CN105739311A - Electromechanical servo system limitation control method based on preset echo state network - Google Patents

Abstract

The invention provides an electromechanical servo system limitation control method based on a preset echo state network. The method comprises the steps that a dynamic model of an electromechanical servo system is established, and the system state, sampling time and control parameters are initialized; linear processing is performed on nonlinear input saturation limitation in the system according to the differential mean value theorem so as to derive an electromechanical servo system model with unknown saturation; control system tracking error and a sliding mode surface are calculated based on a preset echo state network control method, and a new error variable is obtained to design a virtual control variable according to preset performance function conversion tracking error; and the virtual control variable is through a high-order differentiator and control input is deigned. The invention provides the high-order dynamic surface sliding mode control method capable of effectively compensating input and output limitation and enhancing neural network approximation performance so that stable and rapid tracking of the system can be realized, and great transient and steady performance can be guaranteed.

Description

Electromechanical servo system constrained control method based on default echo state network

Technical field

The invention belongs to the control method of electromechanical servo system, relate to a kind of electromechanical servo system constrained control method based on default echo state network, particular with input-bound and the control method exporting limited electromechanical servo system.

Background technology

Along with Power Electronic Technique, computer science, modern control theory, the fast development of material technology and stepping up of motor manufacturing technology level, at industrial control fields such as precise numerical control machine, industrial robot, electronics processing and detection equipment, laser process equipment, printing machinery, package packing machine, tailoring machinery, the production automations, non-linear electromechanical servo system is widely applied.The automatic control system that non-linear electromechanical servo system is is control object with mechanical parameters such as displacement, angle, power, torque, speed and acceleration.But, limited link, limited including input-bound and output, it is widely present in electromechanical servo system, frequently can lead to the transient state steady-state behaviour of control system and decline, or even unstability.Therefore, in controller design process, it is necessary to consider negative effect that limited link brings and compensate, how to realize electromechanical servo system quickly accurately control had become as a hot issue.

Default capabilities control method is applied relatively broad in output constrained system.In control process, it is proposed that one describes rate of convergence, the default capabilities function of maximum overshoot and steady-state error, and is used to conversion output error.The key problem in technology of the method is how to select default capabilities function that primal system is converted to error converting system, thus ensureing the transient state steady-state behaviour of system.For the saturated input of the unknown existed in system, traditional saturation compensation method is usually sets up saturated inversion model or approximate inverse model, and by estimating saturated bound parameter designing adaptive controller, to compensate saturated impact.But, in the nonlinear systems such as electromechanical servo system, saturated inversion model often not easily accurately obtains.Thus, based on Order Derivatives in Differential Mid-Value Theorem through line linearity so that it is become a simple time-varying system, it is to avoid ancillary relief.

Neutral net, due to its good approximation capability, is often used to estimate, be similar to unknown function., but feedforward neural network is essentially static network, can only realize static non linear mapping relations, is not suitable for the real-time identification of dynamical system.Recurrent neural network is a kind of dynamic network, can reflect system dynamic characteristic well, but owing to its training method is complicated, be rarely applied to reality. due to simplicity and the rapidity of echo state network training so that it is become the important channel solving a lot of problems.The maximum feature of echo state network is that hidden layer is made up of the intrinsic nerve unit of a large amount of (hundreds of to several thousand) partially connected, is called state reserve pool.Thus this network has extremely strong short term memory capacity and training algorithm is simple.But on control field, the application of echo state network is few, it is necessary to exploitation and research.

Summary of the invention

Cannot effective compensation input-bound in order to what overcome existing electromechanical servo system, do not consider transient performance, and the deficiency that static neural network Approximation effect is good etc., the present invention provides a kind of electromechanical servo system constrained control method based on default echo state network, consider the existence of input and output limitation problem, have employed echo state network and approach unknown function, it is achieved that limited electromechanical servo system tracing control, it is ensured that system has good transient state steady-state behaviour.

Technical scheme in order to solve the proposition of above-mentioned technical problem is as follows:

A kind of electromechanical servo system constrained control method based on default echo state network, comprises the following steps:

Step 1, sets up the dynamic model of electromechanical servo system, and process is as follows:

The dynamic model expression-form of 1.1 electromechanical servo systems is

$m\stackrel{\·\·}{x}+f(\stackrel{\‾}{x},t)+d(\stackrel{\‾}{x},t)=k_{0}v\left(u\right)---\left(1\right)$

Wherein, x is position；M is inertia；k₀It it is force constant；It it is state variable；It is frictional force；It is model a produced BOUNDED DISTURBANCES, derives from coupled characteristic, measure noise, electronic interferences and other uncertain factors；U is the control input voltage of motor；V (u), for saturated, is expressed as:

$v\left(u\right)=sat\left(u\right)=\left\{\begin{array}{cc}{v}_{max}\mathrm{sgn}\left(u\right),& \left|u\right|\≥v_{max}\\ u,& \left|u\right|\≤v_{max}\end{array}\right.---\left(2\right)$

