Novel integral sliding mode control for small-scale unmanned helicopters

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Journal of the Franklin Institute 356 (2019) 2668–2689 www.elsevier.com/locate/jfranklin

Novel integral sliding mode control for small-scale unmanned helicopters Tao Jiang∗, Defu Lin, Tao Song Beijing Key Laboratory of UAV Autonomous Control, Beijing Institute of Technology, Haidian District, Beijing 100081, People’s Republic of China Received 22 July 2017; received in revised form 14 December 2018; accepted 24 January 2019 Available online 1 February 2019

Abstract Integral sliding mode (ISM) control which consists of a nominal control and a sliding-mode motion control, provides a nice framework for high tracking performance and good disturbance reduction. Our work develops ISM to attenuate the adverse effect of mismatched perturbations. By properly choosing sliding-manifold surface, the elimination of disturbances on control outputs enables to be achieved. Additionally, the chattering of sliding-mode control part is attenuated based on second-order sliding mode idea. Then, the proposed novel ISM control scheme is applied to address trajectory tracking problem for helicopters under perturbations. Approximated input-output linearization is implemented, such that the obtained linearized model is suitable for applying the proposed ism control. The stability of the closed-loop system for helicopter and its convergence to zeros of tracking errors are demonstrated by Lyapunov theory analysis. Several comparison simulations illustrate the effectiveness and superiority of the proposed methods. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Unmanned helicopters have received wide interest over the past decades due to their capabilities, such as vertical taking off and landing from unprepared sites, broad envelope of ∗

Corresponding author. E-mail addresses: [email protected] (T. Jiang), [email protected] (D. Lin), [email protected] (T. Song). https://doi.org/10.1016/j.jfranklin.2019.01.035 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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flight that ranges from hovering to cruising, potential to fly at low altitudes, and highly agile maneuvering in tightly constrained environments [1,2]. Unmanned helicopters have been widely applied in defense, rescue, surveillance, supervision, and other fields. However, controlling unmanned helicopters, especially small-scale ones, which are more sensitive to manipulation and external disturbances, is difficult. Such difficulty arises from their complex aerodynamics, strong dynamic couplings, and significant parameter and model uncertainties [1–4]. In the field of unmanned aircraft vehicle, designing high-performance controller for helicopters bears importance. Traditional linear control method is based on the model linearized around the trim points, including PID [1], linear quadratic regulation [5], H∞ control theory [4–6], and gain scheduling controllers [7], that may utilize synthesis techniques. These techniques are advantageous to project realization, whereas they suffer from performance degradation when helicopters leave away from their designed trim points or execute aggressive maneuvers. In the last few decades, various nonlinear control approaches were developed to tackle these limitations, which are divided into full nonlinear and linearized methods [3]. Full nonlinear methods mainly refer to backstepping (BS) based control [8] due to the hierarchical framework of helicopters’ dynamics. For guaranteeing the robustness performance, BS controllers are often combined with other disturbance compensation method, such as integral control [9], adaptive control [10,11], and disturbance observer (DO) [12,13], etc. Compared with full nonlinear methods, linearized methods provide a framework to connect linear control methods with nonlinear models. When applying linearized methods to a helicopter model, approximate input-output feedback linearization is initially achieved through choosing position and yaw angle as outputs [14]. However, unmatched disturbances occur in the linearized dynamics, causing some difficulties in implementation of exact trajectory tracking. In [15], model predictive control (MPC) combined with DO guarantees the global asymptotic stability of tracking errors. Fang et al. [16] developed a composite controller based on H∞ control and feedforward technique, which can attenuate the mismatched disturbances from the state variables. A novel sliding mode control (SMC) based on the estimation of disturbances was proposed in [17], which provides excellent tracking performance and robustness. As a variable structure control technique, SMC can eliminate the effect of the parameter uncertainties and perturbations by applying a switching control law to alter the plant dynamics [18]. In the sliding phase, the system response remains invariant for disturbances and the nominal control performance is recovered. However, during the reaching phase, the invariance of SMC is not guaranteed and the system response is sensitive to perturbations. Additionally, the imperfect implementation of high-frequency switching in SMC results in chattering at control responses, which may cause the tracking performance degradation and the loss of actuators. Integral sliding mode (ISM) enables to eliminate the reaching phase by enforcing the sliding mode in an entire system response so that the invariance of SMC is ensured from the initial time instant [19–21]. Due to these advantages, ISM has been widely used in practical applications, such as helicopters [22–26], mobile robot [27], hypersonic vehicles [28], etc. Conventional ISM [19–21] methods select sliding manifolds based on projection matrix approach such that no amplification of unmatched disturbances occurs. Then other robust techniques, such as H∞ control [20], or MPC [29,30], can be combined with ISM method to robustify against perturbations. Unfortunately, given the existence of unmatched perturbations, tracking performance degradations still occur. For helicopter control, the unmatched disturbances (the external force disturbances generated by wind gusts) have much more effect on

