# Observer Based Optimal Vibration Control of a Full Aircraft System Having Active Landing Gears and Biodynamic Pilot Model

Authors: Hakan Yazici; Mert Sever
Journal: Shock and Vibration
 (2016)

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## Abstract

This paper deals with the design of an observed based optimal state feedback controller having pole location constraints for an active vibration mitigation problem of an aircraft system. An eleven-degree-of-freedom detailed full aircraft mathematical model having active landing gears and a seated pilot body is developed to control and analyze aircraft vibrations caused by runway excitation, when the aircraft is taxiing. Ground induced vibration can contribute to the reduction of pilot’s capability to control the aircraft and cause the safety problem before take-off and after landing. Since the state variables of the pilot body are not available for measurement in practice, an observed based optimal controller is designed via Linear Matrix Inequalities (LMIs) approach. In addition, classical LQR controller is designed to investigate effectiveness of the proposed controller. The system is then simulated against the bump and random runway excitation. The simulation results demonstrate that the proposed controller provides significant improvements in reducing vibration amplitudes of aircraft fuselage and pilot’s head and maintains the safety requirements in terms of suspension stroke and tire deflection.


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## Body

## 1. Introduction

The landing of an aircraft is the most critical operation, since it directly affects the safety, comfort, taxiing performance, and aircraft fatigue life [1]. During landing, ground induced vibrations cause the safety and comfort problem, hence need to mitigate quickly. It is well known that most of the nonfatal accidents occur during the landing [2]. The landing gear system is the intermediate element between the aircraft fuselage and the runway [3]. Therefore, a successfully designed landing gear system should be able to provide the aircraft stability and maneuverability after the landing impact [4].Dynamic and impact loads in aircrafts resulting from landing and uneven runway surfaces are recognized as a significant factor in causing fatigue damage [5]. Additionally, runway irregularities can contribute to reduction of pilot’s capability and the accidents may occur because of this problem [1, 6]. Therefore, landing gear systems have to attenuate the ground induced vibrations to improve the passenger-crew safety and comfort.Design of landing gear systems has been studied for a long time [7–9]. For all aircraft, landing gears are equipped with passive damping devices that suppress the ground induced vibrations [10]. Well-designed passive landing gears can be able to provide the required safety and comfort and absorb the energy during the touchdown [11]. However, the vibration damping performance of the passive landing gear system is not ensured since the passive damper devices cannot be tuned for every single runway’s characteristics [12]. Therefore, semiactive and active landing gears systems are currently being studied and experimented to overcome this adaptability problem [13, 14].In literature, many linear and nonlinear control methods have been proposed for active and semiactive landing gear systems [15, 16]. Ghiringhelli and Gualdi have designed a hybrid semiactive controller that combines a nonlinear PID term to mitigate the ground induced vibrations [17]. Gharapurkar et al. have designed a semiactive magnetorheological landing gear system by the use of Linear Quadratic Regulator (LQR) and H∞ control methods for a three-degree-of-freedom aircraft model considering the bounce, the pitch, and the roll motions [18]. Sateesh and Maiti have developed a torsional MR damper to enhance the stability of the nose landing gear system due to the ground induced lateral excitation [19].Among the passive, semiactive, and active landing gears, active landing gears can provide the best comfort and safety performance. Therefore, active landing gears have received much more attention in the last decades. Ross and Edson have designed an active landing gear system to mitigate the landing loads and ground induced vibrations under various runway irregularities in their brief study [20]. Freymann has applied an experimental and analytical study to show the efficiency of active landing gears in reducing landing loads and ground induced vibrations under different runway excitation [21]. A full aircraft model having active landing gears has been considered by Sivakumar and Haran, and PID controllers have been designed to attenuate the ground induced vibrations for passenger comfort and safety while the aircraft is taxiing [22].All the papers discussed above add some significant contribution to the literature. However, the pilot body vibration analysis in terms of safety and comfort performance requirements has not been considered using detailed full aircraft mathematical model having an integrated biodynamic pilot model so far which is the main motivation factor of this study.In this paper, landing gear control strategy for a detailed full aircraft with biodynamic pilot model is proposed to improve vibration suppression performance on pilot ride comfort, ride safety, and fuselage acceleration when the aircraft is subjected to runway disturbance during the taxiing. An integrated pilot’s seat and detailed full aircraft model that includes Abbas et al.’s [23] four-degree-of-freedom biodynamic seated pilot model is presented. This detailed mathematical model provides a platform to analyze ride comfort performance in terms of pilot’s head acceleration responses during taxiing under different type runway excitation. Based on the detailed aircraft mathematical model, an optimal controller is designed to attenuate the fuselage and pilot’s head vibrations. Because the state variables of biodynamic pilot body model are not available for measurement in practice, an observer based state feedback optimal controller having pole location constraints is developed using LMIs approach. Additionally, classical LQR controller is designed to demonstrate the effectiveness of the proposed observer based controller. Finally, numerical simulation studies are conducted to evaluate the performance of the designed controller under bump and random type runway excitation.Rest of the paper is organized as follows: Section2 describes the dynamic model of a full aircraft system having active landing gears and biodynamic pilot model. The design of an observer based state feedback optimal controller having pole location constraints is presented in Section 3. Simulation results with discussions are given in Section 4. Finally, Section 5 concludes the paper.Notation. A fairly standard notation is used throughout the paper. R stands for the set of real numbers, and Rn×n is the set of n×n dimensional real matrices. “trace” stands for the standard trace operator. The identity and null matrices are denoted by I and 0, respectively. X>0(≥,<0) denotes that X is a positive-definite (positive semidefinite, negative definite) matrix. The notation “⊗” denotes the Kronecker product. Finally, diag⁡M1,…,Mn stands for a diagonal matrix with elements M1,…,Mn appearing on its diagonal.


## 2. Mathematical Modelling of a Full Aircraft System Having Active Landing Gears and Biodynamic Pilot Model

In this study, an eleven-degree-of-freedom detailed full aircraft model is shown in Figure1, including active landing gears and a pilot seat suspension system, and an integrated four-degree-of-freedom biodynamic pilot body model is provided. The mathematical model consists of full vehicle dynamics and vertical vibrations of seated pilot body. In this model, aircraft dynamics include vertical motions of fuselage and tires, angular motions of fuselage as known as roll, and pitch.Figure 1
Full aircraft vibration model having active landing gears and biodynamic pilot body subject to runway excitation.In Figure1,  mf is the fuselage body mass which is connected to the three masses of the front, rear left, and rear right landing gears. mt1, mt2, and mt3 are the front, rear left, and rear right landing gear masses, respectively. ms is the pilot’s seat mass, and the pilot body is composed of four-mass segment, that is, head and neck mh, upper torso mu, lower torso ml, and thigh and pelvis mp. The arms and legs are combined with the upper torso and thigh, respectively [23]. zt1(t), zt2(t), and zt3(t) are the vertical displacements of front, rear left, and rear right tires. zf(t) is bounce motion of the fuselage and ϕ(t) and θ(t) are the roll and pitch angular positions which are defined with respect to the body fixed coordinate frame. zs(t) is the vertical displacement of pilot’s seat. zp(t), zl(t), zu(t), and zh(t) are defined as vertical displacements of thigh and pelvis, lower torso, upper torso, and head, respectively. zr1(t), zr2(t), and zr3(t) are the runway displacement inputs. cs1,ks1, cs2,ks2, and cs3,ks3 are the damping and stiffness of the front landing gear system, rear left landing gear system, and rear right landing gear system, respectively. ct1,kt1, ct2,kt2, and ct3,kt3 stand for the damping and stiffness of the front tires, rear left tires, and rear right tires, respectively. Seat suspension stiffness and damping elements are kfs,cfs. Stiffness and damping elements between seat and pelvis are ksp,csp, pelvis and lower torso are kpl,cpl, pelvis and upper torso are kpu,cpu, lower torso and upper torso are klu,clu, and upper torso and head are kuh,cuh, respectively. F1(t), F2(t), and F3(t) represent the active control forces that are applied to the front landing gear, rear left landing gear, and rear right landing gear of modelled aircraft, respectively. lf and lr denote the distance from Center of Gravity (CG) to the front and rear landing gear along xs axis. lsl and lsr denote the distances between CG and rear left landing gear and rear right landing gear yf direction. Finally, lsf and lss denote the distances between CG and seat along xf and yf directions.


