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Stabilization and tracking control of X–Z inverted pendulum with sliding-mode control
ISA Transactions 51 (2012) 763–770
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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
abstract
Article history: Received 27 April 2012 Received in revised form 26 May 2012 Accepted 12 June 2012 Available online 10 July 2012
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.
Keywords: X–Z inverted pendulum Stabilization Tracking control Sliding-mode control
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 challenging 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.
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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 guarantees 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,
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J.-J. Wang / ISA Transactions 51 (2012) 763–770
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: 2
x€ ¼
Mmly_ sin y þ ðM þ m cos2 yÞF x mF z sin y cos ys þ d1 MðM þ mÞ
ð1Þ
2
Mmly_ cos ymF x sin y cos y þðM þ m sin2 yÞF z g þ d2 z€ ¼ MðM þ mÞ F x cos y þ F z sin y þd3 y€ ¼ Ml
ð2Þ
differentiable. Then the state equations (1)–(3) can be rewritten as 2
e€x ¼
Mmly_ sin y þðM þm cos2 yÞF x mF z sin y cos y x€ d þ d1 MðM þ mÞ
e€z ¼
Mmly_ cos ymF x sin y cos y þ ðM þm sin2 yÞF z gz€ d þ d2 MðM þ mÞ
2
ð5Þ
y€ ¼
F x cos y þF z sin y þ d3 Ml
ð6Þ
First, we assume that d1 ¼ d2 ¼ d3 ¼ 0. Defining exp ¼ ex þl sin y and ezp ¼ ez þ lðcos y1Þ, we can obtain that e_ xp ¼ e_ x þ ly_ cos y, 2 e_ zp ¼ e_ z ly_ sin y. Then e€ xp ¼ e€ x þ ly€ cos yly_ sin y and e€ zp ¼ e€ z 2 ly€ sin yly_ cos y can be acquired. Based on Eqs. (4)–(6), we can obtain the following equations: e€ xp ¼
2 2 F x sin y þ F z sin y cos yMly_ sin y x€ d M þm
ð7Þ
e€ zp ¼
2 F x sin y cos y þ F z cos2 yMly_ cos y gz€ d Mþm
ð8Þ
y€ ¼
F x cos y þF z sin y Ml
ð9Þ
Defining exm ¼ exp , ezm ¼ ezp , uxz ¼ ðF x sin y þ F z cos yÞ=ðM þ mÞ and uy ¼ F x cos y þ F z sin y=Ml, then we can obtain the following equations: e€ xm ¼ uxz sin y þ
ð3Þ
_ z_ Þ, ðx, € z€ Þ is the position, speed, acceleration of the pivot where (x,z), ðx, 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 1 r x r1, 1 rz r1, 30 r F x r30, 30 rF z 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 F z ¼ ðM þ mÞg. We define ex ¼ xxd , ez ¼ zzd , where xd and zd are desired signals. And we assume that xd and zd are no less than two times
ð4Þ
e€ zm ¼ uxz cos y
2 M ly_ sin y þ x€ d M þm
2 M ly_ cos ygz€ d M þm
y€ ¼ uy
ð10Þ
ð11Þ ð12Þ
From the definition of the uxz and uy , Fx and Fz can be obtained as the following equations: F x ¼ ðM þmÞuxz sin yMluy cos y
ð13Þ
F z ¼ ðM þ mÞuxz cos y þMluy sin y
ð14Þ
2 Defining um ¼ uxz ðM=ðM þmÞÞly_ , we can obtain the following equations:
e€ xm ¼ um sin y þ x€ d
ð15Þ
e€ zm ¼ um cos ygz€ d
ð16Þ
y€ ¼ uy
ð17Þ
Remark 2.1. If let xd ¼0 and zd ¼0, 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
Fig. 1. Structure 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. Because
J.-J. Wang / ISA Transactions 51 (2012) 763–770
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Fig. 2. Figure of sign and tanh function. (a) Figure of the tanh function. (b) Figure of the sign function.
