- Email: [email protected]
Research on One Type of Saturated Nonlinear Stabilization Control Method of X-Z Inverted Pendulum
Research on One Type of Saturated Nonlinear Stabilization Control Method of Z Inverted Pendulum X -Z Jia-Jun WANG1
Dong-Liang LIU1
Bao-Jun WANG2
Abstract: X-Z inverted pendulum not only has nonminimum phase and underactuated performance that normal inverted pendulum has, but also has more control freedoms. The equivalence relationship between X-Z inverted pendulum and planar vertical takeoff and landing (PVTOL) aircraft is found through definite states transformation. The saturated nonlinear control method original from the control method of PVTOL aircraft is used to the stabilization control of the X-Z inverted pendulum based on the equivalence relationship between them. The stabilization method is compared with the PID (Proportion integration differentiation) control through simulation. And this proofs the effectiveness of the stabilization method. Keywords: phase
X-Z inverted pendulum, planar vertical takeoff and landing (PVTOL) aircraft, stabilization, nonminimum
The control of the inverted pendulum is a well known problem in control theory and has been studied excessively in control literatures[1−3] . It provides many challenges to control design. The inverted pendulum is nonlinear, underactuated and nonminimum phase. The X-Z inverted pendulum is free to move in the horizontal and vertical plane with horizontal and vertical forces which is demonstrated by Maravall et al.[4−5] . The traditional inverted pendulum is viewed as the special case of the X-Z inverted pendulum. Compared with the classical inverted pendulum, the X-Z inverted pendulum has more versatility and is more like the actual control object in reality. The moving of the high buildings in the earthquake, the launching of the rocket and all such motions can be modeled as a X-Z inverted pendulum directly or indirectly. The X-Z inverted pendulum has three control freedoms and two control forces. Three freedoms include the pendulum angle freedom, the horizontal and vertical position moving freedoms. The horizontal and vertical motion are coupled. So the control of the X-Z inverted pendulum is more difficult than that of the traditional inverted pendulum. When the X-Z inverted pendulum moves in the horizontal and vertical plane, it has some common properties as the planar vertical take off and landing (PVTOL) aircraft. Unfortunately, no paper gives the underlying mechanisms between them. It seems interesting that if we can discover the equivalence between the systems, then the control scheme of the X-Z inverted pendulum can be simplified significantly and save lots of design efforts. Recently, the control of the PVTOL aircraft has received lots of attention from the control community[6−9] . Geometric nonlinear control techniques cannot be directly applied to flight control due to the presence of unstable zeros. The approximate input-output linearization was applied to solve the nonminimum phase problem of the V/STOL aircraft in [6]. A very novel decoupling transformation method was given in [7], which can realize the decoupling of the PVTOL Manuscript received May 6, 2011; accepted September 3, 2011 Supported by Natural Science Foundation of Zhejiang Province (LY 12E07001) Recommended by Associate Editor Shao-Yuan LI Citation: Jia-Jun Wang, Dong-Liang Liu, Bao-Jun Wang. Research on one type of saturated nonlinear stabilization control method of XZ inverted pendulum. Acta Automatica Sinica, 2013, 39(1): 92−96 1. School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China 2. Institute of Information Technology, Zhejiang Institute of Communications, Hangzhou 311112, China
aircraft with strong input coupling. A nonlinear outputfeedback controller was developed to force a PVTOL aircraft to globally asymptotically track a reference trajectory generated by a reference model, where the control development was based on a global exponential observer and the backstepping technique[8] . Based on the method in [7], simple tracking controllers were designed for the PVTOL aircraft with bounded inputs in [9], which ensured semiglobal asymptotic stabilization of the PVTOL aircraft. In the research of the PVTOL aircraft control, we find some minor shortcomings in the transformation of the roll angle in [7] and [9], which will be given in the later part of the paper. We establish the equivalence between the X-Z inverted pendulum and the PVTOL aircraft. So it is feasible that the stabilization method of the PVTOL aircraft can be applied to the stabilization of the X-Z inverted pendulum, and this is the motivation of the paper. In Section 1, the transformations of the X-Z inverted pendulum and the PVTOL aircraft models are given. In Section 2, the analysis of the shortcomings existing in [7] and [9] is given. In Section 3, the stabilization of the XZ inverted pendulum is given based on the method proposed in [7] and [9]. Numerical simulations are provided in Section 4 to illustrate the effectiveness of the stabilization controllers. And some conclusions are given in Section 5.
1 1.1
Model transformations Model transformations of the x -zz inverted pendulum
The X-Z inverted pendulum on a pivot driven by one horizontal and one vertical control force is shown in Fig. 1. In Fig. 1, the control action is based on the horizontal and vertical displacements of the pivot. The state equations of the X-Z inverted pendulum are given in [10] as the following equations: x ¨=
M mlθ˙2 sin θ + (M + m cos2 θ)Fx − mFz sin θ cos θ , M (M + m) (1) ˙2
z¨ =
2
M mlθ cos θ−mFx sin θ cos θ + (M + m sin θ)Fz − g, M (M + m) (2)
Jia-Jun WANG et al./ Research on One Type of Saturated Nonlinear Stabilization Control Method of · · ·
−Fx cos θ + Fz sin θ θ¨ = , (3) Ml where (x, z), (x, ˙ z), ˙ (¨ x, z¨) are 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 masses of the pivot and the pendulum respectively, g is the acceleration constant due to gravity, Fx is the horizontal force and Fz is the vertical force. We assume that the inertia of the pendulum is negligible.
