A Robust Dynamic Inversion Technique For Asymptotic Tracking Control Of .
A Robust Dynamic Inversion Technique for Asymptotic Tracking Control ofan AircraftIlker Tanyer, Enver Tatlicioglu, and Erkan ZergerogluAbstract— In this paper, a tracking controller is developedfor an aircraft model subject to uncertainties in the dynamicsand additive state-dependent nonlinear disturbance-like terms.In the design of the controller, dynamic inversion techniqueis utilized in conjuction with a robust term. Only the outputof aircraft dynamics is utilized in the controller design andacceleration measurements are not required. Lyapunov basedstability analysis is used to prove global asymptotic tracking.I. I NTRODUCTIONDynamic inversion (DI) technique, which is a controldesign approach for nonlinear systems, was firstly developedfor, and generally used in aerospace systems [1], [2]. Themain idea of this technique is to transform the nonlinearsystem to a linear time invariant system by making change ofvariables and after using an appropriate control input, to drivethe linear aircraft dynamics to a reference model [3], [4],[5], [6], [7], [8]. Guarino et al. utilized DI technique in servocontrol design and also compared it with traditional feedbackcontrollers [3]. In [4], Oppenheimer and Doman applied theDI technique to stabilize an unstable, non-minimum phasehypersonic aircraft system. In [5], a DI based controller wasproposed for Wiener systems. In [6], a DI based methodwas developed for finite time stability of a class of nonlinearsystems where the input matrix was full rank. In [7], anautonomous flight control system was developed for a smallscale unmanned helicopter based on approximate DI method.In [8], closed–loop stability of a six degree-of-freedomnonlinear air-to-air missile was ensured with a DI basedcontroller. A DI based aircraft controller was developed forautonomous operation of a linear Yamaha RMAX helicopterin [9]. DI based controllers were also applied to experimentalsystems as in [10] and [11].In the literature, DI technique is generally utilized whensystem dynamics is known. However, in some cases, andspecifically for flight systems, exact dynamics is not available. When the system dynamics is subject to uncertainties(be it structured or unstructured), DI based algorithms canhave difficulty in compensating for these uncertainties dueto the increase in inversion error. Another reason for theincrease of inversion error is the uncertainties in the inputmatrix. To avoid the increase in inversion error, uncertaintiesmust be compensated by fusing the DI technique withadaptive and/or robust techniques. Some past research wasI. Tanyer and E. Tatlicioglu are with the Department of Electrical &Electronics Engineering, Izmir Institute of Technology, Urla, Izmir, 35430Turkey. ilkertanyer@iyte.edu.trE. Zergeroglu is with the Department of Computer Engineering, GebzeInstitute of Technology, Gebze, Kocaeli, 41400 Turkey.devoted to fusing DI technique with robust controllers. In[12], a robust DI method based on sliding mode controlwas proposed for tracking control of an unpowered flyingvehicle. Yamasaki et al. proposed a robust DI controllerfor tracking control of an unmanned aerial vehicle (UAV)[13]. In [14], A stochastic robust nonlinear control approachfused with DI technique was applied to a highly nonlinearcomplicated aircraft model. In [15], a nonlinear dynamicinversion controller is combined with a PI controller to linearize the dynamics of UAVs. Some other past research fusedadaptive control techniques with DI to compensate for linearly parameterizable uncertainties. For example, in [16], DItechnique based null-space injection controller, and in [17],an adaptive DI based switching control methodology wasproposed to compensate for structured uncertainties. In [18],DI was used in conjuction with a nonlinear model referenceadaptive controller (MRAC) based on neural networks. Chenet al. proposed an adaptive dynamic inversion (ADI) basedfeedback linearization control for a flexible spacecraft [19].To compensate for modeling errors and external disturbances,Wang and Stengel designed an ADI controller for a miniatureUAV [20]. In [21], Calise and Rysdek proposed an ADIcontroller which was a combination of adaptive feedforwardneural networks with feedback linearization. Lavretsky andHovakimyan designed a direct MRAC augmented with aDI controller [22]. ADI based controllers, while compensating for structured uncertainties, mostly failed to addressunstructured uncertainties. To compensate for both structuredand unstructured uncertainties, neural networks were utilizedin conjuction with ADI based controllers [23], [24], [25],[26] and [27]. However, in these works, while boundednessof the tracking error was ensured, asymptotic tracking waslost. Recently, Shin et al. developed a position trackingcontrol system for a rotorcraft-based unmanned aerial