基于滑模干擾觀測(cè)器的反演終端滑模飛行控制

王堅(jiān)浩, 胡劍波, 張 亮, 張鵬濤, 宋 敏

(1. 空軍工程大學(xué)裝備管理與無人機(jī)工程學(xué)院, 陜西 西安 710051;2. 空軍工程大學(xué)空管領(lǐng)航學(xué)院, 陜西 西安 710051)

0 引 言

近年來,將滑?？刂芠1-2]和反演設(shè)計(jì)[3-5]兩種非線性控制方法相結(jié)合的滑模反演控制方法受到了大量學(xué)者的關(guān)注[6-11],并在電力系統(tǒng)控制[12-14]、導(dǎo)彈制導(dǎo)控制一體化設(shè)計(jì)[15-16]、航空航天飛行控制[17-19]等領(lǐng)域得到了廣泛應(yīng)用。文獻(xiàn)[16]針對(duì)制導(dǎo)控制一體化控制律設(shè)計(jì)問題,提出了一種滑模反演控制方法,通過設(shè)計(jì)自適應(yīng)誤差滑模面克服了模糊自適應(yīng)系統(tǒng)逼近誤差和不確定干擾,但由于滑?？刂剖腔诓淮_定性的上界設(shè)計(jì)的,因此要求不確定干擾上界已知或采用自適應(yīng)律對(duì)上界進(jìn)行估計(jì),然而不確定性上界已知條件往往因先驗(yàn)知識(shí)缺乏難以滿足,而自適應(yīng)項(xiàng)的引入容易引起抖振;文獻(xiàn)[17]采用自適應(yīng)策略在線估計(jì)模型的不確定干擾,將自適應(yīng)滑模反演控制方法應(yīng)用于航天飛行器姿態(tài)控制;文獻(xiàn)[18]采用反演滑?？刂品椒▽?duì)超低空重裝空投的控制問題進(jìn)行研究,在每一步遞推設(shè)計(jì)過程中均采用滑?？刂品椒ㄔO(shè)計(jì)虛擬控制律,為了減弱抖振,采取邊界層內(nèi)的正則化方法,僅能保證每一步跟蹤誤差收斂至指定的跟蹤誤差帶內(nèi),多步遞推后容易造成較大的累計(jì)誤差。文獻(xiàn)[16-18]基于線性滑模設(shè)計(jì)滑模面,因此系統(tǒng)所有狀態(tài)只能趨近平衡點(diǎn);此外,在系統(tǒng)設(shè)計(jì)過程中也并未考慮傳統(tǒng)反演控制存在的“計(jì)算膨脹”問題,因此控制律包含虛擬控制的導(dǎo)數(shù)使得控制器實(shí)現(xiàn)較為復(fù)雜。文獻(xiàn)[19]采用快速終端滑模面提高系統(tǒng)的收斂速度和穩(wěn)態(tài)跟蹤精度,并結(jié)合動(dòng)態(tài)面控制設(shè)計(jì)基于非線性干擾觀測(cè)器的反演終端滑模控制方法,但在非線性干擾觀測(cè)器設(shè)計(jì)中參數(shù)選取較為困難,若選取不當(dāng)非線性干擾觀測(cè)器不僅不能起到補(bǔ)償作用[20-23]。此外,動(dòng)態(tài)面控制[24-25]實(shí)質(zhì)上基于濾波器設(shè)計(jì)的一種方法,但對(duì)于濾波器設(shè)計(jì)參數(shù)選擇要求較高,且有效抑制噪聲的能力較弱。

本文研究存在氣動(dòng)參數(shù)攝動(dòng)和力矩干擾不確定性的戰(zhàn)斗機(jī)機(jī)動(dòng)飛行控制問題?；诳焖俳K端滑模反演方法設(shè)計(jì)控制器,結(jié)合滑模微分器獲取虛擬控制律的導(dǎo)數(shù),避免計(jì)算復(fù)雜性問題,并基于滑模微分器設(shè)計(jì)滑模干擾觀測(cè)器(sliding mode disturbance observer,SMDO),實(shí)現(xiàn)對(duì)模型不確定性的平滑估計(jì)和補(bǔ)償,提高單純反演終端滑模控制精度,最后通過某戰(zhàn)斗機(jī)姿態(tài)跟蹤機(jī)動(dòng)飛行進(jìn)行了仿真驗(yàn)證。

