CN106292279A - Electric machine position servo systems by output feedback control method based on nonlinear observer - Google Patents

Abstract

The invention discloses a kind of output feedback ontrol method of electric machine position servo system based on nonlinear observer, belong to motor servo control field, the method comprises the following steps: set up the mathematical model of electric machine position servo system；Design Nonlinear Observer, the output feedback controller based on nonlinear observer of design electric machine position servo system.The present invention at only location status it is known that and in the case of Velocity-acceleration the unknown, it is provided that a kind of output feedback ontrol method of electric machine position servo system based on nonlinear observer, decrease hardware cost, be conducive in Practical Project apply；Controller designed by the present invention, has taken into full account the non-linear friction characteristic of system and outer interference etc., and ensures that system mode tends to balance state in finite time, improves the tracking performance of system.

Description

Electric machine position servo systems by output feedback control method based on nonlinear observer

Technical field

The present invention relates to electric machine position servo system regions, a kind of motor position based on nonlinear observer is watched Dress system output feedback ontrol method.

Background technology

Motor servo system has that response is fast, easy to maintenance, transmission efficiency is high and the energy obtains the outstanding advantages such as convenient, It is widely used in each key areas, such as robot, lathe, Aero-Space etc..Along with the fast development in these fields, to motor The requirement of servosystem tracking performance is more and more higher, and the performance of system is closely related with the design of controller.Motor servo system System is a typical nonlinear system, can face many modeling uncertainties and include that structure is not during design controller Definitiveness and unstructured uncertainty, these factors may the desired control performance of severe exacerbation, cause undesirable control Precision processed, produces limit cycles oscillations and even makes designed controller unstable, so that the design of controller becomes difficulty.

At present for the control of motor servo system, method based on classical three close-loop control is still industry and national defence Main method, it is based on linear control theory, the most successively design current ring, speed ring and position ring, each ring Control strategy mostly uses pid correction and modification thereof.But it is as the continuous progress of industry and national defence technical merit, tradition Three close-loop control method based on linear theory the most gradually can not meet the high performance demands of system, becomes limiting motor servosystem One of bottleneck factor of development.In order to improve the tracking performance of electric system, many advanced gamma controllers are ground Study carefully, such as Robust Adaptive Control, adaptive robust control, sliding formwork control etc..But, the full shape used in all said methods State feedback, say, that in motor control, except needs position signalling, in addition it is also necessary to speed and acceleration signal. But during many is applied, understand owing to reducing the needs of cost, only positional information.Additionally, serious measurement noise would generally Pollute the speed and acceleration signal surveyed, and then deteriorate the full-state feedback device realizing performance.Therefore, in the urgent need to setting Count the most practical nonlinear object feedback control strategy.

Summary of the invention

It is an object of the invention to provide the output feedback of a kind of electric machine position servo system based on nonlinear observer Control method.

The technical scheme realizing the object of the invention is: a kind of electric machine position servo system based on nonlinear observer defeated Go out feedback, comprise the following steps:

Step 1, sets up the mathematical model of electric machine position servo system；

Step 2, designs Nonlinear Observer；

Step 3, the output feedback controller based on nonlinear observer of design electric machine position servo system.

Compared with prior art, the remarkable advantage of the present invention is:

(1) present invention at only location status it is known that and in the case of Velocity-acceleration the unknown, it is provided that a kind of based on non-thread Property observer the output feedback ontrol method of electric machine position servo system, reduce hardware cost, advantageously in actual work Journey is applied.

(2) controller designed by the present invention, has taken into full account the non-linear friction characteristic of system and outer interference etc., and And ensure that system mode tends to balance state in finite time, improve the tracking performance of system.

Accompanying drawing explanation

Fig. 1 is the output feedback ontrol method flow of present invention electric machine position servo based on nonlinear observer system Figure.

Fig. 2 is motor servo system schematic diagram of the present invention.

Fig. 3 is two kinds of controller track following instruction schematic diagrams.

Fig. 4 is system mode x under controller action designed by the present invention₂Estimated value time history plot.

Fig. 5 is two kinds of controller tracking error time history plots.

Fig. 6 is its control input time history plot of the controller designed by the present invention.

