CN105892297A - Control algorithm of self-adaptive fractional order dynamic sliding mode - Google Patents

Abstract

The invention discloses a control algorithm of a self-adaptive fractional order dynamic sliding mode. The control algorithm of the self-adaptive fractional order dynamic sliding mode comprises the following steps: establishing a mathematical model of a controlled system according to a parameter matrix of one or multiple system parameters, a system state vector, a control force vector and a tracking error calculated by the system state vector; constructing a fractional order sliding mode surface equation by using the tracking error and a fractional derivative of the tracking error; establishing a fractional order dynamic sliding mode surface equation by using the fractional order sliding mode surface equation and a derivative equation of the fractional order sliding mode surface equation; establishing a fractional order self-adaptive law equation based on parameter matrixes of the system parameters by using the fractional order dynamic sliding mode surface equation; designing a structure and a calculation equation of the control force vector, and calculating and outputting the control force vector to the controlled system. According to the invention, a fractional calculus algorithm and dynamic sliding mode control are combined to improve the control effect and the system parameter identification effect of the traditional integer order sliding mode control, and reduce the control force chattering in the sliding mode control.

Description

A kind of adaptive fractional rank dynamic sliding mode control algorithm

Technical field

The invention belongs to automatic control system field, be specifically related to a kind of adaptive fractional rank dynamic sliding mode control algorithm.

Background technology

Generally, during sliding formwork controls, the design of sliding-mode surface is all to use the ratio of error, integration, the combination of differential, Qi Zhongji Dividing the exponent number with differential is all integer.Owing to the differential of fractional calculus and the exponent number of integration can be carried out adjusting, with biography System integer rank calculus is compared, the calculus exponent number item that fractional calculus is many can be regulated, owing to fractional order sliding formwork controls Many adjustable exponent number degree of freedom, controlling effect can improve to some extent.

Summary of the invention

Controlling power present in sliding formwork control tremble shake phenomenon to overcome, it is dynamic that the present invention proposes a kind of adaptive fractional rank Sliding mode control algorithm, is combined fractional calculus algorithm with dynamic sliding mode control, improves conventional integer rank sliding formwork The control effect controlled and Parameter identification effect, and reduction sliding formwork control control power and trembles the phenomenon of shake.

Realizing above-mentioned technical purpose, reach above-mentioned technique effect, the present invention is achieved through the following technical solutions:

A kind of adaptive fractional rank dynamic sliding mode control algorithm, comprises the following steps:

According to 1 or the parameter matrix of multiple systematic parameter, system mode vector, control force vector, by system mode The tracking error that vector calculates, sets up the mathematical model of controlled system；

Utilize tracking error and Fractional Derivative structure fractional order sliding-mode surface equation thereof；

Utilize fractional order sliding-mode surface equation and derivative equation thereof, set up fractional order Dynamic sliding mode face equation；

Utilize fractional order Dynamic sliding mode face equation, set up the fractional order adaptive law of parameter matrix based on each systematic parameter Equation；

Design controls structure and the accounting equation of force vector, and calculates and export control force vector to controlled system.

The mathematical model of described controlled system particularly as follows:

$\stackrel{\·\·}{q}+D\stackrel{\·}{q}+Kq=u-2\mathrm{\Ω}\stackrel{\·}{q}---\left(1\right)$

In formula:D, K, Ω are Comprise the parameter matrix of systematic parameter, d_xx、d_yyIt is respectively X-axis and the damped coefficient of Y-axis, d_xyFor X, the coefficient of coup between Y-axis, q For system mode vector, comprising x, y, x, y are mass displacement in X, Y-axis, and u is for controlling force vector, u_x、u_yFor in X, Y-axis Control power；

Described tracking error is e=q_d-q, the derivative of tracking error isWherein q_dFor Setting signal.

Described fractional order sliding-mode surface equation is:

$S=c_1\stackrel{\·}{e}+c_{2}e+c_3{D}^{\mathrm{\α}-1}e---\left(2\right)$

In formula: c₁, c₂, c₃It is the parameter of sliding-mode surface equation, is all positive number, D^α-1E is the Fractional Derivative of tracking error, α takes any rational number.

