CN111487870B - Design method of adaptive inversion controller in flexible active suspension system - Google Patents
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Abstract
The invention discloses a design method of a self-adaptive inversion controller in a flexible active suspension system, which comprises the following steps: building an active suspension model; step two: designing a self-adaptive inversion controller based on a neural network; aiming at a two-degree-of-freedom flexible active suspension system containing unknown nonlinear dynamics, in the control design of an automobile active suspension system, a BL function is selected to construct a control condition meeting the system stability, and meanwhile, a second-order auxiliary system is introduced to realize the compensation of asymmetric control saturation; the adaptive inversion controller based on the neural network is designed and obtained through the method, and hardware loop experiments show that the method can better attenuate the vertical vibration of the sprung mass in the automobile active suspension system and simultaneously improve the grounding performance of the automobile active suspension system under sine and impact excitation compared with the traditional PD and LQR control in the control design of the automobile active suspension system.
Description
Technical Field
The invention belongs to the field of automatic control of automobiles, and particularly relates to a design method of an adaptive inversion controller in a flexible active suspension system.
Background
The automobile suspension system is a key part of an automobile, the shimmy and the comfort of the automobile can be improved in the using process, the suspension system is used for attenuating the vibration transmitted to an automobile body by road interference, and also must ensure the dynamic stroke, the grounding property and other time domain constraint conditions of the suspension, the research of an advanced active control method is crucial to realize the vibration control of the suspension system, and in the research process of the automobile suspension system, a certain research result is obtained at present: for example, network control of an active suspension of an automobile is developed, and communication burden is reduced through an event trigger mechanism; a self-organized fuzzy controller is provided to control the active suspension system, and the stability and the comfort of the automobile suspension system are improved to a certain extent;
when the automobile runs, the mass of the automobile body generally changes along with the weight and the load of a driver, and at the moment, the perturbation of the automobile needs to be considered; sunwei et al propose a nonlinear suspension adaptive inversion control strategy, and estimate the change of the vehicle body mass by an adaptive method; on the basis, the saturation of the control force is compensated by adding the state constraint compensation module, a self-adaptive road control method of the active suspension is researched, and a relatively ideal effect is achieved;
however, because the actual suspension system contains unknown nonlinear dynamics, ideal feedback linearization is difficult to implement, and an expanded state observer is adopted in the prior art to estimate the unknown dynamics of the system, so that the unknown dynamics is required to be continuously bounded by Lipschitz; although a control method of self-adaptive fuzzy and neural network is provided for the uncertainty of a suspension system, most researches of the method are mainly numerical simulation and only stay on the level of analog simulation, and related hardware loop experiments are rare;
therefore, in the design process of an automobile active suspension system, a control method which can not only consider the unknown nonlinear dynamics and the parameter perturbation and asymmetric control saturation effect of the system, but also can consider the hard nonlinear characteristics such as friction force, clearance and the like contained in the active suspension system and can meet the condition of continuous and bounded Lipschitz is urgently needed, so that the automobile active suspension system which can compensate the asymmetric saturation phenomenon of the control force, can better attenuate the vertical vibration of the sprung mass and can improve the grounding performance and has good stability is designed.
Disclosure of Invention
Aiming at a two-degree-of-freedom flexible active suspension system containing unknown nonlinear dynamics, in the control design of an automobile active suspension system, a Barrier-Lyapunov (BL) function is selected to construct a control condition meeting the system stability, a second-order auxiliary system is introduced to realize the compensation of asymmetric control saturation, and the self-adaptation law based on error convergence is utilized to estimate the weight of a neural network and the quality of an automobile body; hardware loop experiments show that in the control design of the automobile active suspension system, compared with the traditional PD and LQR control, the method can better attenuate the vertical vibration of the sprung mass in the automobile active suspension system under sine and impact excitation, and simultaneously improve the grounding performance.
In order to achieve the purpose, the technical scheme of the invention is realized as follows:
a design method of an adaptive inversion controller in a flexible active suspension system comprises the following steps:
the method comprises the following steps: building flexible active suspension model
In a two-degree-of-freedom flexible active suspension system containing unknown nonlinear dynamics, a flexible active suspension is built, and the dynamic relation of the model is as follows:
step two: designing adaptive inversion controller based on neural network
According to the flexible active suspension model established in the first step, a BL function is introduced to establish a control condition meeting the system stability, a second-order auxiliary system is introduced to compensate asymmetric control saturation in the flexible active suspension system, and a neural network-based adaptive inversion controller is designed through a saturation adaptive inversion control method to improve the control performance of the active suspension.
Preferably, the building process of the flexible active suspension model in the step one is as follows:
s1, arranging in a flexible active suspension model: m is s Is a sprung mass; m is u Is an unsprung mass; z is a radical of s Is the vertical displacement at the center of mass of the sprung mass, z u Is the vertical displacement at the unsprung mass centroid; z is a radical of r Exciting the unevenness of the road surface; f s Force generated for a non-linear spring of a suspension, F d A force generated for the damper; f t Elastic force generated for tire rigidity, F b A damping force generated for damping; u is an active control force generated by a direct current servo motor in the active suspension;
s2, the dynamic relation of the flexible active suspension model is as follows:
wherein: f Δ Is an unknown disturbance force caused by friction and control errors, and F Δ Bounded but not continuously bounded with respect to Lipschitz, in the flexible active suspension model, the spring force F s And a damping force F d Is not measurable and belongs to unknown nonlinear dynamics, m s Is a parameter perturbation term;
s3, setting the state variable of the system as x 1 ＝z s ，
x 3 ＝z u ，
Then the state space equation of the controlled system equation (1) is:
preferably, the step of designing the adaptive inversion controller based on the neural network in the second step includes:
s1, firstly, considering the saturation phenomenon of an active control force u, introducing a second-order auxiliary system to compensate a control error caused by saturation, and defining the saturation condition of the control force u;
s2, defining a tracking error e 1 ＝x 1 -x r -λ 1 ，e 2 ＝x 2 Alpha, and selecting a BL function to construct a control condition meeting the stability of the flexible active suspension system;
s3, designing a virtual control law f by using a self-adaptive neural network algorithm to enable a variable e 1 、e 2 、
Asymptotically converges to the origin or near the origin,
is an estimator;
s4, designing an adaptive law based on error convergence to estimate the NN weight and the sprung mass;
and S5, analyzing the stability of the flexible active suspension system by utilizing a Lyapunov function and carrying out classification discussion on an error convergence set.