Wherein sgn (u), for unknown nonlinear function；v_maxFor unknown parameter of saturation, meet v_max> 0；

1.2 definition x₁=x,Formula (1) is rewritten as

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=-\frac{f(\stackrel{\‾}{x},t)+d(\stackrel{\‾}{x},t)}{m}+\frac{k_0}{m}v\left(u\right)\\ y=x_1\end{array}\right.---\left(3\right)$

Wherein, y is system output trajectory；

Step 2, according to Order Derivatives in Differential Mid-Value Theorem, carries out linearization process by saturated for the non-linear input in system, derives the electromechanical servo system model saturated with the unknown, including following process；

2.1 pairs of saturated models carry out smooth treatment

$\begin{array}{c}g\left(u\right)=v_{max}\×\tanh \left(\frac{u}{v_{\max }}\right)\\ =v_{\max }\×\frac{e^{u/v_{\max }}-e^{u/v_{\max }}}{e^{u/v_{\max }}+e^{-u/v_{\max }}}\end{array}---\left(4\right)$

Then

V (u)=sat (u)=g (u)+dsat (u) (5)

Wherein, dsat (u) represents the error existed between smooth function and saturated model；

2.2 according to Order Derivatives in Differential Mid-Value Theorem, there is δ ∈ (0,1) and makes

$g\left(u\right)=g\left(u_0\right)+g_{u_{\mathrm{\ξ}}}(u-u_0)---\left(6\right)$

Whereinu₀∈(0,u)；

Select u₀=0, formula (6) is rewritten as

$g\left(u\right)=g_{u_{\mathrm{\ξ}}}u---\left(7\right)$

Formula (3), by formula (5) and formula (7), is rewritten as following equivalents by 2.3:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=a\left(\stackrel{\‾}{x}\right)+b\left(\stackrel{\‾}{x}\right)u\\ y=x_1\end{array}\right.---\left(8\right)$

Wherein

Step 3, calculating control system tracking error, sliding-mode surface and transformed error, process is as follows:

The tracking error of 3.1 definition control systems, sliding-mode surface is

$\left\{\begin{array}{c}e=y-y_d\\ s_1=e+\mathrm{\λ}\∫edt\end{array}\right.---\left(9\right)$

Wherein, y_dCan leading desired trajectory for second order, λ is constant, and λ > 0；

3.2 obtain new transformed error ε according to sliding-mode surface₁；

$\begin{array}{c}{\mathrm{\ϵ}}_1=\frac{1}{2}\ln \frac{\underset{\‾}{\mathrm{\α}}\left(t\right)+S\left({\mathrm{\ϵ}}_1\right)}{\stackrel{\‾}{\mathrm{\α}}\left(t\right)-S\left({\mathrm{\ϵ}}_1\right)}\\ =T_1(\frac{s_1\left(t\right)}{{\mathrm{\ρ}}_1\left(t\right)},\stackrel{\‾}{\mathrm{\α}}\left(t\right),\underset{\‾}{\mathrm{\α}}\left(t\right))\end{array}---\left(10\right)$

Wherein ρ₁T the expression formula of () is

ρ₁(t)=(ρ₀-ρ_∞)e^-lt+ρ_∞(11)

Parameter ρ₀> ρ_∞> 0 and l > 0；WithαT the derivative expressions of () is

ParameterSize and initially need design；Function S () expression formula is

$S\left(\mathrm{\ϵ}\right)=\frac{\stackrel{\‾}{\mathrm{\α}}\left(t\right)\exp \left(\mathrm{\ϵ}\right)-\underset{\‾}{\mathrm{\α}}\left(t\right)\exp (-\mathrm{\ϵ})}{\exp \left(\mathrm{\ϵ}\right)+\exp (-\mathrm{\ϵ})}---\left(13\right)$

Wherein, ε is transformed error variable；

The derivation of 3.3 pairs of formulas (10) obtains:

${\stackrel{\·}{\mathrm{\ϵ}}}_1=r_1(x_2-{\stackrel{\·}{y}}_d+\mathrm{\λ}e)+{\mathrm{\υ}}_1---\left(14\right)$

Wherein

3.4 design virtual controlling amounts

Wherein, k₁For constant, and k₁> 0；Function Q () is Nussbaum function, and selection expression formula is

WhereinAdaptive law be designed as

3.5 allow virtual controlling amountBy High-Order Sliding Mode differentiator

$\left\{\begin{array}{c}{\stackrel{\·}{\mathrm{\β}}}_{1,1}={\mathrm{\ω}}_{1,1}\\ {\mathrm{\ω}}_{1,1}=-{\mathrm{\μ}}_{1,1}{\left|{\mathrm{\β}}_{1,1}-{\stackrel{\‾}{\mathrm{\β}}}_1\right|}^{\frac{1}{2}}sign({\mathrm{\β}}_{1,1}-{\stackrel{\‾}{\mathrm{\β}}}_1)+{\mathrm{\β}}_{2,1}\\ {\stackrel{\·}{\mathrm{\β}}}_{2,1}={\mathrm{\ω}}_{2,1}\\ {\mathrm{\ω}}_{2,1}=-{\mathrm{\μ}}_{2,1}sign({\mathrm{\β}}_{2,1}-{\mathrm{\ω}}_{1,1})\end{array}\right.---\left(18\right)$

