Discontinuous Homogeneous Control

Then the sliding order is a number of continuous total time derivatives of σ including the zero one in the vicinity of the sliding mode.. Thus, the rth order sliding mode is determined b

Arie Levant and Lela Alelishvili

Applied Mathematics Department, Tel-Aviv University

{levant,lela}@post.tau.ac.il

Control under heavy uncertainty conditions remains among the main topics ofthe control theory The sliding-mode control [50, 53, 10] is one of the maintools in the field This approach is based on exactly keeping a properly chosenconstraint by means of control switching of high (theoretically infinite) frequency.Although very robust and accurate, the approach has two basic restrictions Thedirect implementation of standard sliding modes requires the relative degree ofthe constraint to be 1, i.e control has to appear explicitly already in the firsttotal time derivative of the constraint function Also, high-frequency controlswitching may cause the so-called chattering effect [14, 15, 16, 33]

High-gain control with saturation is used to overcome the chattering effectapproximating the sign-function in a boundary layer around the switching man-ifold [45], also the sliding-sector method [17] was proposed to control disturbedlinear time-invariant systems The sliding-mode order approach [24, 28] consid-ered in this chapter is capable to treat successfully both the chattering and therelative-degree restrictions preserving the sliding-mode features and improvingits practical accuracy

High order sliding mode (HOSM) is actually a movement on a discontinuityset of a dynamic system understood in Filippov’s sense [12] The sliding ordercharacterizes the dynamics smoothness degree in the vicinity of the mode Letthe task be to provide for keeping a constraint given by equality of a smooth

function σ to zero Then the sliding order is a number of continuous total time derivatives of σ (including the zero one) in the vicinity of the sliding mode Thus, the rth order sliding mode is determined by the equalities

Standard sliding mode is used with the relative degree 1 of the constraint

function σ In that case ˙σ = h(t, x) + g(t, x)u, where g(t, x) = 0 Then the sliding

G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 71–95, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

mode can be locally obtained by means of the simplest control u = −k sign σ, if g

is positive and k > 0 is sufficiently large The general case situation is very much

alike The sliding order of the standard sliding mode is 1, for ˙σ is discontinuous.

Asymptotically stable HOSMs appear in many systems with traditionalsliding-mode control and are deliberately introduced in systems with dynamicalsliding modes [43, 36] While finite-time-convergent arbitrary-order sliding-modecontrollers are still theoretically studied [28, 13, 23, 34], 2-sliding controllersare already successfully implemented for the solution of real problems [5, 7, 11,

26, 44, 39, 19, 46, 47, 48, 49]

Construction of r-sliding controllers, r ≥ 3, is more difficult due to the

high dimension of the problem Thus, only a few families of such controllers

[27, 28, 23, 30, 29, 35, 41] are known The finite-time-convergent controllers

(r-sliding controllers) [28, 23, 30] require actually only the knowledge of the system

relative degree r The finite-time-stable exact tracking is lost with alternative controllers developed in [49] and [3] for r = 3 and r = 2 respectively Many new

results on HOSM controllers are based on their homogeneity properties whichproved to be very useful It is natural to construct new finite-time convergentHOSM controllers basing on the homogeneity-based approach [29] On the otherhand, recently published non-homogeneous controllers [23, 41] lack the highestaccuracy characterizing other HOSM controllers The aim of this paper is to sum-marize a number of recent results in one frame of homogeneous discontinuouscontrol

Most known HOSM controllers possess specific homogeneity properties The

corresponding homogeneity of r-sliding controllers is called the rth-order sliding

homogeneity [29] The homogeneity makes the convergence proofs of the HOSMcontrollers standard and provides for the highest possible asymptotic accuracy

in the presence of measurement noises, delays and discrete measurements [29]

In particular, the controllers provide in finite time for keeping σ ≡ 0, if the surements of σ are exact, and for σ proportional to the maximal measurement error otherwise With discrete measurements and τ being the sampling interval, the accuracy σ = O(τ r) is assured This asymptotic accuracy is proved to bethe best possible with discontinuous control [24] An output-feedback controllerwith the same asymptotical accuracy is obtained, when a recently developed

mea-robust exact homogeneous differentiator of the order r − 1 [25, 28] is included as

a standard part of the homogeneous r-sliding controller.