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Remark 1. From above state transformations, the standard backstepping control procedure can be directly used to prove that a globally stabilizing static-feedback law exists for (12) ∼ (14)[7] .
1.2
Model transformations of the PVTOL aircraft
The simplified model of the PVTOL aircraft in Fig. 2 is represented by the following equations as in [6−9]: x ¨ = −u1 sin θ + εu2 cos θ,
(15)
z¨ = u1 cos θ + εu2 sin θ − g,
(16)
θ¨ = u2 ,
(17)
where (x, z) and θ denote the position of the aircraft center of mass and roll angle respectively, u1 and u2 are the vertical control force and rotational moment, g is the acceleration constant due to gravity and ε (ε = 0) is the coupling constant between the roll moment and the lateral force.
Fig. 1
The X-Z inverted pendulum
Defining xp = x + l sin θ and zp = z + l(cos θ − 1), we can obtain that x˙ p = x˙ + lθ˙ cos θ, z˙p = z˙ − lθ˙ sin θ. And then ¨ + lθ¨ cos θ − lθ˙2 sin θ and z¨p = z¨ − lθ¨ sin θ − lθ˙2 cos θ x ¨p = x can be acquired. Based on (1) ∼ (3), x ¨p , z¨p , and θ¨ can be rewritten as: x ¨p =
Fx sin2 θ + Fz sin θ cos θ − M lθ˙2 sin θ , M +m
(4)
z¨p =
Fx sin θ cos θ + Fz cos2 θ − M lθ˙2 cos θ − g, M +m
(5)
−Fx cos θ + Fz sin θ . θ¨ = Ml
(6)
z cos θ , and Defining x1 = −xp , z1 = zp , uxz = Fx sinMθ+F +m −Fx cos θ+Fz sin θ , they leads to the following equations: uθ = Ml
x ¨1 = − uxz sin θ +
M lθ˙2 sin θ, M +m
M z¨1 = uxz cos θ − lθ˙2 cos θ − g, M +m θ¨ = uθ .
(7)
The PVTOL aircraft
x ¨1 = − u ¯1 sin θ,
(18)
z¨1 = u ¯1 cos θ − g,
(19)
θ¨ = u2 . (8) (9)
From the definition of uxz and uθ , Fx and Fz can be expressed by the following equations: Fx = (M + m)uxz sin θ − M luθ cos θ,
(10)
Fz = (M + m)uxz cos θ + M luθ sin θ.
(11)
Defining u1 = uxz − MM lθ˙2 and u2 = uθ , we can have +m the following equations: x ¨1 = − u1 sin θ,
(12)
z¨1 = u1 cos θ − g,
(13)
θ¨ = u2 .
Fig. 2
Defining the change of coordinates x1 = x − ε sin θ, z1 = z + ε(cos θ − 1), and u ¯1 = u1 − εθ˙2 , we have the following equations:
(14)
(20)
Remark 2. Comparing (12) ∼ (14) with (18) ∼ (20), we conclude that the model of the X-Z inverted pendulum and the PVTOL aircraft model are equivalent through certain state transformations. This demonstrates that the control method designed for the PVTOL aircraft can be applied to the control of the X-Z inverted pendulum directly and vice versa. In the following section, we will use the stabilization method of PVTOL aircraft proposed in [7] and [9] for the stabilization of the X-Z inverted pendulum.
2
Analysis of the shortcomings existing in [7] and [9]
Before the stabilization design of the X-Z inverted pendulum, the shortcomings of the stabilization methods proposed in [7] and [9] applied to the PVTOL aircraft stabilization are analyzed.
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Defining x2 = x˙ 1 , z2 = z˙1 , ξ1 = θ, and ξ2 = θ˙ in (12) ∼ (14), then we can have: x˙ 1 = x2 ,
(21)
x˙ 2 = − u1 sin θ,
(22)
z˙1 = z2 ,
(23)
z˙2 = u1 cos θ − g,
(24)
ξ˙1 = ξ2 ,
(25)
ξ˙2 = u2 .
(26)
Theorem 2 in [7] points out that there exists a smooth static-state feedback in explicit form that asymptotically and locally exponentially stabilizes any desired configuration of the PVTOL aircraft model given by (21) ∼ (26) with zero steady-state performance. With (21) ∼ (26), we restrict the control u1 and u2 as 0 ≤ u1 ≤ U1 and 0 ≤ |u2 | ≤ U2 , where U1 > g and U2 > 0 are given constants. To realize (x1 , z1 ) = (0, 0), two auxiliary variables r1 and r2 are given in [7] as following: r1 = c11 x1 + c12 x2 ,
(27)
r2 = c0 tanh(c21 z1 + c22 z2 ),
(28)
where ci1 , ci2 for i = 1, 2 are coefficients of a Hurwitz polyx −x nomial, 0 < c0 < g, and tanh = eex −e . To control variable +e−x ξ1 , the reference variable ξ1d is given as following: −r1 ξ1d = arctan . (29) r2 + g From the explicit definition of the roll angle θ in [11], we know θ ∈ [0, 2π] or θ ∈ [−π, π]. That is to say ξ1d should also be defined in the same range as θ. Whereas the definition of ξ1d shows ξ1d ∈ [− π2 , π2 ], then we can say that the definition of ξ1d decreases the control range of θ in [7]. The arctan function figure is given in Fig. 3.