vehicle(RUAV) by using robust integral of the signum of the error(RISE) feedback and neural network feedforward terms [28],[29]. Different from typical neural network based robustcontrollers, this method guaranteed semi-global asymptotictracking. In [30], MacKunis et al. fused the robust controllerin [31], [32] with DI technique to achieve asymptotic outputtracking for aircraft systems with an uncertain input matrixand subject to additive unknown nonlinear disturbances.However, the signum of the time derivative of the outputwas utilized (i.e., acceleration information was required) inthe design of the controller. Acceleration measurements arewidely used in aircraft systems for system identification orcontrol design. While acceleration measurements are available for some aircraft systems, utilizing these measurements978-1-4673-5769-2/13/ 31.00 2013 IEEE
in control design may not be preferred from control theoryperspective. Additionally, although accelerometers may beseen as good and practical solutions in system identificationand control applications, there are several reasons for notusing them in some applications. Firstly, aside from onerousness in implementation, one needs to deal with sensor–relatedissues such as calibration and possible sensor failures. Oneway to avoid calibration requirements and sensor failures is,if possible, not to use them. For some cases, using themmay be considered as redundant due to their costs. Whilethe costs of sensors are decreasing rapidly, using them stilladds to the cost of the overall system. Furthermore, asidefrom these, it should also be noted that using an additionalsensor complicates the sensing system.In this paper, model reference tracking control of anuncertain aircraft model subject to uncertainties is discussed.Specifically, the state and the input matrices are considered tobe uncertain, and the dynamics is subject to an additive statedependent nonlinear disturbance-like terms. Furthermore, toremove the need for acceleration measurements, we considerthat only the output of the aircraft being available for controldevelopment. In the design of the controller, the robustintegral of the sign of the error component in [31], [32]is utilized. Since the input matrix of the aircraft systemis considered to be uncertain, a matrix decomposition isutilized in the development of the error system. The controldesign is based on Lyapunov based design and analysistechniques, and global asymptotic stability of the trackingerror is ensured.II. A IRCRAFT M ODELFollowing aircraft model is considered [4], [16], [33], [34]ẋ Ax f (x, t) Bu , y Cx(1)where x(t) Rn is the state vector, A Rn n is thestate matrix, B Rn m is the input matrix, y(t) Rmis the output, C Rm n is the output matrix, u(t) Rmis the control input, and f (x, t) Rn is a state-dependentnonlinear disturbance-like term representing gravity, inertialcoupling and nonlinear gust modeling. The above model isassumed to satisfy the following properties.Assumption 1: The model in (1) is controllable.Assumption 2: Thestate-dependentnonlineardisturbance-like term f (x, t) is continuously differentiableand bounded up to its first order time derivative (i.e.,f (x, t) C 1 and f (x, t), f (x, t) L ).III. C ONTROL D ESIGNThe control design objective is to develop a robust controllaw that ensures that the output of the aircraft model y (t)tracks the output of a reference model that will be designedsubsequently, and additionally, all closed-loop signals arerequired to remain bounded. In the subsequent development,C is assumed to be known, while A, B, and f (x, t) areconsidered to be uncertain, thus, will not be utilized in thecontrol design. The subsequent development is derived basedon the assumption that only the output y (t) is measurable.The reference model is represented asẋm Am xm Bm um , ym Cxm(2)where xm (t) Rn is the reference state vector, Am Rn n is the reference state matrix, Bm Rn m is thereference input matrix, um (t) Rm is the reference input,ym (t) Rm is the reference output, and C is the sameoutput matrix in (1). The reference state matrix Am is chosento be Hurwitz, and the reference input um (t) and its timederivative are designed as bounded functions. Linear analysistools can then be utilized along with these assumptions toprove that xm (t), ẋm (t), ẍm (t) and thus, ym (t), ẏm (t),ÿm (t) are bounded functions.To quantify the tracking control objective, an output tracking error, denoted by e (t) Rm , is defined ase , y ym C(x xm ).