CN104485866A - Motor indirect adaptive robust output feedback control method based on high-order slip-form differentiator - Google Patents

Abstract

The invention provides a motor indirect adaptive robust output feedback control method based on a high-order slip-form differentiator. The method comprises the following steps: step I: establishing a motor position servo system module; step II: designing a motor indirect adaptive robust output feedback controller based on the high-order slip-form differentiator; step III: regulating parameters in a robust controller based on finite time interference estimation and enabling the system to meet the control performance indexes. According to the motor indirect adaptive robust output feedback control method based on the high-order slip-form differentiator, the motor position servo system module is established according to the characteristics of the motor position servo system, the designed motor indirect adaptive robust output feedback controller based on the high-order slip-form differentiator is used for estimating the state of the system and designing the controller, so that the problems of parameter uncertainty and uncertain nonlinearity of the motor servo system can be effectively solved while the influences of measurement noises on the controller are avoided.

Description

Based on the motor indirect self-adaptive robust output feedback control method of High-Order Sliding Mode differentiator

Technical field

The present invention relates to a kind of control method, be specifically related to a kind of indirect self-adaptive robust output feedback control method of electric machine position servo system.

Background technology

Electric energy is directly changed into rectilinear motion mechanical energy by linear electric motors, without any need for the transmission device of intermediate conversion mechanism.There is the advantages such as starting thrust is large, transmission stiffness is high, dynamic response is fast, positioning precision is high, haul distance is unrestricted, be thus widely used in the industry.

But, be very difficult for motor servo system designs high performance controller, because designer runs into a lot of model uncertainties possibly, comprise the non-linear of the non-modeling such as parameter uncertainty and Uncertain nonlinear.These uncertain factors may the severe exacerbation control performance that can obtain, thus causes low control precision, and limit cycle is shaken, not even stability.For known non-linear, feedback linearization technical finesse can be passed through.But how accurately the Mathematical Modeling of no matter kinematic nonlinearity and parameter identification, all can not obtain the whole non-linear behavior of actual non linear system and definite parameter, and then perfectly compensate.All the time there is the parameter error that can not simulate with clear and definite function and non-modeling non-linear.

In order to improve the tracking performance of electric system, the gamma controller of many advanced persons is applied in electric system control, as Robust Adaptive Control, and adaptive robust control (ARC), sliding formwork control etc.

But all said methods all carry out Controller gain variations based on overall-finished housing, in motion control, not only need position signalling, also need speed and/or acceleration signal.But in many real systems, limit by mechanical structure, volume, weight and cost, often only positional information is known.In addition, even if speed and acceleration signal can obtain, also there is serious measurement noises, and then the performance that deterioration full-state feedback device can obtain.These practical problems existing in nonlinear Control application, result in PID and control still to be in leading position in Motor Control Field so far.But under the new demand in modern industry epoch, PID is more and more difficult to meet the high performance control day by day pursued.Therefore, in the urgent need to design nonlinear object feedback control strategy.In linear system, this problem can utilize separate design principle to solve, and namely to considerable controlled linear system, design point feedback controller and state observer just can obtain the output feedback controller of system respectively.But in non linear system, because separation principle is no longer set up, the Stabilization utilizing output feedack to realize system is exactly a very difficult problem, in recent years, the output-feedback control problem of non linear system obtains to be paid close attention to widely.System is only had to export to be how to realize control system under scalable condition calm to be the important problem of control theory one.

Summary of the invention

The present invention, when only having location status known, for parameter certainty simultaneous in electric machine position servo system and Uncertain nonlinear problem, proposes a kind of motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator.

Above-mentioned purpose of the present invention is realized by the technical characteristic of independent claims, and dependent claims develops the technical characteristic of independent claims with alternative or favourable mode.

For reaching above-mentioned purpose, the present invention proposes a kind of motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator, and its realization comprises the following steps:

Step one, set up electric machine position servo system model

According to Newton's second law, the kinetic model equation of motor inertia load is:

$m\stackrel{\·\·}{y}=k_{f}u-b\stackrel{\·}{y}-f(y,\stackrel{\·}{y},t)---\left(1\right)$

In formula, y represents angular displacement, and m represents inertia load, k _frepresent torque coefficient, u is Systematical control input, and b represents viscosity friction coefficient, and f represents other non-modeling interference, comprises non-linear friction, external disturbance and Unmarried pregnancy.

(1) formula is write as state space form, as follows:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2={\mathrm{\θ}}_{1}u-{\mathrm{\θ}}_2{x}_2-d(x,t)\end{array}\right.---\left(2\right)$

Wherein x=[x ₁, x ₂] ^trepresent the state vector of position and speed.Parameter θ=[θ ₁, θ ₂] ^t, wherein θ ₁=k _f/ m, θ ₂=b/m, d (x, t)=f/m table represents concentrates interference;

Below suppose always to set up:

Suppose 1: structural uncertainty θ meets:

$\mathrm{\θ}\∈{\mathrm{\Ω}}_{\mathrm{\θ}}\stackrel{\mathrm{\Δ}}{=}\{\mathrm{\θ}:{\mathrm{\θ}}_{\min }\≤\mathrm{\θ}\≤{\mathrm{\θ}}_{\max }\}---\left(3\right)$

Wherein θ _min=[θ _1min, θ _2min] ^t, θ _max=[θ _1max, θ _2max] ^t, they are all known, θ in addition _1min> 0, θ _2min> 0.

