CN105759616A - Servo system finite time control method considering dead zone characteristic - Google Patents

Abstract

A servo system finite time control method considering dead zone characteristics, comprising: establishing a servo system model, and initializing a system state and controller parameters; approximate compensation is carried out on the dead zone link; designing a nonlinear extended state observer; determining observer parameters by using a pole allocation method; and designing a full-order sliding mode controller. Designing an extended state observer for estimating a system state and uncertain items including system friction and external disturbance, and determining an observer gain parameter by adopting a pole allocation method; and a full-order sliding mode controller is designed, so that the tracking error of the system is ensured to be fast and stable and converged to a zero point, the problem of buffeting is weakened, and the fast and stable control of a servo system is realized. The method solves the problem that the external disturbance states such as system friction and the like are not measurable, compensates the influence of a nonlinear dead zone link existing in the system, improves the buffeting problem and the problem of long convergence time existing in the common sliding mode method, enhances the anti-interference performance of the system, and realizes the fast and stable tracking of the system on the expected signal.

Description

Servo system finite time control method considering dead zone characteristic

Technical Field

The invention relates to a servo system finite time control method considering dead zone characteristics, which is suitable for controlling servo systems with nonlinear dead zone links and requiring to quickly and accurately track expected signals.

Background

A servo system (ServoSystem) is a servo system in which a motor is used as a power driving element, and is widely applied to various fields such as aerospace, robots, fire control and the like. However, most servo systems have non-linear links such as friction and dead zones, which not only affect the tracking accuracy and dynamic characteristics of the system, but also even cause instability of the system in severe cases, thereby affecting the normal operation of the system. The dead zone characteristic is caused by the fact that the motor stops rotating or hardly rotates due to friction and the like in a low-speed state, so that the input of an actuator is equal to 0, the control is nonlinear, and the dynamic characteristic of the system is seriously influenced, and the system deviates from a desired signal. Therefore, how to approximately compensate the nonlinear dead zone link effectively and linearize the input of the actuator, thereby improving the control effect, has become one of the key problems to be solved urgently in the control of the servo system.

Disclosure of Invention

In order to overcome the defects that the partial state and disturbance of the system in the prior art are not measurable, the convergence time of a common sliding mode control method is long, and the problem of buffeting is easily generated, and the influence of the nonlinear dead zone link, friction and other external disturbances of the system is approximately compensated, the invention provides a servo system finite time control method based on an extended state observer, which is used for weakening the influence of the dead zone and the friction on the system, shortening the convergence time of a controller and weakening the buffeting problem. An Extended State Observer (ESO) is adopted to estimate the unmeasured states such as system friction and external disturbance, the dead zone link is compensated, meanwhile, a full-order sliding mode controller is designed to obtain a control quantity, and the system can track the expected signal quickly and stably.

The technical scheme proposed for solving the technical problems is as follows:

a servo system finite time control method considering dead zone characteristics comprises the following steps:

step 1, establishing a servo system model shown in a formula (1), and initializing a system state and control parameters;

wherein, theta_m，The state variables respectively represent the position and the rotating speed of the output shaft of the motor; j and D are equivalent moment of inertia converted to the motor shaft and equivalent damping coefficient; k_tIs the motor torque constant; v (u) is a control amount having a dead zone characteristic,b_ir,b_ilthe upper and lower boundaries of the dead zone characteristic are respectively, and u is the output quantity of the controller; t is the loaded friction torque translated to the motor shaft and the disturbance component of the friction;

step 2, carrying out approximate processing on the dead zone characteristics;

now expand pair g_ir(u),g_il(u) is defined as follows:

$\left\{\begin{array}{cc}{g}_{ir}\left(u\right)=g_{ir}^{\′}\left(b_{ir}\right)(u-b_{ir})& u\∈(b_{il},b_{ir}\]\\ g_{il}\left(u\right)=g_{il}^{\′}\left(b_{il}\right)(u-b_{il})& u\∈\[b_{il},b_{ir})\end{array}\right.---\left(2\right)$

according to the median theorem of differentiation, there is ξ_il∈(-∞,b_il),ξ_ir∈(b_ir, + ∞) makes the following hold:

$\left\{\begin{array}{cc}{g}_{ir}\left(u\right)=g_{ir}^{\′}\left({\mathrm{\ξ}}_{ir}\right)(u-b_{ir})& u\∈\[b_{ir},+\mathrm{\∞})\\ g_{il}\left(u\right)=g_{il}^{\′}\left({\mathrm{\ξ}}_{il}\right)(u-b_{il})& u\∈(-\mathrm{\∞},b_{il}\]\end{array}\right.---\left(3\right)$

wherein, ξ_il＝ξ_lu+(1-ξ_l)b_il,0＜ξ_l＜1，ξ_ir＝ξ_ru+(1-ξ_r)b_ir,0＜ξ_r< 1, then the dead band characteristic v (u) is rewritten to

