CN103645638A - Design method for robustness controller of under-actuated vehicle - Google Patents

Abstract

The invention provides a robustness tracking controller for a three-freedom-degree under-actuated vehicle. By taking into consideration the characteristic of under-actuation incapability of realizing asymptotic stabilization, a method enabling an error to tend to be a minimum value is adopted so that the problem of selection of a Lyapunov function is solved. By use of a backstepping method and a function mapping technology, the robustness controller is designed, and the controller, when there is environment disturbance to the vehicle, can start from any one initial state (including the initial speed of a turn bow being zero), and automatically tracks any one smooth locus, and realizes quite good self-adaptive tracking effects.

Description

A kind of method for designing of owing to drive the robust controller of aircraft

Technical field

The invention belongs to robot and control control technology field, relate to a kind of method for designing of owing to drive the robust contrail tracker of aircraft.

Background technology

Owe drive system and refer to that independent variable number is less than the nonlinear system of degree of freedom in system number, it need to complete by less driver complicated motion control task; In most cases Underactuated Mechanical Systems is Nonholonomic Constraints Systems, and showing again general smooth state feedback control simultaneously cannot quelling character.Therefore, owe the control of drive system and the theory of tracking has caused increasing non-linear researcher's interest.

Aspect practical application, owe the minimizing of drive system due to driver, have lightweight, cost is low, less energy consumption, the many merits such as easy to maintenance; On the other hand, in occasions such as controller failures, full drive system can become owes drive system, owes drive system fault-tolerant ability and has also caused numerous concerns.In recent years, owe the new focus that drive system becomes a lot of research fields (as Aero-Space, boats and ships, vehicle, robot etc.).

Yet, in actual applications, for robot system, because its working environment is ever-changing, in working control, will face more uncertain factor, thereby its motion control requires to have stronger Disturbance Rejection ability; The control method with height robustness is one of effective way solving such problem.

Therefore, design a kind ofly reasonably for owing the robust controller of drive system, there is important using value.

Summary of the invention

Technical matters to be solved by this invention is that design a kind ofly can make aircraft in any original state for owing to drive the robust controller of aircraft track following, exists in the situation of environmental perturbation, realizes good tracking effect.

The present invention mainly comprises following content:

(1) set up the dynamic model of owing to drive aircraft;

(2) system of selection of Lyapunov;

(3) design of robust controller;

(4) selection of controller parameter.

Accompanying drawing explanation

Fig. 1 aircraft Three Degree Of Freedom definition figure.

The tracking effect figure of Fig. 2 system during without external disturbance.

The tracking effect figure of system when Fig. 3 has external disturbance.

Embodiment

The present invention be take Three Degree Of Freedom and is owed to drive aircraft as research object (as shown in Figure 1).

1. the foundation of system model

Three Degree Of Freedom owes to drive the mathematical model of aircraft to be:

$\begin{array}{c}\stackrel{\·}{\mathrm{\η}}=J\left(\mathrm{\η}\right)V\\ M\stackrel{\·}{V}=-C\left(V\right)V-\mathrm{DV}+\mathrm{\τ}+R\end{array}.---\left(1\right)$

η=[x y ψ] wherein ^t, position and the direction of representative system in rectangular coordinate system; V=[u v r] ^t, u, v and r represent respectively aircraft swaying, surging and turn the speed of bow; τ=[τ _u0 τ _r] ^t, preflow push power τ _uturn bow moment τ _rcontrol inputs as system; M=diag (m _xy, m _xy, m _ψ), m _xyfor the quality of aircraft, m _ψfor moment of inertia; D=diag (d _u, d _v, d _r), d _u, d _v, d _rfor fluid damping coefficient; R (t)=[R _u(t), R _v(t), R _r(t)] ^t, R _u(t), R _vand R (t) _r(t) environmental perturbation producing for wave, wind, ocean current, and have | R _u(t) |≤R _u? _max< ∞, | R _v(t) |≤R _v? _max< ∞, | R _r(t) |≤R _r? _max< ∞; $J\left(\mathrm{\&eta;}\right)=\left[\begin{array}{ccc}\cos \mathrm{\&psi;}& -\sin \mathrm{\&psi;}& 0\\ \sin \mathrm{\&psi;}& \cos \mathrm{\&psi;}& 0\\ 0& 0& 1\end{array}\right],$ $C\left(v\right)={\mathrm{<figure-callout\; id="0"\; label="m\; xy"\; filenames="HSA0000098944860000012.png,HSA0000098944860000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_{\mathrm{xy}}\left[\begin{array}{ccc}0& 0& -\mathrm{<figure-callout\; id="0"\; label="v"\; filenames="HSA0000098944860000012.png,HSA0000098944860000013.png"\; state="\{\{state\}\}">v</figure-callout>}\\ 0& 0& u\\ v& -\mathrm{<figure-callout\; id="0"\; label="u"\; filenames="HSA0000098944860000012.png,HSA0000098944860000013.png"\; state="\{\{state\}\}">u</figure-callout>}& 0\end{array}\right].$ Obviously, because aircraft lacks side direction propelling power, thereby be to owe to drive.

