Paper No 072 ROBUST CONTROL SYNTHESIS FOR AN UNMANNED HELICOPTER

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1 Paper No 072 ROBUS CONROL SYNHESIS FOR AN UNMANNED HELICOPER V. I. Garkushenko S.S. Vinogradov and G.N. Barakos upolev Kazan National Research echnical University 0 Karl Marx St. Kazan 420 Russian Federation lvi@au.kstu-kai.ru School of Engineers University of Liverpool Liverpool L69 3GH U.K. G.Barakos@liverpool.ac.uk Abstract Controller synthesis for unmanned helicopters with minimum initial information about their parameters of their mathematical models is considered in this paper. he unknown parameters and system nonlinearity are considered as external disturbances. wo methods are proposed to solve this problem: design of the controller using feedback with compensation for disturbances estimated using observers or design of fuzzy controllers based on the approach of Mamdani and results of the controller of the first method. he paper presents a comparative study of the Raptor helicopter dynamics with the proposed control laws and with wind disturbances. Nomenclature pitch roll and yaw angles (deg) PID the proportional-integral-derivative controller pqr roll pitch and yaw rates (deg/s) MR R main rotor tail rotor u v w longitudinal lateral and normal NED north-east-down coordinate system velocity components (m/s) lon longitudinal and lateral cyclic angles lat col ped W a b s s LQR linear quadratic regulator main and tail rotor collective UAV an unmanned aerial vehicle wind actions in the body coordinate system (m/s) longitudinal and lateral flapping angle of main rotor (rad) GPS global positioning system. INRODUCION he study of UAV control systems is motivated by the complexity of their mathematical model their large number of experimentally determined parameters as well as the demanding requirements for their operation. here are various approaches for the design of UAV control systems however the ones preferred for practical implementation employ parameters that are easy to estimate. Such control laws include traditional PID controllers that are not effective in conditions of uncertainty and in the presence of interconnected control channels []. herefore autopilots are offered based on modified control laws like PID with tuned coefficients and control constraints [] robust controllers with feedback on the state vector H [2] LQR [3] μ - controllers [4] with disturbance compensation [5 6] non-linear

2 control laws using backstepping [7] dynamic inversion [8] and adaptive controllers with adjustable regulation coefficients [9]. It should be noted that the adjustment of the coefficients of the PID controller requires effort and does not eliminate the problem of helicopter control with added wind disturbances. Robust controllers H LQR and μ - controllers reduce the effect of external disturbances on the controlled output but do not eliminate their influence and usually require measurements of the state vector of the system. o improve the stability of the helicopter a controller with disturbance compensation is used in this paper. he assessment of the disturbance is carried out using an observer [5]. Non-linear control laws built using backstepping do not require adjustment of controller parameters but depend on the adopted mathematical models. In addition they are complex to implement and sensitive to parameters and external disturbances. A method of dynamic inversion is free of these shortcomings [8] and does not require an exact model of the helicopter. A model of the torque of the main and tail rotors however are needed for the implementation of the controller and measurement of accelerations are also required. Adaptive controllers use in addition to the main control laws varying parameters under changing external conditions. he efficiency of such algorithms depends on the speed of convergence of the adaptation processes that finally depends on the employed mathematical model of the helicopter and uncontrollable dynamics. Recently fuzzy control algorithms have gained popular. his is due to the fact that their use for complex system does not require accurate mathematical description. In addition such systems are able to maintain their performance despite the varying parameters of the system and the effect on it of external disturbances. here are two main approaches to the construction of a helicopter fuzzy controller as in the akagi-sugeno [0] and Mamdani [] works. In the akagi-sugeno approach knowledge of the model helicopter [0] is required while Mamdani's approach uses only information about the input and output signals which can be obtained during flight experiments []. Further to improve the quality of the controller use of neural and neuro-fuzzy concepts as well as various combinations of these are proposed. In this paper use of the Raptor helicopter offers an example for the synthesis of control laws based on its dynamic model and minimum initial information about its parameters. At the same time the unrecorded dynamics of the helicopter the unknown parameters and nonlinearities are considered as external disturbances and the control law is formed using observers of disturbances. o simplify the control algorithm a fuzzy controller based on the Mamdani approach is also designed. A comparative analysis of the helicopter dynamics for the developed control laws under the influence of wind disturbances is presented in the results section of the paper. 2. FORMULAION OF HE PROBLEM he unmanned Raptor helicopter is considered here. Its non-linear mathematical model and its parameters are identified in [22]. he helicopter dynamic equations take the form: () F V V m m S J M J where V [ u v w] and [ q p r] are the linear and angular velocity vectors in the body coordinate system; are the Euler angles; F is the aerodynamic force vec- F g