### 2.1. Equilibrium Equations of Motion

The dynamic equilibrium equations of motion for the full aircraft having active landing gear systems, an integrated pilot’s seat and biodynamic pilot model, are derived by Lagrange method. The general Lagrange equation can be written as(1)ddt∂EK∂q˙i-∂EK∂qi+∂EP∂qi+∂ED∂q˙i=Qi,where EK, EP, and ED are the kinetic energy, potential energy, and dissipative energy of the system, respectively. qi is the generalized coordinates and Qi is the external forces of the system. The generalized coordinates and forces for the modeled full aircraft system can be written as (2)qt=zt1tzt2tzt3tϕtθtzftzstzptzltzutzhtT,(3)Qt=-F1t-F2t-F3tlslF2t-lsrF3tlrF2t+F3t-lfF1tF1t+F2t+F3t00000T.In order to obtain the kinetic energy, potential energy, and dissipative energy of the full aircraft system, deflections of each stiffness and damping element are defined as follows:(4)dt1t=zt1t-zr1t,(5)dt2t=zt2t-zr2t,(6)dt3t=zt3t-zr3t,(7)ds1t=zft-lfsin⁡θt-zt1t≅zft-lfθt-zt1t,(8)ds2t=zft+lrsin⁡θt+lslsin⁡ϕt-zt2t≅zft+lrθt+lslϕt-zt2t,(9)ds3t=zft+lrsin⁡θt-lsrsin⁡ϕt-zt3t≅zft+lrθt-lsrϕt-zt3t,(10)dfst=zst-zft+lsfθt-lssϕt,(11)dspt=zpt-zst,(12)dplt=zlt-zpt,(13)dput=zut-zpt,(14)dlut=zut-zlt,(15)duht=zht-zut.By the use of (4)–(15), kinetic, potential, and dissipative energy functions of the full aircraft system can be written as(16)EK=12mt1z˙t12t+12mt2z˙t22t+12mt3z˙t32t+12Ixϕ˙2t+12Iyθ˙2t+12mfz˙f2t+12msz˙s2t+12mpz˙p2t+12mlz˙l2t+12muz˙u2t+12mhz˙h2t,EP=12kt1dt12t+12kt2dt22t+12kt3dt32t+12ks1ds12t+12ks2ds22t+12ks3ds32t+12kfsdfs2t+12kspdsp2t+12kpldpl2t+12kpudpu2t+12kludlu2t+12kuhduh2t,ED=12ct1d˙t12t+12ct2d˙t22t+12ct3d˙t32t+12cs1d˙s12t+12cs2d˙s22t+12cs3d˙s32t+12cfsd˙fs2t+12cspd˙sp2t+12cpld˙pl2t+12cpud˙pu2t+12clud˙lu2t+12cuhd˙uh2t.The energy functions (16) can be used to construct the equilibrium equations of motion by substituting them into Lagrange equation (3) for each generalized coordinate. The governing equilibrium equations of motion for the full aircraft having active landing gear systems, an integrated pilot’s seat and biodynamic pilot model, are obtained using Lagrange method as follows: For front landing gear motion,(17)mt1z¨t1t=-kt1zt1t-zr1t-ks1zt1t-zft+lfθt-ct1z˙t1t-z˙r1t-cs1z˙t1t-z˙ft+lfθ˙t-F1t.For rear left landing gear motion,(18)mt2z¨t2t=-kt2zt2t-zr2t-ks2zt2t-zft-lrθt-lslϕt-ct2z˙t2t-z˙r2t-cs2z˙t2t-z˙ft-lrθ˙t-lslϕ˙t-F2t.For rear right landing gear motion,(19)mt3z¨t3t=-kt3zt3t-zr3t-ks3zt3t-zft-lrθt+lsrϕt-ct3z˙t3t-z˙r3t-cs3z˙t3t-z˙ft-lrθ˙t+lsrϕ˙t-F3t.For roll motion of the fuselage,(20)Ixϕ¨t=-lslks2lslϕt+zft+lrθt-zt2t-lsrks3lsrϕt-zft-lrθt+zt3t-lsskfslssϕt-zst+zft-lsfθt-lslcs2lslϕ˙t+z˙ft+lrθ˙t-z˙t2t-lsrcs3lsrϕ˙t-z˙ft-lrθ˙t+z˙t3t-lsscfslssϕ˙t-z˙st+z˙ft-lsfθ˙t+lslF2t-lsrF3t.For pitch motion of the fuselage,(21)Iyθ¨t=-lfks1lfθt-zft+zt1t-lrks2lrθt+zft+lslϕt-zt2t-lrks3lrθt+zft-lsrϕt-zt3t-lsfkfslsfθt+zst-zft-lssϕt-lfcs1lfθ˙t-z˙ft+z˙t1t-lrcs2lrθ˙t+z˙ft+lslϕ˙t-z˙t2t-lrcs3lrθ˙t+z˙ft-lsrϕ˙t-z˙t3t-lsfcfslsfθ˙t+z˙st-z˙ft-lssϕ˙t+lrF2t+F3t-lfF1t.For bounce motion of the fuselage,(22)mfz¨ft=-kfszft-zst-lsfθt+lssϕt-ks1zft-lfθt-zt1t-ks2zft+lrθt+lslϕt-zt2t-ks3zft+lrθt-lsrϕt-zt3t-kt1zft-zt1t-kt2zft-zt2t-kt3zft-zt3t-cfsz˙ft-z˙st-lsfθ˙t+lssϕ˙t-cs1z˙ft-lfθ˙t-z˙t1t-cs2z˙ft+lrθ˙t+lslϕ˙t-z˙t2t-cs3z˙ft+lrθ˙t-lsrϕ˙t-z˙t3t-ct1z˙ft-z˙t1t-ct2z˙ft-z˙t2t-ct3z˙ft-z˙t3t+F1t+F2t+F3t.For the vertical motion of the pilot’s seat,(23)msz¨st=-kfszst-zft+lsfθt-lssϕt-kspzst-zpt-cfsz˙st-z˙ft+lsfθ˙t-lssϕ˙t-cspz˙st-z˙ptFor the vertical motion of the pilot’s pelvis,(24)mpz¨pt=-kspzpt-zst-kplzpt-zlt-kpuzpt-zut-cspz˙pt-z˙st-cplz˙pt-z˙lt-cpuz˙pt-z˙ut.For the vertical motion of the pilot’s lower torso,(25)mlz¨lt=-kplzlt-zpt-kluzlt-zut-cplz˙lt-z˙pt-cluz˙lt-z˙ut.For the vertical motion of the pilot’s upper torso,(26)muz¨ut=-kpuzut-zpt-kluzut-zlt-kuhzut-zht-cpuz˙ut-z˙pt-cluz˙ut-z˙lt-cuhz˙ut-z˙ht.For the vertical motion of the pilot’s head,(27)mhz¨ht=-kuhzht-zut-cuhz˙ht-z˙ut.The above equations of motions can be written in the matrix form as(28)Msq¨t+Csq˙t+Ksqt=Fsut+Eswt,where q(t) is the generalized coordinates, u(t) is the control force, Fs∈Rn×m gives the location of the controller, w(t)∈Rp is the disturbance input, and Es∈Rn×p is a matrix that weights the disturbances. Finally, Ms,Cs,Ks∈Rn×n are the mass, damping, and stiffness matrices of the modelled aircraft system, respectively. These matrices are given in Appendix.Using the definitionxt=q(t)q˙(t)T, system (28) can be expressed in state space form as(29)x˙t=Axt+B1wt+B2ut,where(30)A=0I-Ms-1Ks-Ms-1Cs,B1=0Ms-1Es,B2=0Ms-1Fsand wt=zr1tzr2tzr3tz˙r1tz˙r2tz˙r3tT is the vector of disturbances and ut=F1tF2tF3tT is the vector of control inputs.


