A New Backstepping Sliding Mode Guidance Law Considering Control Loop Dynamics

Transcription

1 pp. 9-6 A New Backstepping liing Moe Guiance Law Consiering Control Loop Dynamics V. Behnamgol *, A. Vali an A. Mohammai 3, an 3. Department of Control Engineering, Malek Ashtar University of Technology * Postal Coe: , Tehran, IRAN vahi_behnamgol@mut.ac.ir In this paper, a new proceure for esigning the guiance law consiering the control loop ynamics is propose. The nonlinear guiance loop entailing a first orer lag as the control loop ynamics is formulate. A new finite time an smooth backstepping sliing moe control scheme is use to guarantee the finite time convergence of relative lateral velocity. Also in the propose algorithm the chattering is remove an a smooth control signal is prouce. Moreover, the target maneuver is consiere as an unmatche uncertainty. Then a robust guiance law is esigne without requiring the precise measurement or estimation of target acceleration. imulation results show that the propose algorithm has better performance as compare to the proportional navigation, augmente PN an the other sliing moe guiance law. Keywor: Guiance law, Control loop ynamics, liing moe control, Chattering Nomenclature 3 X tate Variables Vector f ( x ) Nonlinear Function w() t Uncertainty liing Variable c Weighting Constant in liing Variable u Equivalent Control eq u Reaching Control r t Reaching Time r Line of ight Angle Line of ight Rate r Relative Range r Closing Velocity r Relative Lateral Velocity A Missile Lateral Acceleration m, At Ac, N Aˆt Target Lateral Acceleration Lateral Commane Acceleration Time Constant of Control Loop Dynamics Boun of Uncertainty Navigation Constant Estimation Of Target lateral Acceleration Controller Gains. PhD tuent (Corresponing Author). Associate Professor 3. Assistant Professor Introuction The basic principle of parallel navigation is to nullify the line of sight rate. In the case of non-maneuvering targets an a zero lag autopilot, proportional navigation guiance law is the optimal solution to the linear homing problem [], []. Using the sliing moe control theory an base on the parallel navigation iea an nonlinear planar engagement kinematics, we can esign a guiance law only with information for the maximum target lateral acceleration. Therefore, the precise measurement or estimation of the target acceleration is not require an the propose guiance law is robust with respect to the target maneuvers [3]. The sliing moe control has been applie to many guiance problems. Zhou et al. propose an aaptive sliing moe guiance law using linearize equations [4]. Babu et al. [5] stuie the guiance law for highly maneuvering targets using the sliing surface of the zero line-of-sight rate base on the Lyapunov metho. In [6], for removing chattering of a ifferent sliing surface base on relative range the line of sight rate is efine. In this reference, the commane acceleration is smoother than the control input in [5], but the calculation is complicate. In [7], base on pure proportional navigation iea, a sliing moe guiance law for elaye LO rate measurement with sliing variables esigne with regar to the Receive: 3. 5., Accepte: 6.. 3

2 / V. Behnamgol an A. Mohammai relative lateral velocity. In [8], the same sliing variable an true proportional navigation basis is use, but the bounary layer scheme ecreases the control precision. A novel sliing moe-base impact time an angle guiance law for engaging a moern warfare ship is presente in [9] an finally in [], guiance law is esigne using the secon orer sliing moe control. In all of the above references, the guiance law is esigne for a case with ieal ynamics, i.e. no elays exist between the commane an interceptor applie acceleration. In an actual situation, ue to flight conitions an unexpecte environmental variables/changes, we cannot expect the ieal performance of the control system. When ieal ynamics are consiere in the esigning proceure, there is no guaranty for the finite time convergence of the LO rate an the chance of ivergence shoul be consiere. This ivergence may severely affect the miss istance an lea to an unsatisfactory performance. To improve the performance, simultaneous esign of the guiance an control loop can be use [, ]. liing moe control theory is known to be robust against parameter uncertainties an external isturbances in nonlinear systems. However, for the sliing surface to be attractive, a switching function must be use in the control law, which causes the chattering of the control signals. The chattering problem in sliing moe control signal is one of the most common hanicaps in real applications. Chattering phenomenon is the result of low control accuracy, high heat losses in electrical power circuits an high wear of moving mechanical parts. It may also excite un moele high frequency ynamics, that egrae the performance of the system an may even lea to instability [3 5]. One approach to eliminate the chattering in control signal is to use a continuous approximation of the iscontinuous sliing moe controller calle the bounary layer approach. In this metho, insie the bounary layer, the iscontinuous switching function is interpolate by a continuous function to avoi iscontinuity of the control signals. However, this metho brings a finite steay state error an leas to tracking within a guarantee precision rather than a perfect tracking [6]. The with of the bounary layer is normally constant, an the larger the bounary layer with, the smoother the control signal. Even though the bounary layer esign alleviates the chattering phenomenon, it no longer rives the system state to the original state, but to a small resiual set aroun the origin. The size of the resiual set is etermine by the with of the bounary layer: the larger the with of the bounary layer, the larger the size of the resiual set. As a consequence, there exists a esign conflict between requirements of the control signals smoothness an the control accuracy. For smoothness of the control signals, a large bounary layer with is preferre, but for a better control accuracy, a small bounary layer with is preferre [6, 7]. The other way is to use higher orer sliing moe. HOM has two important features that make it a better choice in esigning the controller. It improves the accuracy of the esign, which is a very important issue, an may provie a continuous control [5, 8]. The HOM generalizes the conventional sliing moe iea, seeking to zero not just the sliing variable, but also some of its time erivatives. In particular, secon orer sliing moes woul provie for the zeroing of the sliing variable an its first time erivative in the finite time, through iscontinuous control action acting on its secon time erivative, being the sliing variable of relative egree or. The problem with this metho is that the erivative of a certain state variable is not available for measurement, an therefore methos have to be use to observe that variable an complicate calculations [8, 9]. The back stepping sliing moe control is esigne in some references. In [], an aaptive backstepping sliing moe control is propose for a class of uncertain nonlinear systems with input saturation. The control law an aaptive upating laws of neural networks are erive in the sense of Lyapunov function, so that the stability can be guarantee even uner the input saturation. This control law is robust against the isturbance, an it can also eliminate the impact of input saturation. [] presents an integrate missile guiance an control law base on aaptive fuzzy sliing moe control. The aaptive nonlinear control law is esigne sing backstepping an sliing moe control techniques. Also in [] a robust chattering free backstepping sliing moe controller is evelope for the attitue stabilization an trajectory tracking control of qua rotor helicopter with external isturbances. The control scheme is evelope with the help of backstepping technique an a sliing surface is introuce in the final stage of the algorithm. To attenuate the chattering problem cause by a iscontinuous switching function, a simple fuzzy system is use. The asymptotical stability of the system can be guarantee since the control law is erive base on Lyapunov theorem. In this paper, a new guiance law is propose to guarantee the finite time convergence of the relative lateral velocity consiering control loop ynamics as the first orer lag. The target maneuvers are moele as boune uncertainties. For the equation, a new backstepping sliing moe guiance law is esigne to guarantee the finite time stability of the system states. Also the finite time convergence is prove using a new sliing conition that leas to ecreasing the chattering with high precision in zeroing the sliing variable. The paper is organize as follows. In section II, the conventional sliing moe control theory is presente an then the equation of motion is formulate. The new back stepping sliing moe

3 A New Backstepping liing Moe Guiance law Consiering... control approach is introuce in section III an then the finite time stability of this algorithm is prove. In section IV, the propose algorithm is use to esign the guiance law. Numerical simulation results are shown in section V, an conclusions are reporte in section VI. MC Design Algorithm an Plant Moeling In this section first, the sliing moe control theory is reviewe an then the equation of guiance Loop with approximation of control loop ynamics is formulate. Conventional liing Moe Control In this section the conventional sliing moe control is presente [3-5]. Consier a nonlinear system ( n ) X f ( x ) u w ( t), w ( t) () where f ( x ) is a known nonlinear part, w() t is a ( n) T boune uncertainty, X [ x x... x ] is the system state an u is the control input. Then sliing variable is n c X t () Where c is a strictly positive constant, X X X an is the esire state. Assume that u in equation X () must be esigne in a way that the following system has the esire properties, an system state reaches the esire amount. The tracking problem for X X is equivalent to making. Conventional first orer sliing moe control makes variable equal to zero in finite time an then maintains the conition for all future time. Typical sliing moe control consists of a reaching moe, uring which the sliing variable moves to the sliing surface, an a sliing moe, uring which the sliing variable is confine to the sliing surface. Also the sliing variable has no variation from the zero sliing surface in a system without uncertainty. In conventional sliing moe control the control input is esigne as follows: u u u, u kign( ) (3) eq r r Where ign (.), enoting the ignum function u eq, is the equivalent control etermine to cancel the known terms on the first erivation of the sliing variable in system without uncertainty. If there is no uncertainty in the system, u ueq will maintain the system on the sliing surface. If uncertainties exist, by consiering V.5 as Lyapunov caniate, a sufficient conition to guarantee the finite time attractiveness of sliing surface for, is to ~ ensure: Vol. 8 / No. 4 / Winter 6 / V (4) Where is a positive constant, which implies that: () t r (5) The sliing moe controller (3) contains the iscontinuous nonlinear function ign (.). This nonlinearity can cause a chattering problem ue to elays or imperfections in the switching evices. Also by selecting the sliing variable as () an using the controller (3), the sliing variable reaches the sliing surface in finite time but the state variables are a asymptotically stable. Equation of Guiance an Control Loops In this section, an integrate moel for the guiance an control loop is formulate. Guiance an control loops are shown in figure, where the outer loop is the guiance loop an the inner loop is the control loop. Fig.. Guiance an Control Loops Consier a two-imensional engagement as shown in figure, where r is the relative range between target an interceptor is the LO angle with respect to a reference axis, r is the relative lateral velocity between interceptor an target, an At an Am represent the target an interceptor acceleration normal to the LO line, respectively. The kinematic relation between the target an interceptor is escribe by a nonlinear equation (6) an the control loop ynamic as a first orer lag is escribe by a linear equation (7) [, 3]. t r r Am, At, (6) ( A m, ) A A m c, t (7) Ac enotes the acceleration comman. The control object is to nullify the relative lateral velocity. Note from equation (6) that At can be treate as the aitive unmatche uncertainties of the system.

4 / V. Behnamgol an A. Mohammai () t r () ( ) Proof: ubstituting equations (8) an (3) in equation () an then with substituting the resulte equation in the introuce sliing conition, (9) yiels k w () Fig.. Interceptor - target engagement geometry New liing Moe Control Algorithm The sliing moe controller (3) contains the iscontinuous nonlinear function ign (.). This nonlinearity can cause a chattering problem ue to elays or imperfections in the switching evices. Therefore, we propose a new sliing moe control by replacing iscontinuous ignum function y ign() x by a new continuous function y x x. The responses of the propose continues function an iscontinues signum function are shown in figure 3. By choosing a large enough k in (), we can now guarantee that (9) is verifie. o allowing k () where enotes the maximum value of w. For proving (), let t r be the time require to hit the sliing surface. Integrating (9) from t to t leas to () t r V () tr ( ) tr t () t t r (3) Y Y=ign(X) Y=X/ X -.4 Y=X/ X -. Therefore, (9) is a finite time conition with finite reaching time (). o, we prove the finite time convergence of X. In other wors, we have a finite time sliing moe controller. By substituting equations () an (8) into equation (3), control input is given as u ueq ( ) (4) X Fig. 3. The responses of propose an signum functions ince the propose MC oes not inclue a signum function, the chattering can be reuce. For this, we use the following theorem. Theorem : The reaching term of control input: u reching k, (8) with () as sliing variable an consiering V.5 as Lyapunov caniate an with satisfying the sliing conition: V (9) with k proviing for the convergence of the state X in finite time: Guiance Law Design In this section, the esign proceure of the nonlinear guiance law is presente for the integrate guiance an control system given by (6) an (7). We approach the esign through a back-stepping sliing moe metho. We start with the equation (6). In this equation the control object is zeroing r an Am, view as the virtual control input. The tools of linearization, exact feeback linearization, Lyapunov reesign, back-stepping, or a combination of them coul be use for the esign of this virtual control input [3]. To procee with the esign of the new sliing moe control that is propose in the previous section, set r (5) an take the virtual control m meq Am as A A K, (6) For, this acceleration will be iscontinuous

5 A New Backstepping liing Moe Guiance law Consiering... an oscillation will occur in. Therefore, with a large value for, we have a smooth signal. Equivalent control Am is chosen to cancel the known terms in eq the equation, that is, A m eq r (7) A sufficient conition to guarantee the finite time attractiveness of is to ensure conition of (9) V, V (8) Where is a strictly positive constant, which implies: ( ) () t (9) reach ( ) In orer to satisfy the sliing conition (8), espite the uncertainty on the ynamics system (6), take V ( r Am, K A, ) eq t () ( K A t ) K Where is the maximum value of At. Therefore, by replacing equations () an (7) in (6), the virtual control or esire interceptor acceleration for nullifying relative lateral velocity yiels r Am r r () r Now, in the secon step we want to esign the commane acceleration Ac in equation (7) so that testate variable Am, reaches esire missile acceleration ( A ) m, an track it. Therefore, for this purpose we can introuce the variable as A A () m m Am Is the esire interceptor acceleration for stabilizing r in finite time represente by the right han of (). It is clear that if ; the variable r approaches the origin in finite time that guarantees the intercept. Now we achieve equivalent commane acceleration Ac, eq, while zeroing the erivative is given by AA m m, r r Am, Ac, r r r (3) m m, Am, Am, Vol. 8 / No. 4 / Winter 6 / 3 To cancel the known term on the right-han of equation (3), the equivalent control is taken as r r Aceq Am r r (4) r Let us take the control input as: A c, Ac, K (5) eq Note that for, this comman acceleration will be iscontinuous an oscillation will occur at. Therefore, with a large value for, we have a smooth commane acceleration. A sufficient conition to guarantee the finite time attractiveness of for, is to ensure: V, V, (6) Where is a strictly positive constant. In orer to satisfy (6), substituting equations (-4) in (5) yiel K (7) By substituting equations (4) an (7) in (5), the commane acceleration is given as r r Ac, Am, r r r (8) A A In this commane acceleration, measurements or estimates of variables, r, r, r an Am, are require an the parameters are an. This comman stabilizes the origin (, ) r, Am Am, in finite time. By applying this (,) commane acceleration, first the interceptor acceleration reaches the esire value () in finite time that is ajustable with varying the value of parameter. Then with varying the value of parameter, we are able to zero the relative lateral velocity in another esire time. imulation Numerical simulations are performe to investigate the performance of the propose algorithm. In this section, we consier situation in which the initial relative istance is 4 km, the closing velocity is m/s, the lateral relative velocity is 3 m/s an the control loop ynamic time constant is equal to.5 s.

6 4 / Comparison with PN Guiance Family In this section, the propose guiance law is compare to the true proportional navigation (TPN) represente by the following guiance comman: Am Nr (9) Where the guiance constant is chosen as N 4 in this stuy an augmente proportional navigation (APN) is represente by the following guiance comman: A mnr A ˆ t, (3) Where  t is the estimation of the target acceleration [, ]. In this case the target maneuvers with acceleration. We apply the propose guiance law (8) with. 5,. 5,,. 7 an.5. Note that in comparing the propose guiance law with PN family, the variables, r an Am, are require, but it is more robust with respect to the target acceleration. Fig. 4 illustrates the commane acceleration an Fig. 5 shows the interceptor acceleration. These figures emonstrate that the maximum magnitues of the propose an APN guiance laws are less than that of the PN law an provie goo tracking of the target acceleration. Note that in APN, estimation of the target acceleration is use in APN guiance law. Fig. 6 shows the relative lateral velocity ( variable). It is clear from this figure that the relative lateral velocity in the propose law converges to zero in finite time, but PN is unable to control this variable. Fig. 7 an Fig. 8 show the relative range, an commane, missile an esire missile acceleration, respectively. Fig. 9 epicts the interceptor an target trajectory in the propose guiance law. It is clear from these figures that in the propose law the missile intercepts with the target in a shorter time in comparison with PN law. Also missile acceleration tracks the esire missile acceleration in the propose Law. The integral of the square acceleration is consiere as energy in this paper. Table, shows that the interception, the time an the require energy in the propose law are less than PN. As a result, the propose guiance law by consiering the control loop ynamics in esigning proceure has an acceptable performance in comparison with the PN guiance law. Table. Intercept Time an Energy Guiance Law Intercept Energy Propose Law TPN.6 47 APN Commane Acceleration (m/s ) 4 3 V. Behnamgol an A. Mohammai Propose Law PN Law APN Law Fig. 4. Interceptor Acceleration with ifferent value for Relative Lateral Velocity (m/s) Relative Range (m) Missile Acceleration (m/s ) APN Law Fig. 5. Missile Acceleration Propose Law PN Law Propose Law PN Law APN Law Fig. 6. Relative Lateral Velocity ( Variable) Propose Law PN Law APN Law Fig. 7. Relative Range

7 A New Backstepping liing Moe Guiance law Consiering... Vol. 8 / No. 4 / Winter 6 / 5 Acceleration (m/s ) Y(m) Commane Acceleration 5 Missile Acceleration Desire Missile Acceleration 5 5 Fig. 8. Commane, Missile an Desire Missile Acceleration in Propose Guiance Law Missile Trajectory Target Trajectory Table. Intercept Time an Energy Guiance Law Intercept Energy Propose Law liing Moe Law.9 57 Commane Acceleration (m/s ) Propose Law liing Moe Law 5 5 Fig.. Commane Acceleration X(m) Fig. 9. Interceptor an Target Trajectory in Propose Guiance Law Comparison with Conventional M Guiance Law In this section, the propose guiance law is compare to the conventional guiance law esigne in [3] an represente by the following guiance comman: Am r ign( r ) (3) Figures an show the commane acceleration an the interceptor acceleration. These figures emonstrate that the maximum magnitue of the propose guiance law is less than that of the conventional sliing moe law an provies a goo tracking of the target acceleration. Fig. shows the relative lateral velocity ( variable) an Fig. 3 shows Line of sight rate. It is clear from these figures that the relative lateral velocity an LO rate in the propose law converge to zero in finite time, but M law is unable to control these variables. Table shows that, the interception time an control effort in the propose law are less than PN. As a result, the propose guiance law by consiering control loop ynamics in the esigning proceure enjoys goo performances compare with the PN guiance law. Missile Acceleration (m/s ) Relative Lateral Velocity (m/s) Fig.. Missile Acceleration Propose Law liing Moe Law Propose Law liing Moe Law 5 5 Fig.. Relative Lateral Velocity ( Variable)

8 6 / V. Behnamgol an A. Mohammai LO Rate (Deg/s) Propose Law liing Moe Law 5 5 Fig. 3. Line of ight Rate The Chattering in Propose Algorithm In this section, we apply the guiance law (8) with ifferent values for an. Note that the parameter is for ajusting the chattering on the sliing surface (relative lateral velocity). The smoothness in the esire acceleration is generate with relation (). Also parameter is for ajusting the chattering on the sliing surface (ifference between applie an the esire missile acceleration) an the smoothness of commane acceleration that is generate using relation (8). Fig. 4 shows the commane acceleration an Fig. 5 shows the liing variable with. 7 an.,.5. As shown in these figures, as the sliing variable reaches zero with., chattering occurs, while with. 5 the commane acceleration is smooth. Also by increasing the value of this parameter, we are able to ajust the magnitue of the commane acceleration as well as the time of zeroing the sliing variable. Table 3 shows the intercept time an energy with ifferent values for. As seen in this Table, by ecreasing the value of, the interception time an the control effort are increase. Fig. 6 shows the esire missile acceleration an Fig. 7 shows the sliing variable, with. 5 an.3,.7. As shown in these figures, since the sliing variable reaches zero with. 3, chattering occurs, while with. 7 the esire missile acceleration is smooth. As seen in table 4, by ecreasing the value of, the interception time an control effort are increase. Table 3. Intercept Time an Energy Value of Intercept Energy Table 4. Intercept Time an Energy Value of Intercept Energy Commane Acceleration (m/s ) V ariable (m /s) D es ire A c c eleration (m /s ) Fig. 4. Commane Acceleration Fig. 5. Variable ( Am Am ) Fig. 6. Desire Missile Acceleration =.5 =. =.5 = =.7 =.3 5 5

9 A New Backstepping liing Moe Guiance law Consiering... Vol. 8 / No. 4 / Winter 6 / 7 Relative Lateral Velocity (m/s) =.7 = Fig. 7. Variable (Relative Lateral Velocity) Conclusion In this paper, the esign of a guiance law by consiering the first orer control loop ynamics is propose. The propose algorithm use a new smooth an finite time sliing moe scheme, with a backstepping proceure that guarantees the finite time stability of the integrate guiance an control system. By applying the esigne guiance law, first the interceptor acceleration reaches the esire value in finite time that is ajustable. Then with varying the value of a parameter we are able to zero the relative lateral velocity in finite time. It is emonstrate via simulations that a higher performance can be achieve using this guiance law. References [] Zarchan, P., Tactical an trategic Missile Guiance, AIAA eries, Vol. 99,, pp [] iouris, G. M., Missile Guiance an Control ystems, pringer, 5, pp [3] Moon, J., Kim, K., an Kim, Y., Design of Missile Guiance Law via Variable tructure Control, Journal of Guiance, Control, an Dynamics, Vol. 4, No. 4,, pp [4] Zhou, D., Mu, C., an Xu, W., Aaptive liing-moe Guiance of a Homing Missile, Journal of Guiance, Control, an Dynamics, Vol., No. 4, 999, pp [5] Babu, K. R., arma, I. G., an wmy, K. N., "witche Bias Proportional Navigation for Homing Guiance Against Highly Maneuvering Target," Journal of Guiance, Contro, an Dynamics, Vol. 7, No. 6, 994, pp [6] Innocenti, M., Nonlinear guiance techniques for agile missiles, Control Engineering Practice 9,, pp [7] Lum, K. Y., Xu, J. X., Abii, K., an Xu, J., liing Moe Guiance Law for Delaye LO Rate Measurement, AIAA Guiance, Navigation, an Control Conference an Exhibit, Honolulu, Hawaii, 8 - August, 8. [8] Zhanxia, Z., Feng, X., Huai, G., The Application of liing Moe Control on Terminal Guiance of Interceptors, IEEE, Thir International Conference on Intelligent Networks an Intelligent ystems, [9] Harl, N., an Balakrishnan,. N., Impact Time an Angle Guiance with liing Moe Control, IEEE Transactions on Control ystems Technology, [] htessel, Y.B., hkolnikov, I.A., an Levant, A., mooth econ-orer liing Moes: Missile Guiance Application, Automatica, Vol. 43, Issue 8, 7, pp [] Der, R.T, A liing Moe Nonlinear Guiance with Navigation Loop Dynamics of Homing Missiles, AIAA Guiance, Navigation an Control Conference, Toronto, Ontario Canaa,. [] Dongk young, C. an JIN, Ch., Aaptive Nonlinear Guiance Law Consiering Control Loop Dynamics, IEEE Transactions on Aerospace an Electronics ystems, Vol. 39, No. 4, 3, pp [3] Khalil, H. K., Nonlinear ystems, Prentice-Hall, Upper ale River, NJ, 996, pp [4] lotine, J. J. E. an Li, W., Applie Nonlinear Control, Prentice-Hall, Upper ale River, NJ, 99, pp [5] Friman, L., Moreno, J. an Iriarte, R. liing Moes after the First Decae of the st Century, pringer,. [6] Min-hin Ch., Yean-Ren H. an Tomizuka, M., A tate-depenent Bounary Layer Design for liing Moe Control, IEEE Transaction On Automatic Control, Vol. 47, No.,, pp [7] Kim, K. J., Park, J. B. an Choi, Y. H., Chattering Free liing Moe Control, ICE-ICAE International Joint Conference 6, Busan, Korea,6, pp [8] Behnamgol, V., Mohammazaman, I., Vali, A.R. an Ghahramani, N. A., Guiance Law Design using Finite Time econ Orer liing Moe Control, Journal of Control, K.N. Toosi University of Technology, Vol. 5, No. 3,, pp [9] Behnamgol, V., Mohamma zaman, I., Vali, A. R. an Fattahi, E., Design of liing Moe Guiance Law using PI liing urface, th Iranian Conference on Electrical Engineering, Tehran, Iran, [] Zhang, H. an Zhang, G., Aaptive Backstepping liing Moe Control for Nonlinear ystems with Input aturation, Transactions of Tianjin University, Vol. 8, No.,, pp [] Rana, M., Wanga, Q., Houa, D. an Dong, Ch., Backstepping Design of Missile Guiance an Control Base on Aaptive Fuzzy liing Moe Control, Chinese Journal of Aeronautics, Vol. 7, No. 3, 4, pp [] Basri, M., Ariffanan, M., Husain, R. an Kumeresan, A., Robust Chattering Free Backstepping liing Moe Control trategy for Autonomous Quarotor Helicopter, International Journal of Mechanical an Mechatronics Engineering, Vol. 4,Issue 3, 4, p. 36. [3] Zhou, D., un, h. an Teo, K. L., Guiance Laws with Finite Time Convergence, Journal of Guiance, Control an Dynamics, Vol. 3, No. 6, 9, pp