INT. J. CONTROL,
1999, VOL. 72, NO. 15, 1343± 1353
Robustness of static sliding mode control for non-linear systems XIAO-YUN LU{ { and SARAH K. SPURGEON{ Robustness of static sliding mode control based on a ne non-linear state space models in regular form is considered. This is necessary because the stability of non-linear systems is usually a local property which can be destroyed by an additive uncertainty. Thus the uncertainty should be appropriately structured. A cone bounded uncertainty structure is assumed in this paper. In general, when a regular form is used to design a static sliding mode control, only matched uncertainty can be expelled. For the stability of the closed-loop system, a minimum phase assumption is necessary. It is shown that the sliding manifold is equivalent to the manifold determined by the zero dynamics and the minimum phase assumption relates to the compactness of the sliding manifold in the ideal sliding mode. It is also shown that the two phase separability, i.e. reaching mode and ideal sliding mode, which holds for linear systems, does not hold generally for non-linear systems. A pertinent example is given. 1.
Introduction
Sliding mode control research has recently developed into two main areas. The ® rst is static sliding mode control based on the work of DeCarlo et al. (1988), Utkin (1992) and Luk’yanov and Dodds (1996) . In this direction, design is based on the regular form of the state space model. A main feature of this approach is that it provides static feedback and chattering appears if discontinuous switching is adopted. This can be avoided by replacing discontinuous switching with a continuous round-o such as the saturation function for practical application. Such a regular form is easy to obtain for linear systems but is di cult to ® nd for non-linear systems, even non-linear a ne systems (Utkin 1992). The second approach is dynamic sliding mode control which is based on di erential (input± output) I-O systems (Sira-Ramirez 1993, Lu and Spurgeon 1997 a,b, 1998 a± d, Lu et al. 1997). A main characteristic of this approach is that it provides naturally dynamic feedback control and thus chattering due to switching will be e ectively ® ltered out. Besides, design based on di erential I-O systems has the advantage of overcoming di culties caused by non-minimum phase phenomenon. It is shown in van der Schaft (1989) that any nonlinear state space model may have a locally equivalent di erential I-O model as long as the outputs are locally observable. Thus the method based on di erential I-O systems provides a non-linear controller design method for a rather general class of non-linear systems. Sliding mode control is generally believed to be robust with respect to uncertainties. For linear systems, the robustness property is well established (Drazenovic 1969, Ryan and Corless 1984). Robustness results also Received 1 October 1998. Revised 31 March 1999. { Control Systems Research, Department of Engineering, University of Leicester, Leicester LE1 7RH, UK. { Author for correspondence. e-mail:
[email protected] . ac.uk
exist for particular types of non-linear systems, as described in Slotine and Coetsee (1986) . For non-linear dynamic sliding mode control, robustness is discussed in Lu and Spurgeon (1997 b, 1998 a). The uncertainties are considered as cone bounded. Under a minimum phase assumption, ultimate boundedness of the closed loop system is obtained. The design parameters depend on the parameters of the uncertainty bounding condition. As long as some speci® ed conditions are satis® ed, robustness is guaranteed. However, such a robustness problem has not yet been discussed in the case of nonlinear static sliding mode control. It is felt that, due to the complexity of non-linear dynamic behaviour, it is improper to take an intuitive feeling as a foundation for robustness. The robustness analysis should be based on several key issues: (1) the types of uncertainties the system is subjected to; (2) the magnitude of the uncertainties; (3) the relationship between design parameters, such as gain, location of zeros, region of attraction, etc., and the structure and magnitude of uncertainties which the system can endure and yet maintain stability. For linear stable systems, any additive bounded uncertainty will produce a bounded state trajectory. For non-linear systems this is not the case in general. Besides, when a system has a triangular structure, as usually results from the sliding mode approach x_
…
1†
x_ 2 …
†
ˆ
f
…
1†
ˆ
f
…
2†
1†
…
x
…
x 1 ;x2
…
…
†
† …
†
†
there is the important peaking phenomenon (Sussman and Kokotovic 1991). This phenomenon does not allow 1 the x -dynamics to be asymptotically stable arbitrarily fast and with arbitrarily high gain. …
†
International Journal of Control ISSN 0020± 7179 print/ISSN 1366± 5820 online # 1999 Taylor & Francis Ltd http://www.tandf.co.uk/JNLS/con.htm http://www.taylorandfrancis.com/JNLS/con.htm
1344
X.-Y . L u and S. K. Spurgeon
The main contribution of this paper is as follows: (1) Robustness of static sliding mode control of non-linear a ne systems is considered when uncertainties are subjected to sector conditions; (2) Pointed out the link between zero dynamics and sliding manifold. (3) The two phase, i.e. reaching mode and sliding mode separability for linear systems is not true any more for non-linear systems because of two factors: (1) (a) local stability; (1) (b) peaking phenomenon caused by the choice of sliding gain. (2) This is shown by an example. Notation: D x denotes the domain which is compact with x 2 D x R n and 0 is in the interior of D x . 2.