## 2.1. Equilibrium Equations of Motion

The dynamic equilibrium equations of motion for the full aircraft having active landing gear systems, an integrated pilot’s seat and biodynamic pilot model, are derived by Lagrange method. The general Lagrange equation can be written as(1)ddt∂EK∂q˙i-∂EK∂qi+∂EP∂qi+∂ED∂q˙i=Qi,where EK, EP, and ED are the kinetic energy, potential energy, and dissipative energy of the system, respectively. qi is the generalized coordinates and Qi is the external forces of the system. The generalized coordinates and forces for the modeled full aircraft system can be written as (2)qt=zt1tzt2tzt3tϕtθtzftzstzptzltzutzhtT,(3)Qt=-F1t-F2t-F3tlslF2t-lsrF3tlrF2t+F3t-lfF1tF1t+F2t+F3t00000T.In order to obtain the kinetic energy, potential energy, and dissipative energy of the full aircraft system, deflections of each stiffness and damping element are defined as follows:(4)dt1t=zt1t-zr1t,(5)dt2t=zt2t-zr2t,(6)dt3t=zt3t-zr3t,(7)ds1t=zft-lfsin⁡θt-zt1t≅zft-lfθt-zt1t,(8)ds2t=zft+lrsin⁡θt+lslsin⁡ϕt-zt2t≅zft+lrθt+lslϕt-zt2t,(9)ds3t=zft+lrsin⁡θt-lsrsin⁡ϕt-zt3t≅zft+lrθt-lsrϕt-zt3t,(10)dfst=zst-zft+lsfθt-lssϕt,(11)dspt=zpt-zst,(12)dplt=zlt-zpt,(13)dput=zut-zpt,(14)dlut=zut-zlt,(15)duht=zht-zut.By the use of (4)–(15), kinetic, potential, and dissipative energy functions of the full aircraft system can be written as(16)EK=12mt1z˙t12t+12mt2z˙t22t+12mt3z˙t32t+12Ixϕ˙2t+12Iyθ˙2t+12mfz˙f2t+12msz˙s2t+12mpz˙p2t+12mlz˙l2t+12muz˙u2t+12mhz˙h2t,EP=12kt1dt12t+12kt2dt22t+12kt3dt32t+12ks1ds12t+12ks2ds22t+12ks3ds32t+12kfsdfs2t+12kspdsp2t+12kpldpl2t+12kpudpu2t+12kludlu2t+12kuhduh2t,ED=12ct1d˙t12t+12ct2d˙t22t+12ct3d˙t32t+12cs1d˙s12t+12cs2d˙s22t+12cs3d˙s32t+12cfsd˙fs2t+12cspd˙sp2t+12cpld˙pl2t+12cpud˙pu2t+12clud˙lu2t+12cuhd˙uh2t.The energy functions (16) can be used to construct the equilibrium equations of motion by substituting them into Lagrange equation (3) for each generalized coordinate. The governing equilibrium equations of motion for the full aircraft having active landing gear systems, an integrated pilot’s seat and biodynamic pilot model, are obtained using Lagrange method as follows: For front landing gear motion,(17)mt1z¨t1t=-kt1zt1t-zr1t-ks1zt1t-zft+lfθt-ct1z˙t1t-z˙r1t-cs1z˙t1t-z˙ft+lfθ˙t-F1t.For rear left landing gear motion,(18)mt2z¨t2t=-kt2zt2t-zr2t-ks2zt2t-zft-lrθt-lslϕt-ct2z˙t2t-z˙r2t-cs2z˙t2t-z˙ft-lrθ˙t-lslϕ˙t-F2t.For rear right landing gear motion,(19)mt3z¨t3t=-kt3zt3t-zr3t-ks3zt3t-zft-lrθt+lsrϕt-ct3z˙t3t-z˙r3t-cs3z˙t3t-z˙ft-lrθ˙t+lsrϕ˙t-F3t.For roll motion of the fuselage,(20)Ixϕ¨t=-lslks2lslϕt+zft+lrθt-zt2t-lsrks3lsrϕt-zft-lrθt+zt3t-lsskfslssϕt-zst+zft-lsfθt-lslcs2lslϕ˙t+z˙ft+lrθ˙t-z˙t2t-lsrcs3lsrϕ˙t-z˙ft-lrθ˙t+z˙t3t-lsscfslssϕ˙t-z˙st+z˙ft-lsfθ˙t+lslF2t-lsrF3t.For pitch motion of the fuselage,(21)Iyθ¨t=-lfks1lfθt-zft+zt1t-lrks2lrθt+zft+lslϕt-zt2t-lrks3lrθt+zft-lsrϕt-zt3t-lsfkfslsfθt+zst-zft-lssϕt-lfcs1lfθ˙t-z˙ft+z˙t1t-lrcs2lrθ˙t+z˙ft+lslϕ˙t-z˙t2t-lrcs3lrθ˙t+z˙ft-lsrϕ˙t-z˙t3t-lsfcfslsfθ˙t+z˙st-z˙ft-lssϕ˙t+lrF2t+F3t-lfF1t.For bounce motion of the fuselage,(22)mfz¨ft=-kfszft-zst-lsfθt+lssϕt-ks1zft-lfθt-zt1t-ks2zft+lrθt+lslϕt-zt2t-ks3zft+lrθt-lsrϕt-zt3t-kt1zft-zt1t-kt2zft-zt2t-kt3zft-zt3t-cfsz˙ft-z˙st-lsfθ˙t+lssϕ˙t-cs1z˙ft-lfθ˙t-z˙t1t-cs2z˙ft+lrθ˙t+lslϕ˙t-z˙t2t-cs3z˙ft+lrθ˙t-lsrϕ˙t-z˙t3t-ct1z˙ft-z˙t1t-ct2z˙ft-z˙t2t-ct3z˙ft-z˙t3t+F1t+F2t+F3t.For the vertical motion of the pilot’s seat,(23)msz¨st=-kfszst-zft+lsfθt-lssϕt-kspzst-zpt-cfsz˙st-z˙ft+lsfθ˙t-lssϕ˙t-cspz˙st-z˙ptFor the vertical motion of the pilot’s pelvis,(24)mpz¨pt=-kspzpt-zst-kplzpt-zlt-kpuzpt-zut-cspz˙pt-z˙st-cplz˙pt-z˙lt-cpuz˙pt-z˙ut.For the vertical motion of the pilot’s lower torso,(25)mlz¨lt=-kplzlt-zpt-kluzlt-zut-cplz˙lt-z˙pt-cluz˙lt-z˙ut.For the vertical motion of the pilot’s upper torso,(26)muz¨ut=-kpuzut-zpt-kluzut-zlt-kuhzut-zht-cpuz˙ut-z˙pt-cluz˙ut-z˙lt-cuhz˙ut-z˙ht.For the vertical motion of the pilot’s head,(27)mhz¨ht=-kuhzht-zut-cuhz˙ht-z˙ut.The above equations of motions can be written in the matrix form as(28)Msq¨t+Csq˙t+Ksqt=Fsut+Eswt,where q(t) is the generalized coordinates, u(t) is the control force, Fs∈Rn×m gives the location of the controller, w(t)∈Rp is the disturbance input, and Es∈Rn×p is a matrix that weights the disturbances. Finally, Ms,Cs,Ks∈Rn×n are the mass, damping, and stiffness matrices of the modelled aircraft system, respectively. These matrices are given in Appendix.Using the definitionxt=q(t)q˙(t)T, system (28) can be expressed in state space form as(29)x˙t=Axt+B1wt+B2ut,where(30)A=0I-Ms-1Ks-Ms-1Cs,B1=0Ms-1Es,B2=0Ms-1Fsand wt=zr1tzr2tzr3tz˙r1tz˙r2tz˙r3tT is the vector of disturbances and ut=F1tF2tF3tT is the vector of control inputs.