WhereinIs expressed as follows

The expression of d (u) is:

$d\left(u\right)=\left\{\begin{array}{cc}-g_{ir}^{\&prime;}\left({\mathrm{\&xi;}}_{ir}\right)b_{ir},& u\&GreaterEqual;b_{ir}\\ -\&lsqb;g_{il}^{\&prime;}\left(b_{il}\right)+g_{ir}^{\&prime;}\left(b_{ir}\right)\&rsqb;u& b_{il}<u<b_{ir}\\ -g_{il}^{\&prime;}\left({\mathrm{\&xi;}}_{il}\right)b_{il},& u\&le;b_{il}\end{array}\right.---\left(6\right)$

then the formula (1) is rewritten as

Step 3, designing a nonlinear extended state observer, wherein the process is as follows:

3.1, let x₁＝θ_m，Then the formula (7) is rewritten as

Wherein x is₁，x₂If the system state is satisfied and u is the output of the controller, the formula (8) is rewritten to

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=a\left(x\right)+bu\end{array}\right.---\left(9\right)$

Wherein,x＝[x₁,x₂]，

3.2, let a (x) be a₀+Δa，b＝b₀+ Δ b, d ═ Δ a + Δ bu, where b is₀And a₀The optimal estimated values of b and a (x) are respectively given according to the system structure; defining an extended state x based on the design idea of the extended observer₃D is equal to or less than l_dWherein l is_d(> 0), then equation (9) is rewritten as the equivalent:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=x_3+a_0+b_{0}u\\ {\stackrel{\&CenterDot;}{x}}_3=h\end{array}\right.---\left(10\right)$

wherein,and k is less than or equal to | h |_d，k_dIs a constant;

3.3, order z_iI is 1,2,3, and is the state variable x in formula (10)_iDefining a tracking error e_ci＝z_i ^*-x_iWherein z is_i ^*For the desired signal, the observation error is e_oi＝z_i-x_iThen, the nonlinear extended state observer expression is designed as follows:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2-{\mathrm{\&beta;}}_{l}g\left(e_{o1}\right)\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-{\mathrm{\&beta;}}_{2}g\left(e_{o1}\right)+a_0+b_{0}u\\ {\stackrel{\&CenterDot;}{z}}_3=-{\mathrm{\&beta;}}_{3}g\left(e_{o1}\right)\end{array}\right.---\left(11\right)$

wherein, β₁,β₂,β₃For observer gain parameters, which are determined by pole allocation, g (e)_o1) Is composed of

1,2,3, n +1, wherein, α_j＝[1,0.5,0.25]，＝1°；

Step 4, applying pole allocation methodDetermining an observer gain parameter β₁，β₂，β₃Taking the value of (A);

let x₁＝e_o1＝z₁-x₁，x₂＝z₂-x₂，x₃＝z₃D, subtracting formula (9) from formula (11) to obtain

$\left\{\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_{l}g\left({\mathrm{\&delta;x}}_1\right)\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_{2}g\left({\mathrm{\&delta;x}}_1\right)\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3=-{\mathrm{\&beta;}}_{3}g\left({\mathrm{\&delta;x}}_1\right)-h\end{array}\right.---\left(12\right)$

Let h be bounded, and g (e)_o1) Is smooth, g (0) being 0, g' (e)_o1) Not equal to 0, according to Taylor's formula, equation (12) is written as

$\left\{\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_l{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_2{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3=-{\mathrm{\&beta;}}_3{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1-h\end{array}\right.---\left(13\right)$

Order toEquation (13) is written as the following form of the state space equation

$\left[\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3\end{array}\right]=\left[\begin{array}{ccc}-l_1& 1& 0\\ -l_2& 0& 1\\ -l_3& 0& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ -1\end{array}\right]h---\left(14\right)$

Designing a compensation matrix

$A=\left[\begin{array}{ccc}-l_1& 1& 0\\ -l_2& 0& 1\\ -l_3& 0& 0\end{array}\right],E=\left[\begin{array}{c}0\\ 0\\ -1\end{array}\right],\mathrm{\&delta;}X=\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\end{array}\right],$

Then equation (14) is written as

$\mathrm{\&delta;}\stackrel{\&CenterDot;}{X}=A\mathrm{\&delta;}X+Eh---\left(15\right)$