The reference angle ξ1d is given as following equation: r1 ξ1d = arctan . (32) r2 + g We can see that ξ1d ∈ [− π2 , π2 ]. Then we can conclude that the definition of ξ1d in [9] has the same decreased range as in [7]. From above analysis, we can find the shortcomings existing in [7] and [9]. 1) The control range of roll angle θ is decreased. The definition of ξ1d does not suit the control of the PVTOL aircraft in the full range of θ. 2) The θ is unreachable on the point of θ = ± π2 because of r2 < g. The example simulation (5.2) in [9] gave the initial value θ = π (the initial range of θ includes the point of θ = π2 ), but the author did not give the way how to deal with the point of θ = ± π2 . So we consider that there may exist problems in the simulation results in [9].
3
Stabilization design of the x -zz inverted pendulum
From the above analysis, we know that the design methods in [7] and [9] have their shortcomings in the control of PVTOL aircraft with the whole range of roll angle. However in the stabilization of the X-Z inverted pendulum in the upper-half plane, the swing-up procedure of the pendulum is neglected. The angle of the X-Z inverted pendulum is in the range of [− π2 , π2 ]. The point of θ = ± π2 is not in the range of the control angle of the X-Z inverted pendulum. So we can directly extend the methods in [7] and [9] to the stabilization of the X-Z inverted pendulum and the shortcomings of the method designed for control of the PVTOL aircraft can be avoided. We assume that all the variables in (21) ∼ (26) are measurable. Like the control input u ¯1 in [7] and [9], with the definition of auxiliary variables r1 and r2 in (30) and (31), u1 can be designed as following: 2 u1 = r12 + (r2 + g)2 . (33) Defining ξ1e = ξ1 − ξ1d , ξ2d = ξ˙1d , and ξ2e = ξ2 − ξ2d , where ξ1d is the same as the definition in (32), we can design the control input u2 as following: u2 = ξ¨1d − q1 ξ1e − q2 ξ2e ,
Fig. 3
The figure of the arctan function
In [9], the auxiliary variables r1 and r2 are given as following equations: r1 = a1 tanh(k1 x1 + k2 x2 ) + a2 tanh(k2 x2 ),
(30)
r2 = − b1 tanh(p1 z1 + p2 z2 ) − b2 tanh(p2 z2 ),
(31)
where ai , bi , ki , and pi for i = 1, 2 are positive constants. In addition, b1 + b2 < g and (a1 + a2 )2 + (b1 + b2 + g)2 ≤ U12 .
(34)
where q1 and q2 are positive constants. With Lemmas 3.1 and 3.2 in [9], it can be easily proved that control inputs u1 and u2 as (33) and (34) can realize the locally exponential stabilization of the X-Z inverted pendulum in the upper-half plane. The stabilization design procedure of the X-Z inverted pendulum is given in detail as following eight steps. 1) Based on Model (1) ∼ (3), the new state variables xp and zp can be selected. Then the state equations (4) ∼ (6) can be established. 2) According to Model (4) ∼ (6), the new state variable x1 and z1 can be selected. With uxz , uθ , then the state equations (7) ∼ (9) can be obtained. 3) With the new control inputs u1 and u2 , the state equations (12) ∼ (14) can be acquired. And state equations (12) ∼ (14) of the inverted pendulum can be rewritten as (21) ∼ (26). 4) Through the selection of the positive constants ai , bi , ki , and pi for i = 1, 2, the auxiliary variables r1 and r2 can be designed as (30) and (31).
Jia-Jun WANG et al./ Research on One Type of Saturated Nonlinear Stabilization Control Method of · · ·
5) The reference ξ1d and the control input u1 can be designed as (32) and (33). 6) With the differentiation of ξ2d , the error ξ2e , and the constants q1 and q2 , the control input u2 can be given as (34). 7) With the control input u1 and u2 , uxz and uθ can be obtained. 8) With the control input uxz and uθ , the actual control input Fx and Fz can be designed as (10) and (11) for the control of the X-Z inverted pendulum. And this is the end of the stabilization design.
4
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but also has good steady-state performance.
Simulation results and analysis
The parameters of the X-Z inverted pendulum are given as M = 1 kg, m = 0.1 kg, l = 0.5 m, and g = 9.8 m/s2 . The initial states are x(0) = 7, x(0) ˙ = 0, z(0) = −4, z(0) ˙ = 0, ˙ θ(0) = 0.3 rad, and θ(0) = 0. To compare with the existing stabilization method of XZ inverted pendulum, three PID controllers as proposed in [10] are firstly designed for the X-Z inverted pendulum. The structure of the stabilization method with three PID controllers is given in Fig. 4, where xd = 0 and zd = 0. And the PID controller parameters are designed as following:
Fig. 6
State variables x and z of the X-Z inverted pendulum
PID1: P1 = 50, I1 = 40, D1 = 3, PID2: P2 = −1.8, I2 = −1, D2 = −0.75, PID3: P3 = 5, I3 = 15, D3 = 16. The simulation results of the X-Z inverted pendulum with PID controllers are illustrated in Figs. 5 ∼ 7.