Preferably, the specific design process of step two S1 is:
(1) according to the type of a flexible active suspension system, considering an asymmetric input saturated self-adaptive NN inversion control law of an active suspension model, and introducing a second-order auxiliary system to compensate a control error caused by saturation, wherein the second-order auxiliary system is as follows:
wherein: u-f, λ 1 And λ 2 Representing state variables of the auxiliary system, and zero initial state λ 1 (0)、λ 2 (0) Is zero; c. C 1 And c 2 Is a normal number of the blood vessel which is,
is the inverse number theta of the sprung mass equal to 1/m s An estimate of (a);
(2) according to an introduced second-order auxiliary system, considering the saturation phenomenon of the control force in the active suspension model, and defining the saturation condition satisfied by the control force u as follows:
wherein: f is the virtual control law to be designed.
Preferably, the specific design process of step two S2 is as follows:
(1) defining a tracking error: e.g. of the type 1 ＝x 1 -x r -λ 1 ，e 2 ＝x 2 -α；
Wherein: x is a radical of a fluorine atom 1 Is a closed loopState variable, x, of system equation (2) r Is the system state x 1 Reference track of (2), x r Is normally zero and alpha is the system state x 2 The virtual trajectory of (2);
(2) let initial tracking error | e 1 (0)|＜δ 1 Wherein δ 1 Is an arbitrarily small positive number;
(3) selecting a BL function, said BL function being expressed as follows:
wherein: v 1 (e 1 ) Representing the BL function, δ 1 Is an arbitrarily small positive number, e 1 Represents a tracking error;
(4) by deriving equation (7), we can obtain:
wherein
k 1 ＞0，e 2 Which is indicative of a tracking error,
equation (8) can be transformed into:
(5) as can be seen from equation (9): if the error e 2 → 0, then
Therefore e 1 Asymptotically converge to zero and satisfy V 1 (e 1 )≤V 1 (e 1 (0) ); get x r Zero, | x can be obtained from step (2) 1 (0)|＜δ 1 When t → ∞, there is | x 1 (t)|＜||λ 1 || ∞ ；
(6) To study | | | λ 1 || ∞ Taking the parameter c according to the change rule of delta u 1 ＝50、c 1 ＝60、
Performing numerical simulation on the formula (6) to obtain a simulation curve, and as can be seen from the simulation curve, when the control error Δ u is 10N, | λ 1 The | | ∞ is still less than 3 mm; it is thus possible to adjust the control parameter c 1 And c 2 To adjust | λ 1 || ∞ Thereby reducing | x 1 (t)|。
Preferably, the specific design process of step two S3 is as follows:
(1) for tracking error e 2 The derivation is carried out to obtain:
wherein phi is-F d -F s -F Δ +u,
Can be seen as-F d -F s -F Δ For unknown nonlinear dynamics, θ is 1/m s Is a perturbation parameter;
(2) designing a self-adaptive neural network algorithm to approach the unknown nonlinear dynamics phi, wherein the neural network algorithm is as follows:
T NN ＝W 1 T φ 1 +ε 1 ＝θ(-F d -F s -F Δ ) (11)
wherein: t is NN An approximation quantity, W, representing unknown nonlinear dynamics 1 Represents the ideal NN weight, phi 1 Representing the vector of the activation function, ε 1 Representing a bounded approximation error, i.e. | ε 1 |＜ε 1N ，ε 1N ＞0；
(3) By substituting formula (11) into formula (10), it is possible to obtain:
wherein Θ is 1 ＝[W 1 T θ] T ，
(4) Definition of
Is composed of
Equation (12) may be rewritten as:
wherein
(5) In order to ensure asymptotic stability of closed-loop system formula (2) and to make the error variable e 1 、e 2 、
Asymptotically converging to the origin or the vicinity of the origin, and designing a virtual control law as follows:
wherein: k is a radical of 2 To control the gain, and k 2 ＞0。
Preferably, the specific design process of step two S4 is as follows:
(1) defining virtual filter variables e 2f 、
The filter variable e 2f 、
The relationship of (A) is as follows:
wherein: k > 0 is a design parameter;
(2) defining a virtual filter matrix P 1 Sum vector Q 1 Satisfies the following relation:
wherein: l > 0 is a design parameter;
(3) the following adaptive law based on error convergence is designed:
wherein: gamma-shaped 1 Is a diagonal matrix, σ > 0 is a learning gain value, and
(4) according to the formula (16), H can be represented by 1 Further decomposing into:
wherein:
ε 1f is epsilon 1 Filter factor of, i.e.
(5) It can be seen from equation (18) that for all bounded active control forces u and state vectors x, there is | | | | Δ 1 ||＜ε 1Nf ，ε 1Nf Is a bounded constant; current regression vector
Upon continued excitation, the matrix P 1 More than 0 is a positive definite matrix, and the minimum eigenvalue satisfies the condition
I.e., complete the design of the active suspension control strategy.
