Adaptive Feedforward Control for Gust-Induced Aeroelastic Vibrations

Transcription

1 aerospace Article Adaptive Feedforward Control for Gust-Induced Aeroelastic Vibrations Yongzhi Wang 1,, Andrea Da Ronch 2, * ID and Maryam Ghandchi Tehrani 2, 1 China Academy of Aerospace Aerodynamics, Beijing , China; ywangcasc@yeah.net 2 Faculty of Engineering and Environment, University of Southampton, Southampton SO17 1BJ, UK; M.Ghandchi-Tehrani@soton.ac.uk * Correspondence: a.da-ronch@soton.ac.uk; Tel.: These authors contributed equally to this work. Received: 7 July 2018; Accepted: 3 August 2018; Published: 10 August 2018 Abstract: This paper demonstrates implementation of an adaptive feedforward controller to reduce structural vibrations on a wing typical section. The aeroelastic model includes a structural nonlinearity, which is modelled in a polynomial form. Aeroelastic vibrations are induced by several gusts and atmospheric turbulence, including discrete one-minus-cosine and a notably good approximation in time-domain to von Kármán spectrum. The control strategy based on adaptive feedforward controller has several advantages compared to standard feedback controller. The controller gains, which are updated in real-time during gust encounter, are found solving a minimization problem using finite impulse responses as basis functions. To make progress with application in aeroelasticity, a single-input single-output controller is designed measuring wing torsional deformation. For both deterministic and random atmospheric shapes, controller was found successful in alleviating aeroelastic vibrations. The impact of control action on unmeasured structural modes was found minimal. Keywords: adaptive feedforward control; nonlinear aeroelasticity; gust loads alleviation; structural vibrations 1. Introduction In flight, aircraft regularly encounter atmospheric turbulence. Turbulence, lightning, hail and or phenomena can lead to injuries and discomfort on board and damage to aircraft [1], resulting in huge cost to airlines. Poor wear detection and analysis can result in poor pilot decision making which could lead to orwise completely avoidable danger to flights [2]. In addition, wear-related delays and cancellations cost airlines millions of dollars and cost countries economies billions of dollars in lost productivity each year [3]. Models of atmospheric turbulence for aircraft gust load analysis (GLA) have been developed over time and are today required by certification authorities. The most common models of turbulence rely on a discrete and continuous representation of gust. More details may be found in Ref. [4]. One of most serious forms of turbulence in flight is clear air turbulence (CAT). In 1966, a National Committee for CAT officially defined CAT as all turbulence in free atmosphere of interest in aerospace applications that is not in or adjacent to visible convective activity. Over time, less formal definitions of CAT have appeared but most comprehensive definition is turbulence encountered outside of convective clouds. CAT was recognized as a problem with advent of high altitude jet operations in 1950 s. CAT is particularly problematic because it is often encountered unexpectedly and frequently without visual clues to warn pilots of hazard [5]. CAT is very difficult to predict accurately, due in part to fact that CAT is spotty in both dimensions and time. Commonly, a turbulent area associated Aerospace 2018, 5, 86; doi: /aerospace

2 Aerospace 2018, 5, 86 2 of 19 with a jet-stream extends in order of 100 to 300 miles elongated in direction of wind, 50 to 100 miles wide and 5000 feet deep. These areas may persist from 30 min to a day. Despite difficulty in forecasting CAT, re are certain rules that have been developed to identify those areas where CAT formation is likely. The control for GLA emerged vigorously in 1964 due to a significant incident. During a routine mission, a B52-H bomber encountered severe turbulence with an estimated peak velocity of 35 m/s [4]. Nearly 80% of ventral fin broke off in flight, resulting in an extensive investigation and several development programs on GLA. A wide range of feedback control strategies were used to minimize adverse effects induced by gusts. The reader is invited to consult, as a sample of vast literature on topic, fundamental activities as those reported in [6,7]. In general, se techniques fed structural response, sensed by accelerometers, to controller that calculated commanded control action. Today, strategies based on linear quadratic regulator and derived from optimal control are considered mature in light of extensive applications in aero-servo-elasticity. The drawback of a feedback-based controller is caused by time delay between sensed information, which must propagate through entire plant before exhibiting effect to be measured and determination of control action. As investigated in [8], this time delay is critical for stability of plant. This major disadvantage may be overcome by a feedforward control strategy when an a-priori knowledge of disturbance is available. In principle, feedforward control can eliminate effects of measured disturbance on process output. In presence of modelling errors, feedforward control can often reduce effect of measured disturbance on output better than that achievable by feedback control alone [9]. An advantage of feedforward control is that re is no time delay between disturbance and control compensation and a corrective action can be taken before output deviates from set point [8,10]. The capability to measure turbulence ahead of aircraft is critical to use of an on-board feedforward controller for GLA. Two types of sensors are available. The first is a light detection and ranging (LIDAR) sensor [11]. As an example, Honeywell offers a family of advanced wear radar systems (IntuVue, Primus, Honeywell International Inc., Morris Plains, NJ, USA) that scan sky ahead of aircraft at 17 different tilt angles. Several airliners have equipped ir fleets with this equipment to ensure smoor, more comfortable and safer flights (This is a relevant subject as reported at: (accessed on 3 August 2018)). Using reference system from a LIDAR sensor, Reference [12] developed an adaptive feedforward controller for GLA of a linear aeroelastic model of F/A-18 aircraft. The orthogonal finite impulse response (FIR) filter was used as adaptive controller, which exhibited a good performance. The second type of sensor is referred to as an alpha probe [13]. Reference [14] presented a single-input single-output (SISO) FIR feedforward controller to suppress gust-induced vibrations of a transport airplane. A similar strategy was extended to multiple-input multiple-output case in [15]. Reference [16] investigated a hybrid strategy obtained by combining feedback and feedforward controls. It was found that hybrid strategy significantly improved reduction in wing bending moment when compared with feedback strategy alone. Flight tests of an adaptive feedforward controller were carried out to alleviate wing bending vibrations [17]. Using a nose boom mounted flight log sensor, feedforward compensation was found to provide a good reduction of dynamic loads and an improvement in ride comfort. The already limited literature on topic becomes scarce when nonlinearities exist in plant to be controlled. Reference [18] investigated impact that structural nonlinearities have on effectiveness of a feedforward controller for a wing typical section. It was found that to suppress gust-induced vibrations of an intrinsically nonlinear plant, control performance was degraded when using a linear representation for internal model. The control strategy performed well when including all nonlinearities in model. Based on this forerunner knowledge, this paper aims at investigating deployment of an adaptive feedforward control strategy for GLA of a nonlinear aeroelastic plant. The test case is for a model of a wing typical section that exhibited sub-critical limit