Background
2.1. Static sliding mode control Static sliding mode control considers the following regular form system representation x_
1†
x_
2†
…
…
where x satis® es
…
ˆ
f
…
1†
ˆ
f
…
2†
1†
…
1 2 x ;x
…
x
…
…
†
1†
n1
R , x
2
…
…
†
2†
;x
…
2†
9
†
=
† ‡
gx
…
…
1†
;x
…
m
R , u2 R
2
m
2†
†
u
…
;
2: 1†
1 2 and g… x ; x …
†
…
†
†
some condititions. This shows the average behaviour of the control (2.3). 2.2. Sliding reachability conditions The sliding reachability condition T s s_ < 0
is essentially based on the quadratic Lyapunov function in the s-coordinate V
ˆ
1 T 2s s
The condition can be speci® ed as s_ ˆ
¡
® … s; t†
)
sT ® … s; t† > 0
…
2:4†
which is more suitable for controller design. This is because the control variable u can be directly solved out from the ® rst equation in (2.4). Di erent speci® cations of the function ® … s; t† will lead to di erent types of sliding reachability conditions as described in Lu and Spurgeon (1997 a,b, 1998 a,c) which encompass almost all practically used sliding reachability conditions as de® ned in DeCarlo et al. (1988) and Utkin (1992) . These sliding reachability conditions have been used for dynamic sliding mode control but are also suitable for static sliding mode control design. A generally used sliding reachabiity condition is ® … s; t†
ˆ
Ks ‡ K0 sign … s† T
2: 3†
where K is a positive de® nite matrix, K0 ˆ ‰ k01 ; . . . ; k0m Š T with k0i 0, and sign … s† ˆ ‰ sign … s1 † ; . . . ; sign … sm † Š . The sign ( . ) function may be replaced with the saturation function sat ( . ). This reachability condition guarantees that s ! 0 at an exponential rate, asymptotically or in ® nite time period. To achieve good transient performance, fast response and no over-shoot, it is important to choose the sliding gain appropriately in practical design. In the sliding reachability condition, let K be the asymptotic gain matrix and the vector K0 the switching gain vector. From the dynamical system point of view, K tends to make the closed-loop system asymptotically stable while K0 tends to make it approach zero in ® nite time. However, due to switching, chattering would appear in practice. Thus ideal sliding would never happen. Instead, the trajectories will cross the sliding manifold in® nitely fast and stay in a neighbourhood of the sliding manifold.