## 3. Controller Design

LMI based controller design has received considerable attention in recent years [24–26]. In this paper, to improve system performance, an optimal observer based H2 controller having pole location constraints is designed. Based on the integrated model, first an optimal full state feedback controller could be designed to improve the pilot’s taxiing performance and essential safety requirements. However, it is well known that full sate feedback approach assumes that all the state variables belonging to the full aircraft system having active landing gears and pilot body are available for measurement. The assumption is questionable, when states of the biodynamic pilot body model are considered. In particular, the lower torso, upper torso, and head displacements and velocities are not practically measurable. Therefore, a full state feedback optimal controller is not suitable to improve active landing gear performance on pilot’s comfort and aircraft safety during taxiing. On the other hand, dynamic output feedback controller could be implemented to solve the unmeasurable states problem; however it is an expensive and difficult solution when the model order is increased by adding frequency weighting performance filters [27]. Therefore, in this study an optimal observer based H2 controller having pole location constraints which is less expensive and more practically reliable is applied to the active landing gear system to fulfill the desired performance requirements. The main control objectives are guaranteed asymptotic stability of the closed loop system and disturbance attenuation in the sense of H2 norm. First, solvability conditions for optimal observed based H2 state feedback controller are expressed in terms of a set of LMIs. Then, additional LMI constraints are added to place the closed loop poles into the desired domains in complex plane.Consider an open-loop state space system given by(31)x˙t=Axt+B1wt+B2ut,zt=C1xt+D11wt+D12ut,yt=C2xt+D21wt+D22ut,where x(t)∈Rn is the state vector, u(t)∈Rm is the control inputs vector, w(t)∈Rp is the disturbance inputs vector acting on the system, z(t)∈Rc is the controlled outputs vector, and y(t)∈Rq is the measured outputs vector. Then, A∈Rn×n, B1∈Rn×p, and B2∈Rn×m are known, real, constant state differential equation matrices and C1∈Rc×n, D11∈Rc×p, and D12∈Rc×m are the matrices to represent controlled outputs and C2∈Rq×n, D21∈Rq×p, and D22∈Rq×m denote the measured outputs with appropriate dimensions.Rest of this section is organized as follows. Section3.1 describes the synthesis of optimal state feedback H2 controller having pole location constraints by the use of LMI regions. Thereafter, design of a full order optimal H2 observer is given in Section 3.2. Finally, observer based state feedback optimal controller design is constructed in Section 3.3.Subscript “c” stands for scalar variables or unknown variables related to state feedback controller and subscript “o” stands for the full order observer.


### 3.1. Optimal State Feedback Controller Design Having Pole Location Constraints

In this section, it is assumed that all the state variables are available for measurement. Hence, a state feedback control law(32)ut=Kxt,which is a linear function of state vector, can be constructed. Here, K∈Rm×n is the state feedback controller gain. Then, the closed loop system can be written using (31) and (32) as (33)x˙t=A+B2Kxt+B1wt,zt=C1+D12Kxt+D11wt.An optimal state feedback controller must ensure stability and specified performance against disturbances at the same time. In the presence of exogenous inputs such as disturbances, minimization of H2 norm is one of the well accepted performance specifications [28]. Here, Tzw2 stands for H2 norm of the transfer function matrices from disturbances w(t) to controlled outputs z(t) for the closed loop system (33).Lemma1 presents a full state feedback H2 controller design.Lemma 1 (see [29]).
For a given nonnegative scalarηc, the closed loop system (33) is asymptotically stable with H2 norm less than ηc, if there exist positive-definite symmetric matrix Xc∈Rn×n and a rectangular matrix Fc∈Rm×n subject to(34)traceQc<ηc2,QcC1Xc+D12FcXcC1T+FcTD12TXc≻0,AXc+B2Fc+XcA+FcTB2T+B1TB1≺0.Then, the control law u(t)=Kx(t)=FcXc-1x(t) is a full state feedback H2 controller associated with H2 norm less than ηc.It is well known that the transient response of a linear system is related to location of its poles in complex plane. Therefore controller design with constraints on closed loop pole locations can ensure satisfactory transient response. Establishing a lower bound for decay rate and an upper bound for damping ratio can be described as domains in complex plane as follows:(35)Hτc=x+jy∣x<-τc<0,Hθ=x+jy∣y<xtan⁡θc<0,where τc is the positive real number that restricts the real part of closed loop poles and θc is the value of the angle to prescribe desired damping ratio as cos⁡θc. Any domain in complex plane can be represented as a LMI region, if there exist a symmetric matrix G∈Rd×d and unstructured matrix M∈Rd×d such that (36)DG,M=s∈C∣G+sM+s-MT≺0,where s is a complex number and superscript s- stands for complex conjugate of s. Then, DG,M is called an LMI region [30]. Hence, closed loop pole location constraints can be formulated in LMI approach by Lemma 2.Lemma 2 (see [30]).
For a given LMI regionDG,M associated with G and M, a matrix N∈Rn×n is DG,M stable if and only if there exists a symmetric positive-definite matrix P∈Rn×n which satisfies(37)G⊗P+M⊗NP+MT⊗NPT≺0,where ⊗ is the Kronecker product.In the light of Lemma2, to place the poles of the A+B2K into a LMI region described by (35), (38)Gτc=2τc,Mτc=1,Gθc=02×2,Mθc=sin⁡θccos⁡θc-cos⁡θcsin⁡θccan be used [30, 31]. Then, the matrices (38) are substituted into (37) and as follows and one obtains(39)2τcP+AP+B2Wc+PAT+WcTB2T≺0,AP+PAT+B2Wc+WcTB2Tsin⁡θcPAT-APc+WcB2T-B2Wccos⁡θcAP-PAT+B2Wc-WcTB2cos⁡θcAP+PAT+B2Wc+WcB2Tsin⁡θc≺0.Here, a change of variable Wc=KP is performed to tackle the bilinear matrix inequality problem. Hence, state feedback controller with pole location constraints can be calculated as K=WcP-1 by the solution of (39).By assuming thatK=WcP-1=FcXc-1H2 controller with pole location constraints can be designed. Fc=Wc and Xc=P are compulsory assumptions, since the constraint WcYc-1=FcXc-1 is nonconvex. Consequently, computation of a H2 state feedback controller gain with pole location constraints can be performed by the following theorem.Theorem 3.
For given positive scalarsηc, τc, and θc, the closed loop system (33) is asymptotically stable with H2 norm less than ηc and closed loop poles are shifted to the left side of -τc with a damping ratio which is smaller than cos⁡(θc), if there exist positive-definite symmetric matrices Xc∈Rn×n and Qc∈Rc×c and a rectangular matrix Fc∈Rm×n subject to(40)AXc+B2Fc+XcAT+FcTB2T+B1TB1≺0,QcC1Xc+D12FcXcC1T+FcTD12TXc≻0,traceQ<ηc,2τcXc+AXc+XcAT+B2Fc+FcTB2T≺0,AXc+XcAT+B2Fc+YcFcTsin⁡θcXcAT-AXc+FcB2T-B2Fccos⁡θcAXc-XcAT+B2Fc-YcFcTcos⁡θcAXc+XcAT+B2Fc+FcB2Tsin⁡θc≺0.Then, the control law u(t)=Kx(t)=FcXc-1x(t) is state feedback H2 controller having pole location constraints. If the minimization of ηc is considered, the resulting state feedback controller is an optimal H2 controller.