The target of controller design is design τ _uand τ _r, make the controlled system can automatically track target track (x _dy _d) ^teven, tracking error (x-x _dy-y _d) ^texistence of Global Stable.For convenient design, we are rewritten as form below by (1):

$\begin{array}{c}\stackrel{\&CenterDot;}{X}=F\left(\mathrm{\&psi;}\right)U\\ \stackrel{\&CenterDot;}{\mathrm{\&psi;}}=r\\ M\stackrel{\&CenterDot;}{U}=-\mathrm{MG}\left(r\right)U-\stackrel{\&OverBar;}{U}{D}_u+{\mathrm{H\&tau;}}_u+R_{\mathrm{uv}}\\ m_{\mathrm{\&psi;}}\stackrel{\&CenterDot;}{r}={-d}_{r}r+{\mathrm{\&tau;}}_r+R_r\end{array}.---\left(2\right)$

X=(x y) wherein ^t, U=(u v) ^t, d _u=(d _u, d _v) ^t, M=diag (m _xym _xy), H=(1 0) ^t, R _uv=(R _ur _v) ^t, $F\left(\mathrm{\&psi;}\right)=\left[\begin{array}{cc}\cos \mathrm{\&psi;}& -\sin \mathrm{\&psi;}\\ \sin \mathrm{\&psi;}& \cos \mathrm{\&psi;}\end{array}\right],$ $G\left(r\right)=\left[\begin{array}{cc}0& -\mathrm{<figure-callout\; id="0"\; label="r\; r"\; filenames="HSA0000098944860000012.png,HSA0000098944860000013.png"\; state="\{\{state\}\}">r</figure-callout>}& \\ r& 0\end{array}\right].X_d={(x_d,y_d)}^T$ For desired trajectory, unique constraint condition is smooth, bounded.Note e=(e ₁, e ₂) ^t=X-X _dfor tracking error.

2. the selection of system Lyapunov function

First be defined as follows:

wherein, β ₁> 0, z ₁=(z ₁₁, z ₁₂) ^t.Z ₂=r-(0 1) a, wherein a=-B ^-1(A+ β ₂(z ₁-δ)), δ=(δ ₁, δ ₂) ^t, δ ₁≠ 0 is arbitrarily small constant,

Select following Lyapunov function:

$V=\frac{1}{2}{e}^{T}e+\frac{1}{{{2\mathrm{\&beta;}}_1}^2}{(z_1-\mathrm{\&delta;})}^T(z_1-\mathrm{\&delta;})+\frac{{\mathrm{\&Gamma;}}_1^{-1}}{2}{\tilde{R}}_{\mathrm{uv}}^T{\tilde{R}}_{\mathrm{uv}}+\frac{m_{\mathrm{\&psi;}}}{2}{z}_2^2+\frac{{\mathrm{\&Gamma;}}_2^{-1}}{2}{\tilde{R}}_r^T{\tilde{R}}_r,---\left(3\right)$

Its derivative along system is:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}={-\mathrm{\&beta;}}_1{e}^{T}e+{\mathrm{\&delta;}}^{T}F{\left(\mathrm{\&psi;}\right)}^{T}e+\frac{1}{{{\mathrm{\&beta;}}_1}^2}{(z_1-\mathrm{\&delta;})}^T(A+\mathrm{Bw})\\ +{\tilde{R}}_{\mathrm{uv}}^T{\mathrm{\&Gamma;}}_1^{-1}({\stackrel{\stackrel{\&CenterDot;}{^}}{R}}_{\mathrm{uv}}-\frac{1}{{{\mathrm{\&beta;}}_1}^2}{\mathrm{\&Gamma;}}_1{M}^{-1}(z_1-\mathrm{\&delta;}))+z_2(-{\hat{d}}_{r}r+{\mathrm{\&tau;}}_r+{\hat{R}}_r-\left(\begin{array}{cc}0& m_{\mathrm{\&psi;}}\end{array}\right)\stackrel{\&CenterDot;}{a})+{\mathrm{\&Gamma;}}_2^{-1}{\tilde{R}}_r^T+({\stackrel{\stackrel{\&CenterDot;}{^}}{R}}_r-{\mathrm{\&Gamma;}}_2{z}_2)\begin{array}{}\end{array}\end{array},---\left(4\right)$