HOSM controllers of a new type were recently developed, called

quasi-continuous Such controllers are feedback functions of σ, ˙σ, , σ (r −1), continuous

everywhere except the very manifold (1) of the r-sliding mode The mode σ ≡ 0

is established after a finite-time transient In the presence of errors in evaluation

of the output σ and its derivatives, a motion in some vicinity of (1) takes place.

Control practically turns out to be a continuous function of time, for in

real-ity trajectories never hit the manifold (1) with r > 1 The proposed controllers [30, 35] are also designed r-sliding homogeneous, which extends all the results of

this chapter to their application They demonstrate significantly less chattering

and seem to be generally superior compared with usual r-sliding controllers.

Effective implementation of the differentiator requires sufficiently high pling frequency, which is not always available In the case of the homogeneouscontrollers successive derivatives can be replaced by means of divided finite dif-ferences producing an output-feedback controller possesing the robustness andthe asymptotic accuracies [24] of the original controller With rare measurementssuch controllers can definitely be a good choice.

sam-The r-sliding control is a discontinuous function of the tracking error σ and

of its real-time-calculated successive derivatives σ, ˙σ, , σ (r −1) The control

switches with infinite frequency when in the sliding mode The resulting tering effect is successfully treated, provided the control derivative is used as anew control input [24, 4, 27, 33], hence artificially increasing the relative degree

chat-The (r + 1)-sliding mode is to be established in that case Unfortunately, such

a simplistic procedure is in general only locally valid, because of the possibleinteraction between the control and its derivative The problem can be solved

by means of the integral homogeneous sliding mode [34] The idea is to design

in advance a transient trajectory to the (r + 1)-sliding mode In such a way any

interaction between the control and its derivative is excluded, since the controlbecomes some predetermined function of time

The recent results [33] show the robustness of homogeneous sliding modecontrollers with respect to the presence of unaccounted for fast stable dynamics

of actuators and sensors

Simulation demonstrates the practical applicability of the results

A differential inclusion ˙x ∈ F (x) is further called a Filippov differential sion, if the vector set F (x) is non-empty, closed, convex, locally bounded and

inclu-upper-semicontinuous [12] The latter condition means that the maximal

dis-tance of the points of F (x) from the set F (y) vanishes when x → y Solutions

are defined as absolutely-continuous functions of time satisfying the inclusionalmost everywhere Such solutions always exist and have most of the well-knownstandard properties except uniqueness [12]

It is said that a differential equation ˙x = f (x) with a locally-bounded

Lebesgue-measurable right-hand side is understood in the Filippov sense, if itssolutions are defined as solutions of a specially constructed Filippov differentialinclusion ˙x ∈ F (x) In the most usual case, when f is continuous almost every- where, the procedure is to take F (x) being the convex closure of the set of all possible limit values of f at a given point x, obtained when its continuity point

y tends to x In the general case approximate-continuity [42] points y are taken (one of the equivalent definitions by Filippov [12]) Values of f on any set of the measure 0 do not influence the Filippov solutions Note that with continuous f

the standard definition is obtained The presence of small delays and ment errors results in the solutions, which converge to the Filippov solutions,when these imperfections vanish [12]

measere-A function f : R n → R (respectively a vector-set field F (x) ∈ R n , x ∈ R n,

or a vector field f : R n → R n ) is called homogeneous of the degree q ∈ R with the dilation d κ : (x1, x2, , x n)→ (κ m1x1, κ m2x2, , κ m n x n ) [2], where m1, ,

m n are some positive numbers (weights, homogeneity degrees), if for any κ > 0 the identity f (x) = κ −q f (d κ x) holds (respectively F (x) = κ −q d −1