3 Aerospace 2018, 5, x FOR PEER REVIEW 3 of 18 lies in derivation of aeroelastic equations in time domain and in numerical Aerospace 2018, 5, 86 3 of 19 implementation of adaptive feedforward controller. The second objective relates to identification of plant dynamics. The last objective assesses effects of proposed controller cycle for GLA. oscillations It is worth in wind noting tunnel that tests. a follow-up The workstudy is built conducted around three flight objectives. tests on a The high-altitude, first objective longendurance derivation unmanned of air aeroelastic vehicle with equations a wing span time of 24 domain m [19]. and in numerical implementation lies in of The adaptive paper feedforward continues with controller. a description The second of objective aeroelastic relatestestbed to identification in Section 2. of Models plant of dynamics. atmospheric The turbulence last objective and assesses gusts are given effects in Section of proposed 3. The control controller ory forunderlying GLA. It is worth adaptive noting that feedforward a follow-up strategy study is conducted reviewed flight in Section tests on4. athen, high-altitude, numerical long-endurance results are presented unmanned in air Section vehicle 5. with Concluding a wing span remarks of 24 are m finally [19]. given in Section 6. The paper continues with a description of aeroelastic testbed in Section 2. Models of atmospheric 2. Aeroelastic turbulence Model and gusts are given in Section 3. The control ory underlying adaptive feedforward In this work, strategy aeroelastic is reviewedtest incase Section is for 4. a Then, wing typical numerical section, results Figure are 1. presented This is modelled in Section as 5. a Concluding two-dimensional remarks symmetric are finally airfoil given section in Section and 6. structural motion is characterized by two degrees of freedom: plunge, h is measured at elastic axis (..) and is positive downward; and 2. Aeroelastic Model pitch rotation, α, is rotation of airfoil section about elastic axis, positive nose-up. As common In this in work, classic aeroelasticity, test is case isairfoil for a wing semi-chord. typical The section, elastic Figure axis 1. is This located is modelled at a distance as a two-dimensional a b from mid-chord, symmetric positive airfoilwhen section and elastic structural axis is aft motion of mid-chord. is characterized The center by twof degrees gravity of (. freedom:.) is located plunge, at h aft is measured of mid-chord. at elastic The axis aeroelastic (e.a.) and is model positive is restrained downward; to and ground pitch rotation, through a α, system is rotation of elastic of springs, airfoil section, and about dampers, elastic axis, and positive (omitted nose-up. in Asfigure common for in clarity), classicacting aeroelasticity, on both degrees b is airfoil of freedom. semi-chord. The nonlinearity The elasticin axis istiffness locatedof at a distance elastic springs a h b fromis modelled mid-chord, in a polynomial positive when form including elastic axis terms is up aft to offifth mid-chord. order. The center of gravity (c.g.) is located Herein, at x α b aft of interest mid-chord. is to analyze The aeroelastic dynamic model response is restrained of system to ground induced through by an aexternal system of disturbance. elastic springs, This Kdisturbance α, K ξ and dampers, is representative C α and C h of (omitted atmospheric in turbulence figure for clarity), and gusts, acting which on both are degrees treated in of more freedom. detail The in nonlinearity Section 3. A inflap is stiffness mounted of at elastic trailing-edge springs isof modelled airfoil, in aat polynomial a distance form aft including of mid-chord, terms upfor to fifth GLA. order. The flap deflection,, is defined relative to airfoil chord. a b h x α b α K ξ K α δ Figure 1. Schematic of an airfoil section with trailing-edge flap; wind velocity is to right and horizontal; e.a... and and c.g.. denote,. denote, respectively, elastic elastic axis axis and and center center of gravity. of gravity. Herein, It is assumed interest that is system to analyze has a horizontal dynamic equilibrium response of position system at (h induced =0, =0, by an =0). external The disturbance. equations of This motion disturbance governing dynamics of two degrees of freedom system are h is representative of atmospheric turbulence and gusts, which are treated in more detail in Section 3. A+ flap is + mounted + h at + trailing-edge h + h = of airfoil, at a distance cb aft h (1) of mid-chord, for GLA. The flap deflection, δ, is defined relative = to airfoil chord. The coupled It is assumed system that of equations systemconsists has a horizontal of structural equilibrium dynamics positionon at (h left-hand = 0, α = side 0, δ and = 0). of The aerodynamic equations loads of motion on governing right-hand dynamics side. The of term two degrees (in kg) ofindicates freedom system total aremass that undergoes a vertical motion, { (in kg m) is first moment of inertia about elastic axis and (in kg m 2 ) second moment m.ḣ of + inertia S.. ( αα + about C h + K h h elastic + βh3 haxis. 3 + βthe h5 h 5) lift, = L (in N/m), is defined positive..... ( upward as usual in aerodynamics, S α h + I α α + whereas C α α + Kh α plunge + βα3 α 3 deflection, + β α5 α 5) (1) = h, Mis positive downward as

4 Aerospace 2018, 5, 86 4 of 19 The coupled system of equations consists of structural dynamics on left-hand side and of aerodynamic loads on right-hand side. The term m (in kg) indicates total mass that undergoes a vertical motion, S α (in kg m) is first moment of inertia about elastic axis and I α (in kg m 2 ) second moment of inertia about elastic axis. The lift, L (in N/m), is defined positive upward as usual in aerodynamics, whereas plunge deflection, h, is positive downward as conventionally done in aeroelasticity. Hence, negative sign in front of lift. The aerodynamic moment, M (in N), is taken about elastic axis. The airfoil semi-chord, b and uncoupled pitch natural frequency, ω α, are used to rewrite Equation (1) in terms of non-dimensional parameters. In non-dimensional form, equations become ξ + x α α + 2ζ ξ ω U ξ + ( ω U ) 2 ( ξ + βξ3 ξ 3 + β ξ5 ξ 5) = 1 x α ξ r 2 + α 1 + 2ζ α a U α + ( 1 U πµ C L ) 2(α + βα3 α 3 + β α5 α 5) (2) = 1 C πµr 2 m a It is worth noting that derivative with respect to physical time, ẋ = dx dt, in Equation (1) is replaced by derivative with respect to non-dimensional time, x = dτ dx = U b dx dt, in Equation (2). Non-dimensional parameters are defined in Appendix A. The aerodynamics is for an incompressible two-dimensional flow [20]. The total aerodynamic loads consist of contributions arising from airfoil motion, flap deflection and encounter with a time-varying gust C L (τ) = C α L (τ) + Cτ L (τ) + Cg L (τ)c m(τ) = C α m(τ) + C τ m(τ) + C g m(τ) (3) The generalization of aerodynamic loads to an arbitrary input time-history is obtained through convolution (or Duhamel integral). For a practical evaluation of Duhamel integral, an exponential approximation is used for Wagner and Küssner functions, which describe, respectively, indicial build-up of circulatory part of lift and lift build-up for penetration into a sharp-edged gust [21]. This implies that governing equations Equation (2) are a set of integro-differential equations (IDEs) when an expression for aerodynamic loads is introduced. State Space Form The governing equations are a set of IDEs which are difficult to solve with state-of--art algorithms, primarily developed for ordinary differential equations (ODEs). The mamatical procedure to convert set of IDEs into a set of ODEs is based on defining additional variables and equations describing evolution of aerodynamics. Following [22], aeroelastic equations in state space form are rewritten as x 1 = x 2 x 2 = p 1H(x) + p 2 P(x) x 3 = x 4 x 4 = p 3H(x) + p 4 P(x) x 5 = x 1 ε 1 x 5 x 6 = x 1 ε 2 x 6 x 7 = x 1 ε 1 x 7 x 8 = x 1 ε 2 x 8 x 9 = δ ε 1x 5 x 10 = δ ε 1x 6 x 11 = w g ε 3 x 11 x 12 = w g ε 4 x 12 (4)