Detailed description of the invention

In conjunction with Fig. 1, Fig. 2, the output feedback of a kind of based on nonlinear observer the electric machine position servo system of the present invention Control method, comprises the following steps:

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

Electric dynamic according to Newton's second law and simplification motor is proportional component, the motion of electric machine position servo system Equation is:

$m\stackrel{\·\·}{y}=k_{f}u-B\stackrel{\·}{y}-F_f\left({\stackrel{\·}{y}}_d\right)+\mathrm{\Δ}---\left(1\right)$

In formula (1), m is inertia load parameter, and y is inertia load displacement, k_fFor torque error constant, u is the control of system System input, B is viscosity friction coefficient,For the non-linear friction model that can model,For speed command, y_dRefer to for position Order, Δ is the uncertain item such as outer interference and the friction that do not models.

Choosing continuous static friction model is:

$F_{fnd}\left(\stackrel{\·}{y}\right)=l_1\tanh \left(l_2{\stackrel{\·}{y}}_d\right)+l_3\[\tanh \left(l_4{\stackrel{\·}{y}}_d\right)-\tanh \left(l_5{\stackrel{\·}{y}}_d\right)\]---\left(2\right)$

L in formula (2)₁、l₂、l₃、l₄、l₅It is known constant；The feature of this continuous static friction model is as follows: 1. this rubs Wiping model can be micro-and about origin symmetry about Time Continuous；2. Coulomb friction characteristic available expressionTable Levy；3. static friction coefficient can use l₁+l₃Value carry out approximate representation；4. expression formulaCan characterize Stribeck effect.

Choosing state variable is: x=[x₁,x₂]^T, then the kinematical equation of electric machine position servo system can be converted into shape State equation form:

$\begin{array}{c}{\stackrel{\·}{x}}_1=x_2-\frac{B}{m}{x}_1\\ {\stackrel{\·}{x}}_2=\frac{k_f}{m}u-\frac{1}{m}{F}_{fnd}\left({\stackrel{\·}{y}}_d\right)+\frac{\mathrm{\Δ}}{m}\end{array}---\left(3\right)$

X in formula (3)₁Represent the displacement of inertia load, x₂Represent another unknowable kinestate.

Step 2, design Nonlinear Observer

To unknown state x₂Estimate, be firstly introduced into Coordinate Conversion system, introducing new state ξ:

ξ=x₂-k₁x₁ (4)

K in formula (4)₁For design parameter.Then to formula (4) the right and left differential simultaneously, and combinatorial formula (3) can obtain ξ is dynamically:

$\stackrel{\·}{\mathrm{\ξ}}=-k_1\mathrm{\ξ}+\frac{k_f}{m}u+k_1\frac{B}{m}{x}_1-k_1^2{x}_1-\frac{1}{m}{F}_{fnd}\left({\stackrel{\·}{y}}_d\right)+\frac{\mathrm{\Δ}}{m}---\left(5\right)$

According to equation (5), designing state observer is:

$\stackrel{\·}{\widehat{\mathrm{\ξ}}}=-k_1\widehat{\mathrm{\ξ}}+\frac{k_f}{m}u+k_1\frac{B}{m}{x}_1-k_1^2{x}_1-\frac{1}{m}{F}_{fnd}\left({\stackrel{\·}{y}}_d\right)---\left(6\right)$

In formula (6)It it is the estimated value of state ξ.

DefinitionFor the estimation difference of state observer, formula (5), (6) the dynamic side of estimation difference can be obtained Cheng Wei:

$\stackrel{\·}{\widetilde{\mathrm{\ξ}}}=-k_1\widetilde{\mathrm{\ξ}}-\frac{\mathrm{\Δ}}{m}---\left(7\right)$

Can obtain according to formula (7):

$\mathrm{\ξ}\left(t\right)\≤e^{-k_{1}t}\mathrm{\ξ}\left(0\right)+\frac{{\mathrm{\δ}}_d}{k}\[1-e^{-k_{1}t}\]---\left(8\right)$

δ in formula (8)_dIt it is a unknown constant.By adjusted design parameter k₁Estimation difference can be made to become in finite time In the least value, therefore state observer has good stable state observation performance.

