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This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies areencouraged to visit:
The problem of swinging up an inverted pendulum on a cart and controlling it around the upright positionhas traditionally been treated as two separate problems. This paper proposes a control strategy that isglobally asymptotically stable under actuator saturation and, in addition, locally exponentially stable.The proposed methodology, which performs swing up and control simultaneously, uses elements frominput–output linearization, energy control, and singular perturbation theory. Experimental results on alaboratory-scale setup are presented to illustrate the approach and its implementation.
The inverted pendulum has been used as a classical control ex-ample for nearly half a century because of its nonlinear, unsta-ble, and nonminimum-phase characteristics. Many control ideashave been illustrated on this system, such as linear feedback stabi-lization (Schaefer & Cannon, 1967), variable structure control (Ya-makita& Furuta, 1992), passivity-based control (Fradkov, Guzenko,Hill & Pogromsky, 1995), energy control (Spong& Praly, 1995), pre-dictive control (Ronco, Srinivasan, Favez & Bonvin, 2001), and hy-brid system control (Guckenheimer, 1995). Interest in the invertedpendulum has been renewed by recent developments in the fieldof underactuated mechanical systems (De Luca, Mattone, & Oriolo,1996; Spong & Praly, 1995).Most techniques available in the literature stabilize the system
locally in a region around the upright position, with nonlineartechniques being used to enlarge the stability region. Even thepapers that address the issue of global stabilization, such asMazencand Praly (1996), are restricted to the region where the pendulum
I A preliminary version of this paper was presented at the 15th IFAC WorldCongress in Barcelona in 2002. This paper was recommended for publication inrevised form by Associate Editor Torkel Glad under the direction of Editor HassanK. Khalil.∗ Corresponding author. Tel.: +41 21 693 3843; fax: +41 21 693 2574.E-mail address: [email protected] (D. Bonvin).
1 Present address: Department of Chemical Engineering, Ecole PolytechniqueMontreal, Montreal, Canada.2 Present address: MeteoSwiss CH-1530 Payerne.
is above the horizontal position. The problem of getting into thestability region, i.e. the swinging up, has typically been consideredseparately (Aström & Furuta, 2000; Spong, 1995). Hence, mostimplementation strategies require appropriate switching betweenswing up and control around the upright position. In contrast, theidea in this paper is to design a controller that can combine swingup and control, thereby leading to a globally stabilizing controller.Though a few papers consider such a combined problem, theydo not address the stabilization of the cart nor the saturation ofthe actuator (Angeli, 2001; Ohsumi & Izumikawa, 1995). Generictechniques, such as model-predictive control, have also beenapplied to the global stabilization of the inverted pendulumproblem (Magni, Scattolini & Aström, 2002).The concepts developed in this paper will use elements of in-
put–output linearization, energy control, and singular perturba-tion theory. As such, the approach is sufficiently general to beapplicable to other underactuated mechanical systems. However,there are two main difficulties regarding its applicability to the in-verted pendulum problem:• There is a singularity when the pendulum is horizontal.Actuator saturation, which in general represents an additionalcomplication for control design, is used here to handle theproblem of singularity. Around a singularity, though the controlinput goes theoretically to infinity, the input is limited by itspractical bound. This complicates the stability analysis, which isresolved by artificially introducing a reference for the angle thattakes a value in the set {−2π, 0, 2π}, all of which refer to thesame physical point, the upright position. However, differentchoices of the reference provide different ways of approachingthe desired equilibrium point.
266 B. Srinivasan et al. / Automatica 45 (2009) 265–269
• The system is nonminimum phase, which indicates thatinput–output linearization leads to unstable internal dynamics.A singular perturbation approach can be used to stabilizethe internal dynamics (Khalil, 1996). The controller gains arechosen so that the system exhibits a two-time-scale behavior- fast dynamics for the pendulum and slow dynamics for thecart. Also, two independent controllers are designed such thatboth subsystems are stable. Under these conditions, singularperturbation theory guarantees the existence of a controllerthat can stabilize the global system.The paper is organized as follows. The next section introduces
the standard inverted pendulummodel and normalizes it to obtaina simpler model that is used for analysis. Section 3 presents theinput–output linearization approach. The stability of the pendulumdynamics is investigated in Section 4, and that of the cart dynamicsin Section 5. Section 6 provides experimental results, and Section 7concludes the paper.
2. Inverted pendulum on a cart
The model of the inverted pendulum on a cart, which can befound for example in Mazenc and Praly (1996), is given by:
mp+ µθ cos θ − µθ2 sin θ = u (1)
µp cos θ + J θ − µg sin θ = 0 (2)where p is the position of the cart, θ the angle between the verticalline and the pendulum (positive clockwise), m the total mass ofthe system, µ = mplp/2 withmp and lp the mass and length of thependulum, J = Jp + mpl2p/4 with Jp the inertia of the pendulum, gthe gravity, and u the force applied to the cart. Note that frictionis neglected in this model. The initial conditions are p(0) = po,p(0) = vp,o, θ(0) = θo, and θ (0) = ωo. Eqs. (1) and (2) can berearranged to give:
θ =mµg sin θ − µ cos θ(u+ µθ2 sin θ)
(mJ − µ2cos2θ)(3)
p =J(u+ µθ2 sin θ)− µ2g sin θ cos θ
(mJ − µ2cos2θ). (4)
The system can be normalized by the introduction of: (i) thenew input v = p/g , (ii) the normalized time τ = t
õg/J , and
(iii) the normalized distance y = µp/J . With the first step, thedynamics read: θ = µg
J (sin θ − v cos θ) and p = gv. The secondstep normalizes the period of oscillation. In the new time scale,d2θdτ2= sin θ − v cos θ , d
2pdτ2=
Jµv, and the unforced period of
oscillation is 2π . The normalized system dynamics read:y = v, y(0) = yo, y(0) = vp,o (5)
θ = sin θ − v cos θ, θ(0) = θo, θ (0) = ωo. (6)Note that the time derivatives in (5) and (6) are with respect to thenormalized time τ . The set of Eqs. (5) and (6) is quite simple andwill be used for analysis in this paper.
3. Feedback linearization
An analysis of the Lie algebra of the system (5) and (6) indicatesthat the inverted pendulum is not full-state feedback linearizable.Hence, the input–output linearization of the θ-dynamics (6) isconsidered:
v =kθθ + kω θ + sin θ
cos θ. (7)
This imposes the dynamics θ + kω θ + kθθ = 0, which is stable forany choice of kω > 0 and kθ > 0. Nevertheless, this methodologyhas two basic drawbacks:• The linearizing feedback (7) has a singularity at θ = 2k+1
2 π , i.e.when the pendulum is horizontal.
Fig. 1. Switching strategy to choose θr in the set {−2π, 0, 2π}.
• The internal dynamics, which correspond to y = v, areunstable.These two problems will be handled in the next two sections.
4. Pendulum stability under actuator saturation
The two problems of singularity of the linearizing feedback andactuator saturation are closely related. For cos θ small, v becomeslarge and may exceed its bound vmax. In such a case, the input v is:v = sat (vus, vmax) (8)with the unsaturated input vus given by:
vus =kθ (θ − θr)+ kω θ + sin θ
cos θ(9)
and
sat(vus, vmax) =
{−vmax for vus ≤ −vmaxvus for − vmax < vus < vmaxvmax for vus ≥ vmax.
The novelty in this paper is the introduction of the artificialreference angle θr and its choice to enforce global stabilization. θr ischosen in the set {−2π, 0, 2π} so as to guarantee (θ − θr)θ ≤ 0, acondition that ensures that the pendulum is moving towards thedesired upright position. Note that all choices of θr refer to thesame physical point, i.e. the upright position, but different choicesprovide different ways of approaching it.
4.1. Switching logic
Let θ be the smallest positive value that satisfies:
vmax cos θ = kθ (θ − θr)+ sin θ . (10)
If the angular position θ can be reached with zero velocity, thelinearizing feedback (9) will then be able to bring the pendulumto the upright position. Hence, for |(θ − θr)| ≤ θ and θ = 0, thelinearizing feedback will take the system to the upright positiondespite actuator saturation.θr is allowed to change value when θ = 0, i.e. every time the
velocity changes sign. The switching is based on the angle at theinstant of switching (denoted as θi for the ith switching). If θi is inZone I of Fig. 1, θr does not change since the linearizing feedback(9) can take the pendulum to the upright position. If θi is in Zone IIwhen θ = 0, then there is too little energy to bring the pendulumto the upright position by rotating it clockwise, so a swing in thecounter-clockwise direction is required. This is achieved by givingan extra encirclement to the reference, i.e. the reference is set at(θr − 2π). A similar argument can be provided for adding 2π tothe referencewhen θi is in Zone III. The number of times θr changesits value depends on the number of swings necessary to bring thependulum to its upright position.Note that the switching occurs when θ changes sign and that
the new set point and input are given so as to give more velocityin the direction of its current movement rather than to reversethe direction of the pendulum. Hence, the next switching can onlyoccur in the other zone, and infinite switching cannot occur.
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Let θ−r,i and θ+
r,i denote the values of the reference before andafter the ith switching, respectively. Note that θ−r,i+1 = θ+r,i. Withθ−r,1 = 0,−π ≤ θo ≤ π , the switching law reads:
θ+r,i =
θ−r,i − 2π −π ≤ (θi − θ
−
r,i) < −θ
θ−r,i −θ ≤ (θi − θ−
r,i) ≤ θ
θ−r,i + 2π θ < (θi − θ−
r,i) < π.
(11)
4.2. Analysis of the switching logic
The stability of the θ-dynamics with the controller (8)–(11)can be analyzed using the Lyapunov approach. However, the maindifficulty arises from the switching of θr , for the analysis of whichconcepts from energy control are used next.
Lemma 1. Consider the θ-dynamics given by (6) and the controller(8)–(11). If kθ > 1 + kω , kω > 0, then the switching of θr will stopin finite time after a finite number of switchings.Proof. Consider the energy in the pendulum and its timederivative:
E =12θ2 + cos θ + 1 (12)
E = θ θ − θ sin θ = −vθ cos θ. (13)
The total energy consists of two parts, the normalized kineticenergy 12 θ
2 and the normalized potential energy (cos θ+1). Whenthe pendulum rests at the downward position (θ = π, θ = 0),the energy is E = 0. The problem of swinging up the pendulumconsists of pumping sufficient energy into the system so as toachieve E = 2, which corresponds to (θ = 0, θ = 0). Also, notethat E does not have a discontinuity evenwhen θr is discontinuous.This proof will show that, if the energy is insufficient to reach
the upright position (pendulum in Zone II or III at zero velocity), theproposed control lawwill pump energy into the systemuntil Zone Iis reached. Note that switching is only necessary if E < (cos θ+1).We first determine an upper bound for |kω θ + sin θ | for the
case where switching is needed. An upper bound for θ can beobtained from the energy E in (12), |θ | ≤ |
√2 (E − 1− cos θ)|.