5 Aerospace 2018, 5, 86 5 of 19 where dependence on non-dimensional time, τ, is omitted for brevity. In a matrix-vector form state space vector is x (τ) = f(x, τ) + B µ µ(τ) + B d d(τ) (5) x = ( α, α, ξ, ξ, w 1, w 2, w 3, w 4, w 5, w 6, w 7, w 8 ) T (6) The function f(x, τ) contains (nonlinear) elastic contributions from springs and is a nonlinear function of state vector, x The term µ(τ) = δ indicates control input (rotation of trailing-edge flap) and d(τ) = w g denotes exogenous external disturbance (atmospheric turbulence and gust). The elements of state space vector w 1 through w g are aerodynamic states and a full derivation is given [22]. 3. Atmospheric Turbulence and Gust Models Models used for prediction of aircraft response need to accommodate those events that are perceived as discrete and usually described as gusts, as well as phenomena described as continuous turbulence [4] Discrete Deterministic Gust The most common model of a discrete deterministic gust, which has evolved over years from isolated sharp-edge gust function in earliest airworthiness requirements, is one-minus-cosine function. Its formulation is w 1 = U ds 2 ( 1 cos ( )) πs H where U ds is design gust velocity and H is gust gradient distance, that is, distance over which gust velocity increases to a maximum value. Airworthiness requirements prescribe a relationship between design gust velocity and gust gradient distance Continuous Atmospheric Turbulence The engineering model of random turbulence at altitude has been developed over many years. It is now widely accepted that it is satisfactory to treat atmospheric turbulence as frozen, homogeneous and isotropic in relatively large patches. The frozen-field assumption, closely allied to Taylor s hyposis, is that turbulent velocities do not change during time of passage of airplane. This is a valid assumption in most cases. Dryden and von Kármán models are considered adequate engineering models to predict correlation and spectra, with weight of experimental evidence favoring latter. A commonly used spectrum that matches experimental data is von Kármán model. The power spectral density (PSD, in m 2 /(s 2 Hz)) for vertical direction, Φ z, according to Military Specification MIL-F-8785C [23], is given by (7) Φ z (Ω) = 2σ2 z L z 1 + 8/3(1.339L z Ω) 2 U (1 + (1.339L z Ω) 2) 11/6 (8) where Ω = ω/u is scaled frequency (in rad/m), σ z turbulence intensity (in m/s), U freestream speed (in m/s) and L z turbulence scale length (in m). The turbulence intensity is related to wind speed at an altitude of 20 ft (6 m) by σ z = 0.1w 20 (9)

6 Aerospace 2018, 5, 86 6 of 19 Values of wind speed are tabulated based on turbulence severity: for light turbulence, w 20 = 15 kn; for moderate turbulence, w 20 = 30 kn; and for severe turbulence, w 20 = 45 kn. Numerical Implementation A time-domain representation of external disturbance is required to simulate non-linear Aerospace dynamics 2018, of5, ax system FOR PEER in REVIEW time-domain. Because of irrational nature of von Kármán spectrum6 of 18 (in frequency-domain, power 11/6 is not an integer), re is no exact analytical formulation in time-domain The approach corresponding used in this work to Equation consists (8). of following steps. First, Fourier transform of a unit variance The approach band-limited used in this white work noise consists signal, of (Ω), following is calculated. steps. First, This is passed Fourierthrough transforma of filter, a (Ω), unit which variance is band-limited defined as white square noise root signal, of X(Ω), PSD isspectrum calculated. in This Equation is passed (8). through Then, a filter, output signal H z (Ω), is which calculated, is defined given as filter square and root input of signal PSD spectrum in Equation (8). Then, output signal is calculated, given filter and input signal (Ω) = (Ω) (Ω) (10) Finally, inverse Fourier transform W g (Ω) of = H z (Ω)X(Ω) output signal, (Ω), is performed to obtain (10) continuous turbulence in time-domain, ( ), with statistical properties matching von Kármán Finally, inverse Fourier transform of output signal, W g (Ω), is performed to obtain spectrum. continuous turbulence in time-domain, w g (t), with statistical properties matching von The procedure requires calculating Fourier transform and its inverse, requiring more Kármán spectrum. computational resources as time window increases but it was found to be more robust and The procedure requires calculating Fourier transform and its inverse, requiring more accurate than or approaches. The method described herein is available in an open-source toolbox computational resources as time window increases but it was found to be more robust and (both in MatLAB and Python), which is referred to as Von Kármán Turbulence Generator (VKTG). accurate than or approaches. The method described herein is available in an open-source toolbox The (both VKTG in MatLAB toolbox and implements Python), which mamatical is referred to as representations Von Kármán of Turbulence random turbulence Generator (VKTG). defined in TheMilitary VKTG toolbox Specification implements MIL-F-8785C mamatical and Military representations Handbook of MIL-HDBK-1797, random turbulence allowing definedfor in dependence Military of Specification root mean MIL-F-8785C square and turbulent Military velocity Handbook and MIL-HDBK-1797, turbulence length allowing scale on foraircraft mission dependence parameters of root and mean wear square conditions. turbulent velocity Figure 2 and shows turbulence that, at length higher scale frequencies, on aircraft mission PSD of parameters VKTG model and wear achieves conditions. a better agreement Figure 2 shows with that, von atkármán higher frequencies, spectrum of Equation PSD of (8) VKTG than off--shelf model achieves MATLAB/SIMULINK a better agreement with model. vonfor Kármán more spectrum information, of Equation reader (8) is than invited off--shelf to refer to Ref. [24] MATLAB/SIMULINK and on-line documentation model. For more ( information, reader (Retrieved is invited 10 to refer August to Ref. 2018)). [24] and on-line documentation ( (Retrieved 10 August 2018)). (a) Time history of vertical gust (b) Power spectral density (PSD) Figure Figure Random Random vertical vertical gust gust intensity intensity using using Von Von Kármán Kármán spectral spectral representation representation (Military (Military Specification: MIL-F-8785C; flight flight speed: speed: U = 280 = m/s; 280 altitude: m/s; altitude: h = 10,000 h m; = 10,000 and turbulence m; and intensity: turbulence intensity: light 10 2 light ; 10 terms 2 ; Simulink terms Simulink and VKTG and denote, VKTG respectively, denote, respectively, Von Kármán Wind Von Kármán Turbulence Wind Turbulence Model blockmodel of MATLAB block andof MATLAB present Von and Kármán present Turbulence Von Generator Kármán implementation.). Turbulence Generator implementation.). 4. Adaptive Feedforward Control A typical block diagram of an adaptive feedforward control algorithm consists of two channels, Figure 3. The disturbance path includes transfer function of physical plant to be controlled,, between external disturbance, and system response,. The control path contains transfer function of adaptive feedforward controller,, to be designed. The sensed reference

