Robust Fault Tolerant Longitudinal Aircraft Control

5th International Conference on Advances in Control and 5th International Conference on Advances in Control and Optimization of Dynamical Systems 5th International Conference on 5th International Conference on Advances Advances in in Control Control and and Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India Available online at www.sciencedirect.com Optimization of Dynamical Systems Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India 5th International Conference on Advances in Control and February 2018. India February 18-22, 18-22, 2018. Hyderabad, Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

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IFAC PapersOnLine 51-1 (2018) 604–609

Robust Robust Robust Robust Robust

Fault Tolerant Longitudinal Fault Tolerant Longitudinal Fault Tolerant Longitudinal Fault Tolerant Longitudinal Aircraft Control Aircraft Control Fault Tolerant Longitudinal Aircraft Control Aircraft Control ∗ ∗∗ Aircraft Control Dinesh Dinesh D. D. Dhadekar Dhadekar ∗∗ S. S. E. E. Talole Talole ∗∗ ∗∗

Dinesh Dinesh D. D. Dhadekar Dhadekar ∗ S. S. E. E. Talole Talole ∗∗ ∗ ∗ ∗∗ Student, D. Department of Aerospace Engineering, ∗ PhD Dinesh Dhadekar S. E. Talole ∗ PhD Student, Department of Aerospace Engineering, ∗ PhD Student, of Aerospace Engineering,India Defence Institute of Department Advanced Technology, Pune-411025,