Preferably, the specific design process of step two S5 is as follows:
(1) to analyze the stability and error convergence set of the flexible active suspension system, the following Lyapunov function was chosen:
wherein: v 1 The function of the BL is represented by,
and Γ 1 Is a diagonal matrix;
by substituting equation (14) into equation (13), the following can be obtained:
by combining equation (20) and equation (18) after derivation of equation (19), the following can be obtained:
wherein: k is a radical of 1 ，k 2 To control the gain, and k 1 ＞0，k 2 ＞0；
(2) General formula(21) The discussion is divided into two cases, the first case: if the neural network algorithm can ideally approximate the nonlinear dynamics of a flexible active suspension system, the approximation error ε 1 0, and Δ 1 When 0, then equation (21) satisfies:
according to Lyapunov stabilization theory, V (t) ≦ V (0), and the error variable e when time t → ∞ 1 、e 2 、
Will converge to the origin and satisfy | e over the entire time domain 1 (t)|＜δ 1 ；
(3) In the second case: when approximation error epsilon 1 When not equal to 0, the young can be obtained by applying the young inequality:
wherein eta and eta 1 Are all positive tuning parameters;
substituting equation (23) into equation (21) can yield:
wherein:
always exists eta 1 Guarantee
To represent
The maximum singular value of;
the Lyapunov function thus satisfies:
according to the Lyapunov stabilization theory, the error variable e can be known 1 、e 2 、
Is eventually consistently bounded and will converge to a set:
wherein:
to represent
The minimum singular value of;
as can be seen from the formula (26), | e 1 (t)|＜δ 1 The same can be satisfied in the equation (26), and the radius of the set depends on the approximation error ε 1 Of (i.e. when ε 1 On → 0, the same conclusion can be reached as in case 1, which is aggregated as follows:
preferably, the method for designing the adaptive inversion controller in the flexible active suspension system further comprises the third step of: testing the performance of the adaptive inversion controller by a hardware loop test experiment, wherein the specific process of the experiment comprises the following steps:
(1) sine wave excitation experiment
Under the excitation of sine action, the self-adaptive inversion control method can well control vibration, and meanwhile under the road excitation of resonance frequency, the comfort and the grounding stability of the self-adaptive inversion control method are superior to those of the traditional PD, LQR and passive control;
(2) square wave impact excitation experiment
Under impact excitation, compared with the traditional PD, LQR and passive control, the adaptive inversion control method has smaller overshoot displacement and acceleration, and has control force output of saturation effect, and the control effect of the adaptive inversion control method is obviously better than that of the traditional PD and LQR control.
The invention has the beneficial effects that: the invention discloses a design method of a self-adaptive inversion controller in a flexible active suspension system, and compared with the prior art, the improvement of the invention is as follows:
(1) aiming at a two-degree-of-freedom flexible active suspension system containing unknown nonlinear dynamics, a Barrier-Lyapunov (BL) function is introduced to construct a control condition meeting the system stability, so that a controller design method with low conservative property is obtained; a second-order auxiliary system is introduced into the design method to realize the compensation of asymmetric control saturation, and a self-adaptive inversion controller based on a neural network is designed;
(2) meanwhile, through a hardware loop experiment, the adaptive inversion controller designed by the adaptive inversion control method is compared with the traditional classical PD control and LQR control, and a sine excitation experiment and a square wave impact excitation experiment prove that the active suspension of the adaptive inversion controller designed by the method can better attenuate the vertical vibration of the sprung mass and improve the grounding performance under sine and impact excitation, so that the effectiveness of the adaptive inversion controller designed by the adaptive inversion control method is further verified, and the superiority of the adaptive inversion control method is proved.
Drawings
Fig. 1 is a diagram of an active suspension model of the flexible active suspension system according to the present invention.
FIG. 2 shows the invention 1 || ∞ And (4) a change rule graph along with the delta u.
Fig. 3 is a diagram showing the vertical displacement of the sprung mass in a sine wave excitation experiment in embodiment 1 of the present invention.
Fig. 4 is a diagram of the sprung mass vertical acceleration of the sine wave excitation experiment in embodiment 1 of the present invention.
Fig. 5 is a suspension dynamic stroke diagram of a sine wave excitation experiment in embodiment 1 of the present invention.
Fig. 6 is a tire dynamic stroke chart of a sine wave excitation experiment in example 1 of the present invention.
Fig. 7 is a motor control force diagram of a sine wave excitation experiment in embodiment 1 of the present invention.
Fig. 8 is a diagram of the vertical displacement of the sprung mass for the square wave shock excitation experiment in embodiment 1 of the present invention.
Fig. 9 is a diagram of the sprung mass vertical acceleration of the square wave shock excitation experiment of embodiment 1 of the present invention.
Fig. 10 is a suspension dynamic stroke diagram of a square wave impact excitation experiment in embodiment 1 of the invention.
FIG. 11 is a graph of the tire stroke during square wave shock excitation experiments in example 1 of the present invention.
Fig. 12 is a motor control force diagram of the square wave impact excitation experiment in embodiment 1 of the present invention.
Wherein: pass stands for passive control and AB control stands for adaptive inversion control of the invention.
