CN104932531A - Optimal input-saturation-resistant control method based on sliding-mode control for quadrotor aircraft - Google Patents

Abstract

The invention discloses an optimal input-saturation-resistant control method based on sliding-mode control for a quadrotor aircraft. A sliding-mode control method is provided by combining the optimal control since actuator saturation exists in the quadrotor aircraft. On the premise of ensuring that a referred performance index function J achieves an optimum value, system sliding-mode surface parameters and switching time are obtained through calculation, a corresponding sliding-mode surface and sliding-mode control law are designed through comparing the switching time, thereby the optimal controller is formed. The optimal input-saturation-resistant control method simplifies the design steps of the controller by solving an inequation, allows the referred control law to be the optimal control under the condition of actuator input saturation according to the performance index function J, effectively increases control precision and response speed of the quadrotor aircraft, and can provide controller design basis for the quadrotor aircraft with actuator input saturation. The optimal input-saturation-resistant control method is used for input-saturation-resistant control of the quadrotor aircraft with parameter uncertainties and external disturbance.

Description

A kind of optimum anti-input Saturation Control method of the quadrotor based on sliding formwork control

Technical field

The present invention relates to a kind of optimum anti-input Saturation Control method of the quadrotor based on sliding formwork control, belong to flying vehicles control technical field.

Background technology

Quadrotor is a kind of by motor-driven rotation, can the aircraft of vertical takeoff and landing.Compared with conventional rotor craft, its structure is more compact, can produce larger lift, and can to cancel out each other anti-twisted moment due to its four rotors, does not thus need special reactive torque slurry.As a kind of unmanned vehicle, due to its distinctive advantage, quadrotor has wide prospect in the field of civil and military.Then, quadrotor has non-linear, strong coupling, disturbs responsive characteristic to external world.Thus need more sane control mode to ensure its flight safety and flight quality.Due to the uncertainty of quadrotor itself and the change of flight environment of vehicle, inevitably there is the problem of parameter uncertainty and external interference in system, thus controller need possess very strong robustness with avoid uncertainty and interference adverse effect is caused to aircraft.

There are some at present for the new method of the robust control theory of quadrotor, are but often difficult in practical application obtain good effect.Wherein, the existence of the input constraint characteristics such as actuator saturation, dead band and time lag, be cause flight control system closed-loop control actual performance to decline very important main cause, actuator saturation is one the most common.Owing to inputting saturated existence, actuator often cannot obtain theoretially optimum value, even causes system unstable.So, anti-input Saturation Control is carried out to quadrotor, to eliminate the saturated harmful effect to quadrotor of input, and then realizes safe flight, there is important economy and social value.

Sliding mode variable structure control is a kind of nonlinear control method.Its control is discontinuous, and in control procedure, the structure of closed-loop system ceaselessly changes, and forces system state to be moved along pre-designed sliding-mode surface, and " cunning " is to state balance point gradually, i.e. Asymptotic Stability.Its topmost advantage is once system state amount arrives sliding-mode surface, and system is not just subject to the impact of Parameters variation and external disturbance.Optimum control then meeting under certain constraint condition, seeks optimal control policy, makes performance index get maximum value or minimum value.Both are widely used in flight control system, control for flight control system provides effective strong robustness.

In order to the saturated impact on flight control system performance of system actuators input can be eliminated, realize the Existence of Global Stable of system, the full flight control system that there is actuator saturation constraint input constraint for a class of king, proposes a kind of model following restructuring control method, improves the tracking performance of flight control system.Guo Yuying, then based on multi-model switching, devises a kind of NEW ADAPTIVE type Adaptive Reconfigurable Control device, eliminates system institute input bound to the impact of system performance, achieves the Existence of Global Stable of system and the asymptotic tracking to system state amount.But these method major parts do not consider that systematic parameter is uncertain and non-linear, for complex structure, parameter uncertainty and the serious quadrotor of nonlinear degree, control effects is poor, even causes the instability of aircraft.