κ F (d κ x), or

f (x) = κ −q d −1

κ f (d κ x) The non-zero homogeneity degree q of a vector field can

always be scaled to ±1 by an appropriate proportional change of the weights m1, , m n

Note that the homogeneity of a vector field f (x) (a vector-set field F (x)) can

equivalently be defined as the invariance of the differential equation ˙x = f (x)

(differential inclusion ˙x ∈ F (x)) with respect to the combined time-coordinate transformation G κ : (t, x) → (κ p t, d κ x), p = −q, where p might naturally be considered as the weight of t.

Examples Let the weights of x1, x2be 3 and 2 respectively Then the function

x2+ x3 is homogeneous of the weight (degree) 6: (κx1)2+ (κx2)3 = κ6(x2+

x3) The differential inequality | ˙x1| + ˙x2 ≤ |x1| 2/3+|x2| corresponds to the

homogeneous differential inclusion

are of the degree -1, the system being finite-time stable

1◦ A differential inclusion ˙x ∈ F (x) (equation ˙x = f(x)) is further called

globally uniformly finite-time stable at 0, if it is Lyapunov stable and for any

R > 0 there exists T > 0 such that any trajectory starting within the disk

||x|| < R stabilizes at zero in the time T

2◦ A differential inclusion ˙x ∈ F (x) (equation ˙x = f(x)) is further called

globally uniformly asymptotically stable at 0, if it is Lyapunov stable and for

any R > 0, ε > 0 exists T > 0 such that any trajectory starting within the disk

||x|| < R enters the disk ||x|| < ε in the time T to stay there forever.

A set D is called dilation retractable if d κ D ⊂ D for any 0 ≤ κ < 1.

3◦ A homogeneous differential inclusion ˙x ∈ F (x) (equation ˙x = f(x)) is further called contractive if there are 2 compact sets D1, D2 and T > 0 such that D2 lies in the interior of D1 and contains the origin; D1 is dilation-retractable;

and all trajectories starting at the time 0 within D1 are localized in D2 at the

negative homogeneous degree is insensitive with respect to small homogeneousperturbations of the right-hand side For the case of continuous differential equa-tions the equivalence of 1◦and 2◦was proved in [8] The equivalence of 1◦and 2◦

was also independently proved for the Filippov discontinuous differential tions in [40]

equa-Let ˙x ∈ F (x) be a homogeneous Filippov differential inclusion Consider the case of “noisy measurements” of x i with the noise magnitude ε i τ m i

˙x ∈ F (x1 + ε1τ m1[−1, 1], , x n + ε n τ m n[−1, 1]), τ > 0.

Taking successively the convex hull at each point x and the closure of the

right-hand-side graph, obtain some new Filippov differential inclusion ˙x ∈ F τ (x).

Theorem 2 [29] Let ˙ x ∈ F (x) be a globally uniformly finite-time-stable geneous Filippov inclusion with the homogeneity weights m1, , m n and the degree −p < 0, and let τ > 0 Suppose that a continuous function x(t) be defined for any t ≥ −τ p and satisfy some initial conditions x(t) = ξ(t), t ∈ [−τ p , 0] Then if x(t) is a solution of the disturbed inclusion ˙x(t) ∈ F τ (x(t + [ −τ p , 0])),

homo-0 < t < ∞ , the inequalities |x i | ≤ γ i τ m i are established in finite time with some positive constants γ i independent of τ and ξ.

