Further results on global stabilization of the PVTOL aircraft Ahmad Hably, Farid Kendoul 2, Nicolas Marchand, and Pedro Castillo 2 Laboratoire d Automatique de Grenoble, ENSIEG BP 46, 3842 Saint Martin d Hères Cedex, FRANCE {ahmad.hably,nicolas.marchand}@inpg.fr 2 Laboratoire Heudiasyc, Université de Technologie de Compiègne, 62 Compiègne, FRANCE {fkendoul,pcastillo}@hds.utc.fr Introduction u v In the last years, the research on the unmanned aerial vehicles (UAVs) with onboard intelligent capabilities has done a lot of progress. This progress was motivated by the enormous military/civil applications of such vehicles accompanied with the technological progress in sensors, actuators, processors and power storage devices. The Planar Vertical Take Off and Landing aircraft (PVTOL) represents a challenging nonlinear system problem that is often considered the first step to design control laws for general aerial vehicles, (see Figure ). It is classiy x Fig.. The PVTOL system fied under the non-minimum phase systems (i.e. nonlinear systems with zero dynamics that are not asymptotically stable). The simplified dynamics of the PVTOL aircraft initially proposed in [3] takes the following form ẍ = sin(θ)u + ɛ cos(θ)v () ÿ = cos(θ)u + ɛ sin(θ)v (2) θ = v (3)
2 Hably, Kendoul, Marchand, Castillo x and y are the position coordinates of the centre of mass of the aircraft in a fixed inertial frame and θ is its inclination with respect to the horizontal axis. The control inputs u and v represent normalized quantities related to the vertical thrust directed upwards with respect to the aircraft and the angular acceleration (rolling moment). The parameter ɛ is a (small) coefficient which characterizes the coupling between the rolling moment and the lateral acceleration. In this paper, ɛ is supposed to be zero. The coefficient - in equation (2) corresponds to the normalized gravitational acceleration. Several methodologies have treated the stabilization of the PVTOL aircraft. An early approach was presented by Hauser et al. [3], who in 992 designed a stabilizing controller using conventional feedback linearization techniques and a non-minimum phase approximation of a PVTOL system. Sepulchre et al. [] used the same approximation, and a backstepping approach to control the translational dynamics through the cascaded roll subsystem. Olfati-Saber in [8] proposed a configuration stabilization for the PVTOL aircraft with a strong input coupling using a smooth static state feedback. Some restrictions have to be considered in the control law design for small aerial vehicles. In fact, saturation nonlinearities are particulary prevalent, where actuator saturation has a significant effect on the overall stability of the aircraft [2]. Several new nonlinear tools have been introduced for analyzing and controlling linear and nonlinear systems with bounded inputs [4, 6, 7, 2, 3, 4]. Some researchers have exploited these techniques for controlling the PVTOL system with bounded inputs. Indeed, the authors in [5] and [5] have developed new control strategies which coped with (arbitrarily) bounded inputs and which provided global convergence to the origin. However, these control strategies suffer from the low convergence speed of some state variables. In this paper, we have used the same change of variables as in [8] and exploited the control technique [6] that is developed for controlling a class of linear systems with bounded inputs. The proposed controller considers the boundedness of control inputs/energy and it results in simple and explicit expressions which are suitable for implementation. The positiveness of the thrust control is explicitly handled. Furthermore, the controller presents good performance (convergence speed) when comparing it to other approaches [, 5, 5]. 2 Control law formulation The PVTOL aircraft system (-3) is decomposed into two subsystems. The first subsystem Σ represents the translational motion equations (-2) and the second one Σ 2 represents the rotational motion equation (3) which evolves independently of x and y. Let z := (z, z 2, z 3, z 4 ) = (x, ẋ, y, ẏ) and η := (η, η 2 ) = (θ, θ). Then subsystem Σ : ż = f(z, η) is written as
Further results on global stabilization of the PVTOL aircraft 3 ż = z 2 ż 2 = u sin(η ) ż 3 = z 4 ż 4 = u cos(η ) (4) and the subsystem Σ 2 : η = g(η) can be expressed as { η = η 2 η 2 = v As in [9], η is considered as an intermediate input for subsystem Σ (4). The idea is to choose a bounded control input v that drives η to a desired angle η d having the following form with (5) η d := arctan( r r 2 + ) (6) r = εσ(z 2 ) ε 2 σ(εz + z 2 ) (7) r 2 = εσ(z 4 ) ε 2 σ(εz 3 + z 4 ) (8) The choice of r and r 2 is based on our previous work [6] in which the global stabilization of multiple integrators with bounded control is studied. ε is a positive parameter lower than (see [6]) and σ(.) is a twice-differentiable saturation function bounded between ±. It is defined for < α < as follows if s < α p (s) = e s 2 + e 2 s + e 3 if s [ α, + α[ σ(s) = s if s [ + α, + α] (9) p 2 (s) = e s 2 + e 2 s e 3 if s ] α, + α] + if s > + α with e = 4α, e 2 = 2 + 2α and e 3 = α2 2α+ 4α. The application of the thrust control input u u = r 2 + (r 2 + ) 2 () with η = η d will transform subsystem Σ (4) into the form of two independent second order chain of integrators. { z = r z 3 = r 2 () For the subsystem Σ 2 (5), let us define φ = (φ, φ 2 ) T = ( θ, θ) T where θ = η η d. The rolling moment v is defined as v = σ β ( η d ) εσ(η 2 η d ) ε 2 σ(ε(η η d ) + (η 2 η d )) (2) where σ β (.) = βσ(.) for a certain positive parameter β.