7 Aerospace 2018, 5, 86 7 of Adaptive Feedforward Control A typical block diagram of an adaptive feedforward control algorithm consists of two channels, Figure 3. The disturbance path includes transfer function of physical plant to be controlled, Aerospace 2018, 5, x FOR PEER REVIEW 7 of 18 H, between external disturbance, w g and system response, x. The control path contains transfer It is function assumed of that adaptive wing typical feedforward section controller, is equipped G c with, to be an designed. onboard LIDAR The sensed to provide reference signal, GLA system ŵ g, iswith fed forward a lead-time to reference adaptive signal controller for vertical to generate gust. The control objective command. of GLA Thecontrol block Gis indicates to adjust weight transfercoefficients function of of physical controller plant to minimize between error control signal input,. ufurrmore, and system response dynamic predicted characteristics by of model, actuator y. Thedriving control input rotation to of physical flap plant were is not commanded accounted. bythis controller. decision was Finally, made based block on with experimental transfer function observation Ĝ represents that an approximation dynamics of ofactuator G. As an was input fast to enough, systemcompared G c, filter to output characteristic u drives times flap toof produce aeroelastic a cancellation plant, force to be and neglected moment. [22]. TheTheir error signal, inclusion e, is is straightforward sum of disturbance [25] and ir paffect output, would x and appear control within path transfer output, y. function. Figure 3. Block diagram of an adaptive feedforward control algorithm. Figure 3. Block diagram of an adaptive feedforward control algorithm. When re is an exact knowledge of system transfer functions, and, setting error to be It null is assumed that wing typical section is equipped with an onboard LIDAR to provide GLA system with a lead-time reference signal for vertical gust. The objective of GLA control is to adjust weight coefficients ( ) = ( ) of + ( ) controller = ( ) to minimize + ( ) = error + signal e. ( ) Furrmore, =0 dynamic (11) characteristics yields ideal of feedforward actuatorcontroller driving rotation of flap were not accounted. This decision was made based on experimental observation that dynamics of actuator was fast enough, compared to characteristic times of aeroelastic = plant, to be neglected [22]. Their inclusion (12) is straightforward As this is rarely [25] and ir case in effect practice, would appear transfer within functions transfer of function plant are G. approximated using system When identification re is an exact (SI) methods. knowledgegenerally, of system system transfer identification functions, Gmethods and H, setting seek to create error toa be mamatical null relationship between output response and input signal. These methods enjoy widespread utilization e(t) = because y(t) + x(t) model = Gu(t) creation + Hw g (t) is straightforward. = (GG ci + H)w g (t) A major = 0 drawback is (11) yields limited validity ideal feedforward of predictions, controller restricted by characteristics of input signal, often referred to as training signal, used to generate model G in first place [26]. Section 5.3 discusses choice of ci = HG 1 (12) training signal based on a chirp input and evaluates accuracy of approximated transfer functions. As this is rarely case in practice, transfer functions of plant are approximated usingonce system identification system transfer (SI) functions methods. Generally, and are system identified, identification block with methods transfer seek function to create ais mamatical approximated relationship as between output response and input signal. These methods enjoy widespread utilization because model creation is straightforward. A major drawback is = (13) limited validity of predictions, restricted by characteristics of input signal, often referred to asthe output training signal, of used block to generate is n calculated model in first place [26]. Section 5.3 discusses choice of training signal based on a chirp input and evaluates accuracy of approximated ( ) = ( ) (14) transfer functions. On disturbance path, system response ( ) is ( ) = ( ) (15) Substituting Equation (13) into Equation (14) and n substituting from resulting expression into Equation (15) yields ( ) ( ) ( ) = ( ) (16)

8 Aerospace 2018, 5, 86 8 of 19 Once system transfer functions G and H are identified, block with transfer function Ĝ is approximated as Ĝ = G (13) The output signal of block Ĝ is n calculated On disturbance path, system response x(t) is u a (t) = Ĝw g (t) (14) x(t) = Hw g (t) (15) Substituting Equation (13) into Equation (14) and n substituting w g from resulting expression into Equation (15) yields x(t) H(t)G 1 u a (t) = G ci u a (t) (16) This relation allows identifying feedforward controller using a map between u a (t) and x(t) [27]. An adaptive filtering strategy to update controller gains is adopted to increase robustness of weight coefficients of controller against errors in measurement of external disturbance and nonlinear effects in system response Control Model The controller is considered as a discrete linear time-invariant system x(t) = G c (q)w g (t) (17) where G c (q) indicates controller transfer operator and q is forward shift operator, qw g (t) = w g (t + 1). The corresponding transfer function, G c (z) with z C, is formulated as G c (z) = n L k B k (z) (18) k=1 A transfer function for a controller of order n is modelled as a sum of coefficients to be determined, L k, multiplied by an adequate choice of basis functions, B k (z). In this work, basis functions are given by a FIR set B k (z) = z k, k = 1, 2,..., n (19) which captures any shape of response up to a frequency limit Exponentially Weighted Recursive Least-Square Algorithm The calculation of coefficients of basis functions, L k, in Equation (18) is performed using an exponentially weighted recursive least-square algorithm. At i-th time step, t i, error between desired response, e(i) and FIR model output, r(i), is introduced ê(i) = e(i) r(i) = e(i) L T (i)φ(i) (20) where L(i) = (L 1 (i), L 2 (i),..., L n (i)) T is unknown vector of coefficients (often referred to as tap weight vector) and Φ(i) = (u a1 (i), u a2 (i),..., u an (i)) T is vector containing output of every basis function. A cost function is n defined as follows ε(i) = N λ N i ê(i) 2, N = 1, 2,..., N T (21) i=1

9 Aerospace 2018, 5, 86 9 of 19 Aerospace 2018, 5, x FOR PEER REVIEW 9 of 18 where N T indicates total number of time steps and λ (0, 1] is forgetting factor. 5. Results The adaptive algorithm to calculate coefficients of basis functions follows steps: The aeroelastic solver has previously been validated in many numerical and experimental studies. Initialization: The reader parameters is invited to are refer initialized to References before entering [21,22,29] for time furr iteration information. loop. L(0) Based = 0 on is previous initialized findings, to a non-dimensional column vector of time zeros. step P(0) of 0.05 = was δ 1 I deemed is inverse sufficient correlation to capture matrix dynamics (where of δ system. = 0.1 in this case). Time The aeroelastic iteration: parameters recursivein algorithm this study ismodel repeated a wind at each tunnel timeconfiguration step, t i, for tested entirein duration low speed of flow at time University vector. For of i Liverpool, = 1, 2,..., Figure N, calculate 4. The experimental model consists of a rigid wing that spans entire width of wind tunnel cross-section and is suspended by a system of springs to π(i) = P(i 1)Φ(i) tunnel walls. The wing cross-section π(i) is based on NACA0018 airfoil; semi-chord is = k(i) = m and pitch natural frequency λ+φ T (i)π(i) is = rad/s. The aeroelastic parameters are summarized in Table 1 and ε(i) procedure = e(i) for L T (i model 1)Φ(i) updating is discussed in Reference [22]. The (22) last two entries in table are L(i) cubic = L(iand 1) quintic + k(i)ε(i) coefficients of polynomial expansion of stiffness in plunge degree of P(i) freedom. = λ 1 P(i The values 1) λindicate 1 k(i)φa T (i)p(i typical hardening 1) non-linearity. The algorithm Table is1. sensitive Aeroelastic to parameters choice of representative forgetting of factor, wind λ. tunnel In thistest work, rig [22]. value is set to 1. The reader is referred to Reference [28] for more details. Aeroelastic Parameter Numerical Value (-) 5. Results The aeroelastic solver has previously been validated in many numerical and experimental studies The reader is invited to refer to References [21,22,29] for furr information. Based on previous findings, a non-dimensional time step of 0.05 was deemed sufficient to capture dynamics of system The aeroelastic parameters in this study model a wind tunnel configuration tested in low speed flow at University of Liverpool, Figure 4. The experimental model consists of a rigid wing that spans entire width of wind tunnel cross-section and638, is suspended by a system of springs to tunnel walls. The wing cross-section is based on NACA0018 airfoil; semi-chord is b = m and The pitch wind natural tunnel frequency test rig exhibited is ω α = an interesting rad/s. non-linear The aeroelastic dynamic parameters behavior, arecharacterized summarizedby in Table occurrence 1 and of procedure sub-critical forlimit model cycle oscillations updating is(lcos). discussed From in Reference a control [22]. standpoint, The lastit two represents entries a inchallenging table aretest cubic case to anddemonstrate quintic coefficients control of and polynomial suppression expansion of gust-induced of stiffness aeroelastic in plunge vibrations degree and of LCOs. freedom. The values indicate a typical hardening non-linearity Frequency-Domain Analysis Figure 4. Experimental test rig of typical wing section [22]. Figure 4. Experimental test rig of typical wing section [22]. First, aeroelastic stability is analyzed. Figure 5 shows damped frequency and damping ratio for varying freestream speed. The solid line in figure is from numerical predictions that were

