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On the Stability of Sliding Mode Control for a Class of

Underactuated Nonlinear Systems

Sergey G. Nersesov, Hashem Ashrafiuon, and Parham Ghorbanian

Abstract A system is considered underactuated if the num-ber of the actuator inputs is less than the number of degrees offreedom for the system. Sliding mode control for underactuatedsystems has been shown to be an effective way to achieve systemstabilization. It involves exponentially stable sliding surfaces sothat when the closed-loop system trajectory reaches the surfaceit moves along the surface while converging to the origin.In this paper, we present a general framework that providessufficient conditions for asymptotic stabilization by a slidingmode controller for a class of underactuated nonlinear systemswith two degrees of freedom. We show that, with the slidingmode controller presented, the closed-loop system trajectoriesreach the sliding surface in finite time. Furthermore, we developa constructive methodology to determine exponential stabilityof the reduced-order closed-loop system while on the slidingsurface thus ensuring asymptotic stability of the overall closed-loop system and provide a way to determine an estimate of thedomain of attraction. Finally, we implement this framework onthe example of an inverted pendulum.

I. INTRODUCTION

Sliding mode control has been shown to be a robustand effective control approach for stabilization of nonlinearsystems [1]. The approach is based on defining exponentiallystable (sliding) surfaces as a function of the system statesand using the Lyapunov theory to ensure that all closed-loop system trajectories reach these surfaces in finite time.Since the closed-loop system dynamics on the surfaces are

exponentially stable, the system trajectories slide along thesurfaces until they reach the origin. Sliding mode controllershave been successfully developed for a variety of problemsinvolving underactuated systems [2], [3], [4]. The authors in[5], [6] introduced a second order sliding mode control ap-proach with application to the inverted pendulum. The slidingmode control law proposed for underactuated systems in [4]categorizes the stabilization problem based on equilibriummanifold. In their work, the authors present a control lawguaranteeing that all closed-loop system trajectories reachthe proposed sliding surfaces in finite time. However, theapproach in [4] establishes stability of the closed-loop systemdynamics on the sliding surfaces only by linearization anddoes not allow to estimate the domain of attraction on thesliding surface guaranteeing exponential convergence of thesystem trajectories.

In this paper, we show finite-time convergence of theclosed-loop system trajectories to a sliding surface using aLyapunov function approach presented in [7]. Furthermore,we develop a methodology to determine a Lyapunov functionfor a class of two degree-of-freedom underactuated nonlinearsystems that guarantees exponential stability of the reduced

This research was supported in part by the Office of Naval Researchunder Grant N00014-09-1-1195.

The authors are with the Department of Mechanic alEngineering, Villanova University, Villanova, PA 19085-1681, USA (sergey.nersesov@villanova.edu;hashem.ashrafiuon@villanova.edu;parham.ghorbanian@villanova.edu).

order closed-loop system while on the sliding surface. De-termining such Lyapunov function allows to estimate thedomain of attraction on the sliding surface guaranteeing thatall closed-loop system trajectories entering this domain willconverge exponentially to the origin while sliding along thesurface. Finally, we apply this framework to the invertedpendulum previously studied in [4] to prove stabilization by asliding mode controller and to show various estimates of thedomain of attraction for different sets of the sliding surfaceparameters.

I I . SLIDING MOD E CONTROL FOR UNDERACTUATED

NONLINEAR SYSTEMS

The system is considered underactuated if the number ofactuator inputs m is less than the number of degrees offreedom n. In this section, we present the sliding modecontrol framework for underactuated nonlinear dynamicalsystems and develop a general methodology to establishexponential stability of the closed-loop system during thesliding phase for two degree-of-freedom Lagrangian systems.To elucidate this approach, consider a two degree-of-freedomLagrangian dynamical system with a generalized positionvector partitioned as q = [qa, qu]

T, where qa R is theactuated coordinate, qu R is the unactuated coordinate,and q D R2. Then the equations of motion for thissystem can be written as

Maa(q(t)) Mau(q(t))Mau(q(t)) Muu(q(t))

qa(t)qu(t)

=

fa(q(t), q(t)) + u(t)

fu(q(t), q(t))

, (1)

where u(t) R is the control input, f(q, q) [fa(q, q), fu(q, q)]