1 飛機(jī)非線性模型

某型戰(zhàn)斗機(jī)姿態(tài)控制系統(tǒng)的數(shù)學(xué)模型[26]為

(1)

式中,Fx,Fy,Fz和L,M,N分別為氣動(dòng)力和氣動(dòng)力矩;m和T分別表示飛機(jī)質(zhì)量和發(fā)動(dòng)機(jī)推力;飛機(jī)狀態(tài)包括迎角α,側(cè)滑角β,滾轉(zhuǎn)角φ,滾轉(zhuǎn)角速度p,俯仰角速度q,偏航角速度r,俯仰角θ和飛行速度V;g表示地球引力常數(shù);Ii(i=1,2,…,9)由慣性力矩常數(shù)計(jì)算得到。

將模型(1)中的Fx,Fy,Fz和L,M,N用氣動(dòng)力系數(shù)和氣動(dòng)力矩系數(shù)表示如下:

(2)

定義y=x1=[α,β,φ]T,x2=[p,q,r]T,x3=θ,u=[δe,δa,δr]T,式(1)可寫成如下形式:

(3)

2 控制器設(shè)計(jì)

由飛機(jī)姿態(tài)控制系統(tǒng)非線性模型(3)可知,不確定干擾的界限難以確切獲知。為此,首先使用SMDO觀測(cè)系統(tǒng)不確定干擾,并采用反演終端滑?？刂七M(jìn)行控制補(bǔ)償,姿態(tài)控制系統(tǒng)結(jié)構(gòu)如圖1所示。為方便控制系統(tǒng)設(shè)計(jì),給出如下假設(shè):

假設(shè)3存在已知正實(shí)數(shù)αm,βm,θm∈R,對(duì)于所有滿足|α|≤αm,|β|≤βm,|θ|≤θm的α,β,θ,g1和g2均可逆。

假設(shè)4存在正實(shí)數(shù)θm∈R,滿足|θ|≤θm<π/2。

首先引入角度回路和角速度回路狀態(tài)跟蹤誤差z1,z2∈R3：

(4)

式中,x1c,x2c分別為控制系統(tǒng)虛擬控制律,且x1c=yc。

圖1 姿態(tài)控制系統(tǒng)結(jié)構(gòu)Fig.1 Block diagram of control system

步驟1采用反演控制方法設(shè)計(jì)虛擬控制律x2c,作為角速度回路的參考指令信號(hào)。

由角度回路狀態(tài)跟蹤誤差z1=x1-yc,則z1的動(dòng)態(tài)系統(tǒng)為

(5)

(6)

(7)

設(shè)計(jì)虛擬控制律為

(8)

式中,k1=diag{k11,k12,k13},k1i>0為設(shè)計(jì)參數(shù)。

(9)

式中,qsmd,psmd為Terminal吸引子設(shè)計(jì)參數(shù),為正奇數(shù),且滿足1/2

定義角度回路的Lyapunov函數(shù)為

(10)

對(duì)V1求導(dǎo)得

(11)

(12)

若角速度回路能夠?qū)崿F(xiàn)精確跟蹤,即z2=0,則有

(13)

以上分析表明,當(dāng)角速度回路狀態(tài)跟蹤誤差z2=0時(shí),在虛擬控制律x2c的作用下,則可以保證狀態(tài)跟蹤誤差z1收斂到原點(diǎn)附近任意小的鄰域內(nèi)。

步驟2設(shè)計(jì)控制律u,確保角速度回路實(shí)現(xiàn)精確跟蹤,即z2=0。

由z2=x2-x2c,則z2的動(dòng)態(tài)系統(tǒng)為

(14)

為了使角速度回路狀態(tài)跟蹤誤差z2在有限時(shí)間內(nèi)收斂到零,設(shè)計(jì)如下快速終端滑模面[19]

(15)

對(duì)S求導(dǎo)得

(16)

(17)

(18)

(19)

定義角速度回路的Lyapunov函數(shù)為

(20)

對(duì)V2求導(dǎo)得

(21)

(22)