Wherein parameter μ_1,1> 0, μ_2,1> 0, β_1,1It it is virtual controlling amountBy the filtered variable that differentiator obtains；

Step 4, design con-trol device inputs, and process is as follows:

4.1 definition error variances

s₂=x₂-β_1,1(19)

4.2 obtain transformed error ε according to the error variance defined₂

$\begin{array}{c}{\mathrm{\ϵ}}_2=\frac{1}{2}\ln \frac{\underset{\‾}{\mathrm{\α}}\left(t\right)+S\left({\mathrm{\ϵ}}_2\right)}{\stackrel{\‾}{\mathrm{\α}}\left(t\right)-S\left({\mathrm{\ϵ}}_2\right)}\\ =T_2(\frac{s_2\left(t\right)}{{\mathrm{\ρ}}_2\left(t\right)},\stackrel{\‾}{\mathrm{\α}}\left(t\right),\underset{\‾}{\mathrm{\α}}\left(t\right))\end{array}---\left(20\right)$

Wherein ρ₂T the expression formula of () is

ρ₂(t)=(ρ₀-ρ_∞)e-^lt+ρ_∞(21) parameter ρ₀> ρ_∞> 0 and l > 0；WithαT the derivative expressions of () is for shown in such as formula (12)；Shown in function S () expression formula such as formula (13)；

Formula (20) derivation is obtained:

${\stackrel{\·}{\mathrm{\ϵ}}}_2=r_2\left(a\right(\stackrel{\‾}{x})+b(\stackrel{\‾}{x})u-{\stackrel{\·}{\mathrm{\β}}}_{1,1})+{\mathrm{\υ}}_2---\left(22\right)$

Wherein

4.3 approach the Nonlinear uncertainty being not directly availableDefine following neutral net

$f\left(\stackrel{\‾}{x}\right)=a\left(\stackrel{\‾}{x}\right)-{\stackrel{\·}{\mathrm{\β}}}_{1,1}=W^{*T}X\left(\stackrel{\‾}{x}\right)+{\mathrm{\η}}^{*}---\left(23\right)$

Wherein, W^*For ideal weight, η^*For neutral net perfect error value, meet | η^*|≤η_N,Expression formula is:

Wherein W_in, W_d, W_fbFor random value；U is controller input；For Gaussian function, expression formula is

$s_i\left(\stackrel{\‾}{x}\right)=\exp \[-\frac{-{(\stackrel{\‾}{x}-{\mathrm{\χ}}_i)}^T(\stackrel{\‾}{x}-{\mathrm{\χ}}_i)}{{\mathrm{\ι}}_i^2}\],i=1,2,...l---\left(25\right)$

WhereinIt is the output of hidden layer i-th node；χ_iIt is the center vector of i-th node Gaussian function, i.e. χ_i=[χ_i1,χ_i2,…χ_il]^T；l_iIt it is the width of i-th node Gaussian function；Y is neutral net output, and expression formula is

$y=G\left(W^{*T}X\right(\stackrel{\‾}{x}\left)\right)---\left(26\right)$

Selection of Function G=1；

4.4 design con-trol device input u:

Wherein,For ideal weight W^*Estimated value,For estimation difference η^*Estimated value.

4.5 design adaptive rates:

$\left\{\begin{array}{c}\stackrel{\·}{\hat{W}}=\stackrel{\·}{\tilde{W}}=\mathrm{\Γ}\[X\left(\stackrel{\‾}{x}\right)r_2{\mathrm{\ϵ}}_2-\mathrm{\σ}\hat{W}\]\\ {\stackrel{\·}{\widehat{\mathrm{\ϵ}}}}_N={\stackrel{\·}{\widetilde{\mathrm{\ϵ}}}}_N=\mathrm{\κ}\[r_2\left|{\mathrm{\ϵ}}_2\right|-\mathrm{\γ}\widehat{\mathrm{\η}}\]\end{array}\right.---\left(28\right)$

Wherein,Γ is adaptive gain matrix, σ, k, and γ is constant, and σ > 0, κ > 0, γ > 0.

The present invention is based on echo state network, default capabilities control method, it is contemplated that there is input, when exporting limited, the constrained control method of design electromechanical servo system, it is achieved the fast and stable of system is followed the tracks of, and ensures the transient state steady-state behaviour of control performance.