Step 3, the output feedback controller based on nonlinear observer of design electric machine position servo system, it is concrete Step is as follows:

The target of controller design is to make the position output x of electric machine position servo system₁Follow the tracks of expectation as precisely as possible Position command x followed the tracks of_1d；

The output feedback controller based on nonlinear observer of design electric machine position servo system is as follows:

Defined variable is as follows:

$\begin{array}{c}{z}_1=x_1-x_{1d}\\ z_2=\widehat{\mathrm{\ξ}}-{\mathrm{\α}}_1\end{array}---\left(9\right)$

Wherein α₁For virtual controlling amount, design as follows:

$\begin{array}{c}{\mathrm{\α}}_1={\mathrm{\α}}_{1a}+{\mathrm{\α}}_{1s}\\ {\mathrm{\α}}_{1a}={\stackrel{\·}{x}}_{1d}+(\frac{B}{m}-k_1)x_1\\ {\mathrm{\α}}_{1s}={\mathrm{\α}}_{1s1}+{\mathrm{\α}}_{1s2}\\ {\mathrm{\α}}_{1s1}=-k_{s1}{z}_1\end{array}---\left(10\right)$

K in formula (10)_s1For design parameter,For speed command, α_1s2Meet following condition:

$\begin{array}{c}{z}_1({\mathrm{\α}}_{1s2}-\widetilde{\mathrm{\ξ}})\≤{\mathrm{\ϵ}}_1\\ z_1{\mathrm{\α}}_{1s2}\≤0\end{array}---\left(11\right)$

Wherein ε₁＞ 0 is a design parameter, provides the α of satisfied (11) at this_1s2A form

$\begin{array}{c}{h}_1\≥{\mathrm{\δ}}_{\mathrm{\ξ}}^2\\ {\mathrm{\α}}_{2s2}=-\frac{h_1}{2{\mathrm{\ϵ}}_1}{z}_1\end{array}---\left(12\right)$

Wherein δ ξ isThe upper bound

Output feedback controller based on nonlinear observer design is as follows:

$\begin{array}{c}u=u_a+u_s\\ u_a=-\frac{m}{k_f}{z}_1+\frac{m}{k_f}{k}_1\widehat{\mathrm{\ξ}}-\frac{B}{k_f}{k}_1{x}_1+\frac{m}{k_f}{k}_1^2{x}_1+\frac{1}{k_f}{F}_{fnd}\left({\stackrel{\·}{y}}_d\right)+\frac{m}{k_f}{\stackrel{\·}{\mathrm{\α}}}_{1c}\\ u_s=u_{s1}+u_{s2}\\ u_{s1}=-\frac{m}{k_f}{k}_{s2}{z}_2\end{array}---\left(13\right)$

Wherein k_s2For design parameter,For α₁Can calculating section in time-derivative.u_s2Meet following condition

$\begin{array}{c}{z}_2(u_{s2}+\frac{\∂{\mathrm{\α}}_1}{\∂x_1}\widetilde{\mathrm{\ξ}})\≤{\mathrm{\ϵ}}_2\\ z_2{u}_{s2}\≤0\end{array}---\left(14\right)$

Wherein ε₂＞ 0 is a design parameter.The α of satisfied (14) is given at this_1s2A form

$\begin{array}{c}{h}_2\≥\left|\frac{\∂{\mathrm{\α}}_1}{\∂x_1}\right|{\mathrm{\δ}}_{\mathrm{\ξ}}^2\\ u_{s2}=-\frac{h_2}{2{\mathrm{\ϵ}}_2}{z}_2\end{array}---\left(15\right)$

The stability of analysis electric machine position servo system:

According to the stability analysis of system in control theory, choosing Lyapunov Equation is:

$V_1=\frac{1}{2}{z}_1^2+\frac{1}{2}{z}_2^2---\left(16\right)$

Use lyapunov stability theory to carry out stability and prove, to (16) formula derivation, and by formula (10), (12), (13), (15) are brought into and can be obtained:

$V_2\left(t\right)\≤e^{-\mathrm{\λ}t}{V}_2\left(0\right)+\frac{{\mathrm{\ϵ}}_1+{\mathrm{\ϵ}}_2}{\mathrm{\λ}}\[1-e^{-\mathrm{\λ}t}\]---\left(17\right)$

Wherein λ is an arithmetic number, such that it is able to make system reach asymptotically stability.

Below in conjunction with specific embodiment, the invention will be further described.

Embodiment

The parameter value of motor servo system is as follows:

M=0.02kg m²,k_f=5N m V^-1, B=10N m rad-1 s^-1,l₁=0.1N m, l₂= 700s·rad^-1,l₃=0.06N m, l₄=15s rad^-1,l₅=1.5s rad^-1

The controller parameter k of the present invention₁=800, k_s1=2000, k_s2=500.

