A Machine Learned Model of a Hybrid Aircraft

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1 1 A Machine Leaned Model of a Hybid Aicaft Bandon Jones, Kevin Jenkins CS229 Machine Leaning, Fall 2016, Stanfod Univesity I. INTRODUCTION Aicaft development pogams ely on aicaft dynamic models fo flight contol design, handling qualities analysis, and vehicle simulation [1]. Expensive flight tests ae pefomed to validate the aicaft dynamic models with espect to the as-built aicaft, and to quantify unfoeseen vehicle flight issues. Flight tests typically consist of tageted system identification flight maneuves that ae pefomed while vehicle state data is collected [2]. Industy methods ely heavily on domain knowledge of aicaft dynamics fo manual piloted data geneation and featue identification [3]. This poject aims to simplify the modeling pocess with a Machine Leaning pipeline that geneates a non-linea vehicle model with minimal eliance on domain knowledge using full datasets thoughout an entie flight envelope. A hybid Vetical Takeoff and Landing (VTOL) aicaft with a fixed-wing with vetical and fowad hove motos is used as a case study. This aicaft configuation is paticulaly suitable to a Machine Leaning appoach due to complex aeodynamic inteactions and wide anging aispeed envelope which does not fit standad classical aicaft models. A. Backgound The dynamics of an aicaft can be most geneally expessed as the nonlinea fist-ode diffeential equation, ẋ = f(x, u) (1) with vehicle state x, and the vehicle state deivative ẋ, and actuato input u. Undestanding the function f fo a given aicaft is citical fo aifame design, contol law design, and simulation. Theefoe, a significant amount of effot is spent studying and modeling the function f fo a specific aicaft. to achieve acceptable pefomance and ae geneally aifame configuation specific. Methods employed include linea egession and maximum likelihood methods to define a linea model of an aicaft [5]. Well established locally linea model stuctues fo conventional aicaft consist of igid-body equations of motion and aeodynamic foces and moments [6] and can be summaized by the Linea Paamete Vaying State-Space model, ẋ = A(V a, ρ, α, β)x + B(V a, ρ, α, β)u. (2) The state-space matices A and B ae scheduled by aispeed V a, ai density ρ, angle-of-attack α, and sideslip β. These fomulations povide tangible insight into the aeodynamics of an aicaft, howeve they may fail to captue non-linea dependencies on othe paametes. A method to identify the Linea Paamete Vaying statespace models was investigated using an Expectation- Maximization technique [7], howeve this method equied specifically choosing scheduling paametes. A global, eal-time, nonlinea aicaft modeling technique was pesented that leveaged fuzzy logic and multivaiate othogonal functions to define elationships between a collection of identified locally linea models [8]. This appoach identified a model acoss a flight envelope, but elied on a-pioi fomulation of aeodynamic coefficients and equations of motion. All these appoaches ae consideed paametic egession methods in that they can be descibed by a finite numbe of paametes. A significant compaison between classical system identification paadigms of maximum likelihood and pediction eo methods with the Machine Leaning kenel-based egulaization techniques was pefomed in [9]. This showed that leaning techniques tailoed to specific featues of dynamic systems can outpefom conventional paametic appoaches fo identification of linea systems. B. Pevious Wok The discipline of extacting the model f(u, u) fom data is efeed to as system identification [4]. Specific system identification methods as applied to aicaft have been studied and applied fo nealy 100 yeas [2]. Howeve, cuent industy techniques still equie domain knowledge, hand analysis, and manual featue selection II. DATA The efeence taining and test data was geneated fom an existing non-linea, high-fidelity aicaft flight simulation that included igid body dynamics, aeodynamics, and actuato models. A total of 37 vehicle states wee ecoded and ae descibed in Table I. These states wee selected with little egad to the elevance

