Discrete-time quasi-sliding-mode control for
a class of nonlinear control systems
S.-D. Xu, Y.-W. Liang and S.-W. Chiou
A quasi-sliding-mode control law is proposed for a class of discrete-time nonlinear control systems concerning system stabilisation and chatter alleviation. An illustrative example is also given to demonstrate the use and benefits of the scheme.
Introduction: It is known that sliding mode control (SMC) schemes possess the benefits of fast response and low sensitivity to system par-ameter uncertainties and disturbances [1, 2]. Therefore, they have been widely applied to control a variety of systems [1 – 6]. On the other hand, many control systems have been mathematically formulated as a discrete-time version because of the popularity and scale of appli-cations of digital computers in technology and industry. Thus, the study of SMC for discrete-time systems has recently attracted consider-able attention[1, 3 – 6]. However, owing to a finite sampling frequency characteristic in discrete-time systems, the system states can only be expected to approach the selected sliding surface and remain around it, instead of remaining on the surface when the system undergoes exter-nal disturbances. Therefore, the so-called quasi-sliding-mode (QSM) concept was introduced and discussed in discrete-time systems [1, 4, 6]. A brief summary is given here. The system states are required to monotonically approach the sliding surface until they enter the vicinity of the surface, and they then remain inside[1]. The vicinity of the sliding surface is called a quasi-sliding-mode band (QSMB). Under this QSM definition, it is noted that the system states are not required to cross the sliding surface, as in the definition given by Gao et al.[4]. In this Letter, we employ the QSM concept to study the stabilisation for a class of discrete-time nonlinear control systems.
Main results: Consider the discrete-time nonlinear control systems: x1½k þ 1 ¼ f1ðx1½k; x2½kÞ ð1Þ
and
x2½k þ 1 ¼ f2ðx1½k; x2½kÞ þ Gðx1½k; x2½kÞu½k þ d½k ð2Þ
Here, x1[ <n1and x2[ <n2are state vectors, u [ <n2denotes system
input, d[k] contains possible uncertainties and/or disturbances, f1, f2
and G are three smooth functions with appropriate dimensions, and f1(0, 0) ¼ 0 and f2(0, 0) ¼ 0. In this study, we say a vector a 0 if
and only if ai0 for each component aiof a. Below we impose three
assumptions for System (1) – (2):
Assumption 1: G(.) is a non-singular matrix for all state vectors. Assumption 2: There exists a function x2[k] ¼f(x1[k]),f(0) ¼ 0, such
that the origin of the reduced order system is x1[k+1] ¼ f1(x1[k],f(x1[k]))
is asymptotically stable (AS).
Assumption 3: dl d[k] du, where dland duare two constant vectors.
Owing to the aforementioned merits of SMC designs, we apply the QSM approach to the controller design. From Assumption 2, we select the sliding surface to be:
s½k ¼ x2½k fðx1½kÞ ¼ 0 ð3Þ
Clearly, the origin of the reduced order dynamics (i.e. set s[k] ;0) is AS by Assumption 2. To derive a suitableffor a sliding surface, several approaches have been proposed. For instance, Kalman and Bertram
[7] used the second method of Lyapunov, while Zheng et al. [5]
adopted the linear matrix inequalities (LMI) technique.
To organise an appropriate QSM controller, we adopt the approach of Bartoszewicz[1]which was developed for linear systems. Suppose that sd[k] has is a desired sliding variable trajectory. One candidate of sd[k]
has the following form[1]: sd½k ¼
kk
k s½0 if k , k
; sd½k ¼ 0 if k k ð4Þ
where kis a positive integer selected by the designer to make the system
states reach the sliding surface in ksteps; the choice depends on the
desired convergence rate to the selected sliding surface and the maximum control magnitude a control system may provide. Once sd[k] is determined, we choose the QSM controller to be:
u½k ¼ Gðx1½k; x2½kÞ1½fðf1ðx1½k; x2½kÞÞ
f2ðx1½k; x2½kÞ d0þ sd½k þ 1
ð5Þ where d0¼ 1/2(dlþ du). Substituting (1), (2) and (5) into (3), we have
ks[k þ 1] 2 sd[k þ 1]k ¼ kd[k] 2 d0k dd, where dd¼ (1/2)kdu2
dlk and k.k is a vector norm. Thus, the system states will approach
the sliding surface and remain around there after ksteps. The constant
ddis the width of the QSMB. It is worth noting that the asymptotic
stab-ility performance can be achieved if d[k] ; 0.
If the change rate of d[k] is limited by the relation kd[k þ 1] 2 d[k]k Dd, where Dd is a known constant and Dd,dd, then the
control given by (5) can be modified as: u½k ¼ Gðx1½k; x2½kÞ1 fðf1ðx1½k; x2½kÞÞ f2ðx1½k; x2½kÞ d0þ sd½k þ 1 P k i¼0 ðs½i sd½iÞ ð6Þ
We now show, by mathematical induction, that ks[k þ 1] 2 sd[k þ
1]k Ddfor all k. From (1) – (3), the modified law (6) and the fact
s[0] ¼ sd[0], we have s[1] 2 sd[1] ¼ d[0] 2 d0 and s[2] 2 sd[2] ¼
d[1] 2 d02 (s[1] 2 sd[1]) ¼ d[1] 2 d[0]. Suppose that s[i] 2 sd[i] ¼
d[i 2 1] 2 d[i 2 2] for all 2 i k. Then, s[k þ 1] 2 sd[k þ 1] ¼
d[k] 2 d02
Pk
i¼0 (s[i] 2 sd[i]) ¼ d[k] 2 d02 (d[0] 2 d0) 2
Pk
i¼2(d[i 2 1] 2 d[i 2 2]) ¼ d[k] 2 d[k 2 1]. Thus, ks[k þ 1] 2
sd[k þ 1]k Dd, as required. Since Dd,dd, it implies that this
QSMB width is smaller than the former one.