Fig. 4
Fig. 7
Control forces Fx and Fz of the X-Z inverted pendulum
The structure of the PID controllers in [10]
Fig. 8
Fig. 5
Angle θ and ξ1d of the X-Z inverted pendulum
Angle θ of the X-Z inverted pendulum
To realize the X-Z stabilization method proposed in this paper, the control restrictions are U1 = 20, U2 = 20. The control constants are a1 = 1, a2 = 3, b1 = 2, b2 = 3, k1 = k2 = 0.1, p1 = p2 = 1, q1 = 0.2 and q2 = 1. The simulation results are given in Figs. 8 ∼ 11. From Figs. 8 ∼ 11, we illustrate that the method proposed in [9] is a very good method for the stabilization of the X-Z inverted pendulum. The XZ inverted pendulum not only has good dynamic response,
Fig. 9
State variables x and z of the X-Z inverted pendulum
Compared with the existing stabilization method for the X-Z inverted pendulum, three advantages of the proposed method in this paper can be concluded.
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1) The methods proposed in [4−5] are based on the linearized model of the X-Z inverted pendulum, whereas the proposed method in this paper is based on the original nonlinear model of the X-Z inverted pendulum. 2) Through comparing the simulation results of the proposed method in this paper with results in [10], we can see that the proposed controllers have better control performance than PID controllers. 3) The proposed method is designed based on the stabilization theory, and the proof of the stabilization is a very easy job. However the method proposed in [4−5] and [10], the proof of stabilization is not an easy task. Furthermore the selection of the control parameters in the proposed method is simpler than that in the PID control.
be researched further.
References [1] ˚ Astr¨ om K J, Furuta K. Swinging up a pendulum by energy control. Automatica, 2000, 36(2): 287−295 [2] Mason P, Broucke M, Piccoli B. Time optimal swing-up of the planar pendulum. IEEE Transactions on Automatic Control, 2008, 53(8): 1876−1886 [3] Srinivasan B, Huguenin P, Bonvin D. Global stabilization of an inverted pendulum-control strategy and experimental verification. Automatica, 2009, 45(1): 265−269 [4] Maravall D, Zhou C J, Alonso J. Hybrid fuzzy control of the inverted pendulum via vertical forces. International Journal of Intelligent Systems, 2005, 20(2): 195−211 [5] Tarn T J, Chen S B, Zhou C. Robotic Welding, Intelligence and Automation. Berlin: Springer-Verlag, 2004. 190−211 [6] Hauser J, Sastry S, Meyer G. Nonlinear control design for slightly non-minimum phase systems: application to V/STOL aircraft. Automatica, 1992, 28(4): 665−679 [7] Olfati-Saber R. Global configuration stabilization for the VTOL aircraft with strong input coupling. IEEE Transactions on Automatic Control, 2002, 47(11): 1949−1952
Fig. 10
Auxiliary variables r1 and r2 of the X-Z inverted pendulum
[8] Do K D, Jiang Z P, Pan J. On global tracking control of a VTOL aircraft without velocity measurements. IEEE Transactions on Automatic Control, 2003, 48(12): 2212− 2217 [9] Ailon A. Simple tracking controllers for autonomous VTOL aircraft with bounded inputs. IEEE Transactions on Automatic Control, 2010, 55(3): 737−743 [10] Wang J J. Simulation studies of inverted pendulum based on PID controllers. Simulation Modelling Practice and Theory, 2011, 19(1): 440−449 [11] Consolini L, Tosques M. On the VTOL exact tracking with bounded internal dynamics via a poincar´e map approach. IEEE Transactions on Automatic Control, 2007, 52(9): 1757−1762
Fig. 11
5
Control forces Fx and Fz of the X-Z inverted pendulum
Conclusions
In this paper, the method proposed in [9] is applied to the stabilization of the X-Z inverted pendulum. The major contributions of this paper are summarized as following. 1) The equivalence between the X-Z inverted pendulum and the PVTOL aircraft is discovered through the state transformations. Based on their equivalence, the control method design for the PVTOL aircraft can be used to the control of the X-Z inverted pendulum and vice versa. 2) The shortcomings of the methods proposed in [7] and [9] were pointed out. Because the definition of the ξ1d , they had their limitations in the control of the PVTOL aircraft for large range of roll angle. 3) The method designed for the PVTOL aircraft was applied to the stabilization of the X-Z inverted pendulum. And good control performance was achieved for the X-Z inverted pendulum. In the design procedure, we found that the control performance is very sensitive to the designed constants ai , bi , ki , pi , and qi , for i = 1, 2. So how to improve the robustness of the controller is an interesting topic which should
Jia-Jun WANG Associate professor at the School of Automation, Hangzhou Dianzi University. He received his Ph. D. degree in power electronics and power drive from Tianjin University in 2003. His research interest covers sliding mode control, backstepping, neural networks and their applications in motion control system. Corresponding author of this paper. E-mail: [email protected] Dong-Liang LIU Associate professor at the School of Automation, Hangzhou Dianzi University. He received his Ph. D. degree in control theory and control engineering from Zhejiang University in 2005. His research interest covers nonlinear control theory, AC servo driver and PV solar inverter. E-mail: [email protected]
Bao-Jun WANG Professor at the Institute of Information Technology, Zhejiang Institute of Communications. He is responsible for the major of network technology of computer. His research interest covers distributed network, operation system, and automation and control system technology. E-mail: [email protected]
Dong-Liang LIU1
Bao-Jun WANG2
