Nonlinear Trajectory Tracking using Vectorial Backstepping Approach

Transcription

1 International Conference on Control Automation an Systems 8 Oct in COEX Seoul Korea Nonlinear rajectory racking using Vectorial Backstepping Approach Hyochoong Bang Sangjong Lee an Haechang Lee 3 Department of Aerospace Engineering Korea Avance Institute of Science an echnology Daejeon Korea (el : ; hcbang@ascl.kaist.ac.kr) Avance Flight Control Department Korea Aerospace Research Institute Daejeon Korea (el : ; albert@kari.re.kr) 3 Air Vehicle System Department Korea Aerospace Research Institute Daejeon Korea (el : ; hclee@kari.re.kr) Abstract: his paper iscusse a nonlinear trajectory problem for stratospheric airship platform an a novel approach of the nonlinear control scheme is applie. arget airship moel is the m stratospheric airship platform capable of flying upto km altitue. Full -DOF nonlinear ynamics of unmanne airship are efine accoring to the target airship s configuration incluing the moving win fiel. Base on this airship moel a backstepping esign formulation for the trajectory control is escribe. he control strategy of vectorial backstepping is applie to erive the control law an global asymptotically stability is prove by Lyapunov stability analysis. Finally numerical simulations have been carrie out to assess the performance of the propose controller. Keywors: Lighter-han-Air Vehicle Stratospheric Airship Platform Nonlinear Control rajectory racking Backstepping Control.. INRODUCION he last ecae has seen substantial avances in nonlinear control ue to theoretical achievements. Consiering the airship s nonlinear characteristics the nonlinear control theory must be appreciate as the efficient solution in esigning a flight control system or control for Stratospheric Airship Platform (SAP). Backstepping approach is a theoretically establishe an wiely use in controlling nonlinear systems an known as a nonlinear recursive esign metho base on a Control Lyapunov Function (CLF) [-3]. It can provie the flexibility to treat nonlinear terms an guarantees the stability of a close-loop system. In aition it has an avantage to allow the robustness in the presence of the matche uncertainties []. his paper proposes a backstepping controller for trajectory control of the m high-altitue unmanne airship. Global asymptotically stability of control law is proven by applying Lyapunov stability analysis. For the convenience of application vectorial formulation for -DOF nonlinear ynamics is introuce an erive. Finally the propose control law is simulate on full -DOF nonlinear airship ynamic equations to show the performance.. ARGE AIRSHIP (VIA-) Especially the stratospheric airship platform (SAP) is being consiere as a new platform which will provie satellite-like missions such as telecommunication broacasting relays environmental observations an surveillance at the stratospheric altitue. As a result of the conceptual sizing the target SAP s length is m its volume is 7 m 3 with its weight 7 kg an the maximum available power for thrust is kw. he buoyancy to weight ratio is.97 an its pressure altitue is km []. VIA- s shape an specification is presente in Fig. an albe. Fig. Isometric view of SAP (VIA-) able Specification of VIA- Length m Envelope volume m 3 7 Hull Area m Buoyancy kg 7 Maximum take-off weight kg 7 Available power of thrust kw 3. NONLINEAR DYNAMIC MODEL he evelopment of the -DOF nonlinear equations of motion for an airship follows a similar way of the fixe-wing aircraft. he major ifferences are ue to its typical features such as buoyancy ae mass an ae inertia terms [-8]. hese are commonly neglecte in formulation of the aircraft ynamics an lea to more nonlinear characteristics in the airship ynamics. 3. Kinematics For airship s equations of motion following vector notations are efine:) state vector compose of velocity vector v = [ u v w] rate vector = [ p q r] ω. ) of position vector = [ x y z ] i i i ν = v ω is an angular η = p θ consists p an attitue vector 9 Authorize license use limite to: Korea Avance Institute of Science an echnology. Downloae on August 9 at 8: from IEEE Xplore. Restrictions apply.

