Stabilization and tracking control of X–Z inverted pendulum with sliding-mode control

1. Stabilization and tracking control of X–Z inverted pendulum with sliding-mode control Jia-Jun Wang School of Automation, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, PR China a r t i c l e i n f o Article history: Received 27 April 2012 Received in revised form 26 May 2012 Accepted 12 June 2012 Available online 10 July 2012 Keywords: X–Z inverted pendulum Stabilization Tracking control Sliding-mode control a b s t r a c t X–Z inverted pendulum is a new kind of inverted pendulum which can move with the combination of the vertical and horizontal forces. Through a new transformation, the X–Z inverted pendulum is decomposed into three simple models. Based on the simple models, sliding-mode control is applied to stabilization and tracking control of the inverted pendulum. The performance of the sliding mode control is compared with that of the PID control. Simulation results show that the design scheme of sliding-mode control is effective for the stabilization and tracking control of the X–Z inverted pendulum. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction The inverted pendulum problem is one of the most important problems in control theory and has been studied excessively in control literatures [1–3]. The inverted pendulum is nonlinear, nonminimum phase and underactuated system which makes it a well established benchmark problem that provides many challen- ging problems to control design. Until recently, there are a lot of literatures on the swing up, stabilization, and tracking control of the traditional inverted pendulum. Beside the wide research on the traditional inverted pendulum, some researchers concentrate their efforts on the other types of inverted pendulums, such like spherical inverted pendulum (which can also be named as X–Y inverted pendulum, planar inverted pendulum or dual-axis inverted pendulum) [4,5,11], X–Z inverted pendulum [6–8] and inverted 3-D pendulum [9]. The X–Z inverted pendulum can move in the vertical plane with horizontal and vertical forces which is first proposed by Maravall [6,7]. The traditional inverted pendulum can be seen as a special case of the X–Z inverted pendulum. Compared with the traditional inverted pendulum, the X–Z inverted pendulum has more versatility and is more like the real control object in reality. In the control of the X–Z inverted pendulum, the problems of stabilization and tracking control are more meaningful than that of the swing up control. This paper concentrates on solving the stabilization and tracking control problems of the X–Z inverted pendulum. In Ref. [6], Maravall constructed a hybrid fuzzy control system that incorporates PD control into a Takagi–Sugeno fuzzy control structure for stabilizing the X–Z inverted pendulum. And in Ref. [7], Maravall designed a PD-like feedback controller that guaran- tees the global stability of the X–Z inverted pendulum by applying Lyapunov’s direct method. The method proposed in Refs. [6,7] is based on the simplified linearized model of the X–Z inverted pendulum. In Ref. [8], Wang applied the PID controllers to the stabilization and tracking control of the X–Z inverted pendulum. And good control performance is achieved with PID controllers. Because PID control method has too many tuning parameters, it is not a easy job to get the proper PID parameters. And further, it is a very difficult task to select the PID parameters to achieve good tracking performance for the fast reference signals. Sliding-mode control (SMC) is one of the effective nonlinear robust control approaches since it provides system dynamics with an invariance property to uncertainties once the system dynamics are controlled in the sliding mode [10,11]. Wai developed an adaptive sliding-mode control for stabilizing and tracking control for the dual-axis inverted-pendulum system, where an adaptive algorithm is investigated to relax the requirement of the bound of lumped uncertainty in the traditional sliding-mode control [12]. And in Ref. [13], a robust fuzzy-neural-network (FNN) control system is implemented to control a dual-axis inverted-pendulum. The FNN controller is used to learn an equivalent control law as in the traditional sliding-mode control, and a robust controller is designed to ensure the near total sliding motion through the entire state trajectory without a reaching phase. Park introduced a coupled sliding-mode control for the inverted-pendulum to realize the swing-up and stabilization [14]. And further more, Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.06.007 E-mail address: wangjiajun@hdu.edu.cn ISA Transactions 51 (2012) 763–770