Described fractional order Dynamic sliding mode face equation, particularly as follows:

$\begin{array}{c}\mathrm{\σ}=\stackrel{\·}{S}+\∂S\\ =c_1\stackrel{\·\·}{e}+c_2\stackrel{\·}{e}+c_3{D}^{\mathrm{\α}}e+(c_1\stackrel{\·}{e}+c_{2}e+c_3{D}^{\mathrm{\α}-1}e)\end{array}---\left(3\right)$

In formula,For Dynamic sliding mode face parameter, it it is a positive count.

The fractional order adaptive law equation of described parameter matrix based on each systematic parameter is respectively as follows:

${\stackrel{\·}{\tilde{D}}}^T=\stackrel{\·\·}{q}{\mathrm{\σ}}^T{c}_1+\stackrel{\·}{q}{\mathrm{\σ}}^T(c_2+\∂c_1)---\left(4\right)$

${\stackrel{\·}{\widetilde{\mathrm{\Ω}}}}^T=\stackrel{\·\·}{q}{\mathrm{\σ}}^{T}2c_1+\stackrel{\·}{q}{\mathrm{\σ}}^{T}2(c_2+\∂c_1)---\left(5\right)$

$\stackrel{\·}{\tilde{K}}=\stackrel{\·}{q}{\mathrm{\σ}}^T{c}_1+{\mathrm{q\σ}}^T(c_2+\∂c_1)---\left(6\right).$

According in adaptive law equationQ, σ and parameter c₁, c₂,Come the parameter square in controller

Battle arrayComplete self-adaptative adjustment.

Structure and the accounting equation of described control force vector be:

$\hat{u}={\hat{u}}_{eq}+u_{sw}---\left(7\right)$

Wherein,

${\hat{u}}_{eq}=\frac{c_1({\stackrel{\·\·\·}{q}}_d-\stackrel{\·}{\hat{f}}-{\stackrel{\·}{\hat{u}}}_{eq})+(c_2+\∂c_1)({\stackrel{\·\·}{q}}_d-\hat{f}-{\stackrel{\·}{\hat{u}}}_{eq})+c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}}{c_2+\∂c_1}---\left(8\right)$

$u_{sw}=\frac{\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)}{c_2+\∂c_1}---\left(9\right)$

Wherein η is the gain of sliding formwork item, is taken as positive count,

In formula (8):

$\hat{f}=-(\hat{D}+2\widehat{\mathrm{\Ω}})\stackrel{\·}{q}-\hat{K}q---\left(10\right)$

In formula (10),It is respectively the estimates of parameters of D, K, Ω.

Described a kind of adaptive fractional rank dynamic sliding mode control algorithm, also includes: set up Lyapunov function and to it Derivation, then the structure of force vector and accounting equation and the fractional order adaptive law of parameter matrix based on each systematic parameter will be controlled Equation is brought into and is calculated.

Described Lyapunov function particularly as follows:

$V=\frac{1}{2}{\mathrm{\σ}}^T\mathrm{\σ}+\frac{1}{2}tr\left({\tilde{D}}^T\tilde{D}\right)+\frac{1}{2}tr\left({\tilde{K}}^T\tilde{K}\right)+\frac{1}{2}tr\left({\widetilde{\mathrm{\Ω}}}^T\widetilde{\mathrm{\Ω}}\right)---\left(11\right)$

Wherein,It is respectively the parameter estimation deviation of D, K, Ω,

Formula (11) is carried out derivation:

$\begin{array}{c}\stackrel{\·}{V}={\mathrm{\σ}}^T\stackrel{\·}{\mathrm{\σ}}+tr\left({\stackrel{\·}{\tilde{D}}}^T\tilde{D}\right)+tr\left({\stackrel{\·}{\tilde{K}}}^T\tilde{K}\right)+tr\left({\stackrel{\·}{\widetilde{\mathrm{\Ω}}}}^T\widetilde{\mathrm{\Ω}}\right)\\ ={\mathrm{\σ}}^T\[c_1({\stackrel{\·\·\·}{q}}_d-\stackrel{\·}{f}-\stackrel{\·}{u})+(c_2+\∂c_1)({\stackrel{\·\·}{q}}_d-f-u)+c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}\]+tr(*)\end{array}---\left(12\right)$

Wherein:

The equation of force vector will be controlledSubstitution formula (12) obtains:

