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SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 2020, VOL. 8, NO. 1, 605–617 https://doi.org/10.1080/21642583.2020.1851805 Active disturbance rejection switching control of quadrotor based on robust differentiator Jiawei Zhao , Hongli Zhang and Xinkai Li Xinjiang University, Urumqi, People’s Republic of China ABSTRACT ARTICLE HISTORY Received 22 April 2020 To solve the problem of trajectory tracking control of a quadrotor unmanned aerial vehicle (UAV) Accepted 12 November 2020 under white noise disturbance, a new active disturbance rejection control (ADRC), an active distur- bance rejection switching control (ADRSC) algorithm of the quadrotor based on a robust differentia- KEYWORDS tor, is proposed. The dynamic model of the quadrotor is stated in three sub-equations, and a control Quadrotor unmanned aerial algorithm is designed. The exact robust differentiator (ERD) replaces the tracking differentiator in vehicle (UAV); robust the traditional ADRC algorithm to improve the accuracy and robustness of the differential signal. diﬀerentiator; active The ADRSC algorithm controls the quadrotor to improve its anti-white-noise disturbance ability and disturbance rejection control (ADRC); trajectory tracking control accuracy. Simulation results show that the quadrotor has an anti-white-noise disturbance control ability of no more than 0.1 dB intensity with this algorithm. Introduction algorithm and the accuracy of the observer by gradually increasing the disturbance. Lee et al. (2009) designed the The quadrotor unmanned aerial vehicle (UAV) is a rotor- craft that can perform three kinds of attitude changes: controller of a quadrotor based on feedback linearization vertical takeoff and landing, pitch roll, and yaw. Com- theory and compared the control performance of two pared to fixed-wing and other single-rotor flight vehi- methods of adaptive sliding mode control and feedback cles, the quadrotor UAV has a simple structure, high pay- linearization control. An SMC algorithm has strong anti- load, and strong maneuverability. The quadrotor UAV has interference and robustness. Other nonlinear SMC algo- the characteristics of under-actuation, nonlinearity, and rithms were designed for the control of a quadrotor UAV strong coupling. These characteristics bring a great dif- (Chen et al., 2015; Dolatabadi & Yazdanpanah, 2015;Han, ficulty to its control, especially under an external distur- 1998;Wangetal., 2016). However, the form of an SMC bance. The body of the quadrotor UAV is light, and it easily algorithm is often complicated, which is not conducive suffers from disturbance. The attitude and position of the to the streamlined design of the control law. There- quadrotor UAV during the state feedback control process fore, it is necessary to find a simple and efficient control also faces white noise disturbance. algorithm that has the characteristics of anti-disturbance Proportion integral derivative (PID) control is widely and robustness. Huang (2019) investigated for a class of used in quadrotor UAVs, but due to the vehicle’s non- discrete time-varying systems with censored measure- ments and parameter uncertainties. Shen (2019) design linear characteristics, PID control cannot produce good a finite-horizon filter such that an upper bound is guaran- system performance in complex environments (Khatoon teed and the filter gain minimizing such an upper bound et al., 2014). Su et al. (2011) proposed a nonlinear PID is subsequently obtained in terms of the solution to a method to control a quadrotor UAV, but the dynamic set of recursive equations. Active disturbance rejection model did not fully consider the dynamic characteristics. control (ADRC) is an anti-disturbance control