2. Park applied the coupled sliding-mode control method to the periodic orbit generation and the robust exponential orbital stabilization of the inverted-pendulum systems [15]. In this paper, we use the sliding-mode control to solve the stabilization and tracking control problems of the X–Z inverted pendulum. The organization of this paper is as follows. In Section 2, the transformation method for the state transformation of the X–Z inverted pendulum is given. In Section 3, a new transformation of the equivalent equations of the X–Z inverted pendulum is designed step by step. And based on the simplified model of the X–Z inverted pendulum, the sliding-mode controller is designed. In Section 4, simulation results and analysis of the sliding-mode control are shown. At last, some conclusions are presented in Section 5. 2. Model transformations The X–Z inverted pendulum on a pivot driven by one horizontal and one vertical control forces is shown in Fig. 1. The control action is based on the X–Z horizontal and vertical displacements of the pivot. The state equations of the X–Z inverted pendulum are given in [8] as the following equations: €x ¼ Mml_y 2 sin yþðMþm cos2 yÞFxÀmFz sin y cos ys MðMþmÞ þd1 ð1Þ €z ¼ Mml_y 2 cos yÀmFx sin y cos yþðMþm sin2 yÞFz MðMþmÞ Àgþd2 ð2Þ €y ¼ ÀFx cos yþFz sin y Ml þd3 ð3Þ where (x,z), ð_x,_zÞ, ð€x,€zÞ is the position, speed, acceleration of the pivot in the xoz coordinate respectively, l is the distance from the pivot to the mass center of the pendulum, M and m are the mass of the pivot and the pendulum respectively, g is the acceleration constant due to gravity, Fx is the horizontal force, Fz is the vertical force, and d1, d2, d3 are outer disturbances. We assume that À1rxr1, À1rzr1, À30rFx r30, À30rFz r30, and the inertia of the pendulum is negligible. From Eqs. (1)–(3), we can know that (1) The X–Z inverted pendulum is a multi-variable, strong coupled, and naturally unstable nonlinear control system. (2) The X–Z inverted pendulum has three control freedoms (x, z, and y) and two control variable (Fx and Fz). It is a underactuated control system. (3) The horizontal force Fx and vertical force Fz affect the control of the pendulum angle simultaneously. When the inverted pendulum system become stable, vertical force Fz ¼ ðMþmÞg. We define ex ¼ xÀxd, ez ¼ zÀzd, where xd and zd are desired signals. And we assume that xd and zd are no less than two times differentiable. Then the state equations (1)–(3) can be rewritten as €ex ¼ Mml_y 2 sin yþðMþm cos2 yÞFxÀmFz sin y cos y MðMþmÞ À€xd þd1 ð4Þ €ez ¼ Mml_y 2 cos yÀmFx sin y cos yþðMþm sin2 yÞFz MðMþmÞ ÀgÀ€zd þd2 ð5Þ €y ¼ ÀFx cos yþFz sin y Ml þd3 ð6Þ First, we assume that d1 ¼ d2 ¼ d3 ¼ 0. Defining exp ¼ ex þl sin y and ezp ¼ ez þlðcos yÀ1Þ, we can obtain that _exp ¼ _ex þl_y cos y, _ezp ¼ _ezÀl_y sin y. Then €exp ¼ €ex þl€y cos yÀl_y 2 sin y and €ezp ¼ €ezÀ l€y sin yÀl_y 2 cos y can be acquired. Based on Eqs. (4)–(6), we can obtain the following equations: €exp ¼ Fx sin 2 yþFz sin y cos yÀMl_y 2 sin y Mþm À€xd ð7Þ €ezp ¼ Fx sin y cos yþFz cos2 yÀMl_y 2 cos y Mþm ÀgÀ€zd ð8Þ €y ¼ ÀFx cos yþFz sin y Ml ð9Þ Defining exm ¼ Àexp, ezm ¼ ezp, uxz ¼ ðFx sin yþFz cos yÞ=ðMþmÞ and uy ¼ ÀFx cos yþFz sin y=Ml, then we can obtain the following equations: €exm ¼ Àuxz sin yþ M Mþm l_y 2 sin yþ €xd ð10Þ €ezm ¼ uxz cos yÀ M Mþm l_y 2 cos yÀgÀ€zd ð11Þ €y ¼ uy ð12Þ From the definition of the uxz and uy, Fx and Fz can be obtained as the following equations: Fx ¼ ðMþmÞuxz sin yÀMluy cos y ð13Þ Fz ¼ ðMþmÞuxz cos yþMluy sin y ð14Þ Defining um ¼ uxzÀðM=ðMþmÞÞl_y 2 , we can obtain the following equations: €exm ¼ Àum sin yþ €xd ð15Þ €ezm ¼ um cos yÀgÀ€zd ð16Þ €y ¼ uy ð17Þ Remark 2.1. If let xd¼0 and zd¼0, comparing Eqs. (15)–(17) with (7) in Ref. [16] and Eq. (2) in Ref. [17], we can conclude that the models of the X–Z inverted pendulum and the planar vertical takeoff and landing (PVTOL) aircraft are equivalent through certain state transformations. This demonstrates that the control method designed for the PVTOL aircraft can also be applied to the control of the X–Z inverted pendulum directly. On the other side, the research on the control of the X–Z inverted pendulum has its important meanings to the control of the PVTOL aircraft and such like nonlinear control systems. 3. Control design of the X–Z inverted pendulum Although in Ref. [16], Olfati-Saber pointed out that there exists a standard backstepping control procedure which can realize the stabilization of the PVTOL aircraft, and such like models. BecauseFig. 1. Structure of the X–Z inverted pendulum. J.-J. Wang / ISA Transactions 51 (2012) 763–770764