10 Aerospace 2018, 5, of 19 Table 1. Aeroelastic parameters representative of wind tunnel test rig [22]. Aeroelastic Parameter Numerical Value (-) ω µ a h x a r a ζ a ζ ξ β ξ β ξ5 638, The wind tunnel test rig exhibited an interesting non-linear dynamic behavior, characterized by occurrence of sub-critical limit cycle oscillations (LCOs). From a control standpoint, it represents a challenging test case to demonstrate control and suppression of gust-induced aeroelastic vibrations and LCOs Frequency-Domain Analysis Aerospace 2018, 5, x FOR PEER REVIEW 10 of 18 First, aeroelastic stability is analyzed. Figure 5 shows damped frequency and damping ratio obtained for varying solving freestream an eigenvalue speed. problem The solid of linearized figure aeroelastic is from numerical model. Two predictions sets of experimental that were obtained data, labelled solving an figure eigenvalue by WT problem Data, are of available: linearized one aeroelastic set resulting model. from Two control setssurface of experimental excitation data, and labelled or in figure from by shaker WTexcitation Data, are available: of plunge one set degree resulting of from freedom. control At surface intermediate excitation speeds, and predictions or fromof shaker damped excitation frequency of agree plunge well degree with of measurements. freedom. At intermediate A reason speeds, for predictions agreement of degrading damped frequency at lower agree speed well and with close measurements. to flutter Aspeed reasonis for complication agreement degrading conducting at lower speed experiments. and close At to lower flutter speeds, speed shaker is excitation complication is used in conducting to excite modes, experiments. which in At turn lower may speeds, affect free shaker vibrations excitation of is used test rig. to excite Close to flutter, modes, a which difficulty in turn is may coalescence affect of free vibrations pitch and of plunge testfrequencies. rig. Close toa flutter, reasonable a difficulty agreement is is coalescence observed ofor pitch aeroelastic and plunge damping frequencies. ratio. ASimulations reasonable predict agreement a flutter is observed speed of for = aeroelastic m/s. damping ratio. Simulations predict a flutter speed of U f = m/s. (a) Damped frequency (b) Damping ratio Figure 5. Dependency of damped frequency, and damping ratio,, with freestream speed; Figure 5. Dependency of damped frequency, ω d and damping ratio, ζ, with freestream speed; aeroelastic parameters from Table ( and rad/s); flutter speed predicted at aeroelastic parameters from Table 1 (b = m and ω α = rad/s); flutter speed predicted m/s. at m/s Time-Domain Analysis 5.2. Time-Domain Analysis The hardening nonlinearity in plunge degree of freedom, Table 1, was found to significantly The hardening nonlinearity in plunge degree of freedom, Table 1, was found to significantly affect dynamics of system. The aeroelastic system exhibits subcritical LCOs and amplitude affect dynamics of system. The aeroelastic system exhibits subcritical LCOs and amplitude and frequency of LCOs were found in good agreement between measurements and predictions. One example is in Figure 6 where response to an initial perturbation in angle of attack is simulated at two subcritical speeds (8 and 13 m/s). The onset of LCOs is at a speed of m/s. The interest in this work is focused at testing control strategy of Section 4 at a speed = 8 m/s below onset point of LCOs.

11 aeroelastic parameters from Table 1 ( = m and = rad/s); flutter speed predicted at m/s Time-Domain Analysis Aerospace 2018, 5, of 19 The hardening nonlinearity in plunge degree of freedom, Table 1, was found to significantly affect dynamics of system. The aeroelastic system exhibits subcritical LCOs and amplitude and frequency of LCOs were found in ingood agreement between measurements and and predictions. One One example example is in isfigure in Figure 6 where 6 where response to to an aninitial perturbation in inangle of attack is simulated at at two twosubcritical speeds (8 and 13 m/s). The onset of LCOs is is at at a a speed of of m/s. m/s. The The interest interest in this work is is focused at at testing control strategy of of Section Section 4 at 4 aat speed a speed U = 8= m/s 8 m/s below below onset onset point of of LCOs. (a) Pitch (b) Plunge Figure 6. Time-domain response to an initial perturbation in angle of attack ( = 6 at two speeds Figure 6. Time-domain response to an initial perturbation in angle of attack (α below flutter speed, predicted at 0 = 6 at two speeds = m/s. below flutter speed, predicted at U f = m/s Model Identification The system transfer function, G, is identified by post-processing input-output relationship. The input, which describes time evolution of trailing-edge flap, is a chirp signal defined as ( u(t) = u 0 + u A sin 2π f ) t where u 0 is a constant value and u A is amplitude of sinusoidal motion. The frequency of sinusoidal motion, f, varies linearly in time (23) ( ) fm f m f = f m + t (24) T and covers frequency range [ f m, f M ] over total simulation time, T. The frequency range is chosen to adequately excite system over desired frequency range. In this work, parameters of chirp signal were set to u 0 = 0.0 and u A = 1.0. The aeroelastic model has two dominant natural frequencies at Hz (ω h = rad/s, plunge) and Hz (ω α = rad/s, pitch). For increasing dynamic pressure, pitch and plunge frequencies tend to coalesce, Figure 5 and so frequency range covered by chirp signal was chosen between f m = 0.01 Hz and f M = 8.0 Hz. The input signal employed for model identification is illustrated in Figure 7. The aeroelastic response induced by control surface is shown in Figure 8. The resonant behavior in both degrees of freedom confirms adequateness of chosen frequency range. A SISO model is considered in this exploratory work. The pitch degree of freedom is taken as measured output, whereas plunge is assumed unmeasurable. The transfer function, G, between control effector and pitch response is identified using prediction error minimization algorithm available in MATLAB. A preliminary study was carried out to ensure independence from number of zeros and poles used in constructing transfer function. It was found that six zeros and seven poles, which are listed in Table 2, provide a good approximation of aeroelastic system composed of 12 states in total. The real part of all poles is negative and resulting transfer function is refore

12 chosen chosen to to adequately excite excite system system over over desired desired frequency range. range. In In this this work, work, parameters of of chirp chirp signal signal were were set set to to = = and and = = The The aeroelastic model model has has two two dominant natural natural frequencies at at Hz Hz ( ( = = rad/s, rad/s, plunge) plunge) and and Hz Hz ( ( = = rad/s, rad/s, pitch). pitch). For For increasing dynamic dynamic pressure, pitch pitch and and plunge plunge frequencies Aerospace tend 2018, 5, 86tend to to coalesce, coalesce, Figure Figure 55 and and so so frequency range range covered covered by by chirp chirp signal signal was 12 was of 19 chosen chosen between between = = Hz Hz and and = = Hz. Hz. The The input input signal signal employed for for model model identification is is illustrated in in Figure Figure The The aeroelastic response induced induced by by control control surface surface is is stable. shown shown Asin confirmed in Figure Figure inthe Figure resonant resonant 9, behavior identified in in both both transfer degrees degrees function of of freedom freedom (with confirms six zeros and adequateness seven poles) of of is identical chosen chosen to frequency transfer function range. range. that relates Laplace transform of system output to Laplace transform of (chirp) input. Figure Figure Figure 7. Chirp Chirp Chirp input input input signal signal signal of of of trailing-edge trailing-edge flap flap used flap used used for for for identification identification of of of aeroelastic aeroelastic aeroelastic model model model ( (U = 8 m/s, ( = u 0 8 = 8 m/s, 0.0 m/s,, u = A = 0.0, ,, = f m = 1.0, , Hz = and = Hz f M Hz and = and 8.0 Hz). = = Hz). Hz). Aerospace 2018, 5, x FOR PEER REVIEW 12 of 18 A SISO model is considered in this exploratory work. The pitch degree of freedom is taken as measured output, whereas plunge is assumed unmeasurable. The transfer function,, between control effector and pitch response is identified using prediction error minimization algorithm available in MATLAB. A preliminary study was carried out to ensure independence from number of zeros and poles used in constructing transfer function. It was found that six zeros and seven poles, which are listed in Table 2, provide a good approximation of aeroelastic system composed of 12 states in total. The real part of all poles is negative and resulting transfer function (a) (a) Pitch Pitch is refore stable. As confirmed in Figure (b) (b) Plunge Plunge 9, identified transfer function (with six zeros and seven poles) is identical to transfer function that relates Laplace Figure Figure Time-domain Time-domain aeroelastic aeroelastic response response to to chirp chirp input input signal signal of of Figure Figure transform of Figure system 8. Time-domain output to aeroelastic Laplace response transform to of chirp (chirp) input signal input. of Figure 7. (a) Magnitude (b) Phase angle Figure 9. Bode plots of system time-response to a chirp input and identified transfer function; Figure 9. Bode plots of system time-response to a chirp input and identified transfer function; Origin indicates original model to be identified. Origin indicates original model to be identified. Table 2. Identified transfer function of single-input single-output (SISO) system. Zeros (-) Poles (-) ± 1.273i ± 0.604i ± 0.394i ± 0.376i