3 仿真與分析

為了驗(yàn)證控制策略的有效性,對(duì)某型戰(zhàn)斗機(jī)姿態(tài)控制系統(tǒng)進(jìn)行閉環(huán)系統(tǒng)仿真,初始條件為:發(fā)動(dòng)機(jī)推力T=60 kN,高度H=10 000 ft,速度V=500 ft/s,仿真指令信號(hào)為

舵面偏轉(zhuǎn)角范圍為δe∈[-25°,25°],δa∈[-21.5°,21.5°],δr∈[-30°,30°]。為避免出現(xiàn)舵面偏轉(zhuǎn)角飽和問題,參考指令信號(hào)yc為指令信號(hào)yd經(jīng)過二階指令參考模型的輸出,即

控制器設(shè)計(jì)參數(shù)為k1=diag[10,10,10],a=diag[1,1,1],b=diag[0.1,0.1,0.1],γ=κ=diag[2,2,2],ρ1=5,ρ2=7,ρ3=3,ρ4=5,滑模微分器參數(shù)為λsmd0=λsmd1=10,qsmd=5,psmd=7。

為了驗(yàn)證控制策略的魯棒性,假設(shè)氣動(dòng)參數(shù)攝動(dòng)60%,滾轉(zhuǎn)通道、俯仰通道和偏航通道分別存在[0.8 1.7 2.1]×104sin(0.5t)(N·m)的力矩干擾。在控制器和滑模微分器參數(shù)取值相同條件下,將有無采用SMDO的反演快速終端滑?？刂撇呗砸约拔墨I(xiàn)[18]提出的基于NDO的反演快速終端滑?？刂撇呗赃M(jìn)行仿真對(duì)比與驗(yàn)證。SMDO參數(shù)為λ10=λ11=λ20=λ21=10,q1=q2=5,p1=p2=7。

仿真結(jié)果如圖2～圖7所示,其中下標(biāo)c表示參考指令信號(hào),下標(biāo)1表示基于SMDO的反演快速終端滑模控制,下標(biāo)2表示基于NDO的反演快速終端滑?？刂?下標(biāo)3表示傳統(tǒng)反演滑模控制。仿真結(jié)果充分表明,當(dāng)沒有加入SMDO時(shí),模型不確定性對(duì)控制精度影響較大,且升降舵偏轉(zhuǎn)角進(jìn)入飽和狀態(tài),方向舵偏轉(zhuǎn)角進(jìn)入臨界飽和狀態(tài)。采用文獻(xiàn)[19]和本文提出的控制策略,均實(shí)現(xiàn)了對(duì)氣動(dòng)參數(shù)攝動(dòng)和力矩干擾的平滑重構(gòu),舵面偏轉(zhuǎn)角平滑且未出現(xiàn)飽和;此外,兩種控制策略相比較,采用本文提出的控制策略,參考指令跟蹤的準(zhǔn)確性更高、過渡過程品質(zhì)更優(yōu)。

圖2 迎角響應(yīng)Fig.2 Responses of attack angle tracking

圖3 側(cè)滑角響應(yīng)Fig.3 Responses of sideslip angle tracking

圖4 滾轉(zhuǎn)角響應(yīng)Fig.4 Responses of roll angle tracking

圖5 升降舵偏轉(zhuǎn)曲線Fig.5 Elevator deflection curves

圖6 副翼偏轉(zhuǎn)曲線Fig.6 Aileron deflection curves

圖7 方向舵偏轉(zhuǎn)曲線Fig.7 Rudder deflection curves

4 結(jié) 論

本文提出了一種基于SMDO的反演終端滑?？刂品椒ú?yīng)用于戰(zhàn)斗機(jī)姿態(tài)跟蹤機(jī)動(dòng)飛行控制問題,該方法采用滑模微分器獲取虛擬控制律的導(dǎo)數(shù),避免計(jì)算復(fù)雜性問題,并基于滑模微分器設(shè)計(jì)了一種新型SMDO,實(shí)現(xiàn)了對(duì)模型不確定性的平滑估計(jì)和補(bǔ)償,有效解決了傳統(tǒng)滑模反演控制魯棒性不強(qiáng)、控制精度不高的問題。數(shù)值仿真表明:在存在較大氣動(dòng)參數(shù)攝動(dòng)和力矩干擾不確定性的情況下,能夠?qū)崿F(xiàn)對(duì)參考軌跡的穩(wěn)定跟蹤。

參考文獻(xiàn)：

[1] DONGMO J E. Loss-of control autonomous flight recovery regime using feedback linearization and high order sliding mode control with exponential observers[C]∥Proc.of the AIAA Guidance, Navigation, and Control Conference, 2015: 1-16.

