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Nonsingular Terminal Sliding Mode Based Passive Fault-Tolerant Control of a 3-DOF Helicopter System⁎
10th IFAC Symposium on Fault Detection, 10th IFAC Symposium on Fault Detection, Supervision and Safetyon for Technical Processes 10th IFAC Symposium Detection, Available online at www.sciencedirect.com Supervision and Safety forFault Technical Processes 10th IFACPoland, Symposium on29-31, Fault Detection, Warsaw, August 2018 Supervision and Safety Technical Warsaw, Poland, Augustfor 29-31, 2018 Processes Supervision and Safety for Technical Processes Warsaw, Poland, August 29-31, 2018 Warsaw, Poland, August 29-31, 2018
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IFAC PapersOnLine 51-24 (2018) 1368–1372
Nonsingular Terminal Sliding Mode Based Nonsingular Terminal Sliding Mode Based Nonsingular Terminal Sliding Mode Based Passive Fault-Tolerant Control of a 3-DOF Nonsingular Terminal Sliding Mode Based Passive Fault-Tolerant Control of a 3-DOF Passive Fault-Tolerant Control of a 3-DOF Helicopter System Passive Fault-Tolerant Control of a 3-DOF Helicopter Helicopter System System Helicopter System ∗∗ YANG Qiwei ∗∗ YANG Rui ∗∗
YANG Qiwei ∗∗ YANG Rui ∗∗ YANG Qiwei ∗ YANG Rui ∗∗ YANGEngineering Qiwei YANG Rui ∗∗ ∗ of Electrical and Automation, Shandong ∗ ∗ College College of Electrical Engineering and Automation, Shandong ∗ University of Science and Technology, Qingdao 266590, China. College of Electrical Engineering and Automation, Shandong ∗ University of Science and Technology, Qingdao 266590, China. College ofofElectrical Engineering andQingdao Automation, Shandong (e-mail: qiwei [email protected]) University Science and Technology, 266590, China. (e-mail: qiwei [email protected]) ∗∗ University of Science and Technology, Qingdao 266590, China. Engineering and Automation, Shandong (e-mail: qiwei [email protected]) ∗∗ College of Electrical ∗∗ Electrical Engineering and Automation, Shandong ∗∗ College of (e-mail: qiwei [email protected]) University of Science and Technology, Qingdao 266590, China. College of Electrical Engineering and Automation, Shandong ∗∗ University Science and Technology, 266590, China. College ofof Engineering andQingdao Automation, Shandong (e-mail: [email protected]) University of Electrical Science and Technology, Qingdao 266590, China. (e-mail: [email protected]) University of Science and Technology, Qingdao 266590, China. (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Abstract: In In this this paper, paper, a a new new passive passive fault-tolerant fault-tolerant controller controller for for 3-DOF 3-DOF helicopter helicopter is is designed. designed. Abstract: In this paper, a new passive fault-tolerant controller for 3-DOF helicopter is designed. The 3-DOF helicopter is an underactuated system, which has three degrees of freedom, however, The 3-DOF helicopter is an underactuated system, which has three degrees of freedom, however, Abstract: In this inputs. paper, aHere new passive fault-tolerant controller forto3-DOF helicopter is input designed. only two control two virtual components related only one control are The 3-DOF helicopter is an underactuated system, which has three degrees of freedom, however, only two control inputs. Here two virtual components related to only one control input are The 3-DOF helicopter is an underactuated system, which has three degrees of freedom, however, introduced, which when combined with other control input decouple the 3-DOF helicopter only two control inputs. Here two virtual components related to only one control input are introduced, which when combined with other control input decouple the 3-DOF helicopter only two control inputs. Here two virtual components related to onlyfault-tolerant one 3-DOF control helicopter input are system into three separate subsystems. For each subsystem, a passive controller introduced, which when combined with other control input decouple the system into three separate subsystems. For eachcontrol subsystem, a decouple passive fault-tolerant controller introduced, which when combined with other inputmethod, the 3-DOF helicopter system into three separate subsystems. For each subsystem, a passiveachieving fault-tolerant controller is constructed based on nonsingular terminal sliding mode attitude angle is constructed based on nonsingular terminal sliding mode method, achieving attitude angle system into three separate subsystems. For each subsystem, a passiveachieving fault-tolerant controller tracking. Finally, simulations are conducted to verify the effectiveness of the proposed scheme. is constructed based on nonsingular terminal sliding mode method, attitude angle tracking. Finally, simulations are conducted to sliding verify the effectiveness of the proposed scheme. is constructed based on nonsingular terminal mode method, achieving attitude angle tracking. Finally, simulations are conducted to verify the effectiveness of the proposed scheme. © 2018, IFAC (International Federation of Automatic Control) by Elsevier Ltd.proposed All rights reserved. tracking. Finally, simulations are conducted to verify theHosting effectiveness of the scheme. Keywords: sliding mode mode control, control, actuator actuator Keywords: passive passive fault-tolerant fault-tolerant control, control, 3-DOF 3-DOF helicopter, helicopter, sliding Keywords: passive fault-tolerant control, 3-DOF helicopter, sliding mode control, actuator failure. failure. Keywords: passive fault-tolerant control, 3-DOF helicopter, sliding mode control, actuator failure. failure.1. INTRODUCTION into 1. INTRODUCTION into two two subsystems. subsystems. One One subsystem subsystem hasn’t hasn’t been been affected affected 1. INTRODUCTION by actuator faults. And state hashasn’t been been estimated by into two subsystems. Onethe subsystem affected by actuator faults. And the state has been estimated by 1. INTRODUCTION two order subsystems. One subsystem hasn’t been by actuator faults. And the state hasthe been estimated by reduced Kalman filtering and state of affected another In recent years, helicopters have been widely applied in into reduced order Kalman filtering and the state of another In recent years, helicopters have been widely applied in by actuator faults. And the state has been estimated by subsystem affected by the faults has been measurable. reduced order Kalman filtering and the state of another In recentfields. years,Compared helicopterswith havefixed beenwing widely applied in subsystem affected by the faults has been measurable. various aircraft, helivarious fields. Compared with fixed wing aircraft, helireduced order Kalman filtering and the state of another In recenthave years, helicopters havefixed widely applied in subsystem Obtained state has used to actuator faults, affected by the has been measurable. various fields. Compared with wing aircraft, helicopters the characteristics ofbeen getting up and down state has been been usedfaults to compute compute actuator faults, copters the characteristics of up and down subsystem affected by the faults has been measurable. various have fields. Compared with fixed wingdirection aircraft, heli- Obtained and designed controller has been reconstructed according Obtained state has been used to compute actuator faults, copters have the characteristics ofingetting getting up and freely down vertically, hanging on and flying the and designed controller has been reconstructed according vertically, hanging on and flying in the direction freely Obtained stateIn has been used to compute actuator faults, copters have the characteristics ofingetting up and down to this value. Qin et al. (2017), an active fault tolerant designed controller has been reconstructed according vertically, hanging on and flying themilitary, direction and so on. So it has been widely used in civilfreely and and to this value. In Qin et al. (2017), an active faultaccording tolerant and so on. So it has been widely used in military, civil and and designed controller has been reconstructed vertically, hanging on and flying in the direction freely method has been designed by combining the results to this value. In Qin et al. (2017), an active fault tolerant and so on. So it has been widely used in military, civil and other fields. For example, when the 5.12 Wenchuan earth- method has been designed by combining the results of of other example, when 5.12 Wenchuan earththis value. In Qin et estimation, al. (2017), anand active fault tolerant and sofields. on. SoFor it has widely used in military, civil and to PD control fault the method has method has and been designed by combining the results of other fields. For example, when the 5.12 Wenchuan earthquake happened in been China, the the rescuers and equipments PD control and fault estimation, and the method has quake happened in the rescuers and equipments method has and been designed by combining the results of other fields. For example, when the 5.12 Wenchuan earthbeen applied to quadrotor. Zeghlache et al. (2017) have PD control fault estimation, and the method has quake happened in China, China, the rescuers and equipments couldn’t get to the disaster area quickly because of the appliedand to quadrotor. Zeghlache etthe al. method (2017) have couldn’t get to the disasterthe arearescuers quicklyand because of the been PD control fault estimation, and has quake happened in China, equipments combined interval type-2 fuzzy logic control approach with been applied to quadrotor. Zeghlache et al. (2017) have couldn’t get toAt the disaster quicklyespecially because of the combined interval type-2 fuzzy logic control approach with blocked roads. this point, area helicopters, heavy blocked this point, helicopters, heavy applied to quadrotor. Zeghlache et an al. (2017) faulthave couldn’t roads. getshowed toAt the disaster area quicklyespecially because of the been sliding mode control, and have designed active combined interval type-2 fuzzy logic control approach with blocked roads. At this