CN106066602A - The implementation method of motor servo system positioner based on expansion error symbol integration robust - Google Patents

Abstract

The present invention relates to electromechanical servo control field, disclose a kind of motor servo system positioner (RISEE) based on expansion error symbol integration robust, belong to electromechanical servo control field.The present invention chooses dc rotating machine positional servosystem as object of study, establishes the nonlinear model of total disturbance of consideration system；By introducing, the external disturbance existed for system based on the robust item expanding error signal integration and Unmarried pregnancy etc. are uncertain has good robustness to designed controller；Motor servo system robust position controller based on expansion error symbol integration designed by the present invention is full-state feedback device, and the position output of motor servo system can be made to have asymptotic tracking performance, and i.e. when the time tends to infinite, tracking error is zero；Controller parameter designed by the present invention is easily dimmable and controls input voltage continuously, is more conducive to apply in engineering reality.

Description

The implementation method of motor servo system positioner based on expansion error symbol integration robust

Technical field

The present invention relates to electromechanical servo control field, watch in particular to a kind of motor based on expansion error symbol integration robust The implementation method of dress system positioner.

Background technology

Response is fast, transmission efficiency is high, easy to maintenance and the energy obtains the outstanding advantages such as convenient owing to having for motor servo system, It is widely used in the key areas such as industry and national defence, such as machine tool feed, rocket gun servo system, robot etc..Along with these are led The development in territory and the continuous progress of technical merit, in the urgent need to high performance motor servo system as support, tradition is based on linearly The control performance that change method obtains gradually can not meet system requirements.There is many model uncertainties in motor servo system, including Parameter uncertainty (such as change, the viscosity friction coefficient etc. that changes with temperature and abrasion of load quality) and uncertain Non-linear (such as outer interference etc.), these probabilistic existence may the desired control performance of severe exacerbation, even make based on being System controller designed by nominal plant model is unstable, therefore becomes the major obstacle of development Dynamic matrix control device.

Usually, Self Adaptive Control can effectively be estimated unknown constant parameter and can improve its tracking accuracy, but when system suffers Big may be unstable when not modeling disturbance.Tradition robust controller, such as sliding mode controller, can be effectively improved whole closed loop System is to not modeling the robustness of disturbance, but controller input can produce jitter phenomenon, is unfavorable in engineering reality application； As automatic disturbance rejection controller (ADRC) can effectively carry out feedforward compensation to big disturbance present in system, but the ADRC proposed Method can only guarantee the tracking error bounded of system.As a whole, Self Adaptive Control and robust control have the excellent of each of which Shortcoming.The Bin Yao of Purdue Univ-West Lafayette USA teaches the team's all uncertainties for nonlinear system, it is proposed that a kind of mathematics Prove strict nonlinear adaptive robust control (ARC) theoretical frame.Its team is based primarily upon mission nonlinear mathematical model and sets Meter gamma controller, for parameter uncertainty, the on-line parameter being designed correctly estimates strategy, to improve the tracing property of system Energy；Uncertain non-linear to contingent outer interference etc., suppressed by strong nonlinear gain feedback control.Due to by force Nonlinear gain feedback control often leads to stronger conservative (i.e. High Gain Feedback), has certain difficulty in engineering uses, and And potential in system big do not model disturbance the tracking performance of system may be made to be deteriorated.In order to compensate disturbing when ARC designs Dynamic, there is scholar to devise ARC method for designing based on extended state observer, and demonstrate from theoretical and experimental results and carried The controller gone out can make system have good tracking performance.But, Nonlinear Design method set forth above can only be true The tracking error bounded of insurance system, such performance may be difficult to meet requirement in the occasion of actual requirements for high precision.This is had Scholar propose robust control based on error symbol integration (RISE) method the system that there is matching disturbance be can ensure that its with Track error goes to zero when stable state, but this controller design method is relative complex and can only ensure whole system half overall situation gradually Near stable.The most appropriate designing can guarantee that whole system asymptotically stable in the large and simple controller are still research at present Focus.