(3)In the subsequent development, the error system will bedesigned based on a filtered tracking error, denoted by r (t) Rm , which is defined asr , ė Λe(4)where Λ Rm m is a constant, positive definite, diagonalcontrol gain matrix. It is noted that since only y (t) isavailable then ė (t) and thus r (t) are not measurable, andcannot be utilized in the control design.Assumption 3: Since the number of states is strictlygreater than the number of outputs (i.e., n m), there maybe some states that can not be observed through the output.The subsequent control development and stability analysisrely on the assumption that the state vector can be partitionedasx xo xu(5)where xo (t) Rn contains the observable states through theoutput, and xu (t) Rn contains the unobservable states.Furthermore, the unobservable states can be partitioned asxu xuρ xuξ(6)where xuρ (t), xuξ (t) Rn contain the unobservable statesthat can be bounded by a function of error signals and aconstant, respectively. Mathematically speaking, followingbounds are assumedkxuρ (t)k c1 kzk and kxuξ (t)k ξxu(7)where c1 , ξxu R are known positive bounding constants T R2m is the combined error signal. Aand z , eT , rTsimilar upper bound can be obtained for the components ofẋu (t) in the sense thatkẋuρ (t)k c2 kzk and kẋuξ (t)k ξẋu(8)where c2 , ξẋu R are known positive bounding constants.Similar to (5), the reference state vector xm (t) can bepartitioned asxm xmo xmu(9)
where xmo (t) Rn contains the entries of the referencestate vector corresponding to the observable states of the statevector, and xmu (t) Rn contains the rest of the entries ofthe reference state vector.After substituting (1)-(3) into (4), following expression canbe obtainedr CAx Ωu Cf CAm xm CBm um Λe (10)where Ω , CB Rm m is an auxiliary constant matrix.Since B is uncertain, then Ω is uncertain as well. Furthermore, we do not know whether or not Ω is symmetric and/orpositive definite. Given these restrictions, we consider theSDU decomposition of Ω as [35]Ω SDU(11)where S Rm m is a symmetric positive-definite matrix,D Rm m is a diagonal matrix with entries 1 and U Rm m is a unity upper triangular matrix. Details of the SDUdecomposition can be found in [36].Remark 1: We evaluated the SDU decomposition of Ωfor different aircraft models in the literature. For all thesemodels, we observed that the diagonal matrix D was equalto identity matrix. However, for the sake of completeness,the subsequent controller will be designed to be applicableto any diagonal matrix D without imposing any restrictions,as long as it is available for control design.After utilizing (11), the time derivative of the filteredtracking error r(t) can be written asṙ CAẋ SDU u̇ C f CAm ẋm CBm u̇m Λė. (12)After premultiplying (12) with M , S 1 Rm m ,following expression can be obtainedM ṙ M [CAẋ C f CAm ẋm CBm u̇m Λė] DU u̇.(13)It is noted that, since S is symmetric and positive-definite,then so is M . An auxiliary signal, denoted by N (x, ẋ, t) Rm is defined asN , M [CAẋ C f CAm ẋm CBm u̇m Λė] e (14)which can be utilized to rewrite the expression in (13) asM ṙ N e DU u̇.(15)The main idea behind partitioning N as in (16)-(19) is tomake use of the following facts.Remark 2: From Assumptions 1 and 3, and the assumption on boundedness of the reference model signals, it can beshown that Nd (t) is a bounded function of time in the sensethat kNd k ζNd t where ζNd R is a positive boundingconstant. Or alternatively, Nd,i ζNd,i t with ζNd,i Rbeing positive bounding constants.Remark 3: The auxiliary error like term in (19) can beupper bounded askÑ k ρ kzk(19)where ρ R is a positive bounding constant.Based on the subsequent stability analysis, the controlinput is designed asZ tu DK[e(t) e(0) Λe(τ )dτ ] DΠ(20)0where Π (t) Rm is an auxiliary filter signal updatedaccording to1Π̇(t) βSgn(e(t)) with Π(0) 0m 1where β Rm m is a constant, positive-definite, diagonal control gain matrix, Sgn (·) denotes the vector signumfunction, and K Rm m is a constant, positive-definite,diagonal control gain matrix and defined asK Im kg Im diag{kd,1 , kd,2 , ., kd,m 1 , 0}u̇ DKr DβSgn(e)where Nd (t) Rby constantsm(16)contains functions that can be boundedNd , M CAẋuξ M C f M CBm u̇m M C(A Am )ẋmo(17)and Ñ (x, ẋ, e, ė) Rm is an auxiliary error-like termdefined as followsÑ,M C[A(ẋo ẋmo ) Aẋuρ Am ẋmu ] (18) M Λ(r Λe) e.(23)where (4) and (21) were utilized. After substituting (23) into(15), following closed-loop error system is obtainedM ṙ Nd Ñ e DU DβSgn(e) D(U Im )DKr Kr(24)where (16) was utilized.Since U is unity upper triangular then U Im is strictlyupper triangular, thus we can rewrite the D(U Im )DKrterm as TD(U Im )DKr ΦT , 0(25)where the entries of Φ (r) R(m 1) 1 are defined asmXj i 1N Nd Ñ(22)with kg , kd,1 , ., kd,m 1 R being positive gains. The timederivative of the control input in (20) is obtained asΦi diThe auxiliary signal N can be partitioned as(21)dj kj Ui,j rj for i 1, ., (m 1).(26)Since di 1 i 1, ., m, following upper bound can beobtained for the entries of ΦmX Φi kj ζUi,j rj ζΦi kzk(27)j i 1where ζUi,j are positive bounding constants satisfyingζUi,j Ui,j i, j. It is important to highlight that ζΦidepends on the control gains ki 1 , ., km .1 Throughout the paper, I and 0nm r will be used to represent an n nstandard identity matrix and an m r zero matrix, respectively.