point, helicopters, especially heavy helicopters, great advantages. sliding mode control, and have designed an active faulthelicopters, showed great advantages. combined interval type-2 fuzzy logic control approach with blocked roads. At this point, helicopters, especially heavy sliding tolerant controller to estimate the state. mode control, and have designed an active faulthelicopters, showed great advantages. controller to estimate state. an active faultAs a typicalshowed complex system, the flight control system of tolerant sliding mode control, and havethe designed helicopters, great advantages. tolerant controller to estimate the state. As aa typical flight system tolerance refers consideration As typical complex system, the flight control control system of of Passive helicopter is complex strongly system, coupled,the nonlinear and multi-input tolerant controller to estimate thethe state. Passive fault fault tolerance refers to to the consideration of of the the helicopter is strongly coupled, nonlinear and multi-input As aoutput(MIMO). typical system, the flight control systemit of types of faults before designing the controller, so Passive fault tolerance refers to the consideration ofthat the helicopter is complex stronglyOnce coupled, nonlinear and multi-input and the helicopters break down, is types of faults before designing the controller, so that and output(MIMO). Once the helicopters break down, it is Passive tolerance refers to the the helicopter is strongly coupled, nonlinear and multi-input offault faults beforebedesigning theconsideration controller, soofthat the controller could not sensitive to the faults, and and output(MIMO). Once the helicopters break down, it is types likely to cause serious consequences. Good fault tolerance the controller could bedesigning not sensitive to the faults, and likely to cause seriousOnce consequences. Goodbreak faultdown, tolerance types of faults before the controller, so that and output(MIMO). the helicopters it is then achieves the purpose of fault tolerance. Chen et the controller could be not sensitive to the faults, and likely to cause consequences. is essential for aserious helicopter controller.Good fault tolerance then achieves the purpose of fault tolerance. Chen et al. al. is essential for helicopter controller. controller could be feedforward notof sensitive to quantum the Chen faults, likely to cause consequences. (2012) have used aa purpose fuzzy and control then achieves the fault tolerance. etand al. is essential for a aserious helicopter controller.Good fault tolerance the (2012) have used fuzzy feedforward and quantum control Fault tolerant control is divided into active fault tolerant then achieves the purpose of fault tolerance. Chen et al. is essential for control a helicopter controller. technology design a feedforward passive fault-tolerant controller, have to used a fuzzy and quantum control Fault tolerant is divided into active fault tolerant (2012) to design a passive fault-tolerant controller, Fault tolerantfault control is divided into(1997)). active fault tolerant and passive tolerant (Patton. Active fault technology (2012) have used a fuzzy feedforward and quantum control technology to design a passive fault-tolerant controller, which can aa small helicopter repair In and fault tolerant (Patton. Active fault Faultpassive tolerant control istodivided into(1997)). active fault tolerant can make make small helicopter repair itself. itself.controller, In Saied Saied and passive fault tolerant (Patton. (1997)). Active fault which tolerance is the ability detect and isolate system faults technology to adesign a passive fault-tolerant et al. (2016), passive fault tolerant control which can make a small helicopter repair itself.algorithm In Saied tolerance is the ability to detect and isolate system faults and passive fault tolerant (Patton. (1997)). Active fault et al. (2016), a passive fault tolerant control algorithm tolerance is the ability to detectparameters and isolateorsystem faults online, readjust the controller change the which can make a small helicopter repair itself. In Saied based on algorithm has been proposed, al. (2016), a passive fault tolerant online, readjust the controller parameters or change the et tolerance isstructure. the ability to et detect and isolate faults on super-twisting super-twisting algorithm hascontrol been algorithm proposed, online, readjust the controller parameters orsystem change the based controller Liu al. (2012) have used the obet al. (2016), a passive fault tolerant control algorithm which can directly aa fault and the on super-twisting algorithm been proposed, controller structure. Liu et al. parameters (2012) haveorused the obonline, to readjust thethe controller change the based which can directly deal deal with with fault has and compensate compensate the controller structure. Liu et al. (2012) the haveinput used the observer estimate fault. Besides, feedback based on loss super-twisting algorithm been proposed, which can directly dealfault with a fault has and compensate the actuator without information. Benrezki et al. server to estimate the fault. Besides, the input feedback controller structure. Liu et al. (2012) have used the ob- actuator loss without fault information. Benrezki et al. server tohas estimate the fault. Besides, the input observer been reconstructed according to the feedback which can directly dealfault withPID a fault and compensate the (2015) have used nonlinear to design a passive faultactuator loss without information. Benrezki et al. observer been reconstructed according the server tohas estimate the(2016) fault. have Besides, the to input (2015) have used nonlinear PID to design Benrezki a passive et faultobserver has been reconstructed according to the feedback feedback information. He et al. designed a fault diagnoactuator loss without fault information. al. tolerant controller, which can achieve the performance of (2015) have used nonlinear PID to design a passive faultinformation. He et al. (2016) have designed a fault diagnoobserver has according the controller, which can achieve the aperformance of information. He etreconstructed al.three (2016) have designedto a fault diagno- tolerant sis module forbeen a real tank. According to the feedback detected (2015) have used nonlinear PID to design passive faultthe actuator in case of failure. tolerant controller, which can achieve the performance of sis module for a real three tank. According to the detected information. He et al. (2016) have designed a fault diagnothe actuator in case of failure. sis module for a real tank. According totothe detected fault, the control law three has been reconstructed achieve the tolerant controller, can achieve the performance of the actuator in casewhich of failure. fault, the control law has been reconstructed to achieve the sis module for a real three tank. According A novel passive fault-tolerant fault, theofcontrol law has been reconstructed achieve the the purpose fault tolerant control. In Jiangtoto etthe al.detected (2005), actuator in case of failure. controller novel passive fault-tolerant controller for for Quanser Quanser 33purpose fault tolerant control. In et al. (2005), fault, theof control has been reconstructed to achieve the A DOF helicopter is proposed. The helicopter, an equipment A novel passive fault-tolerant controller for Quanser 3purpose of fault law tolerant control. In Jiang Jiang et al. divided (2005), a linear MIMO discrete-time system has been DOF helicopter is proposed. The helicopter, an equipment apurpose linear of MIMO discrete-time system has been divided A novel passive fault-tolerant controller for Quanser 3fault tolerant control. In Jiang al. divided (2005), DOF with two motors controlling three degrees of freedom, is helicopter is proposed. The helicopter, an equipment a linear MIMO discrete-time system has et been with two motorsis controlling three degrees of freedom, is DOF helicopter proposed. The helicopter, an equipment a This linear MIMO discrete-time system has been divided an underactuated system. So it’s vital to decouple the with two motors controlling three degrees of freedom, is was supported by Shandong University of Science and an underactuated system. Sothree it’s degrees vital to ofdecouple the This work work was supported by Shandong University of Science and with two motors controlling freedom, is helicopter system. One of the inputs is decoupled into Technology Postgraduate Technology Innovation Project (SDKDYan underactuated system. So it’s vital to decouple the This work was supported by Shandong University of Science and helicopter system. system. One of So the it’s inputs istodecoupled into Technology Postgraduate Technology Innovation Project (SDKDY an underactuated vital decouple the This work was supported by Shandong University of Science and C170354) helicopter system. One of the inputs is decoupled into Technology Postgraduate Technology Innovation Project (SDKDYC170354) helicopter system. One of the inputs is decoupled into Technology C170354) Postgraduate Technology Innovation Project (SDKDY-
C170354) 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 1368Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 1368Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 1368 10.1016/j.ifacol.2018.09.556 Copyright © 2018 IFAC 1368
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two virtual inputs, and then the control law is designed based on a nonsingular terminal sliding mode method respectively. Passive fault tolerant controller is designed to maintain good performance when an actuator fault occurs. The validity of the passive fault-tolerant controller is verified by simulation. The structure of this paper is organized as follows. In section 2, the model of 3-DOF helicopter is given. Passive fault-tolerant control scheme and a new passive faulttolerant controller is proposed in section 3. Finally, the validity of the design is verified by simulation in section 4.
