facilitate mathematical derivation;
represents oneN i ×N iOne pair of dimensionsA diagonal matrix with corner elements of V; i represents an identity matrix;
based on the above conditions, the designed time-varying controller gain is Kl(t)＝Ul(t)Y-1So that the vehicle active suspension system (4) is mean square stable.
Further, considering the vehicle suspension behavior, the following damping c is givensAnd stiffness ksParameters are as follows: c. Cs，11500 n.s/m, cs,22500 n.s/m, cs,33500 n.s/m; k is a radical ofs，1K 15000 n/ms，225000 n/m, ks，335000 n/m; other parameters of the vehicle suspension system are ms320 kg, m t40 kg, kt200 kg/m;
the dynamics based on the discrete suspension system is governed by a heterogeneous high semi-Markov chain, the system is randomly switched among three sub-modes, and the upper bound of the residence time is respectively as follows:
the time-varying transition probability of the embedded heterogeneous Markov chain is considered as
And
and nominal values are respectively
Generating a probability matrix of
The invention has the following beneficial technical effects:
the invention is an active suspension system based on a random switching system, and abandons some ideal assumptions existing in the existing research of the active suspension system, so that the designed suspension system has practicability. The observation mode dependent controller designed by the control algorithm of the random switching system under the non-ideal condition can effectively solve the problems of instability, jitter and the like of the suspension system caused by random switching of damping and rigidity in different sub-modes on the premise that the suspension system does not depend on a sensor to collect information, and improves the comfort and the practicability of the active suspension of the vehicle.
According to the invention, a suspension system is modeled into a heterogeneous high semi-Markov random switching system, and damping and rigidity in a vehicle active suspension dynamic equation can be randomly switched among a plurality of submodes to adapt to different road conditions. The method for modeling the active suspension under the non-ideal condition by using the non-homogeneous high semi-Markov chain can well model general conditions that the system mode cannot be accurately known, the residence time probability density function is partially unknown and the mode transfer probability is time-varying, greatly improves the applicability of the designed active suspension system, and has generality and application value compared with the existing method. The control rule and the control method provided by the invention have certain practical significance for designing other active suspensions and solving the control problems of other suspensions.
According to the invention, the active suspension system is modeled into the inhomogeneous high semi-Markov random switching system according to the actual situation, the traditional ideal assumption is abandoned, the use cost of the suspension system is reduced, and the comfort and the practicability of the suspension system are improved. The invention overcomes some ideal assumptions of the existing active suspension system modeled by a random process, realizes that the active suspension system can adapt to different road surfaces without depending on a road condition sensor under the non-ideal condition, and solves a plurality of practical application problems of the existing active suspension.
Drawings
FIG. 1 is a vehicle suspension system.
FIG. 2 is a state response curve for each state of the vehicle suspension system.
FIG. 3 is a probability density function
The vehicle suspension system state response curve under unknown.
Figure 4 is a state response curve for a vehicle suspension system without regard to a non-homogeneous semi-markov chain.
In FIGS. 2-4, the horizontal axis represents sample time; FIG. 2 is a vertical axis of a suspension state response curve and a system mode, respectively; the vertical axis of fig. 3-4 represents the condition response curve of the suspension system.
Detailed Description
With reference to fig. 1 to 4, the following explanation is made on the modeling and control method of the active suspension system of the vehicle based on the hidden semi-Markov switching, which realizes the random switching under the non-ideal condition to adapt to different road conditions, and includes the following steps:
the method comprises the following steps: establishing a vehicle active suspension kinetic equation
Wherein x1And x3Respectively representing the displacement of the tire from the balance point and the displacement of the suspension from the balance point; x is the number of2And x4Representing unsprung mass velocity and sprung mass velocity, respectively. u is the actuator output force of the suspension system; m issAnd mtRespectively representing the sprung and unsprung masses; k is a radical oftIs the coefficient of compression of the vehicle tire; c. CsAnd ksThe damping and stiffness of the suspension system are indicated, respectively.