2 θ = [ φ θ ψ ]. he position of airship is expresse in the earth-fixe inertial frame while the velocities are expresse in the boy-fixe frame. Hence kinematic equations can be efine as Eq. (). η = R( θ) ν () R( θ ) is the transformation matrix from the boy-fixe frame to the earth-fixe frame an can be written as Eq. (). R ( θ) η 3 3 R( θ) = () 3 3 Rθ ( θ) where cθ cψ cφsψ + sφsθcψ sφsψ + cφsθcψ Rη ( θ ) = cθ sψ cφcψ sφsθ sψ sφcψ cφsθ sψ + + sθ sφcθ cφcθ sφ tanθ cφ tanθ Rθ ( θ ) = cφ sφ (3) sφ secθ cφ secθ s( ) an c( ) esignate the sine an cosine function. 3. Nonlinear Dynamic Equations of Motion For moeling the flight ynamics of the airship we mae assumptions similar to those for an aircraft; i.e. the vehicle is assume to be perfectly rigi boy an flying at a mean flight spee. However the center of volume (CV) not the center of gravity (CG) is chosen as the origin of the airship boy axes [8]. he buoyancy force ae mass an inertia terms are significantly couple an exists in equations compare to the aircraft nonlinear equations of motion. he ae mass an ae inertia effects arise ue to the airship's mass being of the same orer of magnitue as the mass of isplace air an are escribe as force an moment with respect to the linear an angular accelerations [79]. he Newtonian approach is summarize in Ref. an 7 an VIA- s ynamic equations of motion are efine base on typical geometric an aeroynamic ata of VIA-. his component-type formulation is useful for conventional controller esign an ynamic inversion approach. However it is more efficient to use the state space form in orer to apply the vectorial backstepping approach []. Finally writing the equations of motions of airship in state space form Mν + C ( νν ) + G ( η ) ( ) R θ η g = U () he inertia matrix M is sum of the inertia of the rigi boy ( M rg ) an the buoyancy air ( M am ). Diagonal terms M a an J a matrices esignate the ae mass an ae inertia respectively. Ma m( rcg ) M = Mrg + M am = () m( cg ) r Ja where m Xu Jx Jxz M a = myv J a = J y m Z w Jxz J z az r cg = az a x () ax m is the mass of the airship an the cross prouct r has istance variables ( a x a z ) form CV ( ) matrix ( cg ) to CG in Fig.. Fig. Geometry relationship between CG CB an CV Coriolis matrix C( ν ) is given in Eq. () which contains the nonlinear forces an moments ue to centrifugal an Coriolis forces. he matrix G( η ) contains the restoring forces an moments forme by the buoyancy an gravity. ( ω ) Ma m( ω )( rcg ) C( ν) = m( cg )( ) ( ) r ω ω J a ( mmb ) I3 3 3 G( η) = (7) { m( rcg ) + mb ( rcb )} where bz r q ( r cb ) = bz b x ( ω ) = r p g = (8) bx q p g he cross prouct matrix ( r cb ) has istance variables ( b x b z ) form CV to CB in Fig.. Finally terms of the right han sie of the Eq. () are external forces Fu an moments u an efine in Eq. (9) : i.e. aeroynamic gravitational buoyant an propulsive forces an moments respectively. Fu( δ δμ δe δr α β ) U = (9) u( δ δμ δe δr α β ) In summary kinematics of Eq. () an ynamics of Eq. () is utilize for the esign of the controller in next section. he state vector ν = v ω is compose of six state variables an the real control inputs are compose of thrust δ tilting angle δ μ elevator δ e an ruer δ r. Especially aeroynamic forces an moments can be calculate from the aeroynamic atabase which was constructe by the win-tunnel test by using the / scale moel. In aition ae mass an moment of inertia were obtaine from the results of the CFD analysis []. One of major nonlinearities of Eq. () is the inertia matrix M exists in front of the time 7 Authorize license use limite to: Korea Avance Institute of Science an echnology. Downloae on August 9 at 8: from IEEE Xplore. Restrictions apply.

