sliding mode control: a survey with applications in math

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Mathematics and Computers in Simulation 81 (2011) 954–979

Sliding mode control: A survey with applications in math

Alessandro Pisano, Elio Usai ∗Dipartimento di Ingegneria Elettrica ed Elettronica, Universitá degli Studi di Cagliari, Piazza d’ Armi, Cagliari, Italy

Available online 21 October 2010

Abstract

The paper presents a brief survey on Variable Structure Control Systems with Sliding Modes. Starting from a general case ofsliding modes in dynamical systems with discontinuous right-hand side, classic approaches to sliding mode control systems areconsidered and some basic results about the control of uncertain systems are given. Then, Higher-Order Sliding Modes are presentedas a tool to remove discontinuity from the control action, to deal with higher relative degree systems and to improve the accuracyof the real sliding mode behavior when the discrete time implementation is considered.

Finally, three applications of the sliding mode control theory to applied math problems are presented: the numerical solution ofconstrained ODEs, the real-time differentiation, and the problem of finding the zeroes of nonlinear algebraic systems. The first isan almost straightforward application of the sliding mode control theory, while the last two are accomplished by computing thesolution of properly defined dynamical systems. Some simulations are reported to clarify the approach.© 2010 IMACS. Published by Elsevier B.V. All rights reserved.Variable structure systems; Sliding modes; Differentiation; Constrained ODE; Zero finding

1. Introduction

Nonlinear dynamical systems have been considered an interesting topic of investigation because of the possiblefollow up of the research results. Actually, real systems are always nonlinear and considering their linear approximationcan impose too strict requirements on their working range or give unfeasible results. Furthermore, nonlinear systemscan even provide better performance than linear ones, and often some nonlinear behavior is intentionally introducedin feedback control systems [57,92,59].

Among nonlinear control systems, switching control systems are quite interesting since they are very simple toimplement [96,110] or even the optimal solution to some control problems [2]. Switching dynamical systems originateinteresting mathematical problems since they are characterized by ODE (Ordinary Differential Equations) with dis-continuous right-hand side and the usual definitions and existence conditions for the solution of a ODE are no longervalid; therefore a proper extension of classic differential equations theory has to be taken into account [46].

Switching systems are characterized by changes in the system dynamics, associated to different sets in the statespace [26]. These sets are separated each other by a border, often named as the guard in the hybrid systems literature[1], and it can happen that the vector fields across the border are directed toward the border itself. In this case a stablesliding mode arises and the border between the sets in the state space, defining different vector fields, is usually referredto as the sliding surface [98,38].

∗ Corresponding author.E-mail addresses: [email protected] (A. Pisano), [email protected] (E. Usai).

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A. Pisano, E. Usai / Mathematics and Computers in Simulation 81 (2011) 954–979 955

In the case of existence of a stable sliding mode, the sliding surface is an invariant set in the state space and underproper conditions the state trajectory is independent from the original systems dynamics [36], i.e., the constrainedmotion presents a semi-group property [101].

Such an invariance property, with respect to matching uncertainties, of sliding mode systems interested controlengineers who considered the opportunity of introducing switchings in the feedback intentionally, so that the closedloop control system has the desired performance, in spite of system uncertainties and external disturbance satisfyingthe matching condition [97,32,106].

In the last two decades a number of books [98,38][5], collections [112,55,107,109,83,40,21], special issues[8,74,53,22], and articles on the topic have been published.

Most of contributions are related to control systems where sliding mode system theory is used for the controllerdesign [105,37,85], possibly in combination with other techniques [9,27,84,80,58], while quite few applications tomathematical problems have been presented [60,77]. Nevertheless, sliding mode theory can have interesting appli-cations in solving ill-posed mathematical problems. Actually, if the variable structure system is represented by anequivalent differential inclusion, sliding mode control can be interpreted as a way of choosing a peculiar function froma set of possible solution (i.e., the solution of a differential inclusion that satisfies a constraint), or as a generator forthe internal model [48].

This feature is exploited in partial or full system inversion for state observation [91,47,94,31] or unknown inputestimation [86] and fault detection and identification [39]. Taking into account the control theory, and in particular thesliding mode approach, the system inversion problem, that in general implies the ill-posed problem of differentiating aknown signal, is regularized by considering a proper dynamic feedback system (one or more integrators) to be stabilizedto zero.

The aim of this paper is to give the reader an overview of the sliding mode control theory, showing the develop-ments from classical sliding modes to the higher-order sliding modes approach, and to present some applications tomathematical problems such as constrained ODE, differentiation, and finding zeroes of algebraic nonlinear systems.

The paper is organized as follows. First, the classic sliding mode control approach is presented in Section 2. Thenhigher-order sliding modes are discussed with some detail in Section 3. Since ideal sliding modes need infinite frequencyswitchings that are not possible in real systems, or when considering numerical algorithms, the effect of discretizationis discussed in Section 4. The considered applications are presented and discussed in Section 5, and finally someconclusions are drawn.

1.1. Notation and definitions

In this section some notations that will be used in the following treatment are given to the readers’ convenience, aswell as few definitions.

Notation 1. Let z = [z1, z2, . . . , zk]T ∈Rk be a vector of arbitrary dimension. Define the vectors sign(z), |z | and zp/q,with p > q odd natural numbers, as follows:

sign(z) = [sign(z1), sign(z2), . . . ,sign(zk)]T,

|z| = [|z1|, |z2|, . . . , |zk|]T ,zp/q =

[zp/q1 , z

p/q2 , . . . , z

p/qk

]T.

Notation 2. Consider a scalar function h(z), h : Rk → R. The gradient vector is denoted as follows:

∇h = ∂h

∂z=[∂h

∂z1, . . . ,

∂h

∂zk

].

956 A. Pisano, E. Usai / Mathematics and Computers in Simulation 81 (2011) 954–979

Notation 3. Consider a vector function h(z, w) of arguments z ∈Rk and w ∈Rl, h : Rk × Rl → Rm. The Jacobian

matrix of h with respect to variable z is denoted as follows

Jhz = ∂h

∂z=

⎡⎢⎢⎢⎢⎢⎣

∂h1

∂z1. . .

∂h1

∂zk...

......

∂hm

∂z1. . .

∂hm

∂zk

.

⎤⎥⎥⎥⎥⎥⎦

The Jacobian matrix Jhw is defined similarly. In case function h depends only on one variable, the notation can be

simplified into Jh, and with an extension of notation

∇h = Jh =

⎡⎢⎢⎢⎢⎣

∇h1

∇h2

...

∇hm

⎤⎥⎥⎥⎥⎦ .

Notation 4. Consider a smooth scalar function h(z), h : Rk → R and a vector function g(z), g : Rk → Rk, then the

projection of the gradient of h along the vector field g is referred to as the Lie derivative of h along g:

Lgh = ∇h · g.

Recursively it can be defined the following function [57,92]

Ligh = ∇(Li−1g h) · g (i = 1, 2, . . .) (i = 1, 2, . . .) ,

with L0gh = h.

This notation can be extended to vector functions h, where each row hj is a scalar function.

Definition 1. A vector field F(x) : Rn → Rn is called homogeneous of degree q∈R with the dilation dκ :

(x1, x2, . . . , xn) �→ (xm11 , xm2

2 , . . . , xmnn ), mi ∈R+, if the following identity holds [70]

F(x) = κ−qd−1κ F(dκx) ∀κ > 0.

Definition 2. A differential inclusion x ∈ F(x) is called homogeneous of degree q if it is invariant with respect to thecombined time-coordinate transformation Dκ : (t, x)� (κ−q, dκx) [70].

2. Sliding modes in discontinuous control systems

Consider a general nonlinear system

x(t) = f(x(t),u(t), t), (1)

where x ∈Rn is the state vector, u ∈Rq is the control input vector, t is time, and f : Rn × Rq × R+ → Rn is a vector

field in the state space.Assume that the state space is divided into 2q subspaces Sk (k = 1, 2, . . . , 2q) by the guard

G = {x : σ(x) = 0}, (2)

where σ : Rn → Rq is a sufficiently smooth vector function. Its ε-vicinity is defined as follows

Vε = {x ∈Rn : ‖σ(x)‖ ≤ ε; ε > 0}. (3)


Define the control vector by a state feedback law such that

u(t) = uk(x) if x ∈Sk, k ∈ {1, 2, . . . , 2q}, (4)

then the following theorem holds.