In the invention, a type of damping c is designedsAnd stiffness ksThe adjustable active suspension of the vehicle can be adjusted, and the damping and rigidity in the suspension system are considered to be randomly switched among three sub-modes so as to adapt to different road conditions.
Step (ii) ofII, secondly: converting a suspension dynamics equation into a nonhomogeneous high semi-Markov random switching system equation and damping c in a vehicle active suspension dynamics equationsAnd stiffness ksCan be randomly switched among a plurality of submodes to adapt to different road conditions:
let x be [ x ]1x2x3x4]TRepresents a state vector of the suspension system (1). Setting the sampling period to be T ═ 0.1 seconds, and furthermore, by a first order euler approximation method, we can obtain the following discrete-time vehicle suspension state space expression:
x(k+1)＝Aix(k)+Biu(k) (2)
wherein i represents the sub-mode in which the system is located, i ∈ N, the mode switching of which is governed by the hidden semi-Markov chain, and
next, we design a state feedback controller for observing mode dependence and residence time dependence to eliminate the effect that the system mode cannot directly obtain on the system stabilizing effect, and the form is as follows:
u(k)＝Kl(t)x(k) (3)
wherein Kl(t)The controller gain that is required, and l (t) e M. Further, combining (2) with (3), we will get a closed-loop heterogeneous hidden semi-Markov random switching system of the form:
wherein
It is considered that in an actual vehicle active suspension system, the sub-mode (i.e. different road conditions currently passing through) of the system and the residence time probability density function in the sub-mode are often not directly available or obtained with great cost (such as adding expensive sensors), and the switching intervals of different damping and stiffness adjusted for different road conditions are not always able to satisfy the geometric distribution. Furthermore, the random switching between different modalities is more dependent on the sampling time. For this reason, the suspension system cannot be directly obtained by adopting the high semi-Markov chain modeling system mode in the invention.
In the present invention, the definition
The transition probabilities for the Markov update chains of the nonhomogeneous hidden semi-Markov system are mathematically expressed as:
defining the residence time probability density function in the current mode to depend on the next target mode and the last switching time, which is defined as
Wherein SnRepresenting the time of residence within a modality, RnIndicating that the nth switching occurs is the modality in which the system is located. From the above two definitions, we get the concept of half horse nucleus in the heterogeneous hidden semi-Markov random switching system. It is worth noting that the hemiequine nuclei are elements
Formed matrix of elements therein
It is worth noting that
The method comprises the information of the system modal jump and the information of the system residence time.
In addition, in the heterogeneous hidden semi-Markov switching system, the invention considers the general situation that the system mode cannot be directly obtained. It is noted that in the hidden semi-Markov switching system, each system mode generates a series of system modesThe number of observation modalities depends on the length of the dwell time of the current system modality. We consider that there is a probability of generation of πil(t)＝Pr{rk *＝l(t)|rkI, where l (t) is the observed modality at the dwell time t within the current modality i, and
in order to design the vehicle active suspension system more general, we consider the hidden semi-Markov switching system to be time homogeneous, and in order to deal with such a system we respectively give a time-varying half horse core
Transition probability of inhomogeneous Markov update chains
And a time-varying dwell time probability density function
Upper and lower bounds of (c):
without loss of generality, the present invention considers that the system true modality cannot be directly obtained, but thanks to the generation probability, the system true modality can be estimated from the modality information generated by one detector. In addition, the invention simultaneously considers the problem of analyzing and controlling the random switching system under the practical condition that the probability density function part of the system residence time is unknown. We assume embedded heterogeneous Markov chains
Can be fully known, so that the time-varying half Markov kernel
Is unknown if and only if the dwell time probability density function
Is unknown, otherwise
Are fully known. We define N ═ Ni a∪Ni uAnd is
Wherein N isi a＝{na(1),na(2),...,na(Na)}。
Step three: stability analysis for suspension systems
The stability system conditions of the suspension system are first given:
random switching system considering discrete time
Wherein r iskIs a system modal signal, k0,k1,...,kn,.. time of system modality switching and k0＝0；
Representing the observation modality. For a given set of constants
If a set of functions V (x (k) exists,rk,ψ(k)):
and three
Function(s)
So that for x (0), r for any initial condition0The following inequalities hold for N:
furthermore, considering that the residence time is not exactly known in the suspension system, we need the following assumptions.