2.2 Dynamic modeling In the modeling process of three degree of freedom helicopter control system, the ground coordinate system is used to analyze the force and torque of the system. Fig. 2 is a simplified schematic diagram of a semi physical simulation platform for three degree of freedom helicopter. In order to simplify the modeling process and results, some secondary factors are neglected and some restrictions and assumptions will be added in the process of modeling: • The mechanical part of a helicopter system is a rigid body that does not deform elastically. • The delay of the digital and analog signal in the system is not considered, besides the helicopter control system is undelayed. • Ignore the motor inertia, and the relationship between the voltage of the motor and the thrust of the motor is a linear proportion. • Neglect the air resistance at a low speed, the motion friction of three rotating axis and the gyroscopic effect caused by rotor rotation.
2. MODEL DESCRIPTION 2.1 Helicopter description The plant of this paper is Quanser three degrees of freedom double rotor helicopter hardware in the loop simulation platform (Apkarian et al. (2006)). As shown in Fig. 1, a universal joint is attached to the base and the frame arm. It allows the frame arm to move freely around the elevation axis and the travel axis. On the end of the frame arm is a helicopter body, which is composed of a rectangular frame hanging at the front end of the frame arm and two propellers mounted on the rectangular frame. The two propellers are driven by two DC motors respectively. The axis of the two motors are parallel and the thrust is perpendicular to the frame. The other end of the frame arm is equipped with a counterweight, which is used to reduce the motor energy required for the hovering of the helicopter body, so that the helicopter body can be able to rise with a small voltage. Two motors drive the propellers to produce lift, simulating the pitching, elevation and traveling of the helicopter, which represents at p , ξ and λ respectively.
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According to the simplified model diagram, in the established ground coordinate system, the dynamic equations of three rotating axis can be established respectively. If the lift thrust of the front and the back propeller vary, the helicopter body will tilt around the pitch axis. Then the dynamic equation of the pitch axis is Jp p¨ = (Ff − Fb )Lh whereas Lh is the distance between each propeller and pitch axis; Jp is the moment of inertia of the pitch axis, Jp = Mh L2h . Mh is the mass of helicopter body. Assuming the helicopter hovers in the air, the lift thrusts of the two propellers are Ff and Fb , respectively. And the change of the thrusts makes the helicopter body pitch. When the torque provided by the vertical ground component Fm = (Ff + Fb )cosp are greater than the equivalent gravity Tg of the helicopter body, the helicopter body rises along the elevation axis. Conversely, the helicopter body will descend. Then the dynamic equation of the elevation axis is
Fig. 1. 3-DOF Helicopter practicality picture (Apkarian et al. (2006)) Due to the restriction of the mechanical structure, the pitch and elevation angles are −45deg ≤ p ≤ +45deg and −27.5deg ≤ ξ ≤ +30deg respectively. Two motors of the helicopter drive two propellers to produce a thrust, and the thrust is approximately proportional to the applied voltage. Assuming Kf is force-thrust constant. The thrusts produced by the motors are described as follows Ff Kf Vf = (1) Fb Kf Vb
Fig. 2. 3-DOF Helicopter simplified schematic diagram. (Apkarian et al. (2006))
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Je ξ¨ = (Ff + Fb )La cosp − Tg cosξ whereas La is the distance between the elevation axis and the helicopter body. Tg is the equivalent gravity moment of the pitch axis, Tg = Mh gLa −Mw gLw . Je is the moment of inertia of the elevation axis, Je = Mh L2a + Mw L2w . Lw is the distance from the elevation axis to the counterweight, Mw is the mass of the counterweight. When the helicopter body rotates around the pitch axis, it will drive the propeller fixed on the helicopter body to tilt to a certain angle. At this moment, the thrusts of the propellers will produce a horizontal component to push the helicopter body to rotate. The equation of the travel axis is as follows ¨ = (Ff + Fb )La cosξsinp Jt λ whereas Jt is the moment of inertia of the travel axis, Jt = Mh L2a + Mh L2h + Mw L2w . In summary, the dynamic model of the 3-DOF helicopter can be obtained as follows Je ξ¨ = u1 La cosp − Tg cosξ Jp p¨ = u2 Lh (2) ¨ = u1 La cosξsinp Jt λ where two control inputs are defined as u1 F + Fb = f u2 F f − Fb
Elevation Elevation & & Travel Travel PFTC PFTC Controller Controller
Desired Desired trajectory trajectory
3-DOF 3-DOF Helicopter Helicopter
Fig. 3. Passive fault tolerant scheme Tg cosξ La w2 = u1 cosξsinp
w1 = u1 cosp −
then it can get u1 .
(
w2 2 Tg cosξ 2 ) + (w1 + ) cosξ La
where S = sgn(w1 +
combine (1) with (3), the control voltage of front and back motor is obtained as u + u 1 2 Vf 2Kf = u1 − u2 V b
2Kf
3. PASSIVE FAULT-TOLERANT CONTROLLER
(5)
deform (5) and square can be shown as follows Tg cosξ 2 ) (u1 cosp)2 = (w1 + La w2 2 ) (u1 sinp)2 = ( cosξ
u1 = S
(3)
Pitch Pitch PFTC PFTC Controller Controller
Pitch Pitch & & Control Control Input Input
u1 =
T cosξ w1 + gLa
cosp
Tg cosξ La ).
(6)
Note that cosp > 0 in
because of the hardware restriction.
According to (5), it can be obtained w2 tanp = T cosξ cosξ(w1 + gLa )
(7)
Donate w1d and w2d as the command signals of w1 and w2 respectively, then from (7) it follow that
3.1 Passive fault tolerant scheme Let ξd , pd and λd as the desired angle of pitch, elevation angle and travel, respectively. The object of this paper is to design u1 and u2 such that ξ → ξd , p → pd and λ → λd in finite time. Assuming that the actuator fault occurs, the missing part of the actuator is a constant. Therefore, the remaining actuators are controlled by subtracting the lost energy from the normal actuator control. ff and fb represent the missing control power of front and back actuator control input respectively. It is concluded that the input equation for actuator faults is as follows u1 + fˆ1 Ff + Fb + fˆ1 = (4) u2 + fˆ2 Ff − Fb + fˆ2 whereas fˆ1 = −ff − fb , fˆ2 = −ff + fb .