3 tor; F mg xx yy zz g sin cos sin cos cos is the gravity force vector; m is the helicopter mass; S is the transformation matrix; J diag{ J J J } is the moment of inertia matrix. M is the aerodynamic moment vector: M K mr Hmr sin bs Lvf tr H tr x (2) M y K mr Hmr sin as M hf M z Pmr Nvf tr Dtr mr where mr tr are the main rotor (MR) and tail rotor (R) thrusts; K is the MR stiffness in ; H mr is the MR hub location above the center of gravity (CG); P mr is the total power of the MR; mr is the MR rotating speed; a b are the longitudinal and s lateral flapping angles of MR; hub location behind the CG; s D tr is the R H is the R hub location above the CG; L vf M hf N vf are the aerodynamic moments generated by fins. he dependence of MR flapping angles to the control actions lon lat is expressed by the following equations [2]: (3) s s bs s lonlon latlat a a q A b A A s s as s lonlon latlat b b p B a B B where A lon and B lat are the ratios of the longitudinal and lateral cyclics of the input signal to the control displacements lon and lat respectively; A lat and B lon are the coefficients of the cross ties; is the time constant; A bs and B as are the coupling effects. In order to simplify the procedure of the controller design its dynamics equations are presented in the following form: X R X3 X 2 (4) X F X X X U W X S X X tr X F X X X U W X F X X U where X x y z is the position vector in the local north-east-down (NED) coordinate system; X2 V X3 X 32 X4 ; X 5 as bs pedint X 52 as bs pedint is the intermediate state of yaw rate feedback controller; U 0 U0 ped col cos U diag g g u 0 0 u0 sin sin ; U2 lon lat ped U ped col ped and col are the normalized rudder servo and collective R X and pitch inputs respectively; 3 32 S X are the rotation and kinematic transformations matrices respectively wind wind wind W u v w are the wind actions in the body coordinate system which is defined as [2]: 2 W 0.5V cos t t p (5) i pmax i 3 where V p max is the maximum amplitude of the gust of wind during the time interval t p. We introduce the state vector X X X X X X and control signals vector U lon lat ped col each of which lies in the range from - to [2]. When balancing the helicopter in hover condition we assume: U lon lat ped col X x0 y0 z a 0 s b s. Equation (4) can be rewritten in terms of deviations from the trimmer condition as:

4 x X X x x x x x lon lat ped col x2 u v w u U U x q p r x x y z x 3 x5 as bs pedint. 4 hen the resulting system of equations represented in a form suitable for the synthesis of control: (6) x R( X3) x2 (7) x2 A ( X3) u0 f (8) x3 S ( X32) x4 (9) x4 A2 x5 f2 (0) x5 B0u f3. Here X X x 3 X X32 X 32 x32 x32 ; u0 u0 col u 0 is the virtual control for the outer loop to move the helicopter relative to the earth coordinate system; u lon lat ped ; f i i 3 are the vectors of generalized disturbances derived from the original equations (4) after isolation the terms A ( X3) u 0 Ax 2 5 Bu 0 where A ( X3) diag g g cos b b is the model parameter; A 2 B 0 are the diagonal matrices of the model parameter. 3. SYNHESIS OF CONROL LAW WIH OBSERVER o achieve the desired helicopter stability it is necessary to provide compensation for the generalized disturbances f i i 3. However as follows from the equations (6)- (0) the f and f 2 cannot be fully compensated. herefore the control law is constructed so that it suppress the generalized disturbances which affect the dynamics of the state vectors x i i 3. o do this we use an observer to construct estimates of the generalized disturbances previously considered in [6]. For the original system represented by the equation x Ax Bu D w () y Cx v where n m R x R is the state vector; u is the control vector w R and v R are the vectors of disturbances and a noise in measurements respectively the observer has the form: (2) xˆ Axˆ B u D wˆ L y Cxˆ s (3) wˆ D L L y Cxˆ where D D D D 2 ; is an adjustable parameter and L L 2 are the matrices of coefficients which have to be determined. Unlike known observers [3-5] using the observer of equations (2) (3) it is possible to achieve the desired accuracy in the estimate of the state vector coordinates and disturbances without significant increase of the coefficients of the observer matrices. his is important in the presence of noise in measurement. o simplify the control law we will hold its synthesis separately for each subsystem. First consider the first subsystem (6) (7) for which we write: d x R X 3 x2 R X 3 x2 dt R X A X u w or x x x R X A X u w l