Detailed Description
In order to make those skilled in the art better understand the technical solution of the present invention, the following further describes the technical solution of the present invention with reference to the drawings and the embodiments;
referring to fig. 1 to 12, a method for designing an adaptive inversion controller in a flexible active suspension system includes:
the method comprises the following steps: building flexible active suspension model
In a two-degree-of-freedom flexible active suspension system containing unknown nonlinear dynamics, a flexible active suspension is built, and the dynamic relation of the model is as follows:
the construction process of the active suspension model comprises the steps of construction of the active suspension model and performance evaluation of the active suspension model:
the concrete process of building the active suspension model comprises the following steps:
s1, researching a two-degree-of-freedom active suspension model shown in the figure 1, wherein the two-degree-of-freedom active suspension model is arranged in the active suspension model: m is s Is a sprung mass; m is u Is an unsprung mass; z is a radical of s Is the vertical displacement at the center of mass of the sprung mass, z u Is the vertical displacement at the unsprung mass centroid; z is a radical of r Exciting the unevenness of the road surface; f s Force generated for a non-linear spring of a suspension, F d A force generated for the damper; f t Elastic force generated for tire rigidity, F b A damping force generated for damping; u is the active control force generated by the DC servo motor in the active suspension;
s2, the dynamic relation of the flexible active suspension model is as follows:
wherein F Δ Is an unknown disturbance force caused by friction and control errors, and F Δ Bounded but not continuously bounded with respect to Lipschitz, in the flexible active suspension model, the spring force F s And a damping force F d Is not measurable, belongs to unknown nonlinear dynamics, and has a parameter perturbation term m because the mass of the vehicle body changes along with the weight of a driver s ，m s Is a parameter perturbation term;
s3, setting the state variable of the system as x 1 ＝z s ，
x 3 ＝z u ，
Then the state space equation of the controlled system equation (1) is:
(II) evaluation of the Performance of the active suspension model (generally, when implementing the control strategy of the active suspension, the following four problems need to be considered)
1. Vibration attenuation: since the vertical motion of the vehicle body is closely related to the comfort of the suspension, the main objective of the controller is to suppress the vibration transmitted to the vehicle body by the road surface disturbance and make it approach the vicinity of the zero point;
2. suspension dynamic stroke: due to the limitation of the chassis structure, the suspension dynamic stroke must move within a specified range, namely, the time domain constraint condition | z is required to be met s -z u |≤z max Wherein z is max A safety threshold allowed for the structure;
3. grounding property: in order to ensure the safety and maneuverability of the vehicle, the dynamic load of the tire is required to be less than the static load thereof, i.e. the time domain constraint condition | F is required to be satisfied t +F b |＜(m s +m u ) g; it is noted that the dynamic load of the tire is proportional to the dynamic travel z of the tire u -z r || ∞ So that it is desirable that the smaller the amplitude is, the better;
4. asymmetric input saturation: limited by the output of the motor, in order to guarantee the stability of the closed loop system, the control force saturation must be considered, taking the following constraints: u. of min ≤u≤u max Wherein u is min And u max Respectively the upper and lower limit values of the control force;
in performance evaluation, the root mean square value and the maximum value of vibration can reflect the quality of vibration attenuation performance, so the mean square value and the maximum value of design output need to be considered, and the design formula is as follows:
|χ(t)| max ＝max{χ(t)|,t∈[0,T]} (4)
wherein χ (T) represents the vibration output and T represents the system response time;
and are marked in the present invention as follows:
||x|| ∞ ＝max(x i )，i＝1…n，
λ min/max (A) the minimum/large eigenvalue of matrix a is represented,
a > 0 indicates that it is a positive definite matrix.
Step two: designing adaptive inversion controller based on neural network
According to the flexible active suspension model established in the step one, a Barrier-Lyapunov (BL) function is introduced to construct a control condition meeting the system stability, meanwhile, a second-order auxiliary system is introduced to realize the compensation of asymmetric control saturation in the flexible active suspension system, and a neural network-based adaptive inversion controller is designed through a saturation adaptive inversion control method to improve the control performance of the active suspension;
the design steps of the adaptive inversion controller based on the neural network comprise (the used method is an adaptive inversion control method):
s1, firstly considering the saturation phenomenon of an active control force u, introducing a second-order auxiliary system to compensate a control error caused by saturation, and defining the saturation condition of the control force u, wherein the specific design process comprises the following steps:
(1) according to the type of a flexible active suspension system, considering an asymmetric input saturated self-adaptive NN inversion control law of an active suspension model, and introducing a second-order auxiliary system to compensate a control error caused by saturation, wherein the second-order auxiliary system is as follows:
wherein u-f, λ 1 And λ 2 Represents a state variable of the auxiliary system, and the zero initial state is zero; c. C 1 And c 2 Is a normal number, and is,
is the inverse number theta of the sprung mass equal to 1/m s An estimate of (a);
(2) according to the introduced second-order auxiliary system, the saturation phenomenon of the control force in the active suspension model is considered, and the saturation condition met by the control force u is defined as follows:
where f is the virtual control law.