13 Aerospace 2018, 5, of 19 Table 2. Identified transfer function of single-input single-output (SISO) system. Zeros (-) Poles (-) ± 1.273i ± 0.604i ± 0.394i ± 0.376i Control for Gust Loads Alleviation Discrete Deterministic Gust First, search for worst-case-gust was carried out for one-minus-cosine family considering a range of gust gradient distance, H/b [10,200]. The search, which is an inexpensive process here for small size of numerical model, employed a standard Kriging interpolation as response surface method [30]. The worst-case-gust was identified to have a gust gradient distance H/b = 20. The gust intensity was set to a constant value, U ds /U = 0.1. Then, aeroelastic response caused by worst-case-gust encounter was simulated with and without adaptive feedforward controller. The comparison is reported in Figure 10. For pitch, response with control reveals a significant reduction in peak-to-peak rotation during and immediately after interaction with gust. A reduction of peak-to-peak rotation up to 50% is recorded. As gust moves away from surface of wing typical section, improvement in loads alleviation vanishes because effective reference input for control path is null. For plunge, re is no improvement in response with control compared to aeroelastic response without control. This is not unexpected because controller is a SISO system, with pitch rotation Aerospace 2018, 5, x FOR PEER REVIEW 13 of 18 being sole measurable quantity for GLA. (a) Pitch (b) Plunge Figure 10. Aeroelastic response due to worst-case one-minus-cosine gust with and without Figure 10. Aeroelastic response due to worst-case one-minus-cosine gust with and without control ( 3.5 m, = 0.8 m/s, = 8.0 m/s). control (H = 3.5 m, U ds = 0.8 m/s, U = 8.0 m/s). Figure 11 shows required control action for GLA and a sample of three weight coefficients of Figure controller. 11 shows The trailing-edge required control flap action is commanded for GLA and during a sample gust of three encounter, weight coefficients whereas it of is unactuated controller. as The trailing-edge gust perturbation flap is commanded moves away during from gust wing encounter, typical whereas section. it is As unactuated gust as propagates gust perturbation along airfoil, moves away pitch from rotation wing increases, typical section. as shown As in Figure gust propagates 10a. To counter along airfoil, increasing pitch pitch rotation rotation, increases, control as shown strategy in Figure commands 10a. To counter trailing-edge increasing flap down, pitch that rotation, is, a positive control control strategy input commands as shown in trailing-edge Figure 11. flap This down, control that action is, a positive has two control effects. input First, as shown lift in increases, Figure 11. increasing This control in action turn has vertical two effects. plunge First, deflection lift increases, which increasing is negative, in see turn Figure vertical 10b, according to sign convention of Section 2. Second, deflecting flap moves center of pressure downstream aft of elastic axis and increased lift causes a pitch-down rotation. This highlights capability of controller to exploit structural flexibility effects to reduce gust-induced aeroelastic vibrations. It is also worth noting that flap rotation and angular speed meet fully physical limitations of experimental test rig: maximum allowed rotation is ±7 and speed

14 Figure 11 shows required control action for GLA and a sample of three weight coefficients of controller. The trailing-edge flap is commanded during gust encounter, whereas it is unactuated as gust perturbation moves away from wing typical section. As gust propagates along airfoil, pitch rotation increases, as shown in Figure 10a. To counter increasing pitch rotation, control strategy commands trailing-edge flap down, that is, a Aerospace 2018, 5, of 19 positive control input as shown in Figure 11. This control action has two effects. First, lift increases, increasing in turn vertical plunge deflection which is negative, see Figure 10b, according plunge deflection to sign which convention is negative, of Section see Figure 2. Second, 10b, deflecting according to flap sign moves convention center of Section pressure 2. downstream Second, deflecting aft of elastic flap moves axis and center increased of pressure lift causes downstream a pitch-down aft of rotation. elastic This axis highlights and increased capability lift causes of a pitch-down controller to rotation. exploit This structural highlights flexibility capability effects of to reduce controller gust-induced to exploit aeroelastic structural flexibility vibrations. effects It is also to reduce worth gust-induced noting that aeroelastic flap rotation vibrations. and It angular is also worth speed noting meet fully that physical flap limitations rotation andof angular experimental speedtest meet rig: fully maximum physical limitations allowed rotation of experimental is ±7 and test speed rig: is limited maximum to 15 allowed Hz. A preliminary rotation is study ±7 and was performed speed is limited to select to 15order Hz. Aof preliminary controller. study It was found performed that to a select controller order of order of 20 controller. was a good It was compromise found that abetween controller cost of order and accuracy. 20 was a good The coefficients compromise of between controller cost and were accuracy. initially trained The coefficients for 10,000 of time-steps controller using were von Kármán initiallyturbulence trained for model. 10,000 time-steps using von Kármán turbulence model. (a) Flap rotation (b) Samples of weight coefficients Aerospace 2018, 5, x FOR PEER REVIEW 14 of 18 Figure 11. Aeroelastic response with control corresponding to Figure 10: (a) Commanded rotation of Figure 11. Aeroelastic response with control corresponding to Figure 10: (a) Commanded rotation of trailing-edge flap; (b) Three samples (out of 20) of weight coefficients of controller Continuous trailing-edge Atmospheric flap; (b) Three Turbulence samples (out of 20) of weight coefficients of controller For Continuous random Atmospheric case, atmospheric Turbulenceturbulence was generated with VKTG toolbox at an altitude h= 200 m with intensity set to moderate. The maximum instantaneous gust intensity For random case, atmospheric turbulence was generated with VKTG toolbox at an altitude was set to 10% of freestream speed, = 8 m/s. The time history of vertical speed of h = 200 m with intensity set to moderate. The maximum instantaneous gust intensity was set turbulence is shown in Figure 12. Figure 13 shows aeroelastic response with and without control. to 10% of freestream speed, U = 8 m/s. The time history of vertical speed of turbulence Similar considerations to those already noted for discrete gust hold true. The controller is effective is shown in Figure 12. Figure 13 shows aeroelastic response with and without control. Similar at reducing considerations pitch to those vibrations alreadythroughout noted for discrete entire gust duration hold true. of The gust controller encounter. is effective at reducing pitch vibrations throughout entire duration of gust encounter. Figure Figure Time Time historyof of random atmospheric turbulence with with PSD PSD equal equal to von to Kármán von Kármán spectrum spectrum (h (h = 200 = 200 m, m, intensity: moderate, U = = 8 8 m/s m/s and and maximum maximum instantaneous instantaneous intensity intensity of 0.8 m/s). of 0.8 m/s).