The technology of the present invention is contemplated that: can not survey for state, and with inputting, exporting limited electromechanical servo system, utilizes Order Derivatives in Differential Mid-Value Theorem to optimize saturated structures, it is proposed to based on the electromechanical servo system of saturated model.In conjunction with echo state network, default capabilities controls and high-order dynamic face sliding formwork controls, and designs a kind of constrained control method of electromechanical servo system.By Order Derivatives in Differential Mid-Value Theorem, make saturated continuously differentiable, approach unknown function again through echo state network, eliminate the ancillary relief that tradition is saturated.Further, utilize default capabilities control method transformed error variable, design virtual controlling amount further according to transformed error.Void controlled quentity controlled variable is obtained by higher differentiation device filtered variable and derivative thereof, improves conventional dynamic face stability and be subject to the deficiencies such as parameter impact, it is achieved that the tenacious tracking of system.The present invention provide one can effective compensation input, export limited, improve the high-order dynamic face sliding-mode control of neutral net approximation capability, it is achieved the stable of system is quickly followed the tracks of, it is ensured that have good transient state steady-state behaviour.

Advantages of the present invention is: avoids input, export the limited impact on system tracing control performance, utilizes dynamic neural network to approach Unknown Model indeterminate, it is achieved the tenacious tracking of system, reach good transient state steady-state behaviour.

Accompanying drawing explanation

Fig. 1 is the non-linear saturated schematic diagram of the present invention；

Fig. 2 is the schematic diagram of the tracking effect of the present invention；

Fig. 3 is the schematic diagram of the tracking error of the present invention；

Fig. 4 is the schematic diagram of the controller input of the present invention；

Fig. 5 is the schematic diagram of the estimation effect of the present invention；

Fig. 6 is the control flow chart of the present invention.

Detailed description of the invention

Below in conjunction with accompanying drawing, the present invention will be further described.

With reference to Fig. 1-Fig. 6, based on the electromechanical servo system constrained control method of default echo state network, comprise the following steps:

Step 1, sets up the dynamic model of electromechanical servo system, and process is as follows:

The dynamic model expression-form of 1.1 electromechanical servo systems is

$m\stackrel{\·\·}{x}+f(\stackrel{\‾}{x},t)+d(\stackrel{\‾}{x},t)=k_{0}v\left(u\right)---\left(1\right)$

Wherein, x is position；M is inertia；k₀It it is force constant；It it is state variable；It is frictional force；It is model a produced BOUNDED DISTURBANCES, derives from coupled characteristic, measure noise, electronic interferences and other uncertain factors；U is the control input voltage of motor；V (u), for saturated, is expressed as:

$v\left(u\right)=sat\left(u\right)=\left\{\begin{array}{cc}{v}_{max}\mathrm{sgn}\left(u\right),& \left|u\right|\≥v_{max}\\ u,& \left|u\right|\≤v_{max}\end{array}\right.---\left(2\right)$

Wherein sgn (u), for unknown nonlinear function；v_maxFor unknown parameter of saturation, meet v_max> 0；

1.2 definition x₁=x,Formula (1) is rewritten as

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=-\frac{f(\stackrel{\‾}{x},t)+d(\stackrel{\‾}{x},t)}{m}+\frac{k_0}{m}v\left(u\right)\\ y=x_1\end{array}\right.---\left(3\right)$

Wherein, y is system output trajectory；

Step 2, according to Order Derivatives in Differential Mid-Value Theorem, carries out linearization process by saturated for the non-linear input in system, derives the electromechanical servo system model saturated with the unknown, including following process；

2.1 pairs of saturated models carry out smooth treatment

$\begin{array}{c}g\left(u\right)=v_{max}\×\tanh \left(\frac{u}{v_{\max }}\right)\\ =v_{\max }\×\frac{e^{u/v_{\max }}-e^{u/v_{\max }}}{e^{u/v_{\max }}+e^{-u/v_{\max }}}\end{array}---\left(4\right)$

Then

V (u)=sat (u)=g (u)+dsat (u) (5)

Wherein, dsat (u) represents the error existed between smooth function and saturated model；

2.2 according to Order Derivatives in Differential Mid-Value Theorem, there is δ ∈ (0,1) and makes

$g\left(u\right)=g\left(u_0\right)+g_{u_{\mathrm{\ξ}}}(u-u_0)---\left(6\right)$

Whereinu₀∈(0,u)；

Select u₀=0, formula (6) is rewritten as

$g\left(u\right)=g_{u_{\mathrm{\ξ}}}u---\left(7\right)$

Formula (3), by formula (5) and formula (7), is rewritten as following equivalents by 2.3:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=a\left(\stackrel{\‾}{x}\right)+b\left(\stackrel{\‾}{x}\right)u\\ y=x_1\end{array}\right.---\left(8\right)$

Wherein

Step 3, calculating control system tracking error, sliding-mode surface and transformed error, process is as follows:

The tracking error of 3.1 definition control systems, sliding-mode surface is

$\left\{\begin{array}{c}e=y-y_d\\ s_1=e+\mathrm{\λ}\∫edt\end{array}\right.---\left(9\right)$