Note that Theorem 2 covers the cases of retarded or discrete noisy measurements

of all or some of the coordinates and any mixed cases In particular, infinitelyextendible solutions certainly exist in the case of noisy discrete measurements ofsome variables or in the constant time-delay case The Theorem conditions donot impose any restrictions on the real noises and delays to be observed in reality.Indeed, in any practical case one has some concrete noises or delay magnitudes

Then the choice of parameters ε i and τ is not unique Indeed, one may always increase τ or each one of ε i keeping the same fixed real system parameters.Mark also that this Theorem provides for the asymptotic accuracy of all knownfinite-time stable continuous homogeneous differential equations with negativedegrees [2]

Solution

Let a Single-Input-Single-Output (SISO) system to be controlled have the form

˙x = a(t, x) + b(t, x)u, x ∈ R n , u ∈ R, (2)

σ : (t, x) −→ σ(t, x) ∈ R,

where σ is the measured output of the system, u is the control Smooth functions

a, b, σ are assumed unknown, the dimension n can also be uncertain The task is

to make σ vanish in finite time by means of a possibly discontinuous feedback and

to keep σ ≡ 0 The solutions are understood in the Filippov sense, and system

trajectories are supposed to be infinitely extendible in time for any bounded

Lebesgue-measurable input In real applications σ can be a deviation of a system

output from some command signal available in real time, or from any auxiliaryconstraint chosen by the system designer Although it is formally not needed,the weakly minimum-phase property is usually required in practice

It is assumed that the relative degree r [18] of the system is constant and

known.That means [18] that the equation

σ (r) = h(t, x) + g(t, x)u, g(t, x) = 0, (3)

holds, with some uncertain h(t, x) = σ (r) | u=0 , g(t, x) = ∂u ∂ σ (r) The uncertaintyprevents immediate reduction of (2) to the standard form (3) Suppose that theinequalities

|σ (r) | u=0 | ≤ C, 0 < K m ≤ ∂

∂u σ

hold for some K m , K M , C > 0 These conditions are satisfied at least locally for

any smooth system (2) having a well-defined relative degree at a given point

with σ = ˙σ = = σ (r −1) = 0 Assume that (4) holds globally Then (3), (4)

imply the differential inclusion

σ (r) ∈ [−C, C] + [K m , K M ]u. (5)The problem is solved in two steps First a bounded feedback control

u = Ψ (σ, ˙σ, , σ (r −1) ), (6)

is constructed, such that all trajectories of (5), (6) converge in finite time to

the origin of the r-sliding phase space σ, ˙σ, , σ (r −1) At the next step the

lacking derivatives are real-time evaluated, producing an output-feedback troller Here and further the right-hand sides of all differential inclusions areenlarged at discontinuity points of control producing Filippov inclusions If it isnot mentioned explicitly, the minimal enlargement is taken Here the Filippov

con-set of limit values of (6) is substituted for u in (5) (see Section 2) The function Ψ

is assumed to be a Borel-measurable function, which provides for the Lebesguemeasurability of composite functions to be obtained in the presence of Lebesgue-measurable noises Actually all functions used in the sliding-mode control theoryare Borel measurable Indeed, any superposition of Borel-measurable functions

is Borel-measurable; the sign function and all continuous functions are Borelmeasurable

Note that the function Ψ has to be discontinuous at the origin Otherwise

u is close to the constant Ψ (0, 0, , 0) in a small vicinity of the origin, and, taking c ∈ [−C, C] and k ∈ [K m , K M ] so that c + kΨ (0, 0, , 0) = 0, achieve that (6) cannot stabilize the dynamic system σ (r) = c + ku Thus, σ (r) is to

be discontinuous along the trajectories of the original system (2), (6), which

means that the r-sliding mode σ ≡ 0 is to be established All known r-sliding

controllers [4, 7, 23, 28, 29, 30, 35, 41] may be considered as controllers for (5)

steering σ, ˙σ, , σ (r −1) to 0 in finite time Inclusion (5) does not “remember”

the original system (2) Thus, such controllers are obviously robust with respect

to any perturbations preserving the system relative degree and (4)

