CN103728882A - Adaptive inversion nonsingular terminal sliding mode control method of micro gyroscope - Google Patents

Abstract

The invention provides an adaptive inversion nonsingular terminal sliding mode control method of a micro gyroscope. An inversion method and a terminal sliding mode are combined, an inversion terminal sliding mode control law is designed according to the Lyapunov stability theory, and therefore, system status can be converged to a balance point within the short limited time. Furthermore, by means of self-adaptive control, angular velocity and other system parameters of the micro gyroscope can be identified. Nonsingular terminal sliding mode control is introduced in consideration of singularity in the existing terminal sliding mode control, whole nonsingular control of the system is achieved, response speed of a controller is completely comparable to the traditional terminal sliding mode, and therefore, the adaptive inversion nonsingular terminal sliding mode control method has important theoretical and practical application value. By means of the adaptive inversion nonsingular terminal sliding mode control method of the micro gyroscope, the rate of convergence and tracking performance are ensured while strong robustness and adaptive ability to external disturbance are achieved.

Description

The nonsingular terminal sliding mode control method of self-adaptation inverting of gyroscope

Technical field

The present invention relates to the control system of gyroscope, specifically the nonsingular terminal sliding mode control method of a kind of self-adaptation inverting of gyroscope.

Background technology

Gyroscope is the fundamental measurement element of inertial navigation and inertial guidance system, and because it is in the huge advantage aspect volume and cost, gyroscope is widely used in Aeronautics and Astronautics, automobile, biomedicine, military affairs and consumer electronics field.But, because design and the error in manufacturing exist and thermal perturbation, can cause the difference between original paper characteristic and design, reduced the performance of gyroscope system.In addition, gyroscope itself belongs to multi-input multi-output system and systematic parameter and has impact uncertain and that be subject to external environment.Compensation foozle and measured angular speed become the subject matter that gyroscope is controlled, and are necessary gyroscope system to carry out dynamic compensation and adjustment.And traditional control method concentrates in the stable control and diaxon frequency matching of driving shaft oscillation amplitude and frequency, can not solve well the defect of gyroscope dynamic equation.

Have in the world various advanced control methods are applied in the middle of the control of gyroscope, typically have adaptive control and sliding-mode control.Adaptive control is in the situation that entirely even not knowing little about it of knowing of the model knowledge of controlled device or environmental knowledge, make system can automatically work in optimum or close to optimum running status, provide high-quality control performance, but the adaptive control to external world robustness of disturbance is very low, easily makes system become unstable.Sliding mode variable structure control be the special nonlinear Control of a class in essence, the uncontinuity of its non-linear behavior for controlling, the difference of this control strategy and other control is that the structure of system is unfixing, but can on purpose constantly change according to system current state according to system in dynamic process, force system according to the state trajectory motion of predetermined sliding mode.The shortcoming of the method is to arrive after sliding-mode surface when state trajectory, is difficult to strictly along sliding-mode surface, towards equilibrium point, slide, but passes through back and forth in sliding-mode surface both sides, thereby produce vibration.

Summary of the invention

The present invention is directed to the micro-gyrotron trajectory track that contains modeling error and uncertain noises controls, a kind of nonsingular terminal sliding mode control method of self-adaptation based on inverting design has been proposed, the nonsingular terminal sliding mode control algolithm of self-adaptation inverting based on the design of Lyapunov stability theory, guarantee the Global asymptotic stability of whole control system, improved the reliability of system and the robustness that parameter is changed.

The technical solution used in the present invention is:

The nonsingular terminal sliding mode control method of self-adaptation inverting of gyroscope, comprises the following steps:

1) mathematical model of structure gyroscope system is:

$\stackrel{\·\·}{q}+(D+2\mathrm{\Ω})\stackrel{\·}{q}+\mathrm{Kq}=u+f---\left(3\right)$

Wherein, the mass that q is gyroscope, in the position vector of driving shaft and sensitive axis diaxon, is the output of gyroscope system; U is the control inputs of gyroscope; D is damping matrix; The natural frequency that K has comprised diaxon and the stiffness coefficient of coupling; Ω is angular speed matrix; F is parameter uncertainty and the external disturbance of system;

2) build the nonsingular terminal sliding mode controller of self-adaptation inverting, obtain the nonsingular terminal sliding mode control law of self-adaptation inverting:

2-1) defining variable X ₁and X ₂, order:

based on inverting designing technique, the kinetics equation of gyroscope (3) is changed to following form:

$\left\{\begin{array}{c}{\stackrel{\·}{X}}_1=X_2\\ {\stackrel{\·}{X}}_2=-(D+2\mathrm{\Ω})X_2-K{X}_1+u+f\end{array}---\left(7\right)\right.$

2-2) definition error variance e ₂for:

e ₂＝X ₂-α ₁ （11）

Q wherein _rfor the ideal position output vector of mass along diaxon, α ₁for virtual controlling amount, and have:

${\mathrm{\&alpha;}}_1=-c_1{\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+{\stackrel{\&CenterDot;}{q}}_r---\left(10\right)$

C wherein ₁for error coefficient, be non-zero normal number, e ₁=X ₁-q _r;

2-3) define nonsingular terminal sliding mode face s _cfor:

$s_c={\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+\frac{1}{\mathrm{\&beta;}}{e}_2^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2}---\left(16\right)$

Wherein, β, p ₁, p ₂for sliding-mode surface parameter, meet: β > 0, p ₁, p ₂for odd number, and 1 < p ₁/ p ₂< 2;