Wherein, y_dCan leading desired trajectory for second order, λ is constant, and λ > 0；

3.2 obtain new transformed error according to sliding-mode surface

$\begin{array}{c}{\mathrm{\ϵ}}_1=\frac{1}{2}\ln \frac{\underset{\‾}{\mathrm{\α}}\left(t\right)+S\left({\mathrm{\ϵ}}_1\right)}{\stackrel{\‾}{\mathrm{\α}}\left(t\right)-S\left({\mathrm{\ϵ}}_1\right)}\\ =T_1(\frac{s_1\left(t\right)}{{\mathrm{\ρ}}_1\left(t\right)},\stackrel{\‾}{\mathrm{\α}}\left(t\right),\underset{\‾}{\mathrm{\α}}\left(t\right))\end{array}---\left(10\right)$

Wherein ρ₁T the expression formula of () is

ρ₁(t)=(ρ₀ρρ_∞)e^-lt+ρ_∞(11)

Parameter ρ₀> ρ_∞> 0 and l > 0；WithαT the derivative expressions of () is

ParameterSize and initially need design；Function S () expression formula is

$S\left(\mathrm{\ϵ}\right)=\frac{\stackrel{\‾}{\mathrm{\α}}\left(t\right)\exp \left(\mathrm{\ϵ}\right)-\underset{\‾}{\mathrm{\α}}\left(t\right)\exp (-\mathrm{\ϵ})}{\exp \left(\mathrm{\ϵ}\right)+\exp (-\mathrm{\ϵ})}---\left(13\right)$

The derivation of 3.3 pairs of formulas (10) obtains:

${\stackrel{\·}{\mathrm{\ϵ}}}_1=r_1(x_2-{\stackrel{\·}{y}}_d+\mathrm{\λ}e)+{\mathrm{\υ}}_1---\left(14\right)$

Wherein

3.4 design virtual controlling amounts

Wherein, k₁For constant, and k₁> 0；Function Q () is Nussbaum function, and selection expression formula is

WhereinAdaptive law be designed as

3.5 allow virtual controlling amountBy High-Order Sliding Mode differentiator

$\left\{\begin{array}{c}{\stackrel{\·}{\mathrm{\β}}}_{1,1}={\mathrm{\ω}}_{1,1}\\ {\mathrm{\ω}}_{1,1}=-{\mathrm{\μ}}_{1,1}{\left|{\mathrm{\β}}_{1,1}-{\stackrel{\‾}{\mathrm{\β}}}_1\right|}^{\frac{1}{2}}sign({\mathrm{\β}}_{1,1}-{\stackrel{\‾}{\mathrm{\β}}}_1)+{\mathrm{\β}}_{2,1}\\ {\stackrel{\·}{\mathrm{\β}}}_{2,1}={\mathrm{\ω}}_{2,1}\\ {\mathrm{\ω}}_{2,1}=-{\mathrm{\μ}}_{2,1}sign({\mathrm{\β}}_{2,1}-{\mathrm{\ω}}_{1,1})\end{array}\right.---\left(18\right)$

Wherein parameter μ_1,1> 0, μ_2,1> 0, κ_1,1It it is virtual controlling amountBy the filtered variable that differentiator obtains；

Step 4, design con-trol device inputs, and process is as follows:

4.1 definition error variances

s₂=x₂-β_1,1(19)

4.2 obtain transformed error according to the error variance defined

$\begin{array}{c}{\mathrm{\ϵ}}_2=\frac{1}{2}\ln \frac{\underset{\‾}{\mathrm{\α}}\left(t\right)+S\left({\mathrm{\ϵ}}_2\right)}{\stackrel{\‾}{\mathrm{\α}}\left(t\right)-S\left({\mathrm{\ϵ}}_2\right)}\\ =T_2(\frac{s_2\left(t\right)}{{\mathrm{\ρ}}_2\left(t\right)},\stackrel{\‾}{\mathrm{\α}}\left(t\right),\underset{\‾}{\mathrm{\α}}\left(t\right))\end{array}---\left(20\right)$

Wherein ρ₂T the expression formula of () is

ρ₂(t)=(ρ₀-ρ_∞)e^-lt+ρ_∞(21)

Parameter ρ₀> ρ_∞> 0 and l > 0；WithαT the derivative expressions of () is for shown in such as formula (12)；Shown in function S () expression formula such as formula (13)；

Formula (20) derivation is obtained:

${\stackrel{\·}{\mathrm{\ϵ}}}_2=r_2\left(a\right(\stackrel{\‾}{x})+b(\stackrel{\‾}{x})u-{\stackrel{\·}{\mathrm{\β}}}_{1,1})+{\mathrm{\υ}}_2---\left(22\right)$

Wherein

4.3 in order to approach the Nonlinear uncertainty being not directly availableDefine following neutral net