4 Hably, Kendoul, Marchand, Castillo 3 Stability results Theorem. Consider the PVTOL aircraft system (4-5) with input saturation bounds u max > and v max >. Then the thrust input u given by () and the rolling moment v of (2) globally stabilize the PVTOL aircraft to the origin. Proof. In the proof, both the local and global stability are studied. Local Stability: Using the definition of σ and σ β, the local stability is proved through the linearization within a neighborhood around the origin. For a := ε 3 and b := ε + ε 2, the PVTOL aircraft system takes the following form z 2 u sin(η ) (ż, η) T = h(z, η) = z 4 u cos(η ) (3) η 2 β η d b(η 2 η d ) a(η η d ) The Jacobian matrix of h(z, η) at the origin is given by A = a b a 2 2ba βa a b 2 βb b (4) For ε < and β < ε2 2 polynomial of A (see the global stability section), the characteristic P (λ) = λi A = (λ 2 + λb + a)(λ 4 + λ 3 (βb + b) + λ 2 (βa + a + b 2 ) + 2baλ + a 2 ) will have stable roots. Thus as a result, the local stability of the proposed controller is established. Global stability: For subsystem Σ 2 (5) the change of coordinates presented in [6] is applied { y = εη + η 2 y 2 = η 2 (5) to obtain { ẏ = εy 2 + v ẏ 2 = v (6) Let us define y d := εη d + η d, y d 2 := η d and v d := η d. The control input v in (2) can be rewritten as v = σ β (v d ) εσ(y 2 y d 2) ε 2 σ(y y d ) (7)
Further results on global stabilization of the PVTOL aircraft 5 In the first part of this section, we will prove that the states y 2 and y will be bounded after a finite time. Let us consider the Lyapunov function V 2 = 2 (y 2 y 2d ) 2 i.e. V2 = (y 2 y 2d )(ẏ 2 ẏ 2d ) = (y 2 y 2d )(v v d ). Knowing that for all δ >, there exists t δ such that for all t > t δ we have ( ) vd = d2 dt 2 arctan r < δ (8) r 2 + Therefore, if y 2 y d 2 > and (β + δ + ε 2 < ε), we have σ β (v d ) v d ε 2 σ(y y d ) β + δ + ε 2 < ε = εσ(y 2 y d 2) (9) From (9), it follows that v v d is of opposite sign of σ(y 2 y d 2) and hence of y 2 y d 2. This ensures that V 2 < and as a result, y 2 will join the interval [ + y d 2, + y d 2] after a finite time t > t δ. As a consequence, v will take the following form v = σ β (v d ) ε(y 2 y d 2) ε 2 σ(y y d ) (2) From equation (6), ẏ can be written as ẏ = εy d 2 + v d + σ β (v d ) ε 2 σ(y y d ) v d (2) For y not in [ + y d, + y d ], suppose that β + δ < ε 2 holds then, as for (9): σ β (v d ) v d < ε 2 σ(y y d ) (22) which means that after a finite time t 2 > t > t δ, y will join the interval [ + y d, + y d ]. Again v is written as v = σ β (v d ) ε(y 2 y d 2) ε 2 (y y d ) (23) For β > δ, the control input v takes the following form t > t 2, subsystem Σ 2 (5) becomes v = v d ε(y 2 y d 2) ε 2 (y y d ) (24) η = η 2 η 2 = η d b(η 2 η d ) a(η η d ) (25) and as a result, η (t) η d (t) as t. Subsystem Σ (4), in the case of η (t) = η d (t), takes the following form z = z 2 z 2 = εσ(z 2 ) ε 2 σ(εz + z 2 ) z 3 = z 4 z 4 = εσ(z 4 ) ε 2 σ(εz 3 + z 4 ) (26) From theorem. in [6], the stability of subsystem (26) is guaranteed for ε <. For β < ε2 2 and ε <, the stability of the overall system is also guaranteed using a theorem in [] for systems in cascade. Therefore the global stabilization is achieved for the PVTOL aircraft and this ends the proof.