Suppose that 2:d (x, t) is bounded, namely

|d(x,t)|≤δ _d(4)

Wherein δ _dknown quantity;

Step 2, design motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator, concrete steps are as follows:

The projection adaptive law structure of step 2 (), tape speed restriction

Order represent the estimation of θ, represent the evaluated error of θ, namely define a projection function as follows:

Wherein ζ ∈ R ^p, Γ (t) ∈ R ^{p × p}a time dependent positive definite symmetric matrices, with represent Ω respectively _θinside and border, represent time other unit's normal vector;

For projection function (5) formula, in controling parameters estimation procedure, default self adaptation speed restriction be used, thus, define a saturation function as follows:

${\mathrm{sat}}_{{\stackrel{\·}{\mathrm{\θ}}}_M}=s_0\mathrm{\ζ},s_0=\left\{\begin{array}{cc}1& \left|\right|\mathrm{\ζ}\left|\right|\≤{\stackrel{\·}{\mathrm{\θ}}}_M\\ \frac{{\stackrel{\·}{\mathrm{\θ}}}_M}{\left|\right|\mathrm{\ζ}\left|\right|}& \left|\right|\mathrm{\ζ}\left|\right|>{\stackrel{\·}{\mathrm{\θ}}}_M\end{array}\right.---\left(6\right)$

Wherein be a limiting speed pre-set, lemma below summarizes the architectural characteristic of parameter estimation algorithm:

Lemma 1: suppose the self adaptation limiting speed using projection type adaptive law below and preset upgrade estimated parameter

$\stackrel{\·}{\widehat{\mathrm{\θ}}}={\mathrm{sat}}_{{\stackrel{\·}{\mathrm{\θ}}}_M}\left({\mathrm{Proj}}_{{\stackrel{\·}{\mathrm{\θ}}}_M}\left(\mathrm{\Γ\τ}\right)\right),\widehat{\mathrm{\θ}}\left(0\right)\∈{\mathrm{\Ω}}_{\mathrm{\θ}}---\left(7\right)$

Wherein τ is auto-adaptive function, and Γ (t) > 0 is that continuous print can micro-positive symmetrical adaptive rate matrix, and adaptive law thus, can obtain following ideal characterisitics:

P1) estimates of parameters is always at the Ω of known bounded _θin collection, namely for any t, always have thus can be obtained by hypothesis 1, ${\mathrm{\θ}}_{i\min }\≤{\widehat{\mathrm{\θ}}}_i\left(t\right)\≤{\mathrm{\θ}}_{i\max },i=\mathrm{1,2},\∀t.$

P2) ${\widetilde{\mathrm{\θ}}}^T[{\mathrm{\Γ}}^{-1}{\mathrm{Proj}}_{\widehat{\mathrm{\θ}}}\left(\mathrm{\Γ\τ}\right)-\mathrm{\τ}]\≤0,\∀\mathrm{\τ}.$

P3) Parameters variation rule is uniformly bounded:

$\stackrel{\·}{\left|\right|\widehat{\mathrm{\θ}}}\left(t\right)\left|\right|\≤{\stackrel{\·}{\mathrm{\θ}}}_M,\∀t.;$

The High-Order Sliding Mode differentiator of step 2 (two), structure motor, estimates system unknown state

First, system model (2) is converted to following form again:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2={\widehat{\mathrm{\θ}}}_{1}u-{\widehat{\mathrm{\θ}}}_2{x}_2+D(x,t)\end{array}\right.---\left(8\right)$

Wherein $D(x,t)=-{\widetilde{\mathrm{\θ}}}_{1}u+{\widetilde{\mathrm{\θ}}}_2{x}_2-d(x,t)$ It is a broad sense interference;

By also be bounded with the known D (x, t) of hypothesis 2, that is:

Wherein θ _m=θ _max-θ _min,

A High-Order Sliding Mode differentiator is designed by formula (8), as follows:

$\begin{array}{c}{e}_0={\hat{x}}_1,e_1={\hat{x}}_2,e_2={\hat{x}}_3\\ {\stackrel{\·}{e}}_0=v_0=-{\mathrm{\λ}}_0{|e_0-x_1|}^{2/3}\mathrm{sgn}(e_0-x_1)+e_1\\ {\stackrel{\·}{e}}_1=v_1=-{\mathrm{\λ}}_1{|e_1-v_0|}^{1/2}\mathrm{sgn}(e_0-v_0)+e_2\\ {\stackrel{\·}{e}}_2=v_2=-{\mathrm{\λ}}_2\mathrm{sgn}(e_2-v_1)\end{array}---\left(10\right)$

Wherein x ₁, x ₂, x ₃represent outgoing position, speed and acceleration respectively, be respectively as x ₁, x ₂, x ₃estimated value, λ ₀, λ ₁, λ ₂for design parameter;

Lemma 1: there is a time T ₁, when time t is greater than time constant T ₁time, wherein ${\tilde{x}}_i={\hat{x}}_i-x_i,i=\mathrm{1,2,3};$

Step 2 (three), design motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator, comprise the steps:

Define one group of function as follows:

$\begin{array}{c}{z}_2={\stackrel{\·}{z}}_1+k_1{z}_1=x_2-x_{2\mathrm{eq}}\\ x_{2\mathrm{eq}}\stackrel{\mathrm{\Δ}}{=}{\stackrel{\·}{x}}_{1d}-k_1{z}_1\end{array}---\left(11\right)$