2-4) for described gyroscope system, the sliding-mode surface that employing formula (16) is described, the nonsingular terminal sliding mode control law of design inverting φ is by four control law u ₀, u ₁, u ₂, u ₃form:

φ＝u ₀+u ₁+u ₂+u ₃ (19)

Wherein,

$u_0=(D+2\mathrm{\&Omega;})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)+K({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(20\right)$

D, K, Ω is respectively three parameter matrixs of gyroscope;

$u_1=-\mathrm{\&beta;}\frac{p_2}{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1}\mathrm{diag}\left({{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2}^{1-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2}\right)\stackrel{\&CenterDot;}{e_1}---\left(21\right)$

$u_2=-\mathrm{\&beta;}\frac{p_2}{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1}\mathrm{diag}\left({{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2}^{1-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2}\right)\frac{s_c}{{{\left|\right|s}_c\left|\right|}^2}\left({e_1}^T{e}_2\right)---\left(22\right)$

$u_3=-\mathrm{\&rho;}\frac{{\left[{{\mathrm{<figure-callout\; id="1"\; label="s\; c\; T"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">s</figure-callout>}}_c}^T\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)\right]}^T}{{\left|\right|{{\mathrm{<figure-callout\; id="1"\; label="s\; c\; T"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">s</figure-callout>}}_c}^T\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)\left|\right|}^2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)\left|\right|---\left(23\right)$

ρ is that the nonsingular terminal sliding mode of inverting is controlled parameter;

2-5) due to three parameter matrix D of gyroscope, K, Ω is unknown, according to Adaptive Control Theory, uses estimated value

the parameter matrix D in substituted (20) respectively, K, Ω, and design the adaptive algorithm of three estimated values, online real-time update estimated value, the described control law u of formula (20) ₀be adjusted into u ' ₀:

${u_0}^{\&prime;}=(\hat{D}+2\widehat{\mathrm{\&Omega;}})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)+\hat{K}({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(25\right)$

2-6) with described step 2-5) control law u' after adjusting ₀replacement step 2-4) the control law u in ₀, be brought in formula (19), obtain the control law φ ' of the nonsingular terminal sliding mode controller of self-adaptation inverting:

φ'＝u' ₀+u ₁+u ₂+u ₃ （26)

2-7) using the control law φ ' of the nonsingular terminal sliding mode controller of self-adaptation inverting as gyroscope system control inputs u, bring in the mathematical model of gyroscope system, realize the tracking of gyroscope system is controlled

Aforesaid gyroscope parameter matrix D, K, the estimated value of Ω

adaptive algorithm based on Lyapunov stability theory, design:

Lyapunov function V is designed to:

$V=\frac{1}{2}{e_1}^{\mathrm{<figure-callout\; id="1"\; label="T\; e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">T</figure-callout>}}{e}_{}{}_1+\frac{1}{2}{s_c}^T{s}_c+\frac{1}{2}\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\tilde{D}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\tilde{K}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\widetilde{\mathrm{\&Omega;}}}^T\right\}---\left(29\right)$

Wherein, the mark of tr () representing matrix; M, N, P is adaptive gain, M=M ^t> 0, N=N ^t> 0, P=P ^t> 0 is symmetric positive definite matrix,

be respectively parameter matrix D, K, the parameter estimating error of Ω,

In order to guarantee Lyapunov function derivative

design gyroscope parameter matrix D, K, the estimated value of Ω

adaptive algorithm be respectively:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{D}}}^T=-\frac{1}{2}M[({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1){s_c}^{T}W+s_c{({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)}^{T}W]\\ {\stackrel{\&CenterDot;}{\hat{K}}}^T=-\frac{1}{2}N[({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r){s_c}^{T}W+s_c{({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)}^{T}W]\\ {\stackrel{\&CenterDot;}{\widehat{\mathrm{\&Omega;}}}}^T=-P[({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1){s_c}^{T}W-s_c{({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)}^{T}W]\end{array}---\left(30\right)\right.$

Wherein, $W=\frac{1}{\mathrm{\&beta;}}\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right).$

Compared with prior art, beneficial effect of the present invention is embodied in: first, inverting design can be resolved into the subsystem that is no more than system exponent number by complicated nonlinear system, then for each subsystem, design respectively Lyapunov function and intermediate virtual controlled quentity controlled variable, " retreat " whole system, until complete the design of whole control law always; Secondly, the introducing that nonsingular terminal sliding mode is controlled, can not only make system state converge to equilibrium point in very short finite time, and solved existing terminal sliding mode and controlled the singularity problem existing, controller response speed can compare favourably with conventional terminal sliding formwork completely, has important theory and actual application value; Finally, when all parameters of gyroscope and angular speed are all regarded unknown variable as, for control and the parameter measurement problem of gyroscope, based on Lyapunov function theory design adaptive algorithm, the online angular velocity of real-time update gyroscope and the estimated value of other systematic parameter.

Accompanying drawing explanation

Fig. 1 is the simplified model schematic diagram of gyroscope system in the present invention;

Fig. 2 is the theory diagram of the nonsingular terminal sliding mode control method of self-adaptation inverting of the present invention;

Fig. 3 is X in specific embodiments of the invention, Y-axis location tracking curve;

Fig. 4 is X in specific embodiments of the invention, Y-axis location tracking graph of errors;

Fig. 5 is the convergence curve of nonsingular terminal sliding mode face sc in specific embodiments of the invention;

Fig. 6 is X in specific embodiments of the invention, Y-axis control inputs response curve;

Fig. 7 is gyroscope systematic parameter d in specific embodiments of the invention _xx, d _xy, d _yyand ω _x ², ω _xy, ω _y ²adaptive Identification curve;

Fig. 8 is gyroscope angular velocity Ω in specific embodiments of the invention _zadaptive Identification curve.