PhD Student, of Aerospace Engineering,India Defence Institute of Department Advanced Technology, Pune-411025, ∗ Defence Institute of Advanced Technology, Pune-411025, (e-mail: [email protected]) PhD Student, Department of Aerospace Engineering,India Defence Institute of Advanced Technology, Pune-411025, India (e-mail: [email protected]) ∗∗ (e-mail: [email protected]) Department of Aerospace Engineering, ∗∗ Professor, Defence Institute of Advanced Technology, Pune-411025, India (e-mail: [email protected]) ∗∗ Professor, Department of Aerospace Engineering, ∗∗ Professor, Department of Aerospace Engineering, Defence Institute of Advanced Technology, Pune-411025, India (e-mail: [email protected]) Professor, Department of Aerospace Engineering, Defence Institute of Advanced Technology, Pune-411025, India ∗∗ Institute of Advanced Technology, Pune-411025, India Defence Professor, of Aerospace Engineering,India Defence Institute ofDepartment Advanced Technology, Pune-411025, Defence Institute of Advanced Technology, Pune-411025, India Abstract: In this paper, robust fault tolerant longitudinal control Abstract: In this paper, robust fault tolerant longitudinal control of of an an aircraft aircraft in in the the presence presence Abstract: In this paper, robust fault tolerant longitudinal control of an aircraft in of external disturbances with actuator fault/failure is proposed. This is accomplished using Abstract: this paper, robust fault tolerant longitudinal control of an aircraft in the the presence presence of external In disturbances with actuator fault/failure is proposed. This is accomplished using of external disturbances with actuator fault/failure is proposed. This is accomplished using nonlinear dynamic inversion (NDI) based controller robustified by uncertainty and disturbance Abstract: In this paper, robust fault tolerant longitudinal control of an aircraft in the presence of external disturbances with actuator fault/failure is proposed. This is accomplished using nonlinear dynamic inversion (NDI) based controller robustified by uncertainty and disturbance nonlinear dynamic inversion (NDI) based controller robustified by uncertainty and disturbance estimator (UDE) technique with control allocation scheme. The control allocation scheme of external disturbances with actuator fault/failure is proposed. This is accomplished using nonlinear dynamic inversion (NDI) based controller robustified by uncertainty and disturbance estimator (UDE) technique with control allocation scheme. The control allocation scheme estimator (UDE) technique with allocation scheme. The control allocation scheme redistributes the command from the primary actuator the actuator if nonlinear inversion (NDI) based robustified byto and disturbance estimator dynamic (UDE) technique with control control allocation scheme. control allocation scheme redistributes the control control command from controller the primary actuatorThe touncertainty the secondary secondary actuator if redistributes the control command from the primary actuator to the secondary actuator if fault/failure occurs in the primary actuator. The UDE technique estimates the effect of external estimator (UDE) technique with actuator. control allocation scheme redistributes the control command from the primary actuatorThe to control the secondary if fault/failure occurs in the primary The UDE scheme. technique estimates theallocation effect actuator of external fault/failure occurs in the primary actuator. The UDE technique estimates the effect of external disturbance and the estimate is used to robustify the nominal controller. The proposed control redistributes the control command from the primary actuator to the secondary actuator if fault/failure occurs in the primary actuator. The UDE technique estimates the effect of external disturbance and the estimate is used to robustify the nominal controller. The proposed control disturbance and the estimate is used to robustify the nominal controller. The proposed control scheme makes the plant less sensitive to external disturbances under fault/failure free as well fault/failure occurs in the primary actuator. The UDE technique estimates the effect of external disturbance and the estimate is used to robustify the nominal controller. The proposed control scheme makes the plant less sensitive to external disturbances under fault/failure free as well scheme makes less sensitive external disturbances under fault/failure free as as fault/failure condition. The of proposed is disturbance andthe theplant estimate is effectiveness used toto the nominal control controller. The proposed scheme makes the plant less sensitive torobustify external disturbances under scheme fault/failure free control as well well as fault/failure condition. The effectiveness of the the proposed control scheme is demonstrated demonstrated as fault/failure condition. The effectiveness of the proposed control scheme is demonstrated through numerical simulation. scheme makes the plant less sensitive to external disturbances under fault/failure free as well as fault/failure condition. The effectiveness of the proposed control scheme is demonstrated through numerical simulation. through numerical simulation. as fault/failure condition. The effectiveness of the proposed control scheme is demonstrated through numerical simulation. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. through numerical simulation. Keywords: Fault Tolerant Control, Keywords: Fault Tolerant Control, Longitudinal Longitudinal Aircraft Aircraft Control, Control, Uncertainty Uncertainty and and Keywords: Fault Tolerant Control, Longitudinal Aircraft Control, Uncertainty Disturbance Estimator, Dynamic Inversion Keywords: Fault Tolerant Control, Longitudinal Aircraft Control, Uncertainty and and Disturbance Estimator, Dynamic Inversion Disturbance Estimator, Dynamic Keywords: Tolerant Control,Inversion Longitudinal Aircraft Control, Uncertainty and DisturbanceFault Estimator, Dynamic Inversion Disturbance Estimator, Dynamic Inversion 1. UDE 1. INTRODUCTION INTRODUCTION UDE technique. technique. Performance Performance verification verification of of the the proposed proposed 1. INTRODUCTION UDE technique. Performance verification of the proposed scheme has been carried out through numerical simulation. 