Furuta Sliding Control

MR5036-NLAC FINAL PROJECT 1

Nonlinear Control of the Furuta PendulumIsmael Minchala Avila, Student Member, IEEE, Ruben Marban Romero, MAT09 Student, ITESM

Abstract—This paper presents three different control systemsfor the Furuta Pendulum. Both linear and nonlinear techniquesare used, resulting in the following control techniques: statefeedback with LQR design for gain G, sliding control mode and asliding adaptive controller. Details on design and implementationin Matlab are discused, as well as results and comparisons onthe performance of every system.

Index Terms—Nonlinear, Linear, Control, Pendulum, Sliding,Adaptive.

I. INTRODUCTION

THE Furuta Pendulum is a classic nonlinear system usedfor research and innovation in control systems due to its

static instability. In figure 1 it’s shown the basic configurationof this rotatory inverted pendulum, which is composed of thefollowing components: a rotating arm, which is driven by amotor with a pendulum mounted on its rim. The pendulummoves as an inverted pendulum in a plane perpendicular to therotating arm. Essentially the control of this system is difficultbecause this system has two degrees of freedom and onlyone control input (sub-actuated system). Such systems restrictapplication of nonlinear controller design techniques, such asfeedback linearization and sliding mode controller design [1].

In this paper three different approaches are presented inorder to maintain the pendulum in vertical position and therotatory arm in a fixed position; the first control systemproposed is a linear state feedback using a linearized model forthe desing and the LQR algorithm from Matlab for obtainingthe gain vector, the second design corresponds to a slidingmode controller using the nonlinear model of the pendulumand finally a sliding adaptive controller is proposed. All theresults are compared to find the best performance possiblefrom these three control approaches for this system.

This paper is organized as the following: section II in-troduces the Furuta Pendulum model. In section III, statefeedback control design for the inverted pendulum is discussedand also simulations results are provided there. Section IVintroduces some basics of VSC control using sliding mode. Insection V the sliding mode control of the Furuta Pendulum isconsidered and simulation results are discussed as well. Finallyadaptive control of rotary inverted pendulum is threaded insection VI.

II. THE FURUTA PENDULUM

A mathematical model of the Furuta Pendulum is presentedin this section. The model is based on the Lagrange Theory,as its shown in [5], only the results are given here, equations(1) and (3). Consider the Furuta pendulum in figure 1. Letthe length of the pendulum be l, the mass of the weight M ,the mass of the pendulum m, its moment of inertia J and themoment of inertia for the arm Jp. The length of the arm is r.

The angle of the pendulum, θ, is defined to be zero when inupright position and positive when the pendulum is movingclockwise. The angle of the arm, ϕ is positive when the armis moving in counter clockwise direction. Further, the centralvertical axis is connected to a DC motor which adds a torqueproportional to the control signal u.

Fig. 1. Schematic picture of the Furuta pendulum.

(Jp +Ml2

) (θ − ϕ2 sin θ cos θ

)+Mrlϕ cos θ − gl

(M +

m

2

)sin θ = 0

Mrlθ cos θ −Mrlθ2 sin θ + 2(Jp +ml2

)θϕ sin θ cos θ +[

J +mr2 +Mr2 +(Jp +ml2

)sin2 θ

]ϕ = u

(1)

And assuming that:

a = Jp +Ml2 b = J +Mr2 +mr2

c = Mrl d = lg(M +

m

2

)(2)

We can rewrite equations (1) as the following:

aθ − aϕ2 sin θ cos θ + cϕ cos θ − d sin θ = 0

cθ cos θ − cθ2 sin θ + 2aθϕ sin θ cos θ +(b+ a sin2 θ

)ϕ = u

(3)

The coefficients that will be used for the design and sim-ulation are: l = 0.413[m], r = 0.235[m], M = 0.01[kg],J = 0.05[kgm2], Jp = 9× 10−4[kgm2] and m = 0.02[kg].


III. STATE FEEDBACK CONTROL

To be able to stabilize the pendulum in the upright position,we need a controller. The most commonly used method todesign a controller is to linearize the nonlinear model and forma linear or nonlinear controller for this model. To do this, weintroduce the state x =

[θ θ ϕ ϕ

]and linearization of the

system (1) around x = [0 0 0 0], which is the uprightposition of the pendulum with zero velocity gives:

x = Ax + Bu

x =

0 1 0 0bd

ab−c2 0 0 0

0 0 0 1− cd

ab−c2 0 0 0

x +

0

− cgab−c2

0ag

ab−c2

u (4)

And reemplazing the coefficients that will be used for thedesign in (4), we have:

x =

0 1 0 0

31.3167 0 0 00 0 0 1

−0.5884 0 0 0

x +

0

−71.2340

191.246

u (5)

The control law can be written:

u = −Kx (6)

Where K is a feeback gain for pole placement in the closedloop dynamics:

x = (A−BK)x (7)

Several techniques for determining K can be used, althoughone of the easiests and more efficients is optimal controlthrough LQR design technique using Matlab. Proponing thefollowing design matrices:

Q =

100 0 0 00 1 0 00 0 10 00 0 0 1

R = 1

We obtain the feedback gain vector K:

K = [−6.2234 − 1.1055 − 0.3162 − 0.2320] (8)

Using the same procedure mentioned above, we’ve obtainedthe following gain K = [−7.5343 − 1.3465 0 − 0.2216].Now let’s analyze the closed loop dynamics of the system.

x = (A−BK)x

|λI− (A−BK)| = 0

s4 + 53.5413s3 + 505.348s2 + 1318.21s = 0

s1 = 0

s2 = −42.3415

s3 = −6.07517

s4 = −5.12461

Fig. 2. Schematic picture of the Furuta pendulum.

As we can see, one of the states doesn’t have its rootin the left side of the complex plane s, and of course thiswill represent instability in the system. Because of θ is thesystem response and all the other states depend on its response,definitely this one won’t be unstable and as a consequenceθ will be stable as well. Therefore the state that definitelywon’t stabilize will be ϕ, that represents the position of thearm that is linked to a DC motor, whose main function isto maintain the pendulum in vertical position. In figure 2 it’sshown the graphical programming in Simulink for simulatingthe pendulum’s state feedback control system. It’s easy tonote from figure 3 that one of the states doesn’t stabilize,just as we implied before, that’s clearly identified as the stateϕ. Another simulation was performed in order to compareour first controller, for this case we use the feedback vectorgain obtained in (8), for which the poles are: s1 = −20.042,s2 = −7.3344, s3 = −3.5 + 0.73i, s4 = −3.5− 0.73i. As wecan see, none of the states is unstable, but all of them in thesimulation are oscillating around the equilibrium point, figure4.

Fig. 3. Control system response, K = [−7.53 − 1.3465 0 − 0.2216].

IV. SLIDING MODE CONTROL

Variable Structure Control (VSC) as is also known SlidingControl, basically consists of a high-speed switching controllaw to drive nonlinear plant’s state trajectory onto a specifiedsurface in the state space and to maintain the plant’s state


Fig. 4. Control system response, K = [−6.223 −1.1055 −0.3162 −0.232].

trajectory on this surface for all subsequent time. Particularly,once the plant’s states reach the surface, all the system’strajectories remain on this surface while every state slides untilit reaches its desired value. On other words, from La Salletheorem, we can state that the surface represents an invariantset, and therefore asymptotic stability will be guaranteed if acorrect sliding surface is chosen.

This surface is called the switching surface because if thestate trajectory of the plant is “above” the surface a controlpath has one gain and a different gain if the trajectory drops“below” the surface. The plant dynamics restricted to thissurface represent the controlled system’s behavior [3]. Thereare two main steps to design a sliding mode controller; thefirst step consists on selecting the sliding surface such that thesystem exhibits the desired behavior in the sliding mode andthe second step is to determine control laws to guarantee thereaching and sliding mode conditions [2]. In order to obtaina control law which allow the plants states to remain on thesurface, s = 0, its a common practice to define the followingpositive definite Lyapunov function V(s):

V (t) =1

2s2 (9)

If we are able to assure that the derivative of V (s) isnegative definite, so the system’s stability will be granted,V (s) = ss ≤ 0. Let’s assume now, a second order systemwith the form, x = f (x, x, t) + bu(t). The control law (10) isan option for the design of a sliding mode controller. There aretwo well defined parts; the first one is regarded with equivalentcontrol, ueq , which will be the responsible for keeping s = 0,and on the other hand the discontinuous term usw which isthe commutation control will be responsible for bringing theplants states into the sliding surface.

u = ueq + usw = ueq − ηsgn(s)− κs (10)

The variable s in equation (10) is called the commutationfunction or sliding surface, and for a second order system isdefined as [2]:

s = e+ λe

e = x− xd (11)

V. SLIDING MODE CONTROLLER DESIGN

From equation (3) we can obtain the following state spacerepresentation:

x =[θ θ ϕ ϕ

]Tx1 = x2

x2 = F1(x) +B1(x)u (12)x3 = x4

x4 = F2(x) +B2(x)u

y(t) = [x1 x3]T

F1(x) =

d sin x1

[a2 sin2 x1+ab

]a

− c2x22

cos x1 sin x1

+x24

cos x1 sin x1

[a2 sin2 x1 + ab

]+ 2acx2x4 cos2 x1 sin x1

a2 sin2 x1 + ab − c2 cos2 x1

B1(x) = −c cos x1


F2(x) =2 sin x1a2x2x4 cos x1 − c sin x1ax2

4cos2 x1 + cd sin x1 cos x1


B2(x) = −a


Its possible to threat the states pair (x1, x2) and (x3, x4)as the states of two subsystems with canonical form. Thecontrol objective is to design a control law for controllingsimultaneously the states (x1, x2) and (x3, x4). Therefore itsnecessary to propound two sliding surfaces:

s1 = e1 + λ1e1 = x2 + λ1x1

s2 = e2 + λ2e2 = x4 + λ2x3 (13)

Since the desired trajectory, and even point of operation isx = [0 0 0 0], e = x. The equivalent control law for everysubsystem is:

s1 = x2 + λ1x1 = 0

ueq1 = −F1](x) + λ1x2B1(x)

(14)

s2 = x4 + λ2x4 = 0

ueq2 = −F2](x) + λ2x3B2(x)

In order to control the two system responses simultaneously,its necessary to define the following control law:

u = ueq1 + ueq2 + usw (15)

Finally, the sliding surface for the system is defined as theweighted sum of the surfaces s1 and s2:

S = αs1 + βs2 (16)

Now lets find the switching control law through Lyapunovsstability theorem.


V (t) =1

2S2

V (t) = S S = S (αs1 + βs2)

V (t) = S [α (x2 + λ1x1) + β (x4 + λ2x3)]

V (t) = S

[α (F1 +B1 (ueq1 + ueq2 + usw) + λ1x2) +β (F2 +B2 (ueq1 + ueq2 + usw) + λ2x3)

]V (t) = S [αB1(x) (ueq2 + usw) + βB2(x) (ueq1 + usw)]

αB1(x)ueq2+βB2(x)ueq1+usw (αB1(x) + βB2(x)) = −ηsgn(S)−κS

usw = −αB1(x) + βB2(x) + ηsgn(S) + κS

αB1(x) + βB2(x)(17)

Therefore, V (t) will be negative definite, thus guaranteeingsystems stability.

V (t) = S [−ηsgn(S)− κS]

V (t) = −η|S| − κS2 (18)

Implementing equation (15) as the control law for thesliding mode controller and using Simulink as the plataformof development, we’ve obtained the results depicted in figures5, 6 and 7.

Fig. 5. Control system response implementing the control law (15).

A phase plane analysis for states x1 and x2 are shown infigures 8 and 9. Figure 8 shows the phase plane of theta angle.The proposed sliding surface had of a straight line structure (asshown in the figure). In order to reach the equilibrium point,the control law forces the system’s response to follow thissliding surface until it is being attracted by the limit cycle’sarea of attraction, which is [0 0] for theta angle.

Figure 9 shows the phase plane simulation response of thesystem when a perturbation signal is applied. The simulationlasts 30 seconds and after 15 seconds of elapsed time aperturbation is applied. Analyzing the figure, it can be easilyobserved that the response of the system is following thesliding surface, then a perturbation is added but the controlsignal forces the system to come back to the sliding surfaceuntil the response of the system is pulled by the area of

Fig. 6. Sliding mode control response for θ. Response in Degrees.

Fig. 7. Sliding mode control response for ϕ. Response in Degrees.

attraction of the limit cycle. As in figure 8, it can be seenthat the equilibrium point of the system is [0 0].

It’s easy to note the better performance of this controlsystem compared to the one developed in section III, forboth state feedback gains K1 and K2. So, as we can seefrom figures 6 and 7, the control system response shows atendency of the system to be in the equilibrium point 0,even when perturbation signals are applied in the responsesignal. A feedback control law, u were selected to verify asliding condition, as it’s shown in equation (17). However,in order to account for the presence of modeling imprecisionand of disturbances, the control law had to be discontinuousacross S(t). Since the implementation of the associated controlswitching is necessarily imperfect (in practice switching is notinstantaneous) this leads to chattering, as it’s clearly seen infigure 7.

VI. SLIDING ADAPTIVE CONTROL DESIGN

Given the structure of the plant’s model, the model responseis determined by the values of certain constants referred to asplant or model parameters. In some applications these parame-ters may be measured or calculated using the laws of physics,properties of materials, etc. In many other applications, thisis not possible, and the parameters have to be deduced byobserving the system’s response to certain inputs [7]. In manyapplications, the structure of the model of the plant may beknown, but its parameters may be unknown and changing


Fig. 8. Phase plane analysis without perturbation.

Fig. 9. Phase plane of the system when a perturbation is applied.

with time because of changes in operating conditions, agingof equipment, etc., rendering off-line parameter estimationtechniques ineffective. The appropriate estimation schemes touse in this case are the ones that provide frequent estimatesof the parameters of the plant model by properly processingthe plant I/O data on-line.

For our desing in the Furuta Pendulum, we’ll assume thatthe mass of the pendulum, M , is unknown. In adaptive control,the control objective can be achieved without requiring theparameters to converge to their true values, therefore the lackof parameter convergence is less crucial than in parameteridentification, so a little deviation of the parameters from theirtrue values could be accepted and in addition, the controllerthat we’re propounding is robust. Due to the preceding ar-guments, we’ve been encouraged to use the linear model ofthe pendulum (4) for designing the adaptation law for ourcontroller.

θ = −wθ − cg

ab− c2u = wθ − bu (19)

w = f(M) = −25.52 (M + 0.01) (M + 0.925)

M + 5.2× 10−3

where b = − cgab−c2 will be known all the time.

ε = θ − θ (20)˙θ = −wθ + bu ; θ(0) = θ0 (21)θ = −amθ + (am − w) θ + bu

θ =1

s+ am[(am − w) θ + bu] (22)

˙θ = −amθ + (am − w) θ + bu

θ =1

s+ am[(am − w) θ + bu] (23)

ε = −wε+ wθ − bu (24)w = w − w

We assume that the adaptive law is of the form:

˙w = f(ε, θ, θ, u

)(25)

V (ε, w) =1

2

(ε2 + w2

)(26)

V = −amε2 + aθε+ af1 (27)

We’re looking for we to converge to w, so ˙w = ˙w. If wechoose f = −εθ, then:

V = −amε2 ≤ 0 (28)

And equation (25) becomes: ˙w = −εθ. In the designingphase, it’s quite usefull what is called adaptation gain, γ, soincluding it in the above equation, our adaptative law becomes:

˙w = −γεθ

w = −1

sγεθ (29)

The proposed structure of the sliding-adaptive controller isbased on 3 blocks, as it’s shown in figure 10

• The “Pendulum” block.• The “Sliding-Adaptive Controller” block• The “Parameter Estimation”.The “Pendulum” block contains the non-linear mathematical

model of the Furuta pendulum shown on equation (3). TheSliding-Adaptive Controller block contains the control law formaintaining the desired output of the Pendulums arm. The“Parameter Estimation” implements the adaptive law, equation(29), for estimating the unknown parameter of the plant, in thiscase the mass of the pendulum M . Adaptation process is asfollows:

• The controllers block sends the control signal to thePendulums block according to the desired equilibriumpoint (0,0,0,0).

• The Parameter Estimations block implements equation(29), which corresponds to the adaptive law for esti-mating the unknown mass of the pendulum. The designparameters are: am which corresponds to a pole place-ment and defines parameter convergence speed, γ thatis as we mentioned before the adaptation gain, which isresponsible of the rigidness of the estimation.


Fig. 10. Sliding-Adaptive Controller Structure.

• The information obtained in the Parameter Estimationblock is then send to the Sliding-Adaptive Controllersblock and the plants control input is updated.

This process is repeated on line in order to adapt the controllaw for variation on the parameters of the Model and to obtainthe best response of the system. Its easy to note from figure 11,as it was shown in figure 10, that our adaptive control systemhas three main parts: adaptive law, which is responsible forgenerating the parameter estimation; the sliding control, whichimplements the control law specified in equation (??) allowingthe variation of the unknown parameter, M, that is obtainedfrom the response of the adaptive law; and the last componentof the control system is the plant, which as long as weve beenworking in this project hasnt suffered any change.

Fig. 11. Simulink implementation of Sliding Adaptive Controller.

In figure 12 we can appreciate the controller performance,after establishing a new value on the mass of the pendulum,M = 1[kg]. Its easy to note from there that the controlsystem’s time response is really close to the behavior of thesliding controller depicted in figure 5, and of course thisis due to the additional improvement made on our robust

controller (sliding controller), through the adaptive law thatdirectly modifies it’s response.

Fig. 12. Time response of the sliding adaptive control system.

In order to establish a performance comparison under ex-treme parameter changes, at least the one we selected for ouradaptive design, the mass M, in figures 13 and 14 we presentthe systems time response comparison for both controllersimplemented: Sliding mode controller and Sliding adaptivecontroller. The mass change for the simulation results wasfrom M = 0.01[Kg] (originally) to M = 92[kg]. Its easy tonote the better performance of the adaptive controller underthis parameter change, for both systems responses x1 (θ) andx3 (ϕ).

Fig. 13. System’s time response comparison for θ.


Fig. 14. System’s time response comparison for ϕ.

VII. CONCLUSION

Non linear plants give an opportunity to develop and applynew control techniques, also those plants give the opportu-nity to apply existing control techniques and evaluate theirperformance. We’ve developed three different approaches forcontrolling the Furuta pendulum. The best performance wasachieved for the Sliding mode controller, although underparameter variation, that in our case was the mass of the pen-dulum M , the best performance was reached for the Slidingadaptive controller, as its shown in figures 13 and 14. Sliding-mode control is one of the effective nonlinear robust controlapproaches since it provides fast system dynamic responseswith an invariance property to uncertainties once the systemdynamics are controlled in the sliding mode.

ACKNOWLEDGMENT

The authors would like to thank their professor, PhD Ro-gelio Soto for his support during the research stage of thisproject.

REFERENCES

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[5] Gfvert, M. D1998E: Derivation of furuta pendulum dynamics. TechnicalReport. Department of Automatic Control, Lund Institute of Technology,Lund, Sweden.

[6] Khanesar, M.A.; Teshnehlab, M.; Shoorehdeli, M.A., Sliding mode con-trol of Rotary Inverted Pendulm, Control & Automation, 2007. MED ’07.Mediterranean Conference on , vol., no., pp.1-6, 27-29 June 2007.

[7] Ioannou P. and J. Sun ’Robust Adaptive Control’ published by PrenticeHall, Inc in 1996.

[8] Rong-Jong Wai; Jeng-Dao Lee; Li-Jung Chang; , ”Development ofadaptive sliding-mode control for nonlinear dual-axis inverted-pendulumsystem,” Advanced Intelligent Mechatronics, 2003. AIM 2003. Proceed-ings. 2003 IEEE/ASME International Conference on , vol.2, no., pp. 815-820 vol.2, 20-24 July 2003.

Ismael Minchala Avila is a MSc student at ITESM,enrolled with research and innovation in the mo-nitoring and advance control cathedra in the sameUniversity.

Rubn Marbn Romero is a MSc student at ITESM,enrolled with research and innovation in the mo-nitoring and advance control cathedra in the sameUniversity.

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Furuta Sliding Control