To this end, parameter β_iIs converted into_iThe requirement for asymptotically stabilizing equation (8) under the influence of the disturbance h is that the eigenvalues of the compensation matrix a all fall on the left half-plane of the complex plane, i.e. the poles of equation (8) are sufficiently negative, whereby the desired pole p is selected according to the pole placement method_i(i is 1,2,3), let parameter l_iSatisfy the requirement of

$|sI-A|=\underset{i=1}{\overset{3}{\&Pi;}}(s-p_i)---\left(16\right)$

I is a unit matrix, and when coefficients of polynomials on the left and right sides with respect to s are equal, a parameter l is obtained₁，l₂，l₃To obtain an expression of the extended state observer as

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2-\frac{l_1}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-\frac{l_2}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)+a_0\left(x\right)+b_{0}u\\ {\stackrel{\&CenterDot;}{z}}_3=-\frac{l_3}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)\end{array}\right.---\left(17\right)$

Step 5, designing a terminal sliding mode controller u based on a full-order terminal sliding mode method, wherein the process is as follows:

5.1, designing a sliding mode as follows:

$s={\stackrel{\&CenterDot;}{e}}_{c2}+{\mathrm{\&lambda;}}_{2}sign\left(e_{c2}\right)|e_{c2}{|}^{c_2}+{\mathrm{\&lambda;}}_{1}sign\left(e_{c1}\right)|e_{c1}{|}^{c_1}---\left(18\right)$

wherein, c₁,c₂And λ₁,λ₂Is a constant, λ₁,λ₂Satisfies the polynomial p²+λ₂p+λ₁The poles of the polynomial are all positioned at the left half of the complex plane,c∈(1-,1),0＜＜1；

5.2, the terminal sliding mode controller is designed as follows:

$\left\{\begin{array}{c}u=b^{-1}(u_{eq}+u_n)\\ u_{eq}=-a_0-{\mathrm{\&lambda;}}_{2}sign\left(e_{c2}\right)|e_{c2}{|}^{c_2}-{\mathrm{\&lambda;}}_{1}sign\left(e_{c1}\right)|e_{c1}{|}^{c_1}\\ {\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n=V\\ V=-(k_d+k_T+\mathrm{\&eta;})sign\left(s\right)\end{array}\right.---\left(19\right)$

wherein T is more than or equal to 0, k_T≥Tl_d，η＞0。

The technical conception of the invention is as follows: the extended state observer (the extended state observer) is a novel nonlinear state observer, and can obtain observers of all states of a system by expanding internal and external disturbances in the system into a new first-order state, then utilizing specific nonlinear error feedback, and then selecting proper observer parameters, wherein the observers also comprise uncertainty of a system model and observed values of unknown disturbance. Therefore, it is possible not only to reproduce the state quantity of the control object but also to estimate the "expanded state" of the uncertainty of the control object model and the real-time value of the disturbance. This is well suited for servo systems where system friction and disturbances are difficult to estimate. However, to date, there is no effective method for determining the parameters of the extended state observer.

The pole allocation method (poleasignment) is a comprehensive principle of moving the poles of a linear steady system to a preset position through the feedback of a proportional link, and the essence of the pole allocation method is to change the free motion mode of the original system by using the proportional feedback so as to meet the design requirements. Because the observation error of the extended state observer can be observed and estimated, the observation error can be regarded as a linear system, and then the characteristic roots of the compensation matrix can all fall on the left half plane of the complex plane through a pole allocation method, so that the whole system is asymptotically stable.

The sliding mode variable structure control method has complete self-adaptability and robustness, once the sliding mode state is entered, the transfer of the system state is not influenced by the change of system parameters and external disturbance any more, but the general sliding mode control has no great advantage in convergence time and has the phenomenon of buffeting generally. The full-order sliding mode can ensure that the system converges to a desired point in a limited time, and meanwhile, the robustness of the system is not influenced. On the basis, how to weaken the buffeting problem of the system while considering the convergence time is the key point for designing the sliding mode controller.

The dynamic performance and the steady-state performance of the system are not high due to the friction force and the external interference in the servo system and the influence of a nonlinear dead zone link. Aiming at a servo system with part of states being undetectable (such as friction) and external disturbance and dead zone characteristics, the servo system full-order sliding mode control method based on the extended state observer is designed by combining the extended state observer and the full-order sliding mode control method, and influences of system friction and dead zone links on the system are eliminated as far as possible. By establishing a new extended state, designing external interference and dead zone links such as the friction of an estimated system of the extended state observer, determining the parameters of the extended state observer by adopting a pole allocation method, and designing a full-order sliding mode controller to enable the system to converge to a desired value within a limited time, thereby realizing the fast and stable control of a servo system.