Suppose that: given a constant Ψ, 0 < Ψ < 1, and an upper bound on the dwell time within the i-mode
Such that:
it is worth noting that in practical suspension systems, to ensure the performance of stochastic systems, the modal switching of the system is not very frequent, which means that the dwell time is not so longIt will be short and limited. The above assumption means that the dwell time probability density function is accumulated from τ -1 to
The sum of psi is not smaller than psi, which also means that psi should be chosen as large as possible to be able to approach the actual situation to the maximum. This assumption is reasonable and necessary since the residence time information is not directly fully available.
Next, we construct a Lyapunov function that relies on characterizing the energy of the suspension system, of the form:
wherein for rke.N, ψ (k) is a function of the residence time t. Based on equation (9), we present a sufficient condition for mean square stabilization for the random switching system (4).
Theorem one for finite constant Δi＞0，
If there is a set of matrices Pi(t)，
And is
So as to
The following inequalities hold:
wherein the content of the first and second substances,
The closed loop system (4) is mean square stable.
And (3) proving that: first, consider the Lyapunov function of the form (9), while letting
in addition to this, the present invention is,
and
is unknown, and for all i e N,
are all provided with
In addition, due to
And is
We can get
Considering the inequality (10) at the same time, we have
Wherein
And due to Pi(1) And Pj(1) Satisfy inequality (10), we have
E{V(x(kn+1),j,ψ(k))}-V(x(kn) I, psi (k)) is less than or equal to 0, which ensures that inequality (7) in lemma one holds.
Furthermore, we can derive from the inequality (11)
The above equation satisfies the inequality (6), and by factoring one and the inequalities (10) and (11), we conclude that the system (4) is mean square stable. After the syndrome is confirmed.
Due to presence of non-convex forms, i.e. matrices
And
the existence of the multiplication-by-multiplication term of (c) can not be directly used for solving the controller. Therefore, the next theorem will be directed to solving this non-convex problem.
And 2, theorem II: consider a non-homogeneous vehicle suspension stochastic switching system (4). Given a finite constant Δi＞0，
If there is a set of positive definite matrices Hi(t,υ)，
And
υ∈[0,t-1]，m∈[1,τ-1]so as to be directed to
The following inequalities hold:
wherein the content of the first and second substances,
and
given in theorem one, the closed-loop system (4) is mean-square stable.
And (3) proving that: first, define
We can get from inequality (13):
thus, can obtain
And due to
In turn, we have
Then, it can be obtained from the formula (14)
Combining the inequalities (12), (17), and (18), we obtain
By adding Pi(1) And
substituted as in the above formulaAnd
we can obtain formula (10).
Further, from formula (15), it can be obtained:
this makes it possible to ensure that inequalities (16) and (19) jointly satisfy expression (11). Thus, the theorem two-fold system (4) is mean square stable. After the syndrome is confirmed.
Step four: designing a state feedback controller dependent on an observation modality
And 4, theorem III: given a finite constant Δi> 0 and
if there is a set of matrices Fi(t,υ)，Fi(τ,m)，
And
wherein, i belongs to the N,
υ∈[0,t]，m∈[0,τ]and a set of matrices Ul(υ+1)Y, V, such that
The following inequalities hold:
wherein the content of the first and second substances,
and is
The gain of the time-varying controller is designed to be Kl(t)＝Ul(t)Y-1The vehicle active suspension system (4) is mean square stable.
And (3) proving that:
first, V is applied-1Performing an equality transformation on the inequality (12), if order
The condition (20) can be guaranteed.
According to formula (13), we have
By applying Schur supplement theory, the formula can be obtained as follows:
due to the fact that
Can guarantee
Using diagonal matrices
Carrying out congruent transformation on the inequalities, we have:
wherein the content of the first and second substances,
using diagonal matrices
Performing congruent transformation on inequality (6-24), if order
The condition (21) is established.