T is the vector of Coriolis, centrifugal,

conservative, and non-conservative forces, and q [qa, qu]T

is the acceleration vector. The inertia matrix is accordinglypartitioned into positive definite elements Maa : R

2 R+and Muu : R

2 R+, and an off-diagonal element Mau :R2 R. We can solve (1) for the accelerations as

qa(t) = (Maa(q))

1(fa(q, q) + u(t)), (2)

qu(t) = (Muu(q))

1[fu(q, q)

Mau(q)(Maa(q))

1u(t)], (3)

where

Maa(q) = Maa(q) Mau(q)(Muu(q))1Mau(q),fa(q, q) = fa(q, q) Mau(q)(Muu(q))1fu(q, q),Muu(q) = Muu(q) Mau(q)(Maa(q))1Mau(q),fu(q, q) = fu(q, q) Mau(q)(Maa(q))1fa(q, q).

We define the sliding surface as a linear combination ofthe actuated and unactuated position and velocity variables

s(q, q) = aqa + aqa + uqu + uqu= aqa + uqu + sr(q), (4)

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where s : R2 R2 R, a, a, u, u R are the surfaceparameters, and sr(q) aqa+uqu. The surface parametersmust be selected in such a way that the reduced-order closed-loop system is asymptotically (or, exponentially) stable whileon the sliding surface. Next, the control law can be calculatedby setting s(q, q) = 0. Substituting for qa and qu from (2),(3) and adding a sign function yields

u(t) =

(Ms(q(t)))

1[fs(q(t), q(t)) + sr(q(t))

+ sign(s(q(t), q(t)))], (5)

where > 0 and

Ms(q) = a(Maa(q))

1

u(Muu(q))1Mau(q)(Maa(q))1, (6)fs(q, q) = a(M

aa(q))

1fa(q, q)

+u(Muu(q))

1fu(q, q). (7)

In (5), we added a sign term to ensure finite-time conver-gence of the closed-loop system trajectories to the slidingsurface, as will become evident later from the proof ofTheorem 2.1. Here > 0 is a constant parameter indicatinghow fast the closed-loop system trajectories reach the sliding

surface s(q, q). Now substituting the control law (5) to (2),(3) yields the closed-loop system dynamics

qa(t) = ufu(q(t), q(t))aMuu(q(t)) uMau(q(t))

Muu(q(t))[sr(q(t)) + sign(s(q(t), q(t)))]aMuu(q(t)) uMau(q(t)) , (8)

qu(t) =afu(q(t), q(t))

aMuu(q(t)) uMau(q(t))+

Mau(q(t))[sr(q(t)) + sign(s(q(t), q(t)))]

aMuu(q(t)) uMau(q(t)) . (9)

While on the sliding surface, the closed-loop dynamics

become

qa(t) = ufu(q(t), q(t))aMuu(q(t)) uMau(q(t))

Muu(q(t))(aqa(t) + uqu(t))aMuu(q(t)) uMau(q(t)) , (10)

qu(t) =afu(q(t), q(t))

aMuu(q(t)) uMau(q(t))+

Mau(q(t))(aqa(t) + uqu(t))

aMuu(q(t)) uMau(q(t)) . (11)

Next, introduce an auxiliary variable

z qa +

u

a qu, (12)

and note that the closed-loop system dynamics (10), (11)while on the sliding surface (4) reduce to

qu(t) =afu(q(t), q(t)) + aMau(q(t))z(t)

aMuu(q(t)) uMau(q(t)) , (13)

z(t) = aa

z(t)

u

a u

a

qu(t). (14)

For the closed-loop system (13), (14) introduce the statevariables y1 = qu, y2 = qu, and y3 = z. Then (13), (14)

can be rewritten in the state space form as

y1(t) = y2(t), y1(0) = y10, t 0, (15)

y2(t) =a

ua

ua

Mau(y1(t), y3(t))y2(t)

aMuu(y1(t), y3(t)) uMau(y1(t), y3(t))

+ 2aa

Mau(y1(t), y3(t))y3(t) + afu(y(t))

aMuu(y1(t), y3(t)) uMau(y1(t), y3(t)),

y2(0) = y20, (16)

y3(t) =

u

a u

a

y2(t) a

ay3(t), y3(0) = y30,

(17)

where y [y1, y2, y3]T.