$\begin{array}{c}\stackrel{\·}{V}={\mathrm{\σ}}^T\[c_1({\stackrel{\·\·\·}{q}}_d-\stackrel{\·}{f}-\stackrel{\·}{u})+(c_2+\∂c_1)({\stackrel{\·\·}{q}}_d-f-u)+c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}\]+tr\left(\stackrel{\·}{*}\right)\\ ={\mathrm{\σ}}^T\[c_1{\stackrel{\·\·\·}{q}}_d-c_1\stackrel{\·}{f}-c_1\stackrel{\·}{u}+(c_2+\∂c_1){\stackrel{\·\·}{q}}_d-(c_2+\∂c_1)f-\\ (c_2+\∂c_1)\frac{1}{c_2+\∂c_1}\left(c_1\right({\stackrel{\·\·\·}{q}}_d-\stackrel{\·}{\hat{f}}-\stackrel{\·}{u})+(c_2+\∂c_1\left)\right({\stackrel{\·\·}{q}}_d-\hat{f}-\stackrel{\·}{\hat{u}})+c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}+\mathrm{\η}\mathrm{sgn}(\mathrm{\σ}\left)\right)\\ +c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}\]+tr\left(\stackrel{\·}{*}\right)\\ ={\mathrm{\σ}}^T\[-c_1\stackrel{\·}{f}+c_1\stackrel{\·}{\hat{f}}-(c_2+\∂c_1)f+(c_2+\∂c_1)\hat{f}-\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)\]+tr\left(\stackrel{\·}{*}\right)\\ ={\mathrm{\σ}}^T\[c_1\stackrel{\·}{\tilde{f}}+(c_2+\∂c_1)\tilde{f}-\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)\]+tr\left(\stackrel{\·}{*}\right)\end{array}---\left(13\right)$

Wherein:

$\begin{array}{c}\tilde{f}=\hat{f}-f\\ =-(D+2\widehat{\mathrm{\Ω}})\stackrel{\·}{q}-\hat{K}q-(-(D-2\mathrm{\Ω})\stackrel{\·}{q}-Kq)\\ =\[(D-\hat{D})+2(\mathrm{\Ω}-\widehat{\mathrm{\Ω}})\]\stackrel{\·}{q}+(K-\hat{K})q\\ -(-\tilde{D}-2\widetilde{\mathrm{\Ω}})\stackrel{\·}{q}-\tilde{K}q\end{array}---\left(14\right)$

Its derivative is:

$\stackrel{\·}{\tilde{f}}=(-\tilde{D}-2\widetilde{\mathrm{\Ω}})\stackrel{\·\·}{q}-\tilde{K}\stackrel{\·}{q}---\left(15\right)$

Bring formula (15) into formula (in 13) to obtain:

$\begin{array}{c}\stackrel{\·}{V}={\mathrm{\σ}}^T\{c_1\[(-\tilde{D}-2\widetilde{\mathrm{\Ω}})\stackrel{\·\·}{q}-\tilde{K}\stackrel{\·}{q}\]+(c_2+\∂c_1)\[(-\tilde{D}-2\widetilde{\mathrm{\Ω}})\stackrel{\·}{q}-\tilde{K}q\]-\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)\}+tr\left(\stackrel{\·}{*}\right)\\ ={\mathrm{\σ}}^T\{\[-c_1\tilde{D}\stackrel{\·\·}{q}-(c_2+\∂c_1)\tilde{D}\stackrel{\·}{q}\]+\[-2c_1\widetilde{\mathrm{\Ω}}\stackrel{\·\·}{q}-2(c_2+\∂c_1)\widetilde{\mathrm{\Ω}}\stackrel{\·}{q}\]+\[-c_1\tilde{K}\stackrel{\·}{q}-(c_2+\∂c_1)\tilde{K}q\]-\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)\}+tr\left(\stackrel{\·}{*}\right)\\ =\[-\stackrel{\·\·}{q}{\mathrm{\σ}}^T{c}_1\tilde{D}-\stackrel{\·}{q}{\mathrm{\σ}}^T(c_2+\∂c_1)\tilde{D}\]+tr\left({\stackrel{\·}{\tilde{D}}}^T\tilde{D}\right)+\[-\stackrel{\·\·}{q}{\mathrm{\σ}}^{T}2c_1\widetilde{\mathrm{\Ω}}-\stackrel{\·}{q}{\mathrm{\σ}}^{T}2(c_2+\∂c_1)\widetilde{\mathrm{\Ω}}\]\\ +tr\left({\stackrel{\·}{\widetilde{\mathrm{\Ω}}}}^T\widetilde{\mathrm{\Ω}}\right)+\[-\stackrel{\·}{q}{\mathrm{\σ}}^T{c}_1\tilde{K}-{\mathrm{q\σ}}^T(c_2+\∂c_1)\tilde{K}\]+tr\left({\stackrel{\·}{\tilde{K}}}^T\tilde{K}\right)\mathrm{\η}\left|\mathrm{\σ}\right|\end{array}---\left(16\right)$