Summary of the invention

Goal of the invention: for above-mentioned prior art, a kind of optimum anti-input Saturation Control method of the quadrotor based on sliding formwork control is proposed, according to the optimal value solving performance index function J, select Optimal Sliding Mode face parameter and sliding formwork control law, form corresponding quadrotor controller, eliminate the impact of actuator saturation for system performance.

Technical scheme: a kind of optimum anti-input Saturation Control method of the quadrotor based on sliding formwork control, when considering that quadrotor exists actuator saturation, in conjunction with optimum control, proposes a kind of sliding-mode control.In guarantee carry under performance index function J reaches the prerequisite of optimal value, calculate system sliding-mode surface parameter and switching time, and then by comparing switching time, designing corresponding sliding-mode surface and sliding formwork control law, finally forming optimal controller.Comprise following concrete steps:

Step 1), obtain the Controlling model of quadrotor:

$\stackrel{\·}{x}=F\left(x\right)+G\left(x\right)u+\mathrm{\Φ}---\left(1\right)$

Wherein, F (x)=[x ₂x ₃f (x)+Δ f] ^t, G (x)=[0 0 g (x)] ^t, Φ=[0 0 d] ^t.In formula: x=[x ₁x ₂x ₃] ^tthe state variable of expression system, represents system displacement, speed respectively, acceleration, u represents that Systematical control inputs, and g (x) and f (x) is the nonlinear equation about system state amount, and wherein g (x) meets | g (x) | and≤σ.The parameter uncertainty that Δ f and d representative system exist and external interference, meet

Step 2), according to the requirement of quadrotor flight safety and flight quality, design as performance index function, in order to embody reaction velocity and the control accuracy of quadrotor.These performance index comprise initial time t ₀, displacement tracking error e ₁.

Step 3), calculate each Optimal Sliding Mode face parameter and sliding-mode surface switching time, comprise the steps:

Step 3.1), consider sliding-mode surface equation form.

$s\left\{\begin{array}{cc}\mathrm{\α}+\mathrm{\βt}+e_3+{\mathrm{Ae}}_2+B{e}_1& t\≤t_f\\ e_3+A{e}_2+{\mathrm{Be}}_1& t>t_f\end{array}\right.---\left(2\right)$

In formula: α, β, A, B are scalar constant to be determined.Alpha+beta t is the Global Sliding Mode factor, is required to meet alpha+beta t _f=0.

T _ffor sliding-mode surface carries out time of switching.

Step 3.2), in conjunction with quadrotor system model and sliding-mode surface equation, performance index function J can be converted into:

$J={\left|e_0\right|}^{\frac{5}{3}}{\left|\mathrm{\β}\right|}^{-\frac{2}{3}}(c^{\frac{1}{3}}+\frac{c^{\frac{4}{3}}}{6}+3c^{-\frac{2}{3}})---\left(3\right)$

Wherein, for by B and t _fthe scalar constant to be determined of common decision.

Step 3.3), the minimum value J of calculation of performance indicators function J _min, shown in (4):

$J_{\min }=3.8{\left|e_{10}\right|}^{\frac{5}{3}}{\left(\mathrm{\σU}\right)}^{-\frac{2}{3}}---\left(4\right)$

Wherein $U=u_{\max }-\max \left\{\frac{\left[\right|\stackrel{\·}{x_{3d}}-f\left(x\right)|+\mathrm{\λ}]}{\left|g\left(x\right)\right|}\right\}$

Step 3.4), according to the minimum value J of performance index function J _min, calculate corresponding Optimal Sliding Mode face parameter and switching time:

c _lop2≈2.7 (5)

${\mathrm{\α}}_{\mathrm{op}}=-e_{10}{\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{2}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{2}{3}}---\left(6\right)$

${\mathrm{\β}}_{\mathrm{op}}=\left(\mathrm{\σU}\right){[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]}^{-1}---\left(7\right)$

$A_{\mathrm{op}}=2{\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{1}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{1}{3}}---\left(8\right)$

$B_{\mathrm{op}}={\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{2}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{2}{3}}---\left(9\right)$

$t_{\mathrm{fop}}={{c_{\mathrm{op}}}^{\frac{2}{3}}\left(\mathrm{\σU}\right)}^{-\frac{1}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{1}{3}}---\left(10\right)$

Step 4), according to step 3.4) Optimal Sliding Mode face t switching time that obtains _fop, carry out time judgement, select corresponding sliding-mode surface.