Theorem 2.1: Consider the nonlinear dynamical system(1) with the feedback control law (5). Assume that Ms(q) =0, q D R2, where Ms() is given by (6), and the reducedorder closed-loop system on the sliding surface given by(15)(17) is asymptotically (or, exponentially) stable. Thenthe closed-loop system (1) with the feedback control law (5)

is asymptotically stable.Proof. Consider a Lyapunov function candidate given by

V(q, q) =1

2s2(q, q), q D, q R2. (18)

It follows from (4), (8), and (9) that

s(q, q) = sign(s(q, q)), q D, q R2. (19)

Thus, the Lyapunov derivative along the closed-loop systemtrajectories of (8), (9) is given by

V(q, q) = s(q, q) s(q, q)

= sign(s(q, q)) s(q, q)= |s(q, q)|=

2 (V(q, q))

1

2 , q D, q R2. (20)

It was shown in [7] that condition (20) guarantees thatthe closed-loop system trajectories converge to the slidingsurface in finite time. Furthermore, it follows from (20)that while the closed-loop system trajectories are on thesliding surface, they will remain on the surface. Since, byassumption, the reduced-order closed-loop system on thesliding surface (15)(17) is asymptotically (or, exponentially)stable, it follows that the closed-loop system (1) and (5) isasymptotically stable which proves the result.

Next, we specialize the result of Theorem 2.1 to the caseof Euler-Lagrange systems whose dynamics are given by

Maa(q(t)) Mau(q(t))Mau(q(t)) Muu(q(t))

qa(t)qu(t)

+

Caa(q(t), q(t)) Cau(q(t), q(t))Cau(q(t), q(t)) Cuu(q(t), q(t))

qa(t)qu(t)

+

Kaa(q(t)) Kau(q(t))Kau(q(t)) Kuu(q(t))

qa(t)qu(t)

=

u(t)

0

.

(21)

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In this case, fu(q, q) in (1) becomes

fu(q, q) =

u

aKau(y1, y3) Kuu(y1, y3)

y1

+

u

aCau(y) Cuu(y)

y2

+

a

a

Cau(y) Kau(y1, y3)

y3, (22)

where qa = y3 ua y1 and qu = y1. Now, with fu(q, q) givenby (22), the reduced-order closed-loop system (15)(17) canbe rewritten as

y(t) = A(y(t))y(t), y(0) = y0, t 0, (23)where

A(y) =

0 1 0A21(y) A22(y) A23(y)

0 ua

ua

aa

,

A21(y) =aKau(y1, y3)u

a Kuu(y1, y3), (24)

A22(y) = Mau(y1, y3)u aMau(y1, y3)ua

+Cau(y)u Cuu(y)a, (25)A23(y) =

2a

aMau(y1, y3) + aCau(y)

aKau(y1, y3). (26)In order to prove exponential stability of the closed-loop

system (23), consider a Lyapunov function candidate V(y) =12

yTP y, y D, where P R33 is such that P > 0 andD {y R3 : Ms(y1, y3) = 0}. The Lyapunov derivativealong the trajectories of (23) can be written as

V(y) =1

2

yT AT(y)P + P A(y) y, y D. (27)

Now, assume there exists a matrix R(y) > 0, y D D,such that

AT(y)P + P A(y) + R(y) = 0, y D. (28)Next, we show that the reduced order closed-loop system(23) is exponentially stable. First, note that since P > 0, itfollows that

1

2min(P)y22 V(y)

1

2max(P)y22, y R3, (29)

where min() and max() are the minimum and the max-imum eigenvalues of a matrix, respectively. Next, since

R(y) > 0, y D D, it follows that

0 infyD

(min(R(y)))I3 R(y), y D, (30)

where I3 R33 is the identity matrix. Hence,

V(y) = 12

yTR(y)y (31)

12

infyD

(min(R(y)))yTy (32)

infyD(min(R(y)))max(P)

V(y) (33)

= V(y), y D, (34)

where inf

yD(min(R(y)))

max(P). Thus, if > 0, then the

reduced order closed-loop system (23) is exponentially stable[8].

Remark 2.1: As will be shown in the next section, inspecific cases when the system dynamics are known it ispossible to add a nonquadratic term to a Lyapunov functioncandidate in order to significantly simplify the matrix R(y)so that it only depends on a part of the state vector y.

