Fixed Time Disturbance Observer Based Sliding Mode Control for a Miniature Unmanned Helicopter Hover Operations in Presence of External Disturbances

This paper presents a novel fixed time sliding mode disturbance observer (FTSMDO) based second-order fixed time sliding mode control (FTSMC) for small scale unmanned helicopter to do hover operations in the presence of external disturbances. Sliding mode control is insensitive to matched uncertainties but sensitive to mismatched uncertainties. The novel FTSMDO exactly estimates the total mismatched uncertainties effecting the sliding mode control performance. To counter the mismatched uncertainties a new sliding surface augmented by the total mismatched disturbance approximated by the FTSMDO is defined. The second-order FTSMC forces the system states to reach the sliding surface in fixed time and then on the sliding surface the system states exponentially converge to the desired equilibrium point. The control performance is compared with the disturbance observer based sliding mode controller (DOB-SMC) and shows superior performance.


I. INTRODUCTION
Miniature helicopters are highly unstable, agile, nonlinear under-actuated systems with significant inter-axis dynamic coupling. They are considered to be much more unstable than fixed-wing unmanned air vehicles, and constant control action is required at all times. However, helicopters are highly flexible aircraft, having the ability to hover, maneuvers accurately and carry heavy loads relative to their own weight [1]. Fixed-wing aircraft are used for application in favorable non-hostile conditions but in adverse conditions, agile miniature helicopters become a necessity. The conditions where a helicopter can perform better than fixed-wing UAVs include military investigation, bad weather, firefighting, search and rescue, accessing remote locations and ship The associate editor coordinating the review of this manuscript and approving it for publication was Lei Wang.
operations. In such conditions, helicopters are subjected to unknown external disturbances such as wind and ground effect. These external disturbances have a significant opposing effect on the helicopter stability and can have disastrous results in extreme cases. So it is essential to design a controller for the helicopter which can effectively reject the effect of these unknown external disturbances.
In the last two decades, there is substantial research about helicopter control problems. Early results showed that classical control methods using Single-Input Single-Output feedback loops for each input exhibit moderate performance since they are unable to coup with the highly coupled multivariable dynamics of the helicopter [2]. Control schemes typically used to maintain stable control of helicopters include PID [3], Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian (LQG) [4], H2 [5], H∞ [6], [7]. The majority of linear controllers designed for VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ unmanned helicopters are based on the H∞ method. An H∞ static output feedback control design method [8] was proposed for the stabilization of a miniature unmanned helicopter at hover. An interesting comparative study between several control methods is given in [9], [10]. There are lots of results about Disturbance Observer-based control techniques [11]- [14]. A direct feed-through simultaneous state and disturbance observer [14] is used where the control and observer gains are obtained using H∞ synthesis but in presence of external disturbances, there is a steady-state error in helicopter translational dynamics. Back-stepping control design techniques are used for linear tracking control of a miniature helicopter without considering external disturbances [15], the control design is based on the linearized model of the helicopter and shows good results in X-plane flight simulator. DOB-SMC [16], [17] is used for controlling magnetic levitation suspension system and small scale unmanned helicopter hover operations respectively. The experimental results showed that the proposed method has excellent robustness in the presence of both matched and mismatched uncertainties. In [18], [19] second-order sliding mode control law is used which provides finite time convergence but convergence time depends on the initial condition. Fixed time consensus control of multi agent and leader follower system is presented in [20], [21]. A novel multi-variable FTSMC method is proposed in [22] where a formula is derived to estimate the fixed convergence time of the controller. The fixed convergence time of the FTSMC does not dependent on the initial condition.
In this paper, a novel fixed time sliding mode disturbance observer based second-order fixed time sliding mode control FTSMDO-FTSMC design technique is presented for small scale unmanned helicopter to do hover operations in the presence of external disturbances. The controller design is based on the linearized state-space model of the helicopter. As in [14], [15]and [17] the linearized model of the helicopter can be divided into two subsystems, such as the longitudinal-lateral subsystem and the heading-heave subsystem. As there is no strong coupling between the two subsystems at hover and limited by the scope of the paper, for hovering only the longitudinal-lateral dynamics are considered for designing the control law. To counteract both matched and mismatched uncertainties a new sliding surface is designed based on the total disturbance estimated by the FTSMDO. The model mismatch and external disturbances are estimated as lumped disturbances and are compensated in the controller design. In [16], [17] the proposed disturbance observer (DOB) asymptotically estimates the model mismatch and external disturbance at each and every channel separately. But in this paper using FTSMDO, the total mismatched disturbance (caused by uncertainties and external disturbances acting through all mismatched channels) effecting the sliding mode control performance is estimated as lumped disturbance, exactly in the fixed time. The rotor flapping dynamics are approximated by the steady-state dynamics of the main rotor which helps reducing con-troller order. Simulink simulations have demonstrated superior performance of the FTSMDO-FTSMC compared to the DOB-SMC.
The proposed control method has two attractive features.
• It is a simple first order fixed time disturbance observer which approximate the total mismatch disturbance acting on an n th order system through all n channels provided it satisfy some assumption.
Other disturbance observers are all n th order for an n th order system.

•
The chattering problem is substantially reduced due to the use of second-order FTSMC. The rest of the paper is organized as follows: The linearized model of the helicopter is derived from the nonlinear model in section 2. FTSMDO is designed in section 3. The proposed controller is derived in detail in section 4. Simulation results are given in section 5 and finally concluding remarks are given in section 6.

A. NONLINEAR DYNAMICS OF THE HELICOPTER
The general 11 th state nonlinear model [23] of the miniature unmanned helicopter is given aṡ T is the vector of the state variables all available for measurement except a and b; u, v andw represents linear velocities in longitudinal, lateral and vertical direction respectively; m is mass of helicopter; g represents acceleration due to gravity; p, q and r represents angular velocities in roll, pitch and yaw axis respectively; φ, θ and ψ are Euler angles of roll, pitch and yaw axes; u c (t) = u lon u lat u col u ped T is the control input vector; d wi ∀ i = 1, 2, · · · 6 are unknown external wind disturbances effecting translational as well as rotational dynamics of helicopter; I xx , I yy and I zz are the rolling moment of inertia, pitching moment of inertia and yawing moment of inertia respectively; a and b are flapping angles of tip-pathplane(TPP) in longitudinal and lateral direction respectively; X mr , Y mr and Z mr are the force components of main rotor trust along x, y and z axis; L mr and M mr are roll and pitch moments  Fig.1. The force components generated by the main rotor trust in x,y and z direction are given as where T is the total trust generated by the main rotor. The moments generated by the main rotor along the x and y direction are calculated as where k β is the torsional stiffness of the main rotor hub; h mr is the main rotor hub height above the center of gravity of the helicopter. The trust of the main rotor is calculated by iteratively solving the equations of trust and the induced inflow velocity [24].
Rk a k col u col (4) where v i is the induced inflow velocity; is rotational speed of the main rotors; ρ is air density; R is main rotor radius; b m is the number of main rotor blades; c m is the chord length of the main rotor; C m lα is coefficient of lift curve slope of the main rotor; is control gain of the servo actuator; k col is linkage gain from the collective actuator to the main blade.

B. LINEARIZED STATE SPACE MODEL OF THE HELICOPTER
To derive the control law, the nonlinear model (1) of the helicopter is linearized at hover condition aṡ At hover condition, the longitudinal-lateral and heading-heave dynamics of the helicopter are weakly coupled with each other and are expressed as two separate sub-systems [15].
where (6) represents longitudinal-lateral subsystem and (7) represents the heading-heave subsystem. For hover operation, the FTSMDO-FTSMC control law is only derived for the longitudinal-lateral subsystem and heading-heave dynamics are regulated at hover condition using PID controllers. The subsystem (6) is expanded aṡ where X u , Y v , M u , M v , M a , L u , L v and L b are stability derivatives; A lon , A lat , B lon and B lat are input derivatives; d ti ∀ i = 1, 2, · · · 6 is the total disturbance including both model mismatch and external disturbances acting at channel i. Approximating the flapping angles a and b by the steady state dynamics of the main rotor [25] as Substituting (10) in (8) gives the reduced order linearized model, written in state space form as followinġ x r = A r x r + B r u c1 + E r d tr (11) where T ; E r is 6 × 6 identity matrix; y r is VOLUME 8, 2020 output vector; Matrices A r , B r and C r are given as where all µ i are positive bounded unknown constants. Assumption 5:The disturbance d ti , ∀ i = 1, 2, . . . , 6 belongs to a class of slow varying disturbances having constant value in steady state such that its derivative is bounded and satisfies lim t→∞ḋ ti = 0.

C. INPUT OUTPUT FEEDBACK LINEARIZATION
To derive the proposed control law for hover operation, first the helicopter reduced dynamics (11) is input output feedback linearized. The system output (12) is simplified as Differentiating y r giveṡ where T Differentiating (15) and taking (assumption 5) results (16) where d tr2 = d t3 d t4 T . Differentiating (16) and takingḋ tr1 andḋ tr2 as zero results ...

III. FIXED TIME DISTURBANCE OBSERVER
In this section FTSMDO is designed to estimate the total mismatched disturbance effecting the performance of FTSMC. In order to design FTSMC for system (11) in presence of mismatched uncertainties the sliding surface is defined as σ = C 1 y r + C 2ẏr +ÿ r (20) where σ = σ s1 σ s2 c 4 ). C 1 and C 2 are designed such that σ s1,s2 = 0 is Hurwitz. Substituting (18) in (20) gives where D = C 2 D 1 + D 2 is the total mismatched uncertainty effecting the performance of the FTSMC and FTSMDO is designed to estimate it.
Theorem 1: Considering the 2nd order system (28), both σ 1 and σ 2 converge uniformly to the origin in fixed time e is the base of natural logarithms, provided the following conditions holds: Proof: Considering the following second order system in presence of bounded disturbance ζ the fixed time in which the states x(t) and y(t) converge uniformly to the origin is given by [22] where > 0; M 1 = k 13 + ζ ; m 1 = k 13 − ζ ; h (k 11 ) = 1/k 11 + (2e/m 1 k 11 ) 1/3 ; e is the base of natural logarithms; k 13 > ζ and k 11 h −1 (k 11 ) > M 1 .
Dynamics of (28) is similar to (30) except ζ is zero. Therefore, by substituting ζ = 0 in (31) the convergence time T f of σ 1 and σ 2 to the origin is simplified as (29).
According to theorem1 as σ 1 and σ 2 goes to zero in fixed time T f , then (26) implies that

IV. CONTROLLER DESIGN
In this section FTSMC is designed to stabilize helicopter to do hover operations in presence of external wind disturbances. The sliding surface augmented with the total mismatched disturbance approximated by FTSMDO is designed as whereD is estimation of D given by the disturbance observer and parameters C 1 and C 2 are same as in (20). Theorem 2: Considering system (11) under the fixed time sliding mode control law The closed loop system is exponentially stable and the system output y r exactly converge to the desired equilibrium point, provided the following conditions holds: and all k 7 , k 8 , k 9 , k 10 , k 11 , k 12 are positive bounded constant. Proof: Differentiating the sliding surface (33) resultṡ Substituting (34) in (35) giveṡ Differentiating (37) and combining with (36) giveṡ As FTSMDO is designed such that after time T f the disturbance approximation error converge to zero so and after time T f , (39) reduces tö Now (41) is Hurwitz and the system output y r is uniformly ultimately exponentially convergent to the desired equilibrium point.

V. SIMULATION RESULTS
In this section evaluation of the proposed controller (34) is presented. Performance of (34) is compared with a DOB-SMC. The sliding surface of the traditional DOB-SMC method is designed as whereD 1 =d tr1 andD 2 = K 1dtr1 + K 2dtr2 . The disturbance observer used to estimate the unknown disturbance vector d tr is designed aṡ whered tr = d t1dt2dt3dt4dt5dt6 T is the disturbance estimation vector, P is a 6 × 1 auxiliary vector and L = diag (l 1 , l 2 , l 3 , l 4 , l 5 , l 6 ) is the observer gain.
Using DOB there is initial peaking at time t 0 in disturbance approximation which causes higher control gain and even can takes the control input to saturation. So to avoid the initial peaking phenomena the observer gain L is designed as follows where Q is any positive number and I is 6 × 6 identity matrix. So L is zero at t 0 and positive elsewhere. It satisfies the condition that −L is Hurwitz. Finally DOB-SMC is designed as Raptor 90 SE radio controlled helicopter is used in these simulations.
It is observed from Fig.2 that DOB-SMC and FTSMDO-FTSMC have almost the same settling time but FTSMDO-FTSMC have reduced control chattering compare to DOB-SMC as shown in Fig.3. Fig.4 shows that the mismatched disturbances (d t1 , d t2 , d t3 , d t4 ) approximated by DOB goes to zero as soon as all the states of the system reach zero as there was no external disturbance applied     on the system. There is small scale matched disturbances (d t5 , d t6 ) in steady state due to the model mismatch caused by order reduction. Fig.5 shows that the total mismatched disturbances approximated by FTSMDO. It goes to zero after initial settling time. Calculating the total mismatched disturbance C 2 D 1 + D 2 using DOB output (d t1 ,d t2 ,d t3 ,d t4 ) and comparing it with FTSMDO output it is clear that contrary to DOB, FTSMDO has low peaking during initial settling time.

B. CASE2 HANDLING MISMATCH UNCERTAINTIES
In second case to compare the mismatched uncertainty handling capacity of both controllers, external wind disturbances d w1 and d w2 are applied on the helicopter system (1) defined as Control parameters are same as case 1. All initial states are zero. In Fig.6 it is observed that both FTSMDO-FTSMC and DOB-SMC suppress the external wind disturbances and bring back u and v to the desired equilibrium point. Fig.7 shows control inputs and confirm that FTSMDO-FTSMC have reduced control chattering. Fig8 shows external wind disturbances approximated by DOB and Fig.9 shows the total mismatched disturbance approximated by FTSMDO.

VI. CONCLUSION
This paper presented a novel FTSMDO based FTSMC for miniature unmanned helicopter to do hover operations in presence of external disturbance. The main contribution is design of FTSMDO which can estimate the total mismatch disturbance (caused by uncertainties and external disturbances acting through all mismatched channels) effecting the sliding mode control performance and defining a new sliding surface (augmented by this total disturbance) to design the FTSMDC. The FTSMDC forces the system states to reach the defined sliding surface and then slides towards the desired equilibrium point in presence of mismatched uncertainties. Simulation results showed superior performance of FTSMDO-FTSMC compered to DOB-SMC.