CN103529706A - Method for controlling error to be converged in fixed time - Google Patents

Abstract

The invention relates to a method for controlling a tracking error to be converged in fixed time, and belongs to the technical field of controlling. The method comprises the steps as follows: a dynamic model of a second-order uncertain system is established; a time-varying sliding mode function and a nonsingular terminal sliding mode function are designed, and a time-varying sliding mode controlled quantity and a nonsingular terminal sliding mode controlled quantity are obtained through respective solving and input into an established system model, so that the error is converged to 0 at the expected moment; and a nonsingular terminal sliding mode control technology is combined to design the controlled quantity, thus, the error in remaining time is kept 0. With the adoption of the method, the expected error convergence time can be set in advance, the system state is always maintained in the sliding mode, and the controlled system has overall robustness on parameter uncertainty and external disturbance.

Description

A kind of error is with the control method of set time convergence

Technical field

The present invention relates to a kind of tracking error that makes with the control method of set time convergence, belong to control technology field.

Background technology

As a class basic skills of robust control, sliding formwork control technology possesses lot of advantages, such as: to parameter change insensitive, can resist external disturbance and fast dynamic response etc.Yet traditional sliding formwork is controlled and only can be guaranteed that system is progressive stable, tracking error is extremely zero at infinite time Convergence.In control operation in real time, Infinite Time convergence property is inadequate often.

Finite time convergence can provide more superior characteristic, for example: rate of convergence faster, higher precision, to uncertain and the better robustness of external disturbance etc.In order to realize the dynamic finite time convergence of system, there is scholar to propose terminal sliding mode control method.After the method can make system dynamically arrive sliding-mode surface, error converges to 0 in finite time.On the basis of this theory, scholars have proposed again fast terminal sliding-mode control, and error convergence speed is further got a promotion.Yet, in terminal sliding mode control procedure, may run into singular problem.In order to overcome this defect, scholars have proposed nonsingular terminal sliding mode control technology.The method can be resolved singular problem in the situation that not adding additional procedure.In recent years, a lot of scholars have carried out combination by artificial intelligence approach and terminal sliding mode control method, thereby when having kept terminal sliding mode control method advantage, make chattering phenomenon obtain good inhibition.

Finite time control problem has obtained the extensive concern of Chinese scholars, yet for the control problem of set time convergence, correlative study is but also few.According to author's understanding, one piece of document that the people such as Laghrouche deliver relates to this field content.Corresponding controller design is divided into two parts, and a part is integral sliding mode control, is used for offsetting polymerization disturbance; Another part is that a class Optimal Feedback is controlled, and guarantees the convergence of error set time.Yet this control law is only for single-input single-output system.Helpless for mimo systems.Therefore, need to propose a kind of control method simple and that tally with the actual situation and solve the problem in this field.

Summary of the invention

The present invention is for solving the control problem of error set time convergence, a kind of control method that error was restrained with the set time based on sliding formwork control technology has been proposed, make error converge to 0 in the moment of expectation, and in conjunction with nonsingular terminal sliding mode control technology, design controlled quentity controlled variable makes error in excess time remain zero.

Technical scheme of the present invention is specific as follows:

Step 1, set up the dynamic model of second order uncertain system:

${\stackrel{\·}{x}}_1=x_2$

${\stackrel{\·}{x}}_2=f\left(x\right)+g\left(x\right)+b\left(x\right)u$

Wherein, x=[x ₁, x ₂] ^tfor system state vector, f (x) and b (x) ≠ 0 are the smooth nonlinear function about x, and g (x) represents uncertain and external disturbance and satisfied || g (x) || and≤l _g, l wherein _g> 0, and u is system control inputs.The reference locus of system is x _1d, the convergence time of expectation is t _f.