Wherein z ₁=x ₁-x _1dt () is output tracking error, k ₁>0 is a feedback oscillator;

Due to G (s)=z ₁(s)/z ₂(s)=1/ (s+k ₁) be a stable transfer function, allow z ₁very little or to level off to zero be exactly allow z ₂very little or level off to zero; Therefore, Controller gain variations is transformed into and allows z ₂little as far as possible or level off to zero;

To formula (11) differential and wushu (8) bring into, can obtain:

${\stackrel{\·}{z}}_2={\widehat{\mathrm{\θ}}}_{1}u-{\widehat{\mathrm{\θ}}}_2{x}_2-{\stackrel{\·}{x}}_{2\mathrm{eq}}+D(x,t)---\left(12\right)$

Controller based on state estimation is as follows:

$\begin{array}{c}u=(u_a+u_s)/{\widehat{\mathrm{\θ}}}_1,u_s=u_{s1}+u_{s2}\\ u_a={\stackrel{\·}{x}}_{2\mathrm{eq}}+{\widehat{\mathrm{\θ}}}_2{\hat{x}}_2\\ u_{s1}=-k_2({\hat{x}}_2-x_{2\mathrm{eq}})\end{array}---\left(13\right)$

Wherein k ₂> 0 is a feedback oscillator;

Wushu (13) brings formula (12) into, can obtain z ₂dynamical equation:

${\stackrel{\·}{z}}_2=-k_2({\hat{x}}_2-x_{2\mathrm{eq}})+u_{s2}+D(x,t)+{\widehat{\mathrm{\θ}}}_2{\tilde{x}}_2---\left(14\right)$

U _s2meet following condition:

z ₂{u _s2-D}≤σ ₁(15a)

z ₂u _s2≤0 (15b)

Wherein σ ₁> 0 is design parameter, the u of formula (15a) and formula (15b) _s2choose as follows: make g be as minor function

Wherein be the upper bound of D (x, t), design following u thus _s2:

$u_{s2}=-k_{s1}{z}_2\stackrel{\mathrm{\Δ}}{=}-g^2{z}_2/\left(4{\mathrm{\σ}}_1\right)---\left(17\right)$

Wherein k _s1it is a non-linear gain;

Step 2 (four), design indirect parameter method of estimation

Suppose 3: system does not exist interference, i.e. D=0;

Rewriting system dynamic model, can obtain following formula for parameter Estimation:

$y={\stackrel{\·}{x}}_2={\mathrm{\θ}}_{1}u-{\mathrm{\θ}}_2{x}_2---\left(18\right)$

Introduce a low pass filter H _f, and filter is applied in formula (18), can obtain:

$y_f=H_f\left[{\stackrel{\·}{x}}_2\right]={\mathrm{\θ}}_1{u}_f-{\mathrm{\θ}}_2{x}_{2f}---\left(19\right)$

Wherein u _f, x _2frepresent respectively and be input as u, x ₂filters H _foutput;

For carrying out parameter Estimation, following defined parameters vector sum regression function:

Can linear regression model (LRM) be obtained by (20):

Definition prediction output error wherein following predicated error model can be obtained by (21):

For the linear regression model (LRM) of static state, use the least-squares estimation algorithm having exponential forgetting factor and covariance and preset, for regressor and the corresponding unknown parameter vector of each group, definition adaptation rate matrix is as follows:

Wherein Γ (0)=Γ ^t(0) > 0, α>=0 is forgetting factor, be the Preset Time of covariance, namely work as λ _min(Γ (t))=ρ _l, ρ _lfor the minimum limit value of default Γ (t) also meets 0 < ρ _l< ρ ₀, λ _min(Γ (t)) is the minimal eigenvalue of Γ (t), and I is unit matrix, and now auto-adaptive function is as follows:

In formula, υ >=0 is normalization factor, and when υ=0, then auto-adaptive function is now non-standard auto-adaptive function.Containing in the parameter Estimation of forgetting factor, when recurrence device is not Persistent Excitation, Γ (t) may unbounded, i.e. λ _max(Γ (t)) → ∞ and cause the saturated of estimator, in order to prevent this from occurring, is revised as (23) formula:

ρ in formula _mthe upper bound of the norm of default Γ (t), and λ _max(Γ (t))≤ρ _m, and 0 < ρ _l< ρ < ρ _m;

Based on this improvement, ρ can be ensured _li < Γ (t) < ρ _mi.

As t>=T ₁time, d=0, if Persistent Excitation (PE) condition meets:

Then parameter converges to true value, namely as t → ∞,

Based on parameter update law defined above, following lemma can be obtained:

Lemma 2: by parameter update law (7), least square auto-adaptive function (23), formula (24) and predicated error model (22) can obtain: ε ∈ L ₂(0, ∞) ∩ L _∞(0, ∞), $\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}\&Element;L_2(0,\&infin;)\&cap;L_{\&infin;}(0,\&infin;),\widetilde{\mathrm{\&theta;}}\&Element;L_{\&infin;}(0,\&infin;).$

Step 3, regulate based on finite time Interference Estimation robust controller in the parameter k of u ₁, k ₂, a, v, λ ₀, λ ₁, λ ₂, σ ₁, make system meet Control performance standard.