4 Homogeneous Sliding-Mode Control

Suppose that feedback (6) imparts homogeneity properties to the closed-loopinclusion (5), (6) Due to the term [−C, C], the right-hand side of (5) can only have the homogeneity degree 0 with C > 0 Indeed, with a positive degree the

right hand side of (5), (6) approaches zero near the origin, which is not possible

with C > 0 With a negative degree it is not bounded near the origin, which contradicts the local boundedness of Ψ Thus, the homogeneity degree of the right-hand side of (5) is to be 0, and the homogeneity degree of σ (r −1) is to be

opposite to the degree of the whole system

Scaling the system homogeneity degree to -1, achieve that the homogeneity

weights of t, σ, ˙σ, , σ (r −1) are 1, r, r − 1, , 1 respectively This homogeneity

is further called the r-sliding homogeneity Denote σ = (σ, ˙σ, , σ (r −1))

Trajec-tories of (5), (6) are preserved by the combined time-coordinate transformation

bounded and takes on all its values in any vicinity of the origin

Almost all known r-sliding controllers, r ≥ 2, are r-sliding homogeneous.

Also the sub-optimal 2-sliding controller [4, 7] is homogeneous in the sense that

(7) preserves trajectories The following recursively built r-sliding homogeneous

Obviously, α is to be negative with ∂u ∂ σ (r) < 0.

Nested-sliding-mode (nested-SM) controller [27, 28]

This controller is based on a complicated switching motion, which can be

qualita-tively described by a sequence of nested sliding modes Let p > r, i = 1, , r −1,

β1, , β r −1 be some positive numbers It is defined by the procedure

N1,r=|σ| (r −1)/r , N

i,r= (|σ| p/r+| ˙σ| p/(r −1) + + |σ (i −1) | p/(r −i+1))(r −i)/p ,

Ψ0,r = signσ, ϕ i,r = σ (i) + β i N i,r Ψ i −1,r , Ψ i,r = signϕ i,r

A list of such controllers is presented in [27, 28] Obviously, N i,r and Ψ i,r are

r-sliding homogeneous functions of the weights r − i and 0 respectively, N is

also a positive-definite function of σ, ˙σ, , σ (i −1) The idea of the convergence

proof is that (9) provides in finite time for the approximate keeping of ϕ r −1,r= 0

(the exact 1-sliding mode is impossible, since Ψ r −2,ris discontinuous) The latter

equation provides in finite time for the approximate keeping of ϕ r −2,r = 0, etc.

The last approximate equality is ϕ 1,r = ˙σ + β1|σ|(r −1)/r signσ = 0 Obviously,

there is an attracting vicinity of the origin (σ, ˙σ, , σ (r −1)) = 0 Thus, the closed

loop inclusion (5), (9) is contractive, and, therefore, finite-time stable according

to Theorem 1

Quasi-continuous controller [30]

In order to reduce the chattering, a controller is designed, which is continuous

everywhere except the r-sliding set σ = ˙σ = = σ (r −1)= 0 Such a controller is

naturally called quasi-continuous, for in practice, in the presence of measurementnoises, singular perturbations and switching delays, the motion takes place in

some vicinity of the r-sliding set and the control actually is a continuous function

of time Let i = 1, , r − 1 Denote

ϕ 0,r = σ, N 0,r=|σ|, Ψ0,r = ϕ 0,r /N 0,r = sign σ,

ϕ i,r = σ (i) + β i N (r −i)/(r−i+1)

i −1,r Ψ i −1,r , N i,r=|σ (i) | + β i N (r −i)/(r−i+1)

i −1,r ,

Ψ i,r = ϕ i,r /N i,r ,

where β1, , β r −1 are positive numbers The following proposition is easily

proved by induction

Proposition 1 Let i = 0, , r − 1 N i,r be positive definite, i.e N i,r = 0 iff

σ = ˙σ = = σ (i) = 0 The inequality |Ψ i,r | ≤ 1 holds whenever N i,r > 0 The function Ψ i,r (σ, ˙σ, , σ (i) ) is continuous everywhere (i.e it can be redefined by continuity) except the point σ = ˙σ = = σ (i) = 0.