It follows from E < 2 that |θ | < |√2 (1− cos θ)| =
|√2 (1− cos(θ − θr))| ≤ |(θ − θr)|. Also, from | sin θ | = | sin(θ −
θr)| ≤ |(θ − θr)|, it follows that, |kω θ+ sin θ | < (1+kω)|(θ− θr)|.From (8) and (9), regardless of whether or not the input v is
saturated, v cos θ and kθ (θ − θr)+ kω θ + sin θ will have the samesign. For kθ > (1+kω), kθ |(θ−θr)| > |kω θ+sin θ |, and thus v cos θwill have the same sign as kθ (θ − θr), i.e. v(θ − θr) cos θ ≥ 0.Since θr is chosen to satisfy (θ − θr)θ < 0, when θ 6= 0, itfollows that (θ − θr) and θ will have opposite signs and thereforeE = −vθ cos θ ≥ 0. Also, E > 0 except for specific time instantswhen θ = 0 and cos(θ) = 0.The proposed controller leads to the following two cases: (i) the
energy is sufficient (E ≥ cos θ+1) and there is nomore switching,or (ii) the energy is insufficient and θr switchings with Ei+1 > Ei.This monotonic increase of energy eventually leads to Case (i) withno more switching. �
4.3. Lyapunov analysis
Theorem 1. Consider the θ-dynamics given by (6) and the controller(8)–(11). If kθ ≥ 1+ kω , kω > 0, then (θ = θr , θ = 0) is a globallyasymptotically stable equilibrium point.Proof. Consider the Lyapunov function candidate V and itsderivative V :
V =12
(kθ (θ − θr)2 + θ2
)(14)
V = kθ (θ − θr)θ + θ sin θ − vθ cos θ (15)
Note thatV is discontinuous as θr switches. In fact,V increaseswithevery switching. However, from Lemma 1, switching stops after afinite time. This Lyapunov analysis is valid once all switchings haveoccurred. We need to consider two cases:• If v is not saturated, vus cos θ = kθ (θ − θr) + kω θ + sin θ andV = −kω θ2 ≤ 0.• With saturation, (15) can be rearranged as follows:
V = (1− α) (kθ (θ − θr)+ sin θ) θ − αkω θ2 (16)
withα = vvus. The switching law for θr guarantees (θ−θr)θ ≤ 0.
From the proof of Lemma 1, kθ |(θ − θr)| ≥ | sin θ | for kθ ≥ 1+kω , kω > 0. It follows that (kθ (θ − θr)+ sin θ) and (kθ (θ − θr))have the same sign. Hence, (kθ (θ − θr)+ sin θ) θ ≤ 0. SinceV in (16) is a convex combination of two negative quantities,V ≤ 0.Using LaSalle’s invariance theorem (Khalil, 1996), the invariant
set corresponds to θ = 0, θ = 0, which requires v = tan θ and,from (6) and (8), θ = θr . Hence, with this controller, (θ = θr ,θ = 0) is a globally asymptotically stable equilibrium point of theθ-dynamics. �
5. Cart stability using singular perturbation
The cart dynamics resulting from input–output linearization(internal dynamics) are unstable. This can be explained intuitivelyas follows: Once the inverted pendulum is in the upright position,there is no reason why the cart velocity needs to be zero;and a non-zero cart velocity makes the cart drift away. In thispaper, stabilization of the internal dynamics is approached usinga singular perturbation framework.
5.1. Time-scale separation
Discarding saturation (v = vus), and considering the control law(9) with the external inputw,
v =kθ (θ − θr)+ kω θ + sin θ + w
cos θ(17)
the system dynamics read:
y =kθ (θ − θr)+ kω θ + sin θ + w
cos θ(18)
θ = −kθ (θ − θr)− kω θ − w. (19)For large values of kθ and kω (relative to the gains used to control
the y-dynamics), the θ-dynamics are fast and will quickly reachtheir stable equilibrium: θ = θr −
wkθand ˙θ = 0. Plugging these
values in (18) gives:
y = − tanw
kθ. (20)
This way, a two-time-scale system has been artificially created.
5.2. Stability of the slow dynamics
The slow y-dynamics (20) can be stabilized using the followingfeedback.
Lemma 2. Consider the y-dynamics given by (20) and the controller
w = kθ tan−1(kyy+ kv y). (21)
In the absence of saturation, if ky, kv > 0, then y = y = 0 isexponentially stable.Proof. Consider the Lyapunov function candidateW
W =12
[y y
]Q[yy
](22)
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268 B. Srinivasan et al. / Automatica 45 (2009) 265–269
with
Q =[(ky + k2y + k
2v) kv
kv (1+ ky)
]and its derivative
W = (ky + k2y + k2v) yy+ kv(yy+ y
2)+ (1+ ky)yy
= −kykv(y2 + y2) ≤ −λ3‖z‖2 (23)
with z = [y y]T. W is positive definite and it can be verified thatλ1‖z‖2 ≤ W ≤ λ2‖z‖2, where λ1,2 are the eigenvalues of thematrix Q , i.e.
λ1,2 =(1+ ky)2 + k2v ±
√(k2y + k2v − 1)2 + 4k2v
2. (24)
Since W and W are bounded by ‖z‖2, the origin is locallyexponentially stable. �
5.3. Stability of the overall system
Standard results of singular perturbation theory are used toprove the stability of the overall system (Khalil, 1996). Theseresults require exponential stability of the two subsystems. Here,exponential stability can only be guaranteed in the neighborhoodof the equilibrium point, i.e. where the input is not saturated.Hence, the following result is based on the fact that, after somefinite time, the system enters Zone I, inwhich the system can reachthe desired equilibrium position without input saturation.
Theorem 2. Consider the dynamics of the inverted pendulum (5) and(6) and the controller
with ymax = π2(π + vmax) and ymax = πvmax. If kθ > 1+ kω , andkθ , kω � ky, kv > 0, then (θ = θr , θ = y = y = 0) is globallyasymptotically stable. In the region where the input is not saturated,convergence is exponential.Proof. If ky, kv → 0, asymptotic stability of the pendulumdynamics is proved in Theorem 1, i.e. (θ − θr) and θ will convergeto zero.With such a controller, an upper bound on the position and the
speed of the cart can be derived. Note that the normalized periodof oscillation is 2π and thus either θ or cos θ changes sign withinπ normalized seconds. Since the maximum acceleration is ymax =vmax, the maximum cart velocity is ymax = πvmax. The maximumnumber of switchings required to swing up the pendulum is givenby π+vmax2 vmax
(Aström & Furuta, 2000). Hence, an upper bound on theposition can be obtained as the product of maximum speed andmaximum time taken for swing up:
ymax = ymaxπ + vmax
2 vmax2π = π2(π + vmax). (28)
If the term kθ tan−1(kyy+ kv y) is introduced in the computation ofθ as in (27), then, after a finite time, the systemwill stay in a regionwhere the input is not saturated. In the absence of saturation,
Table 1System parameters and controller gains.
m 0.3235 kg kθ 13µ 1.3625× 10−3 kg m kω 1.6J 1.5265× 10−4 kg m2 ky 0.25g 9.81 m/s2 kv 0.4Km 0.3 kg m/V s2
Fig. 2. Pendulum angle θ and cart position p (Vmax = 3.25 V).
exponential stability of the slow (reduced) cart dynamics is provenin Lemma 2. Exponential stability of the fast (boundary-layer)pendulum dynamics in the unsaturated case can be proven with aLyapunov function similar to (22), where (y, y, ky, kv) are replacedby (θ, θ , kθ , kω). Hence, the proof of the theorem follows directlyfrom the stability result of singularly perturbed systems (referto Khalil (1996), Theorem 9.3, p 380). �
Though the present result is existential in nature, a bound onky, kv as a function of kθ , kω can be obtained. This is not undertakenhere since the expressions are quite involved. Instead, the gains arechosen so that the state matrix of the linearized dynamics aroundthe upright position is stable.
6. Experimental results
The proposed global stabilization strategy is tested experimen-tally on a laboratory-scale setup. The unnormalized version of thecontrol law (25) is used:
vus =1cos θ
(kθ (θ − θr)+
√Jµgkω θ + sin θ+
kθ tan−1(µ
Jkyp+
õ
g Jkv p))
(29)
uus =vg(mJ − µ2 cos2 θ)+ µ2g sin θ cos θ − Jµθ2 sin θ
J
Vm = sat(uus/Km, Vmax) (30)
where uus is the unsaturated force applied to the cart, Vm thevoltage applied to the motor, Vmax the maximum voltage, and Kmthe ratio between the linear force acting on the cart and the voltageapplied to the motor. The parameters used for implementationare given in Table 1. Note that arbitrary gains as proposed bytheory could not be used on the experimental setup due to modeluncertainties. Hence, the controller gains were detuned in order toincrease robustness.Two cases for the bound Vmax are considered: (i) Vmax = 3.25 V,
and (i) Vmax = 2 V. The former is the true physical limit, whilethe latter is artificially imposed to test the performance of thecontroller with less energy. Experimental results with Vmax =3.25 V are shown in Figs. 2 and 3, and with Vmax = 2 V in Figs. 4and 5.