So the system model when the fault occurs can be shown as follows Je ξ¨ = (u1 + fˆ1 )La cosp − Tg cosξ Jp p¨ = (u2 + fˆ2 )Lh ¨ = (u1 + fˆ1 )La cosξsinp Jt λ
Because the system is an underactuated system, two control voltages(Ff , Fb ) need to control three attitude angles (ξ , p and λ). So two virtual inputs are introduced
pd = tan−1 (
w2d
cosξ(w1d + which is the tracking signal of p.
Tg cosξ La )
)
(8)
The whole passive fault tolerant scheme is divided into the following part: (see figure 3) • design w1d and w2d , such that ξ → ξd and p → pd when the actuator fails. • From w1d and w2d , u1 and pd can be obtained accoding to (6) and (8) respectively. • Design u2 so that the p → pd . 3.2 Design passive fault-tolerant controller Define the tracking error e as follows eξ ξ − ξd e = e p = p − pd eλ λ − λd
where pd is from (8).
Tracking error rate follows that ˙ ˙ e˙ ξ ξ − ξd e˙ = e˙ p = p˙ − p˙ d e˙ λ λ˙ − λ˙ d
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(9)
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and the sliding surface is defined as 1 m /n eξ + e˙ ξ ξ ξ βξ sξ 1 mp /np s = sp = ep + e˙ p (10) β p sλ 1 mλ /nλ eλ + e˙ βλ λ with β∗ > 0 is a design constant. m∗ and n∗ are positive odd integers.(∗ being ξ, p and λ) The control law of the design is as follows Je nξ 2−mξ /nξ − (βξ e˙ ξ + (ηξ + lf,ξ )sgn(sξ ) − ξ¨d ) La mξ J w1d np 2−mp /np p u2 = − (βp e˙ p + (ηp + lf,p )sgn(sp ) − p¨d ) L m h p w2d J nλ 2−mλ /nλ t ¨ e˙ λ + (ηλ + lf,λ )sgn(sλ ) − λd ) − (βλ La mλ (11) Fig. 4. Elevation angle ξ (dotted line)(deg) and desired angle ξd (solid line)(deg) versus time (sec) La ˆ with 1 < m∗ /n∗ < 2 , η∗ > 0, Je f1 cosp ≤ lf,ξ , Rewrite (19) into (22), it follows that Lh fˆ ≤ l , La cosξsinp fˆ ≤ l , l > 0. Jp
2
f,p
Jt
1
f,λ
f,∗
nξ +mξ
Theorem 1. When the actuator fails, the designed controller (11) can still track the desired trajectory.
Proof of Theorem 1: The proof is divided into two steps: the first step is proved to be sξ → 0 in finite time; the second step is proved to be eξ → 0 in finite time.
Constructing an Lyapunov function 1 V1 = s2ξ 2 Differentiate (12) with respect to time, then V˙ 1 = sξ s˙ ξ
(12) (13)
and differentiate sξ with respect to time, then 1 mξ mξ /nξ −1 s˙ ξ = e˙ ξ + e˙ e¨ξ (14) β ξ nξ ξ Differentiate the first equation of (9) and substituting in w1d , it follows that nξ 2−mξ /nξ e˙ + (ηξ + lf,ξ )sgn(sξ )) (15) e¨ξ = −(βξ mξ ξ Substituting (15) into (14), it follows that s˙ ξ = −(ηξ + lf,ξ )sgn(sξ ) (16) So it follows that V˙ 1 = sξ s˙ ξ = −(ηξ + lf,ξ )|sξ | < 0 (17)
V˙ 2 = (−βξ )nξ /mξ (2V2 ) 2mξ (23) Since V2 is positive definite, q and p are positive odd integers, V˙ 2 is negative definite. The same principle proves u2 and w2 . So far the proof of Theorem 1 is completed. 4. SIMULATION In this section, simulation results demonstrate the effectiveness of the designed passive fault-tolerant controller when the actuator fails. The designed passive fault-tolerant controllers parameters (see (11)) are shown in Table 1. Besides, helicopter hardware parameters are shown in Table 2. Table 1. PFTC Controller parameter simulation parameter β∗ m∗ n∗ η∗
symbol Vf , V b
ξ
Kf g Mh Mw Lw La Lh
(21) (22)
pitch controller 0.1 5 3 0.5
travel controller 0.4 5 3 0.2
Table 2. Helicopter hardware specifications (Apkarian et al. (2006))
When sξ = 0 , the first equation of (10) can be rewritten as 1 m /n eξ = − e˙ ξ ξ ξ (18) βξ Define the Lyapunov function as follows 1 V2 = e2ξ (19) 2 Differentiate (19) with respect to time, then (20) V˙ 2 = eξ e˙ ξ and (18) can be rewritten as follows e˙ ξ = (−βξ eξ )nξ /mξ Substituting (21) into (20), it follows that (n +m )/m V˙ 2 = (−βξ )nξ /mξ e ξ ξ ξ
elevation controller 0.4 5 3 0.5
description the front and back motors voltage propeller force-thrust constant gravity acceleration the mass of the helicopter the mass of the counterweight distance between elevation axis to the counterweight distance between elevation axis to helicopter distance between pitch axis to each motor
value [-24,+24]
unit V
0.1188
N/V
9.81 1.15
m/s2 kg
1.87
kg
0.47
m
0.66
m
0.178
m
Figure 4-6 display designed passive fault-tolerant controller is still able to track the desired trajectory when
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Simulation results of the designed passive fault tolerant controller verify that the passive fault-tolerant controller can still track the specified target when actuator failure occurs. 5. CONCLUSION In this paper, a new design method of passive faulttolerant controller is proposed for a helicopter system. The helicopter is a dual input three output system, and one of the inputs is decoupled into two virtual inputs. Two virtual input and another input are controlled based on nonlinear terminal sliding mode method, respectively. When the actuator fails, the designed passive fault-tolerant controller can still maintenance the helicopter better performance. The validity of the designed passive fault-tolerant controller is verified by simulation. Fig. 5. Pitch angle p (dotted line)(deg) and desired angle pd (solid line)(deg) versus time (sec)
REFERENCES
Fig. 6. Travel angle λ (dotted line)(deg) and desired angle λd (solid line)(deg) versus time (sec)
J. Apkarian. 3-DOF helicopter reference manual. Quanser Consulting Inc, Canada. 2006. R. R. Benrezki, M. Tadjine, F. Yacef, and O. Kermia. Passive fault tolerant control of quadrotor UAV using a nonlinear PID. In Robotics and Biomimetics (ROBIO), 2015 IEEE International Conference on (pp. 1285-1290). IEEE, 2015. F. Chen, B. Jiang, and G. Tao. Fault self-repairing flight control of a small helicopter via fuzzy feedforward and quantum control techniques. Cognitive Computation, 4(4):543-548, 2012. B. Jiang, and F. N. Chowdhury. Fault estimation and accommodation for linear MIMO discrete-time systems. IEEE Transactions on Control Systems Technology, 13(3):493-499, 2005. X. He, Z. Wang, L. Qin, and D. Zhou. Active faulttolerant control for an internet-based networked threetank system. IEEE Transactions on Control Systems Technology, 24(6):2150-2157, 2016. L. Liu, Y. Shen, and E. H. Dowell. Integrated adaptive fault-tolerant H∞ output feedback control with adaptive fault identification. Journal of Guidance Control and Dynamics, 35(3):881, 2012. R. J. Patton. Fault-tolerant control: the 1997 situation. IFAC Proceedings Volumes, 30(18):1029-1051, 1997. L. Qin, X. He, R. Yan, and D. Zhou. Active Fault-Tolerant Control for a Quadrotor with Sensor Faults. Journal of Intelligent & Robotic Systems, 1-19, 2017. M. Saied, B. Lussier, I. Fantoni, H. Shraim, and C. Francis. Passive fault-tolerant control of an octorotor using super-twisting algorithm: Theory and experiments. In Control and Fault-Tolerant Systems (SysTol), 2016 3rd Conference on (pp. 361-366). IEEE, 2016. S. Zeghlache, T. Benslimane, and A. Bouguerra. Active fault tolerant control based on interval type-2 fuzzy sliding mode controller and non linear adaptive observer for 3-DOF laboratory helicopter. ISA transactions, 71: 280-303, 2017.