Embodiment

Above-mentioned explanation is only general introduction of the present invention, in order to better understand technological means of the present invention, and can be implemented according to the content of instructions, below in conjunction with accompanying drawing and preferred embodiment, the nonsingular terminal sliding mode control method of gyroscope self-adaptation inverting proposing according to the present invention is elaborated.

One, build the mathematical model of gyroscope system

As shown in Figure 1, according to the Newton's law in rotation system, consider manufacturing defect and mismachining tolerance, then process by the nondimensionalization of model, the lumped parameter mathematical model that obtains actual gyroscope is:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+D\stackrel{\&CenterDot;}{q}+\mathrm{Kq}=u-2\mathrm{\&Omega;}\stackrel{\&CenterDot;}{q}+d---\left(1\right)$

Wherein, $q=\left[\begin{array}{c}x\\ y\end{array}\right]$ For the mass of the gyroscope position vector at driving shaft and sensitive axis diaxon, be the output of gyroscope system; $u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right]$ Control inputs for gyroscope diaxon; $d=\left[\begin{array}{c}{d}_x\\ d_y\end{array}\right]$ External disturbance effect for diaxon; $D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}& d_{\mathrm{xy}}\\ d_{\mathrm{xy}}& d_{\mathrm{yy}}\end{array}\right]$ For damping matrix, wherein, d _xx, d _yyfor the ratio of damping of diaxon, d _xyfor Coupling Damping coefficient; $K=\left[\begin{array}{cc}{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="x"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">x</figure-callout>}}^2& {\mathrm{\&omega;}}_{\mathrm{xy}}\\ {\mathrm{\&omega;}}_{\mathrm{xy}}& {\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">y</figure-callout>}}^2\end{array}\right]$ For the natural frequency that comprised diaxon and the stiffness coefficient of coupling, wherein, $\sqrt{\frac{k_{\mathrm{xx}}}{{{\mathrm{m\&omega;}}_0}^2}}\&RightArrow;{\mathrm{\&omega;}}_x,\sqrt{\frac{k_{\mathrm{yy}}}{{{\mathrm{m\&omega;}}_0}^2}}\&RightArrow;{\mathrm{\&omega;}}_y,\frac{k_{\mathrm{xy}}}{{{\mathrm{m\&omega;}}_0}^2}\&RightArrow;{\mathrm{\&omega;}}_{\mathrm{xy}},$ ω ₀for the natural frequency of diaxon, k _xx, k _yyfor the stiffness coefficient of diaxon, k _xystiffness coefficient for coupling; $\mathrm{\&Omega;}=\left[\begin{array}{cc}0& {-\mathrm{\&Omega;}}_z\\ {\mathrm{\&Omega;}}_{\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="HDA0000454131760000021.png,HDA0000454131760000022.png"\; state="\{\{state\}\}">z</figure-callout>}}& 0\end{array}\right]$ For angular speed matrix, Ω _zfor the angular speed in gyroscope working environment, it is a unknown quantity.

The parameter uncertainty of taking into account system and external disturbance, can be shown as gyroscope system table following form according to the mathematical modulo pattern (1) of gyroscope:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+(D+2\mathrm{\&Omega;}+\mathrm{\&Delta;\&Omega;})\stackrel{\&CenterDot;}{q}+(K+\mathrm{\&Delta;K})q=u+d---\left(2\right)$

In formula, Δ D is the uncertainty of the unknown parameter of inertial matrix D+2 Ω, and Δ K is the uncertainty of the unknown parameter of inertial matrix K.

Further, formula (2) can be write as:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+(D+2\mathrm{\&Omega;})\stackrel{\&CenterDot;}{q}+\mathrm{Kq}=u+f---\left(3\right)$

In formula, f represents parameter uncertainty and the external disturbance of system, meets:

$f=d-\mathrm{\&Delta;D}\stackrel{\&CenterDot;}{q}-\mathrm{\&Delta;Kq}---\left(4\right)$

Generally, to the uncertainty of system unknown parameter and external disturbance, can do following hypothesis:

$\left|\right|f\left(t\right)\left|\right|<{\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="HDA0000454131760000021.png,HDA0000454131760000022.png"\; state="\{\{state\}\}">b</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\left|\right|q\left|\right|+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">b</figure-callout>}}_2\left|\right|\stackrel{\&CenterDot;}{q}\left|\right|---\left(5\right)$

In formula, b ₀, b ₁, b ₂for positive unknown constant.