From the above technical solution of the present invention shows that, the motor indirect self-adaptive robust output feedback control method based on high-order synovial membrane differentiator that the present invention proposes, for the feature of electric machine position servo system, establishes electric machine position servo system model; The motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator of the present invention's design, service system state to be estimated and for Controller gain variations, avoid measurement noises on the impact of controller simultaneously, effectively can solve parameter uncertainty and the Uncertain nonlinear problem of motor servo system, under above-mentioned disturbed condition, Systematical control precision meets performance index; This invention simplifies Controller gain variations, simulation result indicates its validity.

As long as should be appreciated that aforementioned concepts and all combinations of extra design described in further detail below can be regarded as a part for subject matter of the present disclosure when such design is not conflicting.In addition, all combinations of theme required for protection are all regarded as a part for subject matter of the present disclosure.

The foregoing and other aspect of the present invention's instruction, embodiment and feature can be understood by reference to the accompanying drawings from the following description more all sidedly.Feature and/or the beneficial effect of other additional aspect of the present invention such as illustrative embodiments will be obvious in the following description, or by learning in the practice of the embodiment according to the present invention's instruction.

Accompanying drawing explanation

Accompanying drawing is not intended to draw in proportion.In the accompanying drawings, each identical or approximately uniform part illustrated in each figure can represent with identical label.For clarity, in each figure, not each part is all labeled.Now, the embodiment of various aspects of the present invention also will be described with reference to accompanying drawing by example, wherein:

Fig. 1 is the typical structure schematic diagram of motor final controlling element.

Fig. 2 is the control strategy figure of the motor indirect self-adaptive robust output feedback control method that the present invention is based on high-order synovial membrane differentiator.

Fig. 3 is the schematic diagram of the additional interference curve of system.

Fig. 4 is the schematic diagram of controller input voltage u-curve.

Fig. 5 is the schematic diagram of parameter Estimation curve.

Fig. 6 is the schematic diagram of location estimation and evaluated error curve.

Fig. 7 is the schematic diagram of velocity estimation and evaluated error curve.

Fig. 8 is the schematic diagram of acceleration estimation and evaluated error curve.

Fig. 9 is the schematic diagram of CONTROLLER DESIGN and PID controller tracking error curve.

Embodiment

In order to more understand technology contents of the present invention, institute's accompanying drawings is coordinated to be described as follows especially exemplified by specific embodiment.

Each side with reference to the accompanying drawings to describe the present invention in the disclosure, shown in the drawings of the embodiment of many explanations.Embodiment of the present disclosure must not be intended to comprise all aspects of the present invention.Be to be understood that, multiple design presented hereinbefore and embodiment, and those designs described in more detail below and execution mode can in many ways in any one is implemented, this should be design disclosed in this invention and embodiment is not limited to any execution mode.In addition, aspects more disclosed by the invention can be used alone, or otherwisely anyly appropriately combinedly to use with disclosed by the invention.

Below in conjunction with accompanying drawing, present embodiment is described, described in present embodiment, a kind of concrete steps of the motor output feedback ontrol method based on state observation are as follows:

Step one, set up electric machine position servo system model;

Step 2, design motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator; And

Step 3, by regulate based on the parameter in the robust controller of finite time Interference Estimation, make system meet Control performance standard.

Shown in accompanying drawing 1-9, describe the exemplary implementation procedure of aforementioned embodiments in detail.

Step one, set up electric machine position servo system model, according to Newton's second law, the kinetic model equation of motor inertia load is:

$m\stackrel{\&CenterDot;\&CenterDot;}{y}=k_{f}u-b\stackrel{\&CenterDot;}{y}-f(y,\stackrel{\&CenterDot;}{y},t)---\left(1\right)$

In formula, y represents angular displacement, and m represents inertia load, k _frepresent torque coefficient, u is Systematical control input, and b represents viscosity friction coefficient, and f represents other non-modeling interference, such as non-linear friction, external disturbance and Unmarried pregnancy.

(1) formula is write as state space form, as follows:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2={\mathrm{\&theta;}}_{1}u-{\mathrm{\&theta;}}_2{x}_2-d(x,t)\end{array}\right.---\left(2\right)$

Wherein x=[x ₁, x ₂] ^trepresent the state vector of position and speed.Parameter θ=[θ ₁, θ ₂] ^t, wherein θ ₁=k _f/ m, θ ₂=b/m, d (x, t)=f/m represents concentrated interference.Generally, due to system parameters m, k _f, b is change, and system is structural uncertainty, although we do not know the specifying information of system, the general information of system can be known.In addition, system also has unstructured uncertainty d (x, t), and obviously it can not specify modeling, but the Unmarried pregnancy of system and disturb always bounded.Thus, below suppose always to set up:

Suppose 1: structural uncertainty θ meets:

$\mathrm{\&theta;}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&theta;}}\stackrel{\mathrm{\&Delta;}}{=}\{\mathrm{\&theta;}:{\mathrm{\&theta;}}_{\min }\&le;\mathrm{\&theta;}\&le;{\mathrm{\&theta;}}_{\max }\}---\left(3\right)$

Wherein θ _min=[θ _1min, θ _2min] ^t, θ _max=[θ _1max, θ _2max] ^t, they are all known, θ in addition _1min> 0, θ _2min> 0.

Suppose that 2:d (x, t) is bounded, namely

|d(x,t)|≤δ _d(4)

Wherein δ _dknown.