Step 2, design finite time control law

The target of design is: system state is from any initial value, at the moment (t of expectation _f) reference locus in tracking, and this constantly after, tracking error remains 0 always.

t>=t _f.Definition tracking error is as follows:

$\begin{array}{cc}{\tilde{x}}_1=x_1-x_{1d}& {\tilde{x}}_2=x_2-{\stackrel{\·}{x}}_{1d}\end{array}$

Step 2.1, becomes sliding formwork function during design

During design, become sliding formwork function as follows:

${\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">S</figure-callout>}}_1=(t_f-t){\tilde{x}}_2+n(1-\frac{1}{e^{{\tilde{x}}_1}})+\left\{\begin{array}{cc}{\mathrm{<figure-callout\; id="2"\; label="At"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">At</figure-callout>}}^2+\mathrm{Bt}+C& t\&le;T\\ 0& t>T^{\&prime;}\end{array}\right.,0\&le;t\&le;t_f$

N > 2,0 < T < t wherein _f, and A, B, C meets following equation:

$C=-t_f{\tilde{x}}_2\left(0\right)-n(1-\frac{1}{e^{{\tilde{x}}_1\left(0\right)}}),A=\frac{C}{T^2},B=-2\mathrm{AT}$

Step 2.2, solves and while obtaining, becomes sliding-mode control law

Solve controlled quentity controlled variable, make system state at t _fconstantly follow the tracks of upper reference locus; While solving by Lyapunov method, become sliding formwork function, obtain becoming when following sliding-mode control law:

$u_1=b^{-1}\left(x\right)[\frac{1}{t_f-t}({\tilde{x}}_2-\frac{n{\tilde{x}}_2}{e^{{\tilde{x}}_1}}-\left\{\begin{array}{cc}2\mathrm{At}+B& t\&le;T\\ 0& t>T\end{array}\right.)-f\left(x\right)+{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}-{\mathrm{\&eta;}}_1\mathrm{sgn}\left(S_1\right)],0\&le;t\&le;t_f$

η wherein ₁> l _g.

Step 2.3, designs nonsingular terminal sliding mode function

Design nonsingular terminal sliding mode function as follows:

$S_2={\tilde{x}}_1+K{\left|{\tilde{x}}_2\right|}^{\mathrm{\&beta;}}\mathrm{sgn}\left({\tilde{x}}_2\right)t>t_f$

K > 0,1 < β < 2 wherein.

Step 2.4, solves and obtains nonsingular terminal sliding mode controlled quentity controlled variable

Solve controlled quentity controlled variable, make tracking error at t > t _fin time, remain 0; By Lyapunov method, solve nonsingular terminal sliding mode function, obtain corresponding nonsingular terminal sliding mode controlled quentity controlled variable:

$u_2=-b^{-1}\left(x\right)(\frac{{\tilde{x}}_2}{\mathrm{K\&beta;}{\left|{\tilde{x}}_2\right|}^{\mathrm{\&beta;}-1}}+f\left(x\right)-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}+{\mathrm{\&eta;}}_2\mathrm{sgn}\left(S_2\right)),t>t_f$

Wherein, η ₂> l _g.

Step 3, the controlled quentity controlled variable u that step 2 is obtained ₁and u ₂(wherein at 0≤t≤t _fin time, select u ₁, at t > t _fin time, select u ₂) system model that input step 1 is set up, can make tracking error at the time Convergence to 0 of expectation.

Beneficial effect

The present invention has three aspects: advantage: 1. the error convergence time of expectation can be given in advance, and 2. the rate of convergence of error can be by regulating the value of parameter n to realize, and n value is larger, and error convergence must be faster.3. system state is always on sliding-mode surface, and controlled system has global robustness to parameter uncertainty and external disturbance.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of the inventive method;

Fig. 2 is system state x in embodiment ₁aircraft pursuit course;

Fig. 3 is system state x in embodiment ₂aircraft pursuit course;

Fig. 4 is steering order response curve in embodiment;

Fig. 5 is sliding formwork function curve in embodiment;

Fig. 6 is the error convergence curve in different n value situations in embodiment;

Embodiment

In order better to illustrate, below in conjunction with accompanying drawing and example, technical scheme is described in further details objects and advantages of the present invention.

1. finite time design of control law.