Abstract: X-Z inverted pendulum not only has nonminimum phase and underactuated performance that normal inverted pendulum has, but also has more control freedoms. The equivalence relationship between X-Z inverted pendulum and planar vertical takeoff and landing (PVTOL) aircraft is found through definite states transformation. The saturated nonlinear control method original from the control method of PVTOL aircraft is used to the stabilization control of the X-Z inverted pendulum based on the equivalence relationship between them. The stabilization method is compared with the PID (Proportion integration differentiation) control through simulation. And this proofs the effectiveness of the stabilization method. Keywords: phase
X-Z inverted pendulum, planar vertical takeoff and landing (PVTOL) aircraft, stabilization, nonminimum
The control of the inverted pendulum is a well known problem in control theory and has been studied excessively in control literatures[1−3] . It provides many challenges to control design. The inverted pendulum is nonlinear, underactuated and nonminimum phase. The X-Z inverted pendulum is free to move in the horizontal and vertical plane with horizontal and vertical forces which is demonstrated by Maravall et al.[4−5] . The traditional inverted pendulum is viewed as the special case of the X-Z inverted pendulum. Compared with the classical inverted pendulum, the X-Z inverted pendulum has more versatility and is more like the actual control object in reality. The moving of the high buildings in the earthquake, the launching of the rocket and all such motions can be modeled as a X-Z inverted pendulum directly or indirectly. The X-Z inverted pendulum has three control freedoms and two control forces. Three freedoms include the pendulum angle freedom, the horizontal and vertical position moving freedoms. The horizontal and vertical motion are coupled. So the control of the X-Z inverted pendulum is more difficult than that of the traditional inverted pendulum. When the X-Z inverted pendulum moves in the horizontal and vertical plane, it has some common properties as the planar vertical take off and landing (PVTOL) aircraft. Unfortunately, no paper gives the underlying mechanisms between them. It seems interesting that if we can discover the equivalence between the systems, then the control scheme of the X-Z inverted pendulum can be simplified significantly and save lots of design efforts. Recently, the control of the PVTOL aircraft has received lots of attention from the control community[6−9] . Geometric nonlinear control techniques cannot be directly applied to flight control due to the presence of unstable zeros. The approximate input-output linearization was applied to solve the nonminimum phase problem of the V/STOL aircraft in [6]. A very novel decoupling transformation method was given in [7], which can realize the decoupling of the PVTOL Manuscript received May 6, 2011; accepted September 3, 2011 Supported by Natural Science Foundation of Zhejiang Province (LY 12E07001) Recommended by Associate Editor Shao-Yuan LI Citation: Jia-Jun Wang, Dong-Liang Liu, Bao-Jun Wang. Research on one type of saturated nonlinear stabilization control method of XZ inverted pendulum. Acta Automatica Sinica, 2013, 39(1): 92−96 1. School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China 2. Institute of Information Technology, Zhejiang Institute of Communications, Hangzhou 311112, China
aircraft with strong input coupling. A nonlinear outputfeedback controller was developed to force a PVTOL aircraft to globally asymptotically track a reference trajectory generated by a reference model, where the control development was based on a global exponential observer and the backstepping technique[8] . Based on the method in [7], simple tracking controllers were designed for the PVTOL aircraft with bounded inputs in [9], which ensured semiglobal asymptotic stabilization of the PVTOL aircraft. In the research of the PVTOL aircraft control, we find some minor shortcomings in the transformation of the roll angle in [7] and [9], which will be given in the later part of the paper. We establish the equivalence between the X-Z inverted pendulum and the PVTOL aircraft. So it is feasible that the stabilization method of the PVTOL aircraft can be applied to the stabilization of the X-Z inverted pendulum, and this is the motivation of the paper. In Section 1, the transformations of the X-Z inverted pendulum and the PVTOL aircraft models are given. In Section 2, the analysis of the shortcomings existing in [7] and [9] is given. In Section 3, the stabilization of the XZ inverted pendulum is given based on the method proposed in [7] and [9]. Numerical simulations are provided in Section 4 to illustrate the effectiveness of the stabilization controllers. And some conclusions are given in Section 5.
1 1.1
Model transformations Model transformations of the x -zz inverted pendulum
The X-Z inverted pendulum on a pivot driven by one horizontal and one vertical control force is shown in Fig. 1. In Fig. 1, the control action is based on the horizontal and vertical displacements of the pivot. The state equations of the X-Z inverted pendulum are given in [10] as the following equations: x ¨=
M mlθ˙2 sin θ + (M + m cos2 θ)Fx − mFz sin θ cos θ , M (M + m) (1) ˙2
z¨ =
2
M mlθ cos θ−mFx sin θ cos θ + (M + m sin θ)Fz − g, M (M + m) (2)
Jia-Jun WANG et al./ Research on One Type of Saturated Nonlinear Stabilization Control Method of · · ·
−Fx cos θ + Fz sin θ θ¨ = , (3) Ml where (x, z), (x, ˙ z), ˙ (¨ x, z¨) are 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 masses of the pivot and the pendulum respectively, g is the acceleration constant due to gravity, Fx is the horizontal force and Fz is the vertical force. We assume that the inertia of the pendulum is negligible.