5 where x R X x d w R X 3 f R X 3 x 2 are the generalized dt disturbances. ) When measuring vectors x and x 2 for example with using GPS and airspeed sensors the control law is adopted in the form: (4) ( 3) r u A X u u R X K x x col col col 0 K2 R X 3 x2 xr ŵ where x r is the move commands vector in the earth coordinate system; K K 2 are the diagonal matrices which are given analytically by direct indicators of quality of transients ŵ is the disturbance estimate which is determined by the observer. hus the control law is found b 0 0 u and desired change r sin sin 0 r r I2 2 u0. r of angles: aking into account the adopted notation we obtain the observer to evaluate the disturbance ŵ : d x ˆ 2 u 0 ŵ L x 2 x ˆ 2 dt (5) d ŵ L L2 x ˆ 2 x2 dt where u0 R X 3diagg g cos sin sin col ; L L 2 are the diagonal matrices with positive elements. Given that the vector x 2 can be measured easy to obtain a reduced observer of third order: P PPx2 u0 (6) ŵ Px 2 where P I 3 L2 L matrix. is the diagonal 2) If the vector x 2 is not measured then in the control law (4) instead of a vector 3 2 R X x its estimate ˆx 2 is uses which together with an estimate ŵ are constructed using the observer: xˆext Aext xˆext Bextur (7) H L y C xˆ ext ext r ext ext where yr x и ur u0 are the observer inputs; xˆ ˆ ˆ 2 ŵ ext x x is the estimate vector; A ext 03 I I Bext 03 I L I6 063 Hext Lext L 2 [03 I3] I 3 L 2 Cext I L L 2 L 2 are the diagonal matrices with positive elements. Considering the measurements of vector x the dimension of the observer (7) can be lowered to a 6th order: Ar Brur Lr yr (8) h C y where ˆ h x2 w 2 r I3 Br 0 2 P G Lr 3 GP P L L G L L L. r A r Cr 2 2 P I 3 G 0 3 P G Note that observers (5) - (8) do not depend on b. Now consider the second subsystem (8) - (0) which similarly can be rewritten as: x x x 3 4 x 4 5

6 5 32 w2 x S X B u where w 2 are the generalized disturbances; x S X x ; B A B diag b b b b i i 24 are the pre-unknown model parameters. When measuring vectors x 3 x 4 the control law is of the form: u B S X u (9) 32 u K x3 K2 S X 32 x4 x3r Kxˆ 3 5 ŵ 2 where x x k x ; x 3r is r r r 3r the command vector of the angles; K K 2 K 3 are the diagonal matrices with positive elements. hus using the control law (9) the desired angles r r or x 3r can be tracked. he vectors ˆx 5 and ŵ 2 can be determined using the observer (7) and (8) at y r S X 32 x4 ur u. In this case xˆ ˆ ˆ 4 5 ŵ ext x x 2 ˆ h x5 w 2 and the observer does not depend on the parameters b i i 24. hus for implementing the control laws (4) (9) is required determine the 4 parameters b i i 4. Note that in [2] for the robust controller synthesis used 29 preunknown model parameters. Parameters b i i 4 can be identified based on the results of the measured speeds values w q p r and set test signals col lon lat ped using observers (6) (8). For this purpose we assume C I A Am A B Bm B for a subsystem n of the form () where A m B m are the initial values of the matrices. hen the subsystem can be rewritten as x Amx Bm u where ( t) Ax( t) Bu( t) Dw( t) are the generalized disturbances is estimated using an observer of the form (2) (3). Consequently if t t then equation is true n A (20) ˆ ( t) x ( t) u ( t) B w () t D Using equation (20) a system of equations is built for discrete time points t i ˆ ( t ) i x ( ti ) u ( t ) i ˆ ( t ) ( ) ( ) A ik x tik u tik ˆ ( ti 2k ) x ( ti 2k ) u ( ti 2k ) B w () t D ˆ ( tink ) x ( tink ) u ( tink ) i i Hi where N n m. he vector of parameters is given by: j j Hi i j i j when applying test signals ut (). For example we can write the equation for the channel of vertical velocity as: where w( t) a w( t) b u ( t) w ( t) a a a 4 3 b b b a 0 b 20.9 ; a b are the unknown pa-

7 rameters u 4 () t is the main rotor collective pitch control w3 () t is a generalized disturbance. As a test the reference velocity w ( t) 0.5sin(5 t) under wind gusts (5) with Vp max 0 m/s and the absence of noise in measurements is used. If the parameters a 0.7 b 0 deviate then identification of parameter a is carried out. Figure shows the process of parameter identification in deviations from the trim values. If noise in the measurement exists then the accuracy of b parameter estimation is enhanced by filtering the signals. c) Figure. Processes of parameter identification in deviations from the balancing values. rim values and parameters were found for each subsystem of () - (3) in hover mode of helicopter [2]: b b b b a) If the vector x is not measured then in the control law (4) K 03 is used and the control is done by the velocities x 2. In the control laws (4) and (9) the values 3 2 K 03 K2 I3 и K 0 I3 K2 3 0 I3 23 K3 3 0 I3 0 5 are used. Obviously in this case for the first isolated subsystem roots of the characteristic equation are s and for the second isolated subsystem s b) Assumed a wind gust (5) with the values V m/s 40 s on each axis of p max 0 t p the coordinate system projected onto the body frame [2]. he coefficients matrices of