Formula (4), (5), (6) are brought into above formula and are obtained:

$\stackrel{\·}{V}=-\mathrm{\η}\left|{\mathrm{\σ}}^T\right|\≤0---\left(17\right).$

Beneficial effects of the present invention:

(1) present invention proposes a kind of adaptive fractional rank dynamic sliding mode control algorithm, the differential of the tracking error of use with The exponent number of integration is the differential of the sliding-mode surface of mark, use and the exponent number of integration is mark, thus, adjustable exponent number degree of freedom Height, controls effect more preferable.

(2) present invention proposes a kind of adaptive fractional rank dynamic sliding mode control algorithm, by fractional calculus algorithm with dynamic State sliding formwork controls to be combined, and improves the control effect of conventional integer rank dynamic sliding mode control, and reduces in sliding formwork control Control power trembles the phenomenon of shake；And devise the fractional order adaptive law equation of parameter matrix based on each systematic parameter, thus greatly Improve greatly Parameter identification effect.

(3) a kind of adaptive fractional rank dynamic sliding mode control algorithm that this method proposes, employs and comprises fractional order item Fractional order Dynamic sliding mode face item, and adaptive algorithm designed in the present invention is all based on Lyapunov stable theory, it is possible to The stability of guarantee system.

Accompanying drawing explanation

Fig. 1 is the flow chart of the adaptive fractional rank dynamic sliding mode control algorithm of the present invention.

Fig. 2 is the schematic diagram of the control system corresponding with the adaptive fractional rank dynamic sliding mode control algorithm of the present invention.

Fig. 3 is that the present invention is embodied as X in example, Y-axis position tracking performance curve.

Fig. 4 is that the present invention is embodied as X in example, Y-axis position tracking error curve.

Fig. 5 is that the present invention is embodied as X in example, and the control force curve of shake is trembled in smooth not the existing of Y-axis.

Fig. 6 is that the present invention is embodied as in example using self adaptation integer rank dynamic sliding mode control and adaptive fractional rank to move The tracking effect comparison diagram that state sliding formwork controls.

Fig. 7 is that the present invention is embodied as in example using self adaptation integer rank dynamic sliding mode control and adaptive fractional rank to move The tracking error comparison diagram that state sliding formwork controls.

Fig. 8 is that the present invention is embodied as in example using self adaptation integer rank dynamic sliding mode control and adaptive fractional rank to move The parameter identification comparison diagram that state sliding formwork controls.

Fig. 9 is that the present invention is embodied as in example using self adaptation integer rank dynamic sliding mode control and adaptive fractional rank to move The angular velocity identification comparison diagram that state sliding formwork controls.

Figure 10 is that the present invention is embodied as in example adaptive sliding-mode observer there is the control force curve trembling shake phenomenon.

Detailed description of the invention

In order to make the purpose of the present invention, technical scheme and advantage clearer, below in conjunction with embodiment, to the present invention It is further elaborated.Should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not used to Limit the present invention.

Below in conjunction with the accompanying drawings the application principle of the present invention is explained in detail.

As it is shown in figure 1, a kind of adaptive fractional rank dynamic sliding mode control algorithm, comprise the following steps:

According to 1 or the parameter matrix of multiple systematic parameter, system mode vector, control force vector, by system mode The tracking error that vector calculates, sets up the mathematical model of controlled system；

Utilize tracking error and Fractional Derivative structure fractional order sliding-mode surface equation thereof；

Utilize fractional order sliding-mode surface equation and derivative equation thereof, set up fractional order Dynamic sliding mode face equation；

Utilize fractional order Dynamic sliding mode face equation, set up the fractional order adaptive law of parameter matrix based on each systematic parameter Equation；

Set up structure and the accounting equation controlling force vector, and calculate and export control force vector to controlled system.