Fig. 7. Voltage of front Vf (solid line)(V) and back motor voltage Vb (dotted line)(V) versus time (sec) actuator failure occurs in 40s. Figure 7 displays the the control voltage of the front and back motor when the fault occurs. 1372
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IFAC PapersOnLine 51-24 (2018) 1368–1372
Nonsingular Terminal Sliding Mode Based Nonsingular Terminal Sliding Mode Based Nonsingular Terminal Sliding Mode Based Passive Fault-Tolerant Control of a 3-DOF Nonsingular Terminal Sliding Mode Based Passive Fault-Tolerant Control of a 3-DOF Passive Fault-Tolerant Control of a 3-DOF Helicopter System Passive Fault-Tolerant Control of a 3-DOF Helicopter Helicopter System System Helicopter System ∗∗ YANG Qiwei ∗∗ YANG Rui ∗∗
YANG Qiwei ∗∗ YANG Rui ∗∗ YANG Qiwei ∗ YANG Rui ∗∗ YANGEngineering Qiwei YANG Rui ∗∗ ∗ of Electrical and Automation, Shandong ∗ ∗ College College of Electrical Engineering and Automation, Shandong ∗ University of Science and Technology, Qingdao 266590, China. College of Electrical Engineering and Automation, Shandong ∗ University of Science and Technology, Qingdao 266590, China. College ofofElectrical Engineering andQingdao Automation, Shandong (e-mail: qiwei [email protected]) University Science and Technology, 266590, China. (e-mail: qiwei [email protected]) ∗∗ University of Science and Technology, Qingdao 266590, China. Engineering and Automation, Shandong (e-mail: qiwei [email protected]) ∗∗ College of Electrical ∗∗ Electrical Engineering and Automation, Shandong ∗∗ College of (e-mail: qiwei [email protected]) University of Science and Technology, Qingdao 266590, China. College of Electrical Engineering and Automation, Shandong ∗∗ University Science and Technology, 266590, China. College ofof Engineering andQingdao Automation, Shandong (e-mail: [email protected]) University of Electrical Science and Technology, Qingdao 266590, China. (e-mail: [email protected]) University of Science and Technology, Qingdao 266590, China. (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Abstract: In In this this paper, paper, a a new new passive passive fault-tolerant fault-tolerant controller controller for for 3-DOF 3-DOF helicopter helicopter is is designed. designed. Abstract: In this paper, a new passive fault-tolerant controller for 3-DOF helicopter is designed. The 3-DOF helicopter is an underactuated system, which has three degrees of freedom, however, The 3-DOF helicopter is an underactuated system, which has three degrees of freedom, however, Abstract: In this inputs. paper, aHere new passive fault-tolerant controller forto3-DOF helicopter is input designed. only two control two virtual components related only one control are The 3-DOF helicopter is an underactuated system, which has three degrees of freedom, however, only two control inputs. Here two virtual components related to only one control input are The 3-DOF helicopter is an underactuated system, which has three degrees of freedom, however, introduced, which when combined with other control input decouple the 3-DOF helicopter only two control inputs. Here two virtual components related to only one control input are introduced, which when combined with other control input decouple the 3-DOF helicopter only two control inputs. Here two virtual components related to onlyfault-tolerant one 3-DOF control helicopter input are system into three separate subsystems. For each subsystem, a passive controller introduced, which when combined with other control input decouple the system into three separate subsystems. For eachcontrol subsystem, a decouple passive fault-tolerant controller introduced, which when combined with other inputmethod, the 3-DOF helicopter system into three separate subsystems. For each subsystem, a passiveachieving fault-tolerant controller is constructed based on nonsingular terminal sliding mode attitude angle is constructed based on nonsingular terminal sliding mode method, achieving attitude angle system into three separate subsystems. For each subsystem, a passiveachieving fault-tolerant controller tracking. Finally, simulations are conducted to verify the effectiveness of the proposed scheme. is constructed based on nonsingular terminal sliding mode method, attitude angle tracking. Finally, simulations are conducted to sliding verify the effectiveness of the proposed scheme. is constructed based on nonsingular terminal mode method, achieving attitude angle tracking. Finally, simulations are conducted to verify the effectiveness of the proposed scheme. © 2018, IFAC (International Federation of Automatic Control) by Elsevier Ltd.proposed All rights reserved. tracking. Finally, simulations are conducted to verify theHosting effectiveness of the scheme. Keywords: sliding mode mode control, control, actuator actuator Keywords: passive passive fault-tolerant fault-tolerant control, control, 3-DOF 3-DOF helicopter, helicopter, sliding Keywords: passive fault-tolerant control, 3-DOF helicopter, sliding mode control, actuator failure. failure. Keywords: passive fault-tolerant control, 3-DOF helicopter, sliding mode control, actuator failure. failure.1. INTRODUCTION into 1. INTRODUCTION into two two subsystems. subsystems. One One subsystem subsystem hasn’t hasn’t been been affected affected 1. INTRODUCTION by actuator faults. And state hashasn’t been been estimated by into two subsystems. Onethe subsystem affected by actuator faults. And the state has been estimated by 1. INTRODUCTION two order subsystems. One subsystem hasn’t been by actuator faults. And the state hasthe been estimated by reduced Kalman filtering and state of affected another In recent years, helicopters have been widely applied in into reduced order Kalman filtering and the state of another In recent years, helicopters have been widely applied in by actuator faults. And the state has been estimated by subsystem affected by the faults has been measurable. reduced order Kalman filtering and the state of another In recentfields. years,Compared helicopterswith havefixed beenwing widely applied in subsystem affected by the faults has been measurable. various aircraft, helivarious fields. Compared with fixed wing aircraft, helireduced order Kalman filtering and the state of another In recenthave years, helicopters havefixed widely applied in subsystem Obtained state has used to actuator faults, affected by the has been measurable. various fields. Compared with wing aircraft, helicopters the characteristics ofbeen getting up and down state has been been usedfaults to compute compute actuator faults, copters the characteristics of up and down subsystem affected by the faults has been measurable. various have fields. Compared with fixed wingdirection aircraft, heli- Obtained and designed controller has been reconstructed according Obtained state has been used to compute actuator faults, copters have the characteristics ofingetting getting up and freely down vertically, hanging on and flying the and designed controller has been reconstructed according vertically, hanging on and flying in the direction freely Obtained stateIn has been used to compute actuator faults, copters have the characteristics ofingetting up and down to this value. Qin et al. (2017), an active fault tolerant designed controller has been reconstructed according vertically, hanging on and flying themilitary, direction and so on. So it has been widely used in civilfreely and and to this value. In Qin et al. (2017), an active faultaccording tolerant and so on. So it has been widely used in military, civil and and designed controller has been reconstructed vertically, hanging on and flying in the direction freely method has been designed by combining the results to this value. In Qin et al. (2017), an active fault tolerant and so on. So it has been widely used in military, civil and other fields. For example, when the 5.12 Wenchuan earth- method has been designed by combining the results of of other example, when 5.12 Wenchuan earththis value. In Qin et estimation, al. (2017), anand active fault tolerant and sofields. on. SoFor it has widely used in military, civil and to PD control fault the method has method has and been designed by combining the results of other fields. For example, when the 5.12 Wenchuan earthquake happened in been China, the the rescuers and equipments PD control and fault estimation, and the method has quake happened in the rescuers and equipments method has and been designed by combining the results of other fields. For example, when the 5.12 Wenchuan earthbeen applied to quadrotor. Zeghlache et al. (2017) have PD control fault estimation, and the method has quake happened in China, China, the rescuers and equipments couldn’t get to the disaster area quickly because of the appliedand to quadrotor. Zeghlache etthe al. method (2017) have couldn’t get to the disasterthe arearescuers quicklyand because of the been PD control fault estimation, and has quake happened in China, equipments combined interval type-2 fuzzy logic control approach with been applied to quadrotor. Zeghlache et al. (2017) have couldn’t get toAt the disaster quicklyespecially because of the combined interval type-2 fuzzy logic control approach with blocked roads. this point, area helicopters, heavy blocked this point, helicopters, heavy applied to quadrotor. Zeghlache et an al. (2017) faulthave couldn’t roads. getshowed toAt the disaster area quicklyespecially because of the been sliding mode control, and have designed active combined interval type-2 fuzzy logic control approach with blocked roads. At this point, helicopters, especially heavy helicopters, great advantages. sliding mode control, and have designed an active faulthelicopters, showed great advantages. combined interval type-2 fuzzy logic control approach with blocked