The invention has the beneficial effects that: the invention combines the extended state observer and the full-order sliding mode control method, designs the servo system full-order sliding mode controller based on the extended state observer, and simultaneously carries out smooth processing on dead zone characteristics and compensates by the observer, thereby realizing the fast and stable position tracking control of the servo system; the influence of the dead zone and friction on the system is weakened, the convergence time of the controller is shortened, and the buffeting problem is weakened; in a simulation experiment, a contrast control method is adopted to highlight the superiority of the method. The invention respectively adopts the following three methods for comparison, namely:

the method comprises the following steps: based on the extended state observer, performing ordinary sliding mode control on a servo system with dead zone link compensation; using s ═ e_c2+λe_c1A slip form face ofWhere λ is 10 and k is 50.

The second method comprises the following steps: based on the extended state observer, performing full-order sliding mode control on a servo system with a dead zone link; the sliding mode surface of the formula (18) and the full-order sliding mode controller of the formula (19) are adopted.

The third method comprises the following steps: based on the extended state observer, performing full-order sliding mode control on a servo system with dead zone link compensation; the sliding mode surface of the formula (18) and the full-order sliding mode controller of the formula (19) are adopted.

Description of the drawings:

FIG. 1(a) shows the system state x of the above three methods₁A response curve;

FIG. 1(b) shows the system state x of the above three methods₂A response curve;

FIG. 2(a) shows the system tracking error e of the above three methods_c1A curve;

FIG. 2(b) shows the system tracking error e of the above three methods_c2A curve;

FIG. 3(a) shows the systematic observation error e of the three methods_o1A curve;

FIG. 3(b) is a diagram showing the systematic observation error e of the above three methods_o2A curve;

FIG. 4 is a graph of system control signal output for the three methods described above;

fig. 5 is a basic flow of the algorithm of the present invention.

The specific implementation mode is as follows:

the invention is further described below with reference to the accompanying drawings.

Referring to fig. 1 to 5, a servo system finite time control method considering a dead zone characteristic includes the steps of:

step 1, establishing a servo system model shown in a formula (1), and initializing a system state and control parameters;

wherein, theta_m，In order to be a state variable, the state variable,respectively representing the position and the rotating speed of an output shaft of the motor; j and D are equivalent moment of inertia converted to the motor shaft and equivalent damping coefficient; k_tIs the motor torque constant; v (u) is a control amount having a dead zone characteristic,b_ir,b_ilthe upper and lower boundaries of the dead zone characteristic are respectively, and u is the output quantity of the controller; t is the loaded friction torque translated to the motor shaft and the disturbance component of the friction;

step 2, carrying out approximate processing on the dead zone characteristics;

now expand pair g_ir(u),g_il(u) is defined as follows:

$\left\{\begin{array}{cc}{g}_{ir}\left(u\right)=g_{ir}^{\&prime;}\left(b_{ir}\right)(u-b_{ir})& u\&Element;(b_{il},b_{ir}\&rsqb;\\ g_{il}\left(u\right)=g_{il}^{\&prime;}\left(b_{il}\right)(u-b_{il})& u\&Element;\&lsqb;b_{il},b_{ir})\end{array}\right.---\left(2\right)$

according to the median theorem of differentiation, there is ξ_il∈(-∞,b_il),ξ_ir∈(b_ir, + ∞) makes the following hold:

$\left\{\begin{array}{cc}{g}_{ir}\left(u\right)=g_{ir}^{\&prime;}\left({\mathrm{\&xi;}}_{ir}\right)(u-b_{ir})& u\&Element;\&lsqb;b_{ir},+\mathrm{\&infin;})\\ g_{il}\left(u\right)=g_{il}^{\&prime;}\left({\mathrm{\&xi;}}_{il}\right)(u-b_{il})& u\&Element;(-\mathrm{\&infin;},b_{il}\&rsqb;\end{array}\right.---\left(3\right)$

wherein, ξ_il＝ξ_lu+(1-ξ_l)b_il,0＜ξ_l＜1，ξ_ir＝ξ_ru+(1-ξ_r)b_ir,0＜ξ_r< 1, then the dead band characteristic v (u) is rewritten to

WhereinIs expressed as follows

The expression of d (u) is:

$d\left(u\right)=\left\{\begin{array}{cc}-g_{ir}^{\&prime;}\left({\mathrm{\&xi;}}_{ir}\right)b_{ir},& u\&GreaterEqual;b_{ir}\\ -\&lsqb;g_{il}^{\&prime;}\left(b_{il}\right)+g_{ir}^{\&prime;}\left(b_{ir}\right)\&rsqb;u& b_{il}<u<b_{ir}\\ -g_{il}^{\&prime;}\left({\mathrm{\&xi;}}_{il}\right)b_{il},& u\&le;b_{il}\end{array}\right.---\left(6\right)$

then the formula (1) is rewritten as

Step 3, designing a nonlinear extended state observer, wherein the process is as follows:

3.1, let x₁＝θ_m，Then the formula (7) is rewritten as

Wherein x is₁，x₂If the system state is satisfied and u is the output of the controller, the formula (8) is rewritten to

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=a\left(x\right)+bu\end{array}\right.---\left(9\right)$

Wherein,x＝[x₁,x₂]，

3.2, let a (x) be a₀+Δa，b＝b₀+ Δ b, d ═ Δ a + Δ bu, where b is₀And a₀The optimal estimated values of b and a (x) are respectively given according to the system structure; defining an extended state x based on the design idea of the extended observer₃D is equal to or less than l_dWherein l is_d(> 0), then equation (9) is rewritten as the equivalent:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=x_3+a_0+b_{0}u\\ {\stackrel{\&CenterDot;}{x}}_3=h\end{array}\right.---\left(10\right)$

wherein,h is less than or equal to kd which is a constant;

3.3, order z_iI is 1,2,3, and is the state variable x in formula (10)_iDefining a tracking error e_ci＝z_i ^*-x_iWherein z is_i ^*For the desired signal, the observation error is e_oi＝z_i-x_iThen, the nonlinear extended state observer expression is designed as follows:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2-{\mathrm{\&beta;}}_{l}g\left(e_{o1}\right)\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-{\mathrm{\&beta;}}_{2}g\left(e_{o1}\right)+a_0+b_{0}u\\ {\stackrel{\&CenterDot;}{z}}_3=-{\mathrm{\&beta;}}_{3}g\left(e_{o1}\right)\end{array}\right.---\left(11\right)$

wherein, β₁,β₂,β₃For observer gain parameters, which are determined by pole allocation, g (e)_o1) Is composed of

1,2,3, n +1, wherein, α_j＝[1,0.5,0.25]，＝1°；

Step 4, determining β observer gain parameters by pole allocation method₁，β₂，β₃Taking the value of (A);

let x₁＝e_o1＝z₁-x₁，x₂＝z₂-x₂，x₃＝z₃D, subtracting formula (9) from formula (11) to obtain

$\left\{\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_{l}g\left({\mathrm{\&delta;x}}_1\right)\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_{2}g\left({\mathrm{\&delta;x}}_1\right)\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3=-{\mathrm{\&beta;}}_{3}g\left({\mathrm{\&delta;x}}_1\right)-h\end{array}\right.---\left(12\right)$

Let h be bounded, and g (e)_o1) Is smooth, g (0) being 0, g' (e)_o1) Not equal to 0, according to Taylor's formula, equation (12) is written as

$\left\{\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_l{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_2{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3=-{\mathrm{\&beta;}}_3{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1-h\end{array}\right.---\left(13\right)$

Order toEquation (13) is written as the following form of the state space equation

$\left[\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3\end{array}\right]=\left[\begin{array}{ccc}-l_1& 1& 0\\ -l_2& 0& 1\\ -l_3& 0& 0\end{array}\right]\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ -1\end{array}\right]h---\left(14\right)$

Designing a compensation matrix

$A=\left[\begin{array}{ccc}-l_1& 1& 0\\ -l_2& 0& 1\\ -l_3& 0& 0\end{array}\right],E=\left[\begin{array}{c}0\\ 0\\ -1\end{array}\right],\mathrm{\&delta;}X=\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\end{array}\right],$

Then equation (14) is written as

$\mathrm{\&delta;}\stackrel{\&CenterDot;}{X}=A\mathrm{\&delta;}X+Eh---\left(15\right)$

To this end, parameter β_iIs converted into_iThe requirement for asymptotically stabilizing equation (8) under the action of the disturbance h is that the eigenvalues of the compensation matrix A all fall on the left half plane of the complex plane, i.e. equation (8)Is sufficiently negative, whereby, according to the pole placement method, the desired pole p is selected_i(i is 1,2,3), let parameter l_iSatisfy the requirement of

$|sI-A|=\underset{i=1}{\overset{3}{\&Pi;}}(s-p_i)---\left(16\right)$

I is a unit matrix, and when coefficients of polynomials on the left and right sides with respect to s are equal, a parameter l is obtained₁，l₂，l₃To obtain an expression of the extended state observer as