[2] ZHANG W J, CHEN H B. Aircraft loss-of-control recovery strategy using high order sliding mode control based on optimal trim condition[C]∥Proc.of the AIAA Guidance, Navigation, and Control Conference, 2016: 1-15.

[3] HUANG P F, WANG D K, MENG Z J, et al. Adaptive postcapture backstepping control for tumbling tethered space robot-target combination[J]. Journal of Guidance, Control and Dynamics, 2016, 39(1): 150-156.

[4] GILS P V, KAMPEN E V, VISSER C D, et al. Adaptive incremental backstepping flight control for a high-performance aircraft with uncertainties[C]∥Proc.of the AIAA Guidance, Navigation, and Control Conference, 2016:1-26.

[5] WU G H, MENG X Y. Nonlinear disturbance observer based robust backstepping control for a flexible air-breathing hypersonic vehicle[J].Aerospace science and Technology,2016,54:174-182.

[6] PENG C C, HSUE ALBEr Y B, CHEN C L. Variable structure based robust backstepping controller design for nonlinear systems[J].Nonlinear Dynamics, 2011, 63(1/2): 253-262.

[7] DAVILA J. Exact tracking using backstepping control design and high-order sliding modes[J].IEEE Trans.on Automatic Control, 2013, 58(8): 2077-2081.

[8] 胡劍波,李飛,魏高樂,等.不確定系統(tǒng)反推滑模變結(jié)構(gòu)理論及其應(yīng)用[J].系統(tǒng)工程與電子技術(shù),2014,36(3):519-526.

HU J B, LI F, WEI G L, et al. Theory and applications of backstepping sliding mode variable structure control for uncertain systems[J].Systems Engineering and Electronics,2014,36(3): 519-526.

[9] LI J P, YANG Y N, HUA C C, et al. Fixed-time backstepping control design for high-order strict-feedback non-linear systems via terminal sliding mode[J].IEE Control Theory and Applications, 2017, 11(8): 1184-1193.

[10] RAHMAN R, HE L, SEPEHRI N. Design and experimental study of a dynamical adaptive backstepping sliding mode control scheme for position tracking and regulating of a low-cost pneumatic cylinder[J].International Journal of Robust and Nonlinear Control, 2016, 26(4): 853-875.

[11] 王堅(jiān)浩,胡劍波.不確定非線性系統(tǒng)的自適應(yīng)反推高階終端滑?？刂芠J].控制與決策,2012,27(3):413-418.

WANG J H, HU J B. Adaptive backstepping high-order terminal sliding mode control for uncertain nonlinear systems[J]. Control and Decision, 2012, 27(3): 413-418.

[12] DHAR S, DASH P K. Adaptive backstepping sliding mode control of a grid interactive PV-VSC system with LCL filter[J]. Sustainable Energy, Grid and Networks,2016, 6:109-124.

[13] DHAR S, DASH P K.A new backstepping finite time sliding mode control of grid connected PV system using multivariable dynamic VSC model[J].Electrical Power and Energy Systems, 2016, 82:314-330.

[14] SU Q Y, QUAN W Z, CAI G W, et al. Improved adaptive backstepping sliding mode control for generator steam values of non-linear power systems[J].IET Control Theory and Applications, 2017, 11(9):1414-1419.

[15] 齊輝,張澤,韓鵬鑫,等.基于反演滑?？刂频膶?dǎo)彈制導(dǎo)控制一體化設(shè)計(jì)[J].系統(tǒng)工程與電子技術(shù),2016,38(3):618-623.

QI H, ZHANG Z, HAN P X, et al. Integrated design of missile guidance and control based on back-stepping and sliding mode control[J].Systems Engineering and Electronics,2016,38(3):618-623.

[16] RAN M P, WANG Q, HOU D L, et al. Backstepping design of missile guidance and control based on adaptive fuzzy sliding mode control[J].Chinese Journal of Aeronautics,2014,27(3):634-642.