$f\left(\stackrel{\‾}{x}\right)=a\left(\stackrel{\‾}{x}\right)-{\stackrel{\·}{\mathrm{\β}}}_{1,1}=W^{*T}X\left(\stackrel{\‾}{x}\right)+{\mathrm{\η}}^{*}---\left(23\right)$

Wherein, W^*For ideal weight, η^*For neutral net perfect error value, meet | η^*|≤η_N,Expression formula is:

Wherein W_in, W_d, W_fbFor a range of random value；U is controller input；For Gaussian function, expression formula is

$s_i\left(\stackrel{\‾}{x}\right)=\exp \[-\frac{-{(\stackrel{\‾}{x}-{\mathrm{\χ}}_i)}^T(\stackrel{\‾}{x}-{\mathrm{\χ}}_i)}{{\mathrm{\ι}}_i^2}\],i=1,2,...l---\left(25\right)$

WhereinIt is the output of hidden layer i-th node；χ_iIt is the center vector of i-th node Gaussian function, i.e. χ_i=[χ_i1,χ_i2,...χ_il]^T；l_iIt it is the width of i-th node Gaussian function；Y is neutral net output, and expression formula is

$y=G\left(W^{*T}X\right(\stackrel{\‾}{x}\left)\right)---\left(26\right)$

Selection of Function G=1；

4.4 design con-trol device input u:

Wherein,For ideal weight W^*Estimated value,For estimation difference η^*Estimated value；

4.5 design adaptive rates:

$\left\{\begin{array}{c}\stackrel{\·}{\hat{W}}=\stackrel{\·}{\tilde{W}}=\mathrm{\Γ}\[X\left(\stackrel{\‾}{x}\right)r_2{\mathrm{\ϵ}}_2-\mathrm{\σ}\hat{W}\]\\ {\stackrel{\·}{\widehat{\mathrm{\ϵ}}}}_N={\stackrel{\·}{\widetilde{\mathrm{\ϵ}}}}_N=\mathrm{\κ}\[r_2\left|{\mathrm{\ϵ}}_2\right|-\mathrm{\γ}\widehat{\mathrm{\η}}\]\end{array}\right.---\left(28\right)$

Wherein,Γ=Γ^T> 0,Γ is adaptive gain matrix, σ, κ, and γ is constant, and σ > 0, κ > 0, γ > 0；

For verifying the effectiveness of institute's extracting method, The present invention gives the contrast of two kinds of control methods:

M1: based on the high-order dynamic sliding-mode surface constrained control method of default echo state network；

M2: based on the common common dynamic surface constrained control method of RBF neural.

Contrasting in order to more effective, the system initialization parameter that it is all consistent that all parameters are arranged is [x₁,x₂]^T=[0,0]^T’k₀=1 ' m=1；Saturated restricted parameters v_max=15；Higher differentiation device parameter μ_1,1=10 ' μ_2,1=5；Default capabilities function parameter ρ₀=1 ' ρ_∝=0.2, l=0.2, Echo state network parameter W_in’W_fb’W₀It is all the random value on interval [-1,1],It is distributed in interval [-4,4] × [-4,4] × [-4,4] × [-6,6]；Self Adaptive Control rate parameter κ=10, γ=0.001 controller parameter k₁=41, k₂=250；

Following the tracks of sine wave input, its expression formula is y_d=3sint.From figures 2 and 3, it will be seen that M2 controller compared by M1 controller, there is tracking velocity faster, less steady-state error and more good transient performance (overshoot is little)；From fig. 4, it can be seen that M1 control input is more smooth compared with M2；From fig. 5, it can be seen that the more common RBF neural of the approximation capability of echo state network is more preferably, estimation difference is less, and dynamic estimation performance is better.Therefore, the present invention provide one can effective compensation input, export limited, improve the high-order dynamic face sliding-mode control of neutral net approximation capability, it is achieved the stable of system is quickly followed the tracks of, it is ensured that have good transient state steady-state behaviour.

The excellent effect of optimization that the embodiment that the present invention provides that described above is shows, the obvious present invention is more than being limited to above-described embodiment, under not necessarily departing from essence spirit of the present invention and the premise without departing from scope involved by flesh and blood of the present invention, it can be done all deformation and be carried out.