IV. S TABILITY A NALYSISTheorem 1: The controller given in (20), (21) ensuresglobal asymptotic tracking in the sense thatke (t)k 0 as t the output was utilized, while in our work only the outputinformation was utilized in the design of the controller.A PPENDIX IB OUNDEDNESS P ROOF(28)provided that the control gain matrices K and β are selectedby using the following procedure:1) For i m, βm is selected according to γ2(29)βm ζNd,m 1 Λmand from i m 1 to i 1, βi are selected accordingto mXγ2 βi ζNd,i ζΨj βj1 (30)Λij i 1where γ2 R is some positive bounding constant andthe subscript i 1, . . . , m denotes the i-th element ofthe vector or the diagonal matrix.2) Control gain kg is chosen big enough to decrease theρ2constant 4k.g3) Choose kd,i , i 1, . . . , (m 1) to decrease thePm 1 ζΦ2iconstant i 1 4kd,i.Proof: The proof of theorem has four subproofs. In thefirst part, boundedness of all the signals under the closedloop operation will be presented (see Appendix I). Secondly,a lemma and its proof (which utilizes the boundedness of theerror signals) will be presented (see Appendix II). The proofofR t this lemma will provide an upper bound on the terms0 ėi (τ ) dτ , which will then be utilized in the next part ofthe proof. In the third part, the positiveness of an auxiliaryintegral term is demonstrated (see Appendix III). Finally, theasymptotic tracking result is proven (see Appendix IV).V. C ONCLUSIONA robust controller was designed for an aircraft modelsubject to uncertainties in the dynamics and additive statedependent nonlinear disturbance-like terms. In the design ofthe controller, a DI technique was used in conjuction withrobust integral of the sign of the error terms to compensatefor the uncertainties in the dynamic model. Lyapunov typestability analysis techniques were utilized to ensure globalasymptotic tracking of the output of a reference model.When compared with the similar studies in the literature,the key contribution of the proposed work is that only theoutput of the aircraft model was utilized in the control designand no acceleration measurements were required. Specifically, the closest work to ours is the work of MacKuniset al. in [30] where adaptive and robust controllers weredesigned for uncertain aircraft models subject to uncertainties in the dynamics and additive state-dependent nonlineardisturbance-like terms. In the design of the controllers, a DItechnique was used in conjuction with robust integral of thesign of the error terms to obtain a similar result. However,in the design of the controllers in [30] the time derivative ofIn this appendix, the boundedness of all the signals underthe closed-loop operation will be demonstrated. Let V1 (z) R be a Lyapunov function defined as1 T1e e rT M r22which can be upper and lower bounded asV1 ,(31)11min{1, Mmin}kzk2 V1 (z) max{1, Mmax}kzk222where Mmin and Mmax denote minimum and maximumeigenvalues of M , respectively. Time derivative of the Lyapunov function can be written asV̇1 eT Λe rT Nd rT Ñ rT DU DβSgn(e)m 1X TrT ΦT , 0 rT r kg rT r kd,i ri2 . (32)i 1After utilizing (27), following upper bound can be obtainedm 1XX T m 1r T ΦT , 0 ri Φi ζΦi ri kzk.i 1(33)i 1After substituting the upper bounds in Remarks 2 and 3, andutilizing (33), following expression can be obtained eT Λe krk2 ζNd krk ζ1 krk ρkrkkzkm 1m 1XX kg krk2 ζΦi ri kzk kd,i ri2(34)V̇1i 1i 1where rT DU DβSgn(e) ζ1 krk was utilized with ζ1 R being a positive bounding constant. After utilizing belowmanipulationsζ1 krk ζNd krk ρkrkkzk kg krk2 ζΦi ri kzk kd,i ri2 1krk2 δ (ζ1 ζNd )2 (35)4δρ2kzk2(36)4kgζΦ2 ikzk2(37)4kd,i i 1, ., (m 1), where δ R is a positive dampingconstant, the right-hand side of (34) can be upper boundedasV̇1 [min{Λmin, (1 δ (ζ1 ζNd )2m 1X ζΦ21ρ2i)} ]kzk24δ4kg4kd,ii 1(38)where Λmin denotes the minimum eigenvalue of Λ. Providedthat the control gains Λ, kg , kd,1 , ., kd,m 1 are selectedsufficiently high, the above expression can be rewritten asV̇1 c1 V1 c2(39)