The mathematical model of described controlled system particularly as follows:

$\stackrel{\·\·}{q}+D\stackrel{\·}{q}+Kq=u-2\mathrm{\Ω}\stackrel{\·}{q}---\left(1\right)$

In formula:D, K, Ω bag Parameter matrix containing systematic parameter, d_xx、d_yyIt is respectively X-axis and the damped coefficient of Y-axis, d_xyFor X, the coefficient of coup between Y-axis, q is System mode vector, comprises x, y, and x, y are mass displacement in X, Y-axis, and u is for controlling force vector, u_x、u_yIt is respectively X, Y-axis The control power of upper input；

Described tracking error is e=q_d-q, the derivative of tracking error isWherein q_dFor Setting signal.

Described fractional order sliding-mode surface equation is:

$S=c_1\stackrel{\·}{e}+c_{2}e+c_3{D}^{\mathrm{\α}-1}e---\left(2\right)$

In formula: c₁, c₂, c₃Being the parameter of sliding-mode surface equation, be all positive number, α takes any rational number, is one and determines Number.

Described fractional order Dynamic sliding mode face equation, particularly as follows:

$\begin{array}{c}\mathrm{\σ}=\stackrel{\·}{S}+\∂S\\ =c_1\stackrel{\·\·}{e}+c_2\stackrel{\·}{e}+c_3{D}^{\mathrm{\α}}e+(c_1\stackrel{\·}{e}+c_{2}e+c_3{D}^{\mathrm{\α}-1}e)\end{array}---\left(3\right)$

In formula,For Dynamic sliding mode face parameter, it it is a positive number.

The fractional order adaptive law equation of described parameter matrix based on each systematic parameter is respectively as follows:

${\stackrel{\·}{\tilde{D}}}^T=\stackrel{\·\·}{q}{\mathrm{\σ}}^T{c}_1+\stackrel{\·}{q}{\mathrm{\σ}}^T(c_2+\∂c_1)---\left(4\right)$

${\stackrel{\·}{\widetilde{\mathrm{\Ω}}}}^T=\stackrel{\·\·}{q}{\mathrm{\σ}}^{T}2c_1+\stackrel{\·}{q}{\mathrm{\σ}}^{T}2(c_2+\∂c_1)---\left(5\right)$

$\stackrel{\·}{\tilde{K}}=\stackrel{\·}{q}{\mathrm{\σ}}^T{c}_1+{\mathrm{q\σ}}^T(c_2+\∂c_1)---\left(6\right).$

According in adaptive law equationQ, σ and parameter c₁, c₂,Come the parameter matrix in controller Complete self-adaptative adjustment.

Due to fractional order sliding-mode surfaceIt comprises the Fractional Derivative of error, and Dynamic sliding mode face σ comprises again fractional order sliding-mode surface, and therefore adaptive law equation is also fractional order form.

Structure and the accounting equation of described control force vector be:

$\hat{u}={\hat{u}}_{eq}+u_{sw}---\left(7\right)$

Wherein,

${\hat{u}}_{eq}=\frac{c_1({\stackrel{\·\·\·}{q}}_d-\stackrel{\·}{\hat{f}}-{\stackrel{\·}{\hat{u}}}_{eq})+(c_2+\∂c_1)({\stackrel{\·\·}{q}}_d-\hat{f}-{\stackrel{\·}{\hat{u}}}_{eq})+c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}}{c_2+\∂c_1}---\left(8\right)$

$u_{sw}=\frac{\mathrm{\η}sgn\left(\mathrm{\σ}\right)}{c_2+\∂c_1}---\left(9\right)$

In formula (9):

$\hat{f}=-(\hat{D}+2\widehat{\mathrm{\Ω}})\stackrel{\·}{q}-\hat{K}q---\left(10\right)$

In formula (10),It is respectively the estimates of parameters of D, K, Ω, It is respectively the parameter estimation deviation of D, K, Ω.

Described a kind of adaptive fractional rank dynamic sliding mode control algorithm, also includes: set up Lyapunov function and to it Derivation, then control power is estimated that the fractional order adaptive law equation of model equation and parameter matrix based on each systematic parameter is brought into Calculate.