Theorem 1. Consider the nonlinear dynamics (1); if a proper ε defining (3) exists such that the control vector (4)satisfies the conditions

sign(Jσ

x (x(t)) · f(x(t),u(t), t)) = −sign(σ),∀x ∈Vε, (5)

then the guard G is an invariant set in the state space and a sliding mode occurs on it.

Proof. Consider the q-dimensional vector

s = σ(x), (6)

usually named sliding variables vector, and define the positive definite function

V (s) = 1

2sT s. (7)

The total time derivative of V is

V (s) = sT s = sT diag{sign(si)}|s|. (8)

Taking into account the implicit function theorem, (1), (4) and (5) then (8) results into

V (s) = −sT diag{sign(si)}|s| = −|s|T |s| < 0. (9)

Therefore V(s) is a Lyapunov function and the origin of the q-dimensional space of variables s is an asymptoticallystable equilibrium point. �

Remark 1. From a geometrical point of view, condition (5) implies that within the neighborhood Vε of G the vectorfield defining the state dynamics (1) is always directed towards G itself. Furthermore, if the magnitude of the controlvector u components is sufficiently large so that |si| ≥ η (i = 1, . . . , q), condition (9) satisfies the classical well-knownreaching condition (that also make s = 0 an invariant set) 1

2ddt |s|2 ≤ η|s| [98]. Therefore, the invariant set G is reached

in a finite time Tr ≤ t0 + |s(t0)|η

[98,92]. s(t0) (|s(t0)| ≤ ξ(ε) < ε) is the sliding variables vector at initial time t0.

Remark 2. From the definition of control u in (4), and taking into account condition (5), it is apparent that the vectorfield f defining the system dynamics (1) is discontinuous across the boundaries of the guardG. Therefore, function f(x(t),u(t), t) has to be Lebesgue integrable on time and solution of (1) exists in the Filippov sense [46]. Control u switchesat infinite frequency when the system performs a sliding mode on G, which is usually named sliding surface [98].

Remark 3. The number of sets Sk partitioning the state space can be less than 2q if a (q + 1)-dimensional controlvector u = [u1, . . ., uq+1]T, ui > 0, is available. In fact the sliding variable space can be partitioned into (q + 1) sets Sk,∩Sk = ∅, defining a simplex. In this case the convergence condition is a little bit different from (5)[18].

It is interesting to analyze the state trajectory when system (1) is constrained on G. A simple approach to the problemis to consider the variable s defined by (6) as the output of the dynamical system (1), in which function σ : Rn → R

q

represents the so-called output transformation. In classic sliding mode control usually condition (5) is assured by aproper choice of the control variables u so that matrix ∂s/∂u has full rank in Vε. Then the overall system dynamics canbe split into the input–output dynamics

s(t) = Jσx (x(t)) · f(x(t),u(t), t) = ϕ(x(t),u(t), t), (10)

and the internal dynamics

w(t) = ψ(w(t), s(t), t), (11)

where w ∈Rn−q is named internal state and ψ : Rn × R+ → Rn−q is a sufficiently smooth vector function.


The relationship between the vector state x and the new state variables s and w is defined by a diffeomorphism : Rn → R

n preserving the origin and defined as follows in a vicinity of the guard G [57,92]:

[sT ,wT ]T = (x) 0 = (0) ∀x ∈Vε. (12)

Theorem 2. Assume that the diffeomorphic transformation (12) holds in the vicinity Vε of the sliding manifold. Thensystem (1), (6) is stabilizable if a unique control u exists such that conditions of Theorem 1 are satisfied, the internaldynamics (11) is Bounded-Input Bounded-State (BIBS) stable and the zero dynamics

w(t) = ψ(w(t), 0, t) (13)

is stable in the Lyapunov sense.

Proof. The proof straightforwardly derives from results of Theorem 1 and the stability of the internal dynamics whenthe system is constrained onto G. �

Remark 4. When the state is constrained onto the sliding surface G, the system behavior is completely defined by thezero dynamics (13)[57], taking into account the invertible relationship (12). That is, only a reduced order dynamicshas to be considered during the sliding motion [98,38]. This “order reduction” property is a peculiar phenomenon invariable structure systems with sliding modes.

2.1. First order sliding mode control

Finding a feedback control (4) such that Theorem 1 holds is quite hard in the general case. Therefore, SlidingMode Control (SMC) of uncertain systems usually refers to systems whose dynamics is affine with respect to control[97,32,98,38] i.e.,

x(t) = A(x(t), t) + B(x(t), t)u(t), (14)

where A : Rn × R+ → Rn is a vector field in the state space, possibly uncertain, and B is a (n × q) matrix of functions

bij(x(t), t) : Rn × R+ → R.When the SMC approach is implemented, the first step of the design procedure is to define a proper system output

s, as in (6), such that the resulting internal dynamics is BIBS stable and, possibly, its zero dynamics is asymptoticallystable. Then the control u is designed such that |s| → 0 in a finite time in spite of possible uncertainties.

Theorem 3. Consider system (14), (6). Assume that the corresponding internal dynamics is BIBS stable, that thenorm of its uncertain drift term A(x(t), t) is upper bounded by a known function F : Rn → R

+, i.e.,

‖A(x, t)‖ ≤ F (x), (15)

and that the known square matrix G(x, t) ≡ Jσx (x) · B(x, t) ∈Rq×q is non singular ∀x ∈Vε, uniformly in time. Then,

the set G in (2) is made finite time stable by means of the control law

u(t) = −(F (x)‖Jσx‖ + η)[G(x, t)]−1 sign(s), η > 0. (16)

Proof. The input–output dynamics of system (14), (6) is

s(t) = Jσx (x) · A(x, t) + G(x, t)u(t), (17)

Consider the positive definite function (7). Considering the time derivative of V along the trajectories of system(17), and taking into account (16), (8) yields

V (s) = sT · (Jσx (x) · A(x, t) − (F (x)‖Jσ

x (x)‖ + η)sign(s)) ≤ −ηsT · sign(s) = −η‖s‖1 < −η‖s‖2 < 0. � (18)

When the system control gain matrix B(x, t) is uncertain, a similar theorem can be proved if some condition aboutB is met.


Theorem 4. Consider system (14), (6) satisfying (15). Assume that the corresponding internal dynamics is BIBSstable, that the uncertain gain matrix B(x, t) and the guard G are such that the square matrix G(x, t) ≡ Jσ

x (x) · B(x, t)is positive definite and a known bound Λm > 0 exists such that

Λm ≤ min{λGi (x, t); i = 1, . . . , q} ∀ x ∈Vε, ∀t (19)

where λGi (x, t) (i = 1, . . ., q) are the eigenvalues of matrix G(x, t). Then, the set G in (2) is made finite time stable by

means of the control law

u(t) = −F (x)‖Jσx (x)‖ + η

Λm

s‖s‖2

, η > 0. (20)

Proof. The proof follows the same steps than previous Theorem 3. Define function V(s) as in (7) and consider (17)and (20) into its time derivative (8). By (19) it results

V (s) = sT (Jσx (x)A(x, t) − F (x)‖Jσ

x (x)‖ + η

ΛmG(x, t)

s‖s‖2

) ≤ − η

Λm‖s‖2sTG(x, t)s ≤ −η‖s‖2 < 0. � (21)

Remark 5. To counteract the uncertainty in the system model the magnitude of the control vector components hasto be sufficiently large. The positive parameter η> 0 is a design parameter guaranteeing that the previously definedreaching condition is met. Several design methods can be found in the technical literature [98,38,107,109,83,40,21].