1. INTRODUCTION UDE technique. Performance verification of the proposed scheme has been carried out through numerical simulation. has out numerical simulation. Fault (FTC) 1. INTRODUCTION UDE technique. Performance verification of the proposed scheme has been been carried carried out through through numerical simulation. Fault tolerant tolerant control control (FTC) deals deals with with the the problem problem of of scheme 2. Fault control (FTC) with problem of achieving acceptable performance the Fault tolerant tolerant control system (FTC) deals deals with the thein of scheme has been 2. MATHEMATICAL MATHEMATICAL MODEL carried out through MODEL numerical simulation. achieving acceptable system performance inproblem the prespres2. achieving acceptable system performance inand thefailures presence of unexpected scenarios as faults 2. MATHEMATICAL MATHEMATICAL MODEL MODEL Fault control (FTC) such deals with thein problem of achieving acceptable system performance the presence oftolerant unexpected scenarios such as faults and failures With an objective of designing aa controller ence of unexpected scenarios such as faults and failures (Harkegard and Glad, 2000; Jiang and Zhao, 2000; Xiao 2. MATHEMATICAL MODELto With an objective of designing controller to track track flight flight achieving acceptable system performance in the presence of unexpected scenarios such as faults and failures (Harkegard and Glad, 2000; Jiang and Zhao, 2000; Xiao With an objective of designing a controller to track flight path angle, assuming that the velocity of aircraft center of (Harkegard and Glad, 2000; Jiang and Zhao, 2000; Xiao et al., 2017; Alwi and Edwards, 2014). Achieving acceptWith an objective of designing a controller to track flight path angle, assuming that the velocity of aircraft center of ence of unexpected scenarios such as faults and failures (Harkegard and Glad, 2000; Jiang and Zhao, 2000; Xiao et al., 2017; Alwi and Edwards, 2014). Achieving accept- path angle, assuming that the velocity of aircraft center of mass is constant then the longitudinal equations of motion et al., 2017; Alwi and Edwards, 2014). Achieving acceptWith an objective of designing a controller to track flight able performance in the presence of external disturbances path angle, assuming that the velocity of aircraft center of (Harkegard 2000; Jiang Zhao,disturbances 2000; Xiao mass is constant then the longitudinal equations of motion et al.,performance 2017; and AlwiGlad, and 2014). Achieving acceptable in theEdwards, presence of and external mass is constant then the longitudinal equations of motion of aircraft are considered as (Romano and Singh, 1990) able performance in the presence of external disturbances path angle, assuming that the velocity of aircraft center when faults/failures occur makes the problem more chalmass is constant then the longitudinal equations of motion of aircraft are considered as (Romano and Singh, 1990) et al., 2017; Alwi and Edwards, 2014). Achieving acceptable performance in the presence when faults/failures occur makes of theexternal problemdisturbances more chal- of aircraft are considered as (Romano and Singh, 1990) of when faults/failures occur makes the problem more is constant then motion lenging to deal with. While there exist many strategies for of aircraft are considered as (Romano equations and Singh,of1990) able performance in the presence of external disturbances when faults/failures occur makes the problem more chalchalgg the longitudinal lenging to deal with. While there exist many strategies for mass lenging to deal with. While there exist many strategies for (cos θθas− cos θθ0 )) and Singh, 1990) γ considered (Romano robust control design, the designs usually do not cater α∆ α− when makes the problem more challengingfaults/failures to deal design, with. occur While there exist many for of aircraft g ∆ − (cos − cos γ˙˙ = = −Z −Zare robust control the designs usually do strategies not cater α α 0 ∆α − V γ robust design, the designs usually do not cater for Vg (cos unexpected happenings such as faults. As it lenging to deal with. While there exist many (cos θθ − − cos cos θθ00 )) γ˙˙˙ = = −Z −Zα robust control control design, the designs usually doof not cater α ∆α − V unexpected happenings such as occurrence occurrence ofstrategies faults. Asfor it g θ = q V unexpected happenings such as ofto faults. As it is well-known, practical systems are subject faults and γθ˙˙˙ = q−Zα ∆α − (cos θ − cos θ0 ) robust control design, the designs usually doof not cater for unexpected happenings such as occurrence occurrence faults. As it is well-known, practical systems are subject to faults and θθ˙ = qq gg V is well-known, practical systems are subject to faults and failures as well as external disturbances and therefore, a =m unexpected happenings such as occurrence of faults. As it is well-known, practical systems are subject to faults and qq˙˙ = ¯¯ α ∆ m ¯¯ q qq + θ − cos θ0 ) + m ¯ failures as well as external disturbances and therefore, a α+ ˙ (cos gm mα = m ∆ + m + ¯ δδee δδee (1) (1) α α q α ˙ (cos θ − cos θ0 ) + m failures as well as external disturbances and therefore, a V design that caters for both the issues simultaneously is θ = q is well-known, practical systems subject faults and failures as well as external disturbances andtotherefore, qq˙˙ = m ¯¯ α ∆ m ¯¯ q qq + θθ − cos θθ0 )) + m ¯ Vg m design that caters for both the are issues simultaneously isa α+ α ˙ (cos e (1) m = m ∆ + m + (cos − cos + m ¯ δδee δδ∆ α α q α ˙ 0 e (1) V design that caters for both the issues simultaneously is where the variables have their usual meanings with = g desirable. In essence, the FTC scheme is required which α failures asInwell as external disturbances and therefore, V design that caters fortheboth the issuesis simultaneously the variables their usual with = desirable. essence, FTC scheme required which isa where m q˙α= m ¯m ∆α=+m m ¯ qhave q++m (cos θ meanings − cos θ0+) + m ¯, δem δ∆ (1) α α ˙ ,usual eα where the variables have their meanings with ∆ desirable. In essence, the FTC scheme is required which is α − , ¯ ˙ Z m ¯ = m m ¯ α = robust against external disturbances. 0 α α α α q q α ˙ δ e where the variables have their usual meanings with ∆ V design that caters for both the issues simultaneously desirable. In essence, the FTC scheme is required which is α − α0 , m ¯ α = mα + m ˙ α Zα , m ¯ q = mq + mα˙ , m ¯ δeα = robust against external disturbances. α α ,, α˙m ¯¯Zvariables m + ˙˙their ¯¯coefficient ¯ robust external m The aerodynamic m ¯¯ δm represents α α α Zα , m q = α ˙, m δ = δe .. = the have with e + e α α δδ− − α00m mα +m m m = m mqq + + ¯∆ desirable. In essence, thedisturbances. FTC scheme is for required which of is where robustofagainst against external disturbances. α Zα ,usual q meanings ˙, m δeeα = m Zα The aerodynamic coefficient m ˙m δe = δm e + mα e represents One the approaches used to cater the effect m + m Z . The aerodynamic coefficient m ¯ represents the control surface effectiveness of elevator and δ is One of the approaches used to cater for the effect of δ α ˙ δ δ e + e = m aerodynamic e represents α δ− α0m, α˙m ¯Zαδsurface +m ˙ α Zα , m ¯coefficient = mq + m ¯ δe the = m m ¯and robustofagainst externaland disturbances. ˙ ,δe δm the effectiveness ofq elevator the e control e . The α e α e is One approaches used to for of external disturbances system uncertainties is One of the the approaches used to cater cater for the the effect effect of elevator the control surface effectiveness of elevator and δ is the deflection. The more details about (1) can be external disturbances and system uncertainties is based based e m + m Z . The aerodynamic coefficient m ¯ represents the control surface effectiveness of elevator and δ is the δ α ˙ δ δ e elevator deflection. The more details about (1) can be e e e external disturbances and system uncertainties is based on uncertainty and One of the approaches used toestimation cater for and the rejection effect of elevator external disturbances and system uncertainties is based deflection. The more about (1) be found in (Romano and 1990). Defining γ, on uncertainty and disturbance disturbance estimation and rejection the control surface effectiveness of elevator and δe11can is= elevator The Singh, more details details about (1)x can be found in deflection. (Romano and Singh, 1990). Defining x =the γ, on and disturbance estimation and rejection g and a of methods been proposed in literexternal disturbances andhave system uncertainties is based on uncertainty uncertainty disturbance estimation and rejection in (Romano and Singh, 1990). Defining x = γ, g x = θ, x = q, a = Z , a = Z α , a = , a = m ¯ and a number number ofand methods have been proposed in the the liter- found 1 2 = θ,inxdeflection. 