CN114114928B - Fixed time self-adaptive event trigger control method for piezoelectric micro-positioning platform - Google Patents
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Abstract
The invention discloses a fixed time self-adaptive event trigger control method of a piezoelectric micro-positioning platform, and aims to solve the problems of long adjustment time, limited control precision and high communication resource loss in the existing control method. The method comprises the following steps: step 1: establishing a model for a piezoelectric micro-positioning platform; step 2: simultaneously observing the undetectable state and the unknown disturbance through a composite observer; step 3: calculating the observation error, tracking error, virtual error and compensation error, constructing Lyapunov function V and deriving to obtainStep 4: virtual control, command filtering, signal compensation and self-adaptive control are performed by using a virtual control law, a command filter, a compensation signal and a self-adaptive law; step 5: and a fixed time self-adaptive event trigger controller is utilized, and the fixed time convergence and event trigger are realized through the controller, so that the high-precision tracking control effect on the output displacement of the piezoelectric micro-positioning platform is realized. The invention is more suitable for practical application and can obtain higher control precision.
Description
Technical Field
The invention belongs to the field of control of intelligent materials and mechanisms thereof, and particularly relates to a fixed time self-adaptive event trigger control method of a piezoelectric micro-positioning platform.
Background
The nano positioning technology has wide development prospect in the engineering application field. The piezoelectric micro-positioning platform has the advantages of nano resolution, quick response, high rigidity and the like, is used as a core device in various micro-nano high-precision positioning systems, and is widely applied to the fields of biological operation, ultra-precise lathes, atomic force microscopes and the like. However, piezoelectric micropositioning platforms, as a smart material driven actuator system, may have hysteresis nonlinearity characteristics created by the smart material itself. Hysteresis nonlinearity has characteristics such as frequency dependence and multivalue mapping nature, is a non-smooth nonlinearity, and the existence of hysteresis characteristic can influence the control effect of piezoelectric micropositioning platform and even lead to the system unstability. In order to reduce the adverse effect of hysteresis nonlinearity on the high-precision positioning of the piezoelectric micro-positioning platform, researchers at home and abroad develop a great deal of research on the hysteresis problem of the piezoelectric micro-positioning platform system. There are two general approaches to dealing with piezoelectric micropositioning platform hysteresis nonlinearity: one is to eliminate the adverse effects of hysteresis by computing or estimating an inverse hysteresis model, and the other is by assuming that the hysteresis term is bounded. However, the former makes the construction of the inverse model difficult due to the complexity of the hysteresis model, and the latter cannot be applied to a nonlinear system with only a measurable output because the hysteresis term cannot be guaranteed to be limited in the actual application process. Therefore, designing an effective control method to eliminate the hysteresis nonlinearity of the piezoelectric micropositioning platform is a problem we have to solve.
In order to improve the positioning accuracy of the piezoelectric micro-positioning platform, students have proposed various control strategies such as sliding mode control, a backstepping method, self-adaptive control and the like. The back-stepping method is a common control method, and the virtual control law and the control law are designed by decomposing a complex nonlinear system into a plurality of subsystems. However, the back-stepping method requires multiple derivations of the virtual control law, which causes calculation explosion problems. To solve this problem, dynamic surface control has been proposed. The dynamic surface control solves the problem of calculation explosion by using a first-order filter, but the addition of the filter generates a filtering error, so that the improvement of control performance is limited. Thus, scholars have proposed command filtering back-step control, which eliminates the computational explosion problem by commanding the filter, and eliminates the filtering error by an error compensation mechanism. The conventional command filter has the problem that the filter constant is limited in value, so that the control capability is affected. It is therefore necessary to design a command filter that can efficiently extract a high quality differential signal and suppress noise.
Most of the current methods applied to the tracking control of the piezoelectric micro-positioning platform can only ensure the asymptotic stability of the system, namely, theoretically, an infinitely long time is required to ensure that the tracking error of the piezoelectric micro-positioning platform system is converged near the balance point. The limited time control can ensure the rapid convergence of the system and the improvement of the control characteristic, and is a control problem with more practical significance. However, the convergence time of the finite time control is severely dependent on the initial state of the system, and for piezoelectric micropositioning stages such high precision positioning systems may cause serious loss of control performance. It is therefore of great practical value to design a fixed time control criterion that is independent of the initial state.
Meanwhile, the existing control method applied to the piezoelectric micro-positioning platform system is based on time triggering, and with the sequential development of a network system and the actual communication requirements during multi-equipment operation, the proposal of event triggering control has strong application significance for efficiently transmitting signals and effectively saving transmission resources.
Therefore, how to combine the fixed time control technology, the event triggering mechanism and the adaptive command filtering back-stepping method to be applied to the piezoelectric micro-positioning platform system with complex input hysteresis so as to realize the high-precision tracking control of the system has not seen the related technology.
Disclosure of Invention
The invention aims to provide a fixed time self-adaptive event trigger control method of a piezoelectric micro-positioning platform, which aims to overcome the defects in the prior art.