where c1 and c2 are some positive bounding constants. From(39), it can be concluded that V1 (t) L , and thus, e(t),r(t) L . The definition of r(t) in (4) can be utilized toprove that ė(t) L . By using (3) and its time derivative,along with the assumption that the reference model signalsbeing bounded, it can be proven that y (t), ẏ (t), x (t), ẋ (t) L . The above boundedness statements and Assumption 2can be utilized along with (1) to prove that u (t) L .From (23), it is easy to see that u̇ (t) L . After utilizingthe above boundedness statements, Assumption 2, and theassumption that the reference model signals being boundedalong with (12), it is clear that ṙ (t) L . Standard signalchasing algorithms can be used to prove that all remainingsignals are bounded.A PPENDIX IIIL EMMA 2 AND ITS PROOFasLemma 2: Let the auxiliary function L(t) R be definedL , rT (Nd DU DβSgn(e)).If the entries of β are selected to satisfy the conditions in (29)and (30), then it can be concluded that P (t) R defined asZ tP , ζb L(τ )dτ.(47)0is nonnegative where ζb R is a positive bounding constant.Proof: The proof can be found in [43].A PPENDIX IVA SYMPTOTIC STABILITYA PPENDIX IIL EMMA 1 AND ITS PROOFLemma 1: Provided that e(t) and ė(t) are bounded, thefollowing expression for the upper bound of the integral ofthe absolute value of the i-th entry of ė(t) can be obtained[41]Z tZ t ėi (τ ) dτ γ1 γ2 ei (τ ) dτ ei (40)t0t0where γ1 , γ2 R are some positive bounding constants.Proof: First, we note that if ei (t) 0 on some interval,then ėi (t) 0 on the same interval, and the inequality (40)yields this qualification. Therefore, without loss of generality,we assume that ei (t) is absolutely greater than zero on theinterval of [t0 , t]. Let T [t0 , t) be the last instant of timewhen ėi (t) changes sign. Then, on the interval [T, t], ėi (t)has a constant sign, henceZ tZ t ėi (τ ) dτ ėi (τ )dτ ei (t) ei (T ) .(41)TTFrom the boundedness of ėi (t), it follows that there exist aconstant γ 0 such that ėi (t) γ, thereforeZ T ėi (τ ) dτ γ(T t0 ).(42)t0On the other hand, by applying the Mean Value Theorem[42], we can obtain the following expressionZ T ei (τ ) dτ (T t0 )ei .(43)t0where ei is some intermediate value of ei (t) on the interval[t0 , T ]. By assumption, ei is bounded away from zero.Therefore, from (42) and (43), we can conclude asZ TZ T ėi (τ ) dτ γ2 ei (τ ) dτ(44)t0t0where γ2 , eγi . Combining the relationships in (41) and(44), we can writeZ tZ t ėi (τ ) dτ ei (t) γ2 ei (τ ) dτ ei (T ) (45)t0t0which yields in (40) with γ1 , sup ei (T ) .(46)PROOFIn this appendix, the asymptotic stability of the outputtracking error is presented.Let V2 (w) R be a Lyapunov function defined asV2 , V1 P(48) Twhere w (t) , eT rT R(2m 1) 1 . It shouldPbe noted that, the non-negativeness of P (t), which is essential to prove that V2 (w) is a valid Lyapunov function, wasproven in Appendix III. The Lyapunov function in (48) canbe upper and lower bounded as follows 11min{1, Mmin}kwk2 V2 (w) max{ Mmax , 1}kwk2 .22Taking the time derivative of the Lyapunov function in (48),substituting (32) and time derivative of (47), and after somestraightforward manipulations, we obtain TV̇2 eT Λe rT Ñ rT ΦT , 0 rT r kg rT r m 1Xkd,i ri2 .