3. the pendulum angle y is coupled with the state variable and the state variable exm and ezm is controlled by control variable um at the same time, the design of the backstepping control is not a easy task. Deﬁning the change of coordinates ext ¼ exm, ezt ¼ ezm, yt ¼ tan y, u1 ¼ Àum cos y, and u2 ¼ ðuy þ2_y 2 tan yÞsec2 y, we can obtain the following equations: €ext ¼ ytu1 þ €xd ð18Þ €ezt ¼ Àu1ÀgÀ€zd ð19Þ €yt ¼ u2 ð20Þ Let ez1 ¼ ezt, _ez1 ¼ ez2, ex1 ¼ ext, _ex1 ¼ ex2, ex3 ¼ yt and ex4 ¼ _yt , then the following equations can be obtained _ez1 ¼ ez2 ð21Þ _ez2 ¼ Àu1ÀgÀ€zd ð22Þ _ex1 ¼ ex2 ð23Þ _ex2 ¼ ex3u1 þ €xd ð24Þ _ex3 ¼ ex4 ð25Þ _ex4 ¼ u2 ð26Þ From state equations (21)–(26), we can know that (1) The X–Z inverted pendulum system can be divided into two subsystems P 1ðez1,ez2Þ and P 2ðex1,ex2,ex3,ex4Þ. P 1 can be seen as a independent linear control system. And subsystem P 2 is affected by subsystem P 1. (2) Subsystem P 2 can be divided into two subsystems P 21ðex1,ex2Þ and P 22ðex3,ex4Þ. P 22 can be seen as a independent linear system. And P 21 is affected be subsystem P 1 and P 22 simultaneously. (3) Because subsystem P 1 and P 22 has its independence, the control design of the X–Z inverted pendulum becomes very easy. Remark 3.1. Through the above transformation, the X–Z inverted pendulum can be seen as three simple linear control systems. There exists some relation between them. And linear or nonlinear control theory can be applied directly to the control of the X–Z inverted pendulum. With the state (21)–(26) the control design procedure of the X–Z inverted pendulum can be given as the following three steps. Step 1. With the subsystem P 1 _ez1 ¼ ez2 ð27Þ _ez2 ¼ Àu1ÀgÀ€zd ð28Þ the control u1 can be designed as a saturated nonlinear control u1 ¼ ÀgÀ€zd þa tanhðk1ez1 þk2ez2Þþb tanhðk2ez2Þ ð29Þ where tanhðxÞ ¼ ðex ÀeÀx Þ=ðex þeÀx Þ, which is given in Fig. 2(a), k1 40, k2 40, a40, b40 are constant, and 9a9þ9b9þ9€zd9og. It is known easily that u1 o0. The convergence of the state ez1 and ez2 can be proved with the Proposition 3.1 in Ref. [17]. Step 2. With the subsystem P 21 _ex1 ¼ ex2 ð30Þ _ex2 ¼ ex3u1 þ €xd ð31Þ ex3u1 can be seen as the control of the subsystem P 21. Because P 21 is a typical linear system, then ex3u1 can be designed as ex3u1 ¼ Àk3ex1Àk4ex2À€xd ð32Þ where k3 40 and k4 40 are constants. From Step 1, we know that u1 a0. Then the desired ex3 can be designed as en x3 ¼ Àk3ex1Àk4ex2À€xd u1 ð33Þ From Eq. (33), en x3 can converge to 9€xd=u19 with ex1-0 and ex2-0. If ex3-xn x3 can obtained, this not only can realize the stabilization of subsystem P 21, but also make the pendulum angle converges to 9€xd=u19. Remark 3.2. Form Eq. (33), we can obtain the following two conclusions. (1) In the stabilization of the X–Z inverted pendulum, because xd ¼0, then we can realize the stabilization of the pendulum with the stable error of the pendulum angle conver- ging to zero. (2) In the tracking control of the X–Z inverted pendulum, because xd a0, and we can know that 9€xd=u19a0. If we make 9€xd=u19 little enough, tracking control of the X–Z inverted pendulum can also be achieved. For example, let xd ¼ 0:25 sinððp=8ÞtÞ, zd ¼ 0:15 sinððp=8ÞtÞ, a ¼ 5, b ¼ 1, the largest stable error of the pendulum angle is actan9maxð€xdÞ=minðu1Þ9 ¼ actan 0:25 Â p 8 Â p 8 9:8À5À1À0:15