6 Hably, Kendoul, Marchand, Castillo 4 Simulation results In this section, we present some simulation results using MATLAB c and SIMULINK c in order to compare the performance of the proposed control law with other controllers: with unbounded inputs [8] and with bounded inputs [, 5, 5]. Two settings for ε were tested, namely ε =.5 and ε =.99. The initial conditions are (z(), η()) = (5,, 5,, π/3, ). This choice enables to compare all the approaches and in particular [] where θ is restricted to ] π/2, π/2[ which is not the case of the proposed control law. The bounds on the thrust u max = and on the rolling moment v max = 5 are the same as the one proposed in [5]. The objective is to stabilize the state variables (z, η) at the origin. In Fig.2 and Fig.3, we traced the evolution of the states of subsystem Σ (4). The settling time is clearly reduced in comparaison with [5] and [5]. In Fig.4, we only present the first 6 seconds of the simulation in order to clearly trace the variations of the states of the rotational subsystem Σ 2 (5). The control inputs proposed in this paper are traced in Fig.5 (we present the first 6 seconds). They respect the saturation bounds (u max, v max ) and their magnitude is smaller than the bounded control approaches [], [5] and [5] respectively. Tuning the parameter ε enables to favour or detriment the convergence speed of (x, y) against θ. 5 Conclusion This paper developed a global stabilizing control law for the planar vertical takeoff and landing aircraft with bounded inputs. The controller is based on global stabilization of multiple integrators with bounded inputs. The proposed control design exploits the technique based on the sum of saturating functions. Thereby, considering the positiveness of the thrust, the boundedness of the two control inputs and improving the controller performance. Indeed, the numerical simulations have showed the effectiveness of the proposed competitive controller. References. P. Castillo, R. Lozano, I. Fantoni, and A. Dzul. Control design for the PV- TOL aircraft with arbitrary bounds on the acceleration. In 4 st IEEE conf. on Decision and Control, CDC 2, 22. 2. M.A. Dornheim. Report pinpoints factors leading to YF-22 crash. Aviation Week and Space Technology, 37(9):53 54, 992. 3. J. Hauser, S. Sastry, and G. Meyer. Nonlinear control design for slightly nonminimum phase systems: Application to V/STOL aircraft. Automatica, 28:665 679, 992. 4. T. Lauvdal and R.M. Murray. Stabilization of a pitch axis flight control experiment with input rate saturation. In AIAA Guidance, Navigation, and Control conference, 999.
Further results on global stabilization of the PVTOL aircraft 7 5. R. Lozano, P. Castillo, and A. Dzul. Global stabilization of the PVTOL: Realtime application to a mini-aircraft. International Journal of Control, 77(8):735 74, May 24. 6. N. Marchand and A. Hably. Global stabilization of multiple integrators with bounded controls. Automatica, 4(2):247 252, 25. 7. R.M. Murray. Geometric approaches to control in the presence of magnitude and rate saturations. The Astrom Symposium on Control, 999. 8. R. Olfati-Saber. Global configuration stabilization for the VTOL aircaraft with strong input coupling. In 39 th IEEE conf. on Decision and Control, CDC, 2. 9. R. Olfati-Saber. Global configuration stabilization for the VTOL aircraft with strong input coupling. IEEE transactions on Automatic Control, 47():949 952, 22.. R. Sepulchre, M. Jankovic, and P. Kokotovic. Constructive Nonlinear Control. Springer-Verlag, 997.. E. Sontag. Smooth stabilization implies coprime factorization. IEEE transactions on automatic control, 34(4):435 443, 989. 2. J.H. Sussmann, D.E. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Transactions on Automatic Control, 39(2), December 994. 3. A.R. Teel. Global stabilization and restricted tracking for multiple integrators with bounded controls. System and control letters, 8(3):65 7, March 992. 4. A.R. Teel. A non-linear small gain theorem for the analysis of control systems with saturation. IEEE transactions on Automatic Control, 4:256 27, 996. 5. A. Zavala, I. Fantoni, and R. Lozano. Global stabilization of a PVTOL aircraft model with bounded inputs. International Journal of Control, 76(8):833 844, 23.
8 Hably, Kendoul, Marchand, Castillo x 5 5 Theorem with ε =.99 5 5 2 25 4 dx/dt 2 2 4 Theorem with ε =.99 6 5 5 2 25 Fig. 2. The evolution of x and its time derivative 6 5 y 4 3 2 Theorem with ε =.99 5 5 2 25 dy/dt 2 3 4 Theorem with ε =.99 5 5 5 2 25 Fig. 3. The evolution of y and its time derivative
Further results on global stabilization of the PVTOL aircraft 9.5 θ.5.5 Theorem with ε =.99 2 3 4 5 6 dθ/dt.5.5 Theorem with ε =.99 2 3 4 5 6 Fig. 4. The evolution of the rotational subsystem Σ 2 2 u.5.5.5 Theorem with ε =.99 2 3 4 5 6.5 v.5.5 Theorem with ε =.99.5 2 3 4 5 6 Fig. 5. The evolution of the thrust input and the rolling moment