which wiil force the trajectories of (2.1) to reach the sliding manifold de® ned by (2.2). Then the dynamics of the closed-loop system will be determined by those of the reduced order system on the sliding manifold (2.2). An equivalent control can be constructed under
2.3. Stability of non-linear triangular systems It is well known from the result in Vidyasagar (1980) that the overall stability of the following triangular nonlinear system
1 2 det ‰ g… x ; x …
†
…
†
Š 6ˆ
0;
8 …
1 2 x ;x …
†
…
†
† 2
D
x
The popularly used sliding mode design approach can be described as follows. Suppose the sliding surface is chosen as 1 2 s… x ; x † …
1†
†
…
†
ˆ
0
…
2: 2†
2†
where s… x ; x † has continuous derivatives with respect to all its arguments. (This assumption will be used throughout this paper.) Then from the sliding reachability condition …
…
s_ T s < 0 the controller can be constructed as 8
ui … x 1 ; x 2 ; t† …
†
…
†
0 ‡
…
†
…
†
…
†
…
†
1 2 1 2 ui … x ; x ; t† ; si … x ; x † < 0 ¡
…
†
…
†
…
†
…
†
i ˆ 1; . . . ; m
…
1345
Robustness of static sliding mode control w_ 1
f
ˆ
w_ 2
f
ˆ
…
…
1†
…
2†
…
9
w1 ; t† w1 ; w2 ; t†
x_
…
=
…
;
2: 5†
1†
f
ˆ
…
"
1†
…
…
w_ 1
f
ˆ
…
1†
…
…
1†
;x
…
…
"
1†
Q … x ; s† …
†
†
#
2†
f
@x 1
n
where wi 2 R i , i ˆ 1; 2; depends on the stability of the following two decoupled systems
…
@ s… x
s_ ˆ
@ s… x
…
ˆ
†
1†
;x
…
…
w1 ; t
1†
f
‰
†
1 1 x ; # … x ; s† † …
…
2†
…
†
…
†
1 Q … x ; s† …
‡
†
…
†
…
…
†
…
†
†
…
w_ 2
f
ˆ
…
2†
…
0; w2 ; t†
w_ 2
f
ˆ
…
f
ˆ
…
1† 2†
9
w1 ; w2 ; t†
…
=
w1 ; w2 ; u; t†
…
;
…
2: 6†
the manipulation of the control variable u will a ect the dynamics of w2 and w1 through w2 . The latter relates to the case of static sliding mode control design based on a regular form. Thus one cannot, in general, expect their stability to be considered separately and unconditionally. This is because of the peaking phenomenon. The choice of controller parameters could retain the stability of the w2 -dynamics, but the trajectory of the w1 could well move outside the stability region for the overall system and cause instability.
"
@ s… x
…
1†
;x
…
2†
@ x… 1†
†
#
f
…
1†
x 1 ; # … x 1 ; s† † …
…
†
…
†
Q… x 1 ; s† …
‡
†
2.4. Sliding mode control design Looking at the sliding mode control design problem from a di erent point of view compared to that described in Utkin (1992) , (2.2) can be considered as a local coordinate transformation 1 2 x ;x …
†
…
†
† !
…
"
uˆ
¡
@ s… x
…
1†
;x
…
@ x… 2†
"
@ s… x
…
1†
†
@ s… x
‡
;x
…
1†
@x
…
2†
@ x… 2†
†
f
1†
…
1†
…
1†
#
¡
1
g… x ; # … x ; s† † …
…
® … s; t†
‡
2†
…
2†
…
;x
…
1†
2†
†
f
…
1 1 x ; # … x ; s† † …
†
x 1 ; # … x 1 ; s† † …
†
…
…
†
#… x
…
ˆ
1†
†
#
†
x_ 1 …
ˆ
f
s_ ˆ
¡
†
…
1†
…
det
@ s… x
…
1†
;x
@ x… 2†
…
2†
†
#
6ˆ
†
…
†
x 1 ; # … x 1 ; s† † …
†
…
†
® … s; t†
® … s; t†
¡
is globally asymptotically stable and thus s ! 0 in ® nite time or asymptotically. (4) In the ideal sliding mode, s 0. This results in a reduced order dynamics (or equivalently, the remainder dynamics)
0
for … x 1 ; x 2 † 2 D x . It is also a necessary condition for the controller construction when the trajectory is outside the sliding surface (2.2). Then the system (2.1) is equivalently transformed as …
†
()
; s†
which is solved from (2.2) by the implicit function theorem if "
†