The method comprises the following steps:
Step 1: according to inherent hysteresis characteristics of the piezoelectric micro-positioning platform, a second-order nonlinear system model with input hysteresis and unknown external disturbance is established for the platform;
Step 2: based on the system model established in the step 1, simultaneously observing an unmeasurable state and unknown disturbance through a composite observer;
Step 3: calculating the observation error, tracking error, virtual error and compensation error, constructing a Lyapunov function V and performing first-order derivation on time to obtain
Step 4: based on the composite observer in step 2 and in step 3The virtual control law, the command filter, the compensation signal and the self-adaptive law are designed by using a back-stepping method, a command filtering technology and a self-adaptive technology and are used for virtual control, command filtering, signal compensation and self-adaptive control;
Step 5: based on the steps 1-4, according to the Lyapunov stabilization theory, the fixed time convergence criterion and the event triggering mechanism, a self-adaptive event triggering controller capable of guaranteeing the system fixed time convergence is obtained, the fixed time convergence and the event triggering are realized through the controller, and the high-precision tracking control effect on the output displacement of the piezoelectric micro-positioning platform is realized;
The specific process of the step1 is as follows:
the second-order nonlinear system model of the piezoelectric micro-positioning platform with input hysteresis and unknown external disturbance is as follows:
wherein x 1,x2 and y are the state variables and outputs of the system, respectively; f 1 (·) and f 2 (·) are unknown nonlinear functions of this system; d 1 (·) and d 2 (·) are unknown external disturbances of this system and satisfy the assumption that the derivatives of the external disturbances are bounded; v=h (u T) is the output of the actuator, u T is the input of the actuator, i.e. the input of the system;
describing a hysteresis in the system using a nonlinear autoregressive moving average model (HT-NARMAX) with hysteresis terms; i.e., v=h (u T) can be represented by the following formula:
Wherein, N (·) is an unknown function representing the input-output mapping relationship of the HT-NARMAX model; h H(t)＝mHuT(t)+hD (t), Representing Duhem operators; m H,a,fH(·),gH (·) is a parameter in the Duhem operator; n u,nh,ny represents the hysteresis of u T,hH, y; p=n u+nh+ny; q is the order of this model; Is a nonlinear function with respect to ζ 1,...,ζp; /(I) Including all unknown parameters in the model; n H = (p+q) +.! /p-! q-! -1; without loss of generality, M 1 = 1 in the present invention;
The two unknown nonlinear functions f 1 (·) and f 2 (·) are approximated by using Takagi-Sugeno fuzzy neural network, then the functions can be written as:
Wherein, Is an ideal weight vector; s i is an input vector of the Takagi-Sugeno fuzzy neural network; χ i is the approximation error and satisfying |χ i|≤χT;χT is a constant; i=1, 2.
Taking equations (2) and (3) into (1), the second order nonlinear system model of the piezoelectric micropositioning stage system with input hysteresis and unknown external disturbances can be rewritten as:
Wherein, by reasonably selecting the parameter l 1,l2, A is ensured to be a strict Hulvitz matrix; that is, for a symmetric positive definite matrix Q, there is a positive definite matrix P such that the equation a T p+pa= -Q holds;
the following symbols are defined: i.e. estimation error = actual quantity-estimated quantity.
The specific process of the step 2 is as follows:
Based on the system model in the step 2, only the measurable problem is output for the piezoelectric micro-positioning platform, and the specific process of designing the composite observer capable of simultaneously observing the system state and unknown external disturbance is as follows:
To ensure convergence of the composite observer, the following assumptions are made:
The derivative of the external disturbance d i (t) is bounded, i.e. satisfies Wherein d m is a known constant;
According to equation (4), the composite observer is designed to:
Wherein v 1,ν2 is an auxiliary variable of the composite observer; n 1,N2 is a design parameter of the composite observer; k 1,K2 denotes a compensation term for the estimation error; d m＝max{dm1,dm2};vd is a positive number.
The specific process of the step 3 is as follows:
Based on the formulas (4) and (5), the state observation error e can be calculated according to the formula (6):
where iota=e κt, kappa is a constant greater than 0, t is time, Is the system observation state;
Based on the equation (4) and the equation (5), the disturbance observation error E D can be calculated by the equation (7):
Is the observed interference of the system;
The tracking error z 1 is:
z1＝ι(y-yr) (8)
where y r is the reference trace.
The virtual error z 2 is:
Wherein ω 2 is the output of the command filter, and the command filter used is shown in formula (13).
The compensation error σ 1,σ2 is:
wherein r 1,r2 is an error compensation signal;
according to equations (6) - (10), the lyapunov function V is designed as:
Derivative of Lyapunov function The method comprises the following steps:
The specific process of the step 4 is as follows:
Step 4.1 tracking differentiator based on the modified acceleration function acts as a command filter:
Wherein ζ (t) is the input signal to the command filter; ζ 1 (t) is an estimated signal of the input signal ζ (t); ζ 2 (t) is the first derivative of ζ 1 (t); p 1i,p2i,qi,si,m1,m2 is a positive design parameter; pi i is a positive odd number; in the tracking differentiator shown in formula (13), T i (·) is the modified acceleration function; for any input signal ζ (t), we can get its estimated signal ζ 1 (t);
In the subsequent step, the input of the formula (13) is a virtual control quantity alpha 1, and the output is command filtering output omega 2;
step 4.2, designing the following virtual control law based on a back-stepping method:
Wherein: k 1,λ1 is a positive design parameter; mu is more than 1,0 < rho is less than 1;
To eliminate the filtering error ω 2-α1 generated between the input α 1 and the output ω 2 of the command filter, the following error compensation signal is designed:
Wherein: k 2,λ2 is a positive design parameter. Is a positive known constant.