(49)i 1After utilizing (36) and (37), the right-hand side of (49) canbe upper bounded as#"m 1X ζΦ2ρ2i kzk2.V̇2 min{λmin (Λ), 1} 4kg4kd,ii 1(50)Provided that the control gains Λ, kg , kd,1 , ., kd,m 1are selected sufficiently high, the below expression can beobtained for the derivative of the Lyapunov functionV̇2 c3 kzk2(51)where c3 is some positive bounding constant. From (48) and(51), it is clear that V2 (w) is nonincreasing and bounded.After integrating (51), it can be concluded that z(t) L2 .Since z(t) L L2 and ż(t) L , from Barbalat’sLemma [42], kz(t)k 0 as t , thus meeting thecontrol objective. Since no restrictions with respect to theinitial conditions of the error signals were imposed on thecontrol gains, the result is global.
R EFERENCES[1] F. Lewis and B. L. Stevens, ”Aircraft Control & Simulation,” NewYork, NY, USA: John Wiley & Sons, 2003.[2] D. Enns, D. Bugajski, R. Hendrick, and G. Stein, ”Dynamic inversion:An evolving methodology for flight control design,” Int. J. of Cont.,vol. 59, no. 1, pp. 71–91, 1994.[3] C. Guarino, L. Bianco, and A. Piazzi, ”A servo control system designusing dynamic inversion,” Cont. Eng. Pra., vol. 10, pp. 847-855, 2002.[4] M. W. Oppenheimer and D. B. Doman, ”Control of an unstable,nonminimum phase hypersonic vehicle model,” IEEE Aerosp. Conf.,Big Sky, MT, USA, 2006, pp. 1-7.[5] Z. Szabo, P. Gaspar, and J. Bokor, ”Tracking design for Wienersystems based on dynamic inversion,” Int. Conf. on Cont. Appl.,Munich, Germany, 2006, pp. 1386-1391.[6] S. Onori, P. Dorato, S. Galeani, and C. Abdallah, ”Finite time stabilitydesign via feedback linearization,” IEEE Conf. on Decision & Cont.,Seville, Spain, 2005, pp. 4915-4920.[7] Z. Zhang, F. Hu, and J. Li, ”Autonomous flight control systemdesigned for small-scale helicopter based on approximate dynamicinversion,” Int. Conf. on Adv. Comp. Cont., 2011, pp. 185–191.[8] C. J. Schumacher and P. P. Khargonekar, ”Stability Analysis of aMissile Control System with a Dynamic Inversion Controller,” J. ofGuidance, Cont. & Dynamics, vol. 21, no. 3, pp. 508-515, 1998.[9] D. Enns and T. Keviczky, ”Dynamic inversion based flight control forautonomous RMAX helicopter,” American Cont. Conf., Minneapolis,MN, USA, 2006, pp. 3916–3923.[10] C. Guowei, A. K. Cai, B. M. Chen, and T. H. Lee, ”Construction,modeling and control of a mini autonomous UAV helicopter,” IEEInt. Conf. Automation & Logistics, 2008, pp. 449–454.[11] K. Peng, M. Dong, B. M. Chen, G. Cai, K. Y. Lum, and T. H.Lee, ”Design and implementation of a fully autonomous flight controlsystem for a UAV helicopter,” Chinese Cont. Conf., 2007, pp.662–667.[12] Z. Liu, F. Zhou, and J. Zhou, ”Flight control of unpowered flyingvehicle based on robust dynamic inversion,” Chinese Cont. Conf.,Heilongjiang, China, 2006, pp. 693-698.[13] T. Yamasaki, H. Sakaida, K. Enomoto, H. Takano, and Y. Baba, ”Robust trajectory-tracking method for UAV guidance using proportionalnavigation,” Int. Conf. on Cont. Autom. Sys., 2007, pp. 1404-1409.[14] Q. Wang and R. F. Stengel, ”Robust nonlinear flight control of a highperformance aircraft,” IEEE Tr. Cont. Sys. Tech., vol. 13, no. 1, pp.15-26, 2005.[15] Z. Xie, Y. Xia, and M. Fu, ”Robust trajectory-tracking method forUAV using nonlinear dynamic inversion,” IEEE Int. Conf. Cyber. &Intel. Sys., 2011, pp. 93-98.[16] A. D. Ngo and D. B. Doman, ”Dynamic inversion-based adaptive/reconfigurable control of the X-33 on ascent,” IEEE Aerosp. Conf.,Big Sky, MT, USA, 2006, pp. 2683-2697.[17] M. D. Tandale and J. Valasek, ”Adaptive dynamic inversion control ofa linear scalar plant with constrained control inputs,” American Cont.Conf., Portland, OR, USA, 2005, pp. 2064-2069.[18] X. J. Liu, F. Lara-Rosano, and C. W. Chan, ”Model-reference adaptivecontrol based on neurofuzzy networks,” IEEE Tr. Sys. Man Cyber. C,Appl. Rev., vol. 34, no. 3, pp. 302-309, 2004.[19] J. Chen, D. Li, X. Jiang, and X. Sun, ”Adaptive feedback linearizationcontrol of a flexible spacecraft,” Conf. Intell. Sys. Design Appl., Jinan,China, 2006, pp. 225-230.[20] Q. Wang and R. F. Stengel, ”A dynamic inversion controller designfor miniature Unmanned Aerial Vehicles,” Consumer Elec., Comm. &Networks, pp. 1916–1921, 2011.[21] A. Calise and R. Rysdyk, ”Nonlinear adaptive flight control usingneural networks,” IEEE Cont. Sys. Mag., vol. 18, no. 6, pp. 14-25,1998.[22] E. Lavretsky and N. Hovakimyan, ”Adaptive compensation of controldependent modeling uncertainties using time-scale separation,” IEEEConf. on Decision & Cont., Seville, Spain, 2005, pp. 2230-2235.[23] J. Leitner, A. Calise, and J. V. R. Prasad, ”Analysis of adaptive neuralnetworks for helicopter flight controls,” J. Guidance Cont. Dynamics,vol. 20, no. 5, pp. 972-979, 1997.[24] Y. Shin, ”Neural network based adaptive control for nonlinear dynamicregimes,” PhD diss., Georgia Inst. Tech., Atlanta, GA, USA, 2005.[25] E. N. Johnson and A. J. Calise, ”Pseudo-control hedging: A newmethod for adaptive control,” Workshop Adv. Guid. Cont. Tech.,Redstone Arsenal, AL, USA, 2000.[26] C. Schumacher and J. D. Johnson, ”PI control of a tailless fighteraircraft with dynamic inversion and neural networks,” American Cont.Conf., 1999, pp. 4173-4177.[27] R. Rysdyk, F. Nardi, and A. J. Calise, ”Robust adaptive nonlinear flightcontrol applications using neural networks,” American Cont. Conf.,1999, pp. 2595-2599.[28] J. Shin, H. J. Kim, K. Youdan, and W. E. Dixon, ”Asymptotic attitudetracking of the rotorcraft-based UAV via RISE feedback and NNfeedforward,” IEEE Conf. Decision & Cont., 2010, pp. 3694–3699.[29] J. Shin, H. J. Kim, K. Youdan, and W. E. Dixon, ”AutonomousFlight of the Rotorcraft-Based UAV Using RISE Feedback and NNFeedforward Terms,” IEEE Tr. Cont. Sys. Tech., vol. 20, no. 5, pp.1392–1399, 2012.[30] W. MacKunis, P. M. Patre, M. K. Kaizer, and W. E. Dixon, ”Asymptotic Tracking for Aircraft via Robust and Adaptive Dynamic InversionMethods,” IEEE Tr. Cont. Sys. Tech., vol. 18, no. 6, pp. 1448–1456,2010.[31] B. Xian, D. M. Dawson, M. S. de Queiroz, and J. Chen, ”A continuousasymptotic tracking control strategy for uncertain nonlinear systems,”IEEE Tr. Autom. Cont., vol. 49, no. 7, pp. 1206-1211, 2004.[32] P. M. Patre, W. MacKunis, C. Makkar, and W. E. Dixon, ”Asymptotictracking for systems with structured and unstructured uncertainties,”IEEE Conf. on Decision & Cont., 2006, pp. 441-446.[33] L. Duan, W. Lu, F. Mora-Camino, and T. Miquel, ”Flight-path trackingcontrol of a transportation aircraft: Comparison of two nonlineardesign approaches,” Digit. Avi. Sys. Conf., Portland, OR, USA, 2006,pp. 1-9.[34] I. Szaszi, B. Kulcsar, G. J. Balas, and J. Bokor, ”Design of FDI filterfor an aircraft control system,” American Cont. Conf., 2002, pp. 42324237.[35] R. R. Costa, L. Hsu, A. K. Imai, and P. Kokotovic, ”Lyapunov-basedadaptive control of MIMO systems,” Automatica, vol. 39, no. 7, pp.1251–1257, 2003.[36] G. Tao, ”Adaptive Control Design and Analysis,” New York, NY,USA: John Wiley & Sons, 2003.