1 (2) The closed-loop system in the … x ; s† -coordinates is of the form
s_ ˆ …
® … s; t†
(1) The sliding mode controller can be readily solved out as
with x2
¡
(3) Proper choice of sliding reachability condition guarantees that
1 x ; s† …
ˆ
From this step, there are the following immediate consequences:
…
…
2: 7†
Now set
The stability of w1 will a ect that of w2 . However, if control action is involved as in w_ 1
> > > > > > > > > > > > > > =
> > > > > > > > > > > > > > ;
1 1 x ; # … x ; s† †
…
g… x 1 ; # … x 1 ; s† † uŠ
‡
and
…
#
2†
@x 2
†
9
1 1 x ; # … x ; s† †
x_
…
1†
ˆ
f
…
1†
…
1 1 x ; # … x ; 0† † …
†
…
†
According to established results on the stability of triangular systems, these determine the stability of the closed-loop system. 2 1 2 It is noted that even if f … x ; x ; t† ‡ 1 2 g… x ; x ; t† u in (2.1) is linear time invariant …
…
†
…
†
†
…
†
…
†
1346
X.-Y . L u and S. K. Spurgeon x
…
A
!
1†
x2 …
…
B
‡
†
x
1†
!
…
u
x2 …
x_
†
†
…
†
…
…
†
The stability of static sliding mode control for a ne non-linear systems should be considered using the stability results of triangular systems in Vidyasager (1980). Note that the dynamics consists of two parts, the sliding part (when s 6ˆ 0) and the reduced order part (when restricted to the manifold s ˆ 0), which cannot be separated. Instead, they should be considered as a whole. This point will become clear after considering the peaking phenomenon. Remark 1:
The next section will show that the remainder dynamics are in fact a type of zero dynamics as used in di erential algebraic control theory (Isidori and Byrnes 1990). 2.5. Zero dynamics and the sliding manifold The zero dynamics in non-linear control design can be motivated from the stability of such triangular systems x_
1†
x_
2†
…
…
1†
n
ˆ
f
…
ˆ
f
…
1† 2†
2†
…
x ;x
…
1†
2†
x ; x ; u† …
1†
…
…
…
†
…
†
=
…
;
2: 8†
1†
x_
2†
…
…
…
†
…
†
ˆ
f~
…
1†
…
†
ˆ
f
…
2†
1 x ; 0†
…
…
2 2 0; x ; u… 0; x
†
…
†
…
†
…
†
†
…
†
†
†
†
x_ 1 …
†
f
ˆ
…
1†
x 1 ; # … x 1 ; s† † …
…
†
…
†
which determines the remainder dynamics in the ideal sliding mode as x_
…
1†
f
ˆ
…
1†
1 1 x ; # … x ; 0† † …
…
†
…
†
It may be concuded that the sliding manifold is in fact determined by a zero dynamics. The sliding manifold determined by the sliding surface in the ideal sliding mode is, in fact, the maximal integral manifold determined by the zero dynamics. It is noted that the 1 dynamics of x in the ideal sliding mode is not controlled. Thus the stability of the closed-loop system resulting from static sliding mode control design depends on the stability of this type of zero dynamics. Similarly, the stability situation can be divided into minimum phase and non-minimum phase cases as in Isidori and Byrnes (1990) , Isidori (1995) and Lu and Spurgeon (1997 b, 1998 b). In the case of linear systems, after coordinate transformation …
†
1 2 x ;x …
†
…
†
† !
…
1 x ; s† …
†
under the assumption that @s @ x… 2†
0
6ˆ
a regular form can be written as x_
…
1†
ˆ
A11 x
…
1†
s_ ˆ A21 x 1 …
†
‡
A12 s
‡
A22 s ‡ B2 u
9 =
;
…
2: 10†
The corresponding zero dynamics are x_
…
1†
ˆ
A11 x
…
1†
There is no wonder that in linear sliding mode design, A11 is supposed to be Hurwitz. It is noted that if the original system is controllable, A11 can always be made Hurwitz. Otherwise, stabilizability is assumed.