Step 4.1), as time t≤t _fop, select sliding-mode surface s ₁, shown in (11):

s ₁＝α+βt+e ₃+Ae ₂+Be ₁(11)

Step 4.2), otherwise, time t > t _fop, select sliding-mode surface s ₂, shown in (12):

s ₁＝e ₃+Ae ₂+Be ₁(12)

Step 5), according to step 3.4) Optimal Sliding Mode face t switching time that obtains _fop, carry out time judgement, select corresponding sliding formwork control law.

Step 5.1), as time t≤t _fop, select sliding formwork control law u ₁, shown in (13):

$u_1=\frac{-{\mathrm{\λ}}_{1}R\left(\stackrel{\·}{s}\right)-{\mathrm{\λ}}_{2}s}{g\left(x\right)}+\frac{\stackrel{\·}{x_{3d}}-f\left(x\right)-\mathrm{\Δf}-d-A{e}_3-B{e}_2-\mathrm{\β}}{g\left(x\right)}---\left(13\right)$

Step 5.2), otherwise, the time select sliding formwork control law u ₂, shown in (14):

$u_2=\frac{-{\mathrm{\λ}}_{1}R\left(\stackrel{\·}{s}\right)-{\mathrm{\λ}}_{2}s}{g\left(x\right)}+\frac{\stackrel{\·}{x_{3d}}-f\left(x\right)-\mathrm{\Δf}-d-A{e}_3-B{e}_2}{g\left(x\right)}---\left(14\right)$

Step 6), by step 5) in the sliding formwork control law u that obtains and quadrotor actuator saturation input u _maxrelatively, judge the success or not of anti-input Saturation Control, quadrotor controller can be formed.As u≤u _max, show to form quadrotor controller; Otherwise, re-start Optimal Sliding Mode parameter and calculating switching time.

Beneficial effect: the optimum anti-input Saturation Control method of a kind of quadrotor based on sliding formwork control that the present invention proposes, considers that quadrotor exists actuator saturation, in conjunction with optimum control, proposes a kind of sliding-mode control.In guarantee carry under performance index function J reaches the prerequisite of optimal value, calculate system sliding-mode surface parameter and switching time, and then by comparing switching time, designing corresponding sliding-mode surface and sliding formwork control law, finally forming optimal controller.There is following concrete advantage:

(1) by solving inequality, being met the sliding formwork control law of input saturation constraints, eliminating the saturated harmful effect for quadrotor of input, realizing anti-input saturation constraints;

(2) by solving the performance index function J optimum of design, obtain corresponding Optimal Sliding Mode face parameter and sliding formwork control law, thus improve control accuracy and the response speed of quadrotor, while anti-input is saturated, improve the flying quality of quadrotor;

(3) application that sliding formwork controls ensure that quadrotor has strong robustness for uncertain and external interference, and the introducing of the sliding-mode surface Global Sliding Mode factor, ensure that the robustness in sliding formwork convergence stage, thus realize Global robust.

Method therefor of the present invention, as a kind of anti-input Saturation Control method of quadrotor, has certain application value, is easy to realize, and control effects is good, effectively can improve control accuracy and the reaction velocity of quadrotor.The method is workable, and application is convenient, reliable.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of method of the present invention;

The experimental provision Qball-X4 quadrotor in order to study quadrotor control of Tu2Shi Quanser company development;

Fig. 3 is the system of axis and the sign convention of Qball-X4;

Fig. 4 is the position control system schematic diagram that Qball-X4 presets in X-direction;

Fig. 5 is displacement tracking curve and the actual displacement aircraft pursuit course of Qball-X4 requirement;

Fig. 6-Fig. 7 is Qball-X4 displacement tracking error e 1 curve and speed tracing error e 2 and acceleration tracking error e3 curve;

Fig. 8 is Qball-X4 actuator input u-curve.