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Remark 1. From above state transformations, the standard backstepping control procedure can be directly used to prove that a globally stabilizing static-feedback law exists for (12) ∼ (14)[7] .
1.2
Model transformations of the PVTOL aircraft
The simplified model of the PVTOL aircraft in Fig. 2 is represented by the following equations as in [6−9]: x ¨ = −u1 sin θ + εu2 cos θ,
(15)
z¨ = u1 cos θ + εu2 sin θ − g,
(16)
θ¨ = u2 ,
(17)
where (x, z) and θ denote the position of the aircraft center of mass and roll angle respectively, u1 and u2 are the vertical control force and rotational moment, g is the acceleration constant due to gravity and ε (ε = 0) is the coupling constant between the roll moment and the lateral force.
Fig. 1
The X-Z inverted pendulum
Defining xp = x + l sin θ and zp = z + l(cos θ − 1), we can obtain that x˙ p = x˙ + lθ˙ cos θ, z˙p = z˙ − lθ˙ sin θ. And then ¨ + lθ¨ cos θ − lθ˙2 sin θ and z¨p = z¨ − lθ¨ sin θ − lθ˙2 cos θ x ¨p = x can be acquired. Based on (1) ∼ (3), x ¨p , z¨p , and θ¨ can be rewritten as: x ¨p =
Fx sin2 θ + Fz sin θ cos θ − M lθ˙2 sin θ , M +m
(4)
z¨p =
Fx sin θ cos θ + Fz cos2 θ − M lθ˙2 cos θ − g, M +m
(5)
−Fx cos θ + Fz sin θ . θ¨ = Ml
(6)
z cos θ , and Defining x1 = −xp , z1 = zp , uxz = Fx sinMθ+F +m −Fx cos θ+Fz sin θ , they leads to the following equations: uθ = Ml
x ¨1 = − uxz sin θ +
M lθ˙2 sin θ, M +m
M z¨1 = uxz cos θ − lθ˙2 cos θ − g, M +m θ¨ = uθ .
(7)
The PVTOL aircraft
x ¨1 = − u ¯1 sin θ,
(18)
z¨1 = u ¯1 cos θ − g,
(19)
θ¨ = u2 . (8) (9)
From the definition of uxz and uθ , Fx and Fz can be expressed by the following equations: Fx = (M + m)uxz sin θ − M luθ cos θ,
(10)
Fz = (M + m)uxz cos θ + M luθ sin θ.
(11)
Defining u1 = uxz − MM lθ˙2 and u2 = uθ , we can have +m the following equations: x ¨1 = − u1 sin θ,
(12)
z¨1 = u1 cos θ − g,
(13)
θ¨ = u2 .
Fig. 2
Defining the change of coordinates x1 = x − ε sin θ, z1 = z + ε(cos θ − 1), and u ¯1 = u1 − εθ˙2 , we have the following equations:
(14)
(20)
Remark 2. Comparing (12) ∼ (14) with (18) ∼ (20), we conclude that the model of the X-Z inverted pendulum and the PVTOL aircraft model are equivalent through certain state transformations. This demonstrates that the control method designed for the PVTOL aircraft can be applied to the control of the X-Z inverted pendulum directly and vice versa. In the following section, we will use the stabilization method of PVTOL aircraft proposed in [7] and [9] for the stabilization of the X-Z inverted pendulum.
2
Analysis of the shortcomings existing in [7] and [9]
Before the stabilization design of the X-Z inverted pendulum, the shortcomings of the stabilization methods proposed in [7] and [9] applied to the PVTOL aircraft stabilization are analyzed.
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Defining x2 = x˙ 1 , z2 = z˙1 , ξ1 = θ, and ξ2 = θ˙ in (12) ∼ (14), then we can have: x˙ 1 = x2 ,
(21)
x˙ 2 = − u1 sin θ,
(22)
z˙1 = z2 ,
(23)
z˙2 = u1 cos θ − g,
(24)
ξ˙1 = ξ2 ,
(25)
ξ˙2 = u2 .
(26)
Theorem 2 in [7] points out that there exists a smooth static-state feedback in explicit form that asymptotically and locally exponentially stabilizes any desired configuration of the PVTOL aircraft model given by (21) ∼ (26) with zero steady-state performance. With (21) ∼ (26), we restrict the control u1 and u2 as 0 ≤ u1 ≤ U1 and 0 ≤ |u2 | ≤ U2 , where U1 > g and U2 > 0 are given constants. To realize (x1 , z1 ) = (0, 0), two auxiliary variables r1 and r2 are given in [7] as following: r1 = c11 x1 + c12 x2 ,
(27)
r2 = c0 tanh(c21 z1 + c22 z2 ),
(28)
where ci1 , ci2 for i = 1, 2 are coefficients of a Hurwitz polyx −x nomial, 0 < c0 < g, and tanh = eex −e . To control variable +e−x ξ1 , the reference variable ξ1d is given as following: −r1 ξ1d = arctan . (29) r2 + g From the explicit definition of the roll angle θ in [11], we know θ ∈ [0, 2π] or θ ∈ [−π, π]. That is to say ξ1d should also be defined in the same range as θ. Whereas the definition of ξ1d shows ξ1d ∈ [− π2 , π2 ], then we can say that the definition of ξ1d decreases the control range of θ in [7]. The arctan function figure is given in Fig. 3.