2 2 to demonstate an algoithm that does not equie specific domain knowledge to povide adequate pedictions. Fo example, moto toques, speeds, and thusts wee selected which ae highly coelated states. Duing nomal cuise o hove flight the vehicle may be at a static flight condition esulting in lage potions of collinea datapoints. Futhemoe, due to the augmented natue of the flight contols system states can be highly coelated. Although it is an objective to be obust to these factos, a test maneuve capability was added to the vehicle flight contol system softwae to stochastically command abitay actuato inputs to ensue the vehicle dynamics wee obsevable. A total of 34 minutes of simulated flight data acoss the aispeed envelope fom hove to cuise flight was collected. III. METHOD The poblem of identifying an aicaft dynamics model can be descibed as a supevised leaning egession poblem. The data is fomulated into a Makov Decision Pocess famewok and a non-paametic method is descibed that is based on Locally Weighted Linea Regession[10]. An advantage of Locally Weighted Linea Regession is it geneates a locally linea and paametic model at a paticula state. This allows the option to extact a locally linea model fo contol stability analysis o manual inspection of the dependance on states. Only the longitudinal states of the aicaft ae consideed fo this investigation to illustate the egession algoithm. A. Model The genealized equation descibing aicaft motion, 1 can be fomulated as a discete Makov Decision Pocess, x (t+1) = f(x, u) x (t) + g(x, u) u (t) (3) with state x (t) R 15 m, actuato input u (t) R 22 m. The objective is to lean the non-linea functions f(x, u) and g(x, u) to minimize the pediction eo of x (t+1). The taining examples (x (i), y (i) ) ae stacked into the matices, [ ] x (t) x X = (m 1) u (t) u (m 1) Y = α (t+1) α (m) θ (t+1) θ (m) θ (t+1) θ(m) V (t+1) ai V (m) ai with input featues X and taget vaiables Y consisting of longitudinal states only, angle of attack α, pitch θ, pitch ate θ, and aispeed V a. (4) B. Selectively Weighted Linea Regession A Selectively Weighted Linea Regession algoithm is poposed. The typical locally-weighted linea egession is used to find the paamete θ R ny n such that, θ = ag min θ ( i w(i) Y (i) θ T 2) (5) with weights, ( X w (i) (i) X 2 ) = exp 2τ 2. (6) This technique uses the same vecto X to establish weightings and to fom the egession coefficients in θ. Selectively weighted egession, by contast, uses featue selection to sepaate states that ae impotant fo local weighting fom those with stong egession infomation. All elements of X that ae not helpful fo weighting o egession can e emoved, then emoves elements individually fom X and X w, to be used in the new minimization fo θ R ny n, θ = ag min θ ( i w (i) Y (i) θ T ) 2 + λ θ 2 2 (7) with weights, w (i) 2 w X w = exp 2τ 2, (8) and an L 2 egulaization constant λ. The L 2 egulaization tem penalizes lage values of θ and impoves machine pecision issues with linealy-dependent columns. The taining samples X and X w contain selected elements fom X, [ X w (i) R nw = [ X (i) R n = j 1 k 1 ] T j 2... j nw ] T. (9) k 2... k n Removing specific featues may allow fo a moe lightweight algoithm that can use fewe coefficients and opeate moe quickly. It also allows fo moe pecise weighting by not down-weighting taining samples that ae fa fom the test enty on un-impotant dimensions. Two methods of featue selection ae pesented, Pinciple Component Analysis was un on the featues X, and only the top linealy uncoelated vaiables ae selected fo the egessos, X A hand smat selection of featues coesponding to only the longitudinal axes of the vehicle was selected fo X w. Fo example, featues on the oll

3 3 Aiplane States (x15) Aispeed Ai Density Angle of Attack Attitude (x3) Rotational Acceleation (x3) Body Acceleation (x3) Velocity (x3) Aiplane Actuatos (x22) Vetical and Fowad Moto Thust (x6) Vetical and Fowad Moto Toque (x6) Vetical and Fowad Moto Speeds (x6) Aeodynamic Contol Suface Deflection (x4) Table I AIRCRAFT STATES and yaw axis wee discaded. It would be the expectation that the hand selected featue set would pedict the longitudinal states with less eo due. A. Algoithm Pefomance IV. RESULTS A summay of the taining and test eo esults fom fou Locally Weighted Linea Regession methods ae pesented in Table II. A total of 34 minutes of data was used, whee 70% of the data was used fo taining and 30% used fo test. Specific τ and λ wee chosen based on paamete sweeps acoss a ange of values. Supisingly, egession methods that did not involve emoving featues woked best. In paticula, the smat emoval of states esulted in the wost pefomance, highlighting potential unintuitive vehicle dynamics whee longitudinal states have dependence on lateal diectional states. The esults show the L 2 egulaization did not impove the oveall test eo, and had wose taining eo fo small sample sets. This esult was paticulaly supising given L 2 egulaization should help the singulaly issues with the many collinea states in the paamete sets duing long staight and level flight segments. B. Taining Size A compaison between each method vesus taining size is pesented in Figue 1. In geneal the taining eos conveged to oughly a stable eo theshold. Howeve, the locally weighted methods showed potential fo futhe decease in eo with a lage taining set. The lage test eo could be evidence of high vaiance, whee additional taining examples would educe eo. C. Featue Removal The smat emoval of states esulted in less eo with fewe taining examples, howeve the eo did not impove with moe taining examples. This demonstates that this method did not leveage the non-standad dynamics of the hybid aicaft obseved in a moe geneal taining set. The lage eo and small gap between test and taining eo is evidence of high bias. Removal of featues using PCA demonstated lowe eo with fewe taining examples, howeve was not able to outpefom the methods that included all featues. D. Time Seies Results and Eo Level Veification An example of the one-step pefomance of Locally Weighted Linea Regession on the dataset with τ = 50 and λ = 10 3 is pesented in Figue 2. This esult shows easonable pedictions of angle of attack, pitch, pitch ate, and aispeed, patially duing peiods with lowe dynamics. Duing the peiod of highe maneuveing, fom 1 sec to 4 sec, pediction eos inceased as much as 10 deg/sec in pitch ate, howeve the othe states wee pedicted well. Oveall this pefomance would be easonable fo simulation analysis. E. Numeical Solution Consideations 1) Nomal Equation: To solve the optimization poblem in Equation 7, the nomal equation, θ = (X W X T + λ) 1 (X W Y T ) (10) was used. Unfotunately, since W R m m as the taining samples gow diectly solving Equation 10 is O(m 2.3 ). As a wokaound, taining examples that esulted weightings < wee emoved which impoved machine-pecision linea independence of columns. Howeve, the emaining columns wee faily simila esulting in matices with low condition numbes. 2) Gadient Descent: To ovecome the complexity incease, low condition numbes, and to allow Locally Weighted linea egession to scale to lage datasets, two gadient descent appoaches wee investigated. Geneally, the gadient of the Locally Weighted linea egession loss function can be fomulated as, θ J (θ) = 2 (X W i,i X θ X W i,i Y ). (11) 3) Stochastic Gadient Descent : Stochastic Gadient Descent was implemented using the update ule, θ θ + α( W i,i θ W i,i Y (i) (12) whee i is a andom taining example. This technique demonstated to be vey sensitive to choice of α, equiing tuning fo obustness acoss diffeent test samples

4 4 Linea Regession Method Taining Test λ τ Eo Samples Eo Samples Locally Weighted Locally Weighted (25 minutes) Weighted with PCA (9.23 minutes) Weighted smat emoval Table II COMPARISON OF LOCALLY WEIGHTED LINEAR REGRESSION METHODS Eo vesus Taining Size weighted weighted idge weighted idge pca weighted "smat" emoval taining eo 10 1 Eo Numbe of Taining Examples Figue 1. Taining and Test Eo Compaison Agessive Maneuveing Time Seies Example Longitudinal States, Actual vesus Estimated Angle of Attack (deg) Pitch Rate (deg/sec) Pitch (deg) Aispeed (m/s) Estimated 20 degees, degees/sec 0 15 metes/sec -20 Figue 2. Time Seies Compaison of Test vesus Pedicted States Time (sec)

5 5 and andom numbe seeds. When highly-weighted samples wee chosen fist by the andom numbe geneato convegence was moe pedictable. 4) Odeed-Gadient Descent: With this technique, the same gadient descent step was pefomed, but the highest-weighted taining sample was used fist. The algoithm stepped though each emaining sample in descending ode until completion. Because many taining samples wee used, it was not necessay to evisit a single sample moe than once. Iteation was halted when the Fobenius nom of the step in the egession coefficient matix, θ (t+1) θ (t) F < tol tol 10 6 (13) o afte 10,000 iteations. This algoithm geneally conveged within 1000 iteations fo taining examples. This technique is simila in context to the numeical technique of thowing out low-weighted taining samples, except that the highe-weighted ones ae used sequentially instead of in a batch. In geneal, this appoach woked a lot bette in tems of obustness, and equied a vey small α V. FUTURE WORK As a next step in the machine leaning pipeline, analysis could be pefomed on the calculated paametic models θ acoss the flight envelope and selected states. Fo example, by sweeping aispeed and aeodynamic suface states one could extact quantiative contol effectiveness estimates. Implementing Locally Weighted Linea Regession to leveage paallel computing would allow use of vey lage datasets that could cove wide anging cones of an aicaft s flight envelope. Finally, unning the algoithm on simulated and actual flight data that includes senso eo and noise is a next step to veify obustness of this egession technique. VI. CONCLUSION A non-paametic aicaft dynamic model was ceated fom simulated flight data using Locally Weighted Linea Regession. Time-seies data of 37 vehicle states fom a high fidelity simulato was fomulated into a discete Makov Decision Pocess famewok, with the objective of pedicting the next state based on the cuent state. The Locally Weighted Linea Regession method was modified to emove paticula featues fom the weights and egessos. In addition, L 2 egulaization was used to aid in singulaity issues of the nomal equation solution, howeve was found to have no impact on oveall eo. Attempts at simplifying the egession poblem by featue emoval did not pove to lowe eo, demonstating that the method is moe accuate when given a lage, moe compehensive featue set. Oveall, locally weighted linea egession povided easonable one-step state pedictions. Stochastic gadient descent techniques wee investigated fo futue scaling to much lage datasets. REFERENCES [1] Mak B Tischle. System identification methods fo aicaft flight contol development and validation. Advances in Aicaft Flight Contol, pages 35 69, [2] Eugene A Moelli and Vladislav Klein. Application of system identification to aicaft at nasa langley eseach cente. Jounal of Aicaft, 42(1):12 25, [3] Vladislav Klein and Eugene A Moelli. Aicaft system identification: theoy and pactice. Ameican Institute of Aeonautics and Astonautics Reston, Va, USA, [4] Lennat Ljung. Pespectives on system identification. Annual Reviews in Contol, 34(1):1 12, [5] Moelli E. Klein, V. Aicaft System Identification, Theoy and Pactice. Ameican Institute of Aeonautics and Astonautics, [6] Antoniewicz R. Kambee K. Duke, E. Deivation and definition of a linea aicaft model. NASA Refeence Publication, (1207), [7] Adian Wills and Bett Ninness. System identification of linea paamete vaying state-space models. Linea paamete-vaying system identification: new developments and tends, pages [8] Jay M Bandon and Eugene A Moelli. Real-time onboad global nonlinea aeodynamic modeling fom flight data. Jounal of Aicaft, pages 1 37, [9] Gianluigi Pillonetto, Fancesco Dinuzzo, Tianshi Chen, Giuseppe De Nicolao, and Lennat Ljung. Kenel methods in system identification, machine leaning and function estimation: A suvey. Automatica, 50(3): , [10] William S Cleveland. Robust locally weighted egession and smoothing scatteplots. Jounal of the Ameican statistical association, 74(368): , 1979.