### 3.2. Optimal Full Order Observer Design Having Pole Location Constraints

In this section, LMI based optimal full order observer design is presented. It is well known that state feedback control approach assumes that all states are available for feedback. However, it is not the case in many real life applications. Thus, observer design is a very important issue and needed in many applications. An observer is another dynamical system which aims to observe the states which are unavailable for measurement. Therefore, a full order observer is given in the form of(41)x^˙t=Ax^t+B2ut+Ly^t-yt,y^t=C2x^t+D22ut,where x^t is the state observation vector and y^t is the observer output vector. Finally, L∈Rn×q is the observer gain. Observation error and dynamics are defines as(42)et=xt-x^t,e˙t=A+LC2et+B1+LD21wt,z^t=C3et.An optimal full order observer design problem can be stated as making the observation error asymptotically stable and least sensitive to the disturbance w(t). Here z^t∈Rk and C3∈Rk×n are the related vector and matrix to choose which observation errors must be less affected by disturbances. Thereafter, full order H2 observer gain can be computed as given in Lemma 4.Lemma 4 (see [29]).
For a given nonnegative scalarηo, observation error (42) is asymptotically stable with H2 norm less than ηo, if there exist symmetric positive-definite matrix Xo∈Rn×n and a rectangular matrix Fo∈Rn×q subject to(43)traceQo<ηo2,XoA+FoC2+ATXo+C2TFoTXoB1+FoD21B1TXo+D21TFoT-I≺0,QoC3C3TXo≻0Then, L=Xo-1Fo is a full order H2 observer gain.In order to improve observation performance, pole location constraints with the LMI regions are added to the full order observer design. By the use of Lemma2 which is given in Section 3.1, poles of the observation error dynamics (42) can be placed into a LMI region with guarantee of decay rate and damping ratio associated with τo and θo, as given in (44)2τoYo+YoA+WoC2+ATYo+C2TWoT≺0,YoA+ATYo+WoC2+C2TWoTsin⁡θoATYo-YoA+C2TWoT-WoC2cos⁡θoYoA-ATYo+WoC2-C2TWoTcos⁡θoYoA+ATYo+WoC2+C2TWoTsin⁡θo≺0,where Yo=P-1 and Wo=YoL. Hence, by assuming that L=Xo-1Fo=Yo-1Wo an optimal full order H2 observer having pole location constraints can be designed. Since, Xo-1Fo=Yo-1Wo is not a convex constraint, Fo=Wo and Xo=Yo are compulsory assumptions. Consequently, full order observer design procedure is formulated in Theorem 5.Theorem 5.
For given positive scalarsηo, τo, and θo, observation error (42) is asymptotically stable with H2 norm less than ηo and observer poles are shifted to the left side of -τo with a damping ratio which is smaller than cos⁡θo, if there exist positive-definite symmetric matrix Xo∈Rn×n and a rectangular matrix Fo∈Rn×q subject to (45)traceQo<ηo2,XoA+FoC2+ATXo+C2TFoTXoB1+FoD21B1TXo+D21TFoT-I≺0,QoC3C3TXo≻0,2τoXo+XoA+SoC2+ATXo+C2TSoT≺0,XoA+ATXo+FoC2+C2TFoTsin⁡θoATXo-XoA+C2TFoT-FoC2cos⁡θoXoA-ATXo+FoC2-C2TFoTcos⁡θoXoA+ATXo+FoC2+C2TFoTsin⁡θo≻0.Then, ηo is an upper bound in terms of the H2 norm for the error dynamics (42), from w(t) to z^(t) for all ≥0, and the observer gain is L=Xo-1Fo. If the minimization of ηo is considered, the resulting observer is an optimal full order H2 observer.


### 3.3. Construction of Observer Based Optimal Controller

In this section, observer based controller structure is introduced. A full order observer can be rewritten as(46)x^˙t=A+LC2x^t+B2+LD22ut-Lyt,where computation of L is formulated by Theorem 5. Here, a full order observer (46) is a dynamical system which takes control input u(t) and measured outputs y(t) as inputs to produce the estimation of the state vector x^t. By combining the state feedback gain matrix K which can be calculated with Theorem 3, an observer based controller(47)x^˙t=A+LC2x^t+B2+LD22ut-Lyt,ut=Kx^t,is obtained. Integrated structure of the state feedback and observer is depicted in Figure 2.Figure 2
Structure of the observer based controller.