algorithm An adaptive sliding mode control (SMC) algorithm was proposed based on immersion and invariant theory, with (Han, 1998). The use of errors and nonlinear functions can verified anti-interference ability under periodic distur- improve the overall performance of control in actual pro- bance (Xia et al., 2017). Chen et al. (2016) designed a back- duction processes. Liu et al. (2015) and Wu et al. (2016) step sliding mode controller and a robust SMC observer researched and applied ADRC algorithms in quadrotors. for a quadrotor UAV and verified the robustness of the In the original ADRC, the tracking differentiator (TD) is CONTACT Jiawei Zhao 784912408@qq.com © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 606 J. ZHAO ET AL. a relatively independent part that is used to extract the compared, and the anti-disturbance and robustness of differential signal and reduce the interference of exter- the control algorithm are verified. nal noise (Han & Wang, 1994). The speed and accuracy of the ADRC is determined by the accuracy of the differ- Dynamic modelling of quadrotor ential signal extracted by the TD and its anti-disturbance performance. Extended state observer (ESO) is a state According to the literature (Wang et al., 2016;Wuetal., observer based on error variation (Han, 1995). An ESO can 2016), the quadrotor dynamic model has the form be categorized as either a linear extended state observer ⎪ x ¨ = u (cos ϕ sin θ cos ψ + sin ϕ sin ψ)/m (LESO) or nonlinear extended state observer (NLESO), each with its own advantages and characteristics. Switch- ⎪ −(κ x ˙/m) + d x x ing control based on ADRC was proposed (Li et al., 2016b); y ¨ = u (cos ϕ sin θ sin ψ − sin ϕ cos ψ)/m ⎪ τ a linear active disturbance rejection control/nonlinear −(κ y/m) + d y y active disturbance rejection control (LADRC/NLADRC) (1) z ¨ = u (cos ϕ cos θ)/m − g − (κ z ˙/m) + d τ z z algorithm combines the advantages of both. Proof of ⎪ ϕ ¨ = lu /J + d − (lκ ϕ/ ˙ J ) ϕ ϕ ϕ ϕ ϕ stability is provided based on the example of a single- ¨ ˙ input single-output (SISO) system. The active distur- ⎪ θ = lu /J + d − (lκ θ/J ) θ θ θ θ θ bance rejection switching control (ADRSC) algorithm is ¨ ˙ ψ = u /J + d − (κ ψ/J ) ψ ψ ψ ψ ψ an enrichment of the ADRC algorithm that is more stable and robust. where m is the quality of the quadrotor UAV; (x, y, z) is Therefore, to improve the anti-disturbance ability of the current position; (ϕ, θ, ψ) corresponds to the roll, the whole control process of the quadrotor, an ADRSC pitch, and yaw (RPY) attitude angles; the three equa- algorithm based on a robust differentiator is proposed. tions of (x, y, z) constitute the position subsystem; the The robust differentiator is used as the scheduling tran- three equations of (ϕ, θ, ψ) constitute the attitude sub- sition process in ADRC. Based on the traditional ADRC, system; (κ , κ , κ , κ , κ , κ ) is the damping coefficient x y z ϕ θ ψ LADRC/NLADRC switching is adopted, and the stability group of the system; l is the length of the quadrotor analysis of the switching control is undertaken. The tra- wing; (d , d , d , d , d , d ) consists of the perturbation x y z ϕ θ ψ jectory tracking control performance of the quadrotor amounts of each channel; and g is the acceleration of based on the traditional differentiator and the robust dif- gravity. The flight principle of the quadrotor is shown in ferentiator in a white noise disturbance environment is Figure 1. Figure 1. Flight principle of quadrotor unmanned aerial vehicle (UAV). SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 607 Four motor inputs are The input matrix is B = diag{b , b } i = 1, 2, 3. where i1 i1 i2 b = 1/J , b = cos ϕ cos θ/m, b = (cos ϕ sin θ cos ψ 32 ψ 31 11 ⎪ u = f + f + f + f + sin ϕ sin ψ)/m, b = l/J , b = (cos ϕ sin θ sin ψ − τ 1 2 3 4 ⎪ 21 ϕ 12 sin ϕ cos ψ)/m, b = l/J . u =−f − f + f + f 22 θ ϕ 1 2 3 4 (2) The external disturbance vectors are d = (d , d ) , 1 x y u =−f + f + f − f ⎪ θ 1 2 3 4 T T T T d = (d , d ) , d = (g , d ) , g = (0, g) , d = (d , d ) . ⎩ 2 ϕ θ 3 3 3 z ψ u = c(f − f + f − f ) ψ 1 2 3 4 O is a second-order zero matrix, and I is a second- 2×2 2×2 order identity matrix. The control vectors are u = where (u , u , u , u ) is the output of the controller, τ ϕ θ ψ T T T (u , u ) , u = (u , u ) , u = (u , u ) . τ τ 2 ϕ θ 3 τ ψ (f , f , f , f ) consists of the lift generated by the rotation 1 2 3 4 of the four motors, and c is the proportional coefficient Remark 2.1: Three decomposed sub-equations facilitate between yaw dynamic torque and lift. When keeping the implementation of the control algorithm when lin- the speeds of the two motors unchanged, changing the ear control law is employed. However, the control is still speed of the other two motors can change the attitude of implemented in the form of a general nonlinear system. the quadrotor UAV. For example, if we keep the speeds of The form of the quadrotor dynamic equation can be motors 1 and 3 unchanged, changing the speed of motors controlled according to the above expression when the 2and4causes f and f to change, so as to realize the pitch 2 4 nonlinear control law is adopted. motion of the quadrotor UAV. Since the quadrotor dynamic model contains two non- Active disturbance rejection switching control integrity constraints, the full six degrees of freedom of based on robust differentiator motion cannot be achieved. By reversing the position information of the expected trajectory of the attitude In this section, an active disturbance rejection switch- (Wang et al., 2018), the attitude angle is tracked to the ing control based on a robust differentiator is proposed. expected (ϕ , θ ) : d d It is used for accurate position trajectory tracking con- trol of the quadrotor. The robust differentiator solves (v − d )(e cos ψ + e sin ψ) ⎪ p p 1 2 = arctan ⎪ θ d the noise interference of the quadrotor, which has a e v − e d 3 p 3 p ⎛ ⎞ strong ability to accurately obtain differential signals and ⎪ (v − d )(−e sin ψ + e cos ψ) filter out external white noise disturbance. The ADRSC p p 1 2 ⎝ ⎠ ϕ = arcsin ⎪ d algorithm enables the quadrotor to track the reference ⎩ T (v − d ) (v − d ) p p p p input {x , y , z , ψ } for trajectory tracking control. d d d d (3) T T where v = (x ¨, y ¨, z ¨ + g) , d = (d , d , d ) .Theunit vec- p p x y z tors {e , e , e } correspond to {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. Robust differentiator 1 2 3 To simplify and improve the efficiency of the control, The traditional higher-order differentiator (Levant, 2005) Equation (1) is decomposed into the following form with- is not generally strong to signal immunity and robust- out loss of generality in the form of the state space. Define ness, so it is necessary to find a higher-order robustness T T T T x = (x, y) , x = (x ˙, y ˙) , x = (ϕ, θ) , x = (ϕ ˙, θ) , x = 1 2 3 4 5 differentiator to replace the traditional TD function. T T (z, ψ) , x = (z ˙, ψ) . The sub-equations of the position, An arbitrary-order exact robust differentiator (ERD) attitude, and altitude, respectively, of the quadrotor are (Levant & Livne, 2012) was proposed. The ERD features optimal asymptotics with respect to input noises and can x ˙ O I x O O 1 2×2 2×2 1 2×2 2×2 = u + d 1 1 be used for numerical differentiation as well. Compared x ˙ A A x B I 2 11 12 2 11 2×2 to the original higher-order differentiator, ERD draws on (4) the design of the higher-order sliding mode. The design x ˙ O I x O O 3 2×2 2×2 3 2×2 2×2 idea of a second-order sliding mode is shown in Figure 2. = + u + d 