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the position tracking of helicopters. To tackle these limitations, the previous works [23,25] usually combine it with the recursive design to eliminate the adverse effect of unmatched disturbances. Our work develops another way to solve these problem through eliminating mismatched perturbations from the output channels. By properly choosing a sliding-manifold surface based on the relative degree idea, mismatched disturbance attenuation is achieved, such that high tracking performance and robustness are guaranteed. A systematic method is developed for the sliding-manifold gain design. Moreover, to attenuate the chattering of the SMC signals, a second-order sliding mode controller [31,32] is introduced to achieve sliding-mode motion. Additionally, to apply ISM for flight control, the linearized model around the trim points is considered [23,26]. Our work applies approximately feedback-linearized for a small-scale helicopter mathematical model subjected to external perturbations and uncertainties. The tracking error dynamics with lumped matched and unmatched disturbances are obtained, which is suitable to larger flight envelop. Then, the proposed ISM method enables to be applied to design the helicopter controller, where the sliding-manifold gain is properly chosen to eliminate the influence of disturbances on output channels. A second-order SMC is introduced to attenuate the chattering of control inputs. The main contributions are listed as follows: (1) A novel form of ISM, by designing, is developed to eliminate the influence of the unmatched perturbations. Its sliding-manifold is properly designed through relative degree idea, such that the effect of disturbances is attenuated on the output channels. Moreover, chattering-free SMC is implemented based on the second-order SMC to drive the state trajectory of the closed-loop system onto the sliding surface. (2) The trajectory tracking problem for small-scale unmanned helicopters subjected to external disturbances and uncertainties are tackled by applying the proposed ISM control. Approximate feedback-linearization technique is conducted to obtain the linearized model which is suitable to applying ISM control. This composite control system builds a feasible control framework to guarantee high tracking performance and robustness. (3) A rigorous proof of stability of the closed-loop system is derived by stability analysis. Some comparative simulation results of helicopter trajectory tracking have illustrated the effectiveness and superiority of our proposed control system. This paper is arranged as follows. Section 2 presents the dynamic model of a smallscale helicopter, and the linearized model is obtained based on the approximated feedback linearization. Section 3 proposes the control method and provides performance analysis. Then, the helicopter control problem is addressed. Section 4 discusses the performed simulations. Finally, Section 5 draws the conclusions. 2. Helicopter model This section presents the nonlinear dynamic model of a small-scale unmanned helicopter. The helicopter is considered a six-degree-of-freedom rigid body model with simplified force and moment generation process. Model uncertainties and external perturbations are considered in the modeling phase. First, two reference frames are defined as follows: earth reference frame (ERF) = {Oxyz}, which is fixed to the earth; body reference frame (BRF) = {Ob xb yb zb }, whose origin is located at the helicopter center of gravity [3,4]. Fig. 1 shows the direction of the two reference frames.

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Tm Tt

b

a O

y (east)

z (down)

zb x (north)

Ob

yb

xb

Fig. 1. Simple illustration of a small-scale helicopter model.

The dynamic model of the small-scale unmanned helicopter can be described as follows [3,4]: P˙ = V

(1)

V˙ = ge3 +

1 R ()F m

(2)

R˙ () = R ()S (ω )

(3)

J ω˙ = −ω × J ω + M

(4)





where P = [x y z]T and V = [u v w]T refer to the helicopter’s position and velocity vector in the ERF, respectively; m is helicopter mass, and g is gravitational acceleration; e3 = [0 0 1]T is a unitary vector; S(·) is a skew-symmetric matrix that corresponds to the vector (·) [8,9]; J represents the approximate inertia matrix written in the following form: ⎡ ⎤ Jxx 0 −Jxz Jyy 0 ⎦ J=⎣ 0 (5) −Jxz 0 Jzz The rotation matrix from BRF to ERF is given as follows: ⎡ ⎤ Cθ Cψ Sφ Sθ Cψ − Cφ Sψ Cφ Sθ Cψ + Sφ Sψ R () = ⎣Cθ Sψ Sφ Sθ Sψ + Cφ Cψ Cφ Sθ Sψ − Sφ Cψ ⎦, −Sθ Sφ Cθ Cφ Cθ

(6) 

where C(·) and S(·) are shorts for cos(·) and sin (·), respectively.  = [φ

θ

ψ ]T represents 

the Euler angles, which include roll, pitch, and yaw angles, respectively. ω = [ p denotes the angular rates in the BRF. Attitude kinematics is expressed as follows: ˙ = ()ω 

q

r ]T

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⎡

1 () = ⎣0 0

Sφ Cθ Cφ Sφ /Cθ

⎤ Cφ Sθ /Cθ −Sφ ⎦ Cφ /Cθ

(7)

In Eqs. (2) and (4), F and M are the external forces and torque exerted on fuselage in BRF, respectively. F = Fb + F

(8)

M = Mb + M

(9)

⎡

⎤ 0 ⎦, 0 Fb = m⎣ −g + Zw w + Zcol δcol

Mb = J (Aω + Buc )

uc = [δcol δlon δlat δ ped ]T is the control input vector, whose elements denote the collective pitch of the main rotor, longitudinal cyclic, and lateral cyclic and collective pitch of the tail rotor, respectively. F and M represent the lumped force and moment disturbance in BRF, respectively, involving external perturbations, parameter variations, and model uncertainties. Constant matrices A and B are expressed as follows [12,15–17]: ⎡ ⎤ −τ Lb −τ La 0 0⎦ A = ⎣−τ Mb −τ Ma 0 0 Nr ⎡ ⎤ 0 Llon Llat 0 Mlon Mlat 0 ⎦, B=⎣ 0 Ncol 0 0 Nped where the coefficients Zw , Zcol , Lb , La , Mb , Ma , Llon , Llat , Mlon , Mlat , Ncol , Nr , and Nped depend on the helicopter structure, which can be obtained via system identification technique. Along Eqs. (8) and (9), helicopter dynamics (2) and (4) can be rewritten as follows: V˙ = ge3 + R ()e3 (−g + Zw w + Zcol δcol ) + F

(10)

ω˙ = −J −1 (ω × J ω ) + (Aω + Buc ) + M,

(11)

where F =

1 R()F , m

and M = J −1 M . 