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2-\frac{l_1}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-\frac{l_2}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)+a_0\left(x\right)+b_{0}u\\ {\stackrel{\&CenterDot;}{z}}_3=-\frac{l_3}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)\end{array}\right.---\left(17\right)$

Step 5, designing a terminal sliding mode controller u based on a full-order terminal sliding mode method, wherein the process is as follows:

5.1, designing a sliding mode as follows:

$s={\stackrel{\&CenterDot;}{e}}_{c2}+{\mathrm{\&lambda;}}_{2}sign\left(e_{c2}\right)|e_{c2}{|}^{c_2}+{\mathrm{\&lambda;}}_{1}sign\left(e_{c1}\right)|e_{c1}{|}^{c_1}---\left(18\right)$

wherein, c₁,c₂And λ₁,λ₂Is a constant, λ₁,λ₂Satisfies the polynomial p²+λ₂p+λ₁The poles of the polynomial are all positioned at the left half of the complex plane,c∈(1-,1),0＜＜1；

5.2, the terminal sliding mode controller is designed as follows:

$\left\{\begin{array}{c}u=b^{-1}(u_{eq}+u_n)\\ u_{eq}=-a_0-{\mathrm{\&lambda;}}_{2}sign\left(e_{c2}\right)|e_{c2}{|}^{c_2}-{\mathrm{\&lambda;}}_{1}sign\left(e_{c1}\right)|e_{c1}{|}^{c_1}\\ {\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n=V\\ V=-(k_d+k_T+\mathrm{\&eta;})sign\left(s\right)\end{array}\right.---\left(19\right)$

wherein T is more than or equal to 0, k_T≥Tl_d，η＞0。

In order to verify the effectiveness and superiority of the method, the invention carries out simulation experiment by comparing the control method, sets the initial conditions and partial parameters in the simulation experiment, namely: system equation where J is 0.5, K_tD is 0.3 as 1. All gain parameters in the extended state observer calculated by a pole allocation method are l₁＝60，l₂＝1200，l₃＝8000，b₀＝10,a₀55. The parameters of the second method and the third method are the same: lambda [ alpha ]₁＝22,λ₂＝160，T＝0.01,k_d+k_T+ η is 180 the expression for the dead band characteristic:

$v\left(u\right)=\left\{\begin{array}{cc}(1-0.3\sin u)(u-0.5)& 0\&GreaterEqual;0.5\\ 0& -0.25<u<0.5\\ (0.8-0.2\cos u)(u+0.25)& u\&le;-0.25\end{array}\right.---\left(20\right)$

as can be seen from fig. 1(a), the third method can basically track the desired signal after almost 0.5s, while the first method barely tracks the desired signal after 1s, which reflects the feature of short convergence time of the full-order sliding mode control method in terms of tracking performance. In the second method, the dead zone characteristic is not compensated, so that the expected signal is tracked only after the system is in a half period, and the tracking error is larger than that in the first method and the third method, which shows the effectiveness of the approximate compensation dead zone method.

As can be seen from fig. 1(b) -4, the system state and the control amount of the first method are always in a larger buffeting amplitude, while the buffeting problem of the second method and the third method is obviously smaller, which shows the superiority of the full-order sliding mode control method in reducing the buffeting phenomenon.

On the whole, the servo system full-order sliding mode control method based on the extended state observer can greatly weaken the buffeting problem of the system, shorten the convergence time, approximately compensate the buffeting problem under the influence of dead zone characteristics, reduce the tracking error of the system and enable the system to track the expected signal more quickly and stably.

Compared with a comparison example of other methods, the method provided by the invention has the advantages that the external interferences such as nonlinear dead zone links, friction and the like in the compensation system can be effectively estimated, the buffeting problem in sliding mode control is greatly reduced, the robustness and anti-interference performance of the system are enhanced, the convergence performance is enhanced, and the system can quickly and stably track the expected signal. It is obvious that the invention is not limited to the above-described examples, but that different systems can be controlled precisely on the basis of the invention.