Step 2, the concrete steps designed based on the motor indirect self-adaptive robust output feedback control device of High-Order Sliding Mode differentiator are as follows:

The projection adaptive law structure of step 2 (), tape speed restriction

Order represent the estimation of θ, represent the evaluated error of θ, namely define a projection function as follows

Wherein ζ ∈ R ^p, Γ (t) ∈ R ^{p × p}a time dependent positive definite symmetric matrices, with represent Ω respectively _θinside and border, represent time other unit's normal vector.

For projection function (5) formula, in controling parameters estimation procedure, default self adaptation speed restriction be used.Thus, a saturation function is defined as follows:

${\mathrm{sat}}_{{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_M}=s_0\mathrm{\&zeta;},s_0=\left\{\begin{array}{cc}1& \left|\right|\mathrm{\&zeta;}\left|\right|\&le;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_M\\ \frac{{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_M}{\left|\right|\mathrm{\&zeta;}\left|\right|}& \left|\right|\mathrm{\&zeta;}\left|\right|>{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_M\end{array}\right.---\left(6\right)$

Wherein it is a limiting speed pre-set.Lemma below summarizes the architectural characteristic of the parameter estimation algorithm will used herein.

Lemma 1: suppose the self adaptation limiting speed using projection type adaptive law below and preset upgrade estimated parameter

$\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}={\mathrm{sat}}_{{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_M}\left({\mathrm{Proj}}_{{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_M}\left(\mathrm{\&Gamma;\&tau;}\right)\right),\widehat{\mathrm{\&theta;}}\left(0\right)\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&theta;}}---\left(7\right)$

Wherein τ is auto-adaptive function, and Γ (t) > 0 is that continuous print can micro-positive symmetrical adaptive rate matrix.Adaptive law thus, can obtain following ideal characterisitics:

P1) estimates of parameters is always at the Ω of known bounded _θin collection, namely for any t, always have thus can be obtained by hypothesis 1, ${\mathrm{\&theta;}}_{i\min }\&le;{\widehat{\mathrm{\&theta;}}}_i\left(t\right)\&le;{\mathrm{\&theta;}}_{i\max },i=\mathrm{1,2},\&ForAll;t.$

P2) ${\widetilde{\mathrm{\&theta;}}}^T[{\mathrm{\&Gamma;}}^{-1}{\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\left(\mathrm{\&Gamma;\&tau;}\right)-\mathrm{\&tau;}]\&le;0,\&ForAll;\mathrm{\&tau;}.$

P3) Parameters variation rule is uniformly bounded.

$\stackrel{\&CenterDot;}{\left|\right|\widehat{\mathrm{\&theta;}}}\left(t\right)\left|\right|\&le;{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_M,\&ForAll;t.$

The High-Order Sliding Mode differentiator of step 2 (two), structure motor, estimates system unknown state.

First, system model (2) can be write as following form again:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2={\widehat{\mathrm{\&theta;}}}_{1}u-{\widehat{\mathrm{\&theta;}}}_2{x}_2+D(x,t)\end{array}\right.---\left(8\right)$

Wherein $D(x,t)=-{\widetilde{\mathrm{\&theta;}}}_{1}u+{\widetilde{\mathrm{\&theta;}}}_2{x}_2-d(x,t)$ It is a broad sense interference.

By also be bounded with the known D (x, t) of hypothesis 2.Namely

Wherein θ _m=θ _max-θ _min,

A High-Order Sliding Mode differentiator can be designed by (8), as follows:

$\begin{array}{c}{e}_0={\hat{x}}_1,e_1={\hat{x}}_2,e_2={\hat{x}}_3\\ {\stackrel{\&CenterDot;}{e}}_0=v_0=-{\mathrm{\&lambda;}}_0{|e_0-x_1|}^{2/3}\mathrm{sgn}(e_0-x_1)+e_1\\ {\stackrel{\&CenterDot;}{e}}_1=v_1=-{\mathrm{\&lambda;}}_1{|e_1-v_0|}^{1/2}\mathrm{sgn}(e_0-v_0)+e_2\\ {\stackrel{\&CenterDot;}{e}}_2=v_2=-{\mathrm{\&lambda;}}_2\mathrm{sgn}(e_2-v_1)\end{array}---\left(10\right)$

Wherein x ₁, x ₂, x ₃represent outgoing position respectively, speed and acceleration, be respectively as x ₁, x ₂, x ₃estimated value, λ ₀, λ ₁, λ ₂for design parameter.

Lemma 1: there is a time T ₁, when time t is greater than time constant T ₁time, wherein ${\tilde{x}}_i={\hat{x}}_i-x_i,i=\mathrm{1,2,3}.$

Step 2 (three), the motor indirect self-adaptive robust output feedback control device designed based on High-Order Sliding Mode differentiator are as follows:

Define one group of function as follows

$\begin{array}{c}{z}_2={\stackrel{\&CenterDot;}{z}}_1+k_1{z}_1=x_2-x_{2\mathrm{eq}}\\ x_{2\mathrm{eq}}\stackrel{\mathrm{\&Delta;}}{=}{\stackrel{\&CenterDot;}{x}}_{1d}-k_1{z}_1\end{array}---\left(11\right)$