Two, build the nonsingular terminal sliding mode controller of self-adaptation inverting, obtain the nonsingular terminal sliding mode control law of self-adaptation inverting

In order to apply inverting design theory, first the mathematical model of gyroscope is converted defining variable X ₁and X ₂:

${\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">X</figure-callout>}}_1=q,{\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">X</figure-callout>}}_2=\stackrel{\&CenterDot;}{q}---\left(6\right)$

Formula (3) can be rewritten into following state equation:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{X}}_1=X_2\\ {\stackrel{\&CenterDot;}{X}}_2=-(D+2\mathrm{\&Omega;})X_2-\mathrm{<figure-callout\; id="1"\; label="K\; X"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">K</figure-callout>}{X}_{}{}_1+u+f\end{array}---\left(7\right)\right.$

The problem that this present invention will solve is the tracking problem of gyroscope, and the target of control is exactly that suitable control law of design makes system output in finite time, reach the tracking completely to desired trajectory.For micro-gyrosystem formula (7), suppose that mass is q along the ideal position output vector of diaxon _r, $q_r=\left[\begin{array}{c}{x}_r\\ y_r\end{array}\right],$ And q _rthere is second derivative,

Inverting control method design procedure is as follows:

The first step:

Definition tracking error e ₁for:

e ₁＝X ₁-q _r (8)

To tracking error e ₁differentiate obtains

${\stackrel{\&CenterDot;}{e}}_1=\stackrel{\&CenterDot;}{q}-{\stackrel{\&CenterDot;}{q}}_r=X_2-{\stackrel{\&CenterDot;}{q}}_r---\left(9\right)$

Design virtual controlling amount α ₁for:

${\mathrm{\&alpha;}}_1={-c}_1{\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+{\stackrel{\&CenterDot;}{q}}_r---\left(10\right)$

C wherein ₁for error coefficient, be non-zero normal number,

Define new error variance e ₂for:

e ₂＝X ₂-α ₁ (11)

Get Lyapunov function V ₁for:

${\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">V</figure-callout>}}_1=\frac{1}{2}{e_1}^T{e}_1---\left(12\right)$

To Lyapunov function V ₁along time t differentiate, obtain:

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_1={e_1}^T{\stackrel{\&CenterDot;}{e}}_1={e_1}^T(X_2-{\stackrel{\&CenterDot;}{q}}_r)\\ ={e_1}^T({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1-{\stackrel{\&CenterDot;}{q}}_r)\\ ={e_1}^T(e_2-c_1{e}_1)\\ {=-c}_1{e_1}^{\mathrm{<figure-callout\; id="1"\; label="T\; e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">T</figure-callout>}}{e}_{}{}_1+{e_1}^T{e}_2\end{array}---\left(13\right)$

If error variance e ₂=0,

${\stackrel{\&CenterDot;}{V}}_1={-c}_1{e_1}^{\mathrm{<figure-callout\; id="1"\; label="T\; e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">T</figure-callout>}}{e}_{}{}_1\&le;0---\left(14\right)$

For this reason, need design control law, guarantee that sliding-mode surface equals zero or levels off to initial point.

Second step:

To formula (11), differentiate obtains:

$\begin{array}{c}{\stackrel{\&CenterDot;}{e}}_2={\stackrel{\&CenterDot;}{X}}_2-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1=-(D+2\mathrm{\&Omega;})X_2-{\mathrm{<figure-callout\; id="1"\; label="KX"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">KX</figure-callout>}}_1+u+f-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1\\ =-(D+2\mathrm{\&Omega;})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)-K({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+u+f-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1\end{array}---\left(15\right)$

In formula (15), there is the control inputs u of gyroscope system.

The 3rd step:

Define nonsingular terminal sliding mode face s _cfor:

$s_c={\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+\frac{1}{\mathrm{\&beta;}}{e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2}---\left(16\right)$

In formula, β, p ₁, p ₂for sliding-mode surface parameter, meet: β > 0, p ₁, p ₂for odd number, and 1 < p ₁/ p ₂< 2.

Get Lyapunov function V ₂for:

${\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">V</figure-callout>}}_2={\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">V</figure-callout>}}_1+\frac{1}{2}{s_c}^T{s}_c---\left(17\right)$

Both sides differentiate obtains:

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_2={\stackrel{\&CenterDot;}{V}}_1+{s_c}^T{\stackrel{\&CenterDot;}{s}}_c\\ ={-c}_1{e_1}^{\mathrm{<figure-callout\; id="1"\; label="T\; e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">T</figure-callout>}}{e}_{}{}_1+{e_1}^{\mathrm{<figure-callout\; id="2"\; label="T\; e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">T</figure-callout>}}{e}_{}{}_2+{s_c}^T[{\stackrel{\&CenterDot;}{e}}_1+\frac{1}{\mathrm{\&beta;}}\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right){\stackrel{\&CenterDot;}{e}}_2]\\ ={-c}_1{e_1}^{\mathrm{<figure-callout\; id="1"\; label="T\; e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">T</figure-callout>}}{e}_{}{}_1+{e_1}^{\mathrm{<figure-callout\; id="2"\; label="T\; e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">T</figure-callout>}}{e}_{}{}_2+{s_c}^T\{{\stackrel{\&CenterDot;}{e}}_1+\frac{1}{\mathrm{\&beta;}}\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)[u-(D+2\mathrm{\&Omega;})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)\\ -K({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+f-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1\left]\right\}\end{array}---\left(18\right)$

The 4th step:

For making

for gyroscope system of the present invention, the sliding-mode surface that employing formula (16) is described, the nonsingular terminal sliding mode control law of design inverting φ is:

φ＝u ₀+u ₁+u ₂+u ₃ (19)

Wherein,

${\mathrm{<figure-callout\; id="0"\; label="u"\; filenames="HDA0000454131760000021.png,HDA0000454131760000022.png"\; state="\{\{state\}\}">u</figure-callout>}}_0=(D+2\mathrm{\&Omega;})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)+K({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(20\right)$