Also here the idea of the convergence proof is that the control successively causes

the aproximate keeping of the equations ϕ r −1,r = 0, , ϕ 0,r= 0

Theorem 3 Provided β1, , β r −1 , α > 0 are chosen sufficiently large in the

list order, both above designs result in the r-sliding homogeneous controller (9) providing for the finite-time stability of (5), (9) with any sufficiently large α The finite-time stable r-sliding mode σ ≡ 0 is established in the system (2), (9) Thus, one does not need to know the exact values of K m , K M , C to apply the controllers in practice Each proper choice of β1, , β r −1 determines a con-

troller family applicable to all systems (2) of the relative degree r, provided α is

large enough Here and further the maximal possible transient time is a locallybounded function of initial conditions (Section 2)

Following are quasi-continuous controllers with r ≤ 4 and simulation-tested

β i Note that the same parameters β ican be used for the nested SM controllers

While the control is a continuous function of time everywhere except the

r-sliding set, it may have infinite derivatives when certain surfaces are crossed.Another quasi-continuous controller family is constructed in [29], generalizedcontrollers are introduced in [35] containing arbitrary functional parameters The

following Theorems are standard consequences [29] of the r-sliding homogeneity

of controller (9) and Theorems 1, 2

Theorem 4 Let the control value be updated at the moments t i , with t i+1 −t i=

τ = const > 0, t ∈ [t i , t i+1 ) (the discrete sampling case) Then controller (9) provides in finite time for keeping the inequalities |σ| < μ0 τ r , | ˙σ| < μ1 τ r −1 , ,

|σ (r −1) | < μ r −1 τ with some positive constants μ0, μ1, , μ r −1 .

That is the best possible accuracy attainable with discontinuous σ (r) [24] Thefollowing result shows robustness of controller (9) with respect to measurementerrors

Theorem 5 Let σ(i) be measured with accuracy η i ε (r −i)/r for some fixed η

i > 0,

i = 1, , r − 1 Then with some positive constants μ i the inequalities |σ (i) | ≤

μ i ε (r −i)/r , i = 0, , r − 1, are established in finite time for any ε > 0.

The convergence time may be reduced changing coefficients β j In particular, one can substitute λ −j σ (j) for σ (j) and λ r α for α , λ > 0, causing convergence time

to be diminished approximately by λ times A set of parameters β j satisfies the

above Theorems, if the differential equations ϕ 1,r = 0, , ϕ r −1,r = 0 are

finite-time stable [35] That indicates the recursive way of choosing the parameters.Note that these equations do not contain uncertainties

Any r-sliding homogeneous controller can be complemented by an (r − 1)th order

differentiator [1, 5, 22, 25, 28, 21, 52] producing an output-feedback controller Due

to the demonstrated robustness of the described controllers with respect to themeasurement errors, the resulting output feedback controller will localy provide

for approximate real [24] r-sliding mode In order to preserve the demonstrated

exactness, finite-time stability and the corresponding asymptotic properties, the

natural way is to calculate ˙σ, , σ (r −1)in real time by means of a robust finite-time

convergent exact homogeneous differentiator [28] Its application is possible due

to the boundedness of σ (r)provided by the boundedness of the feedback function

Ψ in (6) Following is the short description of the differentiator.

Arbitrary-order real-time exact robust differentiation

Suppose that it is known that the input signal is compounded of a smooth signal

f0(t) to be differentiated and a noise being a bounded Lebesgue-measurable

function of time Both signals are unknown and only their sum is available It

is proved that if the base signal f0(t) has (r-1)th derivative with Lipschitz’s constant L > 0, the best possible kth order differentiation accuracy is d k L k/r

ε (r −k)/r , where d k > 1 may be estimated [20, 25] Moreover, it is proved that

such a robust exact differentiator really exists [25, 28]

The aim is to find real-time robust estimations of f0(t), f0(t), , f0(p) (t), being

exact in the absence of measurement noise and continuously depending on thenoise magnitude The differentiator is recursively constructed and has the form

˙z0= ν0, ν0=−λ0 L p+11 |z0 − f(t)| p

p+1 sign(z0− f(t)) + z1 ,

˙z i = ν i , ν i=−λ i L p−i+11 |z i − ν i −1 | p−i+1 p−i sign(z i − ν i −1 ) + z i+1 ,

.