S2, defining a tracking error e 1 ＝x 1 -x r -λ 1 ，e 2 ＝x 2 Alpha, and selecting a BL function to construct a control condition meeting the stability of the flexible active suspension system, wherein the specific process of designing is as follows:
(1) defining a tracking error: e.g. of the type 1 ＝x 1 -x r -λ 1 ，e 2 ＝x 2 -a, wherein x 1 Is a state variable of a controlled system formula (2), x r Is the system state x 1 Reference track of (2), x r Is normally zero and alpha is the system state x 2 The virtual trajectory of (2);
(2) let initial tracking error | e 1 (0)|＜δ 1 Wherein δ 1 Is an arbitrarily small positive number (since in suspension systems, x 1 Is set to zero, and x is set r Is zero, state quantity lambda 1 Is also zero, so 1 Any small positive number can meet the requirement);
(3) selecting a BL function, said BL function being expressed as follows:
wherein: v 1 (e 1 ) Representing the BL function, δ 1 Is an arbitrarily small positive number, e 1 Represents a tracking error;
(4) by deriving equation (7), we can obtain:
wherein
k 1 ＞0，e 2 Which is indicative of a tracking error,
equation (8) can be transformed into:
(5) as can be seen from equation (9): if the error e 2 → 0, then
Therefore e 1 Asymptotically converge to zero and satisfy V 1 (e 1 )≤V 1 (e 1 (0) ); get x r Zero, | x is obtained from step (2) 1 (0)|＜δ 1 When t → ∞, there is | x 1 (t)|＜||λ 1 || ∞ ；
(6) To study | | | λ 1 || ∞ Taking parameter c according to the change rule of delta u 1 ＝50、c 1 ＝60、
The numerical simulation of the formula (6) is performed to obtain a simulation curve (as shown in fig. 2), and it can be seen from the simulation curve that | λ |, when the control error Δ u is 10N 1 || ∞ Still less than 3 mm; it is thus possible to adjust the control parameter c 1 And c 2 To adjust | λ 1 || ∞ Thus decreasing | x 1 (t)|。
S3, designing a virtual control law f by utilizing a self-adaptive neural network algorithm to enable a variable e 1 、e 2 、
Asymptotically converges to the origin or its vicinity,
the specific process for designing the estimator is as follows:
(1) for tracking error e 2 The derivation is carried out to obtain:
wherein phi is-F d -F s -F Δ +u,
Can see-F d -F s -F Δ For unknown nonlinear dynamics, θ is 1/m s Is a perturbation parameter;
(2) designing a self-adaptive neural network algorithm to approach the unknown nonlinear dynamics phi, wherein the neural network algorithm is as follows:
T NN ＝W 1 T φ 1 +ε 1 ＝θ(-F d -F s -F Δ ) (11)
wherein: t is NN An approximation quantity, W, representing unknown nonlinear dynamics 1 Represents the ideal NN weight, phi 1 Representing the vector of the activation function, ε 1 Representing a bounded approximation error, i.e. | ε 1 |＜ε 1N ，ε 1N ＞0；
(3) By substituting formula (11) into formula (10), it is possible to obtain:
wherein Θ is 1 ＝[W 1 T θ] T ，
(4) Definition of
Is theta 1 ＝[W 1 T θ] T Equation (12) may be rewritten as:
wherein
(5) In order to ensure asymptotic stability of the closed-loop system (2) and to make the error variable e 1 、e 2 、
Asymptotically converging to the origin or the vicinity of the origin, and designing a virtual control law as follows:
wherein k is 2 To control the gain, and k 2 0 (see S5 below for a detailed demonstration procedure).
S4, designing an adaptive law based on error convergence to estimate the NN weight and the sprung mass, wherein the specific design process comprises the following steps:
(1) defining virtual filter variables e 2f 、
The filter variable e 2f 、
The relationship of (A) is as follows:
wherein k > 0 is a design parameter;
(2) defining a virtual filter matrix P 1 Sum vector Q 1 Satisfies the following relation:
wherein l > 0 is a design parameter;
(3) the following adaptive law based on error convergence is designed:
wherein: gamma-shaped 1 Is a diagonal matrix, σ > 0 is a learning gain value, and
(4) according to the formula (16), H can be represented by 1 Further decomposing into:
wherein
ε 1f Is epsilon 1 Filter factor of, i.e.
(5) It can be seen from equation (18) that for all bounded active control forces u and state vectors x, there is | | | | Δ 1 ||＜ε 1Nf ，ε 1Nf Is a bounded constant; current regression vector
Upon continued excitation, the matrix P 1 More than 0 is a positive definite matrix, and the minimum eigenvalue satisfies the condition
Namely, the design of the active suspension control strategy is completed, and the stability of the active suspension is further discussed below;
s5, analyzing the stability of the flexible active suspension system by utilizing a Lyapunov function and carrying out classification discussion on an error convergence set, wherein the specific design process comprises the following steps:
(1) to analyze the stability and error convergence set of the flexible active suspension system, the following Lyapunov function was chosen:
wherein: v 1 The function of the BL is represented by,
and Γ 1 Is a diagonal matrix;
by substituting equation (14) into equation (13), the following can be obtained:
after deriving equation (19), equation (20) and equation (18) are combined to obtain:
wherein: k is a radical of formula 1 ，k 2 To control the gain, and k 1 ＞0，k 2 ＞0；
(2) Equation (21) is discussed in two cases, the first case: if the neural network algorithm can ideally approximate the nonlinear dynamics of a flexible active suspension system, the approximation error ε 1 0, and Δ 1 When 0, then equation (21) satisfies:
according to Lyapunov stabilization theory, V (t) ≦ V (0), and the error variable e when time t → ∞ 1 、e 2 、
Will converge to the origin and satisfy | e over the entire time domain 1 (t)|＜δ 1 ；
(3) In the second case: when approximation error epsilon 1 When not equal to 0, the young can be obtained by applying the inequality of young:
wherein eta and eta 1 Are all positive tuning parameters;
substituting equation (23) into equation (21) yields:
wherein
Always exists eta 1 Guarantee
To represent
The maximum singular value of;
the Lyapunov function thus satisfies:
according to the theory of Lyapunov stability,known error variable e 1 、e 2 、
Is eventually consistently bounded and will converge to a set:
wherein:
to represent
The minimum singular value of;
as can be seen from the formula (26), | e 1 (t)|＜δ 1 The same can be satisfied in the equation (26), and the radius of the set depends on the approximation error ε 1 Of (i.e. when ε 1 On → 0, the same conclusion can be reached as in case 1, which is aggregated as follows:
according to the universal approximation principle of the neural network, the approximation error is very small, so that x is reduced 1 (t) convergence to a set of residuals near the origin; stability certification is complete and the system state can converge to an arbitrarily small set.