2 2 x ˙ A A x B I 4 21 22 4 21 2×2 The second-order differentiator (Bartolini et al., 1999)is (5) ⎪ r˙ = v , v =−λ |r − f (t)| sign (r − f (t)) + r 0 0 0 0 0 0 1 x ˙ O I x O 1 5 2×2 2×2 5 2×2 = + u + d (6) 2 r˙ = v , v =−λ |r − v | sign (r − v ) + r 3 3 1 1 1 1 1 0 1 0 2 x ˙ A A x B ⎪ 6 31 32 6 31 r˙ =−λ sign (r − v ) 2 2 2 1 The system matrices are (7) A = diag{a , a }, A = diag{a , a }. where a = where r , r , r is the observation of the second-order i1 i1 i2 i2 i3 i4 i1 0 1 2 a = 0, a =−κ /m, a =−κ /m, a =−κ /m, a = differentiator and f (t), v , v is a noise-disturbing input i2 13 x 14 y 33 z 23 0 1 −lκ /J , a =−lκ /J , a =−κ /J , i = 1, 2, 3. signal. ϕ ϕ 24 θ θ 34 ψ ψ 608 J. ZHAO ET AL. where L is the Lipschitz constant, L > 0, and the value of L is proportional to the change of the disturbance. The selection of the parameter λ is λ = 1, 1, 2, 3, 5, 8, ··· or λ = 1.1, 1.5, 3, 5, 8, 12, ··· . Lemma 3.1: (Filippov, 2013): The parameters being prop- erly chosen, the following equalities are true in the absence of input noises after a finite time of a transient process: (i) r = f (t); r = v = f (t), i = 1, ··· , n (10) 0 0 i i−1 Lemma 3.2: (Levant, 2003): Let the input noise satisfy the inequality |f (t) − f (t)|≤ ε. Then the following inequalities are established in finite time for some positive constants μ , Figure 2. Second-order sliding mode. v , depending exclusively on the parameters of the differen- tiator: Remark 3.2: The second-order differentiator is widely (i) (n−i+1)/(n+1) |r − f (t)|≤ μ ε , i = 0, ··· , n used in engineering applications. For the position infor- i i (i+1) (n−i)/(n+1) mation of a quadrotor, the first-order signal is its speed, |v − f (t)|≤ v ε , i = 0, ··· , n − 1 i i and the second-order signal is its acceleration. How to (11) accurately extract the differential signal is to ensure that If the solution of the dynamic system (10) is stable in the the extraction process of the differential signal is not or sense of Lyapunov, and then selects a reasonable parameter less affected by external disturbances. in (11) guarantees the homogeneity of the robust differen- tiator and convergence. Similarly, a second-order differentiator can be general- ized to an nth-order differentiator. The only requirement Active disturbance rejection switching control is that the resulting systems be homogeneous in the The ADRC algorithm is mainly composed of TD, ESO, and sense described below. The nth-order differentiator is ⎧ a feedback control law (Chen et al., 2014). The traditional r˙ = v ⎪ 0 0 TD is replaced by ERD in this paper’s ADRC algorithm. ⎪ n ⎪ n+1 v =−λ |r − f (t)| sign (r − f (t)) + r The ADRSC algorithm selects one of the two control ⎪ 0 0 0 0 1 states, LESO and NLESO, in different control processes r˙ = v ⎪ 1 1 n−1 ⎪ according to the actual operation of the system. If the v =−λ |r − v | sign (r − v ) + r 1 1 1 0 1 0 2 tracking performance of the system hardly changes with ⎪ . the disturbance amplitude, then the LESO method will improve the control efficiency. When the parameters of r˙ = v ⎪ n−1 n−1 ⎪ the system are disturbed, NLESO can suppress the neg- ⎪ 2 v =−λ |r − v | sign (r − v ) + r ⎪ n−1 n−1 n−1 n−2 n−1 n−2 n ative impact of complex disturbances on the control r˙ =−λ sign (r − v ) n n n n−1 process. (8) The error convergence rate of LESO is faster than that where {r , ··· , r } is the observation of the differentia- 0 k of NLESO when the total operating time t < T (transition tor and {v , ··· , v , f (t)} is the noise-disturbing input 0 n process time) or total disturbance |z (t)| > M (M is the n+1 signal of Equation (8). For any ξ, γ ∈ R, define ω = upper disturbance value) or the ESO error |e| > 1. |ω| sign(ω) if γ> 0or ξ = 0. Define the switching condition as Let ξ = sign(ξ ).For f (t), f : R → R is n times con- (n) tinuously differentiable, n ≥ 0, f being the Lipschitz 1, if t > T ∧|e| > 1 ∧|z (t)| > M n+1 q = (12) (n+1) constant with |f |≤ L.The nth-order ERD can be 0, else described by the Filippov differential theory (Filippov, When q = 1, LESO is 2013), 1 n ⎪ e = z − y ⎪ n+1 n+1 r˙ =−λ L r − f (t) + r ⎪ ⎪ 0 n 0 1 z ˙ = z − β e ⎪ 1 2 01 2 n−1 n+1 n+1 ⎪ r˙ =−λ L r − f (t) + r 1 n−1 0 2 ⎨ ⎨ z ˙ = z − β e 2 3 02 (13) . (9) . . ⎪ . n 1 ⎪ n+1 n+1 r˙ =−λ L r − f (t) + r z ˙ = z − β e + bu n−1 1 0 n ⎪ n n+1 0n r˙ =−λ L sign (r − f (t)) z ˙ =−β e n 0 0 n+1 0(n+1) SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 609 The control law under LESO is Designing ESO for sub-equations of the quadrotor: e = z − y u = k (v − z ) (14) 0 i i i z = z − β · ϕ (e) 1 2 1 1 (18) i=1 z = z − β · ϕ (e) + bu 2 3 2 2 z =−β · ϕ (e) 3 3 3 When the switching condition q = 0, NLESO is else if q = 1 e = z − y ⎪ 1 e = z − y ⎪ 1 z ˙ = z − β · ϕ (e) ⎪ 1 2 1 1 z ˙ = z − β e 1 2 01 ⎪ (19) z ˙ = z − β · ϕ (e) z ˙ = z − β e + bu 2 3 2 2 2 3 02 (15) z ˙ =−β e ⎪ . 3 03 q is the switching condition of the i-th sub-equation, z ˙ = z − β · ϕ (e) + bu ⎪ n n+1 n n i for i(i = 1, 2, 3) corresponding to the sub-equations of z ˙ =−β · ϕ (e) n+1 n+1 n+1 the position, attitude, and altitude, respectively, of the quadrotor. Then the virtual control law of sub-Equations where (4), (5), (6) is ⎛ ⎞ ϕ (e) = fal(e, α , δ ) i i i i k fal(v − z , α , δ ) 1 1 1 1 1(x,ϕ,z) ⎪ ⎜ ⎟ α −3 α −2 i i 3 i 2 ⎪ ⎜ ⎟ ⎪ +k fal(v − z , α , δ ) (α − 1)δ e − (α − 1)δ e sign (e) ⎪ 2 2 2 2 i i 2(x,ϕ,z) i i ⎪ ⎜ ⎟ ⎪ , q = 1 ⎪ ⎜ ⎟ α −1 i ⎨ k fal(v − z , α , δ ) = 1 1 1 1 +δ e , |e|≤ δ ⎝ ⎠ i 1(y,θ,ψ) u = ⎪ oi ⎩ α ⎪ +k fal(v − z , α , δ ) |e| sign (e) ,|e| >δ 2 2 2 2 i ⎪ 2(y,θ,ψ) i i (16) k (v − z ) + k (v − z ) ⎪ 1 1 2 2 1(x,ϕ,z) 2(x,ϕ,z) ⎪ , q = 0 i i k (v − z ) + k (v − z ) 1 1 2 2 1(y,θ,ψ) 2(y,θ,ψ) The control law under NLESO is (20) where the internal gain of the control law is k , 1(x,ϕ,z) 1 i i k , k , k (i = 1, 2, 3). u = k fal(v − z , α , δ ) (17) 0 i i i i i 2(x,ϕ,z) 1(y,θ,ψ) 2(y,θ,ψ) i=1 Then the ADRSC law of the sub-equation is u − z 0i 3 where the output of LESO and NLESO is z (i = 1, ··· , n + i u = (i = 1, 2, 3) (21) 1); β (i = 1, ··· , n + 1) is the observation gain of LESO; oi β (i = 1, ··· , n + 1) is the observation gain of NLESO; e is The block diagram of the ADRSC structure is shown in the observation error; ϕ (e)(i = 1, ··· , n + 1) is the non- Figure 3. The robust differentiator is located before the linearerrorfunction; α , δ , α , δ (i = 1, ··· , n + 1) is a tun- attitude control law and is used to obtain the estimated i i i i ˆ ˆ ˆ ˆ ˆ ˙ ˙ ˆ ¨ ¨ able parameter in the fal function in Equation (17); b is the values (ϕ ˙, θ, ψ) and (ϕ ¨, θ, ψ) of the differential signals of ˆ ˆ ˆ ˙ ˙ ˙ external gain of the control law; and k is the internal gain the attitude angle (ϕ, θ, ψ). The estimated values (x, y, z) ˆ ˆ ˆ ¨ ¨ ¨ of the control law. and (x, y, z) of the differential signal (x, y, z) are solved Figure 3. Block diagram of control structure. 610 J. ZHAO ET AL. when the robust differentiator is located before the posi- When i = 1, Equation (19) is rewritten as tion control law. ESO can improve the anti-disturbance ⎧ e = z − x ⎪ 1 1 ability of system state variables in the control process. Whether to use LESO or NLESO is determined according z ˙ = z − β · γ (e) · ϕ (e) 1 2 1 1 1 (26) to the switching condition q . For position control, when ⎪ i z ˙ = z − β · γ (e) · ϕ (e) + B u 2 3 2 2 1 11 1 the switching conditions q and q satisfy q = q = 1, 1 3 1 3 z ˙ =−β · γ (e) · ϕ (e) 3 3 3 1 then switching to LESO constitutes LADRC. When the switching conditions are q = 0or q = 0, then switching 1 3 where β = (β , β ) , i = 1, 2, 3. i ix iy to NLESO constitutes NLADRC. For attitude control, q = 2 Substituting Equations (22) and (23) in Equation (26), q = 1 corresponds to LESO, and q = 0or q = 0 cor- 3 2 3 Z = A Z + B u ˜ respond to NLESO. The above ERD, ESO, and control law 1 3 1 2 (27) together constitute an ADRSC algorithm to implement ˙ ˜ z =−β · γ (e)u 3 3 3 quadrotor UAV control. where B = (β , β ) , and the virtual input is u ˜ =−ϕ (e). 