3(Romano 0 = α ,Singh, 01 =details α α0 , a 1 = V (1) 21can α ,, elevator The more about be found and 1990). Defining x = γ, x = q, a Z a Z , a = m ¯ g 2 3 0 α 01 α 0 1 2 α and a number of methods have been proposed in the literature for the purpose. In (Youcef-Toumi and Ito, 1990), a V on disturbance estimation rejection g , bae2 = anduncertainty a for number ofand methods have been proposed in the literx = θ, x = q, a = Z , a = Z α , a = = m ¯ a = a α , a = m ¯ , a = m , x = θ , and m ¯ ature the purpose. In (Youcef-Toumi andand Ito, 1990), a found 2 3 0 α 01 α 0 1 α 02 2 0 3 q 4 α ˙ 20 0 δ Singh, 1990). V , baex e, a0m a01m= == ¯δγ, 2 = 21= α a02 =θ,ain2xα3(Romano ¯=and xα20α0=, aDefining θ10 ,=and m ¯m 0 ,=aq, 3 = q ,Zaα4, = α ˙ ,Z ature for the purpose. In (Youcef-Toumi and Ito, 1990), aa x V e, robust control method using time delay control (TDC) for g and a number of methods have been proposed in the literature for the purpose. In (Youcef-Toumi and Ito, 1990), a02 = a α , a = m ¯ , a = m , x = θ , and b = m ¯ , further, (1) can be written in nonlinear state space model robust control method using time delay control (TDC) for x 2 0 3 q 4 α ˙ 20 0 e δ e, = θ, x = q, a = Z , a = Z α , a = , a = m ¯ a = a α , a = m ¯ , a = m , x = θ , and b = m ¯ 2 3 0 α 01 α 0 1 2 α 02 2 0 3 q 4 α ˙ 20 0 e δ further, (1) can be written in nonlinear state space model e V robust control method using time delay control (TDC) for system with dynamics proposed. To enhance ature for the unknown purpose. (Youcef-Toumi and Ito, 1990),for a further, robust control method In using timeis control (TDC) (1) can be written in nonlinear state space model form as system with unknown dynamics isdelay proposed. To enhance a = a α , a = m ¯ , a = m , x = θ , and b = m ¯ further, (1) can be written in nonlinear state space model 02 2 0 3 q 4 α ˙ 20 0 e δ as e, system with dynamics is proposed. To enhance the disturbance rejection ability to overcome certain robust control method using timeand (TDC) for form system with unknown unknown dynamics isdelay proposed. To enhance form as the disturbance rejection ability and tocontrol overcome certain further, (1) can be written in nonlinear state space model form as the rejection ability and to overcome certain drawbacks with the an uncertainty and system withassociated unknown dynamics is proposed. To enhance the disturbance disturbance rejection ability and to certain drawbacks associated with the TDC, TDC, anovercome uncertainty and form − a1 (cos x2 − cos x20 ) x =as−a −a00 (x (x22 − −x x11 )) + +a a01 x˙˙ 11 = drawbacks associated with the TDC, an uncertainty and 01 − a1 (cos x2 − cos x20 ) disturbance estimator (UDE) is presented by Zhong the disturbance rejection ability and to overcome certain drawbacks associated with the TDC, an uncertainty −a (x − x ) + a − a1 (cos x2 − cos x20 ) ˙x˙ 1 = x disturbance estimator (UDE) is presented by Zhong and 0 2 1 01 −a 1= 0 (x2 − x1 ) + a01 − a1 (cos x2 − cos x20 ) = x x disturbance estimator (UDE) presented by Zhong and Rees (2004). The approach has dedrawbacks associated with theis uncertainty disturbance estimator (UDE) isTDC, presented by further Zhong and = −a x33 (x − x ) + a − a (cos x − cos x ) x˙˙˙ 22 = Rees (2004). The UDE UDE approach hasanbeen been further dex 1 01 1 2 20 x Rees The approach has further de3 0 2 veloped and applied in various the x x˙˙ 1223 = disturbance is applications presented byin Zhong and Rees (2004). (2004). The UDE UDE approach has been been further de3(x2 − x1 ) − a02 + a3 x3 + a1 a4 (cos x2 − cos x20 ) x a veloped and estimator applied in(UDE) various applications in the literliter2 x ˙˙ 3 = = a 2 (x2 − x1 ) − a02 + a3 x3 + a1 a4 (cos x2 − cos x20 ) veloped and applied in various applications in the literature (Talole and Phadke, 2008; Phadke and Talole, 2012; = x x Rees (2004). ThePhadke, UDE approach has been further developed and applied in various applications in the 2012; litera23(x (x2 − x1 ) − a02 + a3 x3 + a1 a4 (cos x2 − cos x20 )) ature (Talole and 2008; Phadke and Talole, x˙ 323 = +b a 2 e δ2e − x1 ) − a02 + a3 x3 + a1 a4 (cos x2 − cos x20(2) ature and Phadke, 2008; Phadke and Talole, Dhadekar and Patre, 2017). In work, robust FTC +b (2) veloped and in various applications in the 2012; literature (Talole (Talole and Phadke, 2008; Phadke anda 2012; e δe − x ) − a Dhadekar andapplied Patre, 2017). In this this work, a Talole, robust FTC x ˙ = a ) 3 2 (x 2e 1 02 + a3 x3 + a1 a4 (cos x2 − cos x20(2) +b δnoted eδ Dhadekar and Patre, 2017). In this work, aa Talole, robust FTC for longitudinal aircraft control is designed wherein the It may be that the system (2) is nonlinear, distur+b (2) e e ature (Talole and Phadke, 2008; Phadke and 2012; Dhadekar and Patre, 2017). In this work, robust FTC for longitudinal aircraft control is designed wherein the It may be noted that the system (2) is nonlinear, disturfor aircraft control is wherein It may be that system is distur+b (2) NDI based FTC controller is using the and without redundant actuator. To e Dhadekar Patre, 2017). In this work, aby for longitudinal longitudinal aircraft control is designed designed wherein It mayfree bee δnoted noted that the the system (2) (2) is nonlinear, nonlinear, disturNDI basedand FTC controller is robustified robustified byrobust usingFTC the bance bance free and without redundant actuator. To maintain maintain NDI based FTC controller is robustified by using the bance free and without redundant actuator. To maintain for longitudinal aircraft control is designed wherein the It may be noted that the system (2) is nonlinear, disturNDI based FTC controller is robustified by using bance free and without redundant actuator. To maintain NDI based FTC controller is robustified using Control) the636 Hosting bance free and without redundant Copyright © IFAC 2405-8963 © 2018 2018, IFAC (International Federation of by Automatic by Elsevier Ltd. All rights reserved.actuator. To maintain Copyright © 2018 IFAC 636 Copyright 2018 636 Peer review© responsibility of International Federation of Automatic Copyright © under 2018 IFAC IFAC 636 Control. 10.1016/j.ifacol.2018.05.101 Copyright © 2018 IFAC 636