In addition, the theory is complemented by Schur and
and
as can be seen from the definitions of (14), (15) and (16), the expressions (22), (23) and (24) can be secured, respectively. Further, the mean square stability of the closed loop system (4) can be ensured by the designed controller which depends on the observation mode and the residence time.
Example (b): when the invention is implemented, the practical parameters of the suspension system are as follows:
considering the vehicle suspension reality, we give the following damping csAnd stiffness ksParameters are as follows: c. Cs，11500 n.s/m, cs,22500 n.s/m, cs,33500 n.s/m; k is a radical ofs，1K 15000 n/ms，225000 n/m, ks，335000 n/m. Other parameters of the vehicle suspension system are considered as ms320 kg, mt40 kg, kt200 kg/m.
Furthermore, we consider that the dynamics of a discrete suspension system are governed by a heterogeneous hidden semi-Markov chain, the system switches randomly between three sub-modes, and its dwell time upper bound is:
the time-varying transition probability of the embedded heterogeneous Markov chain is considered as
And
and nominal values are respectively
Generating a probability matrix of
The effects of the present invention were verified as follows:
from fig. 2 we see that in the case of simultaneous switching of damping and stiffness in the vehicle suspension, the control we have designed enables all states of the system to converge eventually to the equilibrium point, even if the suspension system modes are not fully accurately available and the dwell time of the system within a certain mode is unknown; fig. 3 further illustrates that the control strategy proposed by the present invention can effectively stabilize a type of active suspension system of a vehicle based on the hidden semi-Markov switching even under the premise that the conditional probability is not ideal. In fig. 4, it is considered that the unknown residence time information in a certain mode directly affects the stability of the suspension system without adopting the control strategy designed by the present invention, and by comparing with fig. 3, the necessity of considering the unknown residence time information in the vehicle suspension system is further proved.
Claims (4)
1. A method for modeling and controlling a vehicle active suspension system based on high semi-Markov switching, the method comprising:
the method comprises the following steps: establishing a vehicle active suspension dynamic equation:
wherein x1And x3Respectively representing the displacement of the tire from the balance point and the displacement of the suspension from the balance point; x is the number of2And x4Respectively representing unsprung mass velocity and sprung mass velocity; u is the actuator output force of the suspension system; m issAnd mtRespectively representing the sprung and unsprung masses; k is a radical oftIs the coefficient of compression of the vehicle tire; c. CsAnd ksRespectively representing the damping and the rigidity of the suspension system;
step two: transfer the suspension dynamics equation toDamping c in equation of active suspension dynamics of vehicle, namely nonhomogeneous high semi-Markov random switching system equationsAnd stiffness ksCan be randomly switched among a plurality of submodes to adapt to different road conditions:
let x be [ x ]1x2x3x4]TRepresenting a state vector of a vehicle active suspension dynamic equation (1), setting a sampling period as T, and obtaining the following discrete time vehicle suspension state space expression by a first-order Euler approximation method
x(k+1)＝Aix(k)+Biu(k) (2)
Equation (2) is an inhomogeneous high semi-Markov random switching vehicle suspension system equation, wherein i represents the sub-mode state of the system, i belongs to N, the mode switching of the system is governed by a high semi-Markov chain, and
next, a state feedback controller for observing mode dependence and residence time dependence is designed to eliminate the influence of the system mode that cannot directly obtain the stabilizing effect of the system, and the form is as follows:
wherein Kl(t)For the controller gain required, and l (t) e M, further, combining (2) with (3) will result in a closed loop heterogeneous high semi-Markov vehicle suspension system of the form:
wherein
l (t) is the observed modality at the dwell time t within the current modality i,and is
l(t)∈M，πil(t)＝Pr{rk *＝l(t)|rkI represents a conditional probability;
time-varying half-horse core in time-non-uniform high semi-Markov vehicle suspension system
Transition probability of inhomogeneous Markov update chains
And a time-varying dwell time probability density function
Upper and lower bounds of (c):
hypothesis Embedded heterogeneous Markov chains
Can be completely known, and simultaneously takes the probability density function of the residence time of the system into consideration