Wherein z ₁=x ₁-x _1dt () is output tracking error, k ₁> 0 is a feedback oscillator.Due to G (s)=z ₁(s)/z ₂(s)=1/ (s+k ₁) be a stable transfer function, allow z ₁very little or to level off to zero be exactly allow z ₂very little or level off to zero.Therefore, Controller gain variations is transformed into and allows z ₂little as far as possible or level off to zero.To formula (11) differential and wushu (8) bring into, can obtain:

${\stackrel{\&CenterDot;}{z}}_2={\widehat{\mathrm{\&theta;}}}_{1}u-{\widehat{\mathrm{\&theta;}}}_2{x}_2-{\stackrel{\&CenterDot;}{x}}_{2\mathrm{eq}}+D(x,t)---\left(12\right)$

Controller based on state estimation is as follows:

$\begin{array}{c}u=(u_a+u_s)/{\widehat{\mathrm{\&theta;}}}_1,u_s=u_{s1}+u_{s2}\\ u_a={\stackrel{\&CenterDot;}{x}}_{2\mathrm{eq}}+{\widehat{\mathrm{\&theta;}}}_2{\hat{x}}_2\\ u_{s1}=-k_2({\hat{x}}_2-x_{2\mathrm{eq}})\end{array}---\left(13\right)$

Wherein k ₂> 0 is a feedback oscillator.

Wushu (13) brings formula (12) into, can obtain z ₂dynamical equation:

${\stackrel{\&CenterDot;}{z}}_2=-k_2({\hat{x}}_2-x_{2\mathrm{eq}})+u_{s2}+D(x,t)+{\widehat{\mathrm{\&theta;}}}_2{\tilde{x}}_2---\left(14\right)$

U _s2meet following condition:

z ₂{u _s2-D}≤σ ₁(15a)

z ₂u _s2≤0 (15b)

Wherein σ ₁> 0 is a design parameter, how to choose the u meeting formula (15a) and formula (15b) _s2, provide an example here, as follows: to make g be as minor function

Wherein it is the upper bound of D (x, t).Design following u thus _s2

$u_{s2}=-k_{s1}{z}_2\stackrel{\mathrm{\&Delta;}}{=}-g^2{z}_2/\left(4{\mathrm{\&sigma;}}_1\right)---\left(17\right)$

Wherein k _s1it is a non-linear gain.

Step 2 (four), design indirect parameter algorithm for estimating

The main task of this part sets up a suitable parameter estimation algorithm, improves the availability of parameter Estimation.

Suppose 3: system does not exist interference, i.e. D=0.

Rewriting system dynamic model, can obtain following formula for parameter Estimation:

$y={\stackrel{\&CenterDot;}{x}}_2={\mathrm{\&theta;}}_{1}u-{\mathrm{\&theta;}}_2{x}_2---\left(18\right)$

Introduce a low pass filter H _f, and filter is applied in formula (18), can obtain

$y_f=H_f\left[{\stackrel{\&CenterDot;}{x}}_2\right]={\mathrm{\&theta;}}_1{u}_f-{\mathrm{\&theta;}}_2{x}_{2f}---\left(19\right)$

Wherein u _f, x _2frepresent respectively and be input as u, x ₂filters H _foutput.For carrying out parameter Estimation, following defined parameters vector sum regression function:

Linear regression model (LRM) can be obtained by (20)

Definition prediction output error wherein following predicated error model can be obtained by (21):

For the linear regression model (LRM) of this static state, have multiple different algorithm for estimating can be used for determining unknown parameter, we use is a kind of least-squares estimation algorithm having exponential forgetting factor and covariance and preset.For regressor and the corresponding unknown parameter vector of each group, the adaptation rate matrix that we can define is as follows:

Wherein Γ (0)=Γ ^t(0) > 0, α>=0 is forgetting factor be the Preset Time of covariance, namely work as λ _min(Γ (t))=ρ _l, ρ _lfor the minimum limit value of default Γ (t) also meets 0 < ρ _l< ρ ₀, λ _min(Γ (t)) is the minimal eigenvalue of Γ (t), and I is unit matrix.Now auto-adaptive function is as follows:

In formula, υ >=0 is normalization factor, and when υ=0, then auto-adaptive function is now non-standard auto-adaptive function.But in the enforcement of the parameter Estimation containing forgetting factor realizes, when recurrence device is not Persistent Excitation, Γ (t) may unbounded.I.e. λ _max(Γ (t)) → ∞ and cause the saturated of estimator.In order to prevent this from occurring, (23) formula is revised as:

ρ in formula _mthe upper bound of the norm of default Γ (t), and λ _max(Γ (t))≤ρ _m, and 0 < ρ _l< ρ < ρ _m.Based on this improvement, we can ensure ρ _li < Γ (t) < ρ _mi.

As t>=T ₁time, d=0.If Persistent Excitation (PE) condition meets:

Then parameter converges to true value, namely as t → ∞,

Based on parameter update law defined above, following lemma can be obtained:

Lemma 2: by parameter update law (7), least square auto-adaptive function (23), formula (24) and predicated error model (22) can obtain: ε ∈ L ₂(0, ∞) ∩ L _∞(0, ∞), $\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}\&Element;L_2(0,\&infin;)\&cap;L_{\&infin;}(0,\&infin;),\widetilde{\mathrm{\&theta;}}\&Element;L_{\&infin;}(0,\&infin;).$

Step 3, by regulating based on the parameter k of u in the robust controller of finite time Interference Estimation ₁, k ₂, α, v, λ ₀, λ ₁, λ ₂, σ ₁, make system meet Control performance standard.