5th International Conference on Advances in Control and Optimization of Dynamical Systems Dinesh D. Dhadekar et al. / IFAC PapersOnLine 51-1 (2018) 604–609 February 18-22, 2018. Hyderabad, India

closed-loop stability in the case of failures in actuators, the control effort can be redistributed to healthier actuators to obtain the desired performance, or at least some level of acceptable performance. To design such a control system, the plant itself must be equipped with redundant control effectors, which can be exploited to achieve fault tolerance (Jiang and Zhao, 2000). In this work, it is assumed that stabilizer can be used for control purpose in the event of occurrence of fault in the elevator actuator. To this end, the redundant system with external disturbance is rewritten as x˙ 1 = f1 (x1 , x2 ) x˙ 2 = x3 (3) x˙ 3 = f2 (x1 , x2 , x3 ) + be δe + bs δs + (be + bs )d(x, u, t) where, f1 (x1 , x2 ) = −a0 (x2 −x1 )+a01 −a1 (cos x2 −cos x20 ), f2 (x1 , x2 , x3 ) = a2 (x2 − x1 ) − a02 + a3 x3 + a1 a4 (cos x2 − cos x20 ), bs is the control surface effectiveness of stabilizer, δs is its deflection and d(x, u, t) is an external disturbance. It is assumed that an external disturbance d(x, u, t) is continuous and satisfies di d(x, u, t) ≤ µi f or i = 0, 1, 2, ...., r dti where µi is a small positive unknown number. The assumption implies that an external disturbance d(x, u, t), and its derivatives up to some finite order r, be bounded. However, the bound is not required to be known. Now rewriting the system (3) as x˙ = f (x) + g(x)u(t) + g(x)d (4) y = h(x)     f1 (x1 , x2 ) 0 0 x3 where f (x) = , g(x) = 0 0 and f2 (x1 , x2 , x3 ) be bs T u = [δe δs ] is the control input vector with δs as T a redundant actuator. The quantity d = [d d] is an external disturbance vector, h(x) = x1 and y is the system output. The input gain g(x) is related to the primary and secondary actuation, i.e., the elevator and stabilizer respectively. The effect of fault on each actuator is modeled as (Alwi and Edwards, 2014) ui (t) = ωi ui (t) for i = 1, 2 (5) where 0 ≤ ωi ≤ 1 are scalars and called as actuator effectiveness gain. If ω1 = 1, the elevator which is a primary actuator is working perfectly, while ω1 = 0 indicates total failure. It is assumed that the stabilizer is working perfectly and there will not be any fault or failure, it means that ω2 is always one. The partial fault in elevator is indicated by 0 < ω1 < 1. The fault/failure of the actuator can be detected by comparing the actuator output which is measurable with the command input to the actuator (Davidson et al., 2001). Accordingly, the system (4) under faulty actuator can be written as x˙ = f (x) + g(x)W u(t) + g(x)d (6) w where W = diag [w1 2 ]. 3. CONTROLLER DESIGN 3.1 Nonlinear Dynamic Inversion The nonlinear dynamic inversion, also known as inputoutput linearization is a well-known approach for designing 637