Step three: hardware Loop test experiment (example 1)
In order to test the performance of the self-adaptive inversion controller designed in the step two, a hardware loop experiment (HIL) is carried out on the self-adaptive inversion controller, a control program adopts C language to carry out S-function programming, a TLC file is connected in series to carry out hardware accelerated operation, and a reference track x is set r Is zero;
in the control parameter setting, the maximum suspension travel is taken as z according to the system's user manual max 0.02cm, maximum control force | u | less than or equal to 10N, adaptive lawZero initial parameter
In the experiment, the precise parameters of the system are not needed, and only the sprung mass displacement x needs to be fed back 1 The vertical displacement of each part is measured by an encoder, the acceleration of the sprung mass is measured by an accelerometer, and the vertical speed is obtained by a filter;
NN activation function vector phi 1 And the adaptive gain vector is:
φ 1 ＝[x 2 -x 4 x 1 -x 3 (x 1 -x 3 ) 3 ] T
Γ 1 ＝diag([50 100 500 1×10 -5 ])
wherein x 1 -x 3 For the suspension stroke, x 2 -x 4 Is its derivative; the parameters of the controller after setting are shown in table 1:
table 1: controller gain parameter
In the experiment, the road surface excitation is generated by a direct current servo motor at the bottom, two typical road surface excitations are considered, the first is sine wave excitation, and the second is square wave impact excitation; setting the gain of the PD controller to k P ＝5，k Q 6; the gain of the LQR controller is K ═ 24.6648.87-0.473.68]；
Thus, vibration is most pronounced when the excitation frequency is close to the natural frequency of the flexible active suspension system, so the waveform of the sine wave is set to z r 0.002 × sin (6 π t) m, the excitation frequency being the same as the natural frequency of the sprung mass;
(1) sine wave excitation experiment:
FIGS. 3-7 show the results of the test under sinusoidal excitation for 15 s; FIG. 3 is a sprung mass displacement response; FIG. 4 is a sprung mass acceleration response; FIG. 5 is a suspension dynamic travel response; FIG. 6 is a tire stroke; FIG. 7 is an active control force generated by the controller; in the figure: pass stands for passive control, AB control stands for adaptive inversion control of the invention; compared with typical PD control and LQR control, the self-adaptive inversion control provided by the invention has a good vibration control effect under the condition of not needing system accurate parameters;
further, the root mean square value of the sprung mass acceleration can be calculated by using the formula (3), wherein the root mean square value in the uncontrolled condition is 0.9813m/s 2 The root mean square value of PD is 0.8882m/s 2 The root mean square value of LQR is 0.3819m/s 2 The root mean square value of the adaptive inversion control is 0.0421m/s 2 (ii) a Compared with PD control and LQR passive control, the acceleration root mean square value of the self-adaptive inversion control is reduced by 95%;
it can also be seen from fig. 5 that the maximum value of the suspension dynamic travel of the adaptive inversion control is less than 0.02cm and less than the passive control; the size of the tire dynamic stroke directly reflects the quality of the grounding stability, and the adaptive inversion control can obtain smaller tire dynamic stroke;
therefore, in summary, under the excitation of the road surface with the resonant frequency, the comfort and the grounding stability of the adaptive inversion control are better than those of the traditional PD, LQR and passive control.
(2) Square wave impact excitation experiment
In order to test the shock resistance of the active suspension system, a group of square waves are adopted to excite the active suspension system, and the excitation time is 15 s; FIGS. 8-12 are test results under impact excitation; in order to verify that the active suspension system has anti-saturation performance, an asymmetric saturation module is connected to a control input end, wherein the maximum value and the minimum value are respectively set as u max ＝1N，u min ＝-3N；
As can be seen from fig. 8 and 9, under the impact excitation, the adaptive inversion control method provided by the present invention has smaller overshoot displacement and acceleration;
as can be seen from fig. 10 and 11, the suspension stroke and the tire stroke of the adaptive inversion control method proposed by the present invention are also less than given limit values;
FIG. 12 is a control force output with saturation effect, and it can be seen that the control effect of the adaptive inversion control method proposed by the present invention is obviously better than that of the conventional PD and LQR controls;
as can be seen from fig. 2, when the control error is not large, | | λ 1 || ∞ The method is a very small value, so that the effectiveness of the adaptive inversion control method provided by the invention is further verified.
By the adaptive inversion control method, a neural network-based adaptive inversion controller is designed for a two-degree-of-freedom flexible active suspension system containing unknown nonlinear dynamics and asymmetric control saturation; the adaptive inversion control method is different from the traditional QL function, the BL function is selected to analyze the control condition of the stability of the closed-loop system, the controller design method with small conservative is obtained, and the asymmetric saturation phenomenon of the control force is compensated by introducing a second-order auxiliary system; the effectiveness of the adaptive inversion controller based on the neural network, which is designed by means of the adaptive inversion control method, is further verified through a hardware loop experiment, and compared with the classical PD and LQR control, the adaptive inversion control method provided by the invention can better attenuate the vertical vibration of the sprung mass and simultaneously improve the grounding performance under the sine and impact excitation.