Embodiment

Below in conjunction with accompanying drawing the present invention done and further explain.

As shown in Figure 1, based on the optimum anti-input Saturation Control method of the quadrotor that sliding formwork controls, consider that quadrotor exists actuator saturation, in conjunction with optimum control, propose a kind of sliding-mode control.In guarantee carry under performance index function J reaches the prerequisite of optimal value, calculate system sliding-mode surface parameter and switching time, and then by comparing switching time, designing corresponding sliding-mode surface and sliding formwork control law, finally forming optimal controller.Comprise following concrete steps:

Step 1), obtain the Controlling model of quadrotor:

$\stackrel{.}{x}=F\left(x\right)+G\left(x\right)u+\mathrm{\Φ}---\left(1\right)$

Wherein, F (x)=[x ₂x ₃f (x)+Δ f] ^t., G (x)=[0 0 g (x)] ^t, Φ=[0 0 d] ^t.In formula: x=[x ₁x ₂x ₃] T represents the state variable of system, represents system displacement respectively, speed, acceleration, u represents that Systematical control inputs, and g (x) and f (x) is the nonlinear equation about system state amount, and wherein g (x) meets | g (x) | and≤σ.The parameter uncertainty that Δ f and d representative system exist and external interference, meet

Step 2), according to the requirement of quadrotor flight safety and flight quality, design as performance index function, in order to embody reaction velocity and the control accuracy of quadrotor.These performance index comprise initial time t ₀, displacement tracking error e ₁.

Step 3), calculate each Optimal Sliding Mode face parameter and sliding-mode surface switching time, comprise the steps:

Step 3.1), consider sliding-mode surface equation form.

$s\left\{\begin{array}{cc}\mathrm{\α}+\mathrm{\βt}+e_3+{\mathrm{Ae}}_2+B{e}_1& t\≤t_f\\ e_3+A{e}_2+{\mathrm{Be}}_1& t>t_f\end{array}\right.---\left(2\right)$

In formula: α, β, A, B are scalar constant to be determined.Alpha+beta t is the Global Sliding Mode factor, is required to meet alpha+beta t _f=0.T _ffor sliding-mode surface carries out time of switching.

Step 3.2), in conjunction with quadrotor system model and sliding-mode surface equation, performance index function J can be converted into:

$J={\left|e_0\right|}^{\frac{5}{3}}{\left|\mathrm{\β}\right|}^{-\frac{2}{3}}(c^{\frac{1}{3}}+\frac{c^{\frac{4}{3}}}{6}+3c^{-\frac{2}{3}})---\left(3\right)$

Wherein, for by B and t _fthe scalar constant to be determined of common decision.

Step 3.3), the minimum value J of calculation of performance indicators function J _min, shown in (4):

$J_{\min }=3.8{\left|e_{10}\right|}^{\frac{5}{3}}{\left(\mathrm{\σU}\right)}^{-\frac{2}{3}}---\left(4\right)$

Wherein $U=u_{\max }-\max \left\{\frac{\left[\right|\stackrel{\·}{x_{3d}}-f\left(x\right)|+\mathrm{\λ}]}{\left|g\left(x\right)\right|}\right\}$

Step 3.4), according to the minimum value J of performance index function J _min, calculate corresponding Optimal Sliding Mode face parameter and switching time:

c _lop2≈2.7 (5)

${\mathrm{\α}}_{\mathrm{op}}=-e_{10}{\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{2}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{2}{3}}---\left(6\right)$

${\mathrm{\β}}_{\mathrm{op}}=\left(\mathrm{\σU}\right){[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]}^{-1}---\left(7\right)$

$A_{\mathrm{op}}=2{\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{1}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{1}{3}}---\left(8\right)$

$B_{\mathrm{op}}={\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{2}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{2}{3}}---\left(9\right)$

$t_{\mathrm{fop}}={{c_{\mathrm{op}}}^{\frac{2}{3}}\left(\mathrm{\σU}\right)}^{-\frac{1}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{1}{3}}---\left(10\right)$

Step 4), according to step 3.4) Optimal Sliding Mode face t switching time that obtains _fop, carry out time judgement, select corresponding sliding-mode surface.