[17] CONG B L, LIU X D, CHEN Z. Backstepping based adaptive sliding mode control for spacecraft attitude maneuvers[J]. Aerospace Science and Technology, 2013, 30(1):1-7.

[18] 楊雨,陸宇平.運(yùn)輸機(jī)超低空重裝空投縱向反步滑?？刂蒲芯縖J].航空學(xué)報(bào),2012,33(12):2301-2312.

YANG Y,LU Y P. Backstepping sliding mode control for super-low altitude heavy cargo airdrop from transport plane[J].Acta Aeronautica et Astronautica Sinica, 2012, 33(12):2301-2312.

[19] 王堅(jiān)浩,胡劍波,張博鋒.應(yīng)用非線性干擾觀測(cè)器的反推終端滑模飛行控制[J].應(yīng)用科學(xué)學(xué)報(bào),2012,30(4):408-414.

WANG J H, HU J B, ZHANG B F. Backstepping terminal sliding mode for flight control based on nonlinear disturbance observer[J].Journal of Applied Sciences,2012,30(4):408-414.

[20] ZHANG Y M, YAN P, ZHANG Z. A disturbance observer-based adaptive control approach for flexure beam nano manipulators[J].Isa Transactions,2016,60(3):206-217.

[21] CHEN W H, YANG J, GUO L, et al. Disturbance-observer-based control and related methods-an overview[J].IEEE Trans.on Industrial Electronics, 2016, 63(2):1083-1095.

[22] XU B. Disturbance observer-based dynamic surface control of transport aircraft with continuous heavy cargo airdrop[J].IEEE Trans.on Systems, Man, and Cybernetics:Systems,2017,47(1):161-170.

[23] ZHANG H G, HAN J, LUO C M, et al. Fault-tolerant control of a nonlinear system based on generalized fuzzy hyperbolic model and adaptive disturbance observer[J].IEEE Trans.on Systems,Man,and Cybernetics:Systems,2017,47(8):2289-2300.

[24] PAN Y P, YU H Y. Dynamic surface control via singular perturbation analysis[J]. Auotmatic, 2015,57(C):29-33.

[25] LI Z F, LI T S, FENG G. Adaptive neural control for a class of stochastic nonlinear time-delay systems with unknown dead zone using dynamic surface technique[J]. International Journal of Robust and Nonlinear Control, 2016, 26(4):759-781.

[26] TAEYOUNG L, YOUDAN K. Nonlinear adaptive flight control using backstepping and neural networks controller[J].Journal of Guidance, Control, and Dynamics, 2001, 24(4):675-682.

[27] 蒲明,吳慶憲,姜長(zhǎng)生,等.高階滑模微分器的分析與改進(jìn)[J].控制與決策,2011,26(8):1136-1140.

PU M, WU Q X, JIANG C S, et al. Analysis and improvement of high-order differentiator[J]. Control and Decision, 2011, 26(8):1136-1140.

[28] HE S M, WANG J, LIN D F. Composite guidance laws using higher order slding mode differentiator and disturbance observer[J]. Journal of Aerospace Engineering,2015, 229(13):2397-2415.

[29] BU X W, WU X Y, MA Z, et al.Novel adaptive neural control of flexible air-breathing hypersonic vehicles based on sliding mode differentiator[J]. Chinese Journal of Aeronautics, 2015,28(4):1209-1216.

[30] SEOKWON L, YOUDAN K. Design of nonlinear observer for strap-down missile guidance law via sliding mode differentiator and extended state observer[C]∥Proc.of the International Conference on Advanced Mechatronic Systems,2016:143-147.

[31] GHANES M, BARBOT J P, FRIDMAN L, et al. A second order sliding mode differentiator with a variable exponent[C]∥Proc.of the American Control Conference,2017:3300-3305.

[32] 卜祥偉,吳曉燕,陳永興,等.基于非線性干擾觀測(cè)器的高超聲速飛行器滑模反演控制[J].控制理論與應(yīng)用,2014,31(11):1473-1479.

BU X W, WU X Y, CHEN Y X, et al. Nonlinear-disturbance-observer-based sliding mode backstepping control of hypersonic vehicles[J].Control Theory & Applications, 2014,31(11):1473-1479.