The foregoing shows and describes the general principles, essential features, and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are given by way of illustration of the principles of the present invention, but that various changes and modifications may be made without departing from the spirit and scope of the invention, and such changes and modifications are within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (6)
1. A design method of an adaptive inversion controller in a flexible active suspension system is characterized by comprising the following steps:
the method comprises the following steps: building flexible active suspension model
In a two-degree-of-freedom flexible active suspension system containing unknown nonlinear dynamics, a flexible active suspension is built, and the dynamic relation of the model is as follows:
step two: designing adaptive inversion controller based on neural network
According to the flexible active suspension model established in the first step, a BL function is introduced to establish a control condition meeting the system stability, a second-order auxiliary system is introduced to compensate asymmetric control saturation in the flexible active suspension system, a self-adaptive inversion controller based on a neural network is designed through a saturation self-adaptive inversion control method, and the control performance of the active suspension is improved;
the construction process of the flexible active suspension model in the first step is as follows:
s1, arranging in a flexible active suspension model: m is s Is a sprung mass; m is u Is an unsprung mass; z is a radical of s Is the vertical displacement at the center of mass of the sprung mass, z u Is the vertical displacement at the unsprung mass centroid; f s Force generated for a non-linear spring of a suspension, F d A force generated for the damper; f t Elastic force generated for tire rigidity, F b Damping force generated for tire damping; u is the active control force generated by the DC servo motor in the active suspension;
s2, the dynamic relation of the flexible active suspension model is as follows:
wherein: in formula (1), F Δ Is an unknown disturbance force caused by friction and control errors, and F Δ Bounded but not continuously bounded with respect to Lipschitz, in the flexible active suspension model, the spring force F s And damping forceF d Is not measurable and belongs to unknown nonlinear dynamics, m s Is a sprung mass;
s3, setting the state variable of the system as x 1 ＝z s ，
x 3 ＝z u ，
Then the state space equation of the controlled system equation (1) is:
the design step of the adaptive inversion controller based on the neural network comprises the following steps:
s1, firstly, considering the saturation phenomenon of an active control force u, introducing a second-order auxiliary system to compensate a control error caused by saturation, and defining the saturation condition of the control force u;
the specific design process of step two S1 is:
(1) according to the type of a flexible active suspension system, considering an asymmetric input saturated self-adaptive NN inversion control law of an active suspension model, and introducing a second-order auxiliary system to compensate a control error caused by saturation, wherein the second-order auxiliary system is as follows:
wherein: u-f, λ 1 And λ 2 Representing state variables of the auxiliary system, and zero initial state λ 1 (0)、λ 2 (0) Is zero; c. C 1 And c 2 Is a normal number, and is,
is the inverse number theta of the sprung mass equal to 1/m s The estimated amount of (a) is,
is a first derivative of a state variable of a second-order auxiliary system (6);
(2) according to the introduced second-order auxiliary system, the saturation phenomenon of the control force in the active suspension model is considered, and the saturation condition met by the control force u is defined as follows:
wherein: f is the virtual control law to be designed, u is the active control force generated by the DC servo motor in the active suspension, u max Maximum active control force, u, generated for a DC servo motor in an active suspension min The minimum active control force generated by a direct current servo motor in the active suspension;
s2, defining a tracking error e 1 ＝x 1 -x r -λ 1 ，e 2 ＝x 2 Alpha, and selecting a BL function to construct a control condition meeting the stability of the flexible active suspension system;
wherein x is r Is the system state x 1 Reference track of (2), x r Is zero, alpha is the system state x 2 Virtual track of (2), x 1 Is the state variable of the controlled system formula (2);
s3, designing a virtual control law f by utilizing a self-adaptive neural network algorithm to enable a variable e 1 、e 2 、
Asymptotically converges to the origin or near the origin,
is an estimator;
s4, designing an adaptive law based on error convergence to estimate the NN weight and the sprung mass;
and S5, analyzing the stability of the flexible active suspension system by utilizing a Lyapunov function and carrying out classification discussion on an error convergence set.
2. The method for designing the adaptive inversion controller in the flexible active suspension system according to claim 1, wherein the specific design process in step two S2 is as follows:
(1) defining a tracking error: e.g. of the type 1 ＝x 1 -x r -λ 1 ，e 2 ＝x 2 -α；
Wherein: x is the number of 1 Is a state variable of the closed-loop system equation (2), x r Is the system state x 1 Reference track of (2), x r Is zero, alpha is the system state x 2 Virtual trajectory of, λ 1 Is a state variable of a closed-loop system formula (6), and the zero initial state is zero;
(2) let initial tracking error | e 1 (0)|＜δ 1 Wherein δ 1 Is an arbitrarily small positive number;
(3) selecting a BL function, said BL function being expressed as follows:
wherein: v 1 (e 1 ) Representing the BL function, δ 1 Is an arbitrarily small positive number, e 1 Represents a tracking error;
(4) by deriving equation (7), we can obtain:
wherein
k 1 ＞0，e 2 Representing the tracking error, x r Is the system state x 1 Reference track of (2), x r Is zero, k 1 A control gain parameter of α;
equation (8) can be transformed into:
(5) as can be seen from equation (9): if the error e 2 → 0, then
Therefore e 1 Asymptotically converge to zero and satisfy V 1 (e 1 )≤V 1 (e 1 (0) ); get x r Zero, | x can be obtained from step (2) 1 (0)|＜δ 1 When t → ∞, there is | x 1 (t)|＜||λ 1 || ∞ ；
