Helicopter Nonlinear Flight Control Using Incremental Nonlinear Dynamic .

θ0 ,θ1s ,θ1cFlappingMotion a0 ,a1 ,b1θ0trAerodyn.MRT �̇θ̇RxVerticalTailFigure 1. Modular structure of the helicopter model.v̇ ṗ sin θ1 f g sin φ cos θ ω vmcos φ cos θTv 1ω̇ Jθ̇ Ωω(m ω Jω)where, denoting cos α and sin α by cα and sα , cψ cθ cψ sθ sφ sψ cφ cψ sθ cφ sψ sφ T sψ cθ sψ sθ sφ cψ cφ sψ sθ cφ cψ sφ sθcθ sφcθ cφ(2)(3)(4)(5)(6)represents the coordinate conversion from the body-fixedto the NED reference frame and 1 sin φ tan θ cos φ tan θ (7)Ω 0cos φ sin φ 0 sin φ/ cos θ cos φ/ cos θthat, when applied to the system, the relations betweena virtual control input and the outputs of the system arereduced to simple integrators. For the resulting linearsystem, a single linear control law can be designed togenerate the virtual control without the need for gainscheduling to tune the controller for different conditionsof the nonlinear system. A detailed explanation of thistechnique is presented, for example, in Ref. 20 or 21.To exemplify the working principle of the NDI, consider a system of order n with the same number m ofinputs u and outputs y and affine in the control inputs.Furthermore, the outputs typically coincide with the control variables and are assumed to be physically similar(for instance, three attitude angles). The extension ofthe theory to more complex systems is rather straightforward. This type of system can be represented as:ẋ f (x) G(x)uy h(x)(8)where f and h are vector fields in Rn and Rm , respectively, and G is a n m control effectiveness matrix.is the transformation matrix from ω to θ̇.The procedure to obtain the dynamic inversion consists simply of consecutive time-differentiations of y untilan explicit dependence on u appears. To each derivativeIII. Control Theoryisassociated a new state vector and the derivative of theThis section summarizes the control theory necessarylastone is given by a nonlinear expression (the virtualfor the implementation of the proposed control system.control) to complete the transformation.If r time-differentiations are required to obtain theA. Nonlinear Dynamic Inversioncontrol dependence, r.m n is known as the total relaThe Nonlinear Dynamic Inversion (NDI) is a design tive degree of the system. Moreover, if r.m n, there aremethodology developed in the late 1970’s to provide con- n r.m degrees of internal dynamics, unobservable to thetrol of nonlinear systems and is applicable to a class of input-output linearization and which must be Boundedsystems known as feedback linearizable.20 Basically, it Input Bounded-Output (BIBO) stable in the region ofallows to generate a control input via nonlinear feed- interest to assure the effectiveness of the controller.21Assuming now h(x) x (thus m n and r 1), theback and using a state diffeomorphism (a smooth bijective nonlinear mapping with a smooth inverse) such first-order time-derivative of y is given by:3 of 10

ẏ ẋ f (x) G(x)u(9)Since an explicit dependence on u was already found, thelinear relation ν ẋ can be imposed if G(x) is invertible(det G(x) 6 0) by selecting: (10)u G(x) 1 ν f (x)Besides performing the linearization of the system, thisinput also allows to decouple the responses of the controlvariables since each component of ν only depends on thesame component of x.To perform the dynamic inversion, all the states ofthe system have to be known by direct measurement orstate estimation, in order to reconstruct f and G. TheNDI relies on an accurate description of these functionsto cancel all the nonlinearities of the system. Nevertheless, if inaccuracies (differences between the mathematical model and the actual physical system) exist, the exactcancellation of the nonlinearities becomes impossible. Toillustrate this situation, consider that the functions aboveare composed by a nominal part which is known (f n andGn ) plus an uncertain term ( f and G). Hence:ẋ ẋ0 f (x) G(x)u x ,u (x x0 ) 00 x G(x0 )(u u0 )(14)For very small time increments (high sampling frequencies of the controller), x x0 is verified and the assumption x x0 0 can be made. Considering this, Eq. (14)is further simplified into:ẋ ẋ0 G(x0 )(u u0 )(15)and the linear relation ν ẋ can again be imposed fordet G(x0 ) 6 0 with:u G(x0 ) 1 (ν ẋ0 ) u0(16)At this point, the main advantage of the INDI canbe identified: the control law does not depend on f anymore, meaning that the controller is completely insensitive to the part of the model that only depends on thestates of the system. More precisely, this information isreplaced by online measurements (or estimations) of thestate derivative ẋ0 and the effectiveness of the controller(11) is dictated by the accuracy of the sensors (or