28TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

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1 8 TH INTERNATIONAL CONGRESS OF O THE AERONAUTICAL SCIENCES AUTOPILOT DESIGN FOR AN AGILE MISSILE USING L ADAPTIVE BACKSTEPPING CONTROL Chang-Hun Lee*, Min-Jea Tahk* **, and Byung-Eul Jun*** *KAIST, **KAIST, ***Agency for Defensee Development ac.kr;mjtahk@fdcl.kat.ac.kr;mountrees@gmail.com Keywords: Agile msile, L adaptive control, backstepping control, msile autopilot Abstract Th paper deals with an agile msile pitch autopilot design using L adaptive backstepping control methodology. The flight phases of the agile msile systems can be classified into three phases; launch, agile turn, and end-game. Th paper focused on the agile turn phase, which a fast 8 degree turn to engage a rear- under the presence of the aerodynamic uncertainties es. To attain a fast response, the hemphere located target after launch l phase, angle-of-atta ack chosen to be the control variable. Since L adaptive control can guarantee robustness against uncertainty andd a fast response during the transient phase, itt suitable for a control methodology of the agile turn. The performances of the t proposed controller are investigated and demonstrated through numerical simulations. Introduction During the course of years, the control systems of the agile msiles have been extensively studied by many researchers [-6]. In general, the flight phases of the agile msile systems const of three different phases; launch, agile turn, and end-game []. Among three t different phases, the controller design of the agile turn phase introduces many challenging problems. The agile turn defined to be a fast 8 degree turn maneuver to engage a rear-hemphere located target after launch phase. The reason, why handling of the agile turn difficult, the presence of a large variation of the aerodynamic uncertainties induced by a high angle-of-atta ack maneuver. Therefore, a robust control approach needed inn order to compensate such the aerodynamic uncertainties. One of solution that the controller predicts the model m uncertainties and adapts the model parameters, whichh called the adaptive control methodology [-6]. Although th approach can have robustness under the presences off the model uncertainties, the transient performance poor due to the adaptation time nullifying the model error. Hence, the conventional adaptive control methodology may not be suitable for a control method of thee agile msile systemss that require a fast transient performance. As a remedy, in th paper, we propose an agile msile controller based on backsteppin ng control methodology with L adaptive scheme. From previous works [8-9], it has been shown that L adaptive scheme can provide the robustness and a a fast t response during the transient phase. In addition, since the normal acceleration control c of the msile systems s faces with the nonminimum phase phenomena, we choose the angle-of-attack for the control variable. Additionally, according to t references [8-9], the commandingg of angle-of-attack more desirable than thatt of normal acceleration for achievingg a fast response. Finally, the performance of the proposed controller demonstrated by numerical simulations. Th paper consts of five sections. In section, a nonlinear msile modell explained. The proposedd controller providedd in section. The simulation results are shown in section 4. In section, we conclude th study. Nonlinear Msile Model

2 CHANG-HUN LEE, MIN-JEA TAHK, AND BYUNG-EUL JUN In th study, a nonlinear model of the longitudinal msile motion considered as shown in Fig.. and the notations of Δ represents the model error caused by the aerodynamic uncertainties in the high angle-of-attack regime. L Adaptive Backstepping Controller. Model Derivation Fig. The msile geometry Under the assumptions that the msile body rigid and the gravity force compensated, the equations of motion are determined as follows: QS α = ( C N ( α, ) (, ) ) M + C N α δ M δ + q () mv QSl lq q = Cm ( α, M) + C (, ) m M + C q m α M δ δ () Iyy V where α, q, and δ denote the angle-of-attack, the pitch rate, and the fin deflection angle, respectively. The notations of m, I yy, S, and l indicate the mass, the pitching moment of inertia, the reference area, and the reference length, respectively. The velocity, the dynamic pressure, and Mach number are denoted by V, Q, and M, respectively. The aerodynamic coefficients in Eqs. () and () are given by the function of the angle-of-attack and Mach number. There coefficients are computed form the predetermined data tables, which are obtained by the wind tunnel test. In the high angle-of-attack regime, the msile undergoes the aerodynamic uncertainties. From Eq. (), we have the following equations under the presence of the aerodynamic uncertainties. x = f+δ f+ x + ( g+δg) u () x = f +Δ f + ( g +Δg) u (4) where, x = α, x = q, u = δ () f = KC, f = K C + C ( ql/v) (6) α N q m m q g = KC α N δ, g = KC q m δ (7) K QS / mv K = QSl / I (8) = α, q ( yy) In order to apply the backstepping control methodology, a strict feedback form of system equation required. In the msile systems, the magnitude of control force ( g+δ g) u in the right hand side of Eq. () can be generally negligible due to ( g +Δ g) u ( g+δg) u. Therefore, Eqs. () and (4) can be rewritten in a strict feedback form as follows: x = f+ x +Δ (9) x = f + gωu+δ () where Δ = Δ f+ ( g+δ g) u, Δ =Δ f () ω = +Δ g / g () In Eqs. (9) and (), Δ and Δ can be regarded as the total model errors due to the aerodynamic uncertainties and the neglecting term of ( g+δ g) u. By introducing the linear parameterization [9], the model uncertainties can be parameterized as follows: Δ = θ x + σ () Δ = θ x + σ (4) Substituting Eqs. () and (4) into Eqs. (9) and () yields the following equation. x = f+ x + θ x + σ () x = f + g ωu+ θ x + σ (6) where θ, θ, and ω are unknown parameters. The adaptation schemes of these variables will be dcussed in section... State Predictor Design In th section, we dcuss the state predictor design. First, let us define the desired error dynamics of state predictor as follows: x = Kx, x = Kx (7)