17. ¼ 0:0106 rad where min and max represent minimum and maximum function. This tracking control method can solve a kind of tracking problem. And this tracking control method has its application scope. Step 3. With the subsystem P 22 _ex3 ¼ ex4 ð34Þ _ex4 ¼ u2 ð35Þ Fig. 2. Figure of sign and tanh function. (a) Figure of the tanh function. (b) Figure of the sign function. J.-J. Wang / ISA Transactions 51 (2012) 763–770 765

18. From Step 2, we know that if ex3-en x3, the stabilization of the X–Z inverted pendulum can be achieved. To realize ex3-en x3, the sliding-mode control design is given in the following. The sliding-mode surface is designed as s ¼ k5xþ _x ð36Þ where x ¼ ex3Àen x3, k5 40 is constant. And the sliding-mode control can be selected as u2 ¼ Àk5 _x þ €e n x3Àk6sÀr signðsÞ ð37Þ where k6 40, r40 are constant, and sign is the sign function as given in Fig. 2(b). It is very easy to obtain that _ss ¼ Àk6s2 Àr9s9. Then the reaching condition of sliding-mode control is achieved. As we know the sign control can introduce the chattering, which can add the energy consumption and reduce the life of the motion part. Then we can applied the saturated function to take place the sign function. The sliding mode control can be redesigned as u2 ¼ Àk5 _x þ €e n x3Àk6sÀr tanhðsÞ ð38Þ From the above stabilization design for the state equations (21)–(26), the control u1 can make the state variable ez1 and ez2 converge to zero. The saturated control can guarantee u1 a0. And u2 can guarantee that ex3-en x3. This can be easily proved through Barbalat’s Lemma [13]. en x3u1 can make ex1 and ex2 approach zero. Because the equivalent relationship between state equations (21)–(26) and (1)–(3), u1, u2 with en x3 can realize the stabilization of the X–Z inverted pendulum. 4. Simulation results and analysis The parameters of the X–Z inverted pendulum are given in Table 1. To show the effectiveness of proposed method in this paper, the sliding-mode control design is compared with the PID controllers that are proposed in Ref. [8]. The structure of the control method with sliding-mode control and PID control is given in Fig. 3(a) and (b) respectively. Referenced from the tuning method proposed in Ref. [8], the PID controller parameters are designed as follows: PID1: P1 ¼ 25, I1 ¼ 15, D1 ¼ 3, PID2: P2 ¼ À1:5, I2 ¼ À0:5, D2 ¼ À0:2, PID3: P3 ¼ 64, I3 ¼ 5, D3 ¼ 6. The parameters of the PID controllers in this paper are better than that in Ref. [8]. The parameters of the sliding-mode control is given as follows r ¼ 5, a ¼ 5, b ¼ 1, k1 ¼0.2, k2 ¼0.45, k3 ¼ 2, k4 ¼1.5, k5 ¼ 20, k6 ¼ 200. Fig. 3. Control structure of sliding-mode control and PID control. (a) Structure of sliding-mode control. (b) Structure of PID control. Fig. 4. Stabilization of the X–Z inverted pendulum with PID control. Table 1 Parameters of the X–Z inverted pendulum. M (kg) m (kg) l (m) gðm=s2 Þ 1 0.1 0.5 9.8 J.-J. Wang / ISA Transactions 51 (2012) 763–770766