filteringẋ f n (x) f (x) (Gn (x) G(x)) uprocesses). Furthermore, the deflections of the controlUnless some type of system identification is applied, only inputs u0 have also to be accurately known.The robustness of this control strategy is evaluatedthe nominal parts are known by the controller, thus:following the same procedure used for the NDI. Assum(12) ing ideal sensors, all the model inaccuracies lie in G (unu Gn (x) 1 (ν f n (x))certainties in f are reflected in ẋ0 ) and the real systemand the application of this input to the real system de- is mathematically described as:scribed by Eq. (11) yields:ẋ ẋ0 (Gn (x0 ) G(x0 )) (u u0 )(17)ẋ f (x) G(x)Gn (x) 1 f n (x) Once again, only the nominal part is known: I n n G(x)Gn (x) 1 ν (13)(18)u Gn (x0 ) 1 (ν ẋ0 ) u0where I n n is the n n identity matrix. As it canbe seen, the linear relation ẋ ν is only recovered for and replacing this control law in Eq. (17) yields: f (x) G(x) 0. Otherwise, the closed-loop system(19)ẋ C ẋ0 (I n n C)νis not linearized, degrading its performance when a linear control law is used to generate ν, thus compromisingwhere C G(x0 )Gn (x0 ) 1 . Despite the relation ẋ νthe stability of the system. This drawback is the mainis only obtained for G(x0 ) 0, even in the presence ofmotivation to develop a more robust version of the NDI,uncertainties, Eq. (19) remains linear and decoupled. Ingenerally known as Incremental NDI.addition, if a linear controller with a diagonal gain matrix K is applied to generate the virtual control fromB. Incremental Nonlinear Dynamic Inversionthe tracking error, ν K (xcom x), with x x0 , theAs the name indicates, instead of determining the to- closed-loop transfer function for each component i of xtal vector u directly, the Incremental Nonlinear Dynamic is given by:xi (s)KiInversion (INDI) is based on the computation of the re (20)(s)xs Kiicomquired control variation at a given moment with respectto the conditions of the system in the instant of timeimmediately before.9, 22 To do so, let Eq. (9) be approximated by the first-order terms of its Taylor seriesexpansion around the conditions of the system at thatinstant (denoted by the subscript 0):As it can be seen, the influence of model uncertaintiesin the overall control system has also disappeared. Thismeans that, when closing the INDI-based control loop,the system can still be linearized, decoupled and controlled as if no model uncertainties existed.4 of 10

The only limitation is associated with the sign of theentries of the control effectiveness matrix G. It can be intuitively understood that these signs have to be correctlyknown, otherwise, instead of compensating for trackingerrors, the controller will tend to increase them and leadto an unstable response. This effect in the stability ofthe system is also reflected in matrix C.C.Pseudo-Control HedgingThe NDI/INDI theory is derived without any consideration on the dynamics of the physical actuators of a system and, in the case of a multi-loop controller, it neglectsthe limitations imposed by the potential lack of separation between their bandwidths. If these effects are nottaken into account, the performance of the overall controller may be severely degraded. These problems cannormally be avoided by reducing the gains of the system,but this option introduces conservatism in the design, reducing the agility of the controller.To overcome the problem of actuator (or inner loops)dynamics, a technique known as Pseudo-Control Hedging(PCH) was introduced in Ref. 8 and successfully appliedto a Unmanned Air Vehicle (UAV) helicopter in Ref. 6.As explained in this reference, the PCH automaticallymoves (hedges) the signals sent to the controller in theopposite direction by an estimate of the amount ν h theplant did not move due to the dynamics of the actuators.This prevents the continued effort to track the originalcommanded references when, for example, saturation effects are being experienced.In order to implement the hedging of the commandedsignals mentioned above, a first-order Reference Model(RM) is adopted.8 As depicted in figure 2, this RM hasa saturation filter to keep the desired references from being physically unfeasible and is especially useful to compute the derivatives of the commanded variables, whichcan be used by the controller as feedforward terms ν rm .K rm is simply a diagonal gain matrix.