˙z p=−λ p L sign(z p − ν p −1)

(10)

The coefficients are easily found by simulation, since the pth order

differentia-tor requires only one parameter to be chosen, if the lower-order differentiadifferentia-torsare already built A set of such parameters is listed further The proof is based

on the introduction of new variables σ i = z i −f (i)

0 (t) Taking f0(p+1) (t) ∈ [−L, L] obtain a differential inclusion Assigning the weight p − i to σ i = z i − f (i)

0 (t)

ob-tain a homogeneous differential inclusion of the degree -1 With properly chosenparameters the inclusion is finite-time stable Following Theorem 2 the followingaccuracy is obtained

|z i − f (i)

0 (t) | ≤ μ i ε (p (p+1) −i+1) , i = 0, , p;

|v i − f (i+1)

0 (t) | ≤ ν i ε (p (p+1) −i) , i = 0, , p − 1.

Exact differentiation is provided with ε = 0 Using recursive high-order

dif-ferentiators the noise propagation is obviously counteracted as compared withthe cascade implementation of first-order differentiators Consider the discrete-

sampling case, when z0(t j)−f(t j ) is substituted for z0 −f(t), with t j ≤ t < t j+1,

t j+1 − t j = τ > 0 Then the following accuracy is obtained

˙z0 = v0, v0=−λ0L 1/r |z0 − σ| (r −1)/r sign(z0 − σ) + z1,

˙z1= v1, v1=−λ1 L 1/(r −1) |z1 − v0| (r −2)/(r−1) sign(z

1− v0 ) + z2,

˙z r −2 = v r −2 , v r −2=−λ r −2 L 1/2 |z r −2 − v r −3 | 1/2 sign(z r −2 − v r −3 ) + z r −1 ,

˙z r −1=−λ r −1 L sign(z r −1 − v r −2 ),

(12)

where Ψ is an r-sliding homogeneous controller, L ≥ C + sup |Ψ|K M, and

pa-rameters λ iof differentiator (12) are adjusted in advance [28] A possible choice

of the differentiator parameters with r ≤ 6 is λ r −1 = 1.1, λ r −2 = 1.5, λ r −3= 3,

λ r −4 = 5, λ r −5 = 8, λ r −6= 12

Taking the homogeneity weight r − i for σ i = z i − σ (i) , i = 0, 1, , r − 1,

obtain a homogeneous differential inclusion (5), (11), (12) of the degree −1.

The corresponding Filippov inclusion is also globally uniformly finite-time

sta-ble Let σ measurements be corrupted by a noise being an unknown bounded

Lebesgue-measurable function of time, then solutions of (2), (11), (12) are finitely extendible in time under the assumptions of Section 3, and the followingTheorems are simple consequences of Theorem 2

in-Theorem 6 Let controller (11) be r-sliding homogeneous and finite-time stable,

and the parameters of the differentiator (12) be properly chosen with respect to the upper bound of |Ψ| Then in the absence of measurement noises the output- feedback controller (11), (12) provides for the finite-time convergence of each trajectory to the r-sliding mode σ ≡ 0, otherwise convergence to a set defined

by the inequalities |σ| < μ0 δ, | ˙σ| < μ1 δ (r −1)/r , , σ (r −1) < μ r −1 δ 1/r is ensured,

where δ is the unknown measurement noise magnitude and μ0, μ1, , μ r −1 are

some positive constants.