19. 4.1. Simulation of stabilization The initial states of the X–Z inverted pendulum are xð0Þ ¼ 0:3, _xð0Þ ¼ 0, zð0Þ ¼ 0:2, _zð0Þ ¼ 0, yð0Þ ¼ ðp=4Þ rad, _yð0Þ ¼ 0, and d1 ¼ d2 ¼ d3 ¼ 0. The structure of the stabilization of the X–Z inverted pendulum is given in Fig. 3(a), where xd ¼ zd ¼ 0. The simulation results of the PID control and sliding-mode control are given in Figs. (4) and (5) respectively. Comparing Figs. (4) and (5), we can ﬁnd that stabilization of the X–Z inverted pendulum with sliding-mode control has better performance than that of with PID control. Sliding-mode control not only has faster response speed, but also has less stable error. Fig. 5. Stabilization of the X–Z inverted pendulum with sliding-mode control. Fig. 6. First case of tracking control of the X–Z inverted pendulum with PID control. J.-J. Wang / ISA Transactions 51 (2012) 763–770 767

20. 4.2. Simulation of tracking control The initial states of the X–Z inverted pendulum are given as xð0Þ ¼ 0:3, _xð0Þ ¼ 0, zð0Þ ¼ 0:2, _zð0Þ ¼ 0, yð0Þ ¼ p=4 rad, _yð0Þ ¼ 0, and d1 ¼ d2 ¼ d3 ¼ 0. In the ﬁrst case of the tracking control, the reference signals are given as xd ¼ 0:25 sin p 8 t ð39Þ zd ¼ 0:15 sin p 8 tÀ p 2 ð40Þ The simulation results of the PID control and sliding-mode control are given in (6) and (7) respectively. In the second case of the tracking control, the reference signals are given as xd ¼ 0:25 sin p 4 t ð41Þ zd ¼ 0:15 sin p 4 tÀ p 2 ð42Þ The simulation results of the PID control and sliding-mode control are given in (8) and (9) respectively. Fig. 7. First case of tracking control of the X–Z inverted pendulum with sliding-mode control. Fig. 8. Second case of tracking control of the X–Z inverted pendulum with PID control. J.-J. Wang / ISA Transactions 51 (2012) 763–770768

21. To test the robustness of the sliding-mode control for the X–Z inverted pendulum, four cases of simulation are given as following. First case of stabilization: M¼1.2, m¼0.15, d1 ¼ d2 ¼ d3 ¼ 0. Second case of stabilization: d1 ¼ d2 ¼ d3 ¼ 0:5 sinð2ptÞ. First case of tracking control with Eqs. (39) and (40): M¼1.2, m¼0.15, d1 ¼ d2 ¼ d3 ¼ 0. Second case of tracking control with Eqs. (39) and (40): d1 ¼ d2 ¼ d3 ¼ 0:5 sinð2ptÞ. The simulation results are given in (10)–(13). Comparing Figs. 6–13, we can ﬁnd that tracking control of the X–Z inverted pendulum with sliding-mode control has better performance than that of with PID control. When tracking the slow reference signals, PID control and sliding-mode control has good tracking performance. While when tracking the fast reference signals, PID control cannot tracking the fast reference signals with required tracking performance. PID control has little adap- tiveness to reference signals. Sliding-mode control has wider Fig. 9. Second case of tracking control of the X–Z inverted pendulum with sliding-mode control. Fig. 10. First case of stabilization with PID and SMC control when M ¼ 1:2 kg and m ¼ 0:15 kg. Fig. 11. Second case of stabilization with PID and SMC control when d1 ¼ d2 ¼ d3 ¼ 0:5 sinð2ptÞ. J.-J. Wang / ISA Transactions 51 (2012) 763–770 769