When the system exhibits a sliding-mode behavior, the discontinuous control (16), or (20), undergoes infinite-frequency switchings. The effect of the discontinuous and infinite-frequency switching control on the system dynamicsis the same as that of the continuous control which allows the state trajectory to remain on the sliding surface [46,98].Considering the non linear system (1) with a scalar input (i.e., q = 1), in [46] it was shown that such a continuousdynamics is a convex combination of the two vector fields f1 = f(x, u1, t) and f2 = f(x, u2, t) defined on S1 and S2,respectively, i.e.,

x(t) = f0(x, t),

f0 = αf1 + (1 − α)f2,

α = ∇σ · f2

∇σ · (f2 − f1).

(22)

The above approach to regularize differential equations with discontinuous right-hand side is called the Filippov’scontinuation method. In the case of multi-input control (4) the continuous vector field f0 allowing the continuation ofthe state trajectory on G is still a convex combination of the 2q vector fields fk = f(x, uk, t) (k = 1, 2, . . ., 2q), i.e.,

x(t) = f0(x, t),

f0 =2q∑k=1

αkfk,

2q∑k=1

αk = 1.

(23)

If the discontinuous right-hand-side of the differential equation defining the system dynamics satisfies some geo-metric conditions, the Filippov’s continuation method can be unambiguously defined on G [18]. Furthermore, onlyrecently explicit formulas for αk are available [33].

Utkin [98] introduced the concept of equivalent control as the continuous control input ueq which is able to maintainthe sliding-mode behavior by nullifying s, i.e., it is the solution of

x(t) = f(x,ueq, t),

s = ∂σ

∂x(x(t),ueq(t), t) · f(x(t),ueq(t), t) = 0.

(24)


With reference to affine systems (14), s takes the simplified expression reported in (17) thus the equivalent controlturns out to be defined as follows

ueq = −[G(x, t)]−1Jσx (x) · A(x, t). (25)

The Filippov’s continuation method and the Utkin’s equivalent control methods give the same continuous controllaw only for affine scalar control [98] and for a limited class of systems nonlinear in the control [6], recently extended[63,20].

The non-singularity condition for the q-dimensional square matrix G(x, t) implies that the control u appears explicitlyin the first derivative of the sliding variable vector s fulfilling a kind of controllability condition, i.e., the vector relativedegree [57] between the sliding and the input variables is [1, 1, . . ., 1]T.

Such a condition could be not satisfied and then the input would not appear in the first total derivative of the slidingvariable vector affecting, instead, its higher derivatives; in this case a Higher Order Sliding Mode (HOSM) can appear[42,49].

3. Higher-order sliding mode control

A second order sliding mode appears when the differential inclusion V(x, t) defining the closed loop dynamics (1),(4) belongs to the tangential space of the sliding manifold G defined as in (2), i.e. [42,64,50],

G2 = {x : σ(x) = σ(x) = 0.}. (26)

This definition can be extended to higher-order sliding manifolds as follows [64]:

Gr ={

x :dk

dtkσ(x) = 0, k = 0, 1, . . . , r − 1

}, (27)

where σ : Rn → Rq is, again, a sufficiently smooth vector function and r (r = 1, 2, . . .) represents the order of the

so-called sliding set. A rth-order sliding mode (r-SM) appears on the sliding manifold G when the state trajectory isconfined on (27).

Definition 3. [64] Let the r-sliding set (27) be non-empty and assume that it is locally an integral set in Filippov’ssense (i.e. it consists of Filippov’s trajectories of the discontinuous dynamic system). Then the corresponding motionsatisfying (27) is called r-Sliding Mode (r-SM) with respect to the constraint function σ.

A Higher-Order Sliding Mode control (HOSMC) system is implemented when the control u is able to constrain thesystem state onto (27) starting from any point in a ε-vicinity of Gr.

HOSMC systems are difficult to design with respect to the general nonlinear dynamics (1) since it is not possibleto trivially extend the definition of the sliding manifold (2) by using (27). In fact only q control variables are availableand condition (5) cannot be guaranteed with respect to the resulting rq variables s, s, . . . , s(r).

An affine time-independent structure for the nonlinear dynamics can be obtained by considering an augmenteddynamics in which the control u is part of an augmented vector state and its time derivative v = u is the actual controlto be designed:

⎡⎢⎢⎢⎣

x

z

u˙x

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎣

x

1

0∂f

∂xn+1(x,u, z) + ∂f

∂x(x,u, z) · x

⎤⎥⎥⎥⎥⎥⎦+

⎡⎢⎢⎢⎢⎢⎣

0

0

1∂f∂u

(x,u, z)

⎤⎥⎥⎥⎥⎥⎦ · v, (28)

in which ∂f/∂u is a full-rank matrix [64]. Therefore, when considering HOSMC, it is usual to refer to affine stationarynonlinear systems

x(t) = f(x) + g(x)u(t), (29)


where x ∈Rn is the state vector (possibly augmented), u ∈Rq is the control vector, possibly the time derivative of theplant input, f : Rn → R

n and g : Rn → Rn × Rq are sufficiently smooth vector fields and matrix, respectively, in the

state space.

Remark 6. For many years the possibility of considering the time derivative of the plant input as the discontinuouscontrol variable, given by the implementation of second-order sliding mode control systems, was considered an eligibleapproach to eliminate the chattering phenomenon in real control systems, since the plant input results to be continuous[42,64,7,43]. Recently, the chattering phenomenon was associated only to the non ideal dynamics of actuators andmeasurement devices that cause finite frequency switchings of the control command, and it was proved that chatteringcannot be eliminated but only attenuated in real-life applications [51,52,23,24,62].

Proposition 1. Given the system dynamics (29), a r-SMC on the manifold (2) can be designed if n ≥ rq, LgLkf σ =

0 (k = 1, 2, . . . , r − 2) and LgLr−1f σ has full rank.

Almost the simplest way to implement a HOSMC system is to consider a proper linear combination of the functionsdefining Gr as the output for system (29) by resorting to the so-called Dynamical sliding mode control [89].

Theorem 5. Consider system (29) and the set (2). Define the system output as

s = σ(r−1) +r−2∑i=0

ciσ(i), (30)

where ci (i = 0, 1, . . ., r − 2) are positive real constants such that the polynomial P(p) = pr−1 +∑r−2i=0 cip

i is Hurwitz.If the corresponding internal dynamics is BIBS stable and the system dynamics fulfill the following conditions:

‖Lkf σ‖ ≤ Λk(x) k = 1, 2, . . . , r,

LgLkf σ = 0 k = 0, 2, . . . , r − 2,

0 < Λm ≤ min{λi[LgLr−1f σ](x, t); i = 1, . . . , q} ∀x ∈V, ∀t,

(31)

whereΛk(x) : Rn → R+ (k = 1, 2, . . . , r) are sufficiently smooth positive functions, andΛm is a constant lower bound

for the eigenvalues of the matrix LgLr−1f σ, then the control law

u(t) = −Λr(x) +

r−2∑i=0

ciΛi+1(x) + η

Λm

s‖s‖2

, (32)

with η> 0, makes the integral manifold (27) asymptotically stable, and a rth-order sliding mode on the manifold G in(2) is established asymptotically.

Proof. Conditions (31) and (32) fulfil Theorem 4 and therefore s = 0 is achieved in a finite time Tr. From that timeinstant on the internal dynamics of variables σ(k), k = 0, 1, . . ., r − 1, is characterized by a linear dynamics whose stablemodes are defined by the Hurwitz polinomial P(p), and therefore σ(k) → 0, k = 0, 1, . . ., r − 1, asymptotically. �

The above approach to higher-order sliding mode control is, practically, an extension of classical first-order slidingmode. In fact the system state is steered onto a linear manifold in the new space defined by the variable σ and its timederivatives up to the (r − 1) th order, and on such a manifold the origin is reached asymptotically.

3.1. Second-order sliding mode control

A possible improvement for achieving the finite time convergence onto the r th-order sliding manifold (27) is toconsider an auxiliary nonlinear manifold such that once the state is confined on it the origin is reached in a finite time.This approach to SMC is strictly related to the finite time control [56]. On the basis of this idea, the terminal slidingmode control has been proposed to achieve a 2-SM behavior [111,45].