Step 1. is set up the dynamic model of second order uncertain system:

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=f\left(x\right)+g\left(x\right)+b\left(x\right)u\end{array}---\left(1\right)$

Wherein, x=[x ₁, x ₂] ^tfor system state vector, f (x) and b (x) ≠ 0 are the level and smooth nonlinear function about x, and g (x) represents uncertain and external disturbance and satisfied || g (x) || and≤l _g, l wherein _g> 0, and u is system control inputs.The reference locus of supposing the system is x _1d, the convergence time of expectation is t _f.

Step 2, design finite time control law

The target of design is: system state is from any initial value, at the moment (t of expectation _f) reference locus in tracking, and this constantly after, tracking error remains 0 always.

t>=t _f.Definition tracking error is as follows:

$\begin{array}{cc}{\tilde{x}}_1=x_1-x_{1d}& {\tilde{x}}_2=x_2-{\stackrel{\&CenterDot;}{x}}_{1d}\end{array}---\left(2\right)$

First, during employing, become sliding-mode control design control law, make error at the time Convergence to 0 of expectation; Then applying nonsingular terminal sliding mode control method makes tracking error remain 0 within the remaining time.

During step 2.1, become sliding formwork design of control law

During design, become sliding formwork function as follows:

S ₁＝σ(t)+α(t) 0≤t≤t _f （3）

Wherein

n > 2,0 < T < t _f, variable when α (t) is, its initial value α ₀with end value α _fneed meet into lower relation:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_0=-\mathrm{\&sigma;}\left(0\right)\\ {\mathrm{\&alpha;}}_f=0\\ {\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_f=0\end{array}\right.---\left(4\right)$

Selecting α (t) is following truncation funcation.

$\mathrm{\&alpha;}\left(t\right)=\left\{\begin{array}{cc}{\mathrm{At}}^2+\mathrm{Bt}+C& t\&le;T\\ 0& t>T\end{array}\right.---\left(5\right)$

Wherein T is switching time.According to formula (5), A, B, C can be determined by following formula:

$C=-\mathrm{\&sigma;}\left(0\right),A=\frac{\mathrm{<figure-callout\; id="2"\; label="C\; T"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">C</figure-callout>}}{T^{}},B=-2\mathrm{AT}---\left(6\right)$

Based on time become sliding formwork function (3), design and become sliding formwork control law when following:

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1=b^{-1}\left(x\right)[\frac{1}{t_f-t}({\tilde{x}}_2-\frac{n{\tilde{x}}_2}{e^{{\tilde{x}}_1}}-\left\{\begin{array}{cc}2\mathrm{At}+B& t\&le;T\\ 0& t>T\end{array}\right.)-f\left(x\right)+{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}-{\mathrm{\&eta;}}_1\mathrm{sgn}\left(S_1\right)],0\&le;t\&le;t_f---\left(7\right)$

η wherein ₁> l _g, n > 2,0 < T < t _f.

Theorem 1. is for the represented uncertain second-order system of formula (1), shown in selecting type (7) time become sliding formwork control law, tracking error

with will be at t _fconstantly converge to 0 simultaneously.

Proof: the Lyapunov function that is defined as follows positive definite:

${\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">V</figure-callout>}}_1=\frac{1}{2}{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">S</figure-callout>}}_1^2---\left(8\right)$

Formula (8) is carried out to differential, can obtain:

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_1=S_1{\stackrel{\&CenterDot;}{S}}_1\\ =S_1(\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}+\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}\left(t\right))\\ =S_1[-{\tilde{x}}_2+(t_f-t){\stackrel{\&CenterDot;}{x}}_2+\frac{n{\tilde{x}}_2}{e^{{\tilde{x}}_1}}+\left\{\begin{array}{cc}2\mathrm{At}+B& t\&le;T\\ 0& t>T\end{array}\right.\\ =S_1[-{\tilde{x}}_2+(t_f-t)(f\left(x\right)-g\left(x\right)+b\left(x\right)u-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d})+\frac{n{\tilde{x}}_2}{e^{{\tilde{x}}_1}}+\left\{\begin{array}{cc}2\mathrm{At}+B& t\&le;T\\ 0& t>T\end{array}\right.\end{array}----\left(9\right)$