Described Lyapunov function particularly as follows:

$V=\frac{1}{2}{\mathrm{\σ}}^T\mathrm{\σ}+\frac{1}{2}tr\left({\tilde{D}}^T\tilde{D}\right)+\frac{1}{2}tr\left({\tilde{K}}^T\tilde{K}\right)+\frac{1}{2}tr\left({\widetilde{\mathrm{\Ω}}}^T\widetilde{\mathrm{\Ω}}\right)---\left(11\right)$

Formula (11) is carried out derivation:

$\begin{array}{c}\stackrel{\·}{V}={\mathrm{\σ}}^T\stackrel{\·}{\mathrm{\σ}}+tr\left({\stackrel{\·}{\tilde{D}}}^T\tilde{D}\right)+tr\left({\stackrel{\·}{\tilde{K}}}^T\tilde{K}\right)+tr\left({\stackrel{\·}{\widetilde{\mathrm{\Ω}}}}^T\widetilde{\mathrm{\Ω}}\right)\\ ={\mathrm{\σ}}^T\[c_1({\stackrel{\·\·\·}{q}}_d-\stackrel{\·}{f}-\stackrel{\·}{u})+(c_2+\∂c_1)({\stackrel{\·\·}{q}}_d-f-u)+c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}\]+tr(*)\end{array}---\left(12\right)$

Wherein:

By the structure and the accounting equation that control force vector it isSubstitution formula (12) obtains:

$\begin{array}{c}\stackrel{\·}{V}={\mathrm{\σ}}^T\[c_1({\stackrel{\·\·\·}{q}}_d-\stackrel{\·}{f}-\stackrel{\·}{u})+(c_2+\∂c_1)({\stackrel{\·\·}{q}}_d-f-u)+c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}\]+tr\left(\stackrel{\·}{*}\right)\\ ={\mathrm{\σ}}^T\[c_1{\stackrel{\·\·\·}{q}}_d-c_1\stackrel{\·}{f}-c_1\stackrel{\·}{u}+(c_2+\∂c_1){\stackrel{\·\·}{q}}_d-(c_2+\∂c_1)f-\\ (c_2+\∂c_1)\frac{1}{c_2+\∂c_1}\left(c_1\right({\stackrel{\·\·\·}{q}}_d-\stackrel{\·}{\hat{f}}-\stackrel{\·}{u})+(c_2+\∂c_1\left)\right({\stackrel{\·\·}{q}}_d-\hat{f}-\stackrel{\·}{\hat{u}})+c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}+\mathrm{\η}\mathrm{sgn}(\mathrm{\σ}\left)\right)\\ +c_3{D}^{\mathrm{\α}+1}e+\∂c_3{D}^{\mathrm{\α}}e+\∂c_2\stackrel{\·}{e}\]+tr\left(\stackrel{\·}{*}\right)\\ ={\mathrm{\σ}}^T\[-c_1\stackrel{\·}{f}+c_1\stackrel{\·}{\hat{f}}-(c_2+\∂c_1)f+(c_2+\∂c_1)\hat{f}-\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)\]+tr\left(\stackrel{\·}{*}\right)\\ ={\mathrm{\σ}}^T\[c_1\stackrel{\·}{\tilde{f}}+(c_2+\∂c_1)\tilde{f}-\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)\]+tr\left(\stackrel{\·}{*}\right)\end{array}---\left(13\right)$

Wherein:

$\begin{array}{c}\tilde{f}=\hat{f}-f\\ =-(D+2\widehat{\mathrm{\Ω}})\stackrel{\·}{q}-\hat{K}q-(-(D-2\mathrm{\Ω})\stackrel{\·}{q}-Kq)\\ =\[(D-\hat{D})+2(\mathrm{\Ω}-\widehat{\mathrm{\Ω}})\]\stackrel{\·}{q}+(K-\hat{K})q\\ -(-\tilde{D}-2\widetilde{\mathrm{\Ω}})\stackrel{\·}{q}-\tilde{K}q\end{array}---\left(14\right)$

Its derivative is:

$\stackrel{\·}{\tilde{f}}=(-\tilde{D}-2\widetilde{\mathrm{\Ω}})\stackrel{\·\·}{q}-\tilde{K}\stackrel{\·}{q}---\left(15\right)$

Bring formula (15) into formula (in 13) to obtain:

$\begin{array}{c}\stackrel{\·}{V}={\mathrm{\σ}}^T\{c_1\[(-\tilde{D}-2\widetilde{\mathrm{\Ω}})\stackrel{\·\·}{q}-\tilde{K}\stackrel{\·}{q}\]+(c_2+\∂c_1)\[(-\tilde{D}-2\widetilde{\mathrm{\Ω}})\stackrel{\·}{q}-\tilde{K}q\]-\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)\}+tr\left(\stackrel{\·}{*}\right)\\ ={\mathrm{\σ}}^T\{\[-c_1\tilde{D}\stackrel{\·\·}{q}-(c_2+\∂c_1)\tilde{D}\stackrel{\·}{q}\]+\[-2c_1\widetilde{\mathrm{\Ω}}\stackrel{\·\·}{q}-2(c_2+\∂c_1)\widetilde{\mathrm{\Ω}}\stackrel{\·}{q}\]+\[-c_1\tilde{K}\stackrel{\·}{q}-(c_2+\∂c_1)\tilde{K}q\]-\mathrm{\η}\mathrm{sgn}\left(\mathrm{\σ}\right)\}+tr\left(\stackrel{\·}{*}\right)\\ =\[-\stackrel{\·\·}{q}{\mathrm{\σ}}^T{c}_1\tilde{D}-\stackrel{\·}{q}{\mathrm{\σ}}^T(c_2+\∂c_1)\tilde{D}\]+tr\left({\stackrel{\·}{\tilde{D}}}^T\tilde{D}\right)+\[-\stackrel{\·\·}{q}{\mathrm{\σ}}^{T}2c_1\widetilde{\mathrm{\Ω}}-\stackrel{\·}{q}{\mathrm{\σ}}^{T}2(c_2+\∂c_1)\widetilde{\mathrm{\Ω}}\]\\ +tr\left({\stackrel{\·}{\widetilde{\mathrm{\Ω}}}}^T\widetilde{\mathrm{\Ω}}\right)+\[-\stackrel{\·}{q}{\mathrm{\σ}}^T{c}_1\tilde{K}-{\mathrm{q\σ}}^T(c_2+\∂c_1)\tilde{K}\]+tr\left({\stackrel{\·}{\tilde{K}}}^T\tilde{K}\right)\mathrm{\η}\left|\mathrm{\σ}\right|\end{array}---\left(16\right)$

Formula (4), (5), (6) are brought into above formula and are obtained:

$\stackrel{\·}{V}=-\mathrm{\η}\left|{\mathrm{\σ}}^T\right|\≤0---\left(17\right).$

By formula (17) it can be seen that designed control algolithm ensure that Lyapunov function derivative is negative semidefinite 's；According to Lyapunov Theory of Stability, it is possible to determine that the stability of controlled system.

As in figure 2 it is shown, be former by control system corresponding for the adaptive fractional rank dynamic sliding mode control algorithm of the present invention Reason figure.

As in Figure 3-5, in an embodiment of the present invention, each parameter of controlled system is set to: ω_x ²=355.3, ω_y ²=532.9, ω_xy=70.99, d_xx=0.01, d_yy=0.01, d_xy=0.002, Ω=0.1, the initial shape of controlled device State takes X₀=[0.6 0 0.6 0], reference locus

In controller design, parameter is set as: c₁=1, c₂=5, c₃=1,α=1.23, η=10.

The present invention can improve control effect and the Parameter identification effect of conventional integer rank dynamic sliding mode control, and subtracts Little sliding formwork controls power in controlling and trembles the phenomenon of shake, specifically can find out from Fig. 3-Figure 10.

The ultimate principle of the present invention and principal character and advantages of the present invention have more than been shown and described.The technology of the industry Personnel, it should be appreciated that the present invention is not restricted to the described embodiments, simply illustrating this described in above-described embodiment and description The principle of invention, without departing from the spirit and scope of the present invention, the present invention also has various changes and modifications, and these become Change and improvement both falls within scope of the claimed invention.Claimed scope by appending claims and Equivalent defines.

Claims (8)

1. an adaptive fractional rank dynamic sliding mode control algorithm, it is characterised in that comprise the following steps:

According to 1 or the parameter matrix of multiple systematic parameter, system mode vector, control force vector, by system mode vector The tracking error calculated, sets up the mathematical model of controlled system；

Utilize tracking error and Fractional Derivative structure fractional order sliding-mode surface equation thereof；

Utilize fractional order sliding-mode surface equation and derivative equation thereof, set up fractional order Dynamic sliding mode face equation；

Utilize fractional order Dynamic sliding mode face equation, set up the fractional order adaptive law side of parameter matrix based on each systematic parameter Journey；

Design controls structure and the accounting equation of force vector, and calculates and export control force vector to controlled system.