3.1.1. Terminal sliding mode controlConsider system (29) and assume that it has a well defined vector relative degree two with respect to the function σ

in (2) defining the sliding surface G, then a 2-SM on G can be reached in finite time by means of a non singular terminalsliding mode control [45].

Theorem 6. Consider system (29) and define its output as

s = σ + 1

βσp/m, (33)

where p and m, p > m, are odd natural numbers. If the system dynamics fulfills the following conditions:

‖L2f σ‖ ≤ Λ2(x),

Lgσ = 0,

LgLfσ is known and non singular ∀x ∈Vε, ∀t,(34)

where Λ2(x) : Rn → R+ is a sufficiently smooth positive function, then the control law

u(t) = −[LgLfσ]−1[βm

pσ2−p/m + (Λ2(x) + η) sign(s)

](35)

with η> 0, guarantees the finite time stability of the integral manifold (26) and a 2nd-order sliding mode is establishedon G.

Proof. The input–output dynamics of system (29), (33) is

s(t) = σ + 1

β

p

mdiag

{σ

2−p/mi

}(L2

f σ + [LgLfσ]

u(t)). (36)

Consider the positive definite function (7). Considering the time derivative of V along the trajectories of system(36), and taking into account (35), (8) is reduced into

V (s) = 1

β

p

msT · diag

{σp/m−1i

}·(L2

f σ + [LgLfσ]

u(t))

≤ − 1

β

p

mη|s|T · σp/m−1 < 0 ∀σ /= 0. (37)

Taking into account (29), (34) and (35) the second time derivative of σ is such that

σ ≤ −η sign(σ) ∀σ = 0; (38)

then V = 0 only in not stationary insulated points and therefore finite time stability is guaranteed. �The terminal SMC [111,45] is a realization of the 2-SMC with prescribed law of variation described in [42,64] and

some slightly different control laws can also be implemented to deal with system uncertainty [45].Differently to the above algorithms which, in some sense, exploit features of classical sliding modes, several control

algorithms that implement 2-SMC directly (the sliding variable is defined by equation (6)) have been presented inthe literature [64,10,88,71]. These algorithms are characterized by complex trajectories in the (σ, σ) plane whichmake difficult to prove their stability by means of Lyapunov techniques. Actually, apart from recent results [79,75],the stability and finite time convergence of most of them have been proved by means of either the analysis of thegeometrical properties of limiting trajectories in the phase plane [64,7] or homogeneity property of the resultingdifferential inclusions [71,76].

Because of the lack of Lyapunov-like tools, most of the control algorithms that implement HOSM were developedwith respect to single input dynamical systems, and their extension to multi input systems requires more restrictiveassumptions on the gain matrix [11]. Therefore, system dynamics (29) and (27) with q = 1 will be considered in thefollowing treatment.

Considering the most used 2-SMC algorithms [10,71] it is useful to refer mainly to two of them: the Super-Twisting[64] and the Generalized Sub-Optimal [17] algorithms. The first can be applied to relative degree one systems and ischaracterized by a dynamic controller that, using only the current value of the system output (the sliding variable),applies a continuous control to the system input while maintaining the discontinuity on the time derivative of the plantinput. The latter can be applied to relative degree two systems and implement a feedback switching logic derived fromthe time-optimal control problem [2] and based on the current and past values of the system output.


3.1.2. The Generalized Sub-Optimal controllerBoth the “twisting” [64] and the “sub-optimal” algorithm [7] can deal with relative-degree two constraint variables.

They are, both, special cases of the general algorithm [17]

u(t) = −α(t)U sign(s− βsM),

α(t) ={

1, if (s− βsM)sM ≥ 0

α∗, if (s− βsM)sM < 0,

β∈ [0; 1),

(39)

where U is the minimum control magnitude, α∗ ≥ 1 is called the modulation factor, β is the anticipation factor, andsM is the value of the sliding variable s corresponding to the most recent local minimum, maximum or horizontal flexpoint (i.e., the value of s at the last time instant tM,k at which s = 0 occurs). sM can be evaluated either by checkingsign(s) or, by inspection of the past values of s(t), or, approximately, by inspection of the first-difference of s(t); in lasttwo cases no information about s is needed. U, α∗, and β are the controller parameters, that must be tuned to assurethe finite time convergence onto the sliding set G2 in (26).

Theorem 7. Consider system (29) and define its output as (6) with q = 1. If the system dynamics fulfill the followingconditions:

|L2f σ| ≤ Λ,

Lgσ = 0,

0 < Γm ≤ LgLfσ ≤ ΓM,

∀x ∈Vε, ∀t, (40)

where Λ, Γm, ΓM ∈R+ are known constants, then the control law (39) with

U >Λ

Γm,

α∗ ∈ [1; +∞) ∩(

2Λ+ (1 − β)ΓMU

(1 + β)ΓmU; +∞

),

(41)

guarantees the finite time stability of the integral manifold (26), and a 2nd-order sliding mode is established on G.

Proof. The second order sliding dynamics of system (29), (6), satisfying (40) and driven by (39)(41), can be definedby the differential inclusion

s∈ [−Λ,+Λ] + [Γm, ΓM] u. (42)

Considering the limiting trajectories of (39)(42) it can be shown, by means of algebraic computations, that thecondition | sM,k+1 |≤ρ | sM,k |, ρ∈ (0, 1), k = 1, 2, . . . (sM,k = s(tk) is the sequence of singular points of s(t)) is guaranteedif (41) is fulfilled. Therefore the kth singular point of variable s is such that

|sM,k| ≤ ρk−1|sM,1|, (43)

and it also results

|sM,k| = maxt ∈ [tk ;tk+1]|s(t)| ≤ ρ((k−1)/2)√

2(1 − β)(ΛMU +Λ)|sM,1|, (44)

so that both sM,k and sM,k tends to 0 as k → ∞.Furthermore, it results that the sequence tk (k = 1, 2, . . .) of time instants at which a local minimum, maximum or

horizontal flex point of σ(t) occurs is such that tk+1 − tk ≤ �(tk − tk−1), k = 2, 3, . . .,�∈ (0, 1). Therefore the convergencetime is upper bounded by a geometrical serie and it is finite:

t∞ ≤ t1 + max{Tc1, Tc2}, (45)


Tc1 = Uα∗Γm + ΓM

α∗ΓmU −Λ

√2(1 − β)|σM,1|ΓMU +Λ

1

1 −√|Λ+ [(1 − β)ΓM − α∗βΓm]U|√

α∗ΓmU −Λ

,

Tc2 = Uα∗ΓM + Γm

α∗ΓMU +Λ

√2(1 − β)|σM,1|ΓmU −Λ

1

1 −√|Λ− [(1 − β)Γm − α∗βΓM]U|√

α∗ΓMU +Λ

.

�

Remark 7. The first of (41) can be referred to as the dominance condition, that is the minimum control authorityneeded to determine the sign of the second time derivative of the sliding variable. The second of (41) represents theconvergence condition, and it determines the convergence rate as well. Generally, these conditions lead to transienttrajectories twisting around the origin of the phase plane of the sliding variable σ. The additional requirement ofmonotonic convergence to zero of the sliding variable may be fulfilled by imposing a stricter inequality than (41):

U >Λ

Γm,

α∗ ∈ [1; +∞) ∩[Λ+ (1 − β)ΓMU

βΓmU; +∞

).

(46)

Remark 8. From Eqs. (45) and (44) it is apparent that the largest the control magnitude U, the shorter the convergencetime and the largest the maximal values of the time derivative of the sliding variable. Nevertheless, a proper choiceof the controller parameters U, α∗ and β allows for, both, the monotonic convergence to zero of the sliding variableand the counteraction of the peaking phenomenon during the reaching phase of the 2-SM behavior. Obviously, strictrequirements on |σ(t)| imply a long reaching phase; at the limit, if β→ 1 then |σ(t)| → 0 and a sliding mode on thesurface σ = 0 tends to be established (refer to the Drift 2-SMC [64]), but this implies that the sliding behavior isreached in an almost infinite time, i.e., t∞ → ∞.

Different 2-SMC algorithms presented in the literature are characterized by different values of the anticipationcoefficient β: β = 0 gives the Twisting control algorithm [64], β = 1/2 gives the Sub-Optimal [7].

Condition (40), that implies the local convergence of the control algorithm, can be relaxed so that a globalconvergence can be assured by means of a proper adaptation of parameter β [13].

3.1.3. The Super-Twisting controllerThe so-called Super-Twisting algorithm is conceptually different from the other 2-SMC algorithms, for two reasons:

first, it depends only on the actual value of the sliding variable, while the others have more information demand. Second,it is effective only for chattering attenuation purposes as far as relative degree one constraints are dealt with.

It is defined by the following dynamic controller [64]:

u(t) = v(t) − λ|s(t)|1/2sign(s(t)),

v(t) = −α sign(s(t)).(47)

where u(t) ∈R is the input of system (29) with q = 1 and s(t) ∈R is the sliding variable (i.e., the system output) (6)measuring the distance of the system from the sliding surface G in (2).

Theorem 8. Consider system (29), define its output as (6) with q = 1 and assume that its trajectories are infinitelyextendible in time for any bounded feedback control. If the system dynamics fulfill the following conditions:

|L2f σ + (LfLgσ + LgLfσ

)u+ L2

gσ u2| ≤ Λ,

0 < Γm ≤ Lgσ ≤ ΓM ;∀x ∈Vε, ∀t, (48)


where Λ, Γm, ΓM ∈R+ are known constants, then the control law (47) with

α >Λ

Γm,

λ2 > 2αΓM +Λ

Γm,

(49)

guarantees the finite time stability of the integral manifold (26), and a 2nd-order sliding mode is established on G.

Proof. Since the available results based on Lyapunov functions are limited to systems with constant control gain g[79] and with zero equivalent control [75], the proof of the theorem is based on the analysis of the differential inclusiondefining the limiting trajectories of system (29), (6), (47) and only its rationale is hereafter reported.

The differential inclusion including system (29), (6), (47) is as follows [64]

s∈ − [αΓm −Λ,αΓM +Λ] sign(s) − λ

2[Γm, ΓM]

s

|s|1/2 . (50)

The trajectories associated to (50) twist around the origin of the (s, s) plane featuring a sequence of singular pointssM,k on the s = 0 axis. As for the Generalized Sub-Optimal controller, such a sequence is contractive if conditions (49)are satisfied, and the encircling time sequence is upper estimated by a geometrical serie so that convergence to thesecond order sliding manifold G2 occurs in a finite time. �

Remark 9. The Super-Twisting controller can be considered as a nonlinear implementation of a classic PI controllerwith better robustness properties.

3.2. Arbitrary order sliding mode controllers

Consider the problem of finite time stabilization of a r th-order sliding mode for system (29) satisfying Proposition 1with q = 1. Because of the difficulties in defining r-sliding controllers with r ≥ 3 only recently some results are availablefor this case [67,68,70,15,19].

The controllers in [15] and [19] are limited to the case r = 3 and only the latter solves the problem for uncertainsystems by means of a hybrid controller that uses only the sign of s, s and s. The controllers in [67,68] are built ina recursive way, so that a r-order controller embed a (r − 1)-order one and they require the availability of the slidingvariable and its time derivatives up to the (r − 1) th order.

Let m be the least common multiple of 1, 2, . . ., r and define the following quantities

Ni,r =(i−1∑k=0

|s(k)|m/(r−k)

)(r−i)/m, i = 1, . . . , r − 1, (51)

and {φ0,r = s

φi,r = s(i) + βiNi,rsign(φi−1,r

), i = 1, . . . , r − 1

. (52)

The following Theorem was proved in [67]:

Theorem 9. Consider system (29)(6), with q = 1, and assume that its trajectories are infinitely extendible in time forany Lebesgue-measurable bounded feedback control. Then, if the following conditions hold for some constants Λr,Λm and ΛM,

‖Lrfσ‖ ≤ Λr,

LgLkf σ = 0, k = 0, 2, . . . , r − 2,

0 < Γm ≤ LgLr−1f σ ≤ ΓM ∀x ∈Vε, ∀t,

(53)

then, with properly chosen positive parameters β1, β2, . . ., βr−1, α, the controller

u = −α sign(φr−1,r(s, s, . . . , s(r−1))), (54)


where r−1,r( · ) is defined in (51) and (52), makes the integral set (27) finite time stable and a r-sliding mode on themanifold G in (2) is established.

Proof. The proof of the theorem [67] is quite cumbersome and only its rationale is here reported for the sake ofbrevity. Define the subsets

Γi = {(s, s, . . . , s(i)) : φi,r = 0}, i = 1, . . . , r − 1,

and let be Γ = ∪ iΓ i. The controller (54) establishes a sliding mode in the continuity points of the discontinuity subsetΓ r−1. Such a sliding mode, described by the differential equation φr−1,r = 0, provides for the finite time establishmentof a 1-sliding mode on �r−2 satisfying φr−2,r = 0. Note that the primary sliding mode disappears when the secondaryone is to appear. The resulting movement on φr−2,r = 0 transfers in finite time into some vicinity of the subset Γ r−3satisfying φr−3,r = 0 and so on. While the trajectory approaches the r-sliding set, Γ contracts to the origin in thecoordinates s, s, . . . , s(r−1).

The lower bound for the controller’s parametersβ1,β2, . . .,βr−1,α are defined by considering the limiting trajectoriesof the sliding dynamics which, under conditions (53), belong to the following differential inclusion

s(r) ∈ [−Λr,+Λr] + [Γm, ΓM] u. � (55)

The above Theorem determines a controller family (54) applicable to all systems of the type (29)(6), with q = 1 andrelative degree r, satisfying (53). Parameters β1, . . ., βr−1 affect the reaching time and are to be chosen sufficientlylarge in index order. Such a parameters βi can be preliminarily chosen for each r in advance, while parameter α mustbe chosen specifically on the knowledge, or estimation, of the boundsΛr, Γm and ΓM of the uncertain dynamics. Thecontroller performance is insensitive to any system perturbation preserving these bounds.

In the following few examples of sliding controllers are reported [67] for relative degree r ≤ 4:

r = 1. u = −α sign(s)

r = 2. u = −α sign(s+ |s|1/2sign s)

r = 3. u = −α sign(s+ 2(|s|3 + |s|2)1/6

sign(s+ |s|2/3sign s)

r = 4. u = −α sign{s(3) + 3(s6 + s4 + |s|3)1/12

sign[s+ (s4 + |s|3)1/6

sign(s+ 0.5|s|3/4sign s)]}

Remark 10. It should be noted that controllers (39), (47) and (54) are homogeneous since they preserve the differentialinclusions (42), (50) and (55), respectively.

4. Accuracy of discrete time implementation of sliding mode control algorithms

Sliding mode control is characterized by infinite frequency switching in the steady state. No real analogue deviceis able to guarantee such a performance and even computer implementation of the control law has to face the limitof any digital device: round-off errors and machine cycle period. These considerations support the investigation onthe guaranteed accuracy of real sliding mode control implementations. Two different problems can be considered: theeffect of discrete-time Sample&Hold implementation of VSC on continuous systems, the possibility of extending thesliding mode concept to discrete-time systems. The first is strictly related to the control of plants by means of digitalcontrollers and therefore is of major interest for control engineers. The second is a kind of idealization of sampled-data systems but it can also be considered as an actual problem when control techniques are used to solve numericalproblems.

If perfectly known discrete-time systems are considered, the dead-beat control approach can be considered as anelective method since it guarantees the reaching of the zero error after a finite number of sampling steps [61,41]. Thismethod does not require any discontinuous right-hand-side difference equation so that the notion of sliding mode isseparated from discontinuity, but it fails in the presence of uncertainty.