In the absence of measurement noises the convergence time is bounded by a

continuous function of the initial conditions in the space σ, ˙σ, , σ (r −1) , z0,

z1, , z r −1 which vanishes at the origin (Theorem 1)

Theorem 7 Under the conditions of Theorem 6 the discrete-measurement

ver-sion of the controller (11), (12) provides in the absence of measurement noises for the inequalities |σ| < μ0 τ r , | ˙σ| < μ1 τ (r −1) , , σ (r −1) < μ r −1 τ for some μ0,

μ1, , μ r −1 > 0.

The asymptotic accuracy provided by Theorem 7 is the best possible with

dis-continuous σ (r)and discrete sampling [24] A Theorem corresponding to the case

of discrete noisy sampling is also easily formulated based on Theorem 2 Theseresults are also valid for the sub-optimal controller [4, 7]

Controller (6) requires availability of σ, ˙σ, , σ (r −1) The most natural way

is to estimate the derivatives by means of finite differences Denote σ (s) i =

σ (s) (t , x(t )) With the noise magnitude being smaller than ε obtain [9] that

Δ k σ(t) = σˆ (k) (t)τ k + o(τ k ) + O(ε),

where ˆσ is the measured value of the output σ, τ is the sampling interval, and

Δ kˆσ(t) is the kth-order backward finite difference Obviously, with small τ and

ε = 0 any reasonably robust r-sliding controller would approximately solve the

problem using such estimations Unfortunately, any small noise seems to destroy

the system Indeed, Δ k σ(t) contains some valuable information on σˆ (k) (t) only with ε being small compared with τ k Since τ is itself assumed to be small, the condition is very restrictive Nevertheless, in the specific case of the r-sliding

homogeneous control the high order finite differences can be successfully used

The reason is that the higher is the derivative order the less sensitive are

r-sliding homogeneous controllers to derivative evaluation errors In the followingthe noises are supposed to be any bounded Lebesgue-measurable functions oftime

Only constant sampling intervals are considered below, though variable

in-tervals can improve the performance [32] Let σ, , , σ (k) , 0 ≤ k ≤ r − 1

be available Let the measurements be carried out at times t i with constant

time step τ > 0 Denote σ (s) i = σ (s) (t i , x(t i )) Let Δ be the backward difference operator, Δσ (s) i = σ (s) i − σ (s)

i −1 , t ∈ [t i , t i+1 ), s = 1, 2, , r − 1 Define

u = Ψ (σ i , ˙σ i , , σ (k) i , Δσ (k) i /τ , , Δ r −k−1 σ (k)

i /τ r −k−1 ). (13)

Theorem 8 [32] Suppose that controller (6) be r-sliding homogeneous and

finite-time stable, and let σ, , , σ (k) , 0 ≤ k < r, be sampled with ment noises of the magnitudes β0ε, β1ε (r −1)/r , , β

measure-k ε (r −k)/r respectively with

some β0, β1, , β k > 0 and the sampling interval τ = ηε 1/r , η > 0 Then there are such positive constants γ0, γ1, , γ r −1 that for any ε > 0 controller (13)

provides in finite time for keeping the inequalities |σ| ≤ γ0 ε, | ˙σ| ≤ γ1 ε (r −1)/r ,

, |σ (r −1) | ≤ γ r −1 ε 1/r with some positive constants γ0, γ1, , γ r −1 .

Note that noises of lower magnitudes automatically satisfy the Theorem

condi-tions Thus, in the absence of noises the accuracy σ = O(τ r ), ˙σ = O(τ r −1), ,

σ (r −1) = O(τ ) is obtained That asymptotic accuracy is the best possible with

discontinuous σ (r) and discrete sampling [24] The main way to use controller(13) is to choose some constant sampling interval providing for acceptable per-formance without noises The performance will be automatically preserved inthe presence of sufficiently small noises It is further demonstrated in the simu-lation Section that with large sampling intervals the resulting performance can

be much better than one obtained with the differentiator

High-order integral sliding mode notion

The notion of integral sliding mode [51] is naturally extended to the high-ordersliding modes Suppose that it is needed to avoid the uncertainty of the transientprocess, and/or some transient time restrictions are present, transient-trajectory