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 , 2 yt ¼ tan y, u1 ¼ um cos y, and u2 ¼ ðuy þ 2y_ tan yÞsec2 y, we can obtain the following equations:
the control u1 can be designed as a saturated nonlinear control u1 ¼ gz€ d þ a tanhðk1 ez1 þ k2 ez2 Þ þ b tanhðk2 ez2 Þ
ð29Þ
e€ xt ¼ yt u1 þ x€ d
ð18Þ
where tanhðxÞ ¼ ðex ex Þ=ðex þ ex Þ, which is given in Fig. 2(a), k1 40, k2 40, a 40, b 4 0 are constant, and 9a9 þ9b9 þ 9z€ d 9 og. 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]. P Step 2. With the subsystem 21
e€ zt ¼ u1 gz€ d
ð19Þ
e_ x1 ¼ ex2
y€ t ¼ u2
ð20Þ
e_ x2 ¼ ex3 u1 þ x€ d
ð30Þ ð31Þ P
Let ez1 ¼ ezt , e_ z1 ¼ ez2 , ex1 ¼ ext , e_ x1 ¼ ex2 , ex3 ¼ yt and ex4 ¼ y_t , then the following equations can be obtained
ex3 u1 can be seen as the control of the subsystem 21 . Because P is a typical linear system, then ex3 u1 can be designed as
e_ z1 ¼ ez2
ð21Þ
ex3 u1 ¼ k3 ex1 k4 ex2 x€ d
e_ z2 ¼ u1 gz€ d
ð22Þ
where k3 4 0 and k4 40 are constants. From Step 1, we know that u1 a 0. Then the desired ex3 can be designed as
e_ x1 ¼ ex2
ð23Þ
21
enx3 ¼
e_ x2 ¼ ex3 u1 þ x€ d
ð24Þ
e_ x3 ¼ ex4
ð25Þ
e_ x4 ¼ u2
ð26Þ
From state equations (21)–(26), we can know that (1) The X–Z inverted pendulum system can be divided into two P P P subsystems 1 ðez1 ,ez2 Þ and 2 ðex1 ,ex2 ,ex3 ,ex4 Þ. 1 can be seen as P a independent linear control system. And subsystem 2 is P affected by subsystem 1 . P (2) Subsystem can be divided into two subsystems P P 2 P ðex3 ,ex4 Þ. 22 can be seen as a independent 21 ðex1 ,ex2 Þ and 22 P P P linear system. And 21 is affected be subsystem 1 and 22 simultaneously. P P (3) Because subsystem 1 and 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.
k3 ex1 k4 ex2 x€ d u1
ð32Þ
ð33Þ
From Eq. (33), enx3 can converge to 9x€ d =u1 9 with ex1 -0 and ex2 -0. If ex3 -xnx3 can obtained, this not only can realize the stabilization P of subsystem 21 , but also make the pendulum angle converges to 9x€ d =u1 9. 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 converging to zero. (2) In the tracking control of the X–Z inverted pendulum, because xd a0, and we can know that 9x€ d =u1 9 a 0. If we make 9x€ d =u1 9 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 p p 0:25 8 8 ¼ 0:0106 rad actan9maxðx€ d Þ=minðu1 Þ9 ¼ actan 9:8510:15
With the state (21)–(26) the control design procedure of the X–Z inverted pendulum can be given as the following three steps. P Step 1. With the subsystem 1
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. P Step 3. With the subsystem 22
e_ z1 ¼ ez2
ð27Þ
e_ x3 ¼ ex4
ð34Þ
e_ z2 ¼ u1 gz€ d
ð28Þ
e_ x4 ¼ u2
ð35Þ
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J.-J. Wang / ISA Transactions 51 (2012) 763–770
From Step 2, we know that if ex3 -enx3 , the stabilization of the X–Z inverted pendulum can be achieved. To realize ex3 -enx3 , the sliding-mode control design is given in the following. The sliding-mode surface is designed as s ¼ k5 x þ x_
ð36Þ
converge to zero. The saturated control can guarantee u1 a 0. And u2 can guarantee that ex3 -enx3 . This can be easily proved through Barbalat’s Lemma [13]. enx3 u1 can make ex1 and ex2 approach zero. Because the equivalent relationship between state equations (21)–(26) and (1)–(3), u1, u2 with enx3 can realize the stabilization of the X–Z inverted pendulum.