Claims (1)

1. A servo system finite time control method considering dead zone characteristics is characterized in that: the control method comprises the following steps:

step 1, establishing a servo system model shown in a formula (1), and initializing a system state and control parameters;

wherein, theta_m，The state variables respectively represent the position and the rotating speed of the output shaft of the motor; j and D are equivalent moment of inertia converted to the motor shaft and equivalent damping coefficient; k_tIs the motor torque constant; v (u) is a control amount having a dead zone characteristic,b_ir,b_ilthe upper and lower boundaries of the dead zone characteristic are respectively, and u is the output quantity of the controller; t is the loaded friction torque translated to the motor shaft and the disturbance component of the friction;

step 2, carrying out approximate processing on the dead zone characteristics;

now expand pair g_ir(u),g_il(u) is defined as follows:

$\left\{\begin{array}{cc}{g}_{ir}\left(u\right)=g_{ir}^{\&prime;}\left(b_{ir}\right)\left(u-b_{ir}\right)& u\&Element;(b_{il},b_{ir}\&rsqb;\\ g_{il}\left(u\right)=g_{il}^{\&prime;}\left(b_{il}\right)\left(u-b_{il}\right)& u\&Element;\&lsqb;b_{il},b_{ir})\end{array}\right.---\left(2\right)$

according to the median theorem of differentiation, there is ξ_il∈(-∞,b_il),ξ_ir∈(b_ir, + ∞) makes the following hold:

$\left\{\begin{array}{cc}{g}_{ir}\left(u\right)=g_{ir}^{\&prime;}\left({\mathrm{\&xi;}}_{ir}\right)\left(u-b_{ir}\right)& u\&Element;\&lsqb;b_{ir},+\mathrm{\&infin;})\\ g_{il}\left(u\right)=g_{il}^{\&prime;}\left({\mathrm{\&xi;}}_{il}\right)\left(u-b_{il}\right)& u\&Element;(-\mathrm{\&infin;},b_{il}\&rsqb;\end{array}\right.---\left(3\right)$

wherein, ξ_il＝ξ_lu+(1-ξ_l)b_il,0＜ξ_l＜1，ξ_ir＝ξ_ru+(1-ξ_r)b_ir,0＜ξ_r< 1, then the dead band characteristic v (u) is rewritten to

WhereinIs expressed as follows

The expression of d (u) is:

$d\left(u\right)=\left\{\begin{array}{cc}-g_{ir}^{\&prime;}\left({\mathrm{\&xi;}}_{ir}\right)b_{ir},& u\&GreaterEqual;b_{ir}\\ -\&lsqb;g_{il}^{\&prime;}\left(b_{il}\right)+g_{ir}^{\&prime;}\left(b_{ir}\right)\&rsqb;u& b_{il}<u<b_{ir}\\ -g_{il}^{\&prime;}\left({\mathrm{\&xi;}}_{il}\right)b_{il},& u\&le;b_{il}\end{array}\right.---\left(6\right)$

then the formula (1) is rewritten as

Step 3, designing a nonlinear extended state observer, wherein the process is as follows:

3.1, let x₁＝θ_m，Then the formula (7) is rewritten as

Wherein x is₁，x₂If the system state is satisfied and u is the output of the controller, the formula (8) is rewritten to

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=a\left(x\right)+bu\end{array}\right.---\left(9\right)$

Wherein,

3.2, let a (x) be a₀+Δa，b＝b₀+ Δ b, d ═ Δ a + Δ bu, where b is₀And a₀The optimal estimated values of b and a (x) are respectively given according to the system structure; defining an extended state x based on the design idea of the extended observer₃D is equal to or less than l_dWherein l is_d(> 0), then equation (9) is rewritten as the equivalent:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=x_3+a_0+b_{0}u\\ {\stackrel{\&CenterDot;}{x}}_3=h\end{array}\right.---\left(10\right)$

wherein,and k is less than or equal to | h |_d，k_dIs a constant;

3.3, order z_iI is 1,2,3, and is the state variable x in formula (10)_iDefining a tracking error e_ci＝z_i ^*-x_iWherein z is_i ^*For the desired signal, the observation error is e_oi＝z_i-x_iThen, the nonlinear extended state observer expression is designed as follows:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2-{\mathrm{\&beta;}}_{1}g\left(e_{o1}\right)\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-{\mathrm{\&beta;}}_{2}g\left(e_{o1}\right)+a_0+b_{0}u\\ {\stackrel{\&CenterDot;}{z}}_3=-{\mathrm{\&beta;}}_{3}g\left(e_{o1}\right)\end{array}\right.---\left(11\right)$

wherein, β₁,β₂,β₃For observer gain parameters, which are determined by pole allocation, g (e)_o1) Is composed of

Wherein, α_j＝[1,0.5,0.25]，＝1°；

Step 4, determining β observer gain parameters by pole allocation method₁，β₂，β₃Taking the value of (A);

let x₁＝e_o1＝z₁-x₁，x₂＝z₂-x₂，x₃＝z₃D, subtracting formula (9) from formula (11) to obtain