A kind of adaptive fractional rank the most according to claim 1 dynamic sliding mode control algorithm, it is characterised in that: described controlled The mathematical model of system particularly as follows:

$\stackrel{\·\·}{q}+D\stackrel{\·}{q}+Kq=u-2\mathrm{\Ω}\stackrel{\·}{q}---\left(1\right)$

In formula:D, K, Ω for comprising are The parameter matrix of system parameter, d_xx、d_yyIt is respectively X-axis and the damped coefficient of Y-axis, d_xyFor X, the coefficient of coup between Y-axis, q is system State vector, comprises x, y, and x, y are respectively mass displacement in X, Y-axis, and u is for controlling force vector, u_x、u_yIt is respectively X, Y-axis The control power of upper input；

Described tracking error is e=q_d-q, the derivative of tracking error isWherein q_dFor Setting signal.

A kind of adaptive fractional rank the most according to claim 2 dynamic sliding mode control algorithm, it is characterised in that: described mark Rank sliding-mode surface equation is:

$S=c_1\stackrel{\·}{e}+c_{2}e+c_3{D}^{\mathrm{\α}-1}e---\left(2\right)$

In formula: c₁, c₂, c₃It is the parameter of sliding-mode surface equation, is all positive number, D^α-1E is the Fractional Derivative of tracking error, and α takes It is arbitrarily rational number.

A kind of adaptive fractional rank the most according to claim 3 dynamic sliding mode control algorithm, it is characterised in that: described mark Dynamic sliding mode face, rank equation, particularly as follows:

$\begin{array}{c}\mathrm{\σ}=\stackrel{\·}{S}+\∂S\\ =c_1\stackrel{\·\·}{e}+c_2\stackrel{\·}{e}+c_3{D}^{\mathrm{\α}}e+(c_1\stackrel{\·}{e}+c_{2}e+c_3{D}^{\mathrm{\α}-1}e)\end{array}---\left(3\right)$

In formula,For Dynamic sliding mode face parameter, it it is a positive number.

A kind of adaptive fractional rank the most according to claim 4 dynamic sliding mode control algorithm, it is characterised in that: described based on The fractional order adaptive law equation of the parameter matrix of each systematic parameter is respectively as follows:

${\stackrel{\·}{\tilde{D}}}^T=\stackrel{\·\·}{q}{\mathrm{\σ}}^T{c}_1+\stackrel{\·}{q}{\mathrm{\σ}}^T(c_2+\∂c_1)---\left(4\right)$

${\stackrel{\·}{\widetilde{\mathrm{\Ω}}}}^T=\stackrel{\·\·}{q}{\mathrm{\σ}}^{T}2c_1+\stackrel{\·}{q}{\mathrm{\σ}}^{T}2(c_2+\∂c_1)---\left(5\right)$

${\stackrel{\·}{\tilde{K}}}^T=\stackrel{\·}{q}{\mathrm{\σ}}^T{c}_1+{\mathrm{q\σ}}^T(c_2+\∂c_1)---\left(6\right).$

A kind of adaptive fractional rank the most according to claim 5 dynamic sliding mode control algorithm, it is characterised in that: described control The structure of force vector and accounting equation be:

$\hat{u}={\hat{u}}_{eq}+u_{sw}---\left(7\right)$

Wherein,

$u_{sw}=\frac{\mathrm{\η}sgn\left(\mathrm{\σ}\right)}{c_2+\∂c_1}---\left(9\right)$

Wherein η is the gain of sliding formwork item, is taken as positive count,

In formula (8):

$\hat{f}=-(\hat{D}+2\widehat{\mathrm{\Ω}})\stackrel{\·}{q}-\hat{K}q---\left(10\right)$

In formula (10),It is respectively the estimates of parameters of D, K, Ω.

A kind of adaptive fractional rank the most according to claim 1 dynamic sliding mode control algorithm, it is characterised in that also include: Set up Lyapunov function and to its derivation, then the structure of force vector and accounting equation and ginseng based on each systematic parameter will be controlled The fractional order adaptive law equation of matrix number is brought into and is calculated.

A kind of adaptive fractional rank the most according to claim 7 dynamic sliding mode control algorithm, it is characterised in that: described Lyapunov function particularly as follows:

$V=\frac{1}{2}{\mathrm{\σ}}^T\mathrm{\σ}+\frac{1}{2}tr\left({\tilde{D}}^T\tilde{D}\right)+\frac{1}{2}tr\left({\tilde{K}}^T\tilde{K}\right)+\frac{1}{2}tr\left({\widetilde{\mathrm{\Ω}}}^T\widetilde{\mathrm{\Ω}}\right)---\left(11\right)$

Wherein,It is respectively the parameter estimation deviation of D, K, Ω.