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controller for nonlinear systems (Slotine et al., 1991). In this approach, a nonlinear controller known as inner loop control is designed to achieve linear input-output relationship. Once linear, any linear design approach can be used to design the outer loop control to achieve desired performance. In the present work, the NDI controller is designed for the nominal system, i.e., only using the primary actuator with a provision for switching-over to secondary when fault/failure occurs. The system with only primary actuator dynamics is written as x˙ = f (x) + g0 (x)u0 (t) + g0 (x)d (7) y = h(x) T

where u0 = δe and g0 (x) = [0 0 be ] . From (7), it can be verified that the relative degree for the considered output is three and thus, differentiating the output thrice yields ... y = L3f h(x) + Lg0 L2f h(x)u0 + d1 (8) where L3f h(x) = f01 x1 + f02 x2 + f03 x3 + f04 , Lg0 L2f h(x) = (−a0 + a1 sin x2 )be , and d1 = Lg0 L2f h(x)d, is a lumped disturbance. The remaining terms are = a0 a2 + a30 − a1 a2 sin x2 = −a30 − a0 a2 + a1 a2 sin x2 = −a20 − a0 a3 + (a1 a3 + a0 a1 ) sin x2 + a1 x3 cos x2 = (−a0 a1 a4 − a20 a1 + a21 a4 sin x2 )(cos x2 − cos x20 ) + a0 (a02 + a0 a01 ) − a02 a1 sin x2 The NDI control using the UDE based estimate of lumped disturbance can be obtained as   1 u0 (t) = uf + ud + ν (9) 2 Lg0 Lf h(x) f01 f02 f03 f04

where

(10) uf = −L3f h(x) and ud is the disturbance rejection component of the controller given as ud = −dˆ1 (11) where dˆ1 is the estimate of d1 . The outer loop control ν, is designed as ... (12) ν = y d − k1 e − k2 e˙ − k3 e¨ where, yd is the desired flight path angle, e = y−yd and k1 , k2 , k3 are the controller gains to be chosen to ensure closedloop stability and desired tracking performance. Using (9)(12) in (8) and assuming accurate disturbance estimation, the closed-loop error dynamics can be obtained as ... e + k3 e¨ + k2 e˙ + k1 e = 0 (13) The controller (9) needs estimate of lumped disturbance and the same has been done by UDE technique as discussed in the next section. 3.2 Uncertainty and Disturbance Estimator UDE is a promising strategy for estimating slow varying disturbances. The UDE approach is based on the assumption that a signal can be accurately approximated and estimated using a filter with the broad enough bandwidth (Zhong and Rees, 2004). Having the estimate of disturbance, the opposite of it can be used in control to negate the effect of the disturbance (Talole and Phadke,

5th International Conference on Advances in Control and 606 Optimization of Dynamical Systems Dinesh D. Dhadekar et al. / IFAC PapersOnLine 51-1 (2018) 604–609 February 18-22, 2018. Hyderabad, India

2008). Following the UDE theory, the lumped disturbance d1 (x, u, t) can be estimated as (14) dˆ1 (x, u, t) = Gf (s)d1 (x, u, t) where Gf (s) is a strictly proper low-pass filter with unity steady-state gain and sufficiently large bandwidth. The estimate dˆ1 (x, u, t) is obtained by passing the lumped disturbance d1 (x, u, t) through an inertial filter Gf (s) in analogy with the TDC approach wherein the estimate is obtained by delaying the plant signals in time. Using (9)(11) in (8) the lumped disturbance can be computed as ... (15) d1 (x, u, t) = y + dˆ1 − ν while from (14), its estimate is obtained as ...  dˆ1 = Gf (s) y + dˆ1 − ν

(16) Specifically for a choice of Gf (s), if the frequency range of the unknown system dynamics and external disturbances is limited by ωf and if d˙1 is not small but d¨1 is, then the choice of Gf (s) can be a second order low-pass filter given by 2τ s + 1 Gf (s) = 2 2 (17) τ s + 2τ s + 1 with τ = 1/ωf > 0. In other words, an ideal bandwidth, ωf , is expected to be large enough to cover the spectrum of the unknown system dynamics and external disturbances (Zhong and Rees, 2004). Also, it is clear that smaller the filter time constant τ , smaller will be the estimation error. Now (16) can be written as ... ˙ ˙ ¨ ˙ + y − ν + 2τ dˆ1 + dˆ1 (18) τ 2 dˆ1 + 2τ dˆ1 + dˆ1 = 2τ (y (4) − ν) and accordingly, the disturbance estimate is obtained as    1 y − νdt) + y˙ − νdt (19) dˆ1 = 2 2τ (¨ τ 3.3 Control Allocation Control allocation enables the redundant actuators to use in an efficient way in order to achieve fault tolerance (Davidson et al., 2001). Hence, the control allocation matrix, N , which enables switching over the control from primary actuator to secondary one is used as (Alwi and Edwards, 2014)   1 N =  (1 − ω1 )be  (20) ω 2 bs Essentially, the control allocation matrix helps in providing the necessary amount of control action through secondary actuator in case of fault/failure in the primary actuator. Therefore, for the potentially fault systems, the total control law is defined as (21) u(t) = N u0 (t) Based on the control allocation matrix, during fault free conditions the primary actuator works perfectly therefore the control effectiveness gain of primary actuator, i.e., ω1 = 1 and the control signal u(t) becomes   u0 (t) u(t) = (22) 0 implying only the primary actuator, i.e., the elevator is used. The secondary actuator takes over the action according to the control allocation matrix if there is any fault/failure situation.