3 AUTOPILOT DESIGN FOR AN AGILE MISSILE USING L ADAPTIVE BACKSTEPPING CONTROL where x xˆ x represents the prediction error. ˆx and x are the predicted state and the true state, respectively. The gains of K and K decide the convergence speed of the prediction error. Substituting Eqs. () and (6) into Eq. (7) provides the following state predictor. ˆx = Kx + f+ x + θ x + σ (8) ˆx = Kx + f + gωu+ θ x + σ (9) Since θ, θ, and ω are unknown parameters in Eqs. (8) and (9), they should be estimated as follows: xˆ = Kx ˆ ˆ + f+ x + θ x + σ () x ˆ = K x + f + g ωu+ ˆ θ x + ˆ σ (). Adaptive Law Design Th section derives the adaptive law based on Lyapunov function. By using Eqs. (), (6), (), and (), the prediction error dynamics in Eq. (7) can be rewritten as follows: x = xˆ x = Kx+ θ x + σ () x = xˆ x = Kx + g ωu+ θ x + σ () where θ ˆ = θ θ, θ = ˆ θ θ, σ = ˆ σ σ σ ˆ = σ σ, and ω = ˆ ω ω. Let us consider the following Lyapunov function. V = ( ) x + x + ω + θ + θ + σ + Γ σ (4) Taking the time-derivative of V yields: V = Kx Kx + C+ C+ C+ C4+ C() where C = θx x + ( / Γ) θθ (6) C = θx x + ( / Γ) θθ (7) C = σx+ ( / Γ) σσ (8) C = σ x + / Γ σσ (9) 4 ωxgu / C = + Γ ωω () In order to guarantee the asymptotic stability of the prediction error, we enforce C through C to be zero. Then, the time-derivative of V always negative definite as follows: V K x K x () = < From the condition of nullifying C to C, the following adaptive law can be obtained. ˆ θ = θ = Γx x, ˆ θ = θ = Γ x x () ˆ σ = σ = Γx, ˆ σ = σ = Γx () ˆ ω = ω = Γxgu (4) where Γ represents the adaptation gain. In the conventional adaptive control, the increase of the adaptation gain introduces the chattering effect. However, L adaptive control relieves th limitation by introducing a low pass filter rejecting the high frequency signal induced by high adaptation gain..4 Control Law Design For convenience, let new residuals be defined as follows: z = x xd, z = x xd () Then, the residual dynamics can be expressed as: z = f+ x ˆ ˆ + θ x + σ x (6) d z = f ˆ ˆ ˆ + gωu+ θ x + σ x d (7) where x d and x d are the desired state values. We consider the following Lyapunov function to derive the outer loop control law in the backstepping methodology. V = z (8) The time-derivative of V can be determined by using Eq. (6) as follows: V = z( f+ x ˆ ˆ + θ x + σ x ) (9) d In order to enforce V <, the desired value of x chosen as: ( ˆ θ ˆ σ ) xd = Kz f+ x + + x d (4) In order to design the inner loop control law, the following Lyapunov function introduced. V = z + z (4) After taking time-derivative of V and substituting Eqs. (6) and (7) into V gives the following result.

4 CHANG-HUN LEE, MIN-JEA TAHK, AND BYUNG-EUL JUN V = z Kz+ z (4) + z( f ˆ ˆ ˆ + gωu+ θ x + σ x d ) From Eq. (4), we can obtain the inner loop control law that satfies V < as follows: u = ( Kz z ˆ ˆ f θ x σ + x d )(4) g ˆ ω In Eqs. (4) and (4), K and K represents the control gain, which are identical values of the state predictor. If the adaptation gains increase, the adaptation parameters of ˆ θ, ˆ θ, ˆ σ, ˆ σ, and ˆω contain the high frequency signals which cause the chattering effect during the transient phase. In L adaptive control methodology, low pass filters are introduced in the inner loop and the outer loop control law in order to cutoff the high frequency signal. Therefore, the uses of the high adaptation gains are possible in th method. By introducing a low pass filter, the outer loop control law can be obtained as follows: x ˆ d = Kz η C + x d (44) where ˆ η ˆ C C s η in the frequency domain. A second-order low pass filter C ( s ) and the variable ˆ η are defined as follows: C ω ( s) = s + ζ ωs+ ω (4) ˆ η = f + ˆ θ x + ˆ σ (46) where ω and ζ represent the design parameters of low pass filter. In a similar way, the inner loop control law modified by using a first-order low pass filter. uc ( s) = C ( s) u (47) where g ˆ ωk C ( s) = (48) s + g ˆ ωk u= ( Kz z ˆ ˆ f θx σ+ x d ) (49) g ˆ ω where k represents the design parameter of a first-order low pass filter. Fig. shows the overall configuration of the proposed controller. Fig. The configuration of controller 4 Simulation Results In order to demonstrate the performance of the proposed controller, numbers of simulations are carried out. In these simulations, a second-order actuator model with ω act = rad/ s, ζ act =.7, and δ = 4 deg/ s considered. The controller parameters are chosen as K = K = and Γ =. The low pass filter parameters are defined as ω = rad/ s, ζ =.7, and k =. 4. Case : Step Input Command In th simulation, the proposed controller tested with a step input command. Figs. and 4 show the step input response of angle-of-attack and the fin deflection angle in the nominal case. Figs. and 6 provide the simulation results under the presence of % model uncertainties during the flight. The results indicate that the proposed controller can provide a good tracking performance even in the presence of the model uncertainties. 4. Case : Agile Turn Scenario In th simulation, the applicability of the proposed method determined under an agile turn scenario (i.e., 8 degree heading reversal turn). The angle-of-attack command [] which obtained from the trajectory optimization to achieve the terminal velocity after agile turn used. Figs. 7 and 8 give the angle-of-attack response and the fin deflection angle during the agile turn. The results show that the proposed controller can maintain a sound tracking performance. Therefore, the proposed method can be applied to the challenging sues of the agile msile systems. 4

5 AUTOPILOT DESIGN FOR AN AGILE MISSILE USING L ADAPTIVE BACKSTEPPING CONTROL Response Command Angle-of-Attack 4 Angle-of-Attack Response Command α ( ) - α ( ) Fig. Angle-of-attack (nominal) Fig. 7 Angle-of-attack (agile turn) Control Input Control Input - -4 δ ( ) - δ ( ) Fig. 4 Fin deflection angle (nominal) Fig. 8 Fin deflection angle (agile turn) Angle-of-Attack Conclusion α ( ) - - Response Command Fig. Angle-of-attack (% uncertainty) Control Input In th paper, we propose the agile msile autopilot based on backstepping control in conjunction with L adaptive scheme. In the proposed method, the commanding of angle-ofattack was considered for accomplhing a fast turn of the msile s heading angle. The simulation results indicated that the proposed controller can provide the sound performance under the presence of the model uncertainties in high angle-of-attack regime. It can also be applicable to the agile turn maneuver. δ ( ) - - Acknowledgments Th research was supported by Agency for Defense Development under the contract UDCD Fig. 6 Fin deflection angle(% uncertainty) References

6 CHANG-HUN LEE, MIN-JEA TAHK, AND BYUNG-EUL JUN [] Kevin A. We and David J. Broy. Agile Msile Dynamics and Control. Journal of Guidance, Control, and Dynamics, Vol., No., pp , 998. [] A. Thukral and M. Innocenti. A Sliding Mode Msile Pitch Autopilot Synthes for High Angle of Attack Maneuvering. IEEE Transactions on Control System Technology, Vol. 6, No., pp. 9-7, 998. [] D. J. Leith, A. Tsourdos, B. A. White and W. E. Leithead. Application of Velocity-based Gainscheduling to Lateral Auto-pilot Design for an Agile Msile. Control Engineering Practice, Vol. 9, pp. 79-9,. [4] H. Buschek. Full Envelope Msile Autopilot Design Using Gain Scheduled Robust Control. Journal of Guidance, Control, and Dynamics, Vol., No., pp. -, 999. [] M. B. McFarland and A. J. Cale. Neural Networks and Adaptive Nonlinear Control of Agile Antiair Msiles. Journal of Guidance, Control, and Dynamics, Vol., No., pp. 47-,. [6] L. Zhong, X. G. Liang, B. G. Cao and J. Y. Cao. An Optimal Backstepping Design for Blended Aero and Reaction-Jet Msile Autopilot. Journal of Applied Sciences, Vol. 6, No., pp. 6-68, 6. [7] E. Slotin. Applied Nonlinear Control. Prentice Hall International, Inc., 99. [8] C. Cao and N. Hovakimyan. L Adaptive Controller for a Class of System with Unknown Nonlinearities: Part I. American Control Conference, Washington, pp , 8. [9] N. Hovakimyan and C. Cao. L Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation. Society for Industrial and Applied Mathematics,. [] C. H. Lee, T. H. Kim, S. M. Ryu and M. J. Tahk. Analys of Turning Maneuver for Agile Msile using Trajectory Optimization. Proceeding of KSAS Spring Conference, Pyeongchang, Korea, pp. 69-7, 9. (in Korean) Copyright Statement The authors confirm that they, and/or their company or organization, hold copyright on all of the original material included in th paper. The authors also confirm that they have obtained permsion, from the copyright holder of any third party material included in th paper, to publh it as part of their paper. The authors confirm that they give permsion, or have obtained permsion from the copyright holder of th paper, for the publication and dtribution of th paper as part of the ICAS proceedings or as individual off-prints from the proceedings. 6