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">u</figure-callout>}}_1=-\mathrm{\&beta;}\frac{p_2}{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1}\mathrm{diag}\left({{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2}^{1-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2}\right)\stackrel{\&CenterDot;}{{\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1}---\left(21\right)$

${\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">u</figure-callout>}}_2=-\mathrm{\&beta;}\frac{p_2}{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1}\mathrm{diag}\left({{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2}^{1-{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2}\right)\frac{s_c}{{{\left|\right|s}_c\left|\right|}^2}\left({e_1}^T{e}_2\right)---\left(22\right)$

${\mathrm{<figure-callout\; id="3"\; label="u"\; filenames="HDA0000454131760000031.png"\; state="\{\{state\}\}">u</figure-callout>}}_3=-\mathrm{\&rho;}\frac{{\left[{s_{\mathrm{<figure-callout\; id="1"\; label="c\; T"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">c</figure-callout>}}}^T\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)\right]}^T}{{\left|\right|{s_{\mathrm{<figure-callout\; id="1"\; label="c\; T"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">c</figure-callout>}}}^T\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)\left|\right|}^2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)\left|\right|---\left(23\right)$

In formula, ρ is that the nonsingular terminal sliding mode of inverting is controlled parameter, meets

for arbitrarily small normal number.

Formula (19) substitution formula (18) is obtained:

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_2={-c}_1{e_1}^T{e}_1+{s_c}^T\left[\frac{1}{\mathrm{\&beta;}}\frac{p_1}{p_2}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)(u_3+f)\right]\\ ={-c}_1{e_1}^T{e}_1+\frac{p_1}{p_2}[{s_c}^T\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)f-\mathrm{\&rho;}|\left|s_c\right|\left|\right|\left|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\right|\left|\right]\\ \&le;{-c}_1{e_1}^T{e}_1-\frac{p_1}{p_2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|(\mathrm{\&rho;}-f)\\ \&le;{-c}_1{e_1}^T{e}_1-\frac{p_1}{p_2}\left|\right|s_2\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|(\mathrm{\&rho;}-b_0-b_1|\left|q\right||-b_2|\left|\stackrel{\&CenterDot;}{q\left|\right|}\right)\\ \&le;-\frac{p_1}{p_2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|(\mathrm{\&rho;}-b_0-b_1|\left|q\right||-b_2|\left|\stackrel{\&CenterDot;}{q}\right|\left|\right)\\ \&le;-\mathrm{\&xi;}\frac{p_1}{p_2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|<0\end{array}$

In formula, || s _c|| ≠ 0.

According to above-mentioned analysis, meet sliding formwork and arrive condition so s _cfrom arbitrary initial state s _c≠ 0 total energy finite time arrives zero.

Due to three parameter D of gyroscope, K, Ω is unknown, so the control law shown in formula (19) cannot be implemented.According to Adaptive Control Theory, three gyroscope parameter matrixs in formula (20) are used respectively to their estimated value

substitute, and design the adaptive algorithm of three estimated values, online real-time update estimated value, so the control law u shown in formula (20) ₀can be adjusted into u' ₀:

${{\mathrm{<figure-callout\; id="0"\; label="u"\; filenames="HDA0000454131760000021.png,HDA0000454131760000022.png"\; state="\{\{state\}\}">u</figure-callout>}}_0}^{\&prime;}=(\hat{D}+2\widehat{\mathrm{\&Omega;}})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)+\hat{K}({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(25\right)$

Control law φ is adjusted into φ ':

φ'＝u' ₀+u ₁+u ₂+u ₃ (26)

Definition D, K, the parameter estimating error of Ω be respectively:

$\begin{array}{c}\tilde{D}=\hat{D}-D\\ \tilde{K}=\hat{K}-K\\ \widehat{\mathrm{\&Omega;}}=\widehat{\mathrm{\&Omega;}}-\mathrm{\&Omega;}\end{array}---\left(27\right)$

Control law φ ' after adjusting, as the control inputs u of gyroscope system, obtains the sliding-mode surface differentiate shown in formula (16):

$\begin{array}{c}{\stackrel{\&CenterDot;}{s}}_c={\stackrel{\&CenterDot;}{e}}_1+\frac{1}{\mathrm{\&beta;}}\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right){\stackrel{\&CenterDot;}{e}}_2\\ ={\stackrel{\&CenterDot;}{e}}_1+\frac{1}{\mathrm{\&beta;}}\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)[u^{\&prime;}-(D+2\mathrm{\&Omega;})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)-K({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+f-{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1]\\ =\frac{1}{\mathrm{\&beta;}}\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right)[(\tilde{D}+2\widetilde{\mathrm{\&Omega;}})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)+\tilde{K}({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+{\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">u</figure-callout>}}_2+{\mathrm{<figure-callout\; id="3"\; label="u"\; filenames="HDA0000454131760000031.png"\; state="\{\{state\}\}">u</figure-callout>}}_3+f]\\ =W[(\tilde{D}+2\widetilde{\mathrm{\&Omega;}})({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)+\tilde{K}({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)+{\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">u</figure-callout>}}_2+{\mathrm{<figure-callout\; id="3"\; label="u"\; filenames="HDA0000454131760000031.png"\; state="\{\{state\}\}">u</figure-callout>}}_3+f]\end{array}---\left(28\right)$

In formula, $W=\frac{1}{\mathrm{\&beta;}}\frac{p_1}{{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2}\mathrm{diag}\left({e_2}^{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1/p_2-1}\right).$

Getting Lyapunov function V is:

$V=\frac{1}{2}{e_1}^{\mathrm{<figure-callout\; id="1"\; label="T\; e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">T</figure-callout>}}{e}_{}{}_1+\frac{1}{2}{s_c}^T{s}_c+\frac{1}{2}\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\tilde{D}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\tilde{K}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\widetilde{\mathrm{\&Omega;}}}^T\right\}---\left(29\right)$

In formula, M, N, P is adaptive gain, M=M ^t> 0, N=N ^t> 0, P=P ^t> 0 is symmetric positive definite matrix; The mark of tr () representing matrix.