15 Figure 12. Time history of random atmospheric turbulence with PSD equal to von Kármán spectrum (h = 200 m, intensity: 10 3 moderate, = 8 m/s and maximum instantaneous intensity of 0.8 m/s). Aerospace 2018, 5, of 19 (a) Pitch (b) Plunge Aerospace 2018, 5, x FOR PEER REVIEW 15 of 18 Figure 13. Aeroelastic response due to a random atmospheric turbulence (Figure 12) with and without Figure 13. Aeroelastic response due to a random atmospheric turbulence (Figure 12) with and assumed control. without that control. sensor measures solely torsional mode. An alternative approach to mitigate effects of a SISO controller for a multi-modal system is to measure a combination of modes. This would To be quantify achieved, for impact present of test adaptive case, by feedforward placing controller sensor at for any GLA location in or presence than of atmospheric To quantify turbulence, impact mean of and adaptive standard feedforward deviation of controller pitch and for GLA plunge intime responses presence of elastic axis. are atmospheric listed in Table turbulence, 3. When confronted mean andwith standard uncontrolled deviation of aeroelastic pitch and response, plunge time statistics responses of are listed pitch inrotation Table 3. are When significantly confronted with improved uncontrolled by control aeroelastic action: response, mean value statistics of reduced pitch Table 3. Statistics of aeroelastic response induced by atmospheric turbulence. by rotation approximately are significantly 80% and improved standard by deviation control action: by approximately mean value 35%. isthe reduced increase by approximately in statistics 80% of andplunge standard is not unexpected, deviation by as approximately Pitch Rotation it was assumed 35%. unmeasurable The increaseplunge for in purpose statistics Deflection of of GLA. plunge The tradeoff unexpected, between as reduction it was assumed in pitch is not Mean unmeasurable statistics Standard and for Deviation increase purpose in Mean plunge of GLA. Standard statistics The trade-off is Deviation yet in between favor of reduction former. inwithout pitch statistics control and increase in plunge statistics is yet in favor of former. For For wing With wing typical control typical section, section, pitch pitch and plunge and plunge are structural are structural modes. The modes. input The to input controller to iscontroller a measure Improvement is of a measure pitch(%) of degree pitch of freedom, degree of that freedom, is, onethat structural is, one mode. structural Therefore, mode. Therefore, it is assumed it is that sensor measures solely torsional mode. An alternative approach to mitigate effects of a SISO Figure controller 14 shows for a multi-modal time evolution system of is tocontrol measure input a combination to achieve of modes. load alleviation This would and bea achieved, sample of for three coefficients present testof case, by controller. placing It is sensor worth mentioning at any location that or flap thanrotation elastic and axis. angular speed Figure meet 14fully shows physical time limitations evolutionof of experimental control input to apparatus achieve (rotation load alleviation up to ±7 and aspeed sample up of to three 15 Hz). coefficients of controller. It is worth mentioning that flap rotation and angular speed meet fully physical limitations of experimental apparatus (rotation up to ±7 and speed up to 15 Hz). (a) Flap rotation (b) Samples of weight coefficients Figure 14. Aeroelastic response with control corresponding to Figure 13: (a) Commanded rotation of Figure 14. Aeroelastic response with control corresponding to Figure 13: (a) Commanded rotation of trailing-edge flap; (b) Three samples (out of 20) of weight coefficients of controller. trailing-edge flap; (b) Three samples (out of 20) of weight coefficients of controller. 6. Conclusions The paper presents implementation of an adaptive feedforward control strategy to alleviate gust-induced aeroelastic vibrations. The test case is a model problem, consisting of a typical wing section undergoing pitch and plunge motions. The aerodynamics are given by classic ory of Theodorsen. The stiffness of springs is modelled in a polynomial form, introducing a nonlinear element to a linear aeroelastic model. The aeroelastic parameters were set to match a wind tunnel test

16 Aerospace 2018, 5, of 19 Table 3. Statistics of aeroelastic response induced by atmospheric turbulence. Pitch Rotation Plunge Deflection Mean Standard Deviation Mean Standard Deviation Without control With control Improvement (%) Conclusions The paper presents implementation of an adaptive feedforward control strategy to alleviate gust-induced aeroelastic vibrations. The test case is a model problem, consisting of a typical wing section undergoing pitch and plunge motions. The aerodynamics are given by classic ory of Theodorsen. The stiffness of springs is modelled in a polynomial form, introducing a nonlinear element to a linear aeroelastic model. The aeroelastic parameters were set to match a wind tunnel test rig and a good comparison between predictions and measurements was found. The effects of structural nonlinearity, for this specific test case, give rise to appearance of subcritical limit cycle oscillations. The demonstration of control strategy is performed on a single point near but below onset of limit cycle oscillations. The control strategy requires an approximate model of aeroelastic plant and this was created using a system identification method retaining nonlinear effects on system response. Although controller was designed to be a single-input single-output system, response with control was found effective to reduce peak-to-peak structural loads during gust encounters. For a discrete one-minus-cosine gust, peak-to-peak deformations were reduced by about 50% with a negligible effect on unmeasured structural mode. For a random atmospheric turbulence encounter, which has statistical properties defined by von Kármán spectrum, feedforward control significantly reduced mean and amplitude of aeroelastic vibrations. Directions for future work include: extension of current work to a multi-input multi-output aeroelastic system; an improved treatment of nonlinearities in creating approximate system transfer functions; implementation of controller on an experimental aerial platform which will be flight tested. Initial flight tests have already been conducted to characterize system dynamic response to forced excitations. Author Contributions: Conceptualization, G.M.; Methodology, D.A.; Software, validation, formal Analysis and writing-original draft preparation, W.Y. Funding: This research was funded by Royal Academy of Engineering (grant numbers: NCRP/1415/51 and ISS1415/7/44) and Engineering and Physical Sciences Research Council (grant number: EP/P006795/1). Acknowledgments: Wang acknowledges financial support received from China Scholarship Council (CSC) to carry out research presented in this work at University of Southampton. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Aeroelastic parameters in dimensional and non-dimensional form and ir definitions are reported below. b C L, C m C h, C α Ch c Cα c airfoil semi-chord lift and pitch moment coefficients viscous damping in plunge and pitch, respectively critical damping in plunge, 2 mk h critical damping in pitch, 2 I α K α