Partially unknown, the fact that N ═ N is definedi a∪Ni uAnd is
Wherein N isi a＝{na(1),na(2),...,na(Na) τ is the dwell time within a certain modality;
definition of
The transition probabilities for the Markov update chains of the nonhomogeneous hidden semi-Markov system are mathematically expressed as:
defining the residence time probability density function in the current mode to depend on the next target mode and the last switching time, which is defined as
Wherein SnRepresenting the time of residence within a modality, RnDenotes that the nth switching occurs in the mode of the system, k denotes the sampling time, knRepresenting the system switching time, i representing the current mode of the system, and j representing the next mode to be processed by the system; the concept of the half horse nucleus in the nonhomogeneous high semi-Markov random switching system is obtained by the two definitions, and the half horse nucleus is an element
Formed matrix of elements therein
The method comprises the steps of obtaining system modal jump information and system residence time information;
step three: and (3) carrying out stability analysis on the vehicle active suspension system: constructing a Lyapunov function which depends on the system mode and the stay time in the mode; the stability criterion of the vehicle active suspension system is pushed down on the premise of ensuring the energy reduction of the suspension system Lyapunov function switching point, so that the stability analysis of the vehicle active suspension system is realized;
step four: designing a state feedback controller dependent on the observation modality: according to the deduced stability criterion of the vehicle suspension system, the existence condition of the state feedback controller depending on the observation mode of the system is deduced through Schur supplementary theory, so that the controller gain capable of stabilizing the suspension system is obtained, and finally modeling and control of the vehicle active suspension system are completed.
2. The method for modeling and controlling the active suspension system of the vehicle based on the high semi-Markov switching as claimed in claim 1, wherein in the third step, the process of analyzing the stability of the active suspension system of the vehicle is as follows:
the stability condition of the vehicle active suspension system is given as follows:
in the formula:
and carrying out stability analysis on the vehicle active suspension system according to the conditions.
3. The method for modeling and controlling an active suspension system of a vehicle based on high semi-Markov switching as claimed in claim 2, wherein in step four, the expression of the state feedback controller dependent on the observation mode is designed as follows:
given a finite constant Δi> 0 and maximum residence time in i-mode
i ∈ N, if there is a set of intermediate variable matrices Fi(t,υ)，Fi(τ,m)，
And
wherein, the system mode i belongs to N, and the system mode
The residence time t is satisfied
The residence time of the vehicle suspension system in the i mode is satisfied
The staying time in a certain mode of the suspension system satisfies upsilon ∈ [0, t ∈ >]The self-defined variable m satisfies m E [0, tau]And a set of intermediate variable matrices Ul(υ+1)Y, V, such that for any suspension system mode i ∈ N, and the intra-mode dwell time is satisfied
The following inequalities hold:
wherein the content of the first and second substances,
and is
Wherein V and Y are intermediate variable matrices to facilitate mathematical derivation;
represents an Ni×NiOne diagonal element of the dimension is a diagonal matrix of V; i represents an identity matrix;
based on the above conditions, the designed time-varying controller gain is Kl(t)＝Ul(t)Y-1So that the vehicle active suspension system (4) is mean square stable.
4. A method for modeling and controlling an active suspension system of a vehicle based on high semi-Markov switching as claimed in claim 1, 2 or 3, wherein the damping c is given as follows taking into account the actual conditions of the vehicle suspensionsAnd stiffness ksParameters are as follows: c. Cs，11500 n.s/m, cs,22500 n.s/m, cs,33500 n.s/m; k is a radical ofs，1K 15000 n/ms，225000 n/m, ks，335000 n/m; other parameters of the vehicle suspension system are ms320 kg, mt40 kg, kt200 kg/m;
the dynamics based on the discrete suspension system is governed by a heterogeneous high semi-Markov chain, the system is randomly switched among three sub-modes, and the upper bound of the residence time is respectively as follows:
the time-varying transition probability of the embedded heterogeneous Markov chain is considered as
And
and nominal values are respectively
Generating a probability matrix of
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