## 3.1. Optimal State Feedback Controller Design Having Pole Location Constraints

In this section, it is assumed that all the state variables are available for measurement. Hence, a state feedback control law(32)ut=Kxt,which is a linear function of state vector, can be constructed. Here, K∈Rm×n is the state feedback controller gain. Then, the closed loop system can be written using (31) and (32) as (33)x˙t=A+B2Kxt+B1wt,zt=C1+D12Kxt+D11wt.An optimal state feedback controller must ensure stability and specified performance against disturbances at the same time. In the presence of exogenous inputs such as disturbances, minimization of H2 norm is one of the well accepted performance specifications [28]. Here, Tzw2 stands for H2 norm of the transfer function matrices from disturbances w(t) to controlled outputs z(t) for the closed loop system (33).Lemma1 presents a full state feedback H2 controller design.Lemma 1 (see [29]).
For a given nonnegative scalarηc, the closed loop system (33) is asymptotically stable with H2 norm less than ηc, if there exist positive-definite symmetric matrix Xc∈Rn×n and a rectangular matrix Fc∈Rm×n subject to(34)traceQc<ηc2,QcC1Xc+D12FcXcC1T+FcTD12TXc≻0,AXc+B2Fc+XcA+FcTB2T+B1TB1≺0.Then, the control law u(t)=Kx(t)=FcXc-1x(t) is a full state feedback H2 controller associated with H2 norm less than ηc.It is well known that the transient response of a linear system is related to location of its poles in complex plane. Therefore controller design with constraints on closed loop pole locations can ensure satisfactory transient response. Establishing a lower bound for decay rate and an upper bound for damping ratio can be described as domains in complex plane as follows:(35)Hτc=x+jy∣x<-τc<0,Hθ=x+jy∣y<xtan⁡θc<0,where τc is the positive real number that restricts the real part of closed loop poles and θc is the value of the angle to prescribe desired damping ratio as cos⁡θc. Any domain in complex plane can be represented as a LMI region, if there exist a symmetric matrix G∈Rd×d and unstructured matrix M∈Rd×d such that (36)DG,M=s∈C∣G+sM+s-MT≺0,where s is a complex number and superscript s- stands for complex conjugate of s. Then, DG,M is called an LMI region [30]. Hence, closed loop pole location constraints can be formulated in LMI approach by Lemma 2.Lemma 2 (see [30]).
For a given LMI regionDG,M associated with G and M, a matrix N∈Rn×n is DG,M stable if and only if there exists a symmetric positive-definite matrix P∈Rn×n which satisfies(37)G⊗P+M⊗NP+MT⊗NPT≺0,where ⊗ is the Kronecker product.In the light of Lemma2, to place the poles of the A+B2K into a LMI region described by (35), (38)Gτc=2τc,Mτc=1,Gθc=02×2,Mθc=sin⁡θccos⁡θc-cos⁡θcsin⁡θccan be used [30, 31]. Then, the matrices (38) are substituted into (37) and as follows and one obtains(39)2τcP+AP+B2Wc+PAT+WcTB2T≺0,AP+PAT+B2Wc+WcTB2Tsin⁡θcPAT-APc+WcB2T-B2Wccos⁡θcAP-PAT+B2Wc-WcTB2cos⁡θcAP+PAT+B2Wc+WcB2Tsin⁡θc≺0.Here, a change of variable Wc=KP is performed to tackle the bilinear matrix inequality problem. Hence, state feedback controller with pole location constraints can be calculated as K=WcP-1 by the solution of (39).By assuming thatK=WcP-1=FcXc-1H2 controller with pole location constraints can be designed. Fc=Wc and Xc=P are compulsory assumptions, since the constraint WcYc-1=FcXc-1 is nonconvex. Consequently, computation of a H2 state feedback controller gain with pole location constraints can be performed by the following theorem.Theorem 3.
For given positive scalarsηc, τc, and θc, the closed loop system (33) is asymptotically stable with H2 norm less than ηc and closed loop poles are shifted to the left side of -τc with a damping ratio which is smaller than cos⁡(θc), if there exist positive-definite symmetric matrices Xc∈Rn×n and Qc∈Rc×c and a rectangular matrix Fc∈Rm×n subject to(40)AXc+B2Fc+XcAT+FcTB2T+B1TB1≺0,QcC1Xc+D12FcXcC1T+FcTD12TXc≻0,traceQ<ηc,2τcXc+AXc+XcAT+B2Fc+FcTB2T≺0,AXc+XcAT+B2Fc+YcFcTsin⁡θcXcAT-AXc+FcB2T-B2Fccos⁡θcAXc-XcAT+B2Fc-YcFcTcos⁡θcAXc+XcAT+B2Fc+FcB2Tsin⁡θc≺0.Then, the control law u(t)=Kx(t)=FcXc-1x(t) is state feedback H2 controller having pole location constraints. If the minimization of ηc is considered, the resulting state feedback controller is an optimal H2 controller.


## 3.2. Optimal Full Order Observer Design Having Pole Location Constraints

In this section, LMI based optimal full order observer design is presented. It is well known that state feedback control approach assumes that all states are available for feedback. However, it is not the case in many real life applications. Thus, observer design is a very important issue and needed in many applications. An observer is another dynamical system which aims to observe the states which are unavailable for measurement. Therefore, a full order observer is given in the form of(41)x^˙t=Ax^t+B2ut+Ly^t-yt,y^t=C2x^t+D22ut,where x^t is the state observation vector and y^t is the observer output vector. Finally, L∈Rn×q is the observer gain. Observation error and dynamics are defines as(42)et=xt-x^t,e˙t=A+LC2et+B1+LD21wt,z^t=C3et.An optimal full order observer design problem can be stated as making the observation error asymptotically stable and least sensitive to the disturbance w(t). Here z^t∈Rk and C3∈Rk×n are the related vector and matrix to choose which observation errors must be less affected by disturbances. Thereafter, full order H2 observer gain can be computed as given in Lemma 4.Lemma 4 (see [29]).
For a given nonnegative scalarηo, observation error (42) is asymptotically stable with H2 norm less than ηo, if there exist symmetric positive-definite matrix Xo∈Rn×n and a rectangular matrix Fo∈Rn×q subject to(43)traceQo<ηo2,XoA+FoC2+ATXo+C2TFoTXoB1+FoD21B1TXo+D21TFoT-I≺0,QoC3C3TXo≻0Then, L=Xo-1Fo is a full order H2 observer gain.In order to improve observation performance, pole location constraints with the LMI regions are added to the full order observer design. By the use of Lemma2 which is given in Section 3.1, poles of the observation error dynamics (42) can be placed into a LMI region with guarantee of decay rate and damping ratio associated with τo and θo, as given in (44)2τoYo+YoA+WoC2+ATYo+C2TWoT≺0,YoA+ATYo+WoC2+C2TWoTsin⁡θoATYo-YoA+C2TWoT-WoC2cos⁡θoYoA-ATYo+WoC2-C2TWoTcos⁡θoYoA+ATYo+WoC2+C2TWoTsin⁡θo≺0,where Yo=P-1 and Wo=YoL. Hence, by assuming that L=Xo-1Fo=Yo-1Wo an optimal full order H2 observer having pole location constraints can be designed. Since, Xo-1Fo=Yo-1Wo is not a convex constraint, Fo=Wo and Xo=Yo are compulsory assumptions. Consequently, full order observer design procedure is formulated in Theorem 5.Theorem 5.
For given positive scalarsηo, τo, and θo, observation error (42) is asymptotically stable with H2 norm less than ηo and observer poles are shifted to the left side of -τo with a damping ratio which is smaller than cos⁡θo, if there exist positive-definite symmetric matrix Xo∈Rn×n and a rectangular matrix Fo∈Rn×q subject to (45)traceQo<ηo2,XoA+FoC2+ATXo+C2TFoTXoB1+FoD21B1TXo+D21TFoT-I≺0,QoC3C3TXo≻0,2τoXo+XoA+SoC2+ATXo+C2TSoT≺0,XoA+ATXo+FoC2+C2TFoTsin⁡θoATXo-XoA+C2TFoT-FoC2cos⁡θoXoA-ATXo+FoC2-C2TFoTcos⁡θoXoA+ATXo+FoC2+C2TFoTsin⁡θo≻0.Then, ηo is an upper bound in terms of the H2 norm for the error dynamics (42), from w(t) to z^(t) for all ≥0, and the observer gain is L=Xo-1Fo. If the minimization of ηo is considered, the resulting observer is an optimal full order H2 observer.


## 3.3. Construction of Observer Based Optimal Controller

In this section, observer based controller structure is introduced. A full order observer can be rewritten as(46)x^˙t=A+LC2x^t+B2+LD22ut-Lyt,where computation of L is formulated by Theorem 5. Here, a full order observer (46) is a dynamical system which takes control input u(t) and measured outputs y(t) as inputs to produce the estimation of the state vector x^t. By combining the state feedback gain matrix K which can be calculated with Theorem 3, an observer based controller(47)x^˙t=A+LC2x^t+B2+LD22ut-Lyt,ut=Kx^t,is obtained. Integrated structure of the state feedback and observer is depicted in Figure 2.Figure 2
Structure of the observer based controller.