In this example, be further advanced by and choose the stability of Lyapunov Equation to the controller of previous designs and system and carry out checking explanation.

Verification system stability:

Theorem 1: by High-Order Sliding Mode differentiator (10), motor indirect self-adaptive robust output feedback control device (13) based on High-Order Sliding Mode differentiator of design has following character:

A. at a time T ₁afterwards, the state that High-Order Sliding Mode differentiator is estimated is accurate, namely the Lyapunov Equation be defined as follows

$V_1=\frac{1}{2}{z}_2^2+\frac{1}{2}{\tilde{x}}_2^2---\left(27\right)$

Meet following inequality

$V\&le;\exp (-\mathrm{\&lambda;}(t-T_1))V\left(0\right)+\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&lambda;}}[1-\exp (-\mathrm{\&lambda;}(t-T_1))]\&le;,\&ForAll;t\&GreaterEqual;T_1.---\left(28\right)$

Prove: to formula (27) differential, and wushu (14) is brought into and can be obtained

${\stackrel{\&CenterDot;}{V}}_1=z_2{\stackrel{\&CenterDot;}{z}}_2+{\tilde{x}}_2{\stackrel{\&CenterDot;}{\tilde{x}}}_2=-k_2{z}_2^2+z_2(u_{s2}-D)+z_2({\mathrm{\&theta;}}_{2n}{\tilde{x}}_2-k_2{\tilde{x}}_2)---\left(29\right)$

Wushu (15a) is brought (29) into and can be obtained

$\stackrel{\&CenterDot;}{V}\&le;-k_2{z}_2^2+{\mathrm{\&sigma;}}_1=-\mathrm{\&lambda;V}+{\mathrm{\&sigma;}}_2---\left(30\right)$

Inequality (28) can be obtained to formula (30) two ends integration.Therefore controller is convergence, and system is stable.

Test under simulated environment below in conjunction with some concrete examples:

Get following parameter in simulations and modeling is carried out to system: m=0.01kgm ², k _f=5, b=1.25Ns/m.

Get controller parameter k ₁=300, k ₂=500, σ ₁=1 × 10 ⁵, v=22, α=2, λ ₀=5, λ ₁=5, λ ₂=6; θ _min=[0,0] ^t, θ _max=[200,1000] ^t, selected be away from the true value of parameter, to examine the effect of adaptive control laws.

Position angle input signal y=0.2sin (π t) [1-e ^-0.01t3], unit rad.

The added outer interference of system as shown in Figure 3.

Control law action effect:

Fig. 4 is controller input voltage u-curve under interference effect, and controller input voltage meets the input range of-10V ~+10V, realistic application.

Fig. 5 is parameter Estimation curve.

Fig. 6 is location estimation and evaluated error curve.

Fig. 7 is velocity estimation and evaluated error curve.

Fig. 8 is acceleration estimation and evaluated error curve.

Fig. 9 is CONTROLLER DESIGN and PID controller tracking error curve.

From above illustrated contrast, the method that the present invention proposes can estimate state value accurately under simulated environment, estimates system parameters more accurately.Compare traditional PID controller, the controller of the present invention's design can obtain good control precision.Simulation result shows that the method that the present embodiment proposes can meet performance index under parameter uncertainty and Uncertain nonlinear impact.

Although the present invention with preferred embodiment disclose as above, so itself and be not used to limit the present invention.Persond having ordinary knowledge in the technical field of the present invention, without departing from the spirit and scope of the present invention, when being used for a variety of modifications and variations.Therefore, protection scope of the present invention is when being as the criterion depending on those as defined in claim.

Claims (4)

1., based on a motor indirect self-adaptive robust output feedback control method for High-Order Sliding Mode differentiator, it is characterized in that, the realization of the method comprises the following steps:

Step one, set up electric machine position servo system model;

Step 2, design motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator; And

Step 3, by regulate based on the parameter in the robust controller of finite time Interference Estimation, make system meet Control performance standard.

2. the motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator according to claim 1, it is characterized in that, in abovementioned steps one, the structure of electric machine position servo system model comprises the following steps:

According to Newton's second law, the kinetic model equation of motor inertia load is:

In formula, y represents angular displacement, and m represents inertia load, k _frepresent torque coefficient, u is Systematical control input, and b represents viscosity friction coefficient, and f represents other non-modeling interference, comprises non-linear friction, external disturbance and Unmarried pregnancy.

(1) formula is write as state space form, as follows:

Wherein represent the state vector of position and speed.Parameter θ=[θ ₁, θ ₂] ^t, wherein θ ₁=k _f/ m, θ ₂=b/m, d (x, t)=f/m represents concentrated interference;

Below suppose always to set up:

Suppose 1: structural uncertainty θ meets:

Wherein θ _min=[θ _1min, θ _2min] ^t, θ _max=[θ _1max, θ _2max] ^t, they are all known, θ in addition _1min> 0, θ _2min> 0;

Suppose that 2:d (x, t) is bounded, namely

|d(x,t)|≤δ _d(4)

Wherein δ _dknown quantity.