$\left\{\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_{1}g\left({\mathrm{\&delta;x}}_1\right)\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_{2}g\left({\mathrm{\&delta;x}}_1\right)\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3=-{\mathrm{\&beta;}}_{3}g\left({\mathrm{\&delta;x}}_1\right)-h\end{array}\right.---\left(12\right)$

Let h be bounded, and g (e)_o1) Is smooth, g (0) being 0, g' (e)_o1) Not equal to 0, according to Taylor's formula, equation (12) is written as

$\left\{\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1={\mathrm{\&delta;x}}_2-{\mathrm{\&beta;}}_1{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2={\mathrm{\&delta;x}}_3-{\mathrm{\&beta;}}_2{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3=-{\mathrm{\&beta;}}_3{g}^{\&prime;}\left({\mathrm{\&delta;x}}_1\right){\mathrm{\&delta;x}}_1-h\end{array}\right.---\left(13\right)$

Order toEquation (13) is written as the following form of the state space equation

$\left[\begin{array}{c}\mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_1\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_2\\ \mathrm{\&delta;}{\stackrel{\&CenterDot;}{x}}_3\end{array}\right]=\left[\begin{array}{ccc}-l_1& 1& 0\\ -l_2& 0& 1\\ -l_3& 0& 0\end{array}\right]\left[\begin{array}{c}{\mathrm{\&delta;x}}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ -1\end{array}\right]h---\left(14\right)$

Designing a compensation matrix

$A=\left[\begin{array}{ccc}-l_1& 1& 0\\ -l_2& 0& 1\\ -l_3& 0& 0\end{array}\right],E=\left[\begin{array}{c}0\\ 0\\ -1\end{array}\right],\mathrm{\&delta;}X=\left[\begin{array}{c}\mathrm{\&delta;}{x}_1\\ {\mathrm{\&delta;x}}_2\\ {\mathrm{\&delta;x}}_3\end{array}\right],$

Then equation (14) is written as

$\mathrm{\&delta;}\stackrel{\&CenterDot;}{X}=A\mathrm{\&delta;}X+Eh---\left(15\right)$

To this end, parameter β_iIs converted into_iThe requirement for asymptotically stabilizing equation (8) under the influence of the disturbance h is that the eigenvalues of the compensation matrix a all fall on the left half-plane of the complex plane, i.e. the poles of equation (8) are sufficiently negative, whereby the desired pole p is selected according to the pole placement method_i(i is 1,2,3), let parameter l_iSatisfy the requirement of

$|sI-A|=\underset{i=1}{\overset{3}{\&Pi;}}(s-p_i)---\left(16\right)$

I is a unit matrix, and when coefficients of polynomials on the left and right sides with respect to s are equal, a parameter l is obtained₁，l₂，l₃To obtain an expression of the extended state observer as

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1=z_2-\frac{l_1}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-\frac{l_2}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)+a_0\left(x\right)+b_{0}u\\ {\stackrel{\&CenterDot;}{z}}_3=-\frac{l_3}{g^{\&prime;}\left(e_{o1}\right)}g\left(e_{o1}\right)\end{array}\right.---\left(17\right)$

Step 5, designing a terminal sliding mode controller u based on a full-order terminal sliding mode method, wherein the process is as follows:

5.1, designing a sliding mode as follows:

$s={\stackrel{\&CenterDot;}{e}}_{c2}+{\mathrm{\&lambda;}}_{2}sign\left(e_{c2}\right)|e_{c2}{|}^{c_2}+{\mathrm{\&lambda;}}_{1}sign\left(e_{c1}\right)|e_{c1}{|}^{c_1}---\left(18\right)$

wherein, c₁,c₂And λ₁,λ₂Is a constant, λ₁,λ₂Satisfies the polynomial p²+λ₂p+λ₁The poles of the polynomial are all positioned at the left half of the complex plane,

5.2, the terminal sliding mode controller is designed as follows:

$\left\{\begin{array}{c}u=b^{-1}(u_{eq}+u_n)\\ u_{eq}=-a_0-{\mathrm{\&lambda;}}_{2}sign\left(e_{c2}\right)|e_{c2}{|}^{c_2}-{\mathrm{\&lambda;}}_{1}sign\left(e_{c1}\right)|e_{c1}{|}^{c_1}\\ {\stackrel{\&CenterDot;}{u}}_n+{\mathrm{Tu}}_n=V\\ V=-(k_d+k_T+\mathrm{\&eta;})sign\left(s\right)\end{array}\right.---\left(19\right)$

wherein T is more than or equal to 0, k_T≥Tl_d，η＞0。