In order to guarantee the derivative of Lyapunov function V

design

adaptive algorithm be respectively:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{D}}}^T=-\frac{1}{2}M[({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1){s_c}^{T}W+s_c{({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)}^{T}W]\\ {\stackrel{\&CenterDot;}{\hat{K}}}^T=-\frac{1}{2}N[({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r){s_c}^{T}W+s_c{({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HDA0000454131760000022.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_1+q_r)}^{T}W]\\ {\stackrel{\&CenterDot;}{\widehat{\mathrm{\&Omega;}}}}^T=-P[({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1){s_c}^{T}W-s_c{({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000454131760000021.png,HDA0000454131760000031.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+{\mathrm{\&alpha;}}_1)}^{T}W]\end{array}---\left(30\right)\right.$

Lyapunov function V, along time t differentiate, and is brought into the parameters adaption algorithm of formula (30), obtains:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}={-c}_1{e_1}^T{e}_1+{e_1}^T{e}_2+{s_c}^T{\stackrel{\&CenterDot;}{s}}_c+\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\stackrel{\&CenterDot;}{\tilde{D}}}^T\right\}+\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\stackrel{\&CenterDot;}{\tilde{K}}}^T\right\}+\mathrm{tr}\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&Omega;}}}}^T\right\}\\ ={-c}_1{e_1}^T{e}_1+{e_1}^T{e}_2+{s_c}^{T}W[(\tilde{D}+2\widetilde{\mathrm{\&Omega;}})(e_2+{\mathrm{\&alpha;}}_1)+\tilde{K}(e_1+q_r)+u_2+u_3+f]\\ +\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\stackrel{\&CenterDot;}{\tilde{D}}}^T\right\}+\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\stackrel{\&CenterDot;}{\tilde{K}}}^T\right\}+\mathrm{tr}\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&Omega;}}}}^T\right\}\\ ={-c}_1{e_1}^T{e}_1+{s_c}^{T}W(u_3+f)+{s_c}^{T}W\tilde{D}(e_2+{\mathrm{\&alpha;}}_1)+\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&Omega;}}}}^T\right\}\\ +{s_c}^{T}W\tilde{K}(e_1+q_r)+\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\stackrel{\&CenterDot;}{\tilde{K}}}^T\right\}+{{2s}_c}^{T}W\widetilde{\mathrm{\&Omega;}}(e_2+{\mathrm{\&alpha;}}_1)+\mathrm{tr}\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&Omega;}}}}^T\right\}\\ ={-c}_1{e_1}^T{e}_1+{s_c}^{T}W(u_3+f)\\ ={-c}_1{e_1}^T{e}_1+\frac{p_1}{p_2}[{s_c}^T\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)f-\mathrm{\&rho;}|\left|s_c\right|\left|\right|\left|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\right|\left|\right]\\ \&le;{-c}_1{e_1}^T{e}_1-\frac{p_1}{p_2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|(\mathrm{\&rho;}-f)\\ \&le;-\frac{p_1}{p_2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|(\mathrm{\&rho;}-b_0-b_1|\left|q\right||-b_2|\left|\stackrel{\&CenterDot;}{q}\right|\left|\right)\\ \&le;-\mathrm{\&xi;}\frac{p_1}{p_2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|<0\end{array}---\left(31\right)$

In formula, || s _c|| ≠ 0.

Thus, based on Lyapunov stability the second method, can judge that designed controller has guaranteed the Global asymptotic stability of system, and make the output tracking error of system in finite time, converge to zero.

Three, Computer Simulation

In order to show more intuitively the validity of the nonsingular terminal sliding mode control method of gyroscope self-adaptation inverting that the present invention proposes, now utilize mathematical software MATLAB/SIMULINK to carry out computer simulation experiment to the present invention.

With reference to existing document, the parameter of choosing gyroscope is:

m＝1.8×10 ^-7kg,k _xx＝63.955N/m,k _yy＝95.92N/m,k _xy＝12.779N/m

d _xx＝1.8×10 ^-6N·s/m,d _yy＝1.8×10 ^-6N·s/m,d _xy＝3.6×10 ^-7N·s/m

Suppose that unknown input angular velocity is Ω _z=100rad/s, reference length is chosen for q ₀=1 μ m, natural frequency ω ₀=1000Hz, after nondimensionalization, three parameter matrixs of gyroscope are:

$D=\left(\begin{array}{cc}0.01& 0.002\\ 0.001& 0.01\end{array}\right],K=\left[\begin{array}{cc}355.3& 70.99\\ 70.99& 532.9\end{array}\right],\mathrm{\&Omega;}=\left[\begin{array}{cc}0& -0.1\\ 0.1& 0\end{array}\right]---\left(32\right)$