νhxcomK rmẋrmRxrmν rmtor that generates the commanded control input ucomand the virtual control associated with the real values ofthe physical inputs u, known from a model of the actuators or measured directly. Regarding again the systemof the previous subsections, the virtual control (ν ẋ)is given by Eq. (9) for the NDI case and thus: ν h f (x) G(x)ucom f (x) G(x)u (21) G(x) (ucom u)For an INDI control law, the virtual control correspondsto Eq. (15) and the pseudo-control hedge is, by analogy:ν h G(x0 ) (ucom u)These vectors are then employed to automatically adjustthe RM from the continuous monitoring of the system.IV.The signal sent to the control system is then the statevector xrm . When no saturations occur, ν h 0 and theRM behaves exactly as a low-pass filter with bandwidthKrmi for the i-th component of xcom . This value shallmatch the one imposed by the remainder control laws sothat the evolution of xrm is not too aggressive nor slowerthan the capabilities of the system.The pseudo-control hedge ν h corresponds simply tothe difference between the required virtual control vec-Controller DesignThe overall control system suggested in this paper isbased on a three-loop architecture to track commandsin terms of ground velocities (Vx , Vy , Vz ) and yaw angle (ψ). Between them, the existence of a time scaleseparation is assumed. This type of assumption is oftencarried out for flight dynamics and control applications.Between two loops, the parameters associated with theouter loop (slow dynamics) are treated as constants inthe inner loop (fast dynamics) and its dynamic inversion is performed assuming that the states controlled bythe inner loop achieve their commanded values instantaneously. The fast variables are thus used as controlinputs to the slow dynamics. A simplified architectureof the multi-loop overall control system is schematizedin figure 3. In this figure, vectors u and x contain thecontrol inputs and state variables introduced in Sec. II.From this figure, it is also possible to observe that notall the control inputs are used to provide tracking of thecontrol variables of the inner loop. Since the cyclic pitchand the collective of the tail rotor are moment generatorswhile the collective of the main rotor is primarily a forceeffector, the latter one is not used to control the angular rates of the vehicle. Instead, its command signal isgenerated by a control law in the navigational loop. Thefollowing subsections present in more detail the internalstructure of the three control loops.A.Figure 2. First-order RM with saturation filter.(22)Rate ControllerAs the name indicates, the control variables of this innerloop are the angular rates of the helicopter:y rot hrot (x) ω(23)In order to apply the INDI technique, this equation hasto be time-differentiated until an explicit dependence onthe control inputs of the system appears. The first-orderderivative corresponds to the rotational dynamics of thevehicle given by Eq. (4), which can be recast as:5 of 10

comθcomRateAttitude ω torsuHelicopterxFigure 3. Simplified architecture of the overall flight control system.the virtual control ν rot K 1 (ω com ω), each componentof the angular rate vector presents a first-order responsewith a time constant directly enforced by the correspondwhere f (x) is the control independent part of the modeling entry of K 11 . Furthermore, a PCH layer as the oneand g(x, u) is the control effectiveness function that comderived in Sec. III.C is implemented to protect the sysprises the moments generated by both rotors, mmr andtem from undesirable effects due to the dynamics of themtr , which depend on the control inputs. For the caseactuators. It makes use of the first-order RM presentedof a fixed-wing aircraft, this function is normally affinein figure 2 to adjust the commanded signals so that thein the control inputs and the system is simplified intoeffort demanded is within the capabilities of the systemthe form of Eq. (8). For helicopters however, the mostand to provide a feedforward term ν rmrot to enhance thecomplex part of the model is contained in g(x, u) due totracking performance. The internal structure of the ratethe aerodynamics of the rotors and it is represented by:controller is depicted in figure 4.At this point, the closed-loop control system is still 1g(x, u) J [mmr (x, u) mtr (x, u)](25)unstable due to the existence of unbounded internal dySince this equation expresses already the influence of the namics. This problem will be automatically solved withcontrols in the system, the rotational virtual control is the introduction of external control loops for further varidefined as ν rot ω̇. Following the same procedure pre- ables. In the end, five degrees of internal dynamics willsented in Sec. III.B and taking into account that only a remain in the system: three associated with the translavector u′ [θ1s θ1c θ0tr ]T is used for rotational control, tional kinematics (Eq. (3)) which are bounded since T isorthonormal and the translational dynamics will be stathe command signal sent to the actuators is given by:bilized and two associated with the rotor induced inflows! 