where x ¼ ex3 ex3 , k5 40 is constant. And the sliding-mode control can be selected as n
4. Simulation results and analysis
u2 ¼ k5 x_ þ e€ x3 k6 sr signðsÞ n
ð37Þ
where k6 4 0, r 40 are constant, and sign is the sign function as given in Fig. 2(b). It is very easy to obtain that s_ s ¼ k6 s2 r9s9. 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 n u2 ¼ k5 x_ þ e€ x3 k6 sr tanhðsÞ
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:
ð38Þ
From the above stabilization design for the state equations (21)–(26), the control u1 can make the state variable ez1 and ez2 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
PID1: PID2: PID3:
P 1 ¼ 25, I1 ¼ 15, D1 ¼ 3, P 2 ¼ 1:5, I2 ¼ 0:5, D2 ¼ 0:2, P 3 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.
J.-J. Wang / ISA Transactions 51 (2012) 763–770
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, z_ ð0Þ ¼ 0, yð0Þ ¼ ðp=4Þ rad, y_ ð0Þ 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
767
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.
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J.-J. Wang / ISA Transactions 51 (2012) 763–770
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, z_ ð0Þ ¼ 0, yð0Þ ¼ p=4 rad, y_ ð0Þ ¼ 0, and d1 ¼ d2 ¼ d3 ¼ 0. In the first case of the tracking control, the reference signals are given as p ð39Þ xd ¼ 0:25 sin t 8 zd ¼ 0:15 sin
p 8
t
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 t 4
p p zd ¼ 0:15 sin t 4 2
ð41Þ
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) 763–770
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: M ¼1.2, m ¼0.15, 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).
769
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 adaptiveness 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.
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J.-J. Wang / ISA Transactions 51 (2012) 763–770
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.
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).
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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
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
abstract
Article history: Received 27 April 2012 Received in revised form 26 May 2012 Accepted 12 June 2012 Available online 10 July 2012
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.
Keywords: X–Z inverted pendulum Stabilization Tracking control Sliding-mode control
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 challenging 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.
E-mail address: [email protected]
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 guarantees 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,
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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: 2
x€ ¼
Mmly_ sin y þ ðM þ m cos2 yÞF x mF z sin y cos ys þ d1 MðM þ mÞ
ð1Þ
2
Mmly_ cos ymF x sin y cos y þðM þ m sin2 yÞF z g þ d2 z€ ¼ MðM þ mÞ F x cos y þ F z sin y þd3 y€ ¼ Ml
ð2Þ
differentiable. Then the state equations (1)–(3) can be rewritten as 2
e€x ¼
Mmly_ sin y þðM þm cos2 yÞF x mF z sin y cos y x€ d þ d1 MðM þ mÞ
e€z ¼
Mmly_ cos ymF x sin y cos y þ ðM þm sin2 yÞF z gz€ d þ d2 MðM þ mÞ
2
ð5Þ
y€ ¼
F x cos y þF z sin y þ d3 Ml
ð6Þ