17 Aerospace 2018, 5, of 19 K h, K α plunge stiffness and torsional stiffness about elastic axis K ξ Non-dimensional plunge stiffness about elastic axis I α second moment of inertia of airfoil about elastic axis m airfoil sectional mass h plunge displacement H, P non-linear terms in state space equations S α first moment of inertia of airfoil about elastic axis t physical time x α airfoil static unbalance, S α /mb r a radius of gyration of airfoil about elastic axis, r 2 a = I α /mb 2 U freestream velocity U f linear flutter speed U reduced velocity, U/bω α τ state space vector w g gust vertical velocity Greek α angle of attack β h3, third and fifth order terms of plunge stiffness β h5 β α3, third and fifth order terms of pitch stiffness β α5 δ Trailing-edge flap deflection ε 1, ε 2 constants in approximation of Wagner function ε 3, ε 4 constants in approximation of Küssner function τ Non-dimensional time, tu /b ω h uncoupled plunging mode natural frequency, K h /m ω α uncoupled pitching mode natural frequency about elastic axis, K α /I α ω ratio of ω ξ /ω α ζ ξ damping ratio in plunge, C ξ /Cξ c ζ α damping ratio in pitch, C α /Cα c ξ Non-dimensional displacement in plunge, h/b τ mass ratio, m/πρb 2 Symbol ( ) differentiation with respect to τ, d( )/dτ References 1. Dempster, J.B.; Arnold, J.I. Flight Test Evaluation of an Advanced SAS for B-52 Aircraft. J. Aircr. 1969, 6, Stough, H.P.; Martzaklis, K.S. Progress in Development of Wear Information Systems for Cockpit; SAE ; U.S. Government: Washington, DC, USA, Available online: semanticscholar.org/1fae/ b575db3eae5d2a754aa936401bf45.pdf (accessed on 9 August 2018). 3. Honeywell International Inc. Anon. IntuVue RDR D Wear Radar Systems 2016; Technical White Paper; Honeywell International Inc.: Morris Plains, NJ, USA, Tantaroudas, N.D.; Da Ronch, A. Nonlinear reduced order aeroservoelastic analysis of very flexible aircraft. In Advanced UAV Aerodynamics, Flight Stability and Control: Novel Concepts, Theory and Applications; Marques, P., Da Ronch, A., Eds.; Wiley-Blackwell: Chichester, UK, 2017; pp ISBN Karpel, M. Design for Active Flutter Suppression and Gust Alleviation Using State-Space Aeroelastic Modeling. J. Aircr. 1982, 19, [CrossRef] 6. Regan, C.D.; Jutte, C.V. Survey of Applications of Active Control Technology for Gust Alleviation and New Challenges for Lighter-Weight Aircraft; NASA TM ; NASA: Washington, DC, USA, Gangsaas, D.; Ly, U.; Norman, D.C. Practical Gust Load Alleviation and Flutter Suppression Control Laws Based on a LQG Methodology. In Proceedings of 19th Aerospace Sciences Meeting, Saint Louis, MO, USA, January 1981.

18 Aerospace 2018, 5, of Zhou, Q.; Li, D.; Da Ronch, A.; Chen, G.; Li, Y. Computational Fluid Dynamics-Based Transonic Flutter Suppression with Control Delay. J. Fluid. Struct. 2016, 66, [CrossRef] 9. Brosilow, C.; Joseph, B. Techniques of Model-Based Control; Prentice Hall: Saddle River, NJ, USA, 2002; p Hahn, K.U.; Schwarz, C. Alleviation of Atmospheric Flow Disturbance Effects on Aircraft Response. In Proceedings of 26th Congress of International Council of Aeronautical Sciences, Anchorage, AK, USA, September Soreide, D.; Bogue, R.K.; Ehernberger, L.J.; Bagley, H. Coherent Lidar Turbulence Measurement for Gust Load Alleviation; Report No.: NASA Technical Memorandum ; Dryden Flight Research Center: Edwards, CA, USA, Zeng, J.; Moulin, B.; De Callafon, R.; Brenner, M. Adaptive Feedforward Control for Gust Load Alleviation. J. Guid. Control Dyn. 2010, 33, [CrossRef] 13. Sleeper, R. Spanwise Measurement of Vertical Components of Atmospheric Turbulence; NASA Report TP-2963; NASA: Washington, DC, USA, Wildschek, A.; Maier, R.; Hoffmann, F. Active Wing Load Alleviation with an Adaptive Feed-Forward Control Algorithm. In Proceedings of AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, CO, USA, August Wildschek, A.; Bartosiewicz, A.; Mozyrska, D. A Multi-Input Multi-Output Adaptive Feed-Forward Controller for Vibration Alleviation on a Large Blended Wing Body Airliner. J. Sound Vib. 2014, 333, [CrossRef] 16. Alama, M.; Hromcikb, M.; Hanisc, T. Active Gust Load Alleviation System for Flexible Aircraft: Mixed Feedforward/Feedback Approach. Aerosp. Sci. Technol. 2015, 41, [CrossRef] 17. Wildschek, A.; Maier, R.; Hahn, K.U.; Leißling, D.; Preß, M.; Zach, A. Flight Test with an Adaptive Feed-Forward Controller for Alleviation of Turbulence Excited Wing Bending Vibrations. In Proceedings of AIAA Guidance, Navigation, and Control Conference, Chicago, IL, USA, August Ghandchi Tehrani, M.; Da Ronch, A. Gust Load Alleviation Using Nonlinear Feedforward Control. In Proceedings of EURODYN 2014: IX International Conference on Structural Dynamics, Porto, Portugal, 30 June Wang, Y.; Li, F.; Da Ronch, A. Adaptive Feedforward Control Design for Gust Loads Alleviation of Highly Flexible Aircraft. In Proceedings of AIAA Atmospheric Flight Mechanics Conference, Dallas, TX, USA, June Theodorsen, T. General Theory of Aerodynamic Instability and Mechanism of Flutter; Report No.: NACA Report Nr. 496; NACA Langley Research Center: Hampton, VA, USA, Da Ronch, A.; Badcock, K.J.; Wang, Y.; Wynn, A.; Palacios, R.N. Nonlinear Model Reduction for Flexible Aircraft Control Design. In Proceedings of AIAA Atmospheric Flight Mechanics Conference, Minneapolis, MN, USA, August Jiffri, S.; Fichera, S.; Mottershead, J.E.; Da Ronch, A. Experimental nonlinear control for flutter suppression in a nonlinear aeroelastic system. J. Guid. Control Dyn. 2017, 40, [CrossRef] 23. Moorhouse, D.J.; Woodcock, R.J. Background Information and User Guide for MIL-F-8785C, Military Specification-Flying Qualities of Piloted Airplanes; DTIC Document; DTIC: Fort Belvoir, VA, USA, Gianfrancesco, M. Functional Modelling and Design of Energy Harvesters for Slender Wing Structures. Master s Thesis, University of Southampton, Southampton, UK, Zhao, Y.; Yue, C.; Hu, H. Gust Load Alleviation on a Large Transport Airplane. J. Aircr. 2016, 53, [CrossRef] 26. Da Ronch, A.; Badcock, K.J.; Khrabrov, A.; Ghoreyshi, M.; Cummings, R. Modeling of Unsteady Aerodynamic Loads. In Proceedings of AIAA Atmospheric Flight Mechanics Conference, Portland, OR, USA, 8 11 August Wang, Y.; Da Ronch, A.; Ghandchi Tehrani, M.; Li, F. Adaptive Feedforward Control Design for Gust Loads Alleviation and LCO Suppression. In Proceedings of 29th Congress of International Council of Aeronautical Sciences, St. Petersburg, Russia, 7 12 September Haykin, S. Adaptive Filter Theory, 3rd ed.; Prentice Hall Information and System Science Series; Prentice Hall: Englewood Cliffs, NJ, USA, 2001; pp

19 Aerospace 2018, 5, of Papaou, E.; Tantaroudas, N.D.; Da Ronch, A.; Cooper, J.E.; Mottershead, J.E. Active Control for Flutter Suppression: An Experimental Investigation. In Proceedings of International Forum on Aeroelasticity and Structural Dynamics, Bristol, UK, June Da Ronch, A.; Ghoreyshi, M.; Badcock, K.J. On Generation of Flight Dynamics Aerodynamic Tables by Computational Fluid Dynamics. Prog. Aerosp. Sci. 2011, 47, [CrossRef] 2018 by authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under terms and conditions of Creative Commons Attribution (CC BY) license (