Digital second-order sliding mode control for uncertain nonlinear systems

Brief Paper

Digital second-order sliding mode control for uncertain

nonlinear systems

Giorgio Bartolini, Alessandro Pisano, Elio Usai*

Dipartimento di Ingegneria Elettrica ed Elettronica, Universita% di Cagliari piazza d'Armi, I-09123 Cagliari, Italy

Received 29 July 1998; revised 11 September 2000; received in "nal form 28 February 2001

Abstract

In this note, we analyze the discrete-time implementation of a second-order sliding mode control (2-SMC) scheme. The treatment is detailed for a simple class of feedback-linearizable nonlinear systems expressed in the Brunowsky normal form. First, it is shown that the direct discretization of a continuous-time 2-SMC scheme guarantees the "nite-time attainment of a motion in an O(¹) boundary layer of the sliding manifold (¹ being the sampling period). Then, a suitable iterative learning procedure, that leads to the asymptotic reduction of the boundary layer to O(¹) is proposed. Simulation results are reported at the end of the paper.  2001 Elsevier Science Ltd. All rights reserved.

Keywords: Digital control; Second-order sliding modes; Uncertain systems; Nonlinear systems

1. Introduction

Direct time-discretization of the "rst-order sliding mode control (1-SMC) algorithms gives rise to piece-wise-constant control signals commuting at high fre-quency between values with opposite sign (ringing behavior). The motion of the system turns out to be con"ned to an O(¹) boundary layer of the sliding mani-fold, where ¹ is the sampling period (quasi-sliding motion (Milosavljevic, 1985)). This regime cannot be acceptable in many applications.

Various approaches have been presented in the litera-ture in order to improve the sliding accuracy and to attenuate the ringing e!ect, by means of some forms of adaptations (Bartolini, Ferrara, & Utkin, 1995; Corradini, 1998), boundary-layer controls (Drakunov, OGzguner, & Su, 1996; Utkin, 1992; Young, OGzguner, & Utkin, 1999; Utkin & Drakunov, 1989) or by introducing approximated sliding motions through the use of sliding sectors (Furuta & Pan, 2000). In particular, in Drakunov

夽_{This paper was not presented at any IFAC meeting. This paper was}

recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Tamer Basar.

* Corresponding author. Tel.: 0675-5876; fax:

#39-07-0675-5900.

E-mail address: [email protected] (E. Usai).

et al. (1996), the boundary layer size was reduced to

O(¹) by using a predictor of the uncertainties.

This paper concerns the analysis of the actual accuracy featured by the discrete-time implementation of basic second-order sliding mode control algorithms (Bartolini, Ferrara, Levant, & Usai, 1999a), the identi"cation of the highest achievable accuracy, and deriving a sensible way to reach it.

To illustrate the proposed approach, we consider non-linear systems represented in the Brunowsky normal form, with non-parametric matched uncertainties. In Section 2, the control problem is formulated, while in Section 3 the direct discretization of a continuous-time 2-SMC scheme is analyzed, and the su$cient conditions for the "nite-time reaching of an O(¹)-vicinity of the sliding manifold are derived. In the subsequent Section 4, an iterative procedure for improving the accuracy up to

O(¹) is described. Some simulation results and

conclud-ing remarks are presented at the end of the paper.

2. Problem formulation

We consider a single-input nonlinear system represent-ed by the model



x
L"f(x(t))#g(x(t))u(t), (1)

uncertainties. The last component of the state vector, xL, is not measurable. The control objective is the asymp-totic state stabilization by using discrete-time control devices.

To this end, we de"ne the sliding quantity as

s(t)"xL\(t)#L\ GcGxG(t),

(2) where cG, i"1, 2,2, n!2, are proper positive constants such that the polynomial P(q)"qL\# L\

GcGqG\ is

a Hurwitz one.

Consider the sliding quantity s as a system output; then, the system relative degree is two, and the associated input}output and internal dynamics (Isidori, 1995) are given, respectively, by sK(t)"
(x(t))#g(x(t))u(t) (3) and
Q"A
#bs(t), (4) where
(x(t))"f (x(t))#L\ GcGxG>(t). (5)
(t)"[x(t), x(t),2, xL\(t)], c"[c, c,2, cL\],

A is a (n!2);(n!2)-matrix in companion form

with the last row coinciding with vector !c and b"[0,2, 0, 1]23RL\.

Since matrix A is Hurwitz, the system internal dynam-ics (4) is input-to-state stable (ISS), and the associated zero dynamics
Q"A
is globally asymptotically stable (GAS). Therefore, the control objective is to reduce to zero the sliding output s, possibly in a "nite time.