Taking into account the uncertainty, a definition of Discrete-time Sliding Mode (DSM) should be given [5]. Severaldefinitions were presented in the literature and often they are used to define the equivalent control ueq,k; among them


we recall the definitions of ueq,k by sk+1 = 0 [35] and by sk+1 − sk = 0 [54] (k = 0, 1, . . .), where sk represents the sampleof s(t) at t = k Ts, Ts being the sampling period.

Nevertheless, much more interesting is the case in which, starting from a continuous sliding mode design, the effectof discretization is analyzed. Furthermore this can give an upper bound of the errors in full discrete-time systems aswell. This problem was considered since early

′80s [73] and it was shown that only a O(Ts) boundary layer of the

sliding surface can be considered as an invariant set for the discrete-time sliding mode, when the classical first-ordersliding mode design is considered. The accuracy can be improved to O(T 2

s ) if a proper learning adaptation is includedin the control law [102].

A great improvement in the accuracy of discrete-time implementation of sliding mode control techniques can beobtained by means of HOSM [64,70].

Theorem 10. Consider system (29)(6), with q = 1, and assume that conditions of Theorem 9 holds. Then, the appli-cation of a stable homogeneous r-order sliding mode controller with discrete time measurement and Sample&Holdimplementation gives the following finite time accuracy∣∣∣s(k)

∣∣∣ ≤ υk τr−k, k = 0, 1, . . . , r − 1, (56)

where τ is either the sampling period or the switching delay, and υk are r positive real constants independent on τ.

Proof. Under the considered assumption the system trajectories are infinitely extendible in time for any Lebesgue-measurable bounded feedback control so that s(t) ∈ Cr−1 and the following condition hold for some positiveconstant �r,∣∣∣∣drsdtr

∣∣∣∣ ≤ �r, ∀x ∈V, ∀t. (57)

Successive application of the Lagrange’s theorem can show that, for any time interval T of length T there existsconstants Kk (k = 0, 1, . . ., r − 1) and a time instant t∗ ∈ T such that the following relationship is verified∣∣∣∣∣d

ps(k)

dtp

∣∣∣∣∣t=t∗

≤ Kk

∣∣∣∣supt ∈ T

s(k)∣∣∣∣ τ−p ∀p = 1, . . . , r − 1 − k. (58)

By integrating s(k) k times (k = 1, 2, . . ., r), taking into account (57) and (58), relationships (56) can be verified withυk =ϒk(�r), where ϒ : R+ → R

+ is a K∞ function. �The above theorem shows that the accuracy of a discrete-time implementation of a sliding mode control system can

be improved by increasing the sliding order r [64], or by adjusting the current value of the discontinuous part of thecontroller, i.e., reducing �r [102,14].

5. Sliding mode approach to some computational problems

The main idea of SMC systems is to design a control, with a suitable authority, which promptly reacts to any,however small, deviation from a prescribed behavior steering the system back to the so-called sliding manifold, wherethe expected operation mode will take place. An advantage of this approach is that the sliding behavior is insensitiveto model uncertainties and/or disturbances which do not steer the system outside from the chosen manifold. Such arobustness property has gained the interest of engineers [106] and a number of applications of sliding mode controltechniques have been presented in the literature. Only few of them are cited here [100,17,111,87,90,93,66,30,82,103,99].

Much less, and often related to control problems, are the applications of sliding mode theory to simulation and to thesolution of numerical problems. Among them we can recall the sliding mode approach to extremum seeking [60,77],the simulation of constrained systems [25,81], the real-time differentiation [65,108], and the solution of nonlinearalgebraic equations [3,4]. In the following the last three applications will be presented in brief.

5.1. Solution of ordinary differential-algebraic equations (DAEs)

DAEs (Differential-Algebraic Equations) represent an appropriate mathematical tool for modelling constrainedmulti-body systems, as well as other physical objects of engineering relevance. Their numerical solution is not easyand requires the use of rather complicated techniques.


The next developments are based on the lagrangian formulation of the dynamic equations for a mechanical multi-body system with holonomic constraints, which consists of a redundant set of n + m differential algebraic equations (nis the number of bodies, m is the dimension of the constraint manifold):⎧⎨

⎩M(q)q + C(q, q)q + g(q) + k(q) = τ + f = τ +[Jφ

q

]Tλ

φ(q) = 0(59)

where q ∈Rn is the vector of the generalized position coordinates, f =[Jφ

q

]Tλ is the vector of the generalized constraint

(reaction) forces, λ∈Rm is the Lagrange multipliers vector, τ ∈Rn is the vector of the generalized input forces, M(q)is the inertia matrix (symmetric and positive definite), C(q, q)q includes the Coriolis and centrifugal effects, and g(q),k(q) are the vector of the gravitational and elastic generalized forces. The smooth m-dimensional constraint manifoldis defined by the second equation of (59), where φ(q) : Rn → R

m is at least twice continuously differentiable.There are two main approaches to the solution of the differential-algebraic system: constraint stabilization approach,

and order reduction approach [104,72,78]. In both methods, the system initial conditions q(t0), q(t0) should be consistentwith the constraint, i.e. the following conditions should be met:

φ(q(t0)) = 0, φ(q(t0), q(t0)) = 0, (60)

Here we describe an alternative method to solve DAEs, which exploits concepts and tools from the systems andcontrol theory. The main idea is to evaluate the Lagrange multipliers vector via a feedback scheme. The Lagrangemultipliers can be understood as the “adjustable input variables” of a fictitious system where the constraint variablesare treated as the system outputs. The control task is to nullify the output variables by properly adjusting the Lagrangemultipliers according to some feedback-based algorithm. It is a multivariable nonlinear stabilization problem that canbe properly addressed via the sliding mode control approach.

The Lagrange multipliers appear explicitly in the second derivative of the constraint function. Differentiating twicethe constraint function achieve

φ = α(q, q, τ) + H(q)λ, (61)

H(q) = Jφq(q)M−1(q)

(Jφ

q(q))T, (62)

with implicitly defined function α(q, q, τ). Define the following sliding variable

s = φ + cφ, c > 0. (63)

If the sliding variable is nullified in finite time then the constraint variable will be asymptotically vanishing accordingto the solution of the differential equation φ + cφ = 0.

The sliding variable dynamics is

s = α(q, q, τ) + cJφq(q)q+ H(q)λ, c > 0 (64)

Since matrix H(q) is positive definite by construction, classical first-order SMC (see Theorem 4) can be applied toevaluate appropriate Lagrange multipliers nullifying the sliding variable vector s in finite time. A simplified realizationof the control algorithm (19) is suggested, with the magnitude of the components of the control vector u = λ beingconstant:

λ = −ΛM0 sign(σ), (65)

Another controller option is to make use of the second-order sliding mode approach using, e.g., the super-twistingalgorithm (see Section 3.1.3). The dynamical control law (47) representing the Super-Twisting algorithm can berewritten in more compact form as follows:

λ = −ΛM1√

|s| sign(s) −ΛM2

∫ t

0sign(s). (66)

A different form for the sliding variable can be also considered:

s = φ. (67)


l1l2

q1

q2

m1

m2

y

x

Fig. 1. Schematic representation of the considered constrained planar manipulator.

According to (61), the constraint variable s =φ has a uniform (vector) relative degree [2, 2, . . ., 2] with respect tothe Lagrange multiplier vector; therefore, the discontinuous “Generalized sub-optimal” algorithm (39) (Section 3.1.2)can be implemented using the sliding variable s =φ.

The suggested three alternatives are now going to be implemented and compared via computer simulations consid-ering an application example.

5.1.1. Constrained robot manipulator exampleConsider a kinematic chain of rigid links with n = 2 degrees of freedom. Let the end-effector be constrained on a

smooth one-dimensional manifold, and let a frictionless interaction occurs with the infinitely stiff rigid environment.The studied simulation example considers a 2-dof planar manipulator whose end effector is constrained on the planey = 0 (see Fig. 1).

Let li and mi (i = 1, 2) be the length and mass of the ith link. The constraint function takes the form

φ(q) = l1 sin (q1) + l2 sin (q1 + q2) = 0. (68)

The torque vector is τ = [τ1 τ2]T = [sin (t), 0]T. Assigning to the robot physical parameters the values l1 = 1, m1 = 1,l2 = 1, m2 = 0.5, the matrices of the differential part of model (59) are derived as follows:

M(q) =[

0.83 + 0.5 cos(q2) 0.16 + 0.25 cos(q2)

0.16 + 0.25 cos(q2) 0.16

], (69)

C(q, q) =[

−1.01q2 sin(q2) 1.01(q1 + q2) sin(q2)

1.01q1 sin(q2) 0

], (70)

g(q) =[

9.8 cos(q1) + 2.45 cos(q1 + q2)

2.45 cos(q1 + q2)

], k(q) =

[0

0

]. (71)

The plots show the simulation results obtained using the first order SMC technique (65) and the Super-Twistingsecond-order SMC algorithm (66). The tuning parameters of the algorithms are set asΛM0 = 300,ΛM1 = 10,ΛM2 = 100.Fig. 2 shows the constraint variable φ using the two approaches. The higher accuracy of the 2-SMC approach isapparent. The plots in Fig. 3 show the Lagrange multipliers. The Lagrange multiplier corresponding to the first-orderSMC approach is discontinuous and high-frequency switching, as expected, while that corresponding to the second-order SMC is a continuous one. To reduce the computational effort of the algorithm, the value of φ is estimated via


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1x 10−3

Time [sec]

The constraint function with the 1−SMC approach

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1 x 10−3 The constraint function with the 2−SMC approach

Fig. 2. The constraint variable using first-order SMC (left) and second-order SMC (right).

backward-difference numerical differentiation. All the simulations have been made by means of MATLAB-Simulink©

with the fixed step ODE5 integration method, with the step Ts = 10−5s.Higher accuracy is obtained using the discontinuous “Generalized sub-optimal” algorithm (39) with the sliding

manifold (67). The tuning parameters have been set as U = 300,β = 0.8,α∗ = 1. The accuracy in the constraint fulfillmentis considerably higher than the previous case, as it can be seen comparing Fig. 4 with the two plots in Fig. 2. A reasonfor the better performance of the last method is that it does not require the estimation of φ.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−400

−300

−200

−100

0

100

200

300

400

Time [sec]

The Lagrange Multiplier with the 1−SMC approach

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−100

−80

−60

−40

−20

0

20

Time [sec]

The Lagrange Multiplier with the 2−SMC approach

Fig. 3. The lagrange multiplier λ using first-order SMC (left) and second-order SMC (right).

0 0.5 1 1.5 2 2.5 3 3.5 4−1

−0.5

0

0.5

1 x 10−5

Time [sec]

The constraint variable using Sub−Optimal 2−SMC

Fig. 4. The constraint variable using sub-optimal second-order SMC with σ =φ.


5.2. Real-time differentiation

The computation of the derivatives of a signal is an old problem in numerical analysis and signal processing. Incontrol theory, real-time differentiators (RTD) play an important role in linear and non-linear output feedback controldesign, especially under model uncertainties [28,29,16,69].

High-gain observers probably constitute the most popular approach to feedback-oriented real-time differentiation[44,95,29] because of their accuracy, simplicity and linear dynamics. The backward-difference method (also known asthe Euler’s formula) gives the simplest method for numerical differentiation. Its use is quite common in engineeringapplications in spite of its poor performance in the presence of measurement noise. Spline interpolation techniques[34] have been also applied to develop numerical differentiation algorithms. Unfortunately, all these methods areintrinsically affected by an estimation error, even in the ideal case.

Nevertheless, the high-gain observers approach presents an interesting regularization procedure for the ill-poseddifferentiation problem. In fact the problem is solved by forcing a proper dynamical system, generally a chain ofintegrators, to track an uncertain signal, so that differentiation is turned into integration.

If the uncertain signal to track has some boundedness properties, the tracking problem can be solved exactly bymeans of the higher-order sliding mode control approach [65,69,43,16,12]. One of the most effective differentiationalgorithms is the Arbitrary order differentiator [69].

Theorem 11. Consider a bounded signal z(t) ∈L(r−1) ⊂ C(r−1), where L(r−1) is the set of functions having a Lipshitz(r − 1)th derivative, i.e.,

|z(t)| ≤ Z0∣∣z(r)(t)∣∣ ≤ Zr

∀t, (72)

where Z0 and Zr are known positive constants. Then the dynamical system

x0 = x1 − λ0|x0 − z|(r/(r+1))sign (x0 − z) ,...

xi = xi+1 − λi|xi − xi−1|((r−i)/(r+1−i))sign (xi − xi−1) , i = 1, . . . , r − 1,...

xr = −λr sign (xr − xr−1) ,

(73)

with λr > Zr and λi > 0 (i = 1, 2, . . ., r) chosen sufficiently large in the given order, implements a r-order finite timedifferentiator and a time instant t∞ exists such that |x0(t) − z(t) | = 0 and |xi(t) − z(i+1)(t)| = 0, i = 0, 1, . . ., r − 1 forany t ≥ t∞.

The proof of the theorem [69] is cumbersome and it is not reported here for the sake of brevity. It is possible toverify that (73) is a homogeneous (r + 1)-sliding mode controller for an uncertain dynamical system having relativedegree (r + 1) with respect to the sliding output s(t) = x0(t) − z(t). Such a higher-order sliding mode controller belongsto the same class of controllers which the controller considered in Theorem 9 belongs to, therefore, the theorem canbe extended to the arbitrary order differentiator.

When considering sampled data systems or noisy signals the following corollaries are verified.

Corollary 1 to Theorem 11. Let Ts > 0 be the constant input sampling period for the rth-order differentiator (73),in the absence of noise. Then some positive constants νi and ρi (i = 0, 1, . . ., r − 1), independent on Tc, exist such thatthe following steady state estimation error are assured:∣∣xi − z(i)

∣∣ ≤ ρiTr−i+1c , i = 0, 1, . . . , r;∣∣xi − z(i+1)

∣∣ ≤ νiTr−ic , i = 0, 1, . . . , r.

(74)

Corollary 2 to Theorem 11. Let d(t) be an additive noise superimposed to the signal to be differentiated z(t) andsuch that | d(t) |≤δ. Then some positive constants μi and ς (i = 0, 1, . . ., r − 1) exist such that the following steady state


−0.15

−0.1

−0.05

0

0.05

Sig

nal t

rack

ing

−1

−0.5

0

0.5

1st d

eriv

ativ

es

0 5 10 15 20−2

−1

0

1

time [s]

2nd d

eriv

ativ

es

Fig. 5. The actual (dashed) and estimated signals.

estimation error are assured:∣∣xi − z(i)∣∣ ≤ μiδ

((r−i+1)/(r+1)), i = 0, 1, . . . , r;∣∣xi − z(i+1)∣∣ ≤ ςiδ

r−i/r+1, i = 0, 1, . . . , r − 1.(75)

The differentiator’s parameters λi, i = 0, 1, . . ., r, define the convergence speed of the differentiator and have to bedefined such that λi > λi+1. They can be evaluated, for any order of differentiation, with respect to Zr = 1 and thereforeadjusted for any Zr: λi = λiZ

(1/(r−i+1))r , where λi is the generic differentiator parameter for Zr = 1.

5.2.1. Double differentiator exampleConsider a double differentiator with the parameters values taken as suggested in [69]

x0 = x1 − 3|x0 − z|(2/(3+1))sign (x0 − z)

x1 = x2 − 1.5|x1 − x0|1/2sign (x1 − x0)

x2 = −1.1 sign (x2 − x1) .

(76)

The signal to be differentiated twice is obtained by triple integration of a sawtooth with range from −1 to +1,frequency 0.5Hz. Simulation has been carried out by means of MATLAB-Simulink with the Euler’s integration method,using a 1ms step size. After 10s the differentiator is switched from the off to the on position, and after a short transientvery good estimates of the first and second derivatives of the available signal are given in the absence of noise (Fig. 5).The estimation errors at steady-state follow exactly the asymptotic accuracy given by (74) (Fig. 6). If an additivewhite noise with magnitude δ= 0.001 is added to the signal to be differentiate the estimation error does not divergeand remains limited according to (75) (Fig. 7), and the following steady-state errors are obtained: |z0 − y | ∞ < 0.002,|z0 − y|∞ < 0.04, and |z1 − y|∞ < 0.41.