9
…
…
1 2 x ;x
†
det
†
x_
†
p
…
…
…
…
where x 2 R , x 2 R . Several current non-linear control design methods produce a closed-loop system in the triangular form (2.8) (Slotine and Coetsee 1986, Isidori and Byrnes 1990, Koktovic et al. 1992, Lu and Spurgeon 1997 b, 1998 a,d). The di erence is that in the work in Isidori and Byrnes (1990) and Koktovic et al. 2 2 1 2 (1992) , the dynamics of x_ ˆ f … x ; x † correspond to the dynamics of uncontrolled states. In Lu and Spurgeon (1997 b, 1998 a), the zero dynamics is a dynamics of the control. After using proper feedback 1 2 u ˆ u… x ; x † , the stability of (2.8) is closely related to the stability of the two following decoupled systems …
…
2 1 from which x ˆ # … x ; s† can be solved out and replaced in the ® rst equation to yield
9
2†
…
1†
…
…
¡
†
f
ˆ
1 2 s ˆ s… x ; x
the closed-loop system is non-linear because 1 1 2 1 1 ‰ @ s… x ; x † Š =@ x Š and the controller u… x ; t† in (2.3) is non-linear. …
1†
=
…
†
††
;
…
2: 9† 2†
according to the work in Vidyasagar (1980) . The x dynamics in the decoupled form are usually de® ned as the zero dynamics when considering asymptotic stability of closed-loop systems. When a system is in regular form (2.1), in the ideal sliding mode, the reduced order system is determined by …
3.
Robustness of static sliding mode control
3.1. Necessity for robustness consideration For stable linear systems, cone bounded additive uncertainty may a ect stability but the trajectories will remain bounded. For a non-linear system, an additive uncertainty can destroy local stability of the system if the magnitude of the uncertainty exceeds some bounds (Hahn 1967, Khalil 1996) and make the system state
1347
Robustness of static sliding mode control escape to in® nity in ® nite time. Thus, to address the robustness of sliding mode control, it is necessary to consider some type of structured uncertainty. It is well known that the e ect of matched uncertainty can be rejected with proper sliding mode control. The unmatched uncertainties should be within a small threshold (Barmish and Leitmann 1982, Ryan and Corless 1984). When only matched uncertainties are present, ultimately bounded closed-loop stability can be achieved with proper manifold selection. The following discussion will show what one can achieve with sliding mode control when matched uncertainties are present. Unmatched uncertainties will be discussed brie¯ y later. 3.2. Robustness to uncertainties Now suppose uncertainties are present in a system whose nominal part is already in regular form x_
1†
x_
2†
…
…
ˆ
f
…
1†
ˆ
f
…
2†
…
1 2 x ;x
†
…
1 2 x ;x
†
…
†
…
†
1 2 where g… x ; x …
†
…
…
†
…
†
1†
t
… †
=
1 2 g… x ; x † u ‡ …
† ‡
†
…
The uncertainties able and satisfy
†
†
…
†
…
2†
x1 ;x2 …
8 …
t
… †
…
;
3: 1†
1†
1 … t† k
»1k x
k
2 … t† k
»2k … x
…
…
†
…
†
D
;x
…
2†
†k ‡
1 2 s ˆ s… x ; x †
…
†
†
…
†
…
@ s… x
…
1†
1†
;x
…
†
…
det 1†
@s @x
…
6ˆ
2†
†
…
Assumption 2: …
1†
ˆ
f
…
1†
…
1 1 x ; # … x ; s† † …
†
…
…
2†
2†
is asymptotically stable when s ˆ 0, where x ˆ 1 # … x ; s† is obtained from the sliding surface under Assumption 1. This is a minimum phase assumption. 1†
Then di erentiate s… x ; x of (3.1) to yield
…
2†
†
f
1 2 x ;x …
2†
…
…
†
…
2†
…
…
†
…
† ‡
1 2 x ;x
…
†
†
…
†
along the trajectories
1†
t
… ††
†
t
… ††
…
1†
@ s… x
…
Lx1
;x
…
2†
@ x… 2†
†
†
Lx2 …
†
x.