First, we assume that d1 ¼ d2 ¼ d3 ¼ 0. Defining exp ¼ ex þl sin y and ezp ¼ ez þ lðcos y1Þ, we can obtain that e_ xp ¼ e_ x þ ly_ cos y, 2 e_ zp ¼ e_ z ly_ sin y. Then e€ xp ¼ e€ x þ ly€ cos yly_ sin y and e€ zp ¼ e€ z 2 ly€ sin yly_ cos y can be acquired. Based on Eqs. (4)–(6), we can obtain the following equations: e€ xp ¼
2 2 F x sin y þ F z sin y cos yMly_ sin y x€ d M þm
ð7Þ
e€ zp ¼
2 F x sin y cos y þ F z cos2 yMly_ cos y gz€ d Mþm
ð8Þ
y€ ¼
F x cos y þF z sin y Ml
ð9Þ
Defining exm ¼ exp , ezm ¼ ezp , uxz ¼ ðF x sin y þ F z cos yÞ=ðM þ mÞ and uy ¼ F x cos y þ F z sin y=Ml, then we can obtain the following equations: e€ xm ¼ uxz sin y þ
ð3Þ
_ z_ Þ, ðx, € z€ Þ is the position, speed, acceleration of the pivot where (x,z), ðx, 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 1 r x r1, 1 rz r1, 30 r F x r30, 30 rF z 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 F z ¼ ðM þ mÞg. We define ex ¼ xxd , ez ¼ zzd , where xd and zd are desired signals. And we assume that xd and zd are no less than two times
ð4Þ
e€ zm ¼ uxz cos y
2 M ly_ sin y þ x€ d M þm
2 M ly_ cos ygz€ d M þm
y€ ¼ uy
ð10Þ
ð11Þ ð12Þ
From the definition of the uxz and uy , Fx and Fz can be obtained as the following equations: F x ¼ ðM þmÞuxz sin yMluy cos y
ð13Þ
F z ¼ ðM þ mÞuxz cos y þMluy sin y
ð14Þ
2 Defining um ¼ uxz ðM=ðM þmÞÞly_ , we can obtain the following equations:
e€ xm ¼ um sin y þ x€ d
ð15Þ
e€ zm ¼ um cos ygz€ d
ð16Þ
y€ ¼ uy
ð17Þ
Remark 2.1. If let xd ¼0 and zd ¼0, 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
Fig. 1. Structure 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. Because
J.-J. Wang / ISA Transactions 51 (2012) 763–770
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Fig. 2. Figure of sign and tanh function. (a) Figure of the tanh function. (b) Figure of the sign function.
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 , 2 yt ¼ tan y, u1 ¼ um cos y, and u2 ¼ ðuy þ 2y_ tan yÞsec2 y, we can obtain the following equations:
the control u1 can be designed as a saturated nonlinear control u1 ¼ gz€ d þ a tanhðk1 ez1 þ k2 ez2 Þ þ b tanhðk2 ez2 Þ
ð29Þ
e€ xt ¼ yt u1 þ x€ d
ð18Þ
where tanhðxÞ ¼ ðex ex Þ=ðex þ ex Þ, which is given in Fig. 2(a), k1 40, k2 40, a 40, b 4 0 are constant, and 9a9 þ9b9 þ 9z€ d 9 og. 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]. P Step 2. With the subsystem 21
e€ zt ¼ u1 gz€ d
ð19Þ
e_ x1 ¼ ex2
y€ t ¼ u2
ð20Þ
e_ x2 ¼ ex3 u1 þ x€ d
ð30Þ ð31Þ P
Let ez1 ¼ ezt , e_ z1 ¼ ez2 , ex1 ¼ ext , e_ x1 ¼ ex2 , ex3 ¼ yt and ex4 ¼ y_t , then the following equations can be obtained
ex3 u1 can be seen as the control of the subsystem 21 . Because P is a typical linear system, then ex3 u1 can be designed as
e_ z1 ¼ ez2
ð21Þ
ex3 u1 ¼ k3 ex1 k4 ex2 x€ d
e_ z2 ¼ u1 gz€ d
ð22Þ
where k3 4 0 and k4 40 are constants. From Step 1, we know that u1 a 0. Then the desired ex3 can be designed as
e_ x1 ¼ ex2
ð23Þ
21
enx3 ¼
e_ x2 ¼ ex3 u1 þ x€ d
ð24Þ
e_ x3 ¼ ex4
ð25Þ
e_ x4 ¼ u2
ð26Þ
From state equations (21)–(26), we can know that (1) The X–Z inverted pendulum system can be divided into two P P P subsystems 1 ðez1 ,ez2 Þ and 2 ðex1 ,ex2 ,ex3 ,ex4 Þ. 1 can be seen as P a independent linear control system. And subsystem 2 is P affected by subsystem 1 . P (2) Subsystem can be divided into two subsystems P P 2 P ðex3 ,ex4 Þ. 22 can be seen as a independent 21 ðex1 ,ex2 Þ and 22 P P P linear system. And 21 is affected be subsystem 1 and 22 simultaneously. P P (3) Because subsystem 1 and 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.