(6) To study | | | λ 1 || ∞ Taking parameter c according to the change rule of delta u 1 ＝50、c 1 ＝60、
Performing numerical simulation on the formula (6) to obtain a simulation curve, and as can be seen from the simulation curve, when the control error Δ u is 10N, | λ 1 || ∞ Still less than 3 mm; it is thus possible to adjust the control parameter c 1 And c 2 To adjust | λ 1 || ∞ Thus decreasing | x 1 (t)|。
3. The design method of the adaptive inversion controller in the flexible active suspension system according to claim 2, wherein the specific design process of step two S3 is as follows:
(1) for tracking error e 2 The derivation is carried out to obtain:
wherein phi is-F d -F s -F Δ +u,
Can see-F d -F s -F Δ For unknown nonlinear dynamics, θ is 1/m s Is a perturbation parameter;
(2) designing a self-adaptive neural network algorithm to approach the unknown nonlinear dynamics phi, wherein the neural network algorithm is as follows:
wherein: t is NN Approximate quantity, W, representing unknown nonlinear dynamics 1 Represents the ideal NN weight, phi 1 Representing the vector of the activation function, ε 1 Representing a bounded approximation error, i.e. | ε 1 |＜ε 1N ，ε 1N ＞0，ε 1N Maximum error representing the nonlinear dynamics of the neural network and the system;
(3) by substituting formula (11) into formula (10), it is possible to obtain:
wherein
(4) Definition of
Is composed of
Equation (12) may be rewritten as:
wherein
(5) In order to ensure asymptotic stability of closed-loop system formula (2) and to make the error variable e 1 、e 2 、
Asymptotically converging to the origin or the vicinity of the origin, and designing a virtual control law as follows:
wherein: k is a radical of 2 To control the gain, and k 2 ＞0。
4. The design method of the adaptive inversion controller in the flexible active suspension system according to claim 3, wherein the specific design process of step two S4 is as follows:
(1) defining virtual filter variables e 2f 、
The filter variable e 2f 、
The relationship of (A) is as follows:
wherein: k > 0 is a design parameter;
(2) defining a virtual filter matrix P 1 Sum vector Q 1 Satisfies the following relation:
wherein: l > 0 is a design parameter;
(3) the following adaptive law based on error convergence is designed:
wherein: gamma-shaped 1 Is a diagonal matrix, σ > 0 is a learning gain value, and
(4) according to the formula (16), H can be represented by 1 Further decomposing into:
wherein:
ε 1f is epsilon 1 Filter factor of, i.e.
(5) It can be seen from equation (18) that for all bounded active control forces u and state vectors x, there is | | | Δ 1 ||＜ε 1Nf ，ε 1Nf Is a boundedA constant; current regression vector
Upon continued excitation, the matrix P 1 > 0 is a positive definite matrix, and the minimum eigenvalue satisfies the condition lambda min (P 1 ) The theta is more than 0, and the design of the active suspension control strategy is finished;
wherein θ represents a positive definite matrix P 1 The minimum eigenvalue of (c).
5. The method for designing the adaptive inversion controller in the flexible active suspension system according to claim 4, wherein the specific design process in step two S5 is as follows:
(1) to analyze the stability and error convergence set of the flexible active suspension system, the following Lyapunov function was chosen:
wherein: v 1 The function of the BL is represented by,
and Γ 1 Is a diagonal matrix;
by substituting equation (14) into equation (13), the following can be obtained:
by combining equation (20) and equation (18) after derivation of equation (19), the following can be obtained:
wherein: k is a radical of 1 ，k 2 To control the gain, and k 1 ＞0，k 2 ＞0；
(2) Equation (21) is discussed in two cases, the first case: if the neural network algorithm can ideally approximate the nonlinear dynamics of a flexible active suspension system, the approximation error ε 1 0, and Δ 1 When 0, then equation (21) satisfies:
according to Lyapunov stabilization theory, V (t) ≦ V (0), and the error variable e when time t → ∞ 1 、e 2 、
Will converge to the origin and satisfy | e over the entire time domain 1 (t)|＜δ 1 ；
(3) In the second case: when approximation error epsilon 1 When not equal to 0, the young can be obtained by applying the young inequality:
wherein eta and eta 1 Are all positive tuning parameters;
substituting equation (23) into equation (21) can yield:
wherein:
always exists eta 1 Ensuring
To represent
The maximum of the singular values of (c),
a square of a maximum error representing a nonlinear dynamics of the neural network with the system;
is | | | Δ 1 The square of the upper bound of | l;
the Lyapunov function thus satisfies:
according to the Lyapunov stabilization theory, the error variable e can be known 1 、e 2 、
Is eventually consistently bounded and will converge to a set:
wherein:
represent
The minimum singular value of;
as can be seen from the formula (26), | e 1 (t)|＜δ 1 The same can be satisfied in the equation (26), and the radius of the set depends on the approximation error ε 1 Of (i.e. when ε 1 On → 0, the same conclusion can be reached as in case 1, which is aggregated as follows:
6. the method for designing the adaptive inversion controller in the flexible active suspension system according to claim 1, wherein the method for designing the adaptive inversion controller in the flexible active suspension system further comprises the following steps: testing the performance of the adaptive inversion controller by a hardware loop test experiment, wherein the specific process of the experiment comprises the following steps:
(1) sine wave excitation experiment
Under the excitation of sine action, the self-adaptive inversion control method can well control vibration, and meanwhile under the road excitation of resonance frequency, the comfort and the grounding stability of the self-adaptive inversion control method are superior to those of the traditional PD, LQR and passive control;
(2) square wave impact excitation experiment
Under impact excitation, compared with the traditional PD, LQR and passive control, the adaptive inversion control method has smaller overshoot displacement and acceleration, and has control force output of saturation effect, and the control effect of the adaptive inversion control method is obviously better than that of the traditional PD and LQR control.
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