Assume what follows:

H: There exist a non-decreasing positive function

M( ) ) : RPR, and positive known constants ¸, G and G, such that


(x(t)))M(x )#¸s
, (6)

0(G)g(x(t)))G, (7)

where x "[x, x,2, xL\]2 is the available part of the state.

Pisano, and Usai (1999b), where the discrete-time control problem was solved assuming the relevant uncertainties to be globally bounded by some known constants, deal-ing with the more realistic case corresponddeal-ing to Assumption H.

Before going on with the present analysis, the follow-ing question should be raised: _{`What is the highest} achievable accuracy when a piecewise-constant control, with sampling period ¹, is used to force a relative-degree-two output variable s(t) to zero?_a.

The control signal able to keep s(t) exactly at the zero value is the so-called_{`equivalent controla, which is often} assumed to be a Lipschitz signal. The accuracy of a piece-wise-constant approximation of a Lipschitz function can-not be higher than O(¹), hence, considering that the relative degree is two, this approximation provides a slid-ing accuracy that cannot be higher than O(¹), whatever be the control strategy adopted.

3. Direct discretization of 2-SMC

Consider system (1), (2) and let s[k]"s(k¹)

(k"0, 1,2), ¹ be the sampling period. The plant input is piecewise-constant (ZOH device), i.e., u(t)"

u[k], t3[k¹,(k#1)¹).

A discrete-time version of the sub-optimal 2-SMC algorithm can be summarized as follows:

Step 1: Apply the control u[k]"!
I#h G sign(s
[0]), 0)k)kK+ (9) with
I"M (x I)#¸SB+, (10) x  I"Px [k]#QSB+, (11)

where h is an arbitrary non-null constant and P, and

Q are proper non-negative constants (see Appendix A),

Fig. 1. The detection of a singular point.

Fig. 2. The optimal parameters.

Any singular point can occur, in general, within an intersampling period (see Fig. 1). The occurrence of the singular values is detected by means of the following peak-detector device, that ensures s(

+G!s+G)O(¹),

(i"1, 2,2) (Bartolini et al., 1998a).

Approximate digital peak-detector: set s[!1]"s(0), s[!2]"0, i"!1. set[k]"(s[k]!s[k!1])(s[k!1]!s[k!2]). If ([k])0) then



Step 2: For kK +G(k)kK+G>, i"1,2,2, apply the

control u[k]"![k];+ G sign(s[k]!s(+G), ;+G" qH!1 G MG H , (13)

whereMGHis the largest positive root of

MGH "M(x G)#¸[qHMG H ¹ #(s( +GMG H (1#H(qH!1)#a¹)], (14) a"(MGH#(qH!1)MGH) (15) in which x G is de"ned according to (A.11); s(+

G and

kK +

G are evaluated by means of the approximate

peak-holder (12),[k] is adjusted according to [k]"



H otherwise, (16)

and the constants qH andH and are set depending on

G/G in accordance with Fig. 2. The values of qH andH, that minimize the boundary layer size are computed by solving a suitable minimization problem (see Appendix A).

The proposed control strategy, and the attained per-formances, are summarized in the following Theorem. Theorem 1. Consider the system's input}output dynamics (3), which verixes Assumptions H and H. Then, the digital

control strategy (9)}(16) guarantees that, after a xnite tran-sient tQ, the following conditions are satisxed:

s(t))O(¹). (17)

Proof. See Appendix A.

4. 2-SMC with asymptotic O(T3

) accuracy

The problem addressed in this section is to identify a procedure that, once the O(¹)-boundary layer has been reached by using the control strategy developed in Section 3, can improve the sliding accuracy by exploiting an on-line estimation of the equivalent control for 2-SMC, i.e., the control that maintains sK"0 when

s"s
"0 (Bartolini, Ferrara, Pisano, & Usai, 1998b),

that is expressed as

u(t)"!
(x(t))g(x(t)). (18)

The practical on-line availability of the equivalent con-trol at the output of a suitable "rst-order "lter, when the system motion is con"ned within a boundary layer of the sliding manifold, was theorized by Utkin (1992). Once the 2-sliding mode has been established by a switching con-trol u(t), the equivalent concon-trol estimate u (t) can be obtained through "rst-order low-pass "ltering

u
(t)#u (t)"u(t). (19)

Neglecting exponentially vanishing terms, provided that

"



k(¹"kO(¹, ₍₂₂₎

which leads to

u (t)!u(t))(¹, (23)

where is a proper constant (Bartolini et al., 1998b). As u(t) is constant between subsequent samplings, the continuous "lter (19) can be discretized, without any error, as follows:



u [k#1]"e\2Ou [k]#(1!e\2O)u[k], k"0, 1, 2,2

(24)

The average control sequence u [k], which, by (23), is an

O((¹)-approximation of the equivalent control, can be

used to perform an approximate feedback-compensation for the uncertainties, and an O((¹) residual uncertainty will remain corresponding. If, during the process, a known reduction of the uncertainties occurs, it is pos-sible to decrease the switching control e!ort accordingly, and, in turns, the "nal boundary layer size will be propor-tionally reduced as well.