8 the observer are obtained using the method of [6] with 0.5 for the first subsystem: L diag L diag he control Surfaces for each channel are shown in Figure 2. At the same time the reduced observer P diag (6) has he matrices of coefficients were obtained for the second subsystem with 0.5 are L diag diag ; 4 diag L L L 2 L L At the same time the reduced observer (8) has: P diag G diag a) 4. FUZZY CONROLLER DESIGN Further simplification of the controller can be obtained by designing the Mamdani fuzzy controller whose setting is performed by the results of the controller with the observers. o increase the speed of the fuzzy controller and to achieve smoothness of the process in each channel two signals at the input are used: the error between the true value and the command and its rate of change. Control law uses seven membership functions for each input and one output. Writing rules for fuzzy controllers of Mamdani type requires to create a database of rules of the form: If Ai j Bi j and Ai j Bi j and and A B hen C n j n j j b) c) where A i j - input variable B i j - compares the value C j - conclusion. he rule base establishes a relationship between the level of the input signal and control output. Moreover each controller consists of forty-nine lines.

9 bust controller obtained in [2] using the H method. he following measurements are used: the linear helicopter velocity in the earth coordinate system the angular velocities and angles. he simulation results are shown in able and suggests that the controller with observers and the fuzzy controller has the best dynamic properties. d) e) f) Figure 2: Control surfaces for the regulators: a) roll angle b) pitch angle c) yaw angle d) longitudinal velocity e) lateral velocity f) vertical velocity. 5. Simulation results Also a comparative analysis of the simulation of the helicopter dynamics is conducted for the pirouette maneuver without wind effects. he maneuver begins from hover mode at a height of 20 m. he maneuver requires movement of the helicopter on a circle with a radius of 0 m. Nose of the helicopter must be constantly sent to the center of the circle throughout the maneuver. able 2 presents the errors for the respective axes in the earth coordinate system and the angle of the course. Figure 3 shows the "pirouette" maneuver for different control laws in the absence of wind. Here the controller with observers has the best dynamic properties and also copes with wind disturbances. able 3 presents the simulation results for the pirouette maneuver in the wind for all channels simultaneously. It follows that the designed controllers successfully cope with the impact of the wind. he effectiveness of the proposed control laws was also confirmed by results obtained for the simulation of a simplified model of the ANSA helicopter as well as by experimental results on a laboratory stand using the Raptor helicopter. his illustrated in Figure 4. Figure 5 shows the experimental processes. A comparative analysis of the simulation of the Raptor helicopter dynamics is conducted in hover mode with and without wind gusts (5) for the control laws (4) (6) (8) (9) the fuzzy controller and the ro-

10 able : Hover mode with wind gust on each channel Maximum modulo error w u wind wind v wind he error Controller with observ servers Fuzzy controller [2] Controller with observers Fuzzy controller [2] Controller with observ servers Fuzzy controller [2] x (m) e-5 6e e-3 8.7e-3 5e-3 y (m).0e-4 3e z (m).2e-4 6e e u (m/s) 9e e-5 2e e-4 4.4e-4 e-3 v (m/s) 7.8e-6 4e e-3 2.4e w(m/s).3e-5 5e e e able 2: he pirouette maneuver without wind gusts Maximum modulo error he controller he fuzzy [2] he error with observers controller x (m) y (m) z (m) able 3: he pirouette maneuver with wind gusts on all channels simultaneously Maximum modulo error he error he controller with observers he fuzzy controller x (m) y (m) z (m) (deg).07.4 (deg)

11 Figure 3: Simulation responses of pirouette for different control laws. 6. CONCLUSIONS Figure 4: the 2DOF Raptor helicopter stand. Using the proposed control laws it is possible to increase the stability of the Raptor helicopter compared [2] and to simplify the procedure for setting the parameters of the model helicopter and the coefficients of the controllers. he obtained control laws are recommended for use in the auto-pilot of full-size helicopter after further refinement. FUURE WORK At the next stage of the work it is planned to test the developed algorithms for the ANSA helicopter using a non-linear model and test the control laws for the Raptor helicopter in flight. ACKNOWLEDGMENS Figure 5: Stabilization of roll angle with disturbance w 0.4 sin 0.4 t in the control channel: - PID with observer of disturbance 2 - fuzzy controller; 3 PID. his work was supported by a grant of the Government the Russian Federation for state support of scientific research on the decision of the Government under a contract of 220 from December G

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