Step 4.1), as time t≤t _fop, select sliding-mode surface s ₁, shown in (11):

s ₁＝α+βt+e ₃+Ae ₂+Be ₁(11)

Step 4.2), otherwise, time t > t _fop, select sliding-mode surface s ₂, shown in (12):

s ₁＝e ₃+Ae ₂+Be ₁(12)

Step 5), according to step 3.4) Optimal Sliding Mode face t switching time that obtains _fop, carry out time judgement, select corresponding sliding formwork control law.

Step 5.1), as time t≤t _fop, select sliding formwork control law u ₁, shown in (13):

$u_1=\frac{-{\mathrm{\λ}}_{1}R\left(s\right)-{\mathrm{\λ}}_{2}s}{g\left(x\right)}+\frac{\stackrel{\·}{x_{3d}}-f\left(x\right)-\mathrm{\Δf}-d-A{e}_3-B{e}_2-\mathrm{\β}}{g\left(x\right)}---\left(13\right)$

Step 5.2), otherwise, time t > t _fop, select sliding formwork control law u ₂, shown in (14):

$u_2=\frac{-{\mathrm{\λ}}_{1}R\left(s\right)-{\mathrm{\λ}}_{2}s}{g\left(x\right)}+\frac{\stackrel{\·}{x_{3d}}-f\left(x\right)-\mathrm{\Δf}-d-A{e}_3-B{e}_2}{g\left(x\right)}---\left(14\right)$

Step 6), by step 5) in the sliding formwork control law u that obtains and quadrotor actuator saturation input u _maxrelatively, judge the success or not of anti-input Saturation Control, quadrotor controller can be formed.As u≤u _max, show to form quadrotor controller; Otherwise, re-start Optimal Sliding Mode parameter and calculating switching time.

The above is only the preferred embodiment of the present invention, it should be pointed out that for those skilled in the art; under the premise without departing from the principles of the invention; can also make some improvements and modifications, these improvements and modifications also should be considered as protection scope of the present invention

The validity of embodiment is described with real case emulation below.

Adopt by Quanser company develop in order to study quadrotor control experimental provision Qball-X4 quadrotor experiment porch as applied research object, Qball-X4 is as shown in Figure 2.Qball-X4 quadrotor, there are six dimension variable i.e. (X, Y, Z, ψ, θ, φ), wherein X, Y, Z are location variable, and ψ is crab angle, and θ is the angle of pitch, and φ is roll angle.Qball-X4 body axis system OXYZ is set up shown in Fig. 3.Assuming that the pitching angle theta of Qball aircraft, roll angle φ, crab angle ψ are zero, Qball to carry out flat flying motion.Without loss of generality, only carry out line motion control in the X-axis here, choose its displacement in the X-axis, speed, acceleration is as system state amount, and Fig. 4 is the position control system schematic diagram that Qball-X4 presets in X-direction.

Qball-X4 meets following state equation:

$\stackrel{\·}{x}=F\left(x\right)+G\left(x\right)u+\mathrm{\Φ}$

F(x)＝[x ₂x ₃x ₁x ₂+Δf] ^T

G(x)＝[0 0 1] ^T

Φ＝[0 0 d] ^T

Consider the parameter uncertainty form that aircraft interior is common and the external interference easily run in flight course (as situations such as flow perturbations), the parameter uncertainty that system of getting in experiment exists is simultaneity factor is subject to external interference d=0.6sin (10t).

Require that the tracking signal of Qball in X-axis is: x _1d=sin (t+2.2), the power supply due to Qball is two joint three core 2500mAh lithium batteries, thus records the restriction requirement of actuator input quantity demand fulfillment u≤10v.

The quantity of state vector getting initial time system is:

x ₀＝[x ₁₀x ₂₀x ₃₀] ^T＝[1.8085 -0.5885 -0.8085] ^T

According to the inventive method, control inputting saturated Qball-X4 quadrotor with actuator, Fig. 5-Fig. 8 is anti-input Saturation Control result.Wherein Fig. 5 is displacement tracking curve and the actual displacement aircraft pursuit course of Qball-X4 requirement; Fig. 6-Fig. 7 is Qball-X4 displacement tracking error e 1 curve and speed tracing error e 2 and acceleration tracking error e3 curve; Fig. 8 is Qball-X4 actuator input u-curve.

As shown in Figure 5, initial time, there is initial error in Qball displacement tracking in the X-axis, and there is parameter uncertainty and external interference simultaneously.Fig. 6 and Fig. 7 shows, in this case, controls and the anti-input Saturation Control method of quadrotor of optimum control based on sliding formwork, can ensure that Qball still can follow the tracks of the displacement of targets of specifying, speed, acceleration fast, make displacement, speed, the tracking error Fast Convergent of acceleration is zero.The Qball-X4 control inputs curve of Fig. 8 then shows, sliding formwork control law, and namely the absolute value of actuator input quantity is less than input constraint value condition all the time, remains in the scope of 10v, and the system that can calculate is under this controls, and it is J that performance index reach minimum value _1min=0.95.Should to control based on sliding formwork and the anti-input Saturation Control method of quadrotor of optimum control not only can be eliminated and inputs the saturated malicious influences for quadrotor, control accuracy and the reaction velocity of aircraft can also be ensured.

Claims (1)

1. based on an optimum anti-input Saturation Control method for the quadrotor of sliding formwork control, it is characterized in that: consider that quadrotor exists actuator saturation, in conjunction with optimum control, propose a kind of sliding-mode control.In guarantee carry under performance index function J reaches the prerequisite of optimal value, calculate system sliding-mode surface parameter and switching time, and then by comparing switching time, designing corresponding sliding-mode surface and sliding formwork control law, finally forming optimal controller.Comprise following concrete steps:

Step 1), obtain the Controlling model of quadrotor:

x＝F(x)+G(x)u+Φ (1)

Wherein, F (x)=[x ₂x ₃f (x)+Δ f] ^t, G (x)=[0 0 g (x)] ^t, Φ=[0 0 d] ^t.In formula: x=[x ₁x ₂x ₃] ^tthe state variable of expression system, represents system displacement, speed respectively, acceleration, u represents that Systematical control inputs, and g (x) and f (x) is the nonlinear equation about system state amount, and wherein g (x) meets | g (x) | and≤σ.The parameter uncertainty that Δ f and d representative system exist and external interference, meet

Step 2), according to the requirement of quadrotor flight safety and flight quality, design as performance index function, in order to embody reaction velocity and the control accuracy of quadrotor.These performance index comprise initial time t ₀, displacement tracking error e ₁.

Step 3), calculate each Optimal Sliding Mode face parameter and sliding-mode surface switching time, comprise the steps:

Step 3.1), consider sliding-mode surface equation form.

$s=\left\{\begin{array}{cc}\mathrm{\α}+\mathrm{\βt}+e_3+{\mathrm{Ae}}_2+{\mathrm{Be}}_1& t\≤t_f\\ e_3+{\mathrm{Ae}}_2+{\mathrm{Be}}_1& t>t_f\end{array}\right.---\left(2\right)$

In formula: α, β, A, B are scalar constant to be determined.Alpha+beta t is the Global Sliding Mode factor, is required to meet alpha+beta t _f=0.T _ffor sliding-mode surface carries out time of switching.