Example: Consider a trailer-truck kinematic model[8]: x1½k þ 1 ¼ x1½k hT L sinðx1½kÞ þ hT l tanðu½kÞ þ d½k ð7Þ x2½k þ 1 ¼ x2½k þ hT L sinðx1½kÞ ð8Þ x3½k þ 1 ¼ x3½k þhT cosðx1½kÞ sin x2½k þ x2½k þ 1 2 ð9Þ
Here, the three states x1[k], x2[k] and x3[k] are the angle difference
between the trailer and the truck, the angle of the trailer, and the vertical position of the rear end of the trailer, respectively, u[k] and d[k] denote the steering angle and the possible disturbances, L is the length of the trailer, l is the length of the truck, T denotes the sampling period, and his the constant speed of the backward movement. The geometry of the system can be found in Tanaka et al.[8]. By letting x1[k] ¼ (x2[k],
x3[k])T, x2[k] ¼ x1[k] and u[k] ¼ tan(u[k]), System (7) – (9) can be put
into the form of (1) – (2). The control objective is to realise the backward movement for the trailer-truck along the horizontal line x3¼ 0 without
any forward movement; that is, to realise x1!0, x2!0 and x3!0.
The parameters and initial states in this example are selected as follows: L ¼ 0.13 m, l ¼ 0.087 m,h¼ 20.1 m/s, T ¼ 0.5 s, x1[0] ¼
(1.571, 1)T and x
2[0] ¼ 0. The function f given in (3) is set to be
linear in the form off(x1[k]) ¼ cTx1[k], where c ¼ (1.3325, 23.9)T
is the vector such that the linearisation of the reduced-order model given in Assumption 2 has eigenvalues at 0.7 and 0.75. The desired sliding variable trajectory sd[k] is taken in the form of (4) with k¼
10. To demonstrate the robustness performance of the proposed schemes, we choose a slow varying disturbance, d[k] ¼ 0.1sin(0.1 k).
Numerical simulations are given inFigs. 1and2. Among these, three different control laws are adopted. Two of them are the QSM controller (5) (labelled by QSMC1) and the modified QSM controller (6) (labelled by QSMC2), while the other is an existing fuzzy scheme[8](labelled by fuzzy). It is observed from Figs. 1a–c that the states by the three schemes exhibit oscillation because of the effect of disturbance; however, the amplitudes of the oscillation by the two QSM schemes are much smaller than those of the fuzzy design. This demonstrates the robustness characteristic of the QSM designs. Fig. 2shows the time response of the sliding variables and their magnified scale by the two QSM schemes. Both of the sliding variables are seen to approach the selected sliding surface within 10 steps, as desired. By direct
inspection, the QSMB widths of the QSMC1 and the QSMC2 schemes are dd’ 0.2 and Dd¼ jd[k] 2 d[k 2 1]j ’ 0.01, respectively, which
agree with the theoretical results. In addition, the proposed QSM schemes do not require the system states to cross the sliding surface at each step (i.e. each sampling instant) when the states are within the QSMB. Therefore, the chattering phenomenon can be greatly alleviated when comparing to the QSM definition given by Gao et al.[4]. By direct calculation based on Fig. 1d, we have (kuk1)QSMC2¼ 0.3946 ,
(kuk1)QSMC1¼ 0.3981 , (kuk1)Fuzzy¼ 0.4771 and (kuk2)QSMC1¼
2.0157 , (kuk2)QSMC2¼ 2.0424 , (kuk2)fuzzy¼ 2.9723, where
kuk1:¼ max
k ju½kj and kuk2:¼
pP
k
u2½k. From this example, it is
concluded that the proposed QSM schemes are not only more robust than the existing fuzzy controller, but are also able to alleviate chatter without creating an extra control burden.
0.5 fuzzy fuzzy fuzzy fuzzy QSMC1 QSMC2 QSMC1 QSMC2 QSMC1 QSMC1 QSMC2 QSMC2 0 x1 [ k ](m) x3 [ k ](r ad) x2 [ k ](r ad) u [ k ](r ad) –0.5 0 50 100 150 200 0 50 100 150 200 k k b a d c 1 2 0 –1 0.5 1 0 –0.5 0.5 0 –0.5
Fig. 1 Time histories
a – c Time history of system states
d Time history of control by the three schemes
2 1 0 –1 0.15 0.10 0.05 0 –0.05 –0.10 0 20 40 60 80 100 120 k 140 160 180 200 a s [ k ] s [ k ] b QSMC1 QSMC2
Fig. 2 Time histories of sliding variables a Time history of sliding variables
b Time history of sliding variables on magnified scale by the two QSM schemes
Conclusions: A quasi-sliding-mode control scheme for system stabilis-ation is proposed for a class of discrete-time nonlinear control systems. It has been shown that the scheme not only achieves the stabilisation performance, but it also alleviates chatter without creating an extra control burden.
#The Institution of Engineering and Technology 2008 16 April 2008
Electronics Letters online no: 20081070 doi: 10.1049/el:20081070
S.-D. Xu, Y.-W. Liang and S.-W. Chiou (Department of Electrical and Control Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30010 Taiwan, Republic of China)
E-mail: ywliang@cn.nctu.edu.tw References
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