2 1 2 1 Combining Equations (25) and (27), we obtain Stability analysis X = A X + A Z − B z 1 1 1 2 1 1 3 Taking the quadrotor sub-equation of position as an ⎪ Z = A Z + B u ˜ 1 3 1 2 example for stability analysis, define X = (x , x ) , Z = 1 1 2 1 (28) T T T (z , z ) , z = (z , z ) , z = (z , z ) , z z ˙ =−β · γ (e)u ˜ 1 2 1 1(x) 1(y) 2 2(x) 2(y) 3 ⎪ 3 3 3 = (z , z ) 3(x) 3(y) e = C X + C Z 1 1 2 2 where z , z , z , z , z , z is the ESO out- 1(x) 1(y) 2(x) 2(y) 3(x) 3(y) put of the quadrotor sub-equation of position. −100 0 1000 where C = , C = . 1 2 According to Equation (21), the control law of the posi- 0 −100 0100 Let Y = A X − B z . By linear transformation of tion sub-equation is 1 1 1 1 3 Equation (28), we obtain u − z 01 3 ⎧ u = (22) B ⎪ Y = A Y + A A Z + B β γ (e)u 1 1 1 1 2 1 1 3 3 (29) Z = A Z + B u ˜ 1 3 1 2 −1 −1 Lemma 3.3: (Li et al., 2016a):Iftheamountofchangeinthe ⎩ e = C A Y + C Z − C A B z 1 1 2 1 1 1 3 1 1 output v of the ESO is small within a small time interval t, then u in the control law of the position sub-equation can Equation (29) is further simplified to be expressed as ˙ ˜ x ˜ = A x ˜ + b u ˜ 1 1 (30) α −1 α −1 α −1 1 1 1 1 2 1 ˜ e = C ˜ u = (k δ + k δ , k δ x + ρω 1(x) 2(x) 1(y) 1(x) 2(x) 1(y) α −1 1 2 T + k δ ) (23) A A A 2(y) 2(y) 1 1 2 where x = (Y , Z ) , A = , b 1 1 1 1 O A 4×4 3 B β γ (e) According to Lemma 3, substitute Equations (22) and 1 3 3 −1 = , ω satisfies ω ˙ = u ˜, ρ = C A B β γ (e), 1 1 3 3 (23) into Equation (4) to obtain 2 −1 C A ˜ 1 and C = . X = A X + B (u − z ) (24) 1 1 1 1 01 3 Deriving the error equation in Equation (30), we get O I O 2×2 2×2 2×2 where A = , B = . ˜ ˜ ˜ ˜ 1 1 e ˙ = C A x ˜ + (C b + ρ)u ˜ (31) 1 1 1 1 A A I 11 12 2×2 Substituting Equation (23) in Equation (24), By linearly transforming Equations (30) and (31), we obtain X = A X + A Z − B z (25) 1 1 1 2 1 1 3 ˙ ˜ ˜ ˜ x = A x + b ω 1 1 (32) α −1 α −1 ˜ ˜ ˜ ˜ O O 2×2 2×2 1 1 1 2 e ˙ = C A x ˜ + (C b + ρ)ω k δ k δ 1 1 1 1 where A = , k = 1(x) 1(x) 2(x) 2(x) , 2 ˜ ˜ (x) k k (x) (y) Remark 3.3: When rank(A ) is full rank, equivalence of ˜ 1 α −1 α −1 k = . 1 1 1 2 (y) k δ k δ 1(y) 1(y) 2(y) 2(y) Equations (30) and (32) is obtained by linear transforma- ϕ (e) α −1 i i Let γ (e) = = δ (i = 1, 2, 3). i tion of Equation (29). That is, when Equation (30) has ϕ (e) i 1 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 611 asymptotic stability, then Equation (32) is also asymptot- ⎛ 00 1 0 −η −η 1x 2x ically stable. 00 0 1 0 0 β β Equation (32) is written as 3x 3x a + μ μ a 0 −a η −a η 21 3 3 11 11 1x 11 2x ⎜ β β 1x 1y ⎜ β β 3y 3y ˙ ˜ ⎜ μ a + μ 0 a 00 x ˜ = Ax ˜ + bω 3 21 3 12 β β 1x 1y (33) β β ˜ 1x 1x ˙ ˜ ˜ ˜ ⎜ e = Cx + re +˜ ρu μ μ 00 0 0 1 1 β β ⎜ 1x 1y β β 1y 1y μ μ 00 0 0 T T ˜ ˜ ˜ ⎜ 1 1 ˜ ˜ ˜ ˜ ˜ ˜ ˜ β β where A = A − b ρ˜ C A , b = b ρ˜ , C = C A 1x 1y 1 1 1 1 1 1 1 1 1 β β T T −1 2x 2x ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ⎜ μ μ 00 −η 0 (A − b ρ˜ C A ), r˜ = C A b ρ˜ ,ρ˜ = (C b + ρ) , ρ˜ = 1 1 1 1 1 1 1 1 1 1 2 2 1x 1 1 β β ⎜ 1x 1y −1 ρ˜ . β β 2y 2y μ μ 00 0 −η 2 2 1y ⎜ β β 1x 1y β β T 2x 2x ˜ μ μ −10 0 η Let p = (x, e) , and simplify Equation (33) to ⎜ 2 2 2x β β 1x 1y β β 2y 2y μ μ 0 −10 −η 2 2 1y p ˙ = A p + b u ˜ (34) β β 1x 1y Q Q 00 0 0 A b 8×1 −η −η 00 ⎟ 1y 1y where A = , b = ,and A is a 10 × 10 Q Q Q ρ˜ C r˜ ⎟ β β β β 3x 3x 3x 3x − μ − μ μ μ ⎟ 3 3 3 3 β β β β 1x 1x 1x 1x square matrix. β β β β 3y 3y 3y 3y −a η − μ −a η − μ μ μ 12 1y 3 12 2y 3 3 3 Define P as the discriminant matrix of the quadrotor β β β β ⎟ 1x 1x 1x 1x β β β β 1x 1x 1x 1x ⎟ sub-equation. 