D
It is further assumed that 1†
@ s… x
…
;x
…
2†
@ x… 2†
#
†
¡
1
Ls
1
¡
Now set 1†
;x
…
2†
1†
†
f
…
1†
…
x1 ;x2 …
f
…
†
2†
…
…
…
† ‡
1†
…
1†
;x
…
2†
†
2†
@x
…
2†
x ;x …
@ s… x
…
†
1 2 g… x ; x † u† …
† ‡
†
…
†
ˆ
¡
® … s; t†
in which the function ® … s; t† is de® ned in the sliding reachability conditions. The controller is then solved out as ˆ
¡
@ s… x
…
1†
…
"
® … s; t† @ s… x
…
‡
2†
;x
@ x… 2†
1†
†
…
1†
@x 2†
;x
…
†
1†
g x ;x …
…
@ s… x
‡
@ x… 2†
f
…
2†
…
;x
…
…
…
2†
†
1†
2†
f
1†
x ;x …
…
#
1
¡
†
…
1†
2†
…
x 1 ;x2 …
†
…
†
†
#
†
Under this feedback control and in the … x 1 ; s† coordinates, the closed-loop system can be written equivalently as x_
…
1†
ˆ
f
s_ ˆ
¡
~ … 2† … t†
…
1†
…
1 1 x ; # … x ; s† † …
†
® … s; t†
@ s… x
…
ˆ
1†
@ s… x
2†
†
1†
…
1†
@x
…
†
‡
~ … 2† … t†
‡ …
@x ‡
…
;x
…
†
…
†
†
† 2
†
…
…
…
…
…
0
2†
x_
1†
1 2 g… x ; x † u ‡
"
3: 2†
for … x ; x † 2 D x , which is a standard requirement in sliding mode design (Utkin 1992). …
†
2†
@x
Assumption 3:
s… x ; x has continuous derivatives with respect to all its arguments and Assumption 1:
…
…
…
1 2 for … x ; x
2† †
…
2†
;x
…
f
…
†
is chosen such that …
1†
@ s… x
@ x… 1†
u …
†
1†
@x
"
Suppose the sliding surface …
…
…
x
l2
2†
;x
From Assumption 1, it can be deduced that there exist L x 1 ; L x 2 > 0 such that
@x † 2
l1
k ‡
1†
3: 1†
‡
@ s… x
are Lebesgue measur-
2 … t†
k
…
1†
…
…
and
1 … t†
…
‡
…
0;
† Š 6ˆ
@ s… x
ˆ
9
satis® es
det ‰ g… x 1 ; x 2 …
…
† ‡
s_j
;x
…
2†
…
2†
†
1†
t
… †
…
2†
t
… †
which is used for stability analysis.
…
1†
t
… †
†
9 > > > > > > > > > > > > =
> > > > > > > > > > > > ;
…
3:3†
1348
X.-Y . L u and S. K. Spurgeon
To consider the robustness problem, the bounds for the uncertainties ~ 2 … t† in the coordinates … x 1 ; s† are estimated as …
k
†
~ … 2† … t† k
…
1†
L x 1 … »1k x …
…
†
l1 †
k ‡
L x 2 … »2k … x 1 ; x 2
‡
…
…
†
†
†
…
†
l2 †
†k ‡
Suppose under the mapping (3.2), the domain D x is mapped into the domain D x 1 ;s . From the continuity property of @ s=@ x 2 in D x 1 ;s , the Taylor series and the intermediate value theorem …
…
…
…
…
†
2
8
k
x
…
2†
"
0. The robust stability (region of attraction) of the closed-loop system will depend on the following factors …
†
such that
…
†
†
3:5†
when … x 1 ; s† 2 D x 1 ;s . In this case, the results concerning the stability of triangular systems in Vidyasagar (1980) cannot be used although the closed-loop system (3.3) is in a triangular form. This is because the results in Vidyasagar (1980) require the system to be continuous whereas here discontinuity may appear due to switching. In addition Vidyasagar (1980) assumes no uncertainty is present. Instead, the results concerning the ultimate boundedness of triangular systems as developed in Lu and Spurgeon (1997 b) should be used. These are quoted in the appendix without proof. From Theorem 2 (see Appendix) , the stability of the closed-loop system is readily deduced.