III . STABILITY ANALYSIS OF AN INVERTED PENDULUM

In this section, we apply the methodology developed inSection II to the example of an inverted pendulum studiedin [4]. The system is shown in Figure 1 and the equationsof motion are given by

(M + m)x(t) + ml(t)cos (t) = ml2(t)sin (t)

+u(t), (35)

mlx(t)cos (t) + ml2(t) = mgl sin (t), (36)

where m and M are pendulum and cart masses, respectively,q() is the cart position, () is the pendulum angularposition, l is the pendulum length, and u(

) is the control

force acting on the cart. Note that equations (35), (36) canbe written in the form of (1) with

Maa(x, ) = M + m,

Mau(x, ) = ml cos ,

Muu(x, ) = ml2,

fa(x,, x, ) = ml2 sin ,

fu(x,, x, ) = mgl sin .

Next, we define the sliding surface as

ax + ax + lu + lu = 0, (x,, x, ) R4, (37)where a, u, a, u are the sliding surface parameters.Then, the sliding mode control law, while the trajectory ison the surface, can be determined from (5) and is given by

u(t) = ml2(t)sin (t)

+mg sin (t)

uMm

+ 1 a cos (t)

u cos (t) a+

(M + m sin2 (t))(ax(t) + ku(t))

u cos (t) a .(38)

Hence, the closed-loop system (35), (36), and (38) on thesurface (37) becomes

x(t) = ug sin (t) + ax(t) + lu(t)u cos (t) a , (39)

(t) =ag sin (t) cos (t)(ax(t) + lu(t))

l(u cos (t) a) . (40)

Next, we show that, while on the sliding surface, theclosed-loop trajectories converge to the zero equilibrium stateand the origin is exponentially stable for (39), (40). To seethis, introduce an auxiliary variable as in (12) given by

z x +lu

a, (41)

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and note that according to (13), (14), the closed-loop systemdynamics (39), (40) while on the sliding surface reduce to

(t) =ag sin (t) a cos (t)z(t)

l(u cos (t) a) , (42)

z(t) = aa

z(t)

lu

a lu

a

(t). (43)

Next, introduce state variables y1 , y2 , y3 z, andparameters

l

u

a u

a

, (44)

a

u, (45)

and rewrite (42), (43) in the state space form as

y1(t) = y2(t), (46)

y2(t) =ag

ultan y1(t) +

alu

y2(t) +2a

lauy3(t)

1

cos y1(t)

, (47)

y3(t) = y2(t) aa

y3(t). (48)

In order to avoid singularity in (47), we consider y1 I {y1 R : arccos() < y1 < arccos()}and define D {y R3 : y1 I}, where y [y1, y2, y3]

T. In this case, 1 cos y1 > 0, y1 I. Next, toshow exponential stability of (46)(48), consider a Lyapunovfunction candidate

V(y) =1

2yTP y ln cos y1

1 , y D, (49)

where > 0, P R33 is such that P > 0, and

P =

p11 p12 p13p12 p22 0

p13 0 p33

. (50)

Note that V(0) = 0 and V(y) > 0, y D , y = 0. Next,using the fact that y1 tan y1 > y

21 , y1 I, it can be easily

shown that, with

=p22ag

ul> 0 (51)

and p12 > 0, the Lyapunov derivative along trajectories of

(46)(48) satisfies

V(y) 12

yTR(y1)y, y D, (52)

where

R(y1) =

R11(y1) R12(y1) R13(y1)

R12(y1) R22(y1) R23(y1)

R13(y1) R23(y1) R33(y1)

(53)

with

R11(y1) =2p12ag

ul

1 cos y1 ,

R12(y1) = p11 + p13 p12alu

1 cos y1

,

R13(y1) =

p13a

a p12

2a

lau

1 cos y1

,

R22(y1) = 2p12 + p22a

lu

1 cos y1

,

R23(y1) = p13 + p33 p222a

lau

1 cos y1

,

R33(y1) = 2p33a

a.