Step 3.2), in conjunction with quadrotor system model and sliding-mode surface equation, performance index function J can be converted into:

$J={\left|e_0\right|}^{\frac{5}{3}}{\left|\mathrm{\β}\right|}^{-\frac{2}{3}}(c^{\frac{1}{3}}+\frac{c^{\frac{4}{3}}}{6}+{3c}^{-\frac{2}{3}})---\left(3\right)$

Wherein, for by B and t _fthe scalar constant to be determined of common decision.

Step 3.3), the minimum value J of calculation of performance indicators function J _min, shown in (4):

$J_{\min }=3.8{\left|e_{10}\right|}^{\frac{5}{3}}{\left(\mathrm{\σU}\right)}^{-\frac{2}{3}}---\left(4\right)$

Wherein $U=U_{\max }-\max \left\{\frac{\left[\right|x_{3d}-f\left(x\right)|+\mathrm{\λ}]}{\left|g\left(x\right)\right|}\right\}$

Step 3.4), according to the minimum value J of performance index function J _min, calculate corresponding Optimal Sliding Mode face parameter and switching time:

c _1op2≈2.7 (5)

${\mathrm{\α}}_{\mathrm{op}}={-e}_{10}{\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{2}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{2}{3}}---\left(6\right)$

${\mathrm{\β}}_{\mathrm{op}}=\left(\mathrm{\σU}\right){[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]}^{-1}---\left(7\right)$

$A_{\mathrm{op}}=2{\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{1}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{1}{3}}---\left(8\right)$

$B_{\mathrm{op}}={\left(c_{\mathrm{op}}\mathrm{\σU}\right)}^{\frac{2}{3}}{\left\{\right[e^{-c_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{-\frac{2}{3}}---\left(9\right)$

$t_{\mathrm{fop}}={c_{\mathrm{op}}}^{\frac{2}{3}}{\left(\mathrm{\σU}\right)}^{-\frac{1}{3}}{\left\{\right[e^{{-c}_{\mathrm{op}}}(c_{\mathrm{op}}-1)+1]\·|e_{10}\left|\right\}}^{\frac{1}{3}}---\left(10\right)$

Step 4), according to step 3.4) Optimal Sliding Mode face t switching time that obtains _fop, carry out time judgement, select corresponding sliding-mode surface.

Step 4.1), as time t≤t _fop, select sliding-mode surface s ₁, shown in (11):

s ₁＝α+βt+e ₃+Ae ₂+Be ₁(11)

Step 4.2), otherwise, time t > t _fop, select sliding-mode surface s ₂, shown in (12):

s ₁＝e ₃+Ae ₂+Be ₁(12)

Step 5), according to step 3.4) Optimal Sliding Mode face t switching time that obtains _fop, carry out time judgement, select corresponding sliding formwork control law.

Step 5.1), as time t≤t _fop, select sliding formwork control law u ₁, shown in (13):

$u_1=\frac{{-\mathrm{\λ}}_{1}R\left(s\right)-{\mathrm{\λ}}_{2}s}{g\left(x\right)}+\frac{x_{3d}-f\left(x\right)-\mathrm{\Δf}-d-{\mathrm{Ae}}_3-{\mathrm{Be}}_2-\mathrm{\β}}{g\left(x\right)}---\left(13\right)$

Step 5.2), otherwise, time t > t _fop, select sliding formwork control law u ₂, shown in (14):

$u_2=\frac{{-\mathrm{\λ}}_{1}R\left(s\right)-{\mathrm{\λ}}_{2}s}{g\left(x\right)}+\frac{x_{3d}-f\left(x\right)-\mathrm{\Δf}-d-{\mathrm{Ae}}_3-{\mathrm{Be}}_2}{g\left(x\right)}---\left(14\right)$

Step 6), by step 5) in the sliding formwork control law u that obtains and quadrotor actuator saturation input u _maxrelatively, judge the success or not of anti-input Saturation Control, quadrotor controller can be formed.As u≤u _max, show to form quadrotor controller; Otherwise, re-start Optimal Sliding Mode parameter and calculating switching time.