1 − μ − μ μ μ 1 1 1 1⎟ β β β β 1x 1x 1x 1x ∂Q ⎟ Let Q(p) = A p + b u ˜, P = (i = 1, 2, 3). β β β β ⎟ Q Q 1y 1y 1y 1y ∂p − μ 1 − μ μ μ ⎟ p=0 1 1 1 1 β β β β 1x 1x 1x 1x β β β β 2x 2x 2x 2x −η − − μ μ 2x 2 2 β β β β Theorem 3.1: When the matrix P satisfies the Hurwitz 1x 1x 1x 1y ⎟ β β β β 2y 2y 2y 2y stability criterion, i.e. the eigenvalues of the matrix P all − −η − μ μ 2y 2 2⎟ β β β β 1x 1x 1x 1y have negative real parts, then Equation (34) is asymptoti- β β β β 2x 2x 2x 2x −η − − μ μ ⎟ 2x 2 2 β β β β 1x 1x 1x 1y cally stable. β β β β 2y 2y 2y 2y η − − μ μ 1y 2 2 β β β β 1x 1x 1x 1y 10×10 Proof: When p = 0and p ˙ = 0, the equilibrium point of (37) Equation (34) can be obtained. Linearize Equation (34) to α −1 α −1 α −1 α −1 1 2 1 2 obtain an equation containing the matrix P , where (η , η , η , η ) = (δ , δ , δ , δ ), 1x 2x 1y 2y 1(x) 2(x) 1(y) 1(y) γ (e) and μ = , (i = 1, 2, 3). p ˙ = P p (35) γ (e) which is Simulation results ⎛ ⎞ ∂Q ∂Q ∂Q 1 1 1 ··· ∂p ∂p ∂p 1 2 n Take the parameters of a set of quadrotor UAVs and use ⎜ ⎟ ∂Q ∂Q ∂Q ⎜ 2 2 2⎟ ··· the position sub-equation as an example to verify the ⎜ ∂p ∂p ∂p ⎟ ∂Q 1 1 n ⎜ ⎟ P = = ⎜ ⎟ stability analysis method. The quadrotor parameter selec- i . . . ∂p . . . p=0 ⎜ ⎟ . . . ⎝ ⎠ tion is as follows in Table 1, where the parameters of ESO ∂Q ∂Q ∂Q n n n ··· are: ∂p ∂p ∂p 1 1 n n=10,p=0 Substituting these parameters in Equation (35), the (36) characteristic polynomial of the matrix P is obtained as When the matrix P satisfies the Hurwitz stability crite- rion, Equation (35) is exponentially convergent, i.e. it is P P 4 P 2 f (λ ) = (λ + 0.5 ± 0.866j) (λ + 0.5) (38) asymptotically stable. 1 1 1 Remark 3.4: In actual control applications, rank(A ) is Table 1. Quadrotor parameters. not a necessary and sufficient condition for full rank. Parameter Unit Value When the value of an element in A is small, replace it with m kg 2 a smaller number ξ and find the pseudo-inverse A of 2 −1 J Ns rad 1.25 ˜ 2 −1 A to avoid the occurrence of a singular matrix that will J Ns rad 1.25 2 −1 J Ns rad 2.5 affect the calculation and to achieve stable control under l m0.2 practical application. −2 g ms 9.8 2 −2 κ , κ , κ Ns rad 0.01 For the quadrotor subroutine equation, the discrimi- x y z 2 −2 κ , κ , κ Ns rad 0.012 ϕ θ ψ nant matrix P is i 612 J. ZHAO ET AL. The control parameters are selected as Obviously, the eigenvalues of Re(λ ) all have negative real parts, which satisfies the Hurwitz stability criterion, i.e. i i i i k = k = 5, k = k = 15, i = 1, 2, 3 1(xyz) 2(xyz) 1(ϕθ ψ ) 2(ϕθ ψ ) the quadrotor position sub-equation selected according White noise perturbations of 0.001, 0.01, and 0.1 dB to the above control parameters is asymptotically stable. were applied to the three sub-equations of the quadrotor. The attitude, altitude sub-equations have stability Compared to the traditional higher-order differentiator, analyses similar to that of the position sub-equation. By i.e. the linear differentiator (LD) (Ibrir, 2004), nonlinear substituting the parameters of ESO into Equation (35), a differentiator (NLD) (Wang et al., 2003b), and hybrid non- characteristic polynomial similar to Equation (38) can be linear differentiator (HND) (Wang et al., 2003a), we verify obtained, which is also the Hurwitz stability criterion. The the performance of the traditional high-order differen- proof is omitted. Reasonable parameter selection causes tial tracker and ERD when facing the white noise flight the attitude, altitude sub-equations to satisfy the Hurwitz of the quadrotor. The following simulation results were stability criterion. obtained. The