22. response speed bandwidth than the PID control. Further more, the sliding-mode controller has more robustness than the PID controllers from the simulation results. From the above design, simulation results and comparison, we can conclude that (1) The state transformation of the X–Z inverted pendulum and the decomposition of the model is right. The decomposition of the X–Z inverted pendulum make the control problems of the X–Z inverted pendulum become very easy. (2) The sliding-control of the X–Z inverted pendulum is effective for the stabilization and tracking control of the X–Z inverted pendulum. The tuning of the parameters of sliding-mode control is more easily than that of the PID control. (3) The sliding-mode control has better stabilization and tracking control performance for the X–Z inverted pendulum than the PID control method. 5. Conclusions In this paper we give the equivalent relationship between the X–Z inverted pendulum and PVTOL aircraft. The research of X–Z inverted pendulum is very meaningful for the control the PVTOL aircraft. Sliding-mode control in this paper can also be applied to the control of the PVTOL aircraft in a limited range. The major contributions of this paper can be summarized as following three points. (1) We give the equivalent relationship between the X–Z inverted pendulum and PVTOL aircraft. This is very important for the control of the X–Z inverted pendulum and PVTOL aircraft. And this is the base for their control method applied to each other. (2) We give a novel transformation, which can decompose the model of the X–Z inverted pendulum into three simple parts. This transformation make the difficult control problem very easy. And based on this transformation, many modern control methods can be applied directly to the control of the X–Z inverted pendulum and PVTOL aircraft. (3) Sliding-mode control is applied to the stabilization and tracking control of the X–Z inverted pendulum. And good control performance is achieved. Acknowledgements The work is supported by Nature Science Foundation of Zhejiang Province(No. LY12E07001). References [1] ˚Aström KJ, Furuta K. Swinging up a pendulum by energy control. Automatica 2000;36(2):287–95. [2] Mason P, Broucke M, Piccoli B. Time optimal swing-up of the planar pendulum. IEEE Transactions on Automatic Control 2008;53(8):1876–86. [3] Srinivasan B, Huguenin P, Bonvin D. Global stabilization of an inverted pendulum- control strategy and experimental verification. Automatica 2009;45(1): 265–9. [4] Liu GY, Neˇsic´ D, Mareels I. Non-linear stable inversion-based output tracking control for a spherical inverted pendulum. International Journal of Control 2008;81(1):116–33. [5] Liu GY, Mareels I, Neˇsic´ D. Decentralized control design of interconnected chains of integrators: a case study. Automatica 2008;44(8):2171–8. [6] Maravall D, Zhou C, Alonso J. Hybrid fuzzy control of the inverted pendulum via vertical forces. International Journal of Intelligent systems 2005;20(2): 195–211. [7] Maravall D. Control and stabilization of the inverted pendulum via vertical forces. In: Tarn TJ, Chen SB, Zhou C, editors. Robotic welding, intelligence and automation. Lecture notes in control and information sciences, vol. 299. Berlin: Springer-Verlag; 2004. p. 190–211. [8] Wang JJ. Simulation studies of inverted pendulum based on PID controllers. Simulation Modelling Practice and Theory 2011;19(1):440–9. [9] Chaturvedi NA, McClamroch NH, Bernstein DS. Asymptotic smooth stabilization of the inverted 3-D pendulum. IEEE Transactions on Automatic Control 2009;54(6):1204–15. [10] K.J. ˚Aström, B. Wittenmark, Adaptive control. New York: Addison- Wesley; 1995. [11] Utkin VI. Sliding mode control design principles and applications to electric drives. IEEE Transactions on Industrial Electronics 1993;40(1):23–36. [12] Wai RJ, Chang LJ. Adaptive stabilizing and tracking control for a nonlinear inverted-pendulum system via sliding-mode technique. IEEE Transactions on Industrial Electronics 2006;53(2):674–92. [13] Wai RJ, Chang LJ. Stabilizing and tracking control of nonlinear dual-axis inverted-pendulum system using fuzzy neural network. IEEE Transactions on Fuzzy Systems 2006;14(1):145–68. [14] Park M, Chwa D. Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method. IEEE Transactions on Industrial Electronics 2009;56(9):3541–55. [15] Park M, Chwa D. Orbital stabilization of inverted-pendulum systems via coupled sliding-mode control. IEEE Transactions on Industrial Electronics 2009;56(9):3556–70. [16] Olfati-Saber R. Global configuration stabilization for the VTOL aircraft with strong input coupling. IEEE Transactions on Automatic Control 2002;47(11): 1949–52. [17] Ailon A. Simple tracking controllers for autonomous VTOL aircraft with bounded inputs. IEEE Transactions on Automatic Control 2010;55(3):737–43. Fig. 12. First case of tracking with PID and SMC control when M ¼ 1:2 kg and m ¼ 0:15 kg. Fig. 13. Second case of tracking with PID and SMC control when d1 ¼ d2 ¼ d3 ¼ 0:5 sinð2ptÞ. J.-J. Wang / ISA Transactions 51 (2012) 763–770770

Stabilization and tracking control of X–Z inverted pendulum with sliding-mode control

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Stabilization and tracking control of X–Z inverted pendulum with sliding-mode control