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4. STABILITY ANALYSIS Taking into account the disturbance estimation error, the closed loop error dynamics using (8)-(12) can be shown as ... y = ν + d˜ (23) ˜ where d is the disturbance estimation error defined as d˜ = d1 − dˆ1 . Following (Talole and Phadke, 2009), the closed loop output tracking error dynamics written in phase variable state space form with state vector as E = T [e e˙ e¨] can be shown as (24) E˙ = Ae E + Be d˜     0 1 0 0 0 1 ; Be = 0 where, Ae = 0 −k1 −k2 −k3 1 Next, from (14) and (17), the disturbance estimation error dynamics can be obtained as 2 ˙ 1 ¨ d˜ = − d˜ − 2 d˜ + d¨1 (25) τ τ From (25), it can be seen that if d˙1 = 0, d˜ will be ultimately bounded and if d¨1 = 0, d˜ will go to zero asymptotically. The choice of second order filter improves the accuracy of estimation by estimating d1 as well as d˙1 . Now the disturbance estimation error dynamics can be written in phase variable state-space form by defining the state vector  T as ϕ = d˜ d˜˙ as

(26) ϕ˙ = Aϕ ϕ + Bϕ d¨1    0 1 0 2 1 where, Aϕ = and Bϕ = 1 − 2 − τ τ In this section, the Lyapunov stability analysis tool is employed to show the convergence property of the proposed UDE based NDI FTC controller. The following assumptions can be made to show the stability analysis of the proposed controller. Assumption 4.1. The lumped disturbance term d1 is bounded and there exists a constant dmax > 0 such that 0 ≤ d1  ≤ dmax Assumption 4.2. All states are available for feedback. Assumption 4.3. The γ˙ and γ¨ are available, i.e., calculated by using analytical method. Assumption 4.4. The desired flight path angle γd is thrice differential and smooth, i.e., γ˙ d and γ¨d are available. Theorem 4.1. Consider the output tracking error dynamics of (24) and disturbance estimation error dynamics of (26). If the control input is chosen as (9), then the composite error dynamics comprising of output tracking error dynamics and disturbance estimation error dynamics is asymptotically stable. Proof 4.1. Using (24) and (26), the composite error dynamics can be written as T η˙ = Aη η + [01×4 1] d¨1 (27) where     E Ae Be 03×1 η= and Aη = (28) ϕ 02×2 02×1 Aϕ It is possible to select the controller gains in such a way that Aη will have eigenvalues at any desired locations in 

5th International Conference on Advances in Control and Optimization of Dynamical Systems Dinesh D. Dhadekar et al. / IFAC PapersOnLine 51-1 (2018) 604–609 February 18-22, 2018. Hyderabad, India

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the left half plane and for τ > 0, the matrix Aη is Hurwitz, i.e., all eigenvalues of Aη have negative real part, if and only if for a positive definite symmetric matrix Q, there exit a positive definite symmetric matrix P that satisfies the Lyapunov equation (Khalil, 2002) given by

well as combined control surface deflection of elevator and stabilizer. Initial conditions of plant γo , αo and θo are taken as 1.5o , 1.5o and 3o respectively. The necessary flight data for the control of flight path angle of aircraft is taken from (Romano and Singh, 1990).

P Aη + ATη P = −Q (29) Moreover, if Aη is Hurwitz then P has the unique solution of (29). Let,  be the smallest eigenvalue of Q. Defining the Lyapunov function as ˜ = ηT P η V (E, d) (30)

