Research ArticleStabilizing a Rotary Inverted Pendulum Based onLogarithmic Lyapunov Function

Jie Wen,1 Yuanhao Shi,1 and Xiaonong Lu2

1School of Computer Science and Control Engineering, North University of China, Taiyuan 030051, China2School of Management, Hefei University of Technology, Hefei 230009, China

Correspondence should be addressed to Jie Wen; wenjie015@gmail.com

Received 28 October 2016; Revised 16 January 2017; Accepted 30 January 2017; Published 27 February 2017

Academic Editor: Enrique Onieva

Copyright © 2017 Jie Wen et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The stabilization of a Rotary Inverted Pendulum based on Lyapunov stability theorem is investigated in this paper. The key ofdesigning control laws by Lyapunov controlmethod is the construction of Lyapunov function. A logarithmic function is constructedas the Lyapunov function and is compared with the usual quadratic function theoretically. The comparative results show that theconstructed logarithmic function has higher numerical accuracy and faster convergence speed than the usual quadratic function.On this basis, the control law of stabilizing Rotary Inverted Pendulum is designed based on the constructed logarithmic functionby Lyapunov control method. The effectiveness of the designed control law is verified by experiments and is compared with LQRcontroller and the control law designed based on the quadratic function. Moreover, the system robustness is analyzed when thesystem parameters contain uncertainties under the designed control law.

1. Introduction

The Rotary Inverted Pendulum, which was proposed byFuruta et al. [1], is a well-known test platform to verify thecontrol theories due to its static instability. Rotary InvertedPendulumhas also significant real-life applications, for exam-ple, aerospace vehicles control [2, 3] and robotics [4–6]. Con-sidering the mentioned facts, Rotary Inverted Pendulum isselected as the controlled system and lots of control problemsare concerned by researchers, such as stabilizing the pendu-lum around the unstable vertical position [7–9], swinging thependulum from its hanging position to its upright verticalposition [10–13], and creating oscillations around its unstablevertical position [14, 15].

For the swing-up and stabilizing control of RotaryInverted Pendulum, a variety of control methods had beenapplied. Jose et al. [16] and Akhtaruzzaman and Shafie [17]all used proportional-integral-derivative (PID) and linearquadratic regulator (LQR) to balance the pendulum in itsupright position, while PD cascade scheme was applied tothe switching-up control of the pendulum and fuzzy-PDregulator to the stabilizing control of the pendulum byOltean[18]. Besides, Hassanzadeh andMobayen used particle swarm

optimization (PSO)method to search and tune the controllerparameters of PID in the control of balancing the pendulumin an inverted position [19].

Except these classical control techniques, several advan-ced control methods are also used to design controllersto stabilize the pendulum in upright vertical position. Forexample, Chen andHuang proposed an adaptive controller tobring the pendulum close to the upright position regardlessof the various uncertainties and disturbances [20]. Has-sanzadeh and Mobayen applied evolutionary approacheswhich include genetic algorithms (GA), PSO, and ant colonyoptimization (ACO) methods to balance the pendulum ininverted position [21]. Hassanzadeh et al. also proposed anoptimum Input-Output Feedback Linearization (IOFL) cas-cade controller utilized GA to balance the pendulum in aninverted position, in which GA method was used to searchand tune the controller parameters [22]. Furthermore, thefuzzy logic control [23],𝐻-∞ control [24], and sliding modecontrol [25] are all used to achieve the same control objective.

On the other hand, the inverted system can be under-actuated and nonholonomic systems. Yue et al. used indi-rect adaptive fuzzy and sliding mode control approachesto achieve simultaneous velocity tracking and tilt angle

HindawiJournal of Control Science and EngineeringVolume 2017, Article ID 4091302, 11 pageshttps://doi.org/10.1155/2017/4091302

2 Journal of Control Science and Engineering

stabilization for a nonholonomic and underactuated wheeledinverted pendulum vehicle [26]. Different from [26], someresearchers focus on the study of the general classes of under-actuated and nonholonomic systems. For example, Mobayenproposed a new recursive terminal sliding mode strategy fortracking control of disturbed chained-form nonholonomicsystems whose reference targets are allowed to converge tozero in finite timewith an exponential rate [27].Mobayen alsoproposed a recursive singularity-free, fast terminal slidingmode control method, which is able to avoid the possiblesingularity during the control phase, which is applied fora finite-time tracking control of a class of nonholonomicsystems [28].

Among various control methods, Lyapunov controlmethod is a simple method of designing control laws and canbe applied to linear systems and nonlinear systems. Particu-larly, the analytic expression of designed control laws can beobtained, which help to analyze the control performances andsystem characteristics.Thuswe use Lyapunov controlmethodto achieve stabilizing control of Rotary Invert Pendulum inthis paper. The key point of Lyapunov control method isthe construction of the Lyapunov function. Aguilar-Ibanezet al. [8] and Turker et al. [9] all selected energy functionas Lyapunov function from the quadratic function and usedLyapunov’s direct method to stabilize Rotary Inverted Pen-dulum. Energy function considered the physical system andwas constructed from physical standpoint, which leads to theapplication of constructed energy function confined to theconsidered physical system. For different physical systems,the energy functions are different and need to be constructedrespectively. Instead of energy function, we construct theLyapunov function from mathematical standpoint. Basedon the quadratic function and Taylor series, a logarithmicfunction is constructed and as Lyapunov function, which isapplicable to not only Rotary Inverted Pendulum but alsothe physical systems with the linear or nonlinear models.Based on the logarithmic Lyapunov function, the controllaws of stabilizing Rotary Inverted Pendulum are designedby Lyapunov method in this paper, that is, balancing thependulum in its upright position. In order to describethe properties of the constructed logarithmic function, therelationships between the usual quadratic function and theconstructed logarithmic function are compared in numericalvalue and convergence speed. The comparative results showthat the logarithmic function has higher numerical accuracyand faster convergence speed, which will be also verified inexperiments. Furthermore, the experiment results also showthat the designed control law by Lyapunov method in thispaper can also achieve swing-up control of Rotary InvertedPendulum. Based on these results, the system robustness isanalyzed when the system parameters contain uncertaintiesunder the designed control law. Thus, the main constructionof this paper is to construct a logarithmic function asLyapunov function and prove that the logarithmic functionhas higher numerical accuracy and faster convergence speed.