PID controller parameter is k_p=1000, k_i=50, k_d=0.1.

Position angle input signal

Fig. 3 is two kinds of controller track following instruction schematic diagrams.

Fig. 4 is system mode x under controller action designed by the present invention₂The time dependent curve of estimated value, from figure In can be seen that its estimated value, gradually close to nominal value, and fluctuates in certain limit near nominal value such that it is able to accurately Ground is by the state estimation of system out.

Fig. 5 is two kinds of time dependent curves of controller tracking error, it can be seen that designed by the present invention, controller is bright Show and be better than PID controller.

Fig. 6 is that its control of the controller designed by the present invention inputs time dependent curve, it can be seen that this Control input signal obtained by invention is continuous, is beneficial to apply in engineering reality.

Claims (4)

1. the output feedback ontrol method of an electric machine position servo system based on nonlinear observer, it is characterised in that bag Include following steps:

Step 1, sets up the mathematical model of electric machine position servo system；

Step 2, designs Nonlinear Observer；

Step 3, the output feedback controller based on nonlinear observer of design electric machine position servo system.

The output feedback ontrol method of electric machine position servo system based on nonlinear observer the most according to claim 1, It is characterized in that, step 1 particularly as follows:

Electric dynamic according to Newton's second law and simplification motor is proportional component, the equation of motion of electric machine position servo system For:

$m\stackrel{\·\·}{y}=k_{f}u-B\stackrel{\·}{y}-F_f\left({\stackrel{\·}{y}}_d\right)+\mathrm{\Δ}---\left(1\right)$

In formula (1), m is inertia load parameter, and y is inertia load displacement, k_fFor torque error constant, u is that the control of system is defeated Entering, B is viscosity friction coefficient,For the non-linear friction model that can model,For speed command, y_dFor position command, Δ is the uncertain item such as outer interference and the friction that do not models；

Choosing continuous static friction model is:

$F_{fnd}\left(\stackrel{\·}{y}\right)=l_1\tanh \left(l_2{\stackrel{\·}{y}}_d\right)+l_3\[\tanh \left(l_4{\stackrel{\·}{y}}_d\right)-\tanh \left(l_5{\stackrel{\·}{y}}_d\right)\]---\left(2\right)$

L in formula (2)₁、l₂、l₃、l₄、l₅Being known constant, the feature of this continuous static friction model is as follows: 1. this friction mould Type can be micro-and about origin symmetry about Time Continuous；2. Coulomb friction characteristic passes through expression formulaCharacterize； 3. static friction coefficient passes through l₁+l₃Value approximate representation；4. expression formulaStribeck can be characterized Effect；

Choosing state variable is: x=[x₁,x₂]^T, then the kinematical equation of electric machine position servo system can be converted into state side Journey form:

$\begin{array}{c}{\stackrel{\·}{x}}_1=x_2-\frac{B}{m}{x}_1\\ {\stackrel{\·}{x}}_2=\frac{k_f}{m}u-\frac{1}{m}{F}_{fnd}\left({\stackrel{\·}{y}}_d\right)+\frac{\mathrm{\Δ}}{m}\end{array}---\left(3\right)$

X in formula (3)₁Represent the displacement of inertia load, x₂Represent another unknowable kinestate.

The output feedback ontrol method of electric machine position servo system based on nonlinear observer the most according to claim 1, It is characterized in that, step 2 particularly as follows:

Design Nonlinear Observer is to unknown state x₂Estimate, be firstly introduced into Coordinate Conversion system, introducing new state ξ:

ξ=x₂-k₁x₁ (4)

K in formula (4)₁For design parameter, then to formula (4) the right and left differential simultaneously, and combinatorial formula (3) can obtain the dynamic of ξ State is:

$\stackrel{\·}{\mathrm{\ξ}}=-k_1\mathrm{\ξ}+\frac{k_f}{m}u+k_1\frac{B}{m}{x}_1-k_1^2{x}_1-\frac{1}{m}{F}_{fnd}\left({\stackrel{\·}{y}}_d\right)+\frac{\mathrm{\Δ}}{m}---\left(5\right)$

According to equation (5), designing state observer is:

$\stackrel{\·}{\widehat{\mathrm{\ξ}}}=-k_1\widehat{\mathrm{\ξ}}+\frac{k_f}{m}u+k_1\frac{B}{m}{x}_1-k_1^2{x}_1-\frac{1}{m}{F}_{fnd}\left({\stackrel{\·}{y}}_d\right)---\left(6\right)$