Next, assume that R(y1) > 0, y1 I I, and note thatin this case

0 infy1I

(min(R(y1)))I3 R(y1), y1 I. (54)

Furthermore, it follows from the Taylor series expansionaround y1 = 0 on the interval y1 I that, for every > 0and > 0, there exists sufficiently large > 0 such that

ln cos y1 1

1

2y21 , y1 I. (55)

In this case,

V(y) =1

2yTP y ln cos y1

1

1

2

yTP y +1

2

y21

=1

2yTP y, y D, (56)

where D {y R3 : y1 I} and

P

p11 + p12 p13p12 p22 0

p13 0 p33

. (57)

Clearly, P > 0 since P > 0 and > 0. Hence,

1

2min(P)y22 V(y)

1

2max(P)y22, y D. (58)

Furthermore, it follows from (54) and (56) that

V(y) 12

yTR(y1)y

12

infy1I

(min(R(y1)))yTy

infy1I(min(R(y1)))max(P)

V(y)

= V(y), y D, (59)

where inf

y1I(min(R(y1)))

max(P). Thus, if > 0, then (46)

(48) is exponentially stable [8] with the domain of attraction

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given by D. Hence, the closed-loop system (39)(40), whileon the sliding surface (37), is exponentially stable. Finally, itfollows from Theorem 2.1 that the closed-loop system (35),(36), and (37) is asymptotically stable.

We use a numerical procedure involving linear matrixinequalities to obtain P R33 in (50) and R() R33 in(53) such that P > 0 and R(y1) > 0, y1 I. The systemdata are given as M = 0.4 kg; m = 0.14 kg; l = 0.215 m;g = 9.807 m/s2. We choose surface parameters as a = 0.02;a = 0.005; u = 1; u = 2.5. In this case, = 0.02 and,from all possible solutions, we choose p11 = 300, p22 = 250,

p33 = 0.1, p12 = 20, p13 = 4 so that P > 0. Here, = 228.07 and I = {y1 R : arccos() + 0.0014 y1 arccos() 0.0014} = [1.55, 1.55]. The plot ofmin(R(y1)) versus y1 I is shown in Figure 2. It canbe seen from the plot that infy1I(min(R(y1))) > 0,and hence, (59) is satisfied with > 0 which impliesexponential stability of (46)(48) with a subset of the domain

of attraction given by D = {y R3 : 1.55 y1 1.55}.Next, consider a different set of surface parameters given

by a = 0.1; a = 0.025; u = 1; u = 2.5 such that =0.1. Choose p11 = 1000, p22 = 300, p33 = 0.7, p12 = 100,

p13 = 10 so that P > 0 and = 1368.41. In this case, I={y1 R : arccos() + 0.04 y1 arccos() 0.04} =[1.43, 1.43]. Figure 3 shows the plot ofmin(R(y1)) versusy1 Iwith infy1I(min(R(y1))) > 0. Hence, a subset ofthe domain of attraction for (46)(48) is given by D = {y R3 : 1.43 y1 1.43}.Finally, consider a = 0.2; a = 0.05; u = 1; u =

2.5 such that = 0.2. Choose p11 = 1000, p22 = 300,p33 = 1, p12 = 100, p13 = 10 so that P > 0 and =2736.83. In this case, I= {y1 R : arccos() + 0.55 y1 arccos()0.55} = [0.82, 0.82]. It can be seen fromFigure 4 that infy1I(min(R(y1))) > 0. Thus, a subset of

the domain of attraction for (46)(48) is given by D = {y R3 : 0.82 y1 0.82}.This analysis justifies the conclusion made in [4] that,

based on the simulation results, |a|

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-1 -0.5 0 0.5 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

y1

min

(R(y1

))

Fig. 4. Minimum eigenvalue of R(y1) versus y1 for a = 0.2,a = 0.05.

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[5] S. Riachy, Y. Orlov, T. Floquet, R. Santiesteban, and J.-P. Richard,Second-order sliding mode control of underactuated mechanical sys-tems I: Local stabilization with application to an inverted pendulum,

International Journal of Robust and Nonlinear Control, vol. 18, no.4-5, pp. 529543, 2008.

[6] R. Santiesteban, T. Floquet, Y. Orlov, S. Riachy, and J.-P. Richard,Second-order sliding mode control of underactuated mechanical sys-tems II: Orbital stabilization of an inverted pendulum with applicationto swing up/balancing control, International Journal of Robust and

Nonlinear Control, vol. 18, no. 4-5, pp. 544556, 2008.[7] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous

autonomous systems, SIAM J. Control Optim., vol. 38, no. 3, pp. 751766, 2000.

[8] W. M. Haddad and V. Chellaboina, Nonlinear Dynamical Systemsand Control. A Lyapunov-Based Approach. Princeton, NJ: PrincetonUniversity Press, 2008.

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