The remaining of this paper is organized as follows.Section 2 introduces the mathematical model of RotaryInverted Pendulum and describes the control objective. Theconstruction and analysis of logarithmic Lyapunov function

Pendulum

Arm

y

O

r

x

l

A

B

z

𝜃 𝜃

𝛼

��

Figure 1: Schematic diagram of Rotary Inverted Pendulum.

as well as the design of control laws and robustness analysesare shown in Section 3.The experiments in Section 4 are usedto verify the effectiveness of the designed control laws andthe analyzed conclusions of logarithmic Lyapunov functionas well as the system robustness. Lastly, Section 5 concludesthe paper.

2. Mathematical Model of RotaryInverted Pendulum

Rotary Inverted Pendulum is composed of a rotating armwhich is driven by amotor and a pendulummounted on arm’srim, whose structure is shown in Figure 1. The pendulummoves as an inverted pendulum in a plane perpendicular tothe rotating arm. 𝛼 and 𝜃 are employed as the generalizedcoordinates to describe the inverted pendulum system. Thependulum moves a given 𝛼 while the arm rotates with anangle of 𝜃.

By applying Newton method or Lagrange method [29],one can get the nonlinear mathematical model of RotaryInverted Pendulum [24, 30]:

(𝐽eq + 𝑚𝑟2) �� + 𝑚𝑙𝑟 sin (𝛼) ��2 − 𝑚𝑙𝑟 cos (𝛼) ��= 𝑇 − 𝐵𝑞��

43𝑚𝑙2�� − 𝑚𝑙𝑟 cos (𝛼) �� − 𝑚𝑔𝑙 sin (𝛼) = 0,(1)

where 𝑉 is voltage and 𝑇 is torque and given as

𝑇 = 𝜂𝑚𝜂𝑔𝐾𝑡𝐾𝑔𝑉 − 𝐾𝑔𝐾𝑚��𝑅𝑚 . (2)

The parameters in (1) and (2) are described in Table 1, whichis the same as [24].

This paper considers the problem of stabilizing RotaryInverted Pendulum; thus 𝜃 and 𝛼 are all relatively small inmost of the control time. For small 𝜃 and 𝛼, cos(𝛼) ≈ 1

Journal of Control Science and Engineering 3

and sin(𝛼) ≈ 𝛼. Placing the approximate expressions into(1) and solving (1), the state space model of Rotary InvertedPendulum can be written as

�� = 𝐴𝑥 + 𝐵𝑢𝑦 = 𝐶𝑥 + 𝐷𝑢, (3)

where 𝑥 = [𝜃 𝛼 �� ��]𝑇 and 𝑢 is input voltage 𝑉, while

𝐴 =[[[[[[[[[[

0 0 1 00 0 0 10 𝑏𝑑𝐸 −𝑐𝐺𝐸 00 𝑎𝑑𝐸 −𝑏𝐺𝐸 0

]]]]]]]]]]

,

𝐵 =[[[[[[[[[[[

00

𝑐𝜂𝑚𝜂𝑔𝐾𝑡𝐾𝑔𝑅𝑚𝐸𝑏𝜂𝑚𝜂𝑔𝐾𝑡𝐾𝑔𝑅𝑚𝐸

]]]]]]]]]]]

𝐶 = diag (1, 1, 1, 1) ,𝐷 = [0 0 0 0]𝑇

(4)

with

𝑎 = 𝐽eq + 𝑚𝑟2,𝑏 = 𝑚𝑙𝑟,𝑐 = 43𝑚𝑙2,𝑑 = 𝑚𝑔𝑙𝐸 = 𝑎𝑐 − 𝑏2,𝐺 = 𝜂𝑚𝜂𝑔𝐾𝑡𝐾𝑚𝐾2𝑔 + 𝐵𝑅𝑚𝑅𝑚 .

(5)

Obviously, vertically upward position (𝛼 = 0) andvertically downward position (𝛼 = 𝜋) of pendulum are allequilibrium position for an arbitrary fixed 𝜃; that is, system(3) has more than one equilibrium state. In order to solve thisproblem, the states feedback technique, whose block diagramis shown in Figure 2, is used tomake system (3) only have oneequilibrium state (𝛼 = 0, 𝜃 = 0). If the feedback control isnoted as 𝑟, then (3) becomes

�� = (𝐴 − 𝐵𝐾) 𝑥 + 𝐵𝑟 = ��𝑥 + 𝐵𝑟𝑦 = (𝐶 − 𝐷𝐾) 𝑥 + 𝐷𝑟 = ��𝑥 + 𝐷𝑟, (6)

where𝐾 is the gain vector of state feedback and 𝑟 = 𝑢+𝐾𝑥 =𝑉 + 𝐾𝑥.

B

−K

A

C

D

∫××r(t)

u(t)

x(t) y(t)×

Figure 2: Block diagram of states feedback.

In the rest of this paper, we focus on the system model(6) instead of (3). The aim of this paper is to design feedbackcontrol 𝑟 by Lyapunov controlmethod so that Rotary InvertedPendulum can be stabilized at the equilibrium position 𝛼 = 0and 𝜃 = 0 from arbitrary position.The control law 𝑢 or inputvoltage𝑉 can be calculated from 𝑟 easily based on 𝑟 = 𝑢+𝐾𝑥and 𝑢 = 𝑉.

3. Design of Control Laws

For linear control systems, lots of methods can be used todesign control laws. In this paper, the control laws of stabi-lizing Rotary Inverted Pendulum are designed by Lyapunovcontrol method, whose motivations are as follows: (i) theprocedure of designing control laws is simple; (ii) the analyticexpression of control laws can be obtained; (iii) the controllaws designed by Lyapunovmethod can guarantee the systemstability. Based on Lyapunov stability theorem, the controllaws can be designed as follows:

(1) Construct a function 𝑉(𝑥, 𝑡) which satisfies the con-ditions of Lyapunov function; that is, 𝑉(𝑥, 𝑡) is oncecontinuously differentiable for 𝑥, 𝑉(𝑥, 𝑡) ≥ 0 and “=”holds if and only if 𝑥 = 𝑥𝑒 which 𝑥𝑒 is the equilibriumstate.

(2) Calculate the first time derivative of 𝑉(𝑥, 𝑡), that is,��(𝑥, 𝑡).(3) Design control laws tomake ��(𝑥, 𝑡) ≤ 0 and “=” holds

if and only if 𝑥 = 𝑥𝑒.3.1. Construction of Logarithmic Lyapunov Function. In clas-sical literatures [31–33], the quadratic function 𝑉𝑞 is usuallyselected as Lyapunov function; that is,

𝑉𝑞 (𝑥, 𝑡) = 𝑥𝑇𝑃𝑥, (7)

where 𝑃 is a nonnegative matrix.In order to obtain higher numerical accuracy and faster

convergence speed, a logarithmic function𝑉𝑙 from quadraticfunction 𝑉𝑞 and Taylor series is constructed as Lyapunovfunction; that is,

𝑉𝑙 (𝑥, 𝑡) = ln (1 + 𝑥𝑇𝑃𝑥) . (8)

4 Journal of Control Science and Engineering

To illustrate𝑉𝑙 is better than𝑉𝑞, the relationships of numericalvalue and convergence speed between𝑉𝑙 and𝑉𝑞 are shown asfollows:

(i) Numerical value: the Taylor series of 𝑉𝑙(𝑥, 𝑡) is𝑉𝑙 (𝑥, 𝑡) = ln (1 + 𝑥𝑇𝑃𝑥) = ∞∑

𝑛=1

(−1)𝑛+1𝑛 (𝑥𝑇𝑃𝑥)𝑛

= 𝑥𝑇𝑃𝑥 − 12 (𝑥𝑇𝑃𝑥)2 + 13 (𝑥𝑇𝑃𝑥)

3 − ⋅ ⋅ ⋅(9)

fromwhich one can see that𝑉𝑞(𝑥, 𝑡) is the first item ofthe Taylor series of𝑉𝑙(𝑥, 𝑡), while𝑉𝑙(𝑥, 𝑡) contains thequadratic term, cubic term, and higher order termsof 𝑥𝑇𝑃𝑥 relative to𝑉𝑞(𝑥, 𝑡). Thus,𝑉𝑙(𝑥, 𝑡) has a highernumerical accuracy than 𝑉𝑞(𝑥, 𝑡).

(ii) Convergence speed: let𝑋 = 𝑥𝑇𝑃𝑥 ≥ 0; then𝑉𝑞𝑉𝑙 =

𝑋ln (1 + 𝑋) ,

��𝑞��𝑙 = 1 + 𝑋.

(10)

Construct a function 𝑓1(𝑋) as𝑓1 (𝑋) = 𝑉𝑞

��𝑞 −𝑉𝑙��𝑙 =

𝑋(1 + 𝑋) ��𝑙 −

ln (1 + 𝑋)��𝑙

= 1𝑉𝑙 ��1 (𝑋) ,

(11)

where ��1(𝑋) = 𝑋/(1 + 𝑋) − ln(1 + 𝑋).If 𝑓2(Γ) = ��1(𝑋) = (Γ−1)/Γ − ln(Γ)with Γ = 1+𝑋 ≥1, then

��2 (Γ) = −Γ − 1Γ2 < 0 (12)

which means 𝑓2(Γ) is a monotone decreasing func-tion, and

max (𝑓2 (Γ)) = 𝑓2 (1) = 0 = max (��1 (0)) ; (13)

thereby,

��1 (𝑋) ≤ max (��1 (𝑋)) = ��1 (0) = 0. (14)

𝑉𝑙 is the constructed logarithmic Lyapunov function,so ��𝑙(𝑥, 𝑡) ≤ 0 for 𝑥 = 0 through designing controllaws. Taking into account (11), one can get that

𝑓1 (𝑋) > 0 for 𝑋 = 0 (15)

whichmeans𝑉𝑞/��𝑞 > 𝑉𝑙/��𝑙 for 𝑥 = 0; namely𝑉𝑙 hasfaster convergence speed than 𝑉𝑞.

According to the above analysis, 𝑉𝑙 has higher numericalaccuracy and faster convergence speed than𝑉𝑞, so the controllaws of stabilizing Rotary Inverted Pendulum are designedbased on 𝑉𝑙 by Lyapunov control method in this paper.

Moreover, the constructed 𝑉𝑙 can be used in nonlinearsystem due to the applicability of Lyapunov control methodand can be as performance function in optimal control toobtain better control effect due to the higher numericalaccuracy and faster convergence speed. Therefore 𝑉𝑙 has agreater range of applications, not limited to Rotary InvertedPendulum and linear systems.

3.2. Design of Control Laws via LyapunovMethod. In order todesign the control laws under the condition of ��𝑙(𝑥, 𝑡) ≤ 0 atany time, we need to calculate the first-order time derivativeof 𝑉𝑙(𝑥, 𝑡)��𝑙 (𝑥, 𝑡) = 11 + 𝑥𝑇𝑃𝑥 (��𝑇𝑃𝑥 + 𝑥𝑇𝑃��) = 11 + 𝑥𝑇𝑃𝑥⋅ (𝑥𝑇 (��𝑇𝑃 + 𝑃��) 𝑥 + (𝐵𝑇𝑃𝑥 + 𝑥𝑇𝑃𝐵) 𝑟)= 11 + 𝑥𝑇𝑃𝑥 (𝑥𝑇 (��

𝑇𝑃 + 𝑃��) 𝑥 + 2𝑥𝑇𝑃𝐵𝑟) = 𝜉𝑀+ 𝜉𝑁𝑟,

(16)

where 𝑀 = 𝑥𝑇(��𝑇𝑃 + 𝑃��)𝑥 and 𝑁 = 2𝑥𝑇𝑃𝐵, 𝜉 = 1/(1 +𝑥𝑇𝑃𝑥).To ensure ��𝑙(𝑥, 𝑡) ≤ 0, the feedback control is designed

as

𝑟 = 𝑟𝑙 = −𝑀𝑁 − 𝑘 ⋅ 𝜉 ⋅ 𝑁, 𝑘 > 0. (17)

Placing (17) into (16), it is easily got that

��𝑙 (𝑥, 𝑡) = −𝑘 (𝜉𝑁)2 < 0 for 𝑥 = 0. (18)

Similarly, the first-order time derivative of 𝑉𝑞(𝑥, 𝑡) is��𝑞 (𝑥, 𝑡) = ��𝑇𝑃𝑥 + 𝑥𝑇𝑃�� = 𝑀 +𝑁𝑢 (19)

and the control law is designed as

𝑟 = 𝑟𝑞 = −𝑀𝑁 − 𝑘𝑁, 𝑘 > 0 (20)

so that ��𝑞(𝑥, 𝑡) = −𝑘𝑁2 < 0 for 𝑥 = 0.Comparing (17) and (20), the first items of 𝑟𝑙 and 𝑟𝑞 are

the samewhile the second item of 𝑟𝑙 contains parameter 𝜉 and𝑟𝑞 does not contain 𝜉. In Section 4, whether the control law𝑟𝑙 can stabilize Rotary Inverted Pendulum will be verified byexperiments, based on which the control performances of 𝑟𝑙,𝑟𝑞, and a LQR controller will be compared.

3.3. Robustness Analyses. Theactual system is usually affectedby the perturbations from environment or other sources,so that the system parameters contain uncertainties. It isreasonable to request that the designed control laws can resist

Journal of Control Science and Engineering 5

the variation of parameters and the effect of perturbations tothe greatest extent, so the controlled systems need strongerrobustness under the control laws. In this subsection, we willinvestigate the system robustness under the designed controllaw 𝑟𝑙. We focus on the effects of 𝐴 and 𝐵 caused by theperturbations.

If the perturbations make 𝐴 become 𝐴 + Δ𝐴, that is, �� +Δ ��, which lead to that𝑀 becomes𝑀+Δ𝑀, according to (17),the designed control law 𝑟𝑙 becomes

��𝑙 = 𝑀 + Δ𝑀𝑁 + 𝑘𝑁 = 𝑀𝑁 + 𝑘𝑁 + Δ𝑀𝑁 = 𝑟𝑙 + Δ𝑀𝑁 (21)

which means the difference between the real control law��𝑙 and the theoretic control law 𝑟𝑙 is Δ𝑀/𝑁 when theperturbations affect 𝐴.

If the perturbations make 𝐵 become 𝐵 + Δ𝐵, which leadto that 𝑁 becomes 𝑁 + Δ𝑁, according to (17), the designedcontrol law 𝑟𝑙 becomes

��𝑙 = 𝑀𝑁 + Δ𝑁 + 𝑘 (𝑁 + Δ𝑁)= 𝑁𝑁 + Δ𝑁 ⋅

𝑀𝑁 + 𝑘𝑁 + 𝑘Δ𝑁= 𝑀𝑁 + 𝑘𝑁 + 𝑘Δ𝑁(1 − 𝑀

𝑘𝑁(𝑁 + Δ𝑁))(22)

which means the difference between the real control law ��𝑙and the theoretic control law 𝑟𝑙 is 𝑘Δ𝑁(1 −𝑀/𝑘𝑁(𝑁+Δ𝑁))when the perturbations affect 𝐵.

From (21) and (22), the two items of 𝑟𝑙 are all affectedwhen the perturbations affect 𝐵, while only one item of 𝑟𝑙is affected when the perturbations affect 𝐴, which meansthe system robustness is better in the most cases when 𝐴contains uncertainties. However, it should be noted that thereal control law is the same as the theoretic control law when𝑀 = 𝑘𝑁(𝑁 + Δ𝑁), and the system robustness is best in thiscase.

4. Experiments

In this section, the values of parameters will be given and thecharacteristics of Rotary Inverted Pendulumwill be analyzed.Then, three experiments are used to investigate the controllaw 𝑟𝑙:

(i) The initial position of Rotary Inverted Pendulumis set to 𝑥0 = [1 𝜋 1 0]𝑇, which verifies that 𝑟𝑙can make the pendulum from vertically downwardposition to vertically upward position.

(ii) The initial position of Rotary Inverted Pendulum isset to 𝑥0 = [1 1 1 0]𝑇, 𝑟𝑙, 𝑟𝑞, and LQR controller𝑢lqr are used to stabilize Rotary Inverted Pendulum,respectively, and then the control performances of 𝑟𝑙,𝑟𝑞, and 𝑢lqr are compared.

(iii) In order to investigate the system robustness underthe designed control law, let the system parameters

Table 1: Descriptions of physical parameters.

Parameter Description𝐽eq Moment of inertia at the load𝑚 Mass of pendulum arm𝑟 Rotating arm length𝑙 Length to pendulums center of mass𝑔 Gravitational constant𝐵𝑞 Viscous damping coefficient𝐾𝑡 Motor torque constant𝐾𝑔 System gear ratio𝐾𝑚 Back-EMF constant𝑅𝑚 Armature resistance𝜂𝑚 Motor efficiency𝜂𝑔 Gear efficiency

Table 2: Values of physical parameters.

𝐽eq 0.0033𝑙 0.1675𝐾𝑡 0.0077𝑅𝑚 2.6𝑚 0.125𝑔 9.81𝐾𝑔 70𝜂𝑚 0.69𝑟 0.215𝐵𝑞 0.0040𝐾𝑚 0.0077𝜂𝑔 0.90

contain uncertainties; that is, the elements 𝑎𝑖𝑗 in 𝐴 or𝑏𝑘 in 𝐵 become ��𝑖𝑗 = (1 + 𝜀)𝑎𝑖𝑗 and ��𝑘 = (1 + 𝜀)𝑏𝑘where 𝜀 represents the uncertainties.Then, the controlperformances with uncertainties are compared withthe case without uncertainties.

4.1. System Parameter Settings. The values of the param-eters in Table 1 are given in Table 2, which is the sameas in [24]. Substituting the values into 𝐴 and 𝐵 in (3),then the eigenvalues of 𝐴 are {0, −17.8671, 7.5266, −4.9783},which means 𝐴 is singular and system (3) is unstable withmore than one equilibrium state due to zero eigenvalueand positive eigenvalues. The rank of the controllabilitymatrix 𝑈𝑐 = [𝐵 𝐴𝐵 𝐴2𝐵 𝐴3𝐵] is 4, that is, full rank,which means the system is controllable. By applying statefeedback technique, system (3) becomes system (6), whichis controllable and stable and has only one equilibriumstate.

The key of state feedback is to calculate the gain vector𝐾. In this section, the desired closed-loop poles are set to

6 Journal of Control Science and Engineering

0 5 10 15 20 25 30

0

5

Time (a.u.)

−5

−10

−15

−20

−25

𝜃

(a) Control law 𝑢 = 𝑟𝑙 − 𝐾𝑥

0 5 10 15 20 25 30

0

0.5

1

1.5

Time (a.u.)

−0.5

𝛼

(b) Feedback control 𝑟𝑙

0 5 10 15 20 25 30Time (a.u.)

104

100

10−2

10−4

102

𝛼

(c) 𝑉𝑞 = 𝑥𝑇𝑃𝑥

0 5 10 15 20 25 30

0

10

Time (a.u.)

−10

−20

−30

−40

𝜃𝛼

𝜃��

(d) System state 𝑥

Figure 3: Results of stability control for vertically downward initial state.

{−1, −2, −3 + 3𝑖, −3 − 3𝑖}; then the closed-loop characteristicequation is

𝑓 (𝑠) = (𝑠 + 1) (𝑠 + 2) (𝑠 + 3 − 3𝑖) (𝑠 + 3 + 3𝑖)= 𝑠4 + 9𝑠3 + 38𝑠2 + 66𝑠 + 36 (23)

so

𝑓 (𝐴) = 𝐴4 + 9𝐴3 + 38𝐴2 + 66𝐴 + 36𝐼. (24)

Then, the gain vector 𝐾 can be calculated as [34]

𝐾 = [0 0 0 1]𝑈−1𝑐 𝑓 (𝐴)= [−0.0307 4.7360 −0.6264 0.4086] . (25)

4.2. Stability Control for Vertically Downward Initial Position.The initial state of Rotary Inverted Pendulum is set to 𝑥0 =[1 𝜋 1 0]𝑇, and the final state is 𝑥𝑓 = [0 0 0 0]𝑇; that is,the initial position of pendulum is vertically downward andthe final position is vertically upward position. The controllaw 𝑟𝑙 in (17) is used to make the pendulum from vertically

downward position to vertically upward position and holdssteady in vertically upward position. The parameters of 𝑟𝑙 are

𝑘 = 0.1,

𝑃 =[[[[[[

1.6090 −8.5878 0.7694 −1.4376−8.5878 147.8866 −11.3186 25.14130.7694 −11.3186 0.9806 −1.9023−1.4376 25.1413 −1.9023 4.4263

]]]]]], (26)

where𝑃 is calculated by solving the Lyapunov equation𝐴𝑇𝑃+𝑃𝐴 = −𝐼. The experiment results are shown in Figure 3,in which Figures 3(a)–3(d) are the curves of control law 𝑢,feedback control 𝑟, function 𝑉𝑞 = 𝑥𝑇𝑃𝑥, and system state 𝑥,respectively.

The results in Figures 3(a) and 3(b) show that 𝑢 and 𝑟 alltend to zero, while the Lyapunov function 𝑉𝑞 in Figure 3(c)is less than 10−3 which is close to zero. One can see fromFigure 3(d) that the system state converges to zero underthe designed control, that is, achieving the control task.Integrating all results, the position of pendulum is fromvertically downward position to vertically upward position

Journal of Control Science and Engineering 7

0 5 10 15

0

5

Time (a.u.)

−5

−10

−15

𝜃𝛼

𝜃��

(a) System state 𝑥 under 𝑟𝑙

10 15Time (a.u.)

0

2

0 5

−2

−4

−6

−8

−10

−12

𝜃 uq𝛼 uq𝜃 uq�� uq

𝜃 ulqr𝛼 ulqr𝜃 ulqr�� ulqr

(b) System state 𝑥 under 𝑟𝑞 and 𝑢lqr

10 15Time (a.u.)

0

5

0 5

0

50

−5

−10

−50

−100

Based on xTPx

Based on LQR

Based on log(1 + xTPx)

(c) Control laws

0500

10001500200025003000350040004500

10 15Time (a.u.)

0 5

Based on xTPx

Based on LQR

Based on log(1 + xTPx)

(d) Energies of control laws

10 15Time (a.u.)

0 5

104

100

10−2

10−4

10−6

10−8

102

Based on xTPx

Based on LQR

Based on log(1 + xTPx)

(e) 𝑉𝑞 = 𝑥𝑇𝑃𝑥

Figure 4: Results of comparative experiments.

under the control law, which realizes not only stability controlbut also swing-up control.

4.3. Comparison of 𝑟𝑙, 𝑟𝑞, and 𝑢𝑙𝑞𝑟 in Stability Control. Theinitial state of Rotary Inverted Pendulum is set to 𝑥0 =

[1 1 1 0]𝑇, while the final state is vertically upwardposition; that is, 𝑥𝑓 = [0 0 0 0]𝑇. In this subsection, theparameters 𝑘 in control laws 𝑟𝑙 and 𝑟𝑞 are 0.1 and 0.0009,respectively, and the parameters 𝑃 in 𝑟𝑙 and 𝑟𝑞 are all identicalwith 𝑃 in (26). The LQR controller 𝑢lqr is designed based on

8 Journal of Control Science and Engineering

the cost functional [24, 35]

𝐽= ∫∞0(𝑥𝑓 − 𝑥 (𝑡))𝑇𝑄(𝑥𝑓 − 𝑥 (𝑡)) + 𝑢 (𝑡)𝑇 𝑅𝑢 (𝑡) 𝑑𝑡

= ∫∞0𝑥 (𝑡)𝑇𝑄𝑥 (𝑡) + 𝑢 (𝑡)𝑇 𝑅𝑢 (𝑡) 𝑑𝑡,

(27)

where 𝑥𝑓 = [0 0 0 0]𝑇.The control laws 𝑟𝑙, 𝑟𝑞, and 𝑢lqr are all used to stabilize

Rotary Inverted Pendulum, and the results are shown inFigure 4, in which Figure 4(a) is the curves of system statesunder 𝑟𝑙; Figure 4(b) is the curves of system states under 𝑟𝑞and 𝑢lqr; Figures 4(c) and 4(d) are the curves of control laws𝑢 and energies ∫𝑡𝑓

0𝑢2𝑑𝑡, respectively, while the curves of𝑉𝑞 =𝑥𝑇𝑃𝑥 under the three control laws are given in Figure 4(e).

From Figures 4(a) and 4(b), one can see that 𝑟𝑙, 𝑟𝑞,and 𝑢lqr can all stabilize Rotary Inverted Pendulum, butthe convergence time of system states under 𝑟𝑙 is less thanthat under 𝑟𝑞 and 𝑢lqr. The comparison result of 𝑟𝑙 and𝑟𝑞 is consistent with the theoretical analysis in Section 3.1.The results in Figures 4(c) and 4(d) show that the controlamplitude and energy consumption of 𝑢lqr are the maximumin the three control laws. The control amplitudes of 𝑟𝑙 and𝑟𝑞 are close while the energy consumption of 𝑟𝑙 is even less,which mean 𝑟𝑙 can stabilize Rotary Inverted Pendulum fasterwith less energy. It should be noted that if𝑄 and 𝑅 in (27) areset as other values, the energy consumption of 𝑢lqr may beless than that of 𝑟𝑙, but the control amplitude of 𝑢lqr is morethan that of 𝑟𝑙. Considering the control amplitude and energyconsumption comprehensively, 𝑟𝑙 is the more recommendedcontrol law to stabilize Rotary Inverted Pendulum. In otherwords, the control law 𝑟𝑙 has the same and even better controleffect than LQR controller. According to the results in [24],the control performance of LQR controller is similar to PIDcontroller and better than𝐻 infinity controller in the stabilitycontrol of Rotary Inverted Pendulum. The designed controllaw 𝑟𝑙 has the same comparative result as LQR controllerbased on the above analyses. In addition, Rojas-Moreno etal. used a FO (Fractional Order) based-LQR controller tostabilize Rotary Inverted Pendulum in [35], from which theFO LQR-based controller has the similar control time andcontrol accuracy to LQR controller but more ability to rejectdisturbances, which is the advantage of the FO LQR-basedcontroller. Instead of better robustness, the designed controllaw 𝑟𝑙 in this paper can ensure the system stability, which isthe property of Lyapunov control method.

In Figure 4(e), the function 𝑉𝑞 under 𝑟𝑙 converges fasterand has higher accuracy than that under 𝑟𝑞, which are alsoconsistent with the results in Figures 4(a) and 4(b) andthe analysis in Section 3.1. When the time is less than 5,the convergence speed of 𝑉𝑞 under 𝑢lqr is faster than thatunder 𝑟𝑙, while the convergence speed of 𝑉𝑞 under 𝑟𝑙 is fasterwhen the time is more than 5. The accuracy comparisonresults have the same conclusion. Comparing Figure 4(c)withFigure 3(a), the control amplitude in Figure 3(a) is more thanthat in Figure 4(c) and the control time in Figure 3(a) is

0 5 10 150

0.2

0.4

0.6

0.8

1

Time (a.u.)

Figure 5: Curve of 𝜉 in control law 𝑟𝑙.

more than that in Figure 4(c), which means swing-up controlrequests more control amplitude. Simultaneously, comparingFigure 4(a) with Figure 3(d), the curves of system state inFigure 4(a) converge to zero before time 10 while that inFigure 3(d) converges to zero after time 25, which mean thecontrol process containing swing-up control takesmore time.In particular, the curve of parameter 𝜉which is the differencebetween 𝑟𝑙 and 𝑟𝑞 is shown in Figure 5, fromwhich one can seethat 𝜉 is time-dependent and from 0 to 1 in control process. If𝜉 = 1, 𝑟𝑙 = 𝑟𝑞, which means control law 𝑟𝑙 has more flexibilitythan control law 𝑟𝑞.4.4. Robustness Experiments. If 𝜀 represents the uncertainties,let ��𝑖𝑗 = (1 + 𝜀)𝑎𝑖𝑗 and ��𝑘 = (1 + 𝜀)𝑏𝑘 with 𝑖, 𝑘 = 3, 4 and𝑗 = 2, 3, where 𝑎𝑖𝑗 and 𝑏𝑘 represent the elements 𝐴(𝑖, 𝑗) and𝐵(𝑘) in matrices 𝐴 and 𝐵 without uncertainties, respectively,while ��𝑖𝑗 and ��𝑘 represent the corresponding elements in 𝐴and 𝐵 with uncertainties, respectively. Under the setting, thecontrol law 𝑟𝑙 is applied to Rotary Inverted Pendulum, and thecurves of𝑉𝑞 in different cases are shown in Figure 6, in whichFigures 6(a)–6(e) show the curves of 𝑉𝑞 in the cases of ��32 =(1+𝜀)𝑎32, ��33 = (1+𝜀)𝑎33, ��43 = (1+𝜀)𝑎43, 𝑏3 = (1+𝜀)𝑏3, and��4 = (1 + 𝜀)𝑏4 with 𝜀 ∈ {0, ±0.02, ±0.04, ±0.06, ±0.08, ±0.1},respectively. The curves of 𝑉𝑞 = 𝑥𝑇𝑃𝑥 in the case of ��42 =(1+ 𝜀)𝑎42 are similar to that in the case of ��32 = (1+ 𝜀)𝑎32 anddo not show in Figure 6.

From Figure 6, the analyses are drawn as follows:

(1) Control performances are better in the cases of 𝑎32,𝑎33, 𝑎42, and 𝑎43 with uncertainties than those 𝑏3 and𝑏4 with uncertainties, which is consistent with therobustness analyses in Section 3.3.

(2) For the case of 𝑎33, when 𝜀 > 0, the controlperformances are worse than that when 𝜀 = 0, whilethe convergence speed is faster at first and slowerthan that when 𝜀 = 0; when 𝜀 < 0, the controlperformances are better than that when 𝜀 = 0, while𝑉𝑞 is not monotone convergence.

(3) For the case of 𝑎43, the control performances of 𝑟𝑙 andthe convergence performances of 𝑉𝑞 when 𝜀 > 0 and𝜀 < 0 are the opposite of that in the case of 𝑎33.

Journal of Control Science and Engineering 9

100

10−5

10−10

10 15Time (a.u.)

0 5

𝜖 = 0

𝜖 = 0.02𝜖 = 0.04

𝜖 = 0.06

𝜖 = 0.08

𝜖 = 0.1

𝜖 = −0.02

𝜖 = −0.04𝜖 = −0.06

𝜖 = −0.08

𝜖 = −0.1

(a) ��32 = (1 + 𝜀)𝑎32

100

10−20

10−10

1510Time (a.u.)

0 5

𝜖 = 0

𝜖 = 0.02𝜖 = 0.04

𝜖 = 0.06

𝜖 = 0.08

𝜖 = 0.1

𝜖 = −0.02

𝜖 = −0.04𝜖 = −0.06

𝜖 = −0.08

𝜖 = −0.1

(b) ��33 = (1 + 𝜀)𝑎33

100

10−20

10−10

10 15Time (a.u.)

0 5

𝜖 = 0

𝜖 = 0.02𝜖 = 0.04

𝜖 = 0.06

𝜖 = 0.08

𝜖 = 0.1

𝜖 = −0.02

𝜖 = −0.04𝜖 = −0.06

𝜖 = −0.08

𝜖 = −0.1

(c) ��43 = (1 + 𝜀)𝑎43

100

10−20

10−10

1510Time (a.u.)

0 5

𝜖 = 0

𝜖 = 0.02𝜖 = 0.04

𝜖 = 0.06

𝜖 = 0.08

𝜖 = 0.1

𝜖 = −0.02

𝜖 = −0.04𝜖 = −0.06

𝜖 = −0.08

𝜖 = −0.1

(d) ��3 = (1 + 𝜀)𝑏3

100

10−20

10−10

1510Time (a.u.)

0 5

𝜖 = 0

𝜖 = 0.02𝜖 = 0.04

𝜖 = 0.06

𝜖 = 0.08

𝜖 = 0.1

𝜖 = −0.02

𝜖 = −0.04𝜖 = −0.06

𝜖 = −0.08

𝜖 = −0.1

(e) ��4 = (1 + 𝜀)𝑏4

Figure 6: Curves of 𝑉𝑞 in different cases.

10 Journal of Control Science and Engineering

(4) For the case of 𝑏3, when 𝜀 > 0, the control perfor-mances are better than that when 𝜀 = 0, while theconvergence speed is lower at first and then fasterthan that when 𝜀 = 0; when 𝜀 < 0, the controlperformances of 𝑟𝑙 are worse than that when 𝜀 = 0,while 𝑉𝑞 is not monotone convergence.

(5) For the case of 𝑏4, the control performances of 𝑢𝑙 andthe convergence performances of 𝑉𝑞 when 𝜀 > 0 and𝜀 < 0 are the opposite of that in the case of 𝑏3

Therefore, when the different system parameters containuncertainties, the system robustness is also different underthe control laws designed by Lyapunov control method. Thebetter system robustness is in the following two cases:

(1) The actual �� is more than the theoretical ��; that is, thecases of 𝑎33 and 𝑏3 contain uncertainties with 𝜀 > 0.

(2) The actual �� is less than the theoretical ��; that is, thecases of 𝑎43 and 𝑏4 contain uncertainties with 𝜀 < 0.

These are consistent with intuition and experience. If theanalytical results can be considered in designing control laws,the designed control laws are expected to have better controlperformance.

5. Conclusions

The control law of stabilizing Rotary Inverted Pendulumis designed based on Lyapunov stability theorem. A loga-rithmic function is constructed as the Lyapunov function,based on which the control law is designed by Lyapunovcontrol method. In particular, the relationships between theconstructed logarithmic function and the usual quadraticfunction in numerical value and convergence speed are ana-lyzed in theory. The results show that the logarithmic func-tion has higher numerical accuracy and faster convergencespeed than the quadratic function, which is also verified inexperiments. Moreover, the control law designed can alsoachieve the swing-up control of Rotary Inverted Pendulum.On this basis, the system robustness when different systemparameters contain uncertainties is investigated. The furtherworks can be considered as follows: (1) Lyapunov controlmethod is applicable to nonlinear system, so the nonlinearmathematical model of Rotary Inverted Pendulum, which ismore accurate to describe system characteristics than linearmodel, can be considered as the controlled system, and thestabilizing control laws can be designed by Lyapunovmethod;(2) the constructed logarithmic function in this paper canbe used in other control methods, such as the logarithmicfunction which was selected as the performance function inoptimal control; (3) the research results for the general classesof the underactuated and nonholonomic systems, such as[27, 28], are applied to Rotary Inverted Pendulum systems.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by 333 Talent Project of NorthUniversity of China andNatural Science Foundation ofNorthUniversity of China (no. XJJ2016032).

References

[1] K. Furuta, M. Yamakita, and S. Kobayashi, “Swing up controlof inverted pendulum,” in Proceedings of the International Con-ference on Industrial Electronics, Control and Instrumentation(IECON ’91), pp. 2193–2198, October-November 1991.

[2] F. Grasser, A. D’Arrigo, S. Colombi, and A. C. Rufer, “JOE: amobile, inverted pendulum,” IEEE Transactions on IndustrialElectronics, vol. 49, no. 1, pp. 107–114, 2002.

[3] N. Phuong, H. Loc, and T. Thuan, “Control of two wheeledinverted pendulum using silding mode technique,” Interna-tional Journal of Engineering Research and Applications, vol. 3,no. 3, pp. 1276–1282, 2013.

[4] N. Muskinja and B. Tovornik, “Swinging up and stabilizationof a real inverted pendulum,” IEEE Transactions on IndustrialElectronics, vol. 53, no. 2, pp. 631–639, 2006.

[5] R. Ortega, M.W. Spong, F. Gomez-Estern, and G. Blankenstein,“Stabilization of a class of underactuated mechanical systemsvia interconnection and damping assignment,” IEEE Transac-tions on Automatic Control, vol. 47, no. 8, pp. 1218–1233, 2002.

[6] H. Mei and Z. He, “Study on stability control for single linkrotary inverted pendulum,” in Proceedings of the InternationalConference on Mechanic Automation and Control Engineering(MACE ’10), pp. 6127–6130, IEEE, Wuhan, China, June 2010.

[7] S. Nair and N. E. Leonard, “A normal form for energy shaping:application to the furuta pendulum,” in Proceedings of the41st IEEE Conference on Decision and Control, pp. 516–521,December 2002.

[8] C. Aguilar-Ibanez, M. S. Suarez-Castanon, and O. O. Gutierres-Frias, “The direct Lyapunov method for the stabilisation of theFuruta pendulum,” International Journal of Control, vol. 83, no.11, pp. 2285–2293, 2010.

[9] T. Turker, H. Gorgun, and G. Cansever, “Lyapunov’s directmethod for stabilization of the Furuta pendulum,” TurkishJournal of Electrical Engineering and Computer Sciences, vol. 20,no. 1, pp. 99–110, 2012.

[10] K. J. A∘strom, J. Aracil, and F.Gordillo, “A family of smooth con-trollers for swinging up a pendulum,”Automatica, vol. 44, no. 7,pp. 1841–1848, 2008.

[11] I. Fantoni and R. Lozano, “Stabilization of the Furuta pendulumaround its homoclinic orbit,” International Journal of Control,vol. 75, no. 6, pp. 390–398, 2002.

[12] V. Sukontanakarn andM. Parnichkun, “Real-time optimal con-trol for rotary inverted pendulum,”American Journal of AppliedSciences, vol. 6, no. 6, pp. 1106–1115, 2009.

[13] P. X. La Hera, L. B. Freidovich, A. S. Shiriaev, and U. Mettin,“New approach for swinging up the furuta pendulum: theoryand experiments,” Mechatronics, vol. 19, no. 8, pp. 1240–1250,2009.

[14] A. S. Shiriaev, L. B. Freidovich,A. Robertsson, R. Johansson, andA. Sandberg, “Virtual-holonomic-constraints-based design ofstable oscillations of Furuta pendulum: theory and experi-ments,” IEEE Transactions on Robotics, vol. 23, no. 4, pp. 827–832, 2007.

Journal of Control Science and Engineering 11

[15] L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, “Generatingself-excited oscillations via two-relay controller,” IEEE Transac-tions on Automatic Control, vol. 54, no. 2, pp. 416–420, 2009.

[16] A. Jose, C. Augustine, S. M. Malola, and K. Chacko, “Perfor-mance study of PID controller and LQR technique for invertedpendulum,”World Journal of Engineering and Technology, vol. 3,no. 2, pp. 76–81, 2015.

[17] M.Akhtaruzzaman andA.A. Shafie, “Modeling and control of arotary inverted pendulum using various methods, comparativeassessment and result analysis,” in Proceedings of the IEEE Inter-national Conference on Mechatronics and Automation (ICMA’10), pp. 1342–1347, IEEE, Xi’an, China, August 2010.

[18] S. E. Oltean, “Swing-up and stabilization of the rotationalinverted pendulum using PD and fuzzy-PD controllers,” Pro-cedia Technology, vol. 12, pp. 57–64, 2014.

[19] I. Hassanzadeh and S. Mobayen, “PSO-based controller designfor rotary inverted pendulum system,” Journal of Applied Sci-ences, vol. 8, no. 16, pp. 2907–2912, 2008.

[20] Y.-F. Chen and A.-C. Huang, “Adaptive control of rotaryinverted pendulum system with time-varying uncertainties,”Nonlinear Dynamics. An International Journal of NonlinearDynamics and Chaos in Engineering Systems, vol. 76, no. 1, pp.95–102, 2014.

[21] I. Hassanzadeh and S. Mobayen, “Controller design for rotaryinverted pendulum system using evolutionary algorithms,”Mathematical Problems in Engineering, vol. 2011, Article ID572424, 17 pages, 2011.

[22] I. Hassanzadeh, S. Mobayen, and A. Harifi, “Input-output feed-back linearization cascade controller using genetic algorithmfor rotary inverted pendulum system,” American Journal ofApplied Sciences, vol. 5, no. 10, pp. 1322–1328, 2008.

[23] M.A.Khanesar,M. Teshnehlab, andM.A. Shoorehdeli, “Slidingmode control of rotary inverted pendulum,” in Proceedings ofthe 15th Mediterranean Conference on Control & Automation,pp. 1–6, July 2007.

[24] V. Sirisha and A. S. Junghare, “A comparative study of con-trollers for stabilizing a rotary inverted pendulum,” Interna-tional Journal of Chaos, Control, Modelling and Simulation, vol.3, no. 1/2, pp. 1–13, 2014.

[25] J. J. Choi and J. S. Kim, “Robust control for rotational invertedpendulums using output feedback sliding mode controller anddisturbance observer,” KSME International Journal, vol. 17, no.10, pp. 1466–1474, 2003.

[26] M.Yue, C.An, Y.Du, and J. Sun, “Indirect adaptive fuzzy controlfor a nonholonomic/underactuated wheeled inverted pendu-lum vehicle based on a data-driven trajectory planner,” FuzzySets and Systems, vol. 290, pp. 158–177, 2016.

[27] S. Mobayen, “Finite-time tracking control of chained-formnonholonomic systems with external disturbances based onrecursive terminal sliding mode method,” Nonlinear Dynamics,vol. 80, no. 1-2, pp. 669–683, 2015.

[28] S. Mobayen, “Fast terminal sliding mode tracking of non-holonomic systems with exponential decay rate,” IET ControlTheory & Applications, vol. 9, no. 8, pp. 1294–1301, 2015.

[29] A. Wu, X. Zhang, and Z. Zhang, “A control system based onthe lagrange modeling method for a single link rotary invertedpendulum,” Engineering Sciences, vol. 7, no. 10, pp. 11–15, 2005.

[30] J. Irfan, J. Rehan, J. Zhao, J. Rizwan, and S. Abdus, “Mathemati-cal model analysis and control algorithms design based on statefeedback method of rotary inverted pendulum,” InternationalJournal of Research in Engineering & Technology, vol. 1, no. 3,pp. 41–50, 2013.

[31] I. R. Petersen and C. V. Hollot, “A Riccati equation approach tothe stabilization of uncertain linear systems,” Automatica, vol.22, no. 4, pp. 397–411, 1986.

[32] E. J. Davison and E. M. Kurak, “A computational methodfor determining quadratic Lyapunov functions for non-linearsystems,” Automatica, vol. 7, pp. 627–636, 1971.

[33] I. R. Petersen, “A stabilization algorithm for a class of uncertainlinear systems,” Systems andControl Letters, vol. 8, no. 4, pp. 351–357, 1987.

[34] K. Xie, Modern Control Theory, Tsinghua University Press,Beijing, China, 2007.

[35] A. Rojas-Moreno, J. Hernandez-Garagatti, O. P.-D. L. Vega, andL. Lopez-Lozano, “FO based-LQR stabilization of the rotaryinverted pendulum,” in Proceedings of the 28th Chinese Controland Decision Conference (CCDC ’16), pp. 4292–4297, IEEE,Yinchuan, China, May 2016.

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