The input sequence u[k] can be de"ned by means of two-components:

u[k]"u[k]#u [k], (25) where u[k] is a switching signal, with a constant

O((¹)-amplitude to be de"ned, and u

[k] is the output of the "lter (24).

Note that, by combining (24) and (25), the actual control u[k] turns out to be expressed as

u[k]"u[k!1]#u[k]!e\2Ou[k!1]. (26) The following Theorem can be proved.

Theorem 2. Consider the system's input}output dynamics (3), which verixes Assumptions H, H and (20). The digital

control strategy in Theorem 1 guarantees that, after a

xnite transient time tQ

, conditions (17) are satisxed. From

t"tQ on, the application of the control (26), in which

Once the reduced boundary layer (29) is reached, the same procedure can be repeated, and this will lead to a further "nite-time contraction of the boundary layer size. The generic iteration consists of the following steps:

E According to the contraction of the boundary layer attained at the previous iteration, reduce the  parameter to improve the accuracy of the equivalent control estimate, thereby reducing the uncertainties.

E According to the reduction of the uncertainties, reduce the switching control amplitude to obtain a further "nite-time contraction of the boundary layer.

The following Theorem 3 summarizes the proposed iterative strategy.

Theorem 3. Consider the system's input}output dynamics (3), which verixes Assumptions H, H and (20). The digital

control strategy in Theorem 1 guarantees that, after a xnite transient time tQ, a motion within the boundary layer (17)

is attained. From t"tQ onwards, the application of the

control (26), in which u[k] and "H are updated as follows: u[k]"!;H+[k]sign







where s(+G and[k] are dexned as in (12),(16), and kH, kH

and k3H are dexned in (C.1) and (C.3), guarantees the

following iterative improvement of accuracy:

s(t))O(¹\\H

), _t*tQ

H> (33)

Fig. 4. The control input.

Fig. 3. The boundary layer contraction across the iteration steps.

5. A simulation example

Consider system (1), with n"3 and

f (x, t)"2#5 sin(2.5t)#3x#2x#x,

g(x, t)"3#sin(3#x#x). (34)

Let x be unavailable for measurements, and the initial conditions are x(0)"[2, 2, 2]. The control task is to re-duce the state vector components to zero, and the sliding manifold is chosen as s"x#2x"0. The sampling period is ¹"10\ s. In the "rst reaching phase, the control amplitude has been kept constant at the value ;+"15. After that, the iteration steps are performed at

t"tQ"3 s and t"tQ"6 s. The controller parameters are set as kO"62, k3"50, kO"4 and k3"80.

In Fig. 3, the iterative improvement of accuracy is pointed out, while Fig. 4 highlights as the control input to be ringing-free after the "rst iteration step, and comes closer and closer to the equivalent control as the iteration step increases.

6. Conclusions

In this paper, the discrete-time implementation of a second-order sliding mode control algorithm has been considered making reference to nonlinear uncertain sys-tems in the Brunowsky normal form. Su$cient condi-tions under which a sliding motion in an O(¹) boundary layer of the desired manifold is established have been derived. Simulations con"rm the e!ectiveness of the proposed method.

Appendix A. Proof of Theorem 1

A.1. Reaching of the xrst singular value

The aim is to prove that the initializing control (9)}(11) guarantees the ful"llment of the reaching-type condition

sK(t)s
(t)(!hs
, which implies that s
converges to zero

after a "nite transient. Since the reaching condition implies a monotonic convergence to zero, by exploiting the ISS property of the internal dynamics (4), and by the de"nition of the sliding variable, condition (11) can be derived (Bartolini et al., 1998a), where P and Q are proper constants depending on the entries of vector c. Moreover, combining (6) and (8), yield (10)}(11), from which, consid-ering (3) and (7), the claim can be demonstrated.

A.2. Contraction property

If the control u sets the sign of sK, then the control law (13)}(16) causes a sequence s+G"s(t+G) of singular

points, (i"1, 2,2). The convergence to the sliding mani-fold is attained if the following contraction property is guaranteed:

s+G>(s+G, i"1, 2,2 (A.1)

With no loss of generality we can assume that the actual singular value s+G is positive. Starting from the

point (s+G,0) in the s O s
plane, and considering the

worst-case system trajectories (i.e. the uncertainties act-ing against the contraction with their maximum e!ort), the subsequent singular point will belong to the interval

*0, q'1# G HG, (A.7) h(q!1)!4'8q  # 4q 



After some manipulations, we can rewrite (A.7) as

h'0, q*1#4 h, (A.8) ' 1 2a(q; h)(!b(q; h)#(b(q; h)!248qa(q; h)), where a(q;h)"[h(q!1)!4], b(q;h)"!16q[h(q!1)!4]. (A.9) Since de"nes the dimension of the boundary layer, it makes sense to search its lower bound

 with respect to

q andH. Using the facilities of Matlab one can solve the

associated nonlinear constrained minimization problem, obtaining the results depicted in Fig. 2. The contraction of the singular points modula takes place untils+

G is

such that'

, and, as does not depend on ¹, then, by (A.5), (17) is directly derived.

To complete the proof,
G must be explicitly evaluated by solving the algebraic loop that derives from the recip-rocal dependence between the control amplitude and the uncertainty bounds. On the basis of (6), and by relying on the worst-case system trajectories, the following chain of inequalities can be written, in which G represents an overestimate of the unknown bound
G

G)M(xG)#¸[(MG#G;+ )¹ #(s( +G(MG#HG;+ )#a¹])MG, (A.10) where x G is given by x G"PM
[kK+ G]#QMs(+G#RM¹ (A.11)

vicinity (17) at the time t"tQ

is reached in a "nite time tQ. Note that s)O(¹) implies thats
)O(¹).

Consider the worst case initial condition on (17), i.e.

s(tQ)"h ¹, s
(tQ)"h ¹(without the loss of general-ity, h_ and h_ can be assumed strictly positive).

The application of the control (22), (26)}(28), leads to

sK(t)"g(x(t))((t)#u[k]), t3[k¹,(k#1)¹), (B.1) where, as long as the condition s
"O(¹) is not lost, one has

(t)"u [k]!u(t))B(¹, t3[k¹,(k#1)¹) (B.2) with B)#=(¹. It is not di$cult to show that

s
remains O(¹) at any t*tQ.

From this point onwards, the proof can be continued as that of Theorem 1, but the solution of system (A.7) is considerably simpli"ed by the fact that, considering (B.2) and (7), the uncertain drift term in (B.1) is upperbounded by the constant GB(¹.

Taking into account the de"nition of the normalized variables q and, the `optimala control amplitude can be computed by (13) as

;

+"(qH!1)B(¹Ok3(¹. (B.3)

Thus, the residual set (29) is reached in a "nite time.

Appendix C. Proof of Theorem 3

The proof of this theorem follows the same guidelines as that of Theorem 2, and the same computations are repeated within each time interval t3¹H,(tQH, tQH>). It

must be proven that an O(¹\\H) boundary layer is attained after a "nite time tQH>.

It is not di$cult to show that (B.1) is still valid, but the term(t) is such that

(t)"u (t)!u(t))kHH#kH¹\\H\ #kH

where kGH, i"1, 2, 3, are proper constants (see Bartolini et al. (1998b)).

If the time constantH is chosen as in (32), then (t) is an O(¹\\H) according to

(t))B¹\\H _(C.2)

and, by (7), the same is for the drift term( ) )g( ) ) in (B.1). The optimal control e!ort can be computed by (13) as ;H

+"(qH!1)B¹\\HO_k3H¹\\H. (C.3) Thus, by (A.5), (7) and (C.2), the residual set (33) is reached in a "nite time.

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Giorgio Bartolini was born in Milano, Italy, in 1944. He graduated in Electrical Engineering at the University of Genova in 1968, where he became associate professor at the Department of Computer, Com-munication and System Sciences (DIST). He is currently a full professor of Control Systems Optimization at the University of Cagliari. His research interests are in the "elds of robust and variable structure con-trol, adaptive concon-trol, robotics and simula-tion of continuous systems. He is the author of about 60 journal papers. He has been responsible of some research projects of the Italian National Research Council (C.N.R.) and the European Community in the "eld of mobile and submarine robotics.

Alessandro Pisano was born in Sassari, Italy, in 1972. He graduated in Electronic Engineering in 1997 at the University of Cagliari, Italy, where he is currently work-ing toward the Ph.D. degree in Electronic Engineering and Computer Sciences in the Department of Electrical and Electronic Engineering (DIEE). His current research interests include nonlinear control, vari-able-structure systems and the application of sliding-mode control to mechanical and electromechanical systems.