## 4. Numerical Simulation Study

In this section, extensive numbers of simulations are carried out to verify the effectiveness and applicability of the proposed controller in reducing the effect of runway irregularities on the modeled full aircraft system and to improve ride quality during taxiing. The full aircraft system parameters used in the simulations are given in Table1, where the full aircraft model with landing gears parameters is borrowed from [22], and the seat suspension and biodynamic pilot body parameters are discussed in [23, 32].Table 1
The parameter values of the proposed full aircraft model and the numerical values of controlled variables coefficients.
Symbol Value Units m t 1 130 [kg] m t 2 260 [kg] m t 3 260 [kg] I x 65 × 103 [kgm2] I y 100 × 103 [kgm2] m f 22 × 103 [kg] m s 15 [kg] m p 36 [kg] m l 5.5 [kg] m u 15 [kg] m h 4.17 [kg] k t 1 1.59 × 106 [N/m] k t 2 1.59 × 106 [N/m] k t 3 1.59 × 106 [N/m] k s 1 6.73 × 105 [N/m] k s 2 4.08 × 105 [N/m] k s 3 4.08 × 105 [N/m] k f s 31 × 103 [N/m] k s p 49340 [N/m] k p l 2 × 104 [N/m] k p u 144 × 103 [N/m] k l u 1 × 104 [N/m] k u h 166990 [N/m] c t 1 4066 [Ns/m] c t 2 4066 [Ns/m] c t 3 4066 [Ns/m] c s 1 1.43 × 105 [Ns/m] c s 2 6.25 × 105 [Ns/m] c s 3 6.25 × 105 [Ns/m] c f s 830 [Ns/m] c s p 2475 [Ns/m] c p l 330 [Ns/m] c p u 909.1 [Ns/m] c l u 200.00 [Ns/m] c u h 310.00 [Ns/m] l f 7.76 [m] l r 1.94 [m] l s l 3.8425 [m] l s r 3.8425 [m] l s f 7 [m] l s s 0.5 [m] r 1 450 — r 2 450 — r 3 150 — r 4 1 — r 5 50 — r 6 50 — r 7 50 —In order to demonstrate proposed controller performance, two typical runway profiles are used as disturbance input which are applied to the aircraft tires. Bump type road profile is considered first to reveal the transient response characteristics which is given by(48)zr1t=0.041-cos⁡ωt,0.2≤t≤10,otherwise,zr2t=0.11-cos⁡ωt,5≤t≤5.80,otherwise,zr3t=0.061-cos⁡ωt,2.6≤t≤3.40,otherwise,where duration of each impulse is 0.8 s and the frequency ω is 7.85 rad/s [22]. Then, very poor random runway profile is used as the worst case disturbance input. The runway disturbance is generated by a shaping filter method which is described in [22, 33, 34]. It is typically specified as random process with a ground displacement Power Spectral Density (PSD) of(49)Sω=2αVσ2ω2+α2V2,where σ2 denotes the runway roughness variance (m2), V is the aircraft longitudinal speed (m/s), and α depends on the type of runway surface (rad/m). Hence, if the aircraft runs with the constant speed, the PSD (49) and the random runway profile signal may be obtained as the output of a first-order linear filter expressed as (50)z˙rt=-αVzrt+ωt,where ω(t) is a white noise process with the spectral density S(ω). The road roughness standard deviations for various runway types are given in Table 2 [22].Table 2
Road roughness and standard deviation [22].
Road class σ (10−3 m) ϕ ( Ω 0 ) (10−6m3), Ω0=1 α (rad/m) A (very good) 2 1 0.127 B (good) 4 4 0.127 C (average) 8 16 0.127 D (poor) 16 64 0.127 E (very poor) 32 256 0.127In this study, runway excitation withE grade very poor random road profile has been considered for the value of longitudinal speed V that is 50 m/s. The modelled bump and very poor random runway disturbances are shown in Figure 3.Figure 3
Bump and random runway disturbance inputs.In this section, first, an observer based optimal state feedback controller is designed for active vibration control of full aircraft system having active landing gears and biodynamic pilot body. Then, to compare the control performance and show the effectiveness of the proposed observer based optimal state feedback controller with classical LQR controller, the LQR is designed and simulation studies are performed. All the simulations and computations are accomplished using Matlab with Simulink. For the solution of the resulting LMIs, Yalmip parser and Sedumi solver are used [35, 36].In order to ensure the responses within permissible limits for the essential safety and comfort requirements, the controlled output vectorz(t) is constructed as(51)zt=r1ϕtr2θtr3zftr4zhtr5ds1tr6ds2tr7ds3tT,where ds1(t), ds2(t), and ds3(t) are suspension strokes and previously given in (7), (8), and (9). In addition, each scalar ri(i=1,2,…,7) given in (51) corresponds to weighting coefficients and their values are given in Table 1. The controlled outputs vector z(t) is described in matrix form as follows:(52)zt=C1xt+D11wt+D12ut,where C1 is the controlled outputs state matrix, D11 is the control input matrix for the controlled outputs, and D12 is the disturbance input scaling matrix for the controlled outputs.To guarantee asymptotic convergence of observed state variables with an optimal observation performance, appropriate measured outputs must be chosen. In the observer design measured outputs are selected as(53)yt=ds1tds2tds3tzst-zftϕtθtzftzstϕ˙tθ˙t⋯z˙ftz˙std˙s1td˙s2td˙s3tz¨t1z¨t2z¨t3ϕ¨tθ¨tz¨ftz¨stT.The measured outputs vector y(t) is described in matrix form as follows:(54)yt=C2xt+D21wt+D22ut,where C2 is the measured outputs state matrix, D21 is the control input matrix for the measured outputs, and D22 is the disturbance input scaling matrix for the measured outputs.In the light of Theorem3, in order to achieve a minimum H2 norm from w(t) to z(t) for all t≥0, one can solve the following optimization problem for the positive-definite symmetric matrices Xc and Qc, and matrix Fc and positive scalar ηc,(55)minηcs.t.40.If the above optimization problem has a feasible solution, a state feedback optimal control law can be constructed as (56)ut=FcXc-1xt.Applying Theorem 3 to this example, ηc is calculated as 28.6555. To improve control performance, the closed loop poles are shifted to the left side of τc=-15 and damping ratios of each pole are enforced to be less than 0.17 which is the maximum damping ratio value of the open-loop poles.Then, Theorem5 is used to construct the observer based controller. To improve observation performance, the observer poles are shifted to the left side of τo1=-10 and right side of τo2=-200 with damping ratios less than 0.7. In order to achieve a minimum H2 norm from w(t) to z^(t) for all t≥0, one can solve the following optimization problem for the positive-definite symmetric matrices Zo and Qo and matrix So and positive scalar ηo,(57)minηos.t.45.If the above optimization problem has a feasible solution, a full order optimal observer gain is computed as(58)L=Xo-1Foand ηo is calculated as 0.2121. Then, the observer based controller gains can be calculated by the use of (56) and (58) as explained in Section 3.3. For brevity, developed observer based optimal state feedback controller is henceforth denoted as Controller 1.To validate and compare the control performance of proposed observer based controller with classical LQR controller, the design equations for a LQR controller can be obtained as in Theorem6.Theorem 6 (see [37]).
LQR controller design problem is to find an optimal full state feedback control law that minimizes the quadratic cost function(59)J=∫0∞xTtQxt+uTtRutdt,where performance weighting matrices are symmetric and positive semidefinite Q=QT≥0 and a symmetric positive-definite matrix R=RT>0. Solution of Algebraic Riccati Equation (ARE) (60)SA+ATS+Q+SB2R-1B2TS=0provides the optimal solution. Then, the LQR control law can be obtained as(61)ut=-R-1B2TSxt.The performance weighting matrices of the LQR controller are constructed for active vibration control of full aircraft system as(62)Q=diag⁡03×11,25,15,5,04×11,5,01×11,R=diag⁡400,40,40.By the use of Theorem 6, LQR control law can be easily calculated. For brevity, designed LQR controller is henceforth denoted as Controller 2.It is well known that the safety and comfort are the essential performance requirements for the active landing gear design problem. The safety of the aircraft mainly depends on the suspension stroke and tire deflection responses, while the comfort level of the pilot depends on the pilot’s head acceleration response. Ride comfort is known to be frequency sensitive. It is defined by ISO2631-1; humans are most sensitive in the frequency band 4–10 Hz for the vertical direction. Therefore, frequency weighting operation must be performed with band pass filterWk(s) on the vertical acceleration time histories as shown in Figure 4 [38, 39].Figure 4
Block diagram of the frequency weighting.Band pass filterWk(s) coefficients are borrowed from [38], to weight the acceleration time histories in the vertical direction. Frequency response of the band pass filter is shown in Figure 5.Figure 5
ISO2631 frequency weighting filter.It is apparently seen that the frequency band 4–10 Hz is the most sensitive region for humans as previously stated. Figure6 shows the time responses of roll, pitch acceleration of fuselage, and weighted vertical acceleration of pilot’s head and fuselage, respectively, for both controlled and uncontrolled cases. When the acceleration response plots of considered aircraft system with uncontrolled and controlled cases are compared, a superior improvement in the mitigation of the maximum amplitude values is obtained by the use of proposed Controller 1 and Controller 2.Figure 6
Controlled and uncontrolled acceleration time responses under bump and random runway excitation.The corresponding peak values of roll, pitch acceleration responses of fuselage, and weighted vertical pilot’s head and fuselage acceleration response are compared in Table3 for bump and E grade very poor random road disturbance inputs, respectively.Table 3
Comparison of peak values of acceleration responses for both controlled and uncontrolled cases.
Bump Random Passive Controller 2 Controller 1 Passive Controller 2 Controller 1 z ¨ f k ( t ) 0.5654 0.2290 0.1939 0.3860 0.4077 0.3017 z ¨ h k ( t ) 1.0303 0.2195 0.1782 2.4260 0.6801 0.4733 ϕ ¨ ( t ) 0.6417 0.2766 0.3164 0.8143 0.3963 0.3406 θ ¨ ( t ) 0.1874 0.0828 0.0667 0.6057 0.2958 0.2442As it can be observed from Table3, Controller 1 achieves good control performance on ride comfort during taxiing in terms of the peak value of pilot head acceleration.To validate and compare the control performance of proposed observer based controller with classical LQR controller, the peak values of the roll, pitch acceleration responses of fuselage, and weighted vertical pilot’s head and fuselage acceleration responses are shown in Figure7 for A grade (very good), B grade (good), C grade (average), D grade (poor), and E grade (very poor) road disturbance inputs.Figure 7
Peak values of acceleration responses under A grade, B grade C grade, D grade, and E grade road class disturbance inputs.Figure7 demonstrates that a superior improvement in the attenuation of the maximum amplitude values is obtained by the use of Controller 1. Note that proposed Controller 1 largely reduces the pilot’s head acceleration compared to Controller 2 and therefore achieves very successful taxiing performance.The landing gear suspension stroke and tire deflection under bump and random runway excitation are compared for the controlled and uncontrolled cases in Figure8. It is observed that the main safety requirements, which are the suspension stroke and road holding, are fulfilled by the proposed Controller 1 and Controller 2.Figure 8
Controlled and uncontrolled bump and random runway responses on suspension stroke and tire deflection properties.Additionally, to evaluate the proposed controller performance on different performance aspects using quantitative results, the peak values of landing gear suspension stroke and tire deflection responses are given in Table4 for the controlled and uncontrolled cases. Table 4 reveals that designed Controller 1 has a satisfactory control performance and achieves a good trade-off, when the different performance requirements are considered.Table 4
Comparison of peak values of landing gear suspension stroke and tire deflection responses for both controlled and uncontrolled cases.
Bump Random Passive Controller 2 Controller 1 Passive Controller 2 Controller 1 d s 1 ( t ) 0.0857 0.0469 0.0471 0.0151 0.0083 0.0078 d s 2 ( t ) 0.1250 0.1256 0.1436 0.0101 0.0087 0.0091 d s 3 ( t ) 0.0850 0.0741 0.0849 0.0105 0.0053 0.0058 z t 1 ( t ) - z r 1 ( t ) 0.0033 0.0007 0.0008 0.0041 0.0028 0.0027 z t 2 ( t ) - z r 2 ( t ) 0.0100 0.0045 0.0041 0.0055 0.0037 0.0031 z t 3 ( t ) - z r 3 ( t ) 0.0093 0.0027 0.0024 0.0053 0.0032 0.0028Figure9 demonstrates the change in control forces for Controller 1 and Controller 2 against bump and random runway irregularities. In this figure, it is seen that the applied control forces could be produced for practical implementation.Figure 9
Time history of the applied control forces.Desired LMI regions and resulting closed loop pole locations obtained by Controller 1 and Controller 2 are shown in Figure10. As it can be observed from Figure 10, all the closed loop poles are successfully placed in the desired LMI regions by Controller 1.Figure 10
Closed loop and observer pole locations.Figure11 indicates the observation error and it is obvious from the responses that the unmeasurable states of the pilot’s body are observed successfully.Figure 11
Observation performance.Figure12 shows the frequency responses of the bounce, roll, and pitch modes of fuselage and vertical of pilot’s head, respectively, for both controlled and uncontrolled cases.Figure 12
Controlled and uncontrolled frequency responses.As expected, high gain responses belong to the uncontrolled system. It is obvious that all modes are successfully suppressed by the use of Controller 1 and Controller 2. Additionally, Controller 1 provides the least gains of system response over the frequency band 4–10 Hz, when compared with the uncontrolled system and Controller 2. Therefore, it is clearly seen that a superior improvement in the ride comfort is achieved by Controller 1.Remark 7.
As it can be observed from simulation results, designed Controller 1 and Controller 2 are both effective in reducing vibration amplitudes. In general, a full state feedback control law provides better control performance against an observer based control law. However, in this study, designed Controller 1 has better disturbance attenuation performance on the acceleration responses without a degradation on tire deflection which directly affects the road holding. Hence, maneuvering and braking responses during the landing are expected to be improved. In addition, frequency responses reveal that Controller 1 has superior vibration mitigation performance over the frequency band 4–10 Hz which drastically influences the ride comfort [38]. Therefore, with this study, a simple easily realizable synthesis method is proposed to obtain observed based optimal control law for full aircraft system having active landing gears and biodynamic pilot body model under runway excitation.


## 5. Conclusions

This paper presents an approach for designing observer based optimal state feedback controller to attenuate vibrations that occurred in full aircraft system. Since the state variables of the pilot’s body are not available for measurement, an observer based optimal controller design has been proposed. In order to improve controller performance, pole location constraints are included in the controller design process. The main importance of this study is to develop an easily realizable synthesis to obtain practically applicable observer based state feedback controller which provides best performance while taking the unmeasurable states into account. At the end of the study, effectiveness of the proposed controller is illustrated through simulations under bump and random runway excitation. Simulation results indicate that the proposed observer based optimal controller having pole location constraint is all effective in improving the comfort of the passenger-crew and provides better road holding performance to enhance the braking and maneuvering performance during taxiing. In addition, actuator dynamics and actuator saturation phenomenon will be considered for the further study on the control of active landing gears.



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*Source: 2150493-2016-10-31.xml*