k3 ex1 k4 ex2 x€ d u1
ð32Þ
ð33Þ
From Eq. (33), enx3 can converge to 9x€ d =u1 9 with ex1 -0 and ex2 -0. If ex3 -xnx3 can obtained, this not only can realize the stabilization P of subsystem 21 , but also make the pendulum angle converges to 9x€ d =u1 9. 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 converging to zero. (2) In the tracking control of the X–Z inverted pendulum, because xd a0, and we can know that 9x€ d =u1 9 a 0. If we make 9x€ d =u1 9 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 p p 0:25 8 8 ¼ 0:0106 rad actan9maxðx€ d Þ=minðu1 Þ9 ¼ actan 9:8510:15
With the state (21)–(26) the control design procedure of the X–Z inverted pendulum can be given as the following three steps. P Step 1. With the subsystem 1
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. P Step 3. With the subsystem 22
e_ z1 ¼ ez2
ð27Þ
e_ x3 ¼ ex4
ð34Þ
e_ z2 ¼ u1 gz€ d
ð28Þ
e_ x4 ¼ u2
ð35Þ
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From Step 2, we know that if ex3 -enx3 , the stabilization of the X–Z inverted pendulum can be achieved. To realize ex3 -enx3 , the sliding-mode control design is given in the following. The sliding-mode surface is designed as s ¼ k5 x þ x_
ð36Þ
converge to zero. The saturated control can guarantee u1 a 0. And u2 can guarantee that ex3 -enx3 . This can be easily proved through Barbalat’s Lemma [13]. enx3 u1 can make ex1 and ex2 approach zero. Because the equivalent relationship between state equations (21)–(26) and (1)–(3), u1, u2 with enx3 can realize the stabilization of the X–Z inverted pendulum.
where x ¼ ex3 ex3 , k5 40 is constant. And the sliding-mode control can be selected as n
4. Simulation results and analysis
u2 ¼ k5 x_ þ e€ x3 k6 sr signðsÞ n
ð37Þ
where k6 4 0, r 40 are constant, and sign is the sign function as given in Fig. 2(b). It is very easy to obtain that s_ s ¼ k6 s2 r9s9. 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 n u2 ¼ k5 x_ þ e€ x3 k6 sr tanhðsÞ
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:
ð38Þ
From the above stabilization design for the state equations (21)–(26), the control u1 can make the state variable ez1 and ez2 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
PID1: PID2: PID3:
P 1 ¼ 25, I1 ¼ 15, D1 ¼ 3, P 2 ¼ 1:5, I2 ¼ 0:5, D2 ¼ 0:2, P 3 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.
J.-J. Wang / ISA Transactions 51 (2012) 763–770
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, z_ ð0Þ ¼ 0, yð0Þ ¼ ðp=4Þ rad, y_ ð0Þ 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
767
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.
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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, z_ ð0Þ ¼ 0, yð0Þ ¼ p=4 rad, y_ ð0Þ ¼ 0, and d1 ¼ d2 ¼ d3 ¼ 0. In the first case of the tracking control, the reference signals are given as p ð39Þ xd ¼ 0:25 sin t 8 zd ¼ 0:15 sin
p 8
t
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 t 4
p p zd ¼ 0:15 sin t 4 2
ð41Þ
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) 763–770
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: M ¼1.2, m ¼0.15, 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).
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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 adaptiveness 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.
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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.
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).
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