3 erivatives of the state vector ue to the ae mass an moment of inertia terms.. BACKSEPPING DESIGN FOR RAJECORY RACKING CONROL Backstepping approach is recursive proceure which allows eriving control law for a nonlinear system base on appropriate Lyapunov function caniate. In this section the control strategy of vectorial backstepping is applie base on the iea of Refs. an. he vectorial backstepping gives an avantage over the integrator backstepping in case of applying to the complicate MIMO system since the control law is erive in few steps. he control law is erive in a vectorial setting through two steps. Global asymptotically stability of the control law is proven by Lyapunov stability analysis.. Step Let s efine the first backstepping variable z an esire state η then the error variable can be z = η η () he iea of backstepping is to choose one vector as virtual control. We suggest the virtual control by introucing the secon backstepping variable z η = R ( θ) ν z + α () where α is a stabilizing function. ime ifferentiation of Eq. () gives z = η η = R( θ) νη () = z + αη Let us consier the Lyapunov function caniate V for the first step. V = zpz (3) where P is a positive efinite esign matrix. o make V negative efinite with respect to z the stabilizing function α can be selecte as α = η Λz () where Λ = Λ > is esign matrix. Substitution of Eq. () into the time erivative of Eq. (3) yiels V = zpz zpλ z () herefore Eqs. () an () are rearrange as follows: z = z Λz () z = R θ ν η + Λ z (7) ( ). Step Let us consier the secon Lyapunov function caniate of Eq. (8) VLF = V+ zpz (8) where P = P > is a esign matrix. After some manipulations using Eqs. () () an (7) the time erivative of z is written as z = R ( θ) νr( θ) M (9) ( ) + ( ) η ( ) + C ν ν G η R θ g U η + Λz hen the erivative of the Lyapunov function V LF is given by V LF = zpz zpλz + z P η + Λ z { η } R ( θ) ν R( θ) M C( ν) ν+ G( η) R ( θ) g + U () o make V LF negative efinite in z an z the brackete term in () must be equals to P Pz Λ z an Λ = Λ >. Finally the control law U can be obtaine as Eq (). U = MR ( θ) Λz P Pz R ( θ) ν+ η () Λ z + Λz + ( ) + ( ) η ( ) C ν ν G η R θ g Substitution of Eq. () into the time erivative of Eq. () yiels z =P Pz Λz () an the time erivative of secon Lyapunov caniate function can be represente as V LF =zpλ zzpλ z (3) It is clear that V LF can be mae negative efinite by choosing the positive efinite weight matrices P P Λ an Λ. hus accoring to Lyapunov stability theory the airship system is globally asymptotically stable []..3 Control allocation It is necessary to fin the real airship control inputs since the obtaine control law of Eq. () regars six control forces an moments to achieve the esire performance. In VIA-A total four real control inputs are use : thrust δ tilting angle δ μ elevator eflection δ e an ruer eflection δ r. Consequently the number of real control inputs reuces by three since VIA- oes not use the tilting angle in flight phase except take-off an laning moe. hus VIA- belongs to an uner-actuate system. Control allocation can be use to istribute the total control effort among the actuators an thrust in case that the number of controlle variables excees the number of actuators. Notice that the aeroynamic erivatives are loae from the aeroynamic atabase. So it is necessary to approximate them as a function of control surface inputs ( δ e δ r ) angle-of-attack α an sieslip angle β as following. CX CY CZ CL CM CN = f ( α β δe δr) () CXq CYp CYr CZq CLp CLr CMq CNp CNr = f ( α ) he control allocation problem with aeroynamic forces an moments can be formulate accoring to the control law of Eq. (). 7 Authorize license use limite to: Korea Avance Institute of Science an echnology. Downloae on August 9 at 8: from IEEE Xplore. Restrictions apply.

4 P 3 q CX + ρ V CXq q + δ cosδ μ 3 u Pq CY + ρ V ( ) CYp p+ CYr r u P 3 q CZ + ρ V CZq q u 3 U = = δ sinδμ () u P 3 u q CL + ρ V ( ) CLp p+ CLr r u P 3 q CM + ρ V CMq q + δ( cosδμ z + sinδ μ x ) P 3 q CN + ρ V ( CNp p+ CNr r ) here is a volume of an airship an P q is a ynamic pressure of the freestream. Solving Eq. () with respect to both control comman U an real control inputs ( δe δr δ ) is not trivial because it has many solutions. Let us efine the real control input vector U = [ δ ] e δr δ then above equation can be rewritten as U = BU () where B contains the terms irectly relate with U. he rest terms are transpose into the right-han sie of Eq. () an yiel U after subtracte from control comman U of Eq. (). he least-square solution is simply given by Eq. (7). U = B B B U (7) ( ). RAJECORY RACKING RESULS In this section numerical simulations are conucte to verify the performance of the propose backstepping controller for trajectories. Numerical simulation coes are base on MALAB/ Simulink environment. he initial conitions for the simulation are given in able an the trajectories are generate from the kinematics of Eq. () cooperate with -DOF nonlinear ynamic equations of Eq. (). Arbitrary control inputs scenario are scheule to construct a helix-shape climb trajectory until SAP coul reach the stratospheric altitue of km. otal simulation time is. hrs ( sec) an no win conitions are consiere. able Initial conitions for results Variable Initial value Variable Initial value V 8.7 m/s ψ eg α.7 eg x i m β eg y i m p eg/s h m q eg/s δ e.3 eg r eg/s δ r eg φ eg δ 78 N θ eg δ μ eg he variables of trajectories are compose of positions an attitue of the refine trajectory results η = ( xi) ( yi) ( zi) φ ref ref ref ref θref ψ ref. he propose backstepping control law is conucte with scheuling esign matrices P P Λ an Λ. hree-imensional trajectory result are shown in Fig. 3 an position results in Fig.. he soli line represents the result of the propose controller an the otte line represents the trajectory. Referring to Figs. an the propose backstepping controller follows the positions successfully. h 3 x i - station-keeping position - launch position y i Fig. 3 hree imensional trajectory results vs. trajectory x i y i h Fig. racking results of position variables vs. s Another results of attitues are shown in Fig. an propose backstepping controller presents accurate attitue following result. Other state variables are shown in Figs. ~ 7 respectively. hey show such a goo performance of the propose tacking controller. 3 7 Authorize license use limite to: Korea Avance Institute of Science an echnology. Downloae on August 9 at 8: from IEEE Xplore. Restrictions apply.

5 Real control inputs of thrust δ tilting angle δ μ elevator eflection δ e an ruer eflection δ r are represente in Fig. 8. Notice that VIA- has no control means for rolling motion because it has not aileron. φ (eg) ψ (eg) θ (eg) V t (m/s) α (eg) β (eg) Fig. racking results of attitue variables vs. s Fig. racking results of flight spee angle-of-attack an sieslip angel vs. s It is har to ientify the level of the error so the errors between the values an the results are rawn in Fig. 9. he position an attitue errors continuously oscillate uring the flight time. his also can be confirme in time histories of Lyapunov function in Fig.. p (eg/s) q (eg/s) r (eg/s) Fig. 7 racking results of angular rate vs. s δ (N) δ e (eg) δ r (eg) δ μ (eg) x Fig. 8 racking results of real control inputs vs. s In Fig. some peak values are exists aroun t = 3.3 hr an. hr. hese are cause by a iscontinuous thrust input as shown in Fig. 8. he root mean square (rms) error is use for compare the error with the value. he rms error of position vector p = [ x ] i yi zi is. m. m an.9ⅹ - m respectively. One of the main source of the position errors are resulte from the attitue errors ue to the transformation relationship of Eq. (). For the attitue vector θ = [ φ θ ψ ] the rms error is. eg.37ⅹ - eg an.37ⅹ - eg respectively. Other rms errors of state variables are summarize in able Authorize license use limite to: Korea Avance Institute of Science an echnology. Downloae on August 9 at 8: from IEEE Xplore. Restrictions apply.

6 able 3 racking errors (rms errors) Variable rms error Variable rms error V.3 m/s φ. eg α.3 eg θ.37e- eg β. eg ψ.37e- eg p.3 eg/s x i. m q. eg/s y i. m r 7.37e- eg/s h.9e- m hese errors are mainly affecte by the pseuo-inverse error as well as the attitue error of itself. Unlike the stabilization problem errors can not converge to zero because the airship s trajectories are continuously changing along the flight time. Δx i (m) Δy i (m) V LF.. -. Error of Position racking x -3 Δh (m) - - Δφ (eg) Δθ (eg).. -. Error of Attitue racking x -3 Δψ (eg) - - Fig. 9 racking errors of position an attitue Fig. ime history of Lyapunov function. CONCLUSIONS AND REMARKS his paper is concerne with the esign of nonlinear controller for the large-scale unmanne airship. he vectorial backstepping approach was applie an the stability analysis for the propose control law was carrie out by using the Lyapunov theory. he nonlinear ynamics an kinematics moeling of the airship was re-erive an rearrange in the form of state space representation. Necessary aeroynamic propulsive an geometric ata are base on the results of win-tunnel test an CFD analysis. Numerical simulation has been carrie out an emonstrates the performance of the propose controller. It keeps track of the given trajectories an state variables. In aition it is shown that the errors are small enough to present its goo performance. Originally the trajectories shoul be generate from the solution of the optimal trajectory problem. Although this paper is not ealing with the optimal trajectories further stuy will be extene to that topic. REFERENCES [] O. Harkegar an S.. Gla A Backstepping Design for Flight Path Angle Control Proceeings of the 39 th IEEE Conference on Decision an Control pp []. Lee an Y. Kim Nonlinear Aaptive Flight Control Using Backstepping an Neural Networks Controller Journal of Guiance Control an Dynamics Vol. No. pp [3] J. R. Azinheira A. Moutinho an E. C. Paiva Airship Hover Stabilization Using a Backstepping Control Approach Journal of Guiance Control an Dynamics Vol. 9 No. pp [] M. Kristic I. Kanellakopoulos an P. Kokotovic Nonlinear an Aaptive Control Design Wiley- Interscience New York 99. [] S. Lee rajectory Optimization an Nonlinear racking Control for Stratospheric Airship Platform Ph.D. Dissertation Department of Aerospace Engineering Korea Avance Institute of Science an echnology Daejeon Korea Feb. 8. [] J. R. Azinheira E. C. Paiva an S. S. Bueno Influence of Win Spee on Airship Dynamics Journal of Guiance Control an Dynamics Vol. No. pp.-. [7] S. B. V. Gomes An Investigation of the Flight Dynamics of Airships with Application to the YEZ-A Ph.D. Dissertation College of Aeronautics Cranfiel University Cranfiel Englan U.K. Oct. 99. [8] G. A. Khoury an J. D. Gillet Airship echnology Cambrige University Press 999. [9] S. Lee D. Kim an H. Bang Feeback Linearization Controller for Semi Station Keeping of the Unmanne Airship Proceeings of AIAA th Lighter-han-Air Sys ech. an Balloon Systems Conferences Arlington Virginia. []. I. Fossen Marine Control Systems st e. Marine Cybernetics AS. []. I. Fossen an A. Grovlen Nonlinear Vectorial Backstepping Design for Global Exponential racking of Marine Vessels in the Presence of Actuator Dynamics Proceeings of the 3 th Conference on Decision & Control pp [] S. Lee H. Lee D. Won an H. Bang Backstepping Approach of rajectory racking Control for the Mi-Altitue Unmanne Airship Proceeings of AIAA Guiance Navigation an Control Conference AIAA South Carolina 7. 7 Authorize license use limite to: Korea Avance Institute of Science an echnology. Downloae on August 9 at 8: from IEEE Xplore. Restrictions apply.