1
…
…
…
L x 1 l1 ‡ L x 2 L s 1 M ‡ L x 2 l2 Š …
†
x_ 1
L x 1 »1 ‡ L x 2 »2Š …
…
x_ 1
l2 Š
l1 †
k ‡
…
but not for
l1 †
k ‡
L x 2 ‰ »2… k x
‡
…
k ‡
L x 2 ‰ »2… k x
‡
sk †
l1 †
k ‡
1 2 L x 2 ‰ »2k … x ; x …
k
†
…
or equivalently for … x ; x
†
L x 1 … »1k x ‡
…
…
…
…
x 1 ; s† …
Ls
k
K0 sign … s†
‡
where K is a positive de® nite matrix to be determined. If 1 … t† ˆ 0, i.e. only matched uncertainties are present then f 1 … x 1 ; # … x 1 ; s† † is continuous. By the inverse Lyapunov theorem and the minimum phase assumption, 1 there exists a Lyapunov function V … x … t† † for
†
2†
…
® 0 … s; t†
ˆ
T T s ® 0 … s; t† > s Ks
there exists M
†
1 s… x ; 0† k …
‡
sk †
k
D
1†
s… x ; 0† k …
†
where ® 0 … s; t† can be any Lebesgue measurable vector function such that
9
1
¡
…
†
D x 1 ;s . It follows that k
…
; h†
There is generally a trade-o between these three factors. The choice of sliding surface under the regularity condition will not a ect the stability of the closed-loop system. However, it will a ect the region of attraction. From the structure of the closed-loop system (3.3), 1 the s-dynamics are independent of the x -dynamics but 1 the x -dynamics depends on the s-dynamics. For the sdynamics, the sliding reachability condition is chosen as …
#
2†
…
From the continuity of s in such that k
1†
@ s… x
1 s… x ; 0† k
1… k
¡
†
†
‡
@ x… 2†
:
†
Thus
†
…
†
"
1 s ˆ s… x ; 0† 1 where … x ; h†
…
…
†
(2) the choice of sliding surfaces (as stated in } 3.1); (3) the choice of sliding gain and other design parameters.
†
(1) the stability of the zero dynamics;
…
†
…
†
Consider the system … 3:1† . Suppose that
(1) The sliding surface is chosen as … 3:2† . (2) Assumptions 1, 2 and 3 are satis® ed. (3) For any given " > 0, choose the sliding reachability condition … 3: 5† with K and K0 satisfying
1349
Robustness of static sliding mode control K ¡ » 2 Im > 0 K0 ¡ … » 1 ¯ x 1 …
(2)
†
l † Im > 0
‡
9
…
;
3: 6†
Then the closed-loop system … 3: 3† is uniformly ultimately bounded by " .
To use Theorem 2 (see Appendix), it is necessary to prove that for the chosen K, the s-dynamics can be rendered into an " -neighbourhood for arbitrarily given " > 0 in ® nite time. Now consider the following Lyapunov function candidate Proof:
V2
1 T 2s s
ˆ
and di erentiate V 2 along the trajectories of (3.3) to obtain V_ 2 j
…
T
ˆ
3: 3†
® 0 … s; t†
K0 sign … s†
s
…¡
¡
sT Ks ