Robust Regulation for a 3-DOF Helicopter via Sliding-Modes Control and Observation Techniques

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1 21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 ThC15.1 Robust Regulation for a 3-DOF Helicopter via Sliding-Modes Control and Observation Techniques Héctor Ríos, Antonio Rosales, Alejandra Ferreira and Alejandro Dávila. Abstract In this paper, two robust control strategies for a 3-DOF Helicopter via sliding-modes techniques are presented. First, quasi-continuous controllers along with a sliding mode differentiator are designed and then the design of classical PID controllers in combination with a second-order sliding mode observer is presented. Both strategies preserve high position regulation accuracy and robustness to model uncertainties and external disturbances. Simulations and experimental results on a 3-DOF Helicopter by Quanser are shown. Index Terms Sliding-Modes, Variable Structure Control, 3- DOF. I. INTRODUCTION Analysis of helicopter models has attracted a lot of attention of researchers (see e.g. [1]). Our aim is to regulate desired angular positions of the prototype, operating under uncertainty conditions and external disturbances. The prototype represents an under-actuated 3-DOF mechanical system, actuated by two motors. This 3-DOF helicopter is a complex system which can be easily damaged. Unfortunately, several electromechanical parameters are not exactly known and/or are subject to large variations during operation, leading to degradation of the prototype. To overcome this drawback, several robust control techniques have been suggested to cope with the uncertainty conditions, ranging from adaptive controllers and VSC schemes [2], [3]. Generally speaking, the main features of VSC are high accuracy, simplicity (of both tuning and implementation), and robustness [2]. VSC schemes are typically based on control signals of the switching type, and the so-called chattering phenomenon, originated by the interaction between parasitic dynamics and high-frequency switching control, arises, which represents the most deleterious implementation drawback of VSC [4]. In order to counteract the chattering phenomenon and to preserve at the same time the main advantages of the original approach (precision, robustness, simplicity, and finite-time convergence) even when stringent dynamic specifications are met for the system, the features of the higher-order sliding-mode-control and observation techniques [5],[6] are exploited in this paper to design two robust strategies based on sliding-modes techniques in a proper combination along with other techniques (PID controllers). Main Contribution: The presence of perturbations and uncertainties in control systems is a daily situation that provides limitations on their functionality. The identification and The authors are with Programa de Maestría y Doctorado en Ingeniería Eléctrica (Control), Universidad Nacional Autónoma de México, UNAM, Ciudad Universitaria, C.P. 451, Mexico, D.F. s: hectorrios@comunidad.unam.mx, manerblind@hotmail.com, da ferreira@yahoo.com, alets davila@yahoo.com compensation of perturbations is of great importance since it adds robustness to the controller. The quasi-continuous controllers along with a sliding mode differentiator and the classical PID controllers in combination with a secondorder sliding mode observer are proposed. Robustness to model uncertainties and external disturbances rejection are studied via simulations and experimental results on a 3-DOF Helicopter platform by Quanser. Structure of the paper: The structure of this paper is as follows. Section II contains the system model formulation and the control problem statement. Section III details the proposed observer/controller structures. Section IV and V present some simulations and experimental results, respectively; and Section VI gives some concluding remarks and future works. II. SYSTEM DESCRIPTION AND PROBLEM STATEMENT A. System Description In this work, a 3-DOF helicopter manufactured by Quanser was employed for the experimental proofs (Fig. 1). Fig DOF Helicopter System. The Helicopter setup consists of a base on which a long arm is mounted. The arm carries the helicopter body on one end and a counterweight on the other end. The arm can tilt on an elevation axis as well as swivel on a vertical (travel) axis. Quadrature optical encoders mounted on these axes measure the elevation and travel of the arm. The helicopter body, which is mounted at the end of the arm, is free to pitch about the pitch axis. The pitch angle is measured via a third encoder. Two motors with propellers mounted on /1/$ AACC 4427

2 the helicopter body can generate a force proportional to the voltage applied to them. The force, generated by the propellers, causes the helicopter body to lift off the ground and to rotate about the pitch axis. The mathematical model of the 3-DOF Helicopter laboratory prototype of the drawn from the manufacturer manual is given by [7]: where ε = K f l a u s T g (1) p = K f l h u d (2) θ = K pl a sin(p) (3) =.91(Kgm 2 ) is the moment of inertia of the system about the elevation axis. u s is the sum of V f and V b, which are the voltages applied to the motors. K f =.5(N/V ) is the force constant of the motor combination. l a =.66(m) is the distance between the base and the helicopter body. T g = (Nm) is the effective gravitational torque. ε is the angular position of the elevation axis. =.364(Kgm 2 ) is the moment of inertia of the helicopter body about the pitch axis. l h is the distance from the pitch axis to either motor. u d is the difference between the voltages V f and V b. p is the angular position of the pitch axis. =.91(Kgm 2 ) is the moment of inertia of the helicopter body about the travel axis. K p =.686(N) is a constant of proportionality of the gravitational force. θ is the angular position of the travel axis. B. Problem Statement Consider (1)-(3) in the general form of a mechanical system in the state-space ẋ 1 = x 2 ẋ 2 = Tg + K f l a K f l h (u + ξ) K pl a sin(x 12 ) }{{} F (t,x 1,x 2,u)+ξ(t,x 1,x 2) y = x 1 (4) where x 1 = [ ε p θ ] T represents the angular positions, x 2 = [ ε ṗ θ ] T the angular velocities, F (t, x 1, x 2, u) is the nominal part of the system dynamics, u = [ u s u d ] T is the control input, while the uncertainties and perturbations are concentrated in the term ξ(t, x 1, x 2 ) = [ ξ 1 ξ 2 ] T. It is assumed that the uncertainties ξ(t, x 1, x 2 ) are Lebesgue measurable, uniformly bounded; ξ(t, x 1, x 2 ) < ξ +, and matched. Augmenting the integral of the regulation error ė = x 1 r, where r = [ r ε r p r θ ] T is the reference of each angular position, with the foregoing equation and applying the change of variables [ ] [ ] e1 x1 r e = = (5) e 2 x 2 we obtain the augmented system ė = e 1 ė 1 = e 2 ė 2 = Tg K pl a sin(x 12 ) + K f l a K f l h (u + ξ) The aim of this paper is to design two robust output controls for system (6) and to make an experimental comparison between them. III. CONTROL DESIGN Now we design two different robust controllers based on sliding-modes techniques for system (6). Firstly, we design a Quasi-Continuous Controller, a recently proposed method based on the so-called arbitrary-order sliding-mode approach [8]. Secondly, PID Controllers are designed along with a sliding-mode Observer for state estimation, identification and compensation of disturbances via equivalent output injection [5]-[9]. A. Quasi-Continuous Controller The quasi-continuous arbitrary-order sliding-mode controller was suggested in [8]-[1] in order to stabilize systems in the form (6) in finite time. Let i = 1,..., n 1 and denote (6) ϕ,n = e, N,n = e (7) Ψ,n = ϕ,n N,n = sign(e ) (8) ϕ i,n = e (i) + β in (n 1)/(n i+1) i 1,n Ψ i 1,n (9) N i,n = e (i) + β i N (n 1)/(n i+1) i 1,n (1) Ψ i,n = ϕ i,n N i,n (11) where β i,..., β n 1 are positive numbers. The quasicontinuous n-sliding controller is ( ) u = αψ n 1,n e, e,..., e (n 1) (12) Provided that the tuning parameters β i,..., β n 1, α > are chosen according to [8]-[1] then the control law defined by (7)-(12) stabilizes system (6) in finite time. Moreover, the control is globally bounded ( u α) and continuous everywhere but the origin of the n-dimensional error space. Now, we design two third-order quasi-continuous controllers, one for u s and other for u d. Considering (6), we can rewrite 4428

3 it component-wise as follows ė 1 = e 11 (13) ė 11 = e 21 (14) ė 21 = T g + K f l a (u s + ξ 1 ) (15) ė 2 = e 12 (16) ė 12 = e 22 (17) ė 22 = K f l h (u d + ξ 2 ) (18) ė 3 = e 13 (19) ė 13 = e 23 (2) ė 23 = K pl a sin(x 12 ) (21) where e 11 = x 11 r ε, e 12 = x 12 r p and e 13 = x 13 r θ. Considering (13)-(15) and (16)-(18), we design u s and u d as follows u s = ë 1 + β 12 ( ė 1 + φ 1 ) 1 2 (ė 1 + φ 1 sign(e 1 )) α s K f l a ë 1 + β 12 ( ė 1 + φ 1 ) 1 2 (22) u d = ë 2 + β 22 ( ė 2 + φ 2 ) 1 2 (ė 2 + φ 2 sign(e 2 )) α d K f l h ë 2 + β 22 ( ė 2 + φ 2 ) 1 2 (23) where φ 1 = β 11 e and φ 2 = β 21 e Provided that the tuning parameters β 11, β 12, α s and β 21, β 22, α d, are chosen according to [8]-[1] then the regulation error converges to zero in finite time even in presence of uncertainties and the perturbations terms ξ 1, ξ 2. The above third-order quasi-continuous controllers require the availability of the derivative of each regulation error (e 11, e 12 ). In order to reconstruct such derivative exactly and in finite time, the sliding-mode differentiator by Levant [6] can be used. The differentiators can be expressed in the following form ż i = v i = z 1i k i z e 1i 1/2 sign(z e 1i ) ż 1i = k 1i sign(z 1i v i ), i = 1, 2. (24) for suitable positive constants k i, and k 1i, i = 1, 2., to be chosen recursively large in the given order [6]. Under the assumption that constants C 1, C 2 exist such that e 1i C i, i = 1, 2, the following equalities are true after a finite time transient process: z ji e (j) =, i = 1, 2., j =, 1. (25) 1i The separation and robustness results relevant to the combined use of the above differentiators and any n-sliding homogenous controller were discussed in [6]. The controllers (22) and (23) allow us to achieve the elevation and pitch reference position. Now, in order to achieve a travel reference position (r θ ) we will design a closed loop controller that commands a desired pitch angle (r p ) in the following form r p = k tp e 13 + k td ė 13 + k ti e 13 dt (26) 4429 where k tp, k td and k ti are chosen considering (19)-(21) and a response frequency analysis, and ė 13 is estimated by means of a differentiator, such as (24). B. PID Controllers with Sliding-Mode Observer This control is designed in two parts, first a nominal PID control and second the identification and compensation of disturbance based on the sliding-mode observer. Considering (13)-(15), the elevation axis constitutes an independent subsystem, then we can propose the following elevation controller u s = k ep e 11 + k ed e 11 + k ei e 11 dt (27) where k ep, k ed and k ei are the proportional, derivative and integral gains control, respectively. Now since the pitch and travel axes are dependent, the travel controller will be used to control the pitch. We can then select the next control input u d = k pp e 12 + k pd e 12 (28) with r p defined as in (26) to command the desired travel position, k pp and k pd are control gains. The last gains are chosen as in (26). Our next goal is to implement a Sliding-Mode Observer [5]-[9] necessary to estimate the states. Moreover, the identification and compensation of disturbances add robustness to the PID controllers. The observer has the form ˆx 1 = ˆx 2 + z 1 ˆx 2 = Tg + K f l a K f l h K pl a sin(x 12 ) }{{} F (t,x 1,ˆx 2,u) u +z 2 (29) where ˆx 1 = [ ] T [ ˆε ˆp ˆθ, ˆx2 = ˆε ˆp ] T ˆθ and z1, z 2 are the correction factors defined as z 1 = λ x sign ( x 1 ) z 2 = αsign ( x 1 ) (3) where α = [ α 1 α 2 α 3 ] T, λ = [ λ1 λ 2 λ 3 ] T, and x 1 = x 1 ˆx 1. Now defining x 2 = x 2 ˆx 2 and considering (4) we can write the error dynamics as x 1 = x 2 λ x sign ( x 1 ) x 2 = F (t, x 1, x 2, ˆx 2, u) + ξ(t) αsign ( x 1 ) (31) where F (t, x 1, x 2, ˆx 2, u) = F (t, x 1, x 2, u + ξ(t, x 1, x 2 )) F (t, x 1, ˆx 2, u), α and λ are chosen considering the next inequalities α > f + [ 2 (α + f + ] )(1 + c) λ > α f + (1 c)

4 The constant f + can be calculated as twice the maximum possible acceleration of the system and c is some chosen constant, < c < 1. If α and λ are chosen considering the above inequalities then the variables of the observer converge in finite time to the system states (see, theorem in [5]). For identification and compensation of disturbances we consider that the convergence time ensures that there exists a time constant t > such that for all t t, we can rewrite (31) as TABLE I MAIN PARAMETER VALUES PID with SMO Quasi-Continuous Gain Value Gain Value Gain Value Gain Value k ep.5398 α 1 3 β 11 1 β 21 1 k ed α 2 14 β 12 2 β 22 2 k ei.756 α 3 63 α s 44 α d 48 k pp λ 1 5 k pd.9121 λ 2 16 k tp.8129 λ k td k ti.976 x 2 F (t, x 1, x 2, ˆx 2, u) + ξ(t) αsign ( x 1 ) However, considering that the observer converges in finite, F (t, x 1, x 2, ˆx 2, u) =,and therefore z eq (t) = ˆξ(x, t) = αsign ( x 1 ) where z eq is the equivalent output injection. Theoretically, the equivalent output injection is the result of an infinite switching frequency of the discontinuous term αsign ( x 1 ). To eliminate the high frequency component we will use the filter of the form τ z eq = z eq (t) z eq (t) (32) where τ is the filter time constant, h << τ << 1, and h is a sampling step. Rewrite z eq as result of the filtering process in the following form z eq = z eq + µ(t) deg. in travel. We considered the matched disturbances ξ 1 and ξ 2 as follows ξ 1 = ( ( ) ) 3π 5 sin 2 t square (4πt) ξ 2 = 3 sin(πt) + (8 sin(2πt) + 1) The simulation time was 5 seconds with initial positions ɛ =, p = and θ = for both controllers. A. Regulation via Quasi-Continuous Controller Fig. 2 shows the regulation error for the Quasi-Continuous Controller, corresponding to elevation, pitch and travel, respectively. The applied control signals are shown in Fig. 3. As can be seen in this figure, u 1 and u 2 are high frequency signals due to the nature of the control. where µ(t) is the difference caused by the filtration and z eq is the filtered version of z eq. Nevertheless, as it is shown in [11] and [4] lim τ h τ z eq (τ, h) = z eq (t) therefore choosing τ and h as h << τ << 1 any bounded disturbance can be identified. Finally, the PID Controllers with Sliding-Mode Observer can be defined as u = u o + u 1 (33) where u represents the nominal PID controllers and u 1 is the disturbances compensator control, defined as [ ] us u = (34) u d [ ] [ ] zeq1 u 1 = = (35) z eq2 ˆξ1ˆξ2 IV. SIMULATION RESULTS The simulations of the above controllers were designed in Matlab-Simulink. The parameters used for the Quasi- Continuous Controller and the PID Controllers with Sliding- Mode Observer (SMO) are given in Table I. Two reference positions were given, ɛ = 5 deg. in elevation and θ = 3 Fig. 2. Regulation error Quasi-Continuous Controller B. Regulation via PID Controllers with SMO The Fig. 4 shows the regulation error for the PID Controllers with Sliding-Mode Observer for the three axes. Fig. 5 contains the control signals for the PID Controllers with the Sliding-Mode Observer, it can be seen that the signals present less oscillations and smaller amplitude than the Quasi-Continuous Controller. However, the regulation errors convergence is slower, see Fig. 4. Considering the simulation results, the most suitable controller for implementation on the 3-DOF Helicopter is the 443

5 Fig. 3. Control signals Quasi-Continuous Controller V. EXPERIMENTAL RESULTS The control strategy was implemented on Matlab Simulink over a dspace-113 data-acquisition card connected to an educational 3-DOF Helicopter by Quanser with a fixed step size of h = 1(µs) and Euler s method to solve the ODE (ordinary differential equation). The experiment is composed of two stages, the first one consists of moving the helicopter from the initial position x 1 (t 1 ) = [ 4 28] T to the origin [ ] T using only the PID controllers with SMO. The second stage consists of maintaining the helicopter in the origin (stable state) in spite of the presence of perturbations and uncertainties, using the PID controllers with SMO and compensation. The parameter values of the controllers are the same as those used in the simulation section with a time constant τ =.13 for the filter described in (32). The development was the following one: starting from a unperturbed stable state, the perturbations signals ξ 1 = sin(2πt) and ξ 2 = sin(2πt) were later added to the motor voltages so that this would act like an exogenous perturbation. Then, the perturbations compensation control is activated. With the activated compensation the amplitude of the perturbations increase. Finally, the compensation is deactivated, leaving the system in the presence of the following perturbations ξ 1 = 5sin(2πt) and ξ 2 = 5sin(2πt). The results appear in the Fig.6-Fig.9. Fig. 4. Regulation error PID Controllers with SMO Fig. 6. Regulation of PID Controllers with SMO Remark. From Fig.9 it is important to note that the disturbances present in the whole system were identified through the SMO by means of a good selection of τ. The high accuracy of the disturbances identification allow us to compensate them with excellent precision. Fig. 5. Control signals PID Controllers with SMO PID Controllers with SMO. This is due to the hardware limitations and the fact that the control signal is continuous. VI. CONCLUSIONS AND FUTURE WORKS The presented robust control strategies preserve high position regulation accuracy and robustness to model uncertainties and external disturbances but the strategy of quasi-continuous control presents better simulation results in position regulation with the disadvantage of the control signals are discontinuous and therefore his implementation 4431

6 identify and compensate all the unmodeled parameters and disturbances on the 3-DOF Helicopter by Quanser. The easy implementation of the PID controllers with SMO on the dspace-113 puts in evidence the robustness of the presented control. In future works the complete experimental setup will be presented. VII. ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from CONACyT (Consejo Nacional de Ciencia y Tecnología) grant 56819, CVU 2754, CVU 26945, CVU and CVU Programa de apoyo a proyectos de investigación e inovación tecnológica (PAPIIT) UNAM, grant Fig. 7. Elevation behavior Fig. 8. Pitch behavior Fig. 9. Disturbance identification REFERENCES [1] A. Isidori, L. Marconi and A. Serrani, Robust nonlinear motion control of a helicopter, IEEE Transaction on Automatic Control, vol. 48, no. 3, pp , 23. [2] K. K. Starkov, L. T. Aguilar, Y. Orlov, Sliding mode control synthesis of a 3-DOF Helicopter Prototype using position feedback, International Workshop on Variable Structure System, pp , 28. [3] B. Andrievsky, D. Peaucelle, and A. L. Fradkov, Adaptive Control of 3-DOF Motion for LAAS Helicopter Benchmark: Design and Experiments, Proceedings of the 27 American Control Conference, pp , New York, 27. [4] V.I. Uktin, Sliding Modes in Control and Optimization, Berlin, Germany: Springer-Verlag, [5] J. Dávila, L. Fridman and A. Levant, Second-Order Sliding- ModeObserver for Mechanical Systems,IEEE Transactions on Automatic Control, Vol.5, No.11, pp , 25. [6] A. Levant, Higher-order sliding modes: differentiation and outputfeedback control, International Journal of Control, Vol.76, pp , 23. [7] 3D Helicopter System, Quanser Co. [Online]. Available [8] A. Levant, Quasi-continuous High-order Sliding-mode Controllers, IEEE Transactions on Automatic Control, Vol.5, pp , 25. [9] J. Dávila, L. Fridman and A. Poznyak, Observation and Identificaction of Mechanical Systems via Second Order Sliding Modes, International Journal of control, Vol.79, No.1, pp , 26. [1] A. Levant, Homogeneous Quasi-continuous High-order Sliding-mode Control, Lecture Notes in Control and Information Sciences, Vol.334, pp , 26. [11] L. Fridman, The problem of chattering: an averaging approach In:Young K.D. and Ozguner U. (eds.) Variable Structure, Sliding Mode and Nonlinear Control, Lecture Notes in Control and Information Science, no. 247, Springer Verlag, pp London, [12] B. Cazzolato, 3 Degree of Freedom Hover Tutorial, The University of Adelaide, 25. [13] C. Edwards, S. Spurgeon, Sliding Mode Control, London: Taylor and Francis, [14] M. Ishutkina, Design and implementation of a supervisory safety controller for a 3-DOF helicopter, Master s Thesis, Massachusetts Institute of Technology, 24. [15] A. Isidori, Nonlinear Control Systems, London: Springer-Verlag [16] H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, Upper Saddle River, New Jersey, 22. [17] Y. S. Zhong, Robust stabilization and disturbance attenuation of a class of MIMO nonlinear systems with multi-operation points Proceedings of the 26th Chinese Control Conference, Vol.3, pp.7-74, 27. [18] Yu Yao, Zhong YiSheng, Robust Tracking Control for a 3-DOF with Multi-operation Points Proceedings of the 27th Chinese Control Conference, pp , 28. is more complicated. A sliding-mode observer along with the PID Controllers were experimentally implemented to 4432