1(Eq. (1)) which, as explained in Sec. II, are stable. g(x, u)′(26)(ν ω̇) uu′com 0rot0 u′x0 ,u0B. Attitude Controllerẏ rot ω̇ f (x) g(x, u)(24)Comparing this equation with Eq. (16), it is possible toverify that, in the case of a system that is not affinein the controls, the information of matrix G(x) is simply replaced by the Jacobian of the control effectivenessg(x, u) with respect to the different control inputs. Forthe model under analysis, the term mtr (x,u) u′x0 ,u0couldbe determined analytically but, because of its complex mmr (x,u)was computed with central finite difity, u′x0 ,u0ferences. Furthermore, due to the fact that angular accelerometers are still not common today, the angular accelerations ω̇ 0 were estimated with backward finite differences and first-order low-pass filters were introducedbefore the actuators to attenuate high frequency oscillations caused by numerical noise.The control law of Eq. (26) allows to linearize anddecouple the response of the helicopter from ν rot to ω.As shown by Eq. (20), if linear proportional controllerswith a diagonal gain matrix K 1 are applied to generateThe attitude controller is constructed externally to therate controller and uses commands in terms of angularrates to track the desired attitude angles of the vehicle.A time scale separation is assumed between the two loopsand the design of this loop is carried out neglecting thedynamics of the inner loop, just like the inner loop INDIneglected the dynamics of the actuators. The outputvector of the attitude controller corresponds thus to theattitude angles:y att hatt (x) θ(27)and following the same procedure as in the previous subsection, when this vector is differentiated with respect totime, Eq. (5) is obtained:ẏ att θ̇ Ωω(28)Since a dependence on the control inputs (the angularrates) has already appeared, the virtual control to linearize and decouple the responses of the helicopter is6 of 10

θ0comν rmrotINDIω comRMω rmK1ων hrotddtν rotω̇ 0 g(x,u) u′x0 ,u0 1θ0u′comActuatorsu′0z 1Helicopterxu′hrot (x) g(x,u) u′x0 ,u0PCHFigure 4. Schematic of the rate control system based on INDI and PCH.defined as ν att θ̇ and the control law to generate thecommanded angular rates is, for det Ω cos θ 6 0:ω com Ω 1 ν att(29)rotor for the vertical speed. Once again, the existenceof a time scale separation is assumed between the translational and rotational dynamics, hence this outer loopassumes that the attitude angles are exactly what theyare commanded to be. In addition, as depicted in figure 3, the commanded value for the yaw angle needs tobe provided externally or, for a flight with no sideslip,Vcomputed as ψcom arctan2 Vxycom .comFor horizontal control, an approach based on the Approximate Dynamic Inversion (ADI) suggested in Ref. 23is adopted. To do so, consider the equation of the translational dynamics written in the NED reference frame: fV̇x01 x (32)ẏ nav V̇y T fy 0 mV̇zgfzNote that this inversion is performed according to thestandard NDI technique since, as there is not a part ofthe model that does not depend on ω and as Ω can bedetermined very accurately, the application of the INDIwould not bring any advantage in this case.The virtual control can again be generated by a proportional controller ν att K 2 (θcom θ) and, in orderto alleviate the time scale separation requirements, allowing the bandwidth of the control loops to be closer, acombined analysis is performed to select the linear gainsof both loops. Taking into account that a first-orderresponse with bandwidth K 1 was imposed to the innerloop, the closed-loop transfer function for each orienta- As a dependence on the attitude angles appears throughmatrix T , define ν nav [νx νy νz ]T [V̇x V̇y V̇z ]T and,tion angle i is given by:assuming fz fx , fy since it corresponds practicallyθi (s)K 1i K 2i(30) to the thrust produced by the main rotor, the following 2θicom (s)s K 1i s K 1i K 2iexpressions are obtained:qfzIt corresponds thus to a second order response for each νx2 νy2 (νz g)2(33)maxis and the diagonal matrices K 1 and K 2 are chosen toνx cos ψcom νy sin ψcomimpose the desired natural frequencies and damping ra(34)θcom arctanνz gtios. In order to further prevent undesirable interactionsν

approach and an adaptive architecture may be required.2 This is in fact the most common strategy adopted in the past few years for helicopter nonlinear flight con-trol:3,4,5 a Nonlinear Dynamic Inversion (NDI) of an ap-proximate model (linearized at a pre-specified trim con-dition) together with adaptive elements to compensate