Then, bring control law formula (7) into formula (9), can obtain:

${\stackrel{\&CenterDot;}{V}}_1=S_1(t_f-t)(-{\mathrm{\&eta;}}_1\mathrm{sgn}\left(S_1\right)+g\left(x\right))\&le;-(t_f-t)({\mathrm{\&eta;}}_1-l_g)\left|{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">S</figure-callout>}}_1\right|\&le;0---\left(10\right)$

Obviously, for S arbitrarily ₁(t) all have

just non-, therefore, obtain V≤V (0).Due to sliding formwork function initial value S ₁(0)=0, therefore obtains V (0)=0.V ₁≤ 0.On the other hand, from formula (8), for any S ₁(t), all there is V ₁>=0.In sum, for t ∈ [0, t _f], there is V ₁≡ 0.Therefore there is S ₁≡ 0.Due at T≤t≤t _ftime α (t)=0, therefore, within this period, have σ (t)=0.

By σ (t)=0, can obtain following relational expression:

${\tilde{x}}_2=\frac{d{\tilde{x}}_1}{\mathrm{dt}}=-\frac{e^{{\tilde{x}}_1}-1}{e^{{\tilde{x}}_1}}\frac{n}{t_f-t}---\left(11\right)$

Through arranging, can obtain:

$\frac{e^{{\tilde{x}}_1}}{e^{{\tilde{x}}_1}-1}d{\tilde{x}}_1=-\frac{n}{t_f-t}\mathrm{dt}---\left(12\right)$

Assumed initial state is

above formula both members is carried out to integration can be obtained

$e^{{\tilde{x}}_1}=1+\frac{e^{{\tilde{x}}_{1r}}-1}{{(t_f-t_1)}^n}{(t_f-t)}^n---\left(13\right)$

Order

known b is constant.By formula (13), can be pushed away with

analytic solution as follows:

$\left\{\begin{array}{c}{\tilde{x}}_1=\ln (1+b{(t_f-t)}^n)\\ {\tilde{x}}_2=\frac{-\mathrm{bn}{(t_f-t)}^{n-1}}{1+b{(t_f-t)}^n}\end{array}\right.---\left(14\right)$

Due to n > 2, by formula (14), can be found out,

with

will be at t=t _fconstantly converge to 0.

Card is finished.

From formula (7), can find out t _f-t appears on the denominator position of controlled quentity controlled variable.Therefore, at t _fconstantly will cause unusual.What at this constantly, formula (7) can equivalence is expressed as:

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1=b^{-1}\left(x\right)[\frac{1}{t_f-t}({\tilde{x}}_2-\frac{n{\tilde{x}}_2}{e^{{\tilde{x}}_1}})-f\left(x\right)+{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}-{\mathrm{\&eta;}}_1\mathrm{sgn}\left(S_1\right)]---\left(15\right)$

Due to now meet formula (14), bring this result into formula (15), can obtain

From formula (16), can find out, as long as n value is greater than 2, singular problem has just been avoided.

Along with t around _f,

with

track will be as the formula (14).Therefore, be easy to following equation:

$\left\{\begin{array}{c}\underset{t\&RightArrow;t_f}{\lim }{\tilde{x}}_1=\underset{t\&RightArrow;t_f}{\lim }\ln (1+b{(t_f-t)}^n)=b{(t_f-t)}^n\\ \underset{t\&RightArrow;t_f}{\lim }{\tilde{x}}_2=\underset{t\&RightArrow;t_f}{\lim }\frac{-\mathrm{bn}{(t_f-t)}^{n-1}}{1+b{(t_f-t)}^n}=-\mathrm{bn}{(t_f-t)}^{n-1}\end{array}\right.---\left(17\right)$

Therefore, approaching t _fin the time of constantly,

with

there is specific track.Further analysis is known, and the rate of convergence of error can be by regulating parameter n obtain.N is larger, and speed of convergence is faster.

By formula (14), can be found out, at t > t _ftime, tracking error will be started from scratch along with the time increases and increases.Therefore need to design another kind of control law makes tracking error remain zero.