starting position of the quadrotor is the origin of a Figure 4 shows the ERD-based attitude angle output Cartesian coordinate system, and the reference trajectory and the traditional differentiator-based attitude angle is selected as output. It can be seen from the figure that the robust πt πt t π differentiator filters out more noise, especially attitude x = cos , y = sin , z = 5 + , ψ = d d d d signal noise, than conventional differentiators. It has 4 4 10 3 Figure 4. Attitude tracking with white noise 0.001 dB intensity. SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 613 Figure 5. Trajectory tracking with white noise of 0.001 dB intensity. a significant effect, and the new differential signal is quadrotor can accurately track the desired trajectory in a smoother, which can provide accurate differential sig- short time. nals to the ADRSC controller. Figure 5 shows the tra- It can be seen from Figure 6 that, taking the position jectory tracking of the white noise 0.001 dB intensity sub-equation as an example, when the error in the ini- based on the ERD method. It can be seen that the error tial stage of control is relatively large, LADRC can well between the desired and actual trajectory is small, and the make the control law quickly approach the vicinity of the 614 J. ZHAO ET AL. desired control law interval. The control is gradually sta- bilized during the middle and rear stages of the control process. To make the control more precise, the NLADRC is used to adjust the error to achieve more accurate and stable control. Figure 7 is an attitude tracking diagram after the white noise intensity is increased to 0.01 dB. It can be seen that the attitude and position output curves of the control algorithm remain relatively stable at this time, and there is a phenomenon of chattering at some nodes. Compared to the conventional differentiator, the ERD acts on the pose output of the quadrotor, and it has better perfor- mance. Figure 8 is a trajectory tracking diagram after the white noise intensity is increased to 0.01 dB. It can be seen that the smoothness of the flight path of the quadrotor at 0.1 dB is also lower than the lower strength of 0.001 dB. Figure 6. White noise 0.001 dB position sub-equipment con- troller output. Figure 7. Attitude tracking and position tracking under white noise of 0.01 dB. SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 615 Figure 8. Trajectory tracking with white noise of 0.01 dB intensity. It can be seen from Figure 9 that the flight path of Conclusion the quadrotor obtained at this time is not smooth, and In this paper, for the anti-disturbance problem of the tra- the smooth performance is not as good as that under jectory tracking control of a quadrotor UAV under white low-intensity white noise. noise disturbance, the quadrotor UAV is controlled by the 616 J. ZHAO ET AL. estimation for a quadrotor UAV. IEEE Transactions on Industrial Electronics, 63(8), 5044–5056. https://doi.org/10.1109/TIE. 2016.2552151 Chen, Z., Sun, M., & Yang, R. (2014). On the stability of linear active disturbance rejection control. Acta Automatica Sinica, 39(5), 574–580. https://doi.org/10.3724/SP.J.1004.2013. Chen, Z., Wang, C., Li, Y., Zhang, Q., & Sun, M. (2015). Con- trol system design based on integral sliding mode of quadrotor. Journal of System Simulation, 27(9), 2181–2186. doi:10.16182/j.cnki.joss.2015.09.034 Dolatabadi, S. H., & Yazdanpanah, M. J. (2015). 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Systems Science & Control Engineering – Taylor & Francis
Published: Jan 1, 2020
Keywords: Quadrotor unmanned aerial vehicle (UAV); robust differentiator; active disturbance rejection control (ADRC); trajectory tracking control
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