The reference angle ξ1d is given as following equation: r1 ξ1d = arctan . (32) r2 + g We can see that ξ1d ∈ [− π2 , π2 ]. Then we can conclude that the definition of ξ1d in [9] has the same decreased range as in [7]. From above analysis, we can find the shortcomings existing in [7] and [9]. 1) The control range of roll angle θ is decreased. The definition of ξ1d does not suit the control of the PVTOL aircraft in the full range of θ. 2) The θ is unreachable on the point of θ = ± π2 because of r2 < g. The example simulation (5.2) in [9] gave the initial value θ = π (the initial range of θ includes the point of θ = π2 ), but the author did not give the way how to deal with the point of θ = ± π2 . So we consider that there may exist problems in the simulation results in [9].
3
Stabilization design of the x -zz inverted pendulum
From the above analysis, we know that the design methods in [7] and [9] have their shortcomings in the control of PVTOL aircraft with the whole range of roll angle. However in the stabilization of the X-Z inverted pendulum in the upper-half plane, the swing-up procedure of the pendulum is neglected. The angle of the X-Z inverted pendulum is in the range of [− π2 , π2 ]. The point of θ = ± π2 is not in the range of the control angle of the X-Z inverted pendulum. So we can directly extend the methods in [7] and [9] to the stabilization of the X-Z inverted pendulum and the shortcomings of the method designed for control of the PVTOL aircraft can be avoided. We assume that all the variables in (21) ∼ (26) are measurable. Like the control input u ¯1 in [7] and [9], with the definition of auxiliary variables r1 and r2 in (30) and (31), u1 can be designed as following: 2 u1 = r12 + (r2 + g)2 . (33) Defining ξ1e = ξ1 − ξ1d , ξ2d = ξ˙1d , and ξ2e = ξ2 − ξ2d , where ξ1d is the same as the definition in (32), we can design the control input u2 as following: u2 = ξ¨1d − q1 ξ1e − q2 ξ2e ,
Fig. 3
The figure of the arctan function
In [9], the auxiliary variables r1 and r2 are given as following equations: r1 = a1 tanh(k1 x1 + k2 x2 ) + a2 tanh(k2 x2 ),
(30)
r2 = − b1 tanh(p1 z1 + p2 z2 ) − b2 tanh(p2 z2 ),
(31)
where ai , bi , ki , and pi for i = 1, 2 are positive constants. In addition, b1 + b2 < g and (a1 + a2 )2 + (b1 + b2 + g)2 ≤ U12 .
(34)
where q1 and q2 are positive constants. With Lemmas 3.1 and 3.2 in [9], it can be easily proved that control inputs u1 and u2 as (33) and (34) can realize the locally exponential stabilization of the X-Z inverted pendulum in the upper-half plane. The stabilization design procedure of the X-Z inverted pendulum is given in detail as following eight steps. 1) Based on Model (1) ∼ (3), the new state variables xp and zp can be selected. Then the state equations (4) ∼ (6) can be established. 2) According to Model (4) ∼ (6), the new state variable x1 and z1 can be selected. With uxz , uθ , then the state equations (7) ∼ (9) can be obtained. 3) With the new control inputs u1 and u2 , the state equations (12) ∼ (14) can be acquired. And state equations (12) ∼ (14) of the inverted pendulum can be rewritten as (21) ∼ (26). 4) Through the selection of the positive constants ai , bi , ki , and pi for i = 1, 2, the auxiliary variables r1 and r2 can be designed as (30) and (31).
Jia-Jun WANG et al./ Research on One Type of Saturated Nonlinear Stabilization Control Method of · · ·
5) The reference ξ1d and the control input u1 can be designed as (32) and (33). 6) With the differentiation of ξ2d , the error ξ2e , and the constants q1 and q2 , the control input u2 can be given as (34). 7) With the control input u1 and u2 , uxz and uθ can be obtained. 8) With the control input uxz and uθ , the actual control input Fx and Fz can be designed as (10) and (11) for the control of the X-Z inverted pendulum. And this is the end of the stabilization design.
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but also has good steady-state performance.
Simulation results and analysis
The parameters of the X-Z inverted pendulum are given as M = 1 kg, m = 0.1 kg, l = 0.5 m, and g = 9.8 m/s2 . The initial states are x(0) = 7, x(0) ˙ = 0, z(0) = −4, z(0) ˙ = 0, ˙ θ(0) = 0.3 rad, and θ(0) = 0. To compare with the existing stabilization method of XZ inverted pendulum, three PID controllers as proposed in [10] are firstly designed for the X-Z inverted pendulum. The structure of the stabilization method with three PID controllers is given in Fig. 4, where xd = 0 and zd = 0. And the PID controller parameters are designed as following:
Fig. 6
State variables x and z of the X-Z inverted pendulum
PID1: P1 = 50, I1 = 40, D1 = 3, PID2: P2 = −1.8, I2 = −1, D2 = −0.75, PID3: P3 = 5, I3 = 15, D3 = 16. The simulation results of the X-Z inverted pendulum with PID controllers are illustrated in Figs. 5 ∼ 7.