Step 2.2 terminal sliding mode design of control law

Design nonsingular terminal sliding mode function as follows:

${\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">S</figure-callout>}}_2={\tilde{x}}_1+K{\left|{\tilde{x}}_2\right|}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\mathrm{sgn}\left({\tilde{x}}_2\right)---\left(18\right)$

Wherein K > 0, p ₁, p ₂for positive odd number and satisfied 1 < p ₁/ p ₂< 2.For convenient statement, by p ₁/ p ₂with β, represent.Formula (18) differential can be obtained

${\stackrel{\&CenterDot;}{S}}_2={\tilde{x}}_2+\mathrm{K\&beta;}{\left|{\tilde{x}}_2\right|}^{\mathrm{\&beta;}-1}{\stackrel{\&CenterDot;}{\tilde{x}}}_2---\left(19\right)$

Due to 1 < β < 2, and at t=t _fconstantly

with

be equal to 0, therefore, by formula (18) and formula (19), can be found out,

consider again formula (3), obviously S ₁(t _f)=0.To S ₁differentiate obtains:

${\stackrel{\&CenterDot;}{S}}_1=-{\tilde{x}}_2+(t_f-t){\stackrel{\&CenterDot;}{\tilde{x}}}_2+\frac{n{\tilde{x}}_2}{e^{{\tilde{x}}_1}}+\left\{\begin{array}{cc}2\mathrm{At}+B& t\&le;T\\ 0& t>T\end{array}\right.---\left(20\right)$

By

equal 0, obviously have therefore, at t _fconstantly, have

therefore, two designed sliding-mode surface S ₁and S ₂at t _fconstantly smoothly join.

Nonsingular terminal sliding mode function (18) for design, is constructed as follows control law:

${\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">u</figure-callout>}}_2=b^{-1}\left(x\right)(-\frac{{\tilde{x}}_2}{\mathrm{K\&beta;}{\left|{\tilde{x}}_2\right|}^{\mathrm{\&beta;}-1}}-f\left(x\right)+{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}-{\mathrm{\&eta;}}_2\mathrm{sgn}\left(S_2\right))---\left(21\right)$

η wherein ₂> l _g.

Theorem 2. is for nonlinear second-order system (1), if as t ∈ [0, t _f] time, shown in employing formula (7) time become sliding formwork control law, as t > t _ftime, the nonsingular terminal sliding mode control law shown in employing formula (21), can obtain drawing a conclusion:

(1) controlled system has global robustness to model uncertainty and external disturbance;

(2) system tracking error will be in t _fconvergence constantly also remains zero after this always.

Proof: the Lyapunov function that is defined as follows positive definite

${\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">V</figure-callout>}}_2=\frac{1}{2}{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">S</figure-callout>}}_2^2---\left(22\right)$

By V ₂carry out differential and formula (21) brought into, obtaining:

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_2=S_2({\tilde{x}}_2+\mathrm{K\&beta;}{\left|{\tilde{x}}_2\right|}^{\mathrm{\&beta;}-1}(f\left(x\right)+g\left(x\right)+b\left(x\right)u-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}))\\ =S_2(-{\mathrm{\&eta;}}_2\mathrm{sgn}\left(S_2\right)+g\left(x\right))\\ \&le;-({\mathrm{\&eta;}}_2-l_g)\left|{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">S</figure-callout>}}_2\right|\&le;0\end{array}---\left(23\right)$

Similar with the proof of theorem 1, can push away to obtain t > t _fin time, V ₂≡ 0, therefore there is S ₂≡ 0.Because theorem 1 has proved [0, the t at t ∈ _f] in the time period, by S ₁therefore ≡ 0, can show that system state is always on sliding-mode surface, and controlled system has global robustness to model uncertainty and external disturbance.

By S ₂≡ 0 can obtain following formula:

${\tilde{x}}_2=-K_1{\tilde{x}}_1^{{\mathrm{\&beta;}}_1}---\left(24\right)$

β wherein ₁=1/ β,

verified in other documents

terminal attractors for formula (24).Consider t=t _fconstantly, have

set up, after this, with

will remain zero.