Claims (1)

1. the electromechanical servo system constrained control method based on default echo state network, it is characterised in that: comprise the following steps:

Step 1, sets up the dynamic model of electromechanical servo system, and process is as follows:

The dynamic model expression-form of 1.1 electromechanical servo systems is

$m\stackrel{\·\·}{x}+f(\stackrel{\‾}{x},t)+d(\stackrel{\‾}{x},t)=k_{0}v\left(u\right)---\left(1\right)$

Wherein, x is position；M is inertia；k₀It it is force constant；It it is state variable；It is frictional force；It is model a produced BOUNDED DISTURBANCES, derives from coupled characteristic, measure noise, electronic interferences and other uncertain factors；U is the control input voltage of motor；V (u), for saturated, is expressed as:

$v\left(u\right)=sat\left(u\right)=\left\{\begin{array}{cc}{v}_{max}\mathrm{sgn}\left(u\right),& \left|u\right|\≥v_{max}\\ u,& \left|u\right|\≤v_{max}\end{array}\right.---\left(2\right)$

Wherein sgn (u), for unknown nonlinear function；v_maxFor unknown parameter of saturation, meet v_max> 0；

1.2 definition x₁=x,Formula (1) is rewritten as

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=-\frac{f(\stackrel{\‾}{x},t)+d(\stackrel{\‾}{x},t)}{m}+\frac{k_0}{m}v\left(u\right)\\ y=x_1\end{array}\right.---\left(3\right)$

Wherein, y is system output trajectory；

Step 2, according to Order Derivatives in Differential Mid-Value Theorem, carries out linearization process by saturated for the non-linear input in system, derives the electromechanical servo system model saturated with the unknown, including following process；

2.1 pairs of saturated models carry out smooth treatment

$\begin{array}{c}g\left(u\right)=v_{\max }\×\tanh \left(\frac{u}{v_{\max }}\right)\\ =v_{\max }\×\frac{e^{u/v_{\max }}-e^{-u/v_{\max }}}{e^{u/v_{\max }}+e^{-u/v_{\max }}}\end{array}---\left(4\right)$

Then

V (u)=sat (u)=g (u)+dsat (u) (5)

Wherein, dsat (u) represents the error existed between smooth function and saturated model；

2.2 according to Order Derivatives in Differential Mid-Value Theorem, there is δ ∈ (0,1) and makes

G (u)=g (u₀)+g_uξ(u-u₀)(6)

Whereinu_ξ=ξ u+ (1_-ξ)u₀, u₀∈(0,u)；

Select u₀=0, formula (6) is rewritten as

$g\left(u\right)=g_{u_{\mathrm{\ξ}}}u---\left(7\right)$

Formula (3), by formula (5) and formula (7), is rewritten as following equivalents by 2.3:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=a\left(\stackrel{\‾}{x}\right)+b\left(\stackrel{\‾}{x}\right)u\\ y=x_1\end{array}\right.---\left(8\right)$

Wherein

Step 3, calculating control system tracking error, sliding-mode surface and transformed error, process is as follows:

The tracking error of 3.1 definition control systems, sliding-mode surface is

$\left\{\begin{array}{c}e=y-y_d\\ s_1=e+\mathrm{\λ}\∫edt\end{array}\right.---\left(9\right)$

Wherein, y_dCan leading desired trajectory for second order, λ is constant, and λ > 0；

3.2 obtain new transformed error ε according to sliding-mode surface₁；

$\begin{array}{c}{\mathrm{\ϵ}}_1=\frac{1}{2}\ln \frac{\underset{\‾}{\mathrm{\α}}\left(t\right)+S\left({\mathrm{\ϵ}}_1\right)}{\stackrel{\‾}{\mathrm{\α}}\left(t\right)-S\left({\mathrm{\ϵ}}_1\right)}\\ =T_1\left(\frac{s_1\left(t\right)}{{\mathrm{\ρ}}_1\left(t\right)},\stackrel{\‾}{\mathrm{\α}}\left(t\right),\underset{\‾}{\mathrm{\α}}\left(t\right)\right)\end{array}---\left(10\right)$

Wherein ρ₁T the expression formula of () is

ρ₁(t)=(ρ₀-ρ_∞)e^-lt+ρ_∞(11)

Parameter ρ₀> ρ_∞> 0 and l > 0；WithαT the derivative expressions of () is

ParameterSize and initially need design；Function S () expression formula is

$S\left(\mathrm{\ϵ}\right)=\frac{\stackrel{\‾}{\mathrm{\α}}\left(t\right)\exp \left(\mathrm{\ϵ}\right)-\underset{\‾}{\mathrm{\α}}\left(t\right)\exp (-\mathrm{\ϵ})}{\exp \left(\mathrm{\ϵ}\right)+\exp (-\mathrm{\ϵ})}---\left(13\right)$

Wherein, ε is transformed error variable；

The derivation of 3.3 pairs of formulas (10) obtains:

${\stackrel{\·}{\mathrm{\ϵ}}}_1=r_1(x_2-{\stackrel{\·}{y}}_d+\mathrm{\λ}e)+{\mathrm{\υ}}_1---\left(14\right)$

Wherein

3.4 design virtual controlling amounts

Wherein, k₁For constant, and k₁> 0；Function Q () is Nussbaum function, and selection expression formula is