In formula (6)It it is the estimated value of state ξ；

DefinitionFor the estimation difference of state observer, formula (5), (6) dynamical equation that can obtain estimation difference is:

$\stackrel{\·}{\widetilde{\mathrm{\ξ}}}=-k_1\widetilde{\mathrm{\ξ}}-\frac{\mathrm{\Δ}}{m}---\left(7\right)$

Can obtain according to formula (7):

$\mathrm{\ξ}\left(t\right)\≤e^{-k_{1}t}\mathrm{\ξ}\left(0\right)+\frac{{\mathrm{\δ}}_d}{k}\[1-e^{-k_{1}t}\]---\left(8\right)$

δ in formula (8)_dIt it is a unknown constant；By adjusted design parameter k₁Estimation difference can be made to tend to very in finite time Little value, therefore state observer has good stable state observation performance.

The output feedback ontrol method of electric machine position servo system based on nonlinear observer the most according to claim 1, It is characterized in that, step 3 particularly as follows:

Defined variable is as follows:

$\begin{array}{c}{z}_1=x_1-x_{1d}\\ z_2=\widehat{\mathrm{\ξ}}-{\mathrm{\α}}_1\end{array}---\left(9\right)$

Wherein, x_1dThe position command followed the tracks of for expectation, α₁For virtual controlling amount, design as follows:

$\begin{array}{c}{\mathrm{\α}}_1={\mathrm{\α}}_{1a}+{\mathrm{\α}}_{1s}\\ {\mathrm{\α}}_{1a}={\stackrel{\·}{x}}_{1d}+(\frac{B}{m}-k_1)x_1\\ {\mathrm{\α}}_{1s}={\mathrm{\α}}_{1s1}+{\mathrm{\α}}_{1s2}\\ {\mathrm{\α}}_{1s1}=-k_{s1}{z}_1\end{array}---\left(10\right)$

K in formula (10)_s1For design parameter,For speed command, α_1s2Meet following condition:

$\begin{array}{c}{z}_1({\mathrm{\α}}_{1s2}-\widetilde{\mathrm{\ξ}})\≤{\mathrm{\ϵ}}_1\\ z_1{\mathrm{\α}}_{1s2}\≤0\end{array}---\left(11\right)$

Wherein ε₁＞ 0 is a design parameter, provides the α of satisfied (11) at this_1s2A form

$\begin{array}{c}{h}_1\≥{\mathrm{\δ}}_{\mathrm{\ξ}}^2\\ {\mathrm{\α}}_{2s2}=-\frac{h_1}{2{\mathrm{\ϵ}}_1}{z}_1\end{array}---\left(12\right)$

Wherein δ_ξIt isThe upper bound；

Output feedback controller based on nonlinear observer design is as follows:

$\begin{array}{c}u=u_a+u_s\\ u_a=-\frac{m}{k_f}{z}_1+\frac{m}{k_f}{k}_1\widehat{\mathrm{\ξ}}-\frac{B}{k_f}{k}_1{x}_1+\frac{m}{k_f}{k}_1^2{x}_1+\frac{1}{k_f}{F}_{fnd}\left({\stackrel{\·}{y}}_d\right)+\frac{m}{k_f}{\stackrel{\·}{\mathrm{\α}}}_{1c}\\ u_s=u_{s1}+u_{s2}\\ u_{s1}=-\frac{m}{k_f}{k}_{s2}{z}_2\end{array}---\left(13\right)$

Wherein k_s2For design parameter,For α₁Can calculating section in time-derivative；u_s2Meet following condition

$\begin{array}{c}{z}_2(u_{s2}+\frac{\∂{\mathrm{\α}}_1}{\∂x_1}\widetilde{\mathrm{\ξ}})\≤{\mathrm{\ϵ}}_2\\ z_2{u}_{s2}\≤0\end{array}---\left(14\right)$

Wherein ε₂＞ 0 is a design parameter；The α of satisfied (14) is given at this_1s2A form

$\begin{array}{c}{h}_2\≥\left|\frac{\∂{\mathrm{\α}}_1}{\∂x_1}\right|{\mathrm{\δ}}_{\mathrm{\ξ}}^2\\ u_{s2}=-\frac{h_2}{2{\mathrm{\ϵ}}_2}{z}_2\end{array}.---\left(15\right)$