Wherein, $B=\left(\begin{array}{cc}\frac{1}{m_{\mathrm{xy}}}& {\mathrm{\&delta;}}_2\\ 0& {-\mathrm{\&delta;}}_1\end{array}\right),w={\left(\begin{array}{cc}{\mathrm{\&tau;}}_u& r\end{array}\right)}^T,$ $A={-M}^{-1}\stackrel{\&OverBar;}{U}{\hat{D}}_u+M^{-1}{\hat{R}}_{\mathrm{uv}}-F{\left(\mathrm{\&psi;}\right)}^T{\stackrel{\&CenterDot;\&CenterDot;}{X}}_d+{\mathrm{\&beta;}}_1{z}_1.$

3. the design of robust controller

Design following Robust Control Law, can be by

and

from

middle cancellation.

${\stackrel{\stackrel{\&CenterDot;}{^}}{R}}_u=\frac{{\mathrm{\&Gamma;}}_1}{m_{\mathrm{xy}}}\frac{1}{{{\mathrm{\&beta;}}_1}^2}\mathrm{proj}({\stackrel{\stackrel{\&CenterDot;}{^}}{R}}_u,(z_{11}-{\mathrm{\&delta;}}_1)),{\stackrel{\stackrel{\&CenterDot;}{^}}{R}}_v=\frac{{\mathrm{\&Gamma;}}_1}{m_{\mathrm{xy}}}\frac{1}{{{\mathrm{\&beta;}}_1}^2}\mathrm{proj}({\stackrel{\stackrel{\&CenterDot;}{^}}{R}}_v,(z_{12}-{\mathrm{\&delta;}}_2)),{\stackrel{\stackrel{\&CenterDot;}{^}}{R}}_r=\mathrm{proj}({\hat{R}}_r,{\mathrm{\&Gamma;}}_2{z}_2),---\left(5\right)$

By in above-mentioned value substitution (4), obtain:

${\stackrel{\&CenterDot;}{V}}_3\&le;{-\mathrm{\&beta;}}_1{e}^{T}e+{\mathrm{\&delta;}}^T{F}^{T}e-{(z_1-\mathrm{\&delta;})}^T\frac{{\mathrm{\&beta;}}_2}{{{\mathrm{\&beta;}}_1}^2}(z_1-\mathrm{\&delta;})-{\mathrm{\&beta;}}_3{z}_2^2.---\left(7\right)$

4. the selection of parameter

To appointing to γ > 0, have:

${\mathrm{\&delta;}}^T{F}^T\left(\mathrm{\&psi;}\right)e\&le;\frac{1}{2\mathrm{\&gamma;}}{\mathrm{\&delta;}}^T\mathrm{\&delta;}+\frac{\mathrm{\&gamma;}}{2}{e}^{T}F\left(\mathrm{\&psi;}\right)F^T\left(\mathrm{\&psi;}\right)e=\frac{\mathrm{\&gamma;}}{2}{e}^{T}e+\frac{1}{2\mathrm{\&gamma;}}{\mathrm{\&delta;}}^T\mathrm{\&delta;},---\left(8\right)$

(7) are changed to:

${\stackrel{\&CenterDot;}{V}}_3\&le;{-e}^T({\mathrm{\&beta;}}_1-\frac{\mathrm{\&gamma;}}{2})e-{(z_1-\mathrm{\&delta;})}^T\frac{{\mathrm{\&beta;}}_2}{{{\mathrm{\&beta;}}_1}^2}(z_1-\mathrm{\&delta;})-{\mathrm{\&beta;}}_3{z}_2^2+\frac{1}{2\mathrm{\&gamma;}}{\left|\right|\mathrm{\&delta;}\left|\right|}^2,---\left(9\right)$

Appoint γ > 0, order to λ > 0 ${\mathrm{\&beta;}}_1-\frac{\mathrm{\&gamma;}}{2}\&GreaterEqual;\frac{\mathrm{\&lambda;}}{2},{\mathrm{\&beta;}}_2\&GreaterEqual;\frac{\mathrm{\&lambda;}}{2},{\mathrm{\&beta;}}_3\&GreaterEqual;\frac{\mathrm{\&lambda;}}{2}{m}_{\mathrm{\&psi;}},$ ?

$\stackrel{\&CenterDot;}{V}\&le;-\mathrm{\&lambda;V}+\frac{{\mathrm{\&lambda;\&Gamma;}}_1^{-1}}{2}{\tilde{R}}_{\mathrm{uv}}^T{\tilde{R}}_{\mathrm{uv}}+\frac{{\mathrm{\&lambda;\&Gamma;}}_2^{-1}}{2}{\tilde{R}}_r^T{\tilde{R}}_r+\frac{1}{2\mathrm{\&gamma;}}{\left|\right|\mathrm{\&delta;}\left|\right|}^2.---\left(10\right)$

According to the character of proj, can obtain

for bounded,

and σ ₁, σ ₂can select arbitrarily small, therefore,

$\stackrel{\&CenterDot;}{V}\&le;-\mathrm{\&lambda;V}+l.---\left(11\right)$

I is error || e|| converges on radius and is

ball territory, system stability.

The simulation result of system keeps track when Fig. 2 and Fig. 3 are respectively without external disturbance and have external disturbance.In figure, dotted portion is desired trajectory, and solid line is partly actual pursuit path.As can be seen from the figure,, even under the impact of external disturbance, this controller still has good tracking effect.Illustrated that this controller has stronger robust shape

Three Degree Of Freedom of the present invention owes to drive aircraft to design a kind of robust tracking controller.Consider to owe to drive and cannot realize asymptotically stable characteristic, taked to make error to be tending towards minimizing method, solved the selection of Lyapunov function.For the external disturbance of bounded, the controller that adopted Function Mapping technical design, has realized good tracking effect and robustness.

Claims (6)

1. for Three Degree Of Freedom, owe to drive a robust tracking controller for aircraft, it is characterized in that, in the uncertain situation of controller parameter, can setting out by arbitrary initial state, automatically follow the tracks of the track of arbitrary smooth, and obtained good tracking effect.

2. according to the Three Degree Of Freedom described in right 1, owe to drive the robust tracking controller of aircraft, it is characterized in that aircraft has following dynamic model:

$\stackrel{\&CenterDot;}{X}=F\left(\mathrm{\&psi;}\right)U$

$\stackrel{\&CenterDot;}{\mathrm{\&psi;}}=r$

$M\stackrel{\&CenterDot;}{U}=-\mathrm{MG}\left(r\right)U-\stackrel{\&OverBar;}{U}{D}_u+{\mathrm{H\&tau;}}_u+R_{\mathrm{uv}}.$

$m_{\mathrm{\&psi;}}\stackrel{\&CenterDot;}{r}={-d}_{r}r+{\mathrm{\&tau;}}_r+R_r$

3. according to the Three Degree Of Freedom described in right 1, owe to drive the robust tracking controller of aircraft, it is characterized in that the following form of being chosen as of Lyapunov function:

$V=\frac{1}{2}{e}^{T}e+\frac{1}{{{2\mathrm{\&beta;}}_1}^2}{(z_1-\mathrm{\&delta;})}^T(z_1-\mathrm{\&delta;})+\frac{{\mathrm{\&Gamma;}}_1^{-1}}{2}{\tilde{R}}_{\mathrm{uv}}^T{\tilde{R}}_{\mathrm{uv}}+\frac{m_{\mathrm{\&psi;}}}{2}{z}_2^2+\frac{{\mathrm{\&Gamma;}}_2^{-1}}{2}{\tilde{R}}_r^T{\tilde{R}}_r.$

4. according to the Three Degree Of Freedom described in right 1, owe to drive the robust tracking controller of aircraft, it is characterized in that having adopted Function Mapping technology in the design of robust controller.

5. according to the Three Degree Of Freedom described in right 1, owe to drive the robust tracking controller of aircraft, it is characterized in that original state is unfettered.

6. according to the Three Degree Of Freedom described in right 1, owe to drive the robust tracking controller of aircraft, it is characterized in that the shape of track is unfettered, only require it is smooth, even and surging and the speed that turns bow be all zero, still have good tracking effect.

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