Card is finished.

Although adopted two kinds of different control strategies in the whole control stage, yet if n > 2 sets up, (be t at control strategy switching instant _f), continuous during steering order.Concrete reason is as described below.At t _fconstantly, if n > 2 sets up, formula (7) can be reduced to

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">u</figure-callout>}}_1=b^{-1}\left(x\right)({\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}-f\left(x\right))---\left(25\right)$

At this constantly, have 1 < γ < 2 and therefore formula (21) can be reduced to

${\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="HDA0000399709910000012.png,HDA0000399709910000021.png"\; state="\{\{state\}\}">u</figure-callout>}}_2=b^{-1}\left(x\right)({\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}-f\left(x\right))---\left(26\right)$

Obviously, u ₁(t _r)=u ₂(t _r).

Because system state is from starting always in sliding-mode surface, overall buffeting problem will be inevitable.In order to reduce controlled quentity controlled variable, buffet, adopted following saturation function to replace switching function sgn (S):

$\mathrm{sat}\left(S\right)=\left\{\begin{array}{cc}{\mathrm{\&epsiv;}}^{-1}S,& \left|S\right|\&le;\mathrm{\&epsiv;}\\ \mathrm{sgn}\left(S\right),& \mathrm{otherwise}\end{array}\right.---\left(27\right)$

Wherein, ε is boundary layer thickness.Boundary layer is thicker, suppress buffeting effect better, yet tracking accuracy also decreases.Therefore need the value of compromise selection ε.

The controlled quentity controlled variable u that step 3. obtains step 2 ₁and u ₂(wherein at 0≤t≤t _fin time, select u ₁, at t > t _fin time, select u ₂) system model that input step 1 is set up, can make tracking error at the time Convergence to 0 of expectation.

2. the validity of the control law that checking the present invention proposes

For different situations, the validity of this invention is verified.First, verify that the control law that this invention proposes can make system tracking error restrain in the expected time; Then, verify that the control law that this invention proposes can be by regulating parameter n to realize the adjustment to error convergence speed.

Taking into account system (1), wherein, f (x)=0.6sin (x ₁+ 2x ₂), g (x)=0.3sin (10t)+0.2cos (0.5x ₁+ 7x ₂), b (x)=0.5sin (x ₁+ x ₂)+1.System starting condition is x ₁₀=-2, x ₂₀=1, the reference locus of expectation is x _1d=-cos (t).Control law parameter is chosen as follows: T=2s, t _r=4s, ε=1e-3, η ₁=η ₂=4, K=10, γ=5/3.

1. the checking control law of carrying can make system tracking error restrain in the expected time

In the present embodiment, n value is set as 3.In Fig. 2 and Fig. 3, provided the aircraft pursuit course of system state amount x1 and x2, from figure, result can be found out, system state is followed the tracks of the reference locus of expectation constantly at t=4s, and after this, the track of system state track and expectation overlaps.Fig. 4 has provided corresponding controlled quentity controlled variable curve, and from figure, curve can be found out, does not occur the phenomenons such as unusual or buffeting in steering order.This is because the n value of selecting is 3, therefore there will not be unusual appearance, has used saturation function in addition, and chattering phenomenon has also obtained good inhibition.In Fig. 5, provided sliding-mode surface function curve, from this result, can find out, owing to having adopted saturation function, sliding formwork function cannot strictly converge to 0, can only be limited in boundary layer.But due to boundary layer thickness very little (ε=1e-3), thereby guaranteed higher tracking accuracy.

2. the situation of different n values

In the present embodiment, select different n values (n=3,4,5) to carry out emulation.Fig. 6 is the state error convergence curve in different n value situations.From simulation result, can find out, n is larger, and the speed of convergence of error is faster, therefore, can obtain different error convergence speed by adjusting parameter n.。

In sum, the control law strong robustness that this invention proposes, can make error restrain at a fixed time, and can regulate speed of convergence, has very high engineering using value.