With this data, simulations are carried out for three cases. In Case-1, the wind gust disturbance is applied at 5 seconds, i.e., during the Phase-1 of flight. In Case-2, the wind gust is applied at 15 seconds which falls in Phase-2 and in Case-3, the wind gust is applied at 25 seconds, i.e., in Phase-3. The simulation results corresponding to Case-1 when the wind gust is applied at 5 seconds are presented in Figs. 1-4. From Fig. 1, it can be seen that the desired flight path angle is successfully tracked in all the three phases by the proposed UDE based NDI-FTC scheme with negligible effect of lumped disturbance. It can also be observed that the flight path angle has deviated from its desired position after 5 seconds due to the saturation in control input caused by the wind gust. The corresponding control input histories for both the actuators are presented in Fig. 2 from where it can be seen that the control input from secondary actuator is zero during Phase-1 as expected in FTC, i.e., when primary actuator is healthy, secondary will produce null control action and vice-versa. In Phase2, i.e., partial fault case, the control allocation scheme has distributed the necessary control over secondary actuator and action has been taken together by both the actuators to compensate the effect of partial fault and lumped disturbance on system performance. It can be seen that the secondary actuator provides exactly half of the control action that the primary actuator generates in fault free case because of 50% fault in the primary actuator. This combined effect of both the actuators makes the output robust against lumped disturbance even in presence of fault in primary actuator. Finally, during Phase-3, i.e., total failure case, the input from the primary actuator is zero and therefore the total control action is taken over by the secondary actuator through control allocation scheme, thus achieving the objective of FTC. During the partial fault and total failure case the control inputs from primary and secondary actuator exceeds the limit of saturation barely by 1o . From Fig. 3, it can be seen that the UDE scheme has successfully estimated the lumped disturbance in all three Phases where dˆ is an estimate of actual disturbance d. The estimation accuracy is degraded due to control input saturation on introduction of wind gust at 5 seconds. Similarly, at every sinusoidal peaks during the partial fault case, the disturbance estimation accuracy is degraded because of the control input saturation. However, the degradation at sinusoidal peaks is not significant as can be verified from the disturbance estimation errors depicted in Fig. 4.

˜ along (29) in view of Assumpand differentiating V (E, d) tion 4.1, one gets ˜ = η T (P Aη + AT P )η + 2η T P [01×3 1]T d¨1 V˙ (E, d) η

T = −η Qη + 2η P [01×3 1] d¨1 T

T

(31)

2

≤ − η + 2 η P  µ Thus, the composite error dynamics vector η is ultimately bounded by and guarantees that 2 P  µ 2 P  µ E =≤ and ϕ =≤ (32)   therefore, η can be kept within a required value for any µ by appropriately choosing the controller gains and τ . 5. SIMULATIONS AND DISCUSSIONS Simulations are carried out for flight time of 30 seconds comprising of three phases. Phase-1 corresponding to 0 < t < 10 seconds considers perfectly working actuator whereas in Phase-2 corresponding to 10 < t < 20 seconds, the actuator has been assumed with partial fault. Lastly, 20 < t < 30 seconds corresponds to Phase-3 wherein total failure of the actuator is assumed. In partial fault case, 50% fault is considered in the actuator behavior whereas if the actuator fails to respond in action, then 100% fault is assumed and represents the total failure case. The discrete gust as part of the lumped disturbance is modeled as (Rong et al., 2012)   0 x < 0  V πx m Vwind = (33) (1 − cos ) 0 ≤ x ≤ dm  2 dm    Vm x > dm

where Vm = 60 m/s represents the gust intensity, dm = 120 m represents the gust scale and x is the horizontal distance traveled. The influence of the crosswind on the aircraft motion is reflected in terms of change of angle of attack of the aircraft and is given as αactual = α + ∆αwind (34) where Vwind ∆αwind = − V and Vwind represent the component of the wind field along pitch axis and V is the velocity of aircraft. In addition to this wind disturbance, an external disturbance of 5 sin(2t) is also considered in the input channel. The desired flight path angle considered is, γd = 30o . The UDE filter time constant, τ is taken as 1 msec. The controller gains ki are computed to ensure settling time of 2 sec and damping ratio of 0.707. In simulations, physical limit for control surface deflection is considered as ±30o for elevator as 639

Next, simulation are carried out for Case-2 as well as Case3 which correspond to application of wind gust disturbance at 15 seconds (partial failure phase) and 25 seconds (total failure case) respectively to prove the superiority of the proposed UDE based NDI-FTC design. The effect of wind gust applied during the partial fault case and total failure case on flight path angle is shown in Fig. 5 and it can be seen that the effect of lumped disturbance is effectively dealt with in both the cases. Hence, the FTC technique succeeded in maintaining the system performance by pro-

5th International Conference on Advances in Control and Optimization of Dynamical Systems Dinesh D. Dhadekar et al. / IFAC PapersOnLine 51-1 (2018) 604–609 608 February 18-22, 2018. Hyderabad, India 40

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5th International Conference on Advances in Control and Optimization of Dynamical Systems Dinesh D. Dhadekar et al. / IFAC PapersOnLine 51-1 (2018) 604–609 February 18-22, 2018. Hyderabad, India

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REFERENCES

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Fig. 10. Disturbance estimation error when wind gust is applied at 25 sec. allocation matrix allows the control to switch over effectively from primary actuator to healthy redundant one when fault/failure occurs. Closed-loop stability of the formulation is established by the Lyapunov stability theory. The effectiveness of UDE in estimation of lumped disturbance and efficacy of the NDI-FTC technique in the presence of lumped disturbance during fault/failure and fault free situations are demonstrated through simulations. In simulations, it has been shown that the unknown lumped disturbance effect has been negated completely even in the presence of faults/failure situations without degrading the system performance significantly which confirms the effectiveness of proposed technique in terms of robustness as well as performance. 641

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