Step 4.3, designing an adaptive law based on an adaptive technology:
Wherein: Λ 1,Λ2, Γ is the positive definite matrix to be designed; γ, τ 1,τ2 is a positive design parameter.
The specific process of the step 5 is as follows:
Step 5.1, the novel fixed time convergence criterion is:
For nonlinear systems X (0) =x 0, f (·) is a nonlinear function of the neighborhood around 0, and the origin is assumed to be the system balance point; for any real number α, β, γ > 0, μ > 1,0 < ρ < 1, iota=e κt, there is a positive definite function V (x) that satisfies the following lyapunov condition:
then the nonlinear system is stable for a fixed time and the adjustment time T f is:
step 5.2 to facilitate the design of the adaptive event triggered controller, an auxiliary function is constructed:
based on equation (21) and the event trigger mechanism, the event trigger controller is designed to:
Wherein: e T(t)＝uT (t) -u (t) are measurement errors in event-triggered control; 0.ltoreq.η < 1, ω > 0, θ > 0, ε > 0 being a design parameter.
The parameters Γ,γ,τ1,Λ1,τ2,Λ2,l1,l2,N1,N2,dm1,dm2,p11,p12,p21,p22,q1,q2,s1,s2,π1,π2,m1,m2,k1,k2,λ1,λ2,μ,ρ,η,ω,θ,ε,κ to be designed in the invention are selected according to the following principle:
1) The selection of the parameter l 1,l2 in the composite observer requires a guaranteed matrix Is a strict Hulvitz matrix, i.e. the choice of l 1,l2 has a great influence on the stability of the system. For this system of piezoelectric micropositioning stages, the parameter of l 1,l2 is chosen as l 1＝1.88,l2 = 0.99; the parameters for parameter N 1,N2,dm1,dm2 related to the interference observations in the composite observer are selected as: n 1＝0.1,N2＝0.05,dm1＝dm2 = 1;
2) The parameters p11,p12,p21,p22,q1,q2,s1,s2,π1,π2,m1,m2 in the command filter are chosen to ensure that the output of the command filter is a high quality differential signal. The comparison of the filtering effects of the filter under different parameters is made available according to the command, for which the parameters are chosen as p11＝p12＝5,p21＝p22＝5,q1＝q2＝1,s1＝s2＝100,π1＝π2＝3,m1＝m2＝1.2;
3) The principle of parameter selection in the virtual control law, the compensation signal and the self-adaptive law is to ensure the stability of the piezoelectric micro-positioning platform system and the convergence of the self-adaptive parameters, so the parameters are selected as ：k1＝0.07,k2＝1.5,λ1＝0.1,λ2＝1,μ＝1021/1001,ρ＝997/1001,Γ＝[0.185,0.002,0.03,0.02]T,γ＝0.4,τ1＝0.05,Λ1＝0.045I7,τ2＝0.1,Λ2＝0.001I7;
4) The selection of the controller parameters under the event triggering mechanism needs to ensure that the system can avoid the gano phenomenon, so the parameters are selected as follows: η=0.0001, ω=0.0001, θ=0.001, ε=0.0001, and κ=0.001.
Bringing formulae (13) - (18), (22) - (24) in step 4 and step 5 into formula (12) in step 3And according to the new fixed time convergence criterion in 5.1: by reasonably selecting design parameters in the controller, the system can be ensured to be stable in fixed time and the closed-loop signal is bounded.
The beneficial effects of the invention are as follows:
Aiming at the tracking control problem of the piezoelectric micro-positioning platform, the invention provides a fixed time self-adaptive event triggering control method of the piezoelectric micro-positioning platform so as to improve the control effect of the piezoelectric micro-positioning platform. The method has the following four advantages: first, a system model with input hysteresis and unknown external disturbances is built, wherein it is proposed to describe the input hysteresis nonlinearity using the HT-NARMAX model. Based on the established system model, a composite observer capable of simultaneously observing an unmeasurable state and an unknown external disturbance is designed. The observer can update parameters of the HT-NARMAX model on line in real time, estimate unknown external disturbance and overcome the defects in the processing means of hysteresis and unknown external disturbance in the prior art. The improved tracking differentiator is used as the command filter, so that the defect that the command filter in the prior art cannot effectively extract high-quality differential signals can be overcome, and the performance of the controller is further improved. Thirdly, the invention provides a novel fixed time convergence criterion, based on which the steady state and dynamic performance of the system can be ensured within a fixed time, and the robustness of the system can be improved. The controller is designed based on an event triggering mechanism, so that the effective combination of event triggering control and fixed time control can be realized, and the convergence rate of the system is improved while the communication loss is reduced.
The method lays a foundation for the precise positioning control application of the piezoelectric micro-positioning platform, and is more close to engineering requirements.
Drawings
FIG. 1 is a control block diagram of a fixed time adaptive event trigger control method of a piezoelectric micropositioning platform of the present invention;
FIG. 2 is a diagram of an experimental setup for a piezoelectric micropositioning stage of the present invention;
FIG. 3 is a schematic diagram of the experimental setup of the piezoelectric micropositioning stage of the present invention;
FIG. 4 is a graph comparing reference trace curves and piezoelectric micro-positioning platform output curves when tracking complex harmonic traces according to the invention;
FIG. 5 is a graph of the error of the reference trajectory curve and the piezoelectric micro-positioning platform output when tracking complex harmonic trajectories of the present invention;
FIG. 6 is a graph comparing time-triggered and event-triggered based control signals of a controller in tracking complex harmonic trajectories according to the present invention;
FIG. 7 is a graph of the adjustment of the adaptive parameter M H in tracking complex harmonic trajectories in accordance with the present invention;
FIG. 8 is a graph of the adjustment of the adaptive parameter Φ 1 in tracking complex harmonic trajectories of the present invention;
fig. 9 is a graph of the adjustment of the adaptive parameter Φ 2 in tracking complex harmonic trajectories of the present invention.
Detailed description of the preferred embodiments
The present invention will be described in further detail with reference to the accompanying drawings and specific examples.
The control block diagram of the fixed time self-adaptive event trigger control method of the piezoelectric micro-positioning platform is shown in figure 1. The method comprises the following steps:
Step 1: according to inherent hysteresis characteristics of the piezoelectric micro-positioning platform, a second-order nonlinear system model with input hysteresis and unknown external disturbance is built for the platform.
The second-order nonlinear system model of the platform is as follows:
Where x 1,x2 and y are the state variables and outputs of the system, respectively. f 1 (·) and f 2 (·) are unknown nonlinear functions of this system. d 1 (·) and d 2 (·) are unknown external perturbations of this system and satisfy hypothesis 1. v=h (u T) is the output of the actuator and u T is the input to the actuator, i.e. the input to the system. Since the hysteresis behavior existing in the actuator can greatly influence the tracking effect of the piezoelectric micro-positioning platform system, a nonlinear autoregressive moving average (HT-NARMAX) model with hysteresis terms is provided to describe the hysteresis part in the system. The HT-NARMAX model enhances the NARMAX model's ability to describe hysteresis nonlinearities by introducing the Duhem operator as an exogenous function into the NARMAX model. I.e., v=h (u T) can be represented by the following formula:
Where N (·) is an unknown function representing the input-output mapping of the HT-NARMAX model. h H(t)＝mHuT(t)+hD (t), The Duhem operator is represented. m H,a,fH(·),gH (·) is a parameter in the Duhem operator. n u,nh,ny is the hysteresis of u T,hH, y. p=n u+nh+ny. q is the order of this model. ψ= [ ψ 2,...,ψnH]T＝[ζ2,...,ζi1...ζiq]T ] is a nonlinear function with respect to ζ 1,...,ζp. M H＝[M2,...,MnH]T contains all unknown parameters in this model. n H = (p+q) +.! /p-! q-! -1. Without loss of generality, M 1 = 1 in the present invention.
Since the unknown nonlinear functions f 1 (·) and f 2 (·) in the system contain uncertainty, the two functions are approximated by using the Takagi-Sugeno fuzzy neural network (TSFNN), and the functions can be written as:
Wherein, Is an ideal weight vector. S i is the input vector of TSFNN. χ i is the approximation error and satisfying |χ i|≤χT.χT is a constant. i=1, 2.
Bringing equations (2) and (3) into (1), the piezoelectric micropositioning stage system can be rewritten as:
Wherein, by reasonably selecting the parameter l 1,l2, A is ensured to be a strict Hulvitz matrix. That is, for a symmetric positive definite matrix Q, there is a positive definite matrix P such that the equation a T p+pa= -Q holds.
Step 2: based on the system model established in the step 1, a composite observer capable of simultaneously observing an undetectable state and unknown disturbance is designed.
To ensure convergence of the composite observer, the following assumptions are made:
Suppose 1: the derivative of the external disturbance d i (t) is bounded, i.e. satisfies . Where d m is a known constant.
According to equation (4), the composite observer is designed to:
Where v 1,ν2 is an auxiliary variable of the composite observer. N 1,N2 is a design parameter of the composite observer. K 1,K2 denotes a compensation term for the estimation error. d m＝max{dm1,dm2}.vd is a positive number.
Step 3: by defining the observation error, tracking error, virtual error and compensation error, the Lyapunov function V is designed and the first order derivation is carried out on time to obtain
Based on the formulas (4) and (5), the state observation error e can be calculated according to the formula (6):
where iota=e κt, kappa is a constant greater than 0, t is time, Is the system observation state;
Based on the equation (4) and the equation (5), the disturbance observation error E D can be calculated by the equation (7):
Is the observed interference of the system;
The observation error of the state is obtained through calculation:
The interference observation error is as follows:
The tracking error is:
z1＝ι(y-yr) (10)
Wherein y r is a reference trace;
the virtual error z 2 is:
Wherein ω 2 is the output of the command filter, and the command filter used is shown in formula (13).
The compensation error σ 1,σ2 is:
wherein r 1,r2 is an error compensation signal;
According to equations (6), (7), (10) - (12), the lyapunov function V is designed as:
Derivative of Lyapunov function The method comprises the following steps:
step 4: based on the composite observer in step 2 and in step 3 Designing a virtual control law, a command filter, a compensation signal and an adaptive law by using a back-stepping method, a command filtering technology and an adaptive technology; for virtual control, command filtering, signal compensation and adaptive control;
4.1 in the design of a back-step control based on command filtering technique, the following tracking differentiator based on an improved acceleration function is proposed as a command filter in the present invention:
Wherein ζ (t) is the input signal to the command filter; ζ 1 (t) is an estimated signal of the input signal ζ (t); ζ 2 (t) is the first derivative of ζ 1 (t); p 1i,p2i,qi,si,m1,m2 is a positive design parameter; pi i is a positive odd number; in the tracking differentiator shown in formula (15), T i (·) is the modified acceleration function; for any input signal ζ (t), we can get its estimated signal ζ 1 (t);
In the subsequent step, the input of the formula (15) is a virtual control quantity alpha 1, and the output is command filtering output omega 2;
4.2 based on the back-stepping method, the following virtual control law is designed:
Wherein: k 1,λ1 is a positive design parameter; mu is more than 1,0 < rho is less than 1;
To eliminate the filtering error ω 2-α1 generated between the input α 1 and the output ω 2 of the command filter, the following error compensation signal is designed:
/>
Wherein: k 2,λ2 is a positive design parameter. Is a positive known constant.
Based on the self-adaptive technology, the self-adaptive law is designed:
Wherein: Λ 1,Λ2, Γ is the positive definite matrix to be designed; γ, τ 1,τ2 is a positive design parameter.
Step 5: based on the steps 1-4, according to the Lyapunov stabilization theory, the fixed time convergence criterion and the event triggering mechanism, a self-adaptive event triggering controller capable of guaranteeing the system fixed time convergence is obtained, and the fixed time convergence and the event triggering are realized through the controller, so that the high-precision tracking control effect on the output displacement of the piezoelectric micro-positioning platform is realized.
5.1 Novel fixed time convergence criteria are:
For nonlinear systems X (0) =x 0, f (·) is a nonlinear function of the neighborhood around 0, and the origin is assumed to be the system balance point. For any real number α, β, γ > 0, μ > 1,0 < ρ < 1, iota=e κt, there is a positive definite function V (x) that satisfies the following lyapunov condition:
then the nonlinear system is stable for a fixed time and the adjustment time T f is:
And (3) proving:
Case1: if V (x) > 1, according to equation (21), The expression can be as follows:
Dividing V μ (x) by both sides of formula (23) and then integrating between intervals [0, t ] yields:
From V (x) > 1:
By design of V μ-1 (x) is less than or equal to 1 for any T is more than or equal to T 1, and then V (x) is less than or equal to 1. /(I)
Case2: if V (x) is less than or equal to 1, according to the formula (21),The expression can be as follows:
When x (t) satisfies The method can obtain:
dividing V ρ (x) by both sides of formula (27), and then integrating between intervals [ t 0, t ] yields:
The method is available according to V (x) is less than or equal to 1:
By design of V 1-ρ (x) is less than or equal to 0 for any t is more than or equal to t 0+T2, and V (x) is a positive function, so that/>
When x (t) satisfiesBased on LaSalle principle of invariance and the above analysis procedure we can obtain:
From the discussion above, it is possible that the nonlinear system is stable for a fixed time, and the adjustment time can be calculated from T f≤Tmax＝T1+T2.
5.2 To facilitate the design of the adaptive event triggered controller, an auxiliary function is constructed:
based on equation (30) and the event trigger mechanism, the event trigger controller is designed to:
wherein: e T(t)＝uT (t) -u (t) are measurement errors in event-triggered control. 0.ltoreq.η < 1, ω > 0, θ > 0, ε > 0 being a design parameter.
Bringing formulae (15) - (20), (31) - (33) in step 4 and step 5 into formula (12) in step 3And partial terms are subjected to unequal transformation through the Young's inequality, so that the method can be obtained: /(I)
Wherein,
Lambda min (Q) is the minimum eigenvalue of matrix Q.Is a positive constant.
The main results of the present invention can be summarized as follows.
Theorem 1: for a second-order piezoelectric micro-positioning platform (1) with input hysteresis and external disturbance, a composite observer (5) is designed, a tracking differentiator (15), a virtual control law (16), a compensation signal (17), adaptive laws (18) - (20), event triggering mechanisms (32) - (33) and a control law (31) are improved, the fixed time adaptive event triggering control method constructed by the invention can ensure that a closed loop system signal is stable in fixed time, and a Zhinor phenomenon does not occur in the designed event triggering controller.
Before analyzing the stability of the control method of the present invention from a theoretical point of view, the following arguments necessary in the proving process were introduced:
lemma 1: assuming the presence of unknown constants Satisfy/>The following inequality holds:
Wherein,
Theory on theorem 1 proves as follows:
According to lemma 1, partial terms in equation (34) may be subjected to unequal transformation:
Wherein, Aed＝min{Adi}。/>I=1, 2.v 1,v2,v3,v4,v5 can be solved according to the unequal transformations of lemma 1.
Bringing formula (36) into formula (34) yields:
Wherein,
α＝Ac+Aov1+Aedv2+Aλv3+AMv4+AΦv5.β＝min{Ao,Aed,Aλ+2ρλi,AΦ,AM,i＝1,2}.
The compensation signal σ i is shown to be stable over a fixed time in equation (37). From σ i＝zi-ri in equation (12), we know that if the compensation signal r i can be guaranteed to be stable for a fixed time, then the stability of z i can also be guaranteed. Design of Lyapunov functionAfter deriving it and substituting the compensation signal r i, it can be obtained:
according to the nature of the command filter Selecting a suitable parameter to satisfy/>Further, the expression (38) can be derived as follows:
/>
wherein, xi 1＝2min{ki },
According to formulas (37) - (39), the system can be ensured to be stable for a fixed time, and the adjustment time can be obtained according to formula (22).
Furthermore, we should demonstrate that event triggering mechanisms are designed to avoid the occurrence of the gano phenomenon.
From the definition of the measurement errors e T(t)＝uT (t) -u (t) in the event-triggered control, it is possible to obtain:
From theorem 1, it follows that: due to Is a function of the bounded signal, so/>Is also bounded. Therefore, there must be a positive constant ω u to satisfy/>Then based on/>And e T(tk) =0, |e T|≤ωu(t-tk) is obtained. Recombination event trigger Condition/>For T k＝tk+1-tk, the inequalityThis is true. When k → infinity,/>Thus, the event trigger interval time T k > 0 holds, i.e., the phenomenon of Zhinox does not occur.
Parameters to be designed in the invention:
Γ,γ,τ1,Λ1,τ2,Λ2,l1,l2,N1,N2,dm1,dm2,p11,p12,p21,p22,q1,q2,s1,s2,π1,π2,m1,m2,k1,k2,λ1,λ2,μ,ρ,η,ω,θ,ε,κ The selection is performed according to the following principle:
1) The selection of the parameter l 1,l2 in the composite observer requires a guaranteed matrix Is a strict Hulvitz matrix, i.e. the choice of l 1,l2 has a great influence on the stability of the system. For this system of piezoelectric micropositioning stages, the parameter of l 1,l2 is chosen as l 1＝1.88,l2 = 0.99; the parameters for parameter N 1,N2,dm1,dm2 related to the interference observations in the composite observer are selected as: n 1＝0.1,N2＝0.05,dm1＝dm2 = 1;
2) The parameters p11,p12,p21,p22,q1,q2,s1,s2,π1,π2,m1,m2 in the command filter are chosen to ensure that the output of the command filter is a high quality differential signal. The comparison of the filtering effects of the filter under different parameters is made available according to the command, for which the parameters are chosen as p11＝p12＝5,p21＝p22＝5,q1＝q2＝1,s1＝s2＝100,π1＝π2＝3,m1＝m2＝1.2;
3) The principle of parameter selection in the virtual control law, the compensation signal and the self-adaptive law is to ensure the stability of the piezoelectric micro-positioning platform system and the convergence of the self-adaptive parameters, so the parameters are selected as ：k1＝0.07,k2＝1.5,λ1＝0.1,λ2＝1,μ＝1021/1001,ρ＝997/1001,Γ＝[0.185,0.002,0.03,0.02]T,γ＝0.4,τ1＝0.05,Λ1＝0.045I7,τ2＝0.1,Λ2＝0.001I7;
4) The selection of the controller parameters under the event triggering mechanism needs to ensure that the system can avoid the gano phenomenon, so the parameters are selected as follows: η=0.0001, ω=0.0001, θ=0.001, ε=0.0001, and κ=0.001.
The following specific examples are presented to demonstrate the benefits of the present invention.
Examples: taking a piezoelectric micro-positioning platform as an example, the fixed time self-adaptive event trigger control method designed by the invention is used for tracking the track of the experimental platform shown in fig. 2 so as to verify the validity of the constructed controller.
① Experiment platform: the validity of the control method was verified using the experimental platform shown in fig. 2. The working principle of the experimental platform is shown in figure 3. The hardware facilities of the experiment platform comprise a computer (programming a controller program and carrying out data processing analysis), a data acquisition card (realizing conversion of digital signals and analog signals), a precise positioning controller (transmitting driving signals of the computer and displacement signals of the piezoelectric micro-positioning platform), the piezoelectric micro-positioning platform (generating displacement and measuring the displacement by a built-in displacement sensor) and a shock insulation platform.
② Setting reference tracking signals and controller parameters: and selecting a complex harmonic signal with variable frequency and amplitude as a reference tracking track of the piezoelectric micro-positioning platform. And the sampling time was set to 0.0001s. The initial value of the parameters of the controller is ：Γ＝[0.185,0.002,0.03,0.02]T,γ＝0.4,τ1＝0.05,Λ1＝0.045I7,τ2＝0.1,Λ2＝0.001I7,l1＝1.88,l2＝0.99,N1＝0.1,N2＝0.05,dm1＝dm2＝1,p11＝p12＝5,p21＝p22＝5,q1＝q2＝1,s1＝s2＝100,π1＝π2＝3,m1＝m2＝1.2,k1＝0.07,k2＝1.5,λ1＝0.1,λ2＝1,μ＝1021/1001,ρ＝997/1001,η＝0.0001,ω＝0.0001,θ＝0.001,ε＝0.0001,κ＝0.001.:
The tracking results of the control method of the present invention obtained through experiments are shown in fig. 4 to 7. As can be seen from fig. 4, under the control method of the present invention, the actual output displacement of the piezoelectric micro-positioning platform can track the desired track well. The desired track and the actual track output shown in fig. 4 are subtracted to obtain the tracking error graph shown in fig. 5. The tracking error under the action of the controller is far smaller than 0.5 mu m, which proves the effectiveness of the controller. Fig. 6 is a diagram of control signals based on a time trigger mechanism and control signals based on an event trigger mechanism. As can be seen from the partial enlarged view in fig. 6, the communication resource of the control signal based on the event trigger mechanism according to the present invention is less than the loss of the control signal based on the time trigger mechanism. Fig. 7-9 are adjustment curves of adaptive parameters in the controller of the present invention. The results of fig. 4-9 are combined to demonstrate that the controller of the present invention provides good control.
Claims (2)
1. A fixed time self-adaptive event trigger control method of a piezoelectric micro-positioning platform is characterized in that: the method comprises the following specific steps:
Step 1: according to inherent hysteresis characteristics of the piezoelectric micro-positioning platform, a second-order nonlinear system model with input hysteresis and unknown external disturbance is established for the platform;
Step 2: based on the system model established in the step 1, simultaneously observing an unmeasurable state and unknown disturbance through a composite observer;
Step 3: calculating the observation error, tracking error, virtual error and compensation error, constructing a Lyapunov function V and performing first-order derivation on time to obtain
Step 4: based on the composite observer in step 2 and in step 3The virtual control law, the command filter, the compensation signal and the self-adaptive law are designed by using a back-stepping method, a command filtering technology and a self-adaptive technology and are used for virtual control, command filtering, signal compensation and self-adaptive control;
Step 5: based on the steps 1-4, according to the Lyapunov stabilization theory, the fixed time convergence criterion and the event triggering mechanism, a self-adaptive event triggering controller capable of guaranteeing the system fixed time convergence is obtained, the fixed time convergence and the event triggering are realized through the controller, and the high-precision tracking control effect on the output displacement of the piezoelectric micro-positioning platform is realized;
the following symbols are defined: I.e. estimation error = actual quantity-estimated quantity;
the second-order nonlinear system model with input hysteresis and unknown external disturbance in the step 1 is as follows:
wherein x 1,x2 and y are the state variables and outputs of the system, respectively; f 1 (·) and f 2 (·) are unknown nonlinear functions of this system; d 1 (·) and d 2 (·) are unknown external disturbances of this system and satisfy the assumption that the derivatives of the external disturbances are bounded; v=h (u T) is the output of the actuator, u T is the input of the actuator, i.e. the input of the system;
Describing a hysteresis part in the system by adopting a nonlinear autoregressive moving average model with hysteresis terms; i.e., v=h (u T) can be represented by the following formula:
Wherein, N (·) is an unknown function representing the input-output mapping relationship using a nonlinear autoregressive moving average model with hysteresis terms; h H(t)＝mHuT(t)+hD (t), Representing Duhem operators; m H,a,fH(·),gH (·) is a parameter in the Duhem operator; n u,nh,ny represents the hysteresis of u T,hH, y; p=n u+nh+ny; q is the order of this model; /(I)Is a nonlinear function with respect to ζ 1,...,ζp; Including all unknown parameters in the model; n H = (p+q) +.! /p-! q-! -1; without loss of generality, M 1 = 1 in the present invention;
The two unknown nonlinear functions f 1 (·) and f 2 (·) are approximated by using Takagi-Sugeno fuzzy neural network, then the functions can be written as:
Wherein, Is an ideal weight vector; s i is an input vector of the Takagi-Sugeno fuzzy neural network; χ i is the approximation error and satisfying |χ i|≤χT;χT is a constant; i=1, 2, x= [ x 1,x2]T;
Taking equations (2) and (3) into (1), the second order nonlinear system model of the piezoelectric micropositioning stage system with input hysteresis and unknown external disturbances can be rewritten as:
Wherein, by reasonably selecting the parameter l 1,l2, A is ensured to be a strict Hulvitz matrix; that is, for a symmetric positive definite matrix Q, there is a positive definite matrix P such that the equation a T p+pa= -Q holds;
Step 2, based on a system model, aiming at the problem that only the piezoelectric micro-positioning platform outputs the measureable problem, the specific process of designing the composite observer capable of simultaneously observing the system state and the unknown external disturbance is as follows:
To ensure convergence of the composite observer, the following assumptions are made:
The derivative of the external disturbance d i (t) is bounded, i.e. satisfies Wherein d m is a known constant;
According to equation (4), the composite observer is designed to:
Wherein v 1,ν2 is an auxiliary variable of the composite observer; n 1,N2 is a design parameter of the composite observer; k 1,K2 denotes a compensation term for the estimation error; d m＝max{dm1,dm2};vd is a positive number;
The specific process of calculating the observation error, the tracking error, the virtual error and the compensation error, constructing the Lyapunov function V and performing first-order derivation on time in the step 3 is as follows:
Based on the formulas (4) and (5), the state observation error e can be calculated according to the formula (6):
where iota=e κt, kappa is a constant greater than 0, t is time, Is the system observation state;
Based on the equation (4) and the equation (5), the disturbance observation error E D can be calculated by the equation (7):
Is the observed interference of the system;
The tracking error z 1 is:
z1＝ι(y-yr) (8)
Wherein y r is a reference trace;
the virtual error z 2 is:
wherein ω 2 is the output of the command filter;
the compensation error σ 1,σ2 is:
wherein r 1,r2 is an error compensation signal;
according to equations (6) - (10), the lyapunov function V is designed as:
Derivative of Lyapunov function The method comprises the following steps:
Step4 derivative of Lyapunov function The specific processes of obtaining the command filter, the virtual control law, the compensation signal and the self-adaptive law are as follows:
Step 4.1 tracking differentiator based on the modified acceleration function acts as a command filter:
Wherein ζ (t) is the input signal to the command filter; ζ 1 (t) is an estimated signal of the input signal ζ (t); ζ 2 (t) is the first derivative of ζ 1 (t); p 1i,p2i,qi,si,m1,m2 is a positive design parameter; pi i is a positive odd number; in the tracking differentiator shown in formula (13), T i (·) is the modified acceleration function; for any input signal ζ (t), we can get its estimated signal ζ 1 (t);
In the subsequent step, the input of the formula (13) is a virtual control quantity alpha 1, and the output is command filtering output omega 2;
step 4.2, designing the following virtual control law based on a back-stepping method:
Wherein: k 1,λ1 is a positive design parameter; mu >1,0< ρ <1;
To eliminate the filtering error ω 2-α1 generated between the input α 1 and the output ω 2 of the command filter, the following error compensation signal is designed:
Wherein: k 2,λ2 is a positive design parameter; δ 1,δ2 is a positive known constant;
Step 4.3, designing an adaptive law based on an adaptive technology:
Wherein: Λ 1,Λ2, Γ is the positive definite matrix to be designed; γ, τ 1,τ2 is a positive design parameter;
step 5, the specific process of the self-adaptive event trigger controller capable of ensuring the convergence of the system fixed time is as follows:
Step 5.1, the novel fixed time convergence criterion is:
For nonlinear systems X (0) =x 0, f (·) is a nonlinear function of the neighborhood around 0, and the origin is assumed to be the system balance point; for any real number α, β, γ >0, μ >1,0< r <1, iota=e κt, there is a positive definite function V (x) that satisfies the following lyapunov condition:
then the nonlinear system is stable for a fixed time and the adjustment time T f is:
step 5.2 to facilitate the design of the adaptive event triggered controller, an auxiliary function is constructed:
based on equation (21) and the event trigger mechanism, the event trigger controller is designed to:
Wherein: e T(t)＝uT (t) -u (t) are measurement errors in event-triggered control; 0.ltoreq.eta <1, omega >0, theta >0, epsilon >0 being a design parameter.
2. A method of fixed time adaptive event triggered control of a piezoelectric micropositioning stage according to claim 1,
Parameters Γ,γ,τ1,Λ1,τ2,Λ2,l1,l2,N1,N2,dm1,dm2,p11,p12,p21,p22,q1,q2,s1,s2,π1,π2,m1,m2,k1,k2,λ1,λ2,μ,r,η,ω,θ,ε,κ to be designed are selected according to the following principles:
1) The selection of the parameter l 1,l2 in the composite observer requires a guaranteed matrix Is a strict Hulvitz matrix, i.e. the selection of l 1,l2 has great influence on the stability of the system; for this system of piezoelectric micropositioning stages, the parameter of l 1,l2 is chosen as l 1＝1.88,l2 = 0.99; the parameters for parameter N 1,N2,dm1,dm2 related to the interference observations in the composite observer are selected as: n 1＝0.1,N2＝0.05,dm1＝dm2 = 1;
2) The parameters p11,p12,p21,p22,q1,q2,s1,s2,π1,π2,m1,m2 in the command filter are chosen to ensure that the output of the command filter is a high quality differential signal; the comparison of the filtering effects of the filter under different parameters is made available according to the command, for which the parameters are chosen as p11＝p12＝5,p21＝p22＝5,q1＝q2＝1,s1＝s2＝100,π1＝p2＝3,m1＝m2＝1.2;
3) The principle of parameter selection in the virtual control law, the compensation signal and the self-adaptive law is to ensure the stability of the piezoelectric micro-positioning platform system and the convergence of the self-adaptive parameters, so the parameters are selected as ：k1＝0.07,k2＝1.5,λ1＝0.1,λ2＝1,μ＝1021/1001,r＝997/1001,Γ＝[0.185,0.002,0.03,0.02]T,γ＝0.4,τ1＝0.05,Λ1＝0.045I7,τ2＝0.1,Λ2＝0.001I7;
4) The selection of the controller parameters under the event triggering mechanism needs to ensure that the system can avoid the gano phenomenon, so the parameters are selected as follows: η=0.0001, ω=0.0001, θ=0.001, ε=0.0001, and κ=0.001.
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