WhereinAdaptive law be designed as

3.5 allow virtual controlling amountBy High-Order Sliding Mode differentiator

$\left\{\begin{array}{c}{\stackrel{\·}{\mathrm{\β}}}_{1,1}={\mathrm{\ω}}_{1,1}\\ {\mathrm{\ω}}_{1,1}=-{\mathrm{\μ}}_{1,1}|{\mathrm{\β}}_{1,1}-{\stackrel{\‾}{\mathrm{\β}}}_1{|}^{\frac{1}{2}}sign\left({\mathrm{\β}}_{1,1}-{\stackrel{\‾}{\mathrm{\β}}}_1\right)+{\mathrm{\β}}_{2,1}\\ {\stackrel{\·}{\mathrm{\β}}}_{2,1}={\mathrm{\ω}}_{2,1}\\ {\mathrm{\ω}}_{2,1}=-{\mathrm{\μ}}_{2,1}sign\left({\mathrm{\β}}_{2,1}-{\mathrm{\ω}}_{1,1}\right)\end{array}\right.---\left(18\right)$

Wherein parameter μ_1,1> 0, μ_2,1> 0, β_1,1It it is virtual controlling amountBy the filtered variable that differentiator obtains；

Step 4, design con-trol device inputs, and process is as follows:

4.1 definition error variances

s₂=x₂-β_1,1(19)

4.2 obtain transformed error ε according to the error variance defined₂

$\begin{array}{c}{\mathrm{\ϵ}}_2=\frac{1}{2}\ln \frac{\underset{\‾}{\mathrm{\α}}\left(t\right)+S\left({\mathrm{\ϵ}}_2\right)}{\stackrel{\‾}{\mathrm{\α}}\left(t\right)-S\left({\mathrm{\ϵ}}_2\right)}\\ =T_2\left(\frac{s_2\left(t\right)}{{\mathrm{\ρ}}_2\left(t\right)},\stackrel{\‾}{\mathrm{\α}}\left(t\right),\underset{\‾}{\mathrm{\α}}\left(t\right)\right)\end{array}---\left(20\right)$

Wherein ρ₂T the expression formula of () is

ρ₂(t)=(ρ₀-ρ_∞)e^-lt+ρ_∞(21)

Parameter ρ₀> ρ_∞> 0 and l > 0；WithαT the derivative expressions of () is for shown in such as formula (12)；Shown in function S () expression formula such as formula (13)；

Formula (20) derivation is obtained:

${\stackrel{\·}{\mathrm{\ϵ}}}_2=r_2\left(a\right(\stackrel{\‾}{x})+b(\stackrel{\‾}{x})u-{\stackrel{\·}{\mathrm{\β}}}_{1,1})+{\mathrm{\υ}}_2---\left(22\right)$

Wherein

4.3 approach the Nonlinear uncertainty being not directly available Define following neutral net

$f\left(\stackrel{\‾}{x}\right)=a\left(\stackrel{\‾}{x}\right)-{\stackrel{\·}{\mathrm{\β}}}_{\mathrm{1,1}}=W^{*T}X\left(\stackrel{\‾}{x}\right)+{\mathrm{\η}}^{*}---\left(23\right)$

Wherein, W^*For ideal weight, η^*For neutral net perfect error value, meet | η^*|≤η_N,Expression formula is:

Wherein W_in, W_d, W_fbFor random value；U is controller input；For Gaussian function, expression formula is

$s_i\left(\stackrel{\‾}{x}\right)=\exp [-\frac{-{(\stackrel{\‾}{x}-{\mathrm{\χ}}_i)}^T(\stackrel{\‾}{x}-{\mathrm{\χ}}_i)}{i_i^2}],i=\mathrm{1,2},...l---\left(25\right)$

WhereinIt is the output of hidden layer i-th node；χ_iIt is the center vector of i-th node Gaussian function, i.e. χ_i=[χ_i1,χ_i2,…χ_il]^T；ι_iIt it is the width of i-th node Gaussian function；Y is neutral net output, and expression formula is

$y=G\left(W^{*T}X\right(\stackrel{\‾}{x}\left)\right)---\left(26\right)$ Selection of Function G=1；

4.4 design con-trol device input u:

Wherein,For ideal weight W^*Estimated value,For estimation difference η^*Estimated value.

4.5 design adaptive rates:

$\left\{\begin{array}{c}\stackrel{\·}{\hat{W}}=\stackrel{\·}{\tilde{W}}=\mathrm{\Γ}\[X\left(\stackrel{\‾}{x}\right)r_2{\mathrm{\ϵ}}_2-\mathrm{\σ}\hat{W}\]\\ {\stackrel{\·}{\widehat{\mathrm{\ϵ}}}}_N={\stackrel{\·}{\widetilde{\mathrm{\ϵ}}}}_N=\mathrm{\κ}\[r_2\left|{\mathrm{\ϵ}}_2\right|-\mathrm{\γ}\widehat{\mathrm{\η}}\]\end{array}\right.---\left(28\right)$

Wherein,Γ=Γ^T> 0,Γ is adaptive gain matrix, σ, κ, and γ is constant, and σ > 0, κ > 0, γ > 0.