5.3. Finding zeroes of nonlinear algebraic systems

Given a function f( · ) of an independent variable x, the problem of finding the value(s) of x such that the functionis nullified is a well know problem in computational analysis. Recently it has been shown that such a problem can bereduced to a regulation problem, i.e., how to design an output feedback controller, static or dynamical, such that theoutput of a proper dynamical system is steered to zero [3,4].


Proposition 2. Consider the nonlinear vector function f : Rn → Rn, the problem of finding the solutions of the

algebraic system f(x) = 0 is equivalent to the regulation problem for the dynamical system{x(t) = u(t),

y(t) = f(x),(77)

where x is the state vector variable and y is the output transformation.

In general the output transformation f(x) of system (77) does not preserve the origin and it is more useful to reducethe dynamic system into its input–output form

y(t) = ∇f(x) · u(t). (78)

It is evident that a solution to the feedback regulation problem stated in Proposition 2 can be found if ∇f(x) is notsingular in a neighborhood X of x, X = {x : ‖x − x‖ < ε, f(x) = 0}, and therefore function f is locally invertible.

−4

−2

0

2

4x 10

−9

sign

. tra

ck. e

rror

−4

−2

0

2

4x 10

−6

1st d

er. e

st. e

rror

15 16 17 18 19 20−4

−2

0

2

4x 10

−3

time [s]

2nd d

er. e

st. e

rror

Fig. 6. The steady-state estimation errors.

−0.1

−0.05

0

0.05

Sig

nal t

rack

ing

−1

−0.5

0

0.5

1st d

eriv

ativ

es

0 5 10 15 20−2

−1

0

1

time [s]

2nd d

eriv

ativ

es

Fig. 7. The ideal (dashed) and estimated signals.


It has been shown, by means of Lyapunov techniques, that several algorithms for finding the zeroes of a nonlinearalgebraic system can be reduced to different control laws u(x) [3,4] which assure the local output stability of (78).

An interesting approach is that of re-formulating the problem in terms of finding a special solution of a differentialinclusion, i.e.,

Problem: Find the solution x(t) of the differential inclusion

x(t) ∈U = {u : ‖u‖ ≤ UM} (79)

such that f(x(t)) ≡ 0.Such a formulation suggest to exploit the properties of VSC with sliding mode in terms of internal model [48]

generator.

Theorem 12. Consider system (77), if the output transformation is such that condition

det (∇f(x)) /= 0 ∀x ∈X (80)

is fulfilled, then the control law

u = −∇T f(x)sign(y) (81)

assures the finite time reaching of f(x) = 0 from any x ∈X.

Proof. Consider the positive definite function

V (y) = ‖y‖1. (82)

Taking into account (77) and (81) its total time derivative is

V (y) = sign(y)T · y = −∥∥∇T f · sign(y)∥∥2

2, (83)

which is definite negative under condition (80), and therefore (82) is a Lyapunov function such that V ≤ −η√V . �

Other variable structure feedback laws are also discussed in [3].

5.3.1. Rosenbrock function minimization example

The Rosenbrock function, g(x) : R2 → R : (x1, x2) �→ a(x21 − x2)

2 + (x1 − b)2, is a well known example of hardoptimization problem. It can be restated as the problem of finding the zeroes of f = ∇g,

f(x) =[f1(x)

f2(x)

]=[

4ax1(x21 − x2) + 2(x1 − b)

−2a(x21 − x2)

]. (84)

It is easy to check that the unique solution of the system f(x) = 0 is x = [b, b2]T

, and that the Jacobian of f is givenby the symmetric matrix

Jf(x) =[

4ax1(3x21 − x2) + 2 −4ax1

−4ax1 2a

]; (85)

having singular points on the line defined by x2 = x21 + (1/2a).

The simulation of the dynamical system (77),(81) particularized for (84) was carried out by means of MATLAB-Simulink® with the following parameters: a = 0.5, b = 1, x1(0) = 0.5, x2(0) = 3, Euler’s integration method with stepsize Tc = 10−4. The simulation result is reported in Fig. 8 where the transient lasts about 4.5 s, and the final accuracyis given by the following errors:

|x1 − x1|∞ ∈ [5.8712 × 10−4, 1.8174 × 10−3],

|x2 − x2|∞ ∈ [2.6567e× 10−3, 3.1573 × 10−3],

f1|∞ ∈ [−5.3827 × 10−3, 2.5932 × 10−3];

f2|∞ ∈ [−1.9164e× 10−3, 9.0744 × 10−4].


0 1 2 3 4 5 6 7 8 9 10−4

−3

−2

−1

0

1

2

3

tempo [s]

x1

x2

f1f2

Fig. 8. The functions (dashed) and their arguments.

0 5 10 15−4

−3

−2

−1

0

1

2

3

tempo [s]

x1

x2

f1f2

Fig. 9. The functions (dashed) and their arguments.

The accuracy can be improved by using a higher-order integration method, but the improvement is not dramatic. Betterresults can be obtained by remembering that, under the given hypothesis, the equivalent control for system (78) is zero.This allows for substituting the sign function with a smoother one such as

u = −∇T f(x)√

|y| sign(y). (86)

In this case (Fig. 9) the transient is slower (about 13 s) but the accuracy is much better

|x1 − x1|∞ ∈ [−8.6746 × 10−7, 8.6768 × 10−7],

|x2 − x2|∞ ∈ [−3.1276e× 10−3, 3.1333 × 10−7],

f1|∞ ∈ [−5.8316 × 10−7, 5.8314 × 10−6];

f2|∞ ∈ [−2.0483 × 10−6, 2.0481 × 10−6].


4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 60.995

0.996

0.997

0.998

0.999

1

1.001

1.002

1.003

1.004

1.005

tempo [s]

x1

x2

Fig. 10. The variable convergence after the commutation.

A good compromise between convergence time and accuracy can be obtained by applying controller (81) first, andthen it commutes to controller (86) at t = 4.5s. A zoom of the plots around the commutation time instant is presentedin Fig. 10, where the new transient is pointed out as well as the improvement of the accuracy.

The influence of the discretization method in the discrete-time implementation of such algorithms has to be investi-gated, since the presence of a constant input during the sampling period suggests the Euler’s backward method but inthe presence of integrators, specially in higher-order sliding mode control, it seems sensible to resort to higher-orderdiscretization methods, such as the bilinear one.

6. Conclusions

A brief survey on Variable Structure Control Systems with Sliding Modes have been presented in a systematicway. It has been shown that different choices of the sliding surface can origin quite different sliding regimes mainlydepending on the relative degree between the sliding variable and the control input.

It has been shown that increasing the relative degree between the sliding variable and the switching control canbe useful to reduce the dimension of the internal dynamics, to obtain a continuous control variable to apply to theplant, and to increase the accuracy in discrete time implementations. The cost related to such advantages is a dramaticincreasing of the complexity of the control design, because of the lack of general Lyapunov techniques for these cases.

After a short recall of some implementations of the sliding mode theory to control problems, the possible applicationof such a theory for solving numerical problem have been discussed and three applications were given with some detail:the solution of constrained ODEs, the real-time differentiation of known signals, and the problem of finding the zeroesof nonlinear systems.

These three problems are reduced to the same framework of differential ordinary equations with discontinuousright-hand side. In particular the problems of real-time differentiation of a variable and the zero finding of nonlinearsystem are regularized transforming them into feedback control problems. The simulations give some detail about themain features of the proposed schemes, and confirm the effectiveness of applying the sliding mode control theory tosome mathematical problems.

Acknowledgements

The authors would like to acknowledge Prof. Bartolini for stimulating and fruitful discussion during years of jointwork. This paper has been partially supported by Italian MIUR Project 630/04 ‘Advanced models and methods for thecontrol, diagnosis and management of combined-cycle power plants under perturbed conditions’.


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