Claims (3)

1. error, with a control method for set time convergence, is characterized in that: specifically wrap

Draw together following steps:

Step 1, set up the dynamic model of second order uncertain system:

${\stackrel{\&CenterDot;}{x}}_1=x_2$

${\stackrel{\&CenterDot;}{x}}_2=f\left(x\right)+g\left(x\right)+b\left(x\right)u$

Wherein, x=[x ₁, x ₂] ^tfor system state vector, f (x) and b (x) ≠ 0 are the smooth nonlinear function about x, and g (x) represents uncertain and external disturbance and satisfied || g (x) || and≤l _g, l wherein _g> 0, and u is system control inputs; The reference locus of system is x _1d, expected convergence time is t _f;

Step 2, design finite time control law

Definition tracking error is as follows:

$\begin{array}{cc}{\tilde{x}}_1=x_1-x_{1d}& {\tilde{x}}_2=x_2-{\stackrel{\&CenterDot;}{x}}_{1d}\end{array}$

Step 2.1, becomes sliding formwork function during design

During design, become sliding formwork function as follows:

$S_1=(t_f-t){\tilde{x}}_2+n(1-\frac{1}{e^{{\tilde{x}}_1}})+\left\{\begin{array}{cc}{\mathrm{At}}^2+\mathrm{Bt}+C& t\&le;T\\ 0& t>T^{\&prime;}\end{array}\right.,0\&le;t\&le;t_f$

N > 2,0 < T < t wherein _f, and A, B, C meets following equation:

$C=-t_f{\tilde{x}}_2\left(0\right)-n(1-\frac{1}{e^{{\tilde{x}}_1\left(0\right)}}),A=\frac{C}{T^2},B=-2\mathrm{AT}$

Step 2.2, solves and while obtaining, becomes sliding-mode control law

While solving by Lyapunov method, become sliding formwork function, make system state at t _fconstantly follow the tracks of upper reference locus, obtain becoming when following sliding-mode control law:

$u_1=b^{-1}\left(x\right)[\frac{1}{t_f-t}({\tilde{x}}_2-\frac{n{\tilde{x}}_2}{e^{{\tilde{x}}_1}}-\left\{\begin{array}{cc}2\mathrm{At}+B& t\&le;T\\ 0& t>T\end{array}\right.)-f\left(x\right)+{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}-{\mathrm{\&eta;}}_1\mathrm{sgn}\left(S_1\right)],0\&le;t\&le;t_f$ η wherein ₁> l _g;

Step 2.3, designs nonsingular terminal sliding mode function

Design nonsingular terminal sliding mode function as follows:

$S_2={\tilde{x}}_1+K{\left|{\tilde{x}}_2\right|}^{\mathrm{\&beta;}}\mathrm{sgn}\left({\tilde{x}}_2\right)t>t_f$

K > 0,1 < β < 2 wherein;

Step 2.4, solves and obtains nonsingular terminal sliding mode controlled quentity controlled variable

By Lyapunov method, solve nonsingular terminal sliding mode function, make tracking error at t > t _fin time, remain 0, obtain corresponding nonsingular terminal sliding mode controlled quentity controlled variable:

$u_2=-b^{-1}\left(x\right)(\frac{{\tilde{x}}_2}{\mathrm{K\&beta;}{\left|{\tilde{x}}_2\right|}^{\mathrm{\&beta;}-1}}+f\left(x\right)-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_{1d}+{\mathrm{\&eta;}}_2\mathrm{sgn}\left(S_2\right)),t>t_f$

Wherein, η ₂> l _g;

Step 3, the controlled quentity controlled variable u that step 2 is obtained ₁and u ₂the system model that input step 1 is set up, makes tracking error at the time Convergence to 0 of expectation, and after this moment, tracking error remains 0 always.

2. a kind of error according to claim 1, with the control method of set time convergence, is characterized in that: in described step 3, at 0≤t≤t _fu in time ₁the system model that input step 1 is set up, at t > t _fu in time ₂the system model that input step 1 is set up.

3. a kind of error according to claim 1, with the control method of set time convergence, is characterized in that: the rate of convergence of error can be by regulating the value of parameter n to realize, and n value is larger, and error convergence must be faster.

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