Wherein, nondimensionalization process is: $\frac{d_{\mathrm{xx}}}{{\mathrm{m\&omega;}}_0}\&RightArrow;d_{\mathrm{xx}},\frac{d_{\mathrm{xy}}}{{\mathrm{m\&omega;}}_0}\&RightArrow;d_{\mathrm{xy}},\frac{d_{\mathrm{yy}}}{{\mathrm{m\&omega;}}_0}\&RightArrow;d_{\mathrm{yy}},\sqrt{\frac{k_{\mathrm{xx}}}{{{\mathrm{m\&omega;}}_0}^2}}\&RightArrow;{\mathrm{\&omega;}}_x,$ $\frac{k_{\mathrm{xy}}}{{{\mathrm{m\&omega;}}_0}^2}\&RightArrow;{\mathrm{\&omega;}}_{\mathrm{xy}},\sqrt{\frac{k_{\mathrm{yy}}}{{{\mathrm{m\&omega;}}_0}^2}}\&RightArrow;{\mathrm{\&omega;}}_y,\frac{{\mathrm{\&Omega;}}_z}{{\mathrm{\&omega;}}_0}\&RightArrow;{\mathrm{\&Omega;}}_z.$

In emulation experiment, the estimation initial value of three parameter matrixs of gyroscope is taken as respectively:

the ideal trajectory of diaxon is taken as respectively: x _r=sin (4.17t), y _r=1.2sin (5.11t); The starting condition of system is taken as: parameter uncertainty and the external disturbance of system are taken as: f=[0.5*randn (1,1); 0.5*randn (1,1)].

If sliding-mode surface parameter is chosen for: p ₁=5, p ₂=3, β=diag (1,1), nonsingular terminal sliding mode face is: s _c=e ₁+ e ₂ ^5/3;

The nonsingular terminal sliding mode of self-adaptation inverting is controlled parameter and is chosen, ρ=100, and error coefficient is chosen:

c ₁＝100；

Adaptive gain M, N, P is taken as: M=N=P=diag (150,150).

Fig. 3 is X, the Y-axis location tracking curve that gyroscope adopts the nonsingular terminal sliding mode control method of self-adaptation inverting to obtain, and dotted line is desired trajectory, and solid line is actual motion track.From figure, can see very intuitively, in the situation that having external disturbance, the actual motion track of gyroscope can be followed the tracks of desired trajectory soon, improved the dynamic perfromance of gyroscope, simultaneous verification the nonsingular terminal sliding mode controller of self-adaptation inverting based on Lyapunov stability theory design can Guarantee control system Global asymptotic stability.Fig. 4 is X, Y-axis location tracking graph of errors, as can be seen from the figure, substantially converges to zero, and keep this motion through very short time error curve.

Fig. 5 is gyroscope X, the nonsingular terminal sliding mode face of Y-axis convergence curve, and s1 is X-axis sliding-mode surface convergence curve, and s2 is Y-axis sliding-mode surface convergence curve.As can be seen from the figure, sliding-mode surface levels off to zero soon,

Showing that system arrives at short notice sliding-mode surface and remains on sliding-mode surface slides.

Fig. 6 is gyroscope X, Y-axis control inputs response curve.Control and to compare with conventional terminal sliding formwork, the singular problem that nonsingular terminal sliding mode is controlled without control inputs occurs, the nonsingular and stability of the overall situation that can Guarantee control system, has important theory and actual application value.

Fig. 7 is the Adaptive Identification curve of gyroscope systematic parameter, and result shows d _xx, d _xy, d _yyand ω _x ², ω _xy, ω _y ²these parameters not only can converge to true value separately soon, and overshoot is also less.Fig. 8 is gyroscope angular velocity Ω _zidentification curve, result shows that angular velocity estimates finally also to converge to its true value.

The above, it is only preferred embodiment of the present invention, not the present invention is done to any large restriction in form, although the present invention discloses as above with preferred embodiment, yet not in order to limit the present invention, any those skilled in the art, do not departing within the scope of technical solution of the present invention, when can utilizing the technology contents of above-mentioned announcement to make a little change or being modified to the equivalent embodiment of equivalent variations, in every case be the content that does not depart from technical solution of the present invention, any simple modification of above embodiment being done according to technical spirit of the present invention, equivalent variations and modification, all still belong in the scope of our bright technical scheme.

Claims (2)

1. the nonsingular terminal sliding mode control method of the self-adaptation inverting of gyroscope, is characterized in that, comprises the following steps:

1) mathematical model of structure gyroscope system is:

$\stackrel{\&CenterDot;\&CenterDot;}{q}+(D+2\mathrm{\&Omega;})\stackrel{\&CenterDot;}{q}+\mathrm{Kq}=u+f---\left(3\right)$

Wherein, the mass that q is gyroscope, in the position vector of driving shaft and sensitive axis diaxon, is the output of gyroscope system; U is the control inputs of gyroscope; D is damping matrix; The natural frequency that K has comprised diaxon and the stiffness coefficient of coupling; Ω is angular speed matrix; F is parameter uncertainty and the external disturbance of system;

2) build the nonsingular terminal sliding mode controller of self-adaptation inverting, obtain the nonsingular terminal sliding mode control law of self-adaptation inverting:

2-1) defining variable X ₁and X ₂, order:

based on inverting design theory, the kinetics equation of gyroscope (3) is changed to following form:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{X}}_1=X_2\\ {\stackrel{\&CenterDot;}{X}}_2=-(D+2\mathrm{\&Omega;})X_2-K{X}_1+u+f\end{array}---\left(7\right)\right.$

2-2) definition error variance e ₂for:

e ₂＝X ₂-α ₁ （11）

Q wherein _rfor the ideal position output vector of mass along diaxon, α ₁for virtual controlling amount, and have:

${\mathrm{\&alpha;}}_1=-c_1{e}_1+{\stackrel{\&CenterDot;}{q}}_r---\left(10\right)$

C wherein ₁for error coefficient, be non-zero normal number, e ₁=X ₁-q _r;

2-3) define nonsingular terminal sliding mode face s _cfor:

$s_c=e_1+\frac{1}{\mathrm{\&beta;}}{e}_2^{p_1/p_2}---\left(16\right)$

Wherein, β, p ₁, p ₂for sliding-mode surface parameter, meet: β > 0, p ₁, p ₂for odd number, and 1 < p ₁/ p ₂< 2;

2-4) for described gyroscope system, the sliding-mode surface that employing formula (16) is described, the nonsingular terminal sliding mode control law of design inverting φ is by four control law u ₀, u ₁, u ₂, u ₃form:

φ＝u ₀+u ₁+u ₂+u ₃ (19)

Wherein,

$u_0=(D+2\mathrm{\&Omega;})(e_2+{\mathrm{\&alpha;}}_1)+K(e_1+q_r)+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(20\right)$

D, K, Ω is respectively three parameter matrixs of gyroscope;

$u_1=-\mathrm{\&beta;}\frac{p_2}{p_1}\mathrm{diag}\left({e_2}^{1-p_1/p_2}\right)\stackrel{\&CenterDot;}{e_1}---\left(21\right)$

$u_2=-\mathrm{\&beta;}\frac{p_2}{p_1}\mathrm{diag}\left({e_2}^{1-p_1/p_2}\right)\frac{s_c}{{{\left|\right|s}_c\left|\right|}^2}\left({e_1}^T{e}_2\right)---\left(22\right)$

$u_3=-\mathrm{\&rho;}\frac{{\left[{s_c}^T\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\right]}^T}{{\left|\right|{s_c}^T\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|}^2}\left|\right|s_c\left|\right|\left|\right|\frac{1}{\mathrm{\&beta;}}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right)\left|\right|---\left(23\right)$

ρ is that the nonsingular terminal sliding mode of inverting is controlled parameter;

2-5) due to three parameter matrix D of gyroscope, K, Ω is unknown, according to Adaptive Control Theory, uses estimated value

the parameter matrix D in substituted (20) respectively, K, Ω, and design the adaptive algorithm of three estimated values, online real-time update estimated value, the described control law u of formula (20) ₀be adjusted into u ' ₀:

${u_0}^{\&prime;}=(\hat{D}+2\widehat{\mathrm{\&Omega;}})(e_2+{\mathrm{\&alpha;}}_1)+\hat{K}(e_1+q_r)+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1---\left(25\right)$

2-6) with described step 2-5) control law u' after adjusting ₀replacement step 2-4) the control law u in ₀, be brought in formula (19), obtain the control law φ ' of the nonsingular terminal sliding mode controller of self-adaptation inverting:

φ'＝u' ₀+u ₁+u ₂+u ₃ （26）

2-7) the control inputs u using the control law φ ' of the nonsingular terminal sliding mode controller of self-adaptation inverting as gyroscope system, brings in the mathematical model of gyroscope system, realizes the tracking of gyroscope system is controlled.

2. the nonsingular terminal sliding mode control method of gyroscope self-adaptation according to claim 1 inverting, is characterized in that, described gyroscope parameter matrix D, K, the estimated value of Ω

adaptive algorithm based on Lyapunov stability theory, design:

Lyapunov function V is designed to:

$V=\frac{1}{2}{e_1}^T{e}_1+\frac{1}{2}{s_c}^T{s}_c+\frac{1}{2}\mathrm{tr}\left\{\tilde{D}{M}^{-1}{\tilde{D}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\tilde{K}{N}^{-1}{\tilde{K}}^T\right\}+\frac{1}{2}\mathrm{tr}\left\{\widetilde{\mathrm{\&Omega;}}{P}^{-1}{\widetilde{\mathrm{\&Omega;}}}^T\right\}---\left(29\right)$

Wherein, the mark of tr () representing matrix; M, N, P is adaptive gain, M=M ^t> 0, N=N ^t> 0, P=P ^t> 0 is symmetric positive definite matrix, be respectively parameter matrix D, K, the parameter estimating error of Ω,

In order to guarantee Lyapunov function derivative

design gyroscope parameter matrix D, K, the estimated value of Ω adaptive algorithm be respectively:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\hat{D}}}^T=-\frac{1}{2}M[(e_2+{\mathrm{\&alpha;}}_1){s_c}^{T}W+s_c{(e_2+{\mathrm{\&alpha;}}_1)}^{T}W]\\ {\stackrel{\&CenterDot;}{\hat{K}}}^T=-\frac{1}{2}N[(e_1+q_r){s_c}^{T}W+s_c{(e_1+q_r)}^{T}W]\\ {\stackrel{\&CenterDot;}{\widehat{\mathrm{\&Omega;}}}}^T=-P[(e_2+{\mathrm{\&alpha;}}_1){s_c}^{T}W-s_c{(e_2+{\mathrm{\&alpha;}}_1)}^{T}W]\end{array}---\left(30\right)\right.$

Wherein, $W=\frac{1}{\mathrm{\&beta;}}\frac{p_1}{p_2}\mathrm{diag}\left({e_2}^{p_1/p_2-1}\right).$

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