Fig. 4
Fig. 7
Control forces Fx and Fz of the X-Z inverted pendulum
The structure of the PID controllers in [10]
Fig. 8
Fig. 5
Angle θ and ξ1d of the X-Z inverted pendulum
Angle θ of the X-Z inverted pendulum
To realize the X-Z stabilization method proposed in this paper, the control restrictions are U1 = 20, U2 = 20. The control constants are a1 = 1, a2 = 3, b1 = 2, b2 = 3, k1 = k2 = 0.1, p1 = p2 = 1, q1 = 0.2 and q2 = 1. The simulation results are given in Figs. 8 ∼ 11. From Figs. 8 ∼ 11, we illustrate that the method proposed in [9] is a very good method for the stabilization of the X-Z inverted pendulum. The XZ inverted pendulum not only has good dynamic response,
Fig. 9
State variables x and z of the X-Z inverted pendulum
Compared with the existing stabilization method for the X-Z inverted pendulum, three advantages of the proposed method in this paper can be concluded.
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1) The methods proposed in [4−5] are based on the linearized model of the X-Z inverted pendulum, whereas the proposed method in this paper is based on the original nonlinear model of the X-Z inverted pendulum. 2) Through comparing the simulation results of the proposed method in this paper with results in [10], we can see that the proposed controllers have better control performance than PID controllers. 3) The proposed method is designed based on the stabilization theory, and the proof of the stabilization is a very easy job. However the method proposed in [4−5] and [10], the proof of stabilization is not an easy task. Furthermore the selection of the control parameters in the proposed method is simpler than that in the PID control.
be researched further.
References [1] ˚ Astr¨ om K J, Furuta K. Swinging up a pendulum by energy control. Automatica, 2000, 36(2): 287−295 [2] Mason P, Broucke M, Piccoli B. Time optimal swing-up of the planar pendulum. IEEE Transactions on Automatic Control, 2008, 53(8): 1876−1886 [3] Srinivasan B, Huguenin P, Bonvin D. Global stabilization of an inverted pendulum-control strategy and experimental verification. Automatica, 2009, 45(1): 265−269 [4] Maravall D, Zhou C J, Alonso J. Hybrid fuzzy control of the inverted pendulum via vertical forces. International Journal of Intelligent Systems, 2005, 20(2): 195−211 [5] Tarn T J, Chen S B, Zhou C. Robotic Welding, Intelligence and Automation. Berlin: Springer-Verlag, 2004. 190−211 [6] Hauser J, Sastry S, Meyer G. Nonlinear control design for slightly non-minimum phase systems: application to V/STOL aircraft. Automatica, 1992, 28(4): 665−679 [7] Olfati-Saber R. Global configuration stabilization for the VTOL aircraft with strong input coupling. IEEE Transactions on Automatic Control, 2002, 47(11): 1949−1952
Fig. 10
Auxiliary variables r1 and r2 of the X-Z inverted pendulum
[8] Do K D, Jiang Z P, Pan J. On global tracking control of a VTOL aircraft without velocity measurements. IEEE Transactions on Automatic Control, 2003, 48(12): 2212− 2217 [9] Ailon A. Simple tracking controllers for autonomous VTOL aircraft with bounded inputs. IEEE Transactions on Automatic Control, 2010, 55(3): 737−743 [10] Wang J J. Simulation studies of inverted pendulum based on PID controllers. Simulation Modelling Practice and Theory, 2011, 19(1): 440−449 [11] Consolini L, Tosques M. On the VTOL exact tracking with bounded internal dynamics via a poincar´e map approach. IEEE Transactions on Automatic Control, 2007, 52(9): 1757−1762
Fig. 11
5
Control forces Fx and Fz of the X-Z inverted pendulum
Conclusions
In this paper, the method proposed in [9] is applied to the stabilization of the X-Z inverted pendulum. The major contributions of this paper are summarized as following. 1) The equivalence between the X-Z inverted pendulum and the PVTOL aircraft is discovered through the state transformations. Based on their equivalence, the control method design for the PVTOL aircraft can be used to the control of the X-Z inverted pendulum and vice versa. 2) The shortcomings of the methods proposed in [7] and [9] were pointed out. Because the definition of the ξ1d , they had their limitations in the control of the PVTOL aircraft for large range of roll angle. 3) The method designed for the PVTOL aircraft was applied to the stabilization of the X-Z inverted pendulum. And good control performance was achieved for the X-Z inverted pendulum. In the design procedure, we found that the control performance is very sensitive to the designed constants ai , bi , ki , pi , and qi , for i = 1, 2. So how to improve the robustness of the controller is an interesting topic which should
Jia-Jun WANG Associate professor at the School of Automation, Hangzhou Dianzi University. He received his Ph. D. degree in power electronics and power drive from Tianjin University in 2003. His research interest covers sliding mode control, backstepping, neural networks and their applications in motion control system. Corresponding author of this paper. E-mail: [email protected] Dong-Liang LIU Associate professor at the School of Automation, Hangzhou Dianzi University. He received his Ph. D. degree in control theory and control engineering from Zhejiang University in 2005. His research interest covers nonlinear control theory, AC servo driver and PV solar inverter. E-mail: [email protected]
Bao-Jun WANG Professor at the Institute of Information Technology, Zhejiang Institute of Communications. He is responsible for the major of network technology of computer. His research interest covers distributed network, operation system, and automation and control system technology. E-mail: [email protected]