Given the desired smooth trajectory Pd = [xd yd zd ]T and yaw angle ψd , this work aims to develop control signals [δcol δlon δlat δ ped ]T for helicopter model with uncertainties and external disturbances, such that its practical trajectory P and yaw angle ψ converge asymptotically to Pd and ψd . 2.1. Approximate feedback linearization Feedback linearization is initially carried out to facilitate controller design and performance analysis. In the helicopter model, exact input–output linearization fails to linearize the complete system and results in unstable zero dynamics [14]. In this subsection, the simplified helicopter model is developed based on approximate feedback linearization. It

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requires the second-time derivative of the main rotor thrust T¨m as the new input, where Tm = m(g − Zw w − Zcol δcol ). The approximated input–output feedback linearization procedure is performed as follows: First, we obtain eP1 = P − Pd . The derivative of e p1 from Eq. (1) is given as follows: e˙P1 = V − P˙d

(12)

Next, let eP2 = V − P˙d along Eq. (10); its derivative is expressed as follows: e˙P2 = ge3 −

1 R ()e3 Tm + F − P¨d m

(13)

Selecting eP3 = ge3 − m1 R()Tm − P¨d , the derivative of eP3 is derived from Eq. (3) as follows: ... 1 1 e˙P3 = − R ()S (ω )e3 Tm − R ()e3 T˙m − P d (14) m m Transition control variables are defined as follows: [Mφ Mθ Mψ ]T = −J −1 (ω × J ω ) + ... (Aω + Buc ). Let eP4 = − m1 R()S(ω)e3 Tm − m1 R()e3 T˙m − P d along Eq. (11); its time derivative is obtained as follows: e˙P4 = fPe (t ) + uP + M p,

(15)

where 1 2 fPe (t ) = − R ()S 2 (ω )e3 Tm − R ()S (ω )e3 T˙m − Pd(4 ) m ⎡ m ⎤ ⎡ ⎤ Tm Mθ Tm M (2 ) 1 1 uP = − R ()⎣−Tm Mφ ⎦, M p = − R ()⎣−Tm M (1 )⎦ m m 0 T¨m The position error subsystem is approximately feedback-linearized by applying T¨M as input. The dynamics of yaw angle errors eψ1 = ψ − ψd is described from Eq. (7) as follows: e˙ψ1 =

Sφ Cφ q+ r − ψ˙ d Cθ Cθ

Selecting eψ2 =

Sφ q Cθ

+

(16) Cφ r Cθ

− ψ˙ d , its derivative is yielded along Eq. (11) as follows:

e˙ψ2 = fψe (t ) + uψ + Mψ ,

(17)

where

  ˙ + Sφ Mθ Cθ − Sφ φr ˙ Cθ + Sφ qSθ θ˙ + +Cφ r Sθ θ˙ Cφ φq fψe (t ) = − ψ¨ d Cθ2 Cφ uψ = Mψ Cθ Sφ Cφ Mψ = M (2 ) + M (3 ) Cθ Cθ Combining Eqs. (12)–(17), the complete input-output feedback linearization of the helicopter dynamics is given as follows: e˙P1 = eP2

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e˙P2 = eP3 + F e˙P3 = eP4 e˙P4 = fPe (t ) + uP + M p e˙ψ1 = eψ2 e˙ψ2 = fψe (t ) + uψ + Mψ ,

(18)

where uP and uψ are considered new inputs for the dynamics system (18). Our objective is changed to design the input [uPT uψ ]T to ensure that the trajectory and yaw-angle errors will converge to zero. Remark 1. In Eq. (18), F , M p, and Mψ are lumped disturbances that act on the helicopter model. M p and Mψ are matched disturbances, whereas F is the unmatched disturbance. The unmatched disturbance F significantly affects tracking precision. Given the existence of unmatched disturbances, various techniques, such as the classic SMC and adaptive control, are restricted. 3. Controller design The proposed controller is attained based on ISM technique, which allows a 2-degrees-offreedom design. Through the sliding-mode motion control, disturbance effects can be completely counteracted in the output. Then, the nominal controller is employed to guarantee the ideal performance of the closed-loop system [20,21]. The section is organized as follows. First, the design process and corresponding performance analysis are presented. Then, the implementation of this method, which aims to solve the problem of unmanned helicopter trajectory tracking, is shown. 3.1. Novel ISM The traditional ISM design only compensates the matched perturbations, but the unmatched one is replaced by another equivalent disturbance [20,21]. Through selecting the suitable sliding manifolds, the equivalent disturbance equals the unmatched one, that is, no amplification of unmatched disturbance exists. In this subsection, a novel ISM is proposed for systems with matched and unmatched perturbations. Sliding manifolds are designed to eliminate disturbances in output channels. A simplified perturbed linear system is considered as follows: x˙(t ) = Ax (t ) + Bu (t ) + Bd d (t ) , y (t ) = Cx (t )

(19)

where x(t ) ∈ Rn is the state; u(t ) ∈ Rm denotes the input; d (t ) ∈ Rl expresses the unknown disturbance vector due to model uncertainties or external disturbances. y(t ) ∈ Rm represents the output; A, B, Bd , and C are system matrices with the appropriate dimensions. The following assumptions are considered: Assumption 1. The pair (A, B ) is controllable, and matrices B and Bd are of full ranks, i.e., rank(B) = m, rank(Bd ) = l. Assumption 2. Full state feedback is available for system (19).

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A control law of the form is considered as follows: u (t ) = u0 (t ) + u1 (t ),

(20)

where u0 (t ) = K x is the nominal control, which is responsible for the performance of the nominal system (i.e., without any perturbations); u1 (t ) denotes the control signal for convergence of ISM. ISM surface is designed as follows: 

t s (x, t ) = G x (t ) − x (0 ) − (21) (Ax (t ) + Bu0 (t ))dt , 0

where G ∈ R is the constant matrix, which is designed to eliminate disturbance Bd d (t ) in the output channels. In this condition, the matrix (GB ) should be invertible. Note that at t = 0, s(x(0), 0 ) = 0. Thus, the system consistently starts at the sliding mode. Along Eq. (19), the derivative of the sliding mode surface is expressed as follows: m×n

s˙(x, t ) = G[(Ax (t ) + Bu (t ) + Bd d (t )) − (Ax (t ) + Bu0 (t ))] = GBu1 (t ) + GBd d (t )

(22)

Assumption 3. Disturbances d (t ) satisfy the condition that their time derivatives are bound by known constant values, i.e., d˙(t ) ≤ Cd , where Cd > 0 is the known constant vector. To guarantee the establishment and maintenance of a sliding manifold M = {x |s(x , t ) = 0}, u1 is selected as follows: t −1 u1 (t ) = (GB ) u1s (t )dt 0 1 u1s (t ) = −α · sign s˙ + β|s| 2 sign (s ) , (23) where α > 0, β > 0 is the controller constant. Theorem 1. With controller Eq. (23), the reachability condition of sliding motion can be guaranteed, that is, state trajectory of the closed-loop system will be globally driven onto the sliding mode in finite time. Proof. Considering controller Eq. (23), the second-order time derivative of sliding mode surface s is given as follows: s¨(x, t ) = GBu˙1 (t ) − GBd d˙(t ) = u1s (t ) − GBd d˙(t ) ∈ u1s (t ) − GBd [−Cd , Cd ]

(24)

Following [31], the second-order SMC u1s with prescribed convergence law is completed. A brief analysis is presented in the following section, and additional detailed proofs are illustrated in [31,32]. 1 First, we prove that the trajectory of inclusion inevitably hits the curve s˙ + β|s| 2 sign (s) = 0 1 due to geometrical reasons. Function fs = s˙ + β|s| 2 sign (s) is differentiated along (23) and (24) as follows: 1 1 f˙s (t ) = u1s (t ) − GBd d˙(t ) + β|s|− 2 s˙ · sign (s ) 2

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 1 1 1 ∈ −α · sign s˙ + β|s| 2 sign (s ) − GBd [−Cd , Cd ] + β|s|− 2 s˙ · sign (s ) 2 1 − 21 ∈ −α · sign ( fs ) − GBd [−Cd , Cd ] + β|s| s˙ · sign (s ) 2 Selecting α and β satisfying α − GBd Cd − 21 β 2 > 0 imply that f˙s sign ( fs ) < const < 0 exists in a vicinity of each point on curve fs = 0. Thus, the sliding-mode motion inevitably hits and stays on curve fs = 0. 1 Then, 1-sliding mode s˙(x, t ) = −β|s| 2 sign (s) holds; this mode guarantees that s converges to the origin in finite time. In classic ISM, control signal u1 is discontinuous and is selected with the following form: (GB )T s (x, t ) , (GB )T s (x, t ) where ρ is a gain sufficiently high to enforce sliding motion. Control command chattering can degrade the performance of the closed-loop system. Chattering-free second-order SMC can attenuate chattering. u1 (t ) = −ρ

Remark 2. The derivative of sliding mode motion s˙ in Eq. (23) is unavailable, but it can be approximately expressed as follows: s (t ) − s (t − T ) s (t ) − s (t − Ts ) ≈ , T Ts where Ts is sample time. The parameter α in Eq. (23) is responsible for disturbance elimination, which should be selected according to the amplitude of perturbations. β in Eq. (23) determines the convergence rate of s when the sliding-mode motion stays on curve fs = 0. However, too large values of these parameters may cause violent chattering of control signals. Thus, it is a trade-off process to properly choose the parameters. Next, equivalent control method is employed to determine motion equations at the sliding manifold . Equivalent control is yielded by solving the equation s˙(x, t ) = 0 in Eq. (22) for u1 :

s˙ = lim

T →0

u1eq (t ) = −(GB )−1 (GBd )d (t )

(25)

Remark 3. Note that for the matching case, Bd = B. In Eq. (25), u1eq (t ) = d (t ) can be calculated. Therefore, any matrix G, which guarantees det (GB ) = 0, can remove the matched disturbances through ISM technique. Substituting u1eq (t ) for u1 (t ) in Eq. (19), sliding motion is given with the following form: x˙(t ) = Ax (t ) + BK x (t ) − B (GB )−1 (GBd )d (t ) + Bd d (t ) y (t ) = Cx (t )

(26)

In traditional ISM, G is selected as G = BT , which can counteract the matched disturbances completely but guarantee no amplification of the effect of unmatched disturbances. Mismatched uncertainties cannot be eliminated completely from the state equation regardless of controller design [33]. We primarily aim to eliminate the influence of disturbances (matched and mismatched) on the output via ISM method. Sliding manifold gain G is designed as follows: G = C (A + BK )−1

(27)

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The following lemma and assumptions provided the condition for the state stability and output convergence. Lemma 1. For system with the following form: x˙(t ) = Ax (t ) + Bu (t )

(28)

if A is a Hurwitz matrix, and u(t ) features constant values in the steady state (i.e., limt→∞ u(t ) = U∞ ), then state x(t ) converges to a constant vector −A−1 BU∞ . Assumption 4. Disturbances d (t ) is bound, i.e., d (t ) ≤ Cb . Assumption 5. Disturbances d (t ) satisfy the condition that they possess constant values in the steady state, i.e., limt→∞ d˙(t ) = 0 and limt→∞ d (t ) = C∞ . Note that Assumption 5 is usually stronger than Assumption 4. When Assumptions 3 and 5 are satisfied, Assumption 4 can be obtained. The following theorem illustrates the main results of our method. Theorem 2. Given that Assumptions 1–3 are satisfied, controller gain K is designed such that matrix (A + BK ) is Hurwitz. The sliding mode surface gain G is given in Eq. (27). Then, the system (19) under control law (20) obtains the following properties. (1) If Assumption 4 is satisfied, all the states are ultimately bounded with tunable ultimate bounds. (2) If Assumption 5 is satisfied, then the state of the closed-loop system is bound and asymptotically stable. Lumped disturbance d (t ) can be completely eliminated from the output channels in the steady state. Proof. Before we provide analysis of convergence of properties, we will prove that the state of the closed-loop system is bound in time interval [0 T ]. During this stage, the closed-loop system is described as follows: x˙(t ) = Ax (t ) + BK x (t ) + Bu1 (t ) + Bd d (t ) From Eq. (23), the upper bound of u1 (t ) when t ∈ [ 0 T ] is (GB )−1 αT , i.e., u1 (t ) ≤ (GB )−1 αT , t ∈ [ 0 T ]. d (t ) ≤ Cd T given that time derivative of d (t ) is bound. Considering that the matrix (A + BK ) is Hurwitz, the state will not escape to infinity in finite time T . Theorem 1 states that the sliding mode motion is achieved in finite time T , that is, the systems will reduce to the sliding mode motion (26) in finite time. (1) Then, considering system (26), the proof of conclusion (1) is provided. For the Hurwitz matrix (A + BK ), positive matrices P and Q exist, satisfying the following equation: (A + BK )T P + P (A + BK ) = −Q The Lyapunov function V (x) = x T Px ≥ 0 is selected. Along Eq. (26) and Assumption 4, one obtains the following:



 V˙ (x ) = x T (A + BK )T P + P (A + BK ) x + 2x T P −B (GB )−1 (GBd ) + Bd d (t )

 = −x T Qx + 2x T P −B (GB )−1 (GBd ) + Bd d (t )

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≤ −λmin (Q )x 2 + 2λmax (Gc )xd (t ) = x (−λmin (Q )x + 2λmax (Gc )Cb ), where λmin (∗) and λmax (∗) denote the minimum and maximum singular values, respectively; matrix Gc = P[−B (GB )−1 (GBd ) + Bd ]. Thus, we obtain the following: V˙ (x ) ≤ 0, if x ≥

2λmax (Gc )Cb λmin (Q )

(Gc )Cb Following [18], the state x is ultimately bounded with boundary {x ≥ 2λλmax }. min (Q) (2) Applying Lemma 1 and Assumption 5, the steady value of state x(t ) is obtained as follows:

 limt→∞ x (t ) = −(A + BK )−1 −B (GB )−1 (GBd ) + Bd C∞

Output is defined as y(t ) = Cx(t ). Hence, its steady value is given as follows:

 lim y (t ) = −C (A + BK )−1 −B (GB )−1 (GBd ) + Bd C∞

t→∞

Applying the expression of G in Eq. (27), the above equation can be rewritten as follows:

 lim y (t ) = −G −B (GB )−1 (GBd ) + Bd C∞

t→∞

= (GBd − GBd )C∞ = 0



Remark 4. In Assumption 5, disturbance is assumed to feature a limiting value. Then, the property of asymptotic convergence of the closed-loop system is established. This disturbance case is impossible in numerous practical engineering systems, such as random gust disturbances. Although many actual cases are out of the scope of our assumptions, the effectiveness of our method in such a case is illustrated via numerical simulation (i.e., complex disturbances including wind disturbances and model uncertainties are introduced in our helicopter simulation). Remark 5. The design of nominal controller K x can be implemented based on many control theories, such as pole placement, H∞ method [20], or MPC [29]. Our work mainly focuses on the sliding-mode motion control design. Hence, the design process of controller gain K is omitted to highlight our main contributions. Remark 6. The function of ISM control term is similar to that of DO [33,34]. Estimations of lumped disturbances in the ISM framework are obtained based on the equivalent control law. Moreover, there exist lots of results about unmatched disturbance elimination, including BS control combined with DO based control [12,35] or adaptive control [10], high-order SMC [17,36], and relative degree approach [37]. Actually, the selection of sliding-manifold surface (27) is a special application for relative degree approach. By properly choosing the sliding-manifold gains, the effect of disturbances is removed from output channels. 3.2. Helicopter controller design In this subsection, the proposed ISM is applied to deal with the unmanned helicopter control problem.

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Assumption 6. The attitude of helicopters constantly lies inside the region |φ| < |θ (t )| < π2 .

2679 π 2

and

Remark 7. Assumption 6 ensures that the attitude kinematic matrix () in (7) is not singular. This assumption is valid because our helicopter is supposed to operate in hovering or low-velocity fight condition. In Section 2, the helicopter model has been approximately feedback-linearized in Eq. (18). The model comprises two parts, namely, the position error dynamics and yaw-angle error dynamics. Position error dynamics is computed as follows:

 e˙P = APe eP + BPe fPe (t ) + uP + BPd dP yeP = CPe eP

(29)

Yaw-angle error dynamics is computed as follows:

 e˙ψ = Aψe eψ + Bψe fψe (t ) + uψ + Bψd dψ yeψ = Cψe eψ

(30)

where

⎡

⎤ ⎡ ⎤ ⎡ ⎤ 03×3 I3×3 03×3 03×3 03×3 03×3 03×3 ⎢03×3 03×3 I3×3 03×3 ⎥ ⎢03×3 ⎥ ⎢ I3×3 03×3 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ APe = ⎢ ⎣03×3 03×3 03×3 I3×3 ⎦, BPe = ⎣03×3 ⎦, BPd = ⎣03×3 03×3 ⎦, 03×3 03×3 03×3 03×3 I3×3 03×3 I3×3 





 F 0 1 0 , Aψe = , Bψe = Bψd = , CPe = I3×3 03×3 03×3 03×3 , dP = MP 0 0 1

 dψ = Mψ , and Cψe = 1 0 . Applying the proposed ISM technique, the control command signals for unmanned helicopters are given as follows: Position error subsystem: t uP = − fPe (t ) + KPe e p + (GP BPe )−1 uP1s (t )dt 0  1 (31) uP1s (t ) = −αP sign s˙P + βP |sP | 2 sign (sP ) Yaw-angle error subsystem:  −1 t uψ = − fψe (t ) + Kψe eψ + Gψ Bψe uψ1s (t )dt 0   1   uψ1s (t ) = −αψ sign s˙ψ + βψ sψ  2 sign sψ

(32)

where Kpe and Kψe are controller gain matrices, (APe + BPe KPe ) and (Aψe + Bψe Kψe ), respectively, which are Hurwitz matrices. The poles of the closed-loop systems constantly lie on the left-half plane. The sliding mode surface sP = GP [eP − eP (0) − ∫t0 (APe eP + BPe ( fPe (t ) + KPe eP ) )dt ] and sψ = Gψ [eψ − eψ (0) − ∫t0 (Aψe eψ + Bψe ( fψe (t ) + Kψe eψ ))dt]; GP = CPe (APe + BPe KPe )−1 and Gψ = Cψe (Aψe + Bψe Kψe )−1 .

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Value

Mass of unmanned helicopter 8.2 kg Acceleration of gravity 9.8 m/s2 The moment of inertia matrix of helicopter diag(0.18 0.34 0.28) kg m2 Linkage gain ratio of Tm to w −0.7615 s−1 Linkage gain ratio of Tm to δcol −131.4125 m/(rad s2 ) −1 Coefficient matrix of  in (9) diag( ⎡ −48.1757 −25.5048 −0.9808) s ⎤ 0 0 1689.5 0 ⎣ Coefficient matrix of u in (9) 0 894.5 0 0 ⎦s−2 −0.3705 0 0 135.8

T Kψe is selected, Gψ = [0 1] = Bψe , regardless of the controller value. Next, practical control signals for the helicopter can be calculated from uP and uψ . According to their definitions, one computes the following: ⎡ ⎡ ⎤ ⎤ Mφ 0 Tm 0 ⎣Mθ ⎦ = −m · ⎣−Tm 0 0⎦ · R−1 () · uP (33) ¨ 0 0 1 Tm

Mψ =

Cθ uψ Cφ

(34)

Then, control commands [δcol δlon δlat δ ped ]T can be obtained according to the defi t nition of [Mφ Mθ Mψ ]T and TM = 0 T¨m dt.   Tm δcol = − − g + Zw w /Zcol (35) m ⎡

⎤ ⎛⎡ ⎤ ⎞ δlon Mφ ⎣ δlat ⎦ = B (2 )\⎝⎣ Mθ ⎦ + J −1 (ω × J ω ) − Aω − B (1 )δcol ⎠, δ ped Mψ

(36)

where B(1) and B(2) are the first- and second-forth column of matrix B, respectively. The following result is given directly by applying Theorem 2. Theorem 3. Considering helicopter dynamic models (1)–(4) under Assumptions 1–6, if the controller is designed as Eqs. (35) and (36), then position and yaw-angle tracking errors of the closed-loop system asymptotically converge to zero. 4. Simulation analysis In this section, numerical simulation results are presented to illustrate the efficiency of our proposed ISM method for small-scale unmanned helicopters. Table 1 summarizes the corresponding helicopter model parameters [12,38], where diag(x1 , x2 , . . . , xn ) denotes diagonal matrix with diagonal elements x1 , x2 , . . . , xn .

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Following [12], lumped disturbances in Eqs. (10) and (11) are introduced in simulation. These disturbances include model uncertainties and external disturbances. ⎡ ⎤ 

V F =  · ⎣⎦ + Bwind dwind , (37) M ω where constant matrix  ∈ R6×9 represents model uncertainty, and Bwind denotes the transformation matrix from airspeed to force and moment. All elements of  are pseudorandom values within the open interval (−0.5 0.5 ). Matrices  and Bwind are given as follows: ⎡ ⎤ −0.239 0.164 −0.055 0.456 0.013 −0.362 −0.096 0.071 −0.185 ⎢0.203 0.351 0.419 0.351 −0.135 0.342 0.491 −0.359 −0.387⎥ ⎢ ⎥ ⎢0.035 0165 0.232 0.265 −0.384 0.254 −0.044 0.357 0.031 ⎥ ⎥ =⎢ ⎢0.311 0.045 0.369 −0.250 −0.303 −0.368 −0.352 0.012 −0.011⎥ ⎢ ⎥ ⎣−0.055 −0.465 0.062 0.290 0.048 −0.451 0.426 −0.454 0.209 ⎦ 0.267 −0.409 −0.088 −0.482 0.057 0.459 0.138 −0.028 0.343 ⎤ ⎡ −0.0505 0 ⎢ 0 −0. 151 ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 ⎥ Bwind = m × ⎢ ⎢ −0.144 0.143 ⎥ ⎢ ⎥ ⎣−0.0561 −0.0585⎦ 0 0.0301 Wind disturbance dwind is assumed to possess wind components along [xb yb ] in BRF. This variable is composed of constant and random components, i.e. dwind = dc + dr . The constant component is set as [5 5]T . Stochastic wind disturbance is modeled by independently excited correlated Gauss–Markov processes [6]:  





d˙u 0 −1/τc du q ˙ +ρ u , dr = ˙ = 0 −1/τc dv qv dv where qu and qv (σqu = σqv = 20ft/s) are independent with zero mean, and τc = 3.2 s is the correlation time of the wind. ρ = 1/2 is the scalar weighting factor. Fig. 2 shows the wind disturbance dwind used in the helicopters’ simulation. To provide sufficient fidelity in the simulation, the flapping dynamics model is considered. Flapping dynamics model is equivalent to the additional dynamics in the servo loop [38]: τ f δ˙¯lon = −τ f q − δ¯lon + δlon τ f δ˙¯lat = −τ f p − δ¯lat + δlat ,

(38)

where δ¯lon and δ¯lat are longitudinal and lateral applied control signals; τ f = 0.1 is the main rotor’s dynamics time constant. 4.1. Description of control laws To demonstrate the superiority of our method, several previous methods are compared with our ISM design. These implementations are illustrated as follows.

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Fig. 2. Wind disturbance.

4.1.1. Pole placement (PP) PP method is one of the most popular methods in control field. According to the flight characteristics of small-scale helicopters, the parameters of the PP controller are selected as follows: ⎡ ⎤ 1800 0 0 2430 0 0 679 0 0 50 0 0 1800 0 0 2430 0 0 679 0 0 50 0 ⎦ KPe = −⎣ 0 0 0 1800 0 0 2430 0 0 679 0 0 50 

Kψe = − 450 45 where the corresponding poles in the position error subsystem and yaw-angle error subsystem are selected as (−1 −4 −15 −30 ) and (−15 −30 ), respectively. 4.1.2. PP+ISM This classic method is proposed in [15]. PP+ISM controller architecture is presented in Eqs. (31) and (32). The nominal controller is the same as the PP controller. Sliding manifolds are designed based on the projection matrix approach such that equivalent disturbances are T T equal to the unmatched ones. Sliding manifold gains are given as GP = BPe and Gψ = Bψe . The second-order sliding mode method is used to achieve sliding-mode motion. The corresponding ISM control parameter is designed as follows: αP = diag(2, 2, 2 ), βP = diag(2, 2, 2 ) αψ = 2, βψ = 2

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4.1.3. PP+output ISM PP+output ISM is our proposed ISM method. Implementation of this method is the same as that of the PP+ISM controller, except for the sliding manifold gains. In our method, disturbances in output channels can be completely cancelled out, guaranteeing exact trajectory tracking. The ISM manifold gains GP and Gψ can be obtained by applying Eq. (27). 4.1.4. H∞ +ISM This technique uses an ISM controller to reject the matched perturbation and design the nominal control using H∞ techniques to attenuate the unmatched ones [21]. The ISM component is the same as the PP+ISM controller. The nominal component applies H∞ control in the position error subsystem (unmatched disturbances occur only in this subsystem). Cost matrix is selected as follows: Q = diag(100, 100, 100, 100, 100, 100, 50, 50, 50, 30, 30, 30 ). H∞ controller is solved as follows: ⎡

KPe

⎤ 865 0 0 1576.4 0 0 654.5 0 0 50.1 0 0 −7 −7 ⎣ ⎦ 0 865 0 0 1576 . 4 −4. 1 × 10 0 654. 5 −1 . 5 × 10 0 50. 1 0 = 0 0 865 0 −4.3 × 10−6 1576.4 0 −1.6 × 10−6 654.5 0 0 50.1

4.1.5. 1st output ISM The difference between the 1st output ISM and our method is the sliding-mode motion control component. Comparison demonstrates that by applying the second-order SMC, chattering phenomenon of control signals can be improved. In this case, classic SM control under boundary layer modification is utilized. ⎧ (GB )T s ⎪ ⎪ ⎨−ρ · , i f (GB )T s ≥ ε (GB )T s u1 (t ) = , ⎪ (GB )T s ⎪ T ⎩−ρ · , i f (GB ) s < ε ε where constant ρ = 10000 and ε = 10−3 . 4.1.6. Backstepping + DO (BSDO) This is one of the classical robust control methods to eliminate the adverse effect of matched and unmatched disturbances, which has been widely used in the practical applications. Following the system (18), applying BSDO method, it gives eP2d,bs = −k1,bs eP1   ˆF eP3d,bs = −k2,bs eP2 − eP2d,bs −    eP4d,bs = −k3,bs eP3 − e p3d,bs   ˆM u p = − fPe (t ) − k4,bs e p4 − e p4d,bs −  eψ2d,bs = −k1ψ,bs eψ1   ˆψ uψ = − fψe (t ) − k2ψ,bs eψ2 − eψ2d,bs −  ˆ F = k f (e˙P2 − eP3 ),  ˆ M = k f (e˙P4 − fPe (t ) − uP ),  ˆ ψ = k f (e˙ψ2 − fψe (t ) − uψ ) Where  s+k f s+k f s+k f are the estimation of the unknown disturbance items F , M p and Mψ , respectively. The control and filter gains are set as: k1,bs = 1, k2,bs = 4, k3,bs = 15, k4,bs = 30, k1ψ,bs = 15, k2ψ,bs = 30 and k f = 2.

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Fig. 3. Trajectory tracking of helicopter. Table 2 the RMS of positon tracking errors. Index

PP

PP+ISM

PP+outpout ISM

Hinf +ISM

BSDO

RMS (m)

2.151

1.954

0.453

3.114

0.584

4.2. Simulation results The desired “8-shape” reference trajectory is described as follows:  T

Pd (t ) = 0 0 −7 1 − e−0.3t for t ≤ 7s ⎤ ⎡  2π ⎢20 1 − cos 23 (t − 7 ) ⎥ ⎢   ⎥ ⎥ for t > 7s Pd (t ) = ⎢ ⎢ 10 sin 4π (t − 7 ) ⎥ ⎣ ⎦  23 −0.3t  −7 1 − e ψd = 0

(39)

Simulation results are illustrated in Figs. 3–8. Fig. 3 shows the three-dimension position response curves of the closed-loop system based on the above methods. Compared with other methods, the proposed ISM guarantees high tracking of the reference trajectory in the presence of model uncertainties and wind disturbance. In Table 2, The root-mean-squares (RMS, eP1 ) of tracking errors are given to quantitatively display tracking performance of these control systems. Given that introducing the traditional ISM shows no evident improvement on tracking performance, we can observe that the unmatched disturbance (lumped force disturbance) exerts more influence on helicopter controller design. H∞ design slightly affects suppression of unmatched disturbances and causes a decline in performance due to difficulties in selecting appropriate performance function. Additionally, the proposed method has the comparable performance with BSDO, even smaller steady errors than BSDO. It proves that by properly choosing the sliding manifold, the effect of mismatched perturbation is attenuated. Fig. 4 shows the three-dimensional position trajectory intuitively, and results agree with this analysis. Figs. 5–7 show the velocity, angle, and angular velocity response. The responses of PP+output ISM fluctuate heavily to rapidly eliminate disturbances. Violent chattering exists at t = 7s due to switching of reference trajectory. Fig. 8 displays the response curves of controller inputs. The control signal curves of our method are smoother than the 1st output

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Fig. 4. Position response of the closed-loop helicopter system.

Fig. 5. Velocity response of the closed-loop helicopter system.

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Fig. 6. Angular response of the closed-loop helicopter system.

Fig. 7. Angular velocity response of the closed-loop helicopter system.

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Fig. 8. Control inputs of helicopters.

ISM with low vibration amplitude. This result confirmed that control signals can be improved by the second-order SM technique. 5. Conclusion This paper proposes a novel ISM control method to deal with the trajectory tracking problem for helicopter control. The controller is developed based on the approximate feedbacklinearized model with lumped matched and unmatched disturbances. Bounded disturbances can be completely eliminated in the output channels by selecting the integral sliding mode manifold suitably. The second-order chattering-free SM control is adopted to attenuate the chattering phenomenon. Some simulation results demonstrate that through the ISM control approach presented in this work, a model-scale unmanned helicopter will exhibit excellent tracking performance in the presence of model uncertainties and gust perturbation. The nominal component of the ISM controller, which guarantees ideal performance of the closed-loop system, can be achieved through mature techniques, such as pole placement, H∞ control method, and MPC. Future work will focus on developing the compound control approach based on ISM and other advanced control techniques. References [1] B. Mettler, Identification Modeling and Characteristics of Miniature Rotorcraft, Kluwer Academic Pub, Boston, 2003. [2] F. Kendoul, Survey of advances in guidance, navigation, and control of unmanned rotorcraft systems, J. Field Robot 29 (2) (2012) 315–378. [3] J. Alvarenga, N.I. Vitzilaios, K.P. Valavanis, M.J. Rutherford, Survey of unmanned helicopter model-based navigation and control techniques, J Intell. Robot. Syst. 80 (1) (2015) 87–138. [4] G.W. Cai, B.M. Chen, Unmanned Rotorcraft Systems, Springer, Heidelberg, 2011. [5] O. Tanner, Modeling, Identification, and Control of Autonomous Helicopters, ETH, 2003.

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