roads. At this point, helicopters, especially heavy sliding tolerant controller to estimate the state. mode control, and have designed an active faulthelicopters, showed great advantages. controller to estimate state. an active faultAs a typicalshowed complex system, the flight control system of tolerant sliding mode control, and havethe designed helicopters, great advantages. tolerant controller to estimate the state. As aa typical flight system tolerance refers consideration As typical complex system, the flight control control system of of Passive helicopter is complex strongly system, coupled,the nonlinear and multi-input tolerant controller to estimate thethe state. Passive fault fault tolerance refers to to the consideration of of the the helicopter is strongly coupled, nonlinear and multi-input As aoutput(MIMO). typical system, the flight control systemit of types of faults before designing the controller, so Passive fault tolerance refers to the consideration ofthat the helicopter is complex stronglyOnce coupled, nonlinear and multi-input and the helicopters break down, is types of faults before designing the controller, so that and output(MIMO). Once the helicopters break down, it is Passive tolerance refers to the the helicopter is strongly coupled, nonlinear and multi-input offault faults beforebedesigning theconsideration controller, soofthat the controller could not sensitive to the faults, and and output(MIMO). Once the helicopters break down, it is types likely to cause serious consequences. Good fault tolerance the controller could bedesigning not sensitive to the faults, and likely to cause seriousOnce consequences. Goodbreak faultdown, tolerance types of faults before the controller, so that and output(MIMO). the helicopters it is then achieves the purpose of fault tolerance. Chen et the controller could be not sensitive to the faults, and likely to cause consequences. is essential for aserious helicopter controller.Good fault tolerance then achieves the purpose of fault tolerance. Chen et al. al. is essential for helicopter controller. controller could be feedforward notof sensitive to quantum the Chen faults, likely to cause consequences. (2012) have used aa purpose fuzzy and control then achieves the fault tolerance. etand al. is essential for a aserious helicopter controller.Good fault tolerance the (2012) have used fuzzy feedforward and quantum control Fault tolerant control is divided into active fault tolerant then achieves the purpose of fault tolerance. Chen et al. is essential for control a helicopter controller. technology design a feedforward passive fault-tolerant controller, have to used a fuzzy and quantum control Fault tolerant is divided into active fault tolerant (2012) to design a passive fault-tolerant controller, Fault tolerantfault control is divided into(1997)). active fault tolerant and passive tolerant (Patton. Active fault technology (2012) have used a fuzzy feedforward and quantum control technology to design a passive fault-tolerant controller, which can aa small helicopter repair In and fault tolerant (Patton. Active fault Faultpassive tolerant control istodivided into(1997)). active fault tolerant can make make small helicopter repair itself. itself.controller, In Saied Saied and passive fault tolerant (Patton. (1997)). Active fault which tolerance is the ability detect and isolate system faults technology to adesign a passive fault-tolerant et al. (2016), passive fault tolerant control which can make a small helicopter repair itself.algorithm In Saied tolerance is the ability to detect and isolate system faults and passive fault tolerant (Patton. (1997)). Active fault et al. (2016), a passive fault tolerant control algorithm tolerance is the ability to detectparameters and isolateorsystem faults online, readjust the controller change the which can make a small helicopter repair itself. In Saied based on algorithm has been proposed, al. (2016), a passive fault tolerant online, readjust the controller parameters or change the et tolerance isstructure. the ability to et detect and isolate faults on super-twisting super-twisting algorithm hascontrol been algorithm proposed, online, readjust the controller parameters orsystem change the based controller Liu al. (2012) have used the obet al. (2016), a passive fault tolerant control algorithm which can directly aa fault and the on super-twisting algorithm been proposed, controller structure. Liu et al. parameters (2012) haveorused the obonline, to readjust thethe controller change the based which can directly deal deal with with fault has and compensate compensate the controller structure. Liu et al. (2012) the haveinput used the observer estimate fault. Besides, feedback based on loss super-twisting algorithm been proposed, which can directly dealfault with a fault has and compensate the actuator without information. Benrezki et al. server to estimate the fault. Besides, the input feedback controller structure. Liu et al. (2012) have used the ob- actuator loss without fault information. Benrezki et al. server tohas estimate the fault. Besides, the input observer been reconstructed according to the feedback which can directly dealfault withPID a fault and compensate the (2015) have used nonlinear to design a passive faultactuator loss without information. Benrezki et al. observer been reconstructed according the server tohas estimate the(2016) fault. have Besides, the to input (2015) have used nonlinear PID to design Benrezki a passive et faultobserver has been reconstructed according to the feedback feedback information. He et al. designed a fault diagnoactuator loss without fault information. al. tolerant controller, which can achieve the performance of (2015) have used nonlinear PID to design a passive faultinformation. He et al. (2016) have designed a fault diagnoobserver has according the controller, which can achieve the aperformance of information. He etreconstructed al.three (2016) have designedto a fault diagno- tolerant sis module forbeen a real tank. According to the feedback detected (2015) have used nonlinear PID to design passive faultthe actuator in case of failure. tolerant controller, which can achieve the performance of sis module for a real three tank. According to the detected information. He et al. (2016) have designed a fault diagnothe actuator in case of failure. sis module for a real tank. According totothe detected fault, the control law three has been reconstructed achieve the tolerant controller, can achieve the performance of the actuator in casewhich of failure. fault, the control law has been reconstructed to achieve the sis module for a real three tank. According A novel passive fault-tolerant fault, theofcontrol law has been reconstructed achieve the the purpose fault tolerant control. In Jiangtoto etthe al.detected (2005), actuator in case of failure. controller novel passive fault-tolerant controller for for Quanser Quanser 33purpose fault tolerant control. In et al. (2005), fault, theof control has been reconstructed to achieve the A DOF helicopter is proposed. The helicopter, an equipment A novel passive fault-tolerant controller for Quanser 3purpose of fault law tolerant control. In Jiang Jiang et al. divided (2005), a linear MIMO discrete-time system has been DOF helicopter is proposed. The helicopter, an equipment apurpose linear of MIMO discrete-time system has been divided A novel passive fault-tolerant controller for Quanser 3fault tolerant control. In Jiang al. divided (2005), DOF with two motors controlling three degrees of freedom, is helicopter is proposed. The helicopter, an equipment a linear MIMO discrete-time system has et been with two motorsis controlling three degrees of freedom, is DOF helicopter proposed. The helicopter, an equipment a This linear MIMO discrete-time system has been divided an underactuated system. So it’s vital to decouple the with two motors controlling three degrees of freedom, is was supported by Shandong University of Science and an underactuated system. Sothree it’s degrees vital to ofdecouple the This work work was supported by Shandong University of Science and with two motors controlling freedom, is helicopter system. One of the inputs is decoupled into Technology Postgraduate Technology Innovation Project (SDKDYan underactuated system. So it’s vital to decouple the This work was supported by Shandong University of Science and helicopter system. system. One of So the it’s inputs istodecoupled into Technology Postgraduate Technology Innovation Project (SDKDY an underactuated vital decouple the This work was supported by Shandong University of Science and C170354) helicopter system. One of the inputs is decoupled into Technology Postgraduate Technology Innovation Project (SDKDYC170354) helicopter system. One of the inputs is decoupled into Technology C170354) Postgraduate Technology Innovation Project (SDKDY-
C170354) 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 1368Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 1368Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 1368 10.1016/j.ifacol.2018.09.556 Copyright © 2018 IFAC 1368
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two virtual inputs, and then the control law is designed based on a nonsingular terminal sliding mode method respectively. Passive fault tolerant controller is designed to maintain good performance when an actuator fault occurs. The validity of the passive fault-tolerant controller is verified by simulation. The structure of this paper is organized as follows. In section 2, the model of 3-DOF helicopter is given. Passive fault-tolerant control scheme and a new passive faulttolerant controller is proposed in section 3. Finally, the validity of the design is verified by simulation in section 4.
2.2 Dynamic modeling In the modeling process of three degree of freedom helicopter control system, the ground coordinate system is used to analyze the force and torque of the system. Fig. 2 is a simplified schematic diagram of a semi physical simulation platform for three degree of freedom helicopter. In order to simplify the modeling process and results, some secondary factors are neglected and some restrictions and assumptions will be added in the process of modeling: • The mechanical part of a helicopter system is a rigid body that does not deform elastically. • The delay of the digital and analog signal in the system is not considered, besides the helicopter control system is undelayed. • Ignore the motor inertia, and the relationship between the voltage of the motor and the thrust of the motor is a linear proportion. • Neglect the air resistance at a low speed, the motion friction of three rotating axis and the gyroscopic effect caused by rotor rotation.
2. MODEL DESCRIPTION 2.1 Helicopter description The plant of this paper is Quanser three degrees of freedom double rotor helicopter hardware in the loop simulation platform (Apkarian et al. (2006)). As shown in Fig. 1, a universal joint is attached to the base and the frame arm. It allows the frame arm to move freely around the elevation axis and the travel axis. On the end of the frame arm is a helicopter body, which is composed of a rectangular frame hanging at the front end of the frame arm and two propellers mounted on the rectangular frame. The two propellers are driven by two DC motors respectively. The axis of the two motors are parallel and the thrust is perpendicular to the frame. The other end of the frame arm is equipped with a counterweight, which is used to reduce the motor energy required for the hovering of the helicopter body, so that the helicopter body can be able to rise with a small voltage. Two motors drive the propellers to produce lift, simulating the pitching, elevation and traveling of the helicopter, which represents at p , ξ and λ respectively.
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According to the simplified model diagram, in the established ground coordinate system, the dynamic equations of three rotating axis can be established respectively. If the lift thrust of the front and the back propeller vary, the helicopter body will tilt around the pitch axis. Then the dynamic equation of the pitch axis is Jp p¨ = (Ff − Fb )Lh whereas Lh is the distance between each propeller and pitch axis; Jp is the moment of inertia of the pitch axis, Jp = Mh L2h . Mh is the mass of helicopter body. Assuming the helicopter hovers in the air, the lift thrusts of the two propellers are Ff and Fb , respectively. And the change of the thrusts makes the helicopter body pitch. When the torque provided by the vertical ground component Fm = (Ff + Fb )cosp are greater than the equivalent gravity Tg of the helicopter body, the helicopter body rises along the elevation axis. Conversely, the helicopter body will descend. Then the dynamic equation of the elevation axis is
Fig. 1. 3-DOF Helicopter practicality picture (Apkarian et al. (2006)) Due to the restriction of the mechanical structure, the pitch and elevation angles are −45deg ≤ p ≤ +45deg and −27.5deg ≤ ξ ≤ +30deg respectively. Two motors of the helicopter drive two propellers to produce a thrust, and the thrust is approximately proportional to the applied voltage. Assuming Kf is force-thrust constant. The thrusts produced by the motors are described as follows Ff Kf Vf = (1) Fb Kf Vb
Fig. 2. 3-DOF Helicopter simplified schematic diagram. (Apkarian et al. (2006))
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Je ξ¨ = (Ff + Fb )La cosp − Tg cosξ whereas La is the distance between the elevation axis and the helicopter body. Tg is the equivalent gravity moment of the pitch axis, Tg = Mh gLa −Mw gLw . Je is the moment of inertia of the elevation axis, Je = Mh L2a + Mw L2w . Lw is the distance from the elevation axis to the counterweight, Mw is the mass of the counterweight. When the helicopter body rotates around the pitch axis, it will drive the propeller fixed on the helicopter body to tilt to a certain angle. At this moment, the thrusts of the propellers will produce a horizontal component to push the helicopter body to rotate. The equation of the travel axis is as follows ¨ = (Ff + Fb )La cosξsinp Jt λ whereas Jt is the moment of inertia of the travel axis, Jt = Mh L2a + Mh L2h + Mw L2w . In summary, the dynamic model of the 3-DOF helicopter can be obtained as follows Je ξ¨ = u1 La cosp − Tg cosξ Jp p¨ = u2 Lh (2) ¨ = u1 La cosξsinp Jt λ where two control inputs are defined as u1 F + Fb = f u2 F f − Fb
Elevation Elevation & & Travel Travel PFTC PFTC Controller Controller
Desired Desired trajectory trajectory
3-DOF 3-DOF Helicopter Helicopter
Fig. 3. Passive fault tolerant scheme Tg cosξ La w2 = u1 cosξsinp
w1 = u1 cosp −
then it can get u1 .
(
w2 2 Tg cosξ 2 ) + (w1 + ) cosξ La
where S = sgn(w1 +
combine (1) with (3), the control voltage of front and back motor is obtained as u + u 1 2 Vf 2Kf = u1 − u2 V b
2Kf
3. PASSIVE FAULT-TOLERANT CONTROLLER
(5)
deform (5) and square can be shown as follows Tg cosξ 2 ) (u1 cosp)2 = (w1 + La w2 2 ) (u1 sinp)2 = ( cosξ
u1 = S
(3)
Pitch Pitch PFTC PFTC Controller Controller
Pitch Pitch & & Control Control Input Input
u1 =
T cosξ w1 + gLa
cosp
Tg cosξ La ).
(6)
Note that cosp > 0 in
because of the hardware restriction.
According to (5), it can be obtained w2 tanp = T cosξ cosξ(w1 + gLa )
(7)
Donate w1d and w2d as the command signals of w1 and w2 respectively, then from (7) it follow that
3.1 Passive fault tolerant scheme Let ξd , pd and λd as the desired angle of pitch, elevation angle and travel, respectively. The object of this paper is to design u1 and u2 such that ξ → ξd , p → pd and λ → λd in finite time. Assuming that the actuator fault occurs, the missing part of the actuator is a constant. Therefore, the remaining actuators are controlled by subtracting the lost energy from the normal actuator control. ff and fb represent the missing control power of front and back actuator control input respectively. It is concluded that the input equation for actuator faults is as follows u1 + fˆ1 Ff + Fb + fˆ1 = (4) u2 + fˆ2 Ff − Fb + fˆ2 whereas fˆ1 = −ff − fb , fˆ2 = −ff + fb .
So the system model when the fault occurs can be shown as follows Je ξ¨ = (u1 + fˆ1 )La cosp − Tg cosξ Jp p¨ = (u2 + fˆ2 )Lh ¨ = (u1 + fˆ1 )La cosξsinp Jt λ
Because the system is an underactuated system, two control voltages(Ff , Fb ) need to control three attitude angles (ξ , p and λ). So two virtual inputs are introduced
pd = tan−1 (
w2d
cosξ(w1d + which is the tracking signal of p.
Tg cosξ La )
)
(8)
The whole passive fault tolerant scheme is divided into the following part: (see figure 3) • design w1d and w2d , such that ξ → ξd and p → pd when the actuator fails. • From w1d and w2d , u1 and pd can be obtained accoding to (6) and (8) respectively. • Design u2 so that the p → pd . 3.2 Design passive fault-tolerant controller Define the tracking error e as follows eξ ξ − ξd e = e p = p − pd eλ λ − λd
where pd is from (8).
Tracking error rate follows that ˙ ˙ e˙ ξ ξ − ξd e˙ = e˙ p = p˙ − p˙ d e˙ λ λ˙ − λ˙ d
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(9)
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and the sliding surface is defined as 1 m /n eξ + e˙ ξ ξ ξ βξ sξ 1 mp /np s = sp = ep + e˙ p (10) β p sλ 1 mλ /nλ eλ + e˙ βλ λ with β∗ > 0 is a design constant. m∗ and n∗ are positive odd integers.(∗ being ξ, p and λ) The control law of the design is as follows Je nξ 2−mξ /nξ − (βξ e˙ ξ + (ηξ + lf,ξ )sgn(sξ ) − ξ¨d ) La mξ J w1d np 2−mp /np p u2 = − (βp e˙ p + (ηp + lf,p )sgn(sp ) − p¨d ) L m h p w2d J nλ 2−mλ /nλ t ¨ e˙ λ + (ηλ + lf,λ )sgn(sλ ) − λd ) − (βλ La mλ (11) Fig. 4. Elevation angle ξ (dotted line)(deg) and desired angle ξd (solid line)(deg) versus time (sec) La ˆ with 1 < m∗ /n∗ < 2 , η∗ > 0, Je f1 cosp ≤ lf,ξ , Rewrite (19) into (22), it follows that Lh fˆ ≤ l , La cosξsinp fˆ ≤ l , l > 0. Jp
2
f,p
Jt
1
f,λ
f,∗
nξ +mξ
Theorem 1. When the actuator fails, the designed controller (11) can still track the desired trajectory.
Proof of Theorem 1: The proof is divided into two steps: the first step is proved to be sξ → 0 in finite time; the second step is proved to be eξ → 0 in finite time.
Constructing an Lyapunov function 1 V1 = s2ξ 2 Differentiate (12) with respect to time, then V˙ 1 = sξ s˙ ξ
(12) (13)
and differentiate sξ with respect to time, then 1 mξ mξ /nξ −1 s˙ ξ = e˙ ξ + e˙ e¨ξ (14) β ξ nξ ξ Differentiate the first equation of (9) and substituting in w1d , it follows that nξ 2−mξ /nξ e˙ + (ηξ + lf,ξ )sgn(sξ )) (15) e¨ξ = −(βξ mξ ξ Substituting (15) into (14), it follows that s˙ ξ = −(ηξ + lf,ξ )sgn(sξ ) (16) So it follows that V˙ 1 = sξ s˙ ξ = −(ηξ + lf,ξ )|sξ | < 0 (17)
V˙ 2 = (−βξ )nξ /mξ (2V2 ) 2mξ (23) Since V2 is positive definite, q and p are positive odd integers, V˙ 2 is negative definite. The same principle proves u2 and w2 . So far the proof of Theorem 1 is completed. 4. SIMULATION In this section, simulation results demonstrate the effectiveness of the designed passive fault-tolerant controller when the actuator fails. The designed passive fault-tolerant controllers parameters (see (11)) are shown in Table 1. Besides, helicopter hardware parameters are shown in Table 2. Table 1. PFTC Controller parameter simulation parameter β∗ m∗ n∗ η∗
symbol Vf , V b
ξ
Kf g Mh Mw Lw La Lh
(21) (22)
pitch controller 0.1 5 3 0.5
travel controller 0.4 5 3 0.2
Table 2. Helicopter hardware specifications (Apkarian et al. (2006))
When sξ = 0 , the first equation of (10) can be rewritten as 1 m /n eξ = − e˙ ξ ξ ξ (18) βξ Define the Lyapunov function as follows 1 V2 = e2ξ (19) 2 Differentiate (19) with respect to time, then (20) V˙ 2 = eξ e˙ ξ and (18) can be rewritten as follows e˙ ξ = (−βξ eξ )nξ /mξ Substituting (21) into (20), it follows that (n +m )/m V˙ 2 = (−βξ )nξ /mξ e ξ ξ ξ
elevation controller 0.4 5 3 0.5
description the front and back motors voltage propeller force-thrust constant gravity acceleration the mass of the helicopter the mass of the counterweight distance between elevation axis to the counterweight distance between elevation axis to helicopter distance between pitch axis to each motor
value [-24,+24]
unit V
0.1188
N/V
9.81 1.15
m/s2 kg
1.87
kg
0.47
m
0.66
m
0.178
m
Figure 4-6 display designed passive fault-tolerant controller is still able to track the desired trajectory when
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IFAC SAFEPROCESS 2018 1372 Warsaw, Poland, August 29-31, 2018
Yang Qiwei et al. / IFAC PapersOnLine 51-24 (2018) 1368–1372
Simulation results of the designed passive fault tolerant controller verify that the passive fault-tolerant controller can still track the specified target when actuator failure occurs. 5. CONCLUSION In this paper, a new design method of passive faulttolerant controller is proposed for a helicopter system. The helicopter is a dual input three output system, and one of the inputs is decoupled into two virtual inputs. Two virtual input and another input are controlled based on nonlinear terminal sliding mode method, respectively. When the actuator fails, the designed passive fault-tolerant controller can still maintenance the helicopter better performance. The validity of the designed passive fault-tolerant controller is verified by simulation. Fig. 5. Pitch angle p (dotted line)(deg) and desired angle pd (solid line)(deg) versus time (sec)
REFERENCES
Fig. 6. Travel angle λ (dotted line)(deg) and desired angle λd (solid line)(deg) versus time (sec)
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Fig. 7. Voltage of front Vf (solid line)(V) and back motor voltage Vb (dotted line)(V) versus time (sec) actuator failure occurs in 40s. Figure 7 displays the the control voltage of the front and back motor when the fault occurs. 1372