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B. Srinivasan et al. / Automatica 45 (2009) 265–269 269
Fig. 3. Input Vm and reference θr (Vmax = 3.25 V).
Fig. 4. Pendulum angle θ and cart position p (Vmax = 2 V).
Fig. 5. Input Vm and reference θr (Vmax = 2 V).
Calculations show that even Vmax = 3.25 V is insufficientto swing up the pendulum in one go. Hence, switching of θr isnecessary in both cases: θr switches 4 times before reaching itsfinal value with Vmax = 3.25 V, and 9 times with Vmax = 2 V.Note that the input changes sign every time the horizontal line iscrossed due to the division by cos θ in (29).The model used for control is very simple and neglects effects
such as static friction. However, the experimental setup has staticfriction, which is evidenced by the fact that the cart remains stuckafter about 10 s, though there is a non-zero voltage applied to themotor (Figs. 2 and 3). This is the reason for the slight offset in cartposition.
7. Conclusions
A global stabilization strategy for an inverted pendulum hasbeen proposed in this paper. It uses actuator saturation to handlesingularity, switching of the reference position to solve the globalstabilization problem, and a singular perturbation approach tostabilize the internal dynamics. Though the proposedmethodologyis tuned to the systemat hand, some of the ideas can be generalizedto the control of nonlinear nonminimum-phase systems. Inparticular, the possibility of using a switched reference to handleinput saturation and a two-time-scale approach to handle theinternal dynamics shows promise.
Acknowledgments
The authors would like to thank Prof. Karl Aström for usefuldiscussions andMr. Philippe Cuanillon for his timely help with theexperimental prototype.
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B. Srinivasan graduated in Electronics and Communica-tion Engineering in 1988. He received his M.Tech. in Elec-tronics Design Technology and Ph.D. in Computer Sci-ence and Automation at the Indian Institute of Science,Bangalore, in 1990 and 1993 respectively. From 1994 to2004, he was a Research Associate at the Laboratoired’Automatique, EPFL, Switzerland. He is now AssociateProfessor in the Department of Chemical Engineering atthe Ecole Polytechnique in Montreal. His research inter-ests include measurement-based static and dynamic opti-mization, and nonlinear control systems.
P. Huguenin received the B.Sc. and M.Sc. degrees inMechanical Engineering from the Ecole PolytechniqueFédérale in Lausanne (EPFL), Switzerland. He then workedfor two years at the Automatic Control Laboratory ofEPFL on remote control of dynamical systems. From2001 to 2005, he completed his Ph.D. work dealing withthe on-line optimization of stochastic processes appliedto electrical discharge machining. He then joined theSwiss Federal Office of Meteorology and Climatology(MeteoSwiss), working on the development of a remotesensing network that feeds measurements into fine grid
Numerical Weather Predictionmodels in the framework of Nuclear Power Plant fornational safety.
D. Bonvin is Professor of Automatic Control at the EcolePolytechnique Fédérale in Lausanne (EPFL), Switzerland,where he also serves as Dean of Bachelor andMaster stud-ies. He received his Diploma in Chemical Engineering fromETH, Zürich, and his Ph.D. degree from the University ofCalifornia, Santa Barbara. He worked in the field of pro-cess control for the Sandoz Corporation in Basel and withthe Systems Engineering Group of ETH Zürich. He joinedthe EPFL in 1989, where his current research interests in-cludemodeling, identification and optimization of dynam-ical systems.