3. the motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator according to claim 2, it is characterized in that, design the motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator in abovementioned steps two, concrete steps are as follows:

The projection adaptive law structure of step 2 (), tape speed restriction

Order represent the estimation of θ, represent the evaluated error of θ, namely define a projection function as follows:

Wherein ζ ∈ R ^p, Γ (t) ∈ R ^{p × p}a time dependent positive definite symmetric matrices, with represent Ω respectively _θinside and border, represent time other unit's normal vector;

For projection function (5) formula, in controling parameters estimation procedure, default self adaptation speed restriction be used, thus, define a saturation function as follows:

Wherein be a limiting speed pre-set, lemma below summarizes the architectural characteristic of parameter estimation algorithm:

Lemma 1: suppose the self adaptation limiting speed using projection type adaptive law below and preset upgrade estimated parameter

Wherein τ is auto-adaptive function, and Γ (t) > 0 is that continuous print can micro-positive symmetrical adaptive rate matrix, and adaptive law thus, can obtain following ideal characterisitics:

P1) estimates of parameters is always at the Ω of known bounded _θin collection, namely for any t, always have thus can be obtained by hypothesis 1,

P2)

P3) Parameters variation rule is uniformly bounded:

The High-Order Sliding Mode differentiator of step 2 (two), structure motor, estimates system unknown state

First, system model (2) is converted to following form again:

Wherein it is a broad sense interference;

By also be bounded with the known D (x, t) of hypothesis 2, that is:

Wherein

A High-Order Sliding Mode differentiator is designed by formula (8), as follows:

Wherein x ₁, x ₂, x ₃represent outgoing position, speed and acceleration respectively, be respectively as x ₁, x ₂, x ₃estimated value, λ ₀, λ ₁, λ ₂for design parameter;

Lemma 1: there is a time T ₁, when time t is greater than time constant T ₁time, wherein

Step 2 (three), design motor indirect self-adaptive robust output feedback control device based on High-Order Sliding Mode differentiator, comprise the steps:

Define one group of function as follows:

Wherein z ₁=x ₁-x _1dt () is output tracking error, k ₁> 0 is a feedback oscillator;

Due to G (s)=z ₁(s)/z ₂(s)=1/ (s+k ₁) be a stable transfer function, allow z ₁very little or to level off to zero be exactly allow z ₂very little or level off to zero; Therefore, Controller gain variations is transformed into and allows z ₂little as far as possible or level off to zero;

To formula (11) differential and wushu (8) bring into, can obtain:

Controller based on state estimation is as follows:

Wherein k ₂> 0 is a feedback oscillator;

Wushu (13) brings formula (12) into, can obtain z ₂dynamical equation:

U _s2meet following condition:

z ₂{u _s2-D}≤σ ₁(15a)

z ₂u _s2≤0 (15b)

Wherein σ ₁> 0 is design parameter, the u of formula (15a) and formula (15b) _s2choose as follows: make g be as minor function

Wherein be the upper bound of D (x, t), design following u thus _s2:

Wherein k _s1it is a non-linear gain;

Step 2 (four), design indirect parameter method of estimation

Suppose 3: system does not exist interference, i.e. D=0;

Rewriting system dynamic model, can obtain following formula for parameter Estimation:

Introduce a low pass filter H _f, and filter is applied in formula (18), can obtain:

Wherein u _f, x _2frepresent respectively and be input as u, x ₂filters H _foutput;

For carrying out parameter Estimation, following defined parameters vector sum regression function:

Can linear regression model (LRM) be obtained by (20):

Definition prediction output error wherein following predicated error model can be obtained by (21):

For the linear regression model (LRM) of static state, use the least-squares estimation algorithm having exponential forgetting factor and covariance and preset, for regressor and the corresponding unknown parameter vector of each group, definition adaptation rate matrix is as follows:

Wherein Γ (0)=Γ ^t(0) > 0, α>=0 is forgetting factor, be the Preset Time of covariance, namely work as λ _min(Γ (t))=ρ _l, ρ _lfor the minimum limit value of default Γ (t) also meets 0 < ρ _l< ρ ₀, λ _min(Γ (t)) is the minimal eigenvalue of Γ (t), and I is unit matrix, and now auto-adaptive function is as follows:

In formula, υ >=0 is normalization factor, and when υ=0, then auto-adaptive function is now non-standard auto-adaptive function.Containing in the parameter Estimation of forgetting factor, when recurrence device is not Persistent Excitation, Γ (t) may unbounded, i.e. λ _max(Γ (t)) → ∞ and cause the saturated of estimator, in order to prevent this from occurring, is revised as (23) formula:

ρ in formula _mthe upper bound of the norm of default Γ (t), and λ _max(Γ (t))≤ρ _m, and 0 < ρ _l< ρ < ρ _m;

Based on this improvement, ρ can be ensured _li < Γ (t) < ρ _mi.

As t>=T ₁time, d=0, if Persistent Excitation (PE) condition meets:

Then parameter converges to true value, namely as t → ∞,

Based on parameter update law defined above, following lemma can be obtained:

Lemma 2: by parameter update law (7), least square auto-adaptive function (23), formula (24) and predicated error model (22) can obtain: ε ∈ L ₂(0, ∞) ∩ L _∞(0, ∞), .

4. the motor indirect self-adaptive robust output feedback control method based on High-Order Sliding Mode differentiator according to claim 3, it is characterized in that, the realization of abovementioned steps three comprises: by regulating the parameter k based on u in the robust controller of finite time Interference Estimation ₁, k ₂, α, v, λ ₀, λ ₁, λ ₂, σ ₁, make system meet Control performance standard.