In summary, the weak point of the control strategy of existing motor servo system mainly have following some:

1. tradition sliding mode controller acts on system and can make control input generation jitter phenomenon；

2. automatic disturbance rejection controller (ADRC) can only guarantee the tracking error bounded of system；

There is High Gain Feedback phenomenon in the most traditional adaptive robust control (ARC).There is high-gain in tradition adaptive robust control The problem of feedback, namely reduces tracking error by increase feedback oscillator.But High Gain Feedback is easily by measuring influence of noise And the high frequency of activating system dynamically and then the tracking performance of system may be reduced, even result in system unstable；

The most traditional adaptive robust control (ARC) is to there is parameter uncertainty and the nonlinear system of uncertainty can only simultaneously Ensure tracking error bounded (i.e. ensureing that tracking error, in the range of a bounded, does not ensures that tracking error goes to zero).Pass The adaptive robust control of system is to there is parameter uncertainty and the nonlinear system of uncertainty can only guarantee the tracking of system simultaneously Error bounded, such performance may be difficult to meet requirement in the occasion of actual requirements for high precision.

5. the design of robust control based on error symbol integration (RISE) device is relative complex and can only ensure whole system half overall situation Asymptotic Stability.

Summary of the invention

The present invention solves that tradition sliding mode controller acts on system and control input can be made to produce jitter phenomenon, automatic disturbance rejection controller (ADRC) can only guarantee that the tracking error bounded of system, traditional adaptive robust control (ARC) exist High Gain Feedback phenomenon And to there is parameter uncertainty and the nonlinear system of uncertainty can only ensure that tracking error bounded (i.e. ensures to follow the tracks of simultaneously Error, in the range of a bounded, does not ensures that tracking error goes to zero), it is simultaneously based on the robust control of error symbol integration (RISE) device design is relative complex and can only ensure the globally asymptotically stable problem of whole system half, proposes a kind of based on expansion The motor servo system robust position controller of error symbol integration.

The present invention solves that the technical scheme that the problems referred to above are taked is as follows:

The implementation method of motor servo system positioner based on expansion error symbol integration robust, comprises the following steps:

2, the implementation method of motor servo system positioner based on expansion error symbol integration robust, it is characterised in that bag Include following steps:

Step one, set up the mathematical model of electric machine position servo system:

In formula (1), J is the rotary inertia of load；Y is the angular displacement of load；K_fFor torque error constant；U is defeated for controlling Enter voltage；For the non-linear friction model that can model, whereinRepresent different friction level,Represent different Shape function vector is used for describing the impact of various non-linear friction,Wherein B is viscosity friction coefficient；It is to include outer interference and do not model the uncertain item of friction；

For dc rotating machine servosystem, defined parameters collection θ=[θ₁,θ₂]^T, wherein θ₁=J/K_f, θ₂=B/K_fRepresent The known nominal value of systematic parameter；Definition system state variables is

The nonlinear model characterized by formula (1), then the state space form of mission nonlinear model can be expressed as:

${\stackrel{\·}{x}}_1=x_2$ (2)

${\mathrm{\θ}}_1{\stackrel{\·}{x}}_2=u-{\mathrm{\θ}}_2{x}_2+f(x,t)$

F (x, t)=d (x, t)/K in formula (2)_fThe modeling indeterminate in real system and parameter error shadow is included for total disturbance Ring；

Assume 1: system mode x₁、x₂Can survey；

Assume 2: total disturbance f (x, t) smooth enough andWherein δ is known constant；

Step 2, for the state equation in formula (2), design motor servo system robust based on expansion error symbol integration Positioner, it specifically comprises the following steps that

Step 2 (one), define one group of variable being similar to switch function and be:

$z_2={\stackrel{\·}{z}}_1+k_1{z}_1,r={\stackrel{\·}{z}}_2+k_2{z}_2---\left(3\right)$

Z in formula (3)₁=x₁-x_1dFor the tracking error of system, k₁、k₂For positive feedback oscillator.Formula (3) draws Enter error signal r of an expansion to obtain extra design freely；

Step 2 (two), design Nonlinear Robust Controller input u so that motor servo system has asymptotic tracking performance

According to formula (3), expansion error signal r can arrange and be:

$r={\stackrel{\·}{x}}_2-{\stackrel{\·\·}{x}}_{1d}+(k_1+k_2)z_2-k_1^2{z}_1---\left(4\right)$

Based on system state equation (2), can obtain:

$\begin{array}{c}{\mathrm{\θ}}_{1}r=u-{\mathrm{\θ}}_1{\stackrel{\·\·}{x}}_{1d}-{\mathrm{\θ}}_2{\stackrel{\·}{x}}_{1d}+f(x,t)\\ +({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)z_2-{\mathrm{\θ}}_1{k}_1^2{z}_1+{\mathrm{\θ}}_2{k}_1{z}_1\end{array}---\left(5\right)$

According to the structure of formula (5), the Nonlinear Robust Controller Design of motor servo system is:

U=u_a+u_s+u_n,u_a=θ^TY_d,

$u_s=-\mathrm{\μ},\mathrm{\μ}=k_r{z}_2+{\∫}_0^t{k}_r{k}_2{z}_{2}dv,---\left(6\right)$

$Y_d={\[{\stackrel{\·\·}{x}}_{1d},{\stackrel{\·}{x}}_{1d}\]}^T$

k_rFor positive feedback gain；u_aFor Feed forward Compensating Control Law based on model；u_sIt is used for ensureing name for nonlinear robust control rule The stability of justice system；u_nFor Robust Control Law based on expansion error symbol r integration, its disturbance being used for processing time-varying, u_n's Value will be given in below step；

Control law in (6) is brought in (5), can obtain:

$\begin{array}{c}{\mathrm{\θ}}_{1}r=-k_r{z}_2-{\∫}_0^t{k}_r{k}_3{z}_{2}dv+u_n+f(x,t)\\ +({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)z_2-({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)z_1\end{array}---\left(7\right)$

Formula (7) is carried out differential can obtain:

$\begin{array}{c}{\mathrm{\θ}}_1\stackrel{\·}{r}=-k_{r}r+{\stackrel{\·}{u}}_n+\stackrel{\·}{f}(x,t)\\ -\[k_2({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)+({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)\]z_2\\ +k_1({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)z_1+({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)r\end{array}---\left(8\right)$

Can design Robust Control Law un according to formula (8) is:

${\stackrel{\·}{u}}_n=-\mathrm{\δ}\mathrm{sgn}\left(r\right)---\left(9\right)$

Wherein sign (r) is defined as:

Owing to signal r is unknown, for the sgn (r) in computing formula (9), defined function g (t) is:

$\begin{array}{c}g\left(t\right)={\∫}_0^{t}r\left(v\right)dv\\ =z_2\left(t\right)-z_2\left(0\right)+k_2{\∫}_0^t{z}_2\left(v\right)dv\end{array}---\left(11\right)$

Due to r (t)=lim_τ→0(g (t)-g (t-τ))/τ, τ can be chosen for the sampling time, and can be seen that according to (11) only needs It is to be understood that symbol sgn (r) of r, therefore have only to know that g (t) increases or reduces and be obtained with sgn (r), wherein Sgn (r)=sgn (g (t)-g (t-τ))；

Step 3, regulation parameter τ (τ ＞ 0), k₁(k₁＞ 0), k₂(k₂＞ 0) and k_r(k_r＞ 0), thus guarantee Whole system is stable, and makes position output y (t) of electro-hydraulic position servo system follow the tracks of desired position command y_d。

The invention has the beneficial effects as follows: the present invention chooses dc rotating machine positional servosystem as object of study, establishes and examines The nonlinear model of total disturbance of worry system；Designed controller is by introducing robust item pin based on expansion error signal integration The external disturbance that there is system and Unmarried pregnancy etc. are uncertain has good robustness；Designed by the present invention based on The motor servo system robust position controller of expansion error symbol integration is full-state feedback device, and can make motor servo system The position output of system has asymptotic tracking performance, and i.e. when the time tends to infinite, tracking error is zero；Control designed by the present invention Device parameter is easily dimmable and controls input voltage continuously, is more conducive to apply in engineering reality.

As long as should be appreciated that all combinations of aforementioned concepts and the extra design described in greater detail below are at such structure Think the most conflicting in the case of can be viewed as the part of subject matter of the disclosure.It addition, theme required for protection All combinations be considered as the part of subject matter of the disclosure.

The foregoing and other aspect that can be more fully appreciated with from the following description in conjunction with accompanying drawing present invention teach that, embodiment and Feature.Feature and/or the beneficial effect of other additional aspect such as illustrative embodiments of the present invention will show in the following description See, or by the practice according to the detailed description of the invention that present invention teach that is learnt.

Accompanying drawing explanation

Accompanying drawing is not intended to drawn to scale.In the accompanying drawings, each identical or approximately uniform ingredient illustrated in each figure Can be indicated by the same numeral.For clarity, in each figure, the most each ingredient is the most labeled.Now, By by example embodiment that various aspects of the invention are described in reference to the drawings, wherein:

Fig. 1 is the dc rotating machine positional servosystem schematic diagram that the present invention is considered.

Fig. 2 is motor servo system robust position controller principle signal based on expansion error symbol integration and flow chart；

Fig. 3 is that controller designed by the present invention (identifying with RISEE in figure) and conventional PID controllers are (with PID mark in figure Know) the time dependent curve synoptic diagram of tracking error of the lower system of effect respectively.

Fig. 4 is the actual control input time dependent curve synoptic diagram of u of electric machine position servo system.

Detailed description of the invention

In order to know more about the technology contents of the present invention, especially exemplified by specific embodiment and coordinate institute's accompanying drawings to be described as follows.

Each side the most with reference to the accompanying drawings to describe the present invention, the embodiment illustrated shown in the drawings of many.The disclosure Embodiment must not be intended to include all aspects of the invention.Should be appreciated that multiple design presented hereinbefore and embodiment, with And describe in more detail below those design and embodiment can in many ways in any one is implemented, this is because Design disclosed in this invention and embodiment are not limited to any embodiment.It addition, aspects more disclosed by the invention can be single Solely use, or otherwise any appropriately combined use with disclosed by the invention.

In conjunction with Fig. 1 to Fig. 2, present embodiment is described, the motor servo based on expansion error symbol integration robust of present embodiment The implementation method of system position controller specifically comprises the following steps that

Step one, setting up the mathematical model of electric machine position servo system, the present invention with dc rotating machine (as shown in Figure 1) is Example, the kinematical equation that can obtain system according to Newton's second law is:

In formula (1), J is the rotary inertia of load；Y is the angular displacement of load；K_fFor torque error constant；U is defeated for controlling Enter voltage；For the non-linear friction model that can model, whereinRepresent different friction level,Represent different Shape function vector is used for describing the impact of various non-linear friction, and the present invention, in order to improve the intelligibility of controller design, The re-examination card controller robustness to Unmarried pregnancy, thus simplify the compensation part of controller, thus use linear friction model, I.e.Wherein B is viscosity friction coefficient；For the uncertain item such as outer interference and the friction that do not models.

Design for making controller is more extensive, for dc rotating machine servosystem, defined parameters collection θ=[θ₁,θ₂]^T, Wherein θ₁=J/K_f, θ₂=B/K_fRepresent the known nominal value of systematic parameter；Definition system state variables isThe nonlinear model characterized by formula (1), then the state space form of mission nonlinear model is permissible It is expressed as:

${\stackrel{\·}{x}}_1=x_2$ (2)

${\mathrm{\θ}}_1{\stackrel{\·}{x}}_2=u-{\mathrm{\θ}}_2{x}_2+f(x,t)$

F (x, t)=d (x, t)/K in formula (2)_fThe modeling indeterminate in real system and parameter error shadow is included for total disturbance Ring.

Assume 1: system mode x₁、x₂Can survey；

Assume 2: total disturbance f (x, t) smooth enough andWherein δ is known constant.

In following controller design, it is assumed that 2 are applied with some constraints to not modeling disturbance.Although friction is typically modeled as Discontinuous function can cause assuming that 2 is somewhat conservative, but do not have which executor can produce discontinuous power and compensate discontinuously The impact of frictional force, therefore still uses some continuous print friction models, it is assumed that 2 meet when System design based on model device designs Practical situation.The design object of controller is to make position export x₁Follow the tracks of ideal trajectory x that expectation is followed the tracks of as much as possible_1d=y_d(t)。

Step 2, for the state equation in formula (2), design motor servo system robust based on expansion error symbol integration Positioner, it specifically comprises the following steps that

Step 2 (one), define one group of variable being similar to switch function and be:

$z_2={\stackrel{\·}{z}}_1+k_1{z}_1,r={\stackrel{\·}{z}}_2+k_2{z}_2---\left(3\right)$

Z in formula (3)₁=x₁-x_1dFor the tracking error of system, k₁、k₂For positive feedback oscillator.We are in formula (3) In introduce error signal r of an expansion to obtain extra design freely.Significantly, since the error letter of expansion Number r depends on accelerationInformation so that it can not be surveyed, be used merely to here assist following controller to design.

Step 2 (two), design Nonlinear Robust Controller input u so that motor servo system has asymptotic tracking performance.

According to formula (3), expansion error signal r can arrange and be:

$r={\stackrel{\·}{x}}_2-{\stackrel{\·\·}{x}}_{1d}+(k_1+k_2)z_2-k_1^2{z}_1---\left(4\right)$

Based on system state equation (2), we can obtain:

$\begin{array}{c}{\mathrm{\θ}}_{1}r=u-{\mathrm{\θ}}_1{\stackrel{\·\·}{x}}_{1d}-{\mathrm{\θ}}_2{\stackrel{\·}{x}}_{1d}+f(x,t)\\ +({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)z_2-{\mathrm{\θ}}_1{k}_1^2{z}_1+{\mathrm{\θ}}_2{k}_1{z}_1\end{array}---\left(5\right)$

According to the structure of formula (5), the Nonlinear Robust Controller of motor servo system can be designed as:

U=u_a+u_s+u_n,u_a=θ^TY_d,

$u_s=-\mathrm{\μ},\mathrm{\μ}=k_r{z}_2+{\∫}_0^t{k}_r{k}_2{z}_{2}dv,---\left(6\right)$

$Y_d={\[{\stackrel{\·\·}{x}}_{1d},{\stackrel{\·}{x}}_{1d}\]}^T$

k_rFor positive feedback gain；u_aFor Feed forward Compensating Control Law based on model；u_sIt is used for ensureing name for nonlinear robust control rule The stability of justice system；u_nFor Robust Control Law based on expansion error symbol r integration, its disturbance being used for processing time-varying, u_n's Value will be given in following design procedure.

Control law in (6) is brought in (5), and we can obtain:

$\begin{array}{c}{\mathrm{\θ}}_{1}r=-k_r{z}_2-{\∫}_0^t{k}_r{k}_3{z}_{2}dv+u_n+f(x,t)\\ +({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)z_2-({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)z_1\end{array}---\left(7\right)$

Formula (7) is carried out differential can obtain:

$\begin{array}{c}{\mathrm{\θ}}_1\stackrel{\·}{r}=-k_{r}r+{\stackrel{\·}{u}}_n+\stackrel{\·}{f}(x,t)\\ -\[k_2({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)+({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)\]z_2\\ +k_1({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)z_1+({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)r\end{array}---\left(8\right)$

Robust Control Law u can be designed according to formula (8)_nFor:

${\stackrel{\·}{u}}_n=-\mathrm{\δ}\mathrm{sgn}\left(r\right)---\left(9\right)$

Wherein sign (r) is defined as:

Owing to signal r is unknown, for the sgn (r) in computing formula (9), defined function g (t) is:

$\begin{array}{c}g\left(t\right)={\∫}_0^{t}r\left(v\right)dv\\ =z_2\left(t\right)-z_2\left(0\right)+k_2{\∫}_0^t{z}_2\left(v\right)dv\end{array}---\left(11\right)$

Due to r (t)=lim_τ→0(g (t)-g (t-τ))/τ, τ can be chosen for the sampling time, can be seen that us according to (11) Having only to know symbol sgn (r) of r, therefore we have only to know that g (t) increases or reduces and are obtained with sgn (r), Wherein sgn (r)=sgn (g (t)-g (t-τ)), so, it is thus achieved that sgn (r) is not required to accelerationInformation, thus compare Obtain r easier.

Step 3, regulation parameter τ (τ ＞ 0), k₁(k₁＞ 0), k₂(k₂＞ 0) and k_r(k_r＞ 0), thus guarantee Whole system is stable, and makes position output y (t) of electro-hydraulic position servo system be accurately tracked by desired position command y_d。

In the disclosure, Lyapunov equation is selected to analyze based on the electric machine position servo system under controller (6) effect steady Qualitative:

Theoretical 1: choose sufficiently large feedback oscillator k₁、k₂、k_rSo that matrix Λ positive definite defined below, then proposition Control law (6) is able to ensure that all signal boundeds of whole closed loop motor servo, and can obtain Global Asymptotic tracking performance, i.e. when Z during t → ∞₁→0.Λ is defined as:

$\mathrm{\Λ}=\left[\begin{array}{ccc}{k}_1& -\frac{1}{2}& -\frac{1}{2}{a}_2\\ -\frac{1}{2}& k_2& -\frac{1-a_1}{2}\\ -\frac{1}{2}{a}_2& -\frac{1-a_1}{2}& k_3\end{array}\right]---\left(12\right)$

Wherein:

$a_1=k_2({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)+({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)---\left(13\right)$

$a_2=k_1({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)---\left(14\right)$

k₃=k_r-(θ₁k₁+θ₁k₂-θ₂) (15)

Choosing Lyapunov equation is:

$V=\frac{1}{2}{z}_1^2+\frac{1}{2}{z}_2^2+\frac{1}{2}{\mathrm{\θ}}_1{r}^2---\left(16\right)$

About the time, formula (16) is carried out derivation can obtain:

$\stackrel{\·}{V}=z_1{\stackrel{\·}{z}}_1+z_2{\stackrel{\·}{z}}_2+{\mathrm{\θ}}_{1}r\stackrel{\·}{r}---\left(17\right)$

Formula (3) and (8) are substituted into formula (17), can obtain:

$\begin{array}{c}\stackrel{\·}{V}=z_1(z_2-k_1{z}_1)+z_2(r-k_2{z}_2)\\ +r\{-k_{r}r+{\stackrel{\·}{u}}_n+\stackrel{\·}{f}(x,t)+({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)r-\\ \[k_2({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)+({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)\]z_2+k_1({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)z_1\}\end{array}---\left(18\right)$

(18) conversion further can be obtained:

$\begin{array}{c}\stackrel{\·}{V}\≤-k_1{z}_1^2-k_1{z}_2^2+z_1{z}_2+z_{2}r\\ -k_r{r}^2+r{\stackrel{\·}{u}}_n+r\stackrel{\·}{f}(x,t)+({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)r^2-\\ \[k_2({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)+({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)\]z_{2}r+k_1({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)z_{1}r\end{array}---\left(19\right)$

By(19) can be arranged further obtain:

$\begin{array}{c}\stackrel{\·}{V}\≤\[k_r-({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)\]r^2\\ -k_1{z}_1^2-k_2{z}_2^2-+z_1{z}_2+z_{2}r-a_1{z}_{2}r+a_2{z}_{1}r\end{array}---\left(20\right)$

It is positive definite matrix according to the Λ defined in formula (12), formula (20) conversion further can be obtained:

$\stackrel{\·}{V}\≤-z^T\mathrm{\Λ}z\≤-{\mathrm{\λ}}_{min}\left(\mathrm{\Λ}\right)(z_1^2+z_2^2+r^2)\stackrel{\mathrm{\Δ}}{=}-W---\left(21\right)$

In formula (21), z is defined as z=[z₁,z₂,r]^T；λ_min(Λ) it is the minimal eigenvalue of matrix Λ.

V ∈ L can be obtained according to formula (21)_∞And W ∈ L₂, synchronous signal z bounded.Therefore, it can draw x and control System input u bounded.Based on z₁、z₂And the dynamical equation of r, the time-derivative bounded of W can be obtained, therefore W unanimously connects Continuous.Thus, W → 0 as t → ∞ can be obtained according to Barbalat lemma, theoretical 1 is i.e. proven.

Effect below in conjunction with an instantiation aforementioned embodiments of this disclosure illustrates.

Motor servo system parameter is: inertia load parameter J=0.5kg m²；Torque error constant K_f=4N m/V；Viscosity Coefficient of friction B=1.6N m s/rad；Disturb d (t)=sin (t) N m outside time-varying, choose δ=1；The position that system expectation is followed the tracks of Instruction is curve x_1d(t)=sin (π t) [1-exp (-t³)]rad。

The parameter of the controller designed by the present invention is chosen for: τ=0.2ms, k₁=300, k₂=60 and k_r=20；PID is controlled Device parameter processed is chosen for: P gain k_P=1000, I gain k_I=600, D gain k_D=2.

Controller action effect: Fig. 3 is the controller (identifying with RISEE in figure) designed by the present invention and traditional PID control The time dependent curve synoptic diagram of tracking error of the lower system of device (identifying with PID in figure) effect respectively, can from figure Going out, under under the controller action designed by the present invention, the tracking error of system is significantly less than PID controller effect, the tracking of system is by mistake Difference and steady track error level off to 0, so that its tracking performance obtains the biggest raising.

Fig. 4 is the control input time dependent curve synoptic diagram of u of electric machine position servo system, it can be seen that this Control input signal obtained by invention is continuous and regular, is conducive to applying in engineering reality

Although the present invention is disclosed above with preferred embodiment, so it is not limited to the present invention.The technical field of the invention Middle tool usually intellectual, without departing from the spirit and scope of the present invention, when being used for a variety of modifications and variations.Therefore, originally The protection domain of invention is when being as the criterion depending on those as defined in claim.

Claims (1)

1. an implementation method for motor servo system positioner based on expansion error symbol integration robust, its feature exists In, comprise the following steps:

Step one, set up the mathematical model of electric machine position servo system:

In formula (1), J is the rotary inertia of load；Y is the angular displacement of load；K_fFor torque error constant；U is defeated for controlling Enter voltage；For the non-linear friction model that can model, whereinRepresent different friction level,Represent different Shape function vector is used for describing the impact of various non-linear friction,Wherein B is viscosity friction coefficient；It is to include outer interference and do not model the uncertain item of friction；

For dc rotating machine servosystem, defined parameters collection θ=[θ₁,θ₂]^T, wherein θ₁=J/K_f, θ₂=B/K_fRepresent The known nominal value of systematic parameter；Definition system state variables is

The nonlinear model characterized by formula (1), then the state space form of mission nonlinear model can be expressed as:

${\stackrel{\·}{x}}_1=x_2$

(2)

${\mathrm{\θ}}_1{\stackrel{\·}{x}}_2=u-{\mathrm{\θ}}_2{x}_2+f(x,t)$

F (x, t)=d (x, t)/K in formula (2)_fThe modeling indeterminate in real system and parameter error shadow is included for total disturbance Ring；

Assume 1: system mode x₁、x₂Can survey；

Assume 2: total disturbance f (x, t) smooth enough andWherein δ is known constant；

Step 2, for the state equation in formula (2), design motor servo system robust based on expansion error symbol integration Positioner, it specifically comprises the following steps that

Step 2 (one), define one group of variable being similar to switch function and be:

$z_2={\stackrel{\·}{z}}_1+k_1{z}_1,r={\stackrel{\·}{z}}_2+k_2{z}_2---\left(3\right)$

Z in formula (3)₁=x₁-x_1dFor the tracking error of system, k₁、k₂For positive feedback oscillator.Formula (3) draws Enter error signal r of an expansion to obtain extra design freely；

Step 2 (two), design Nonlinear Robust Controller input u so that motor servo system has asymptotic tracking performance

According to formula (3), expansion error signal r can arrange and be:

$r={\stackrel{\·}{x}}_2-{\stackrel{\·\·}{x}}_{1d}+(k_1+k_2)z_2-k_1^2{z}_1---\left(4\right)$

Based on system state equation (2), can obtain:

$\begin{array}{c}{\mathrm{\θ}}_{1}r=u-{\mathrm{\θ}}_1{\stackrel{\·\·}{x}}_{1d}-{\mathrm{\θ}}_2{\stackrel{\·\·}{x}}_{1d}+f(x,t)\\ +({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)z_2-{\mathrm{\θ}}_1{k}_1^2{z}_1+{\mathrm{\θ}}_2{k}_1{z}_1\end{array}---\left(5\right)$

According to the structure of formula (5), the Nonlinear Robust Controller Design of motor servo system is:

U=u_a+u_s+u_n,u_a=θ^TY_d,

$u_s=-\mathrm{\μ},\mathrm{\μ}=k_r{z}_2+{\∫}_0^t{k}_r{k}_2{z}_{2}dv,---\left(6\right)$

$Y_d={\[{\stackrel{\·\·}{x}}_{1d},{\stackrel{\·}{x}}_{1d}\]}^T$

k_rFor positive feedback gain；u_aFor Feed forward Compensating Control Law based on model；u_sIt is used for ensureing name for nonlinear robust control rule The stability of justice system；u_nFor Robust Control Law based on expansion error symbol r integration, its disturbance being used for processing time-varying, u_n's Value will be given in below step；

Control law in (6) is brought in (5), can obtain:

$\begin{array}{c}{\mathrm{\θ}}_{1}r=-k_r{z}_2-{\∫}_0^t{k}_r{k}_3{z}_{2}dv+u_n+f(x,t)\\ +({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)z_2-({\mathrm{\θ}}_1{k}_1^2+{\mathrm{\θ}}_2{k}_1)z_1\end{array}---\left(7\right)$

Formula (7) is carried out differential can obtain:

$\begin{array}{c}{\mathrm{\θ}}_1\stackrel{\·}{r}=-k_{r}r+{\stackrel{\·}{u}}_n+\stackrel{\·}{f}(x,t)\\ -\[k_2({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)+({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_1{k}_2)\]z_2\\ +k_1({\mathrm{\θ}}_1{k}_1^2-{\mathrm{\θ}}_2{k}_1)z_1+({\mathrm{\θ}}_1{k}_1+{\mathrm{\θ}}_1{k}_2-{\mathrm{\θ}}_2)r\end{array}---\left(8\right)$

Robust Control Law u can be designed according to formula (8)_nFor:

${\stackrel{\·}{u}}_n=-\mathrm{\δ}\mathrm{sgn}\left(r\right)---\left(9\right)$

Wherein sign (r) is defined as:

Owing to signal r is unknown, for the sgn (r) in computing formula (9), defined function g (t) is:

$\begin{array}{c}g\left(t\right)={\∫}_0^{t}r\left(v\right)dv\\ =z_2\left(t\right)-z_2\left(0\right)+k_2{\∫}_0^t{z}_2\left(v\right)dv\end{array}---\left(11\right)$

Due to r (t)=lim_τ→0(g (t)-g (t-τ))/τ, τ can be chosen for the sampling time, and can be seen that according to (11) only needs It is to be understood that symbol sgn (r) of r, therefore have only to know that g (t) increases or reduces and be obtained with sgn (r), wherein Sgn (r)=sgn (g (t)-g (t-τ))；

Step 3, regulation parameter τ, τ ＞ 0, k₁、k₁＞ 0；k₂、k₂＞ 0 and k_r、k_r＞ 0, thus guarantee whole System stability, and make position output y (t) of electro-hydraulic position servo system follow the tracks of desired position command y_d。