[37] W. Mackunis, ”Nonlinear Control for Systems Containing InputUncertainty via a Lyapunov-Based Approach,” Ph.D. dissertation,University of Florida, Gainesville, FL, USA, 2009.[38] W. MacKunis, M. K. Kaiser, P. M. Patre, and W. E. Dixon, ”Asymptotic tracking for aircraft via an uncertain dynamic inversion method,”American Cont. Conf., Seattle, WA, USA, 2008, pp. 3482-3487.[39] W. MacKunis, M. K. Kaiser, P. M. Patre, and W. E. Dixon, ”Adaptivedynamic inversion for asymptotic tracking of an aircraft referencemodel,” AIAA Guid. Nav. Cont. Conf., Honolulu, HI, USA, 20
[13]. In [14], A stochastic robust nonlinear control approach fused with DI technique was applied to a highly nonlinear complicated aircraft model. In [15], a nonlinear dynamic inversion controller is combined with a PI controller to lin-earize the dynamicsof UAVs. Some other past research fused adaptive control techniques with DI to compensate .
Abstract Dynamic rupture inversion is a powerful tool for learning why and how faults fail, but much more work has been done in developing inversion methods than evaluating how well these methods work. This study examines how well a nonlinear rupture inversion method recovers a set of known dynamic rupture parameters on a synthetic fault based .
A Nonlinear Dynamic Inversion Predictor-Based Model Reference Adaptive Controller for a Generic Transport Model Stefan F. Campbell and John T. Kaneshige T. A non-linear dynamic inversion control law is then: . applied to the fast dynamic inversion illustrated in Fig. 2.
by dynamic model inversion is well known and has been applied to the control of high angle of attack ghter aircra [ , ]. e primary drawback of dynamic inversion for aircra ight control is the need for high- delity nonlinear model which must be inverted in real time. However, it is di cult to obtain the exact aircra dynamic model in practice. e .
Adaptive Incremental Nonlinear Dynamic Inversion for Attitude Control of Micro Air Vehicles Ewoud J. J. Smeur, Qiping Chu,† and Guido C. H. E. de Croon‡ Delft University of Technology, 2629 HS Delft, The Netherlands DOI: 10.2514/1.G001490 Incremental nonlinear dynamic inversion is a sensor-based control approach that promises to provide .
When the nonlinear control method, dynamic inversion, is straightforwardly applied to nonminimum phase systems, it results in exact tracking but the unstable zero dynamics remains an unstable part in the -loop system. closed Therefore, the nonminimum phase of an HSV character prevents the application of standard dynamic inversion and all
cients are treated as unknowns. A collection of nonlinear controllers are studied, specifically an ideal dynamic inversion controller and two switching dynamic inversion controllers. A dynamic inversion controller is modified with an adaptive term that learns the aerodynamic e ects on the equation of motion.
of adaptive Neural Nets to add robustness to the nonlinear dynamic inversion control law [9, 10, 11]. In this paper, the artificial neural network using Radial Basis Function [RBF] is applied in order to improve the robustness of the classical dynamic inversion. This can be done by incorporating this
Providing better graphic representation of design concepts. Modify the architectural design review process to ensure greater municipal input. Consultation with the local development industry and with municipal staf. Although the format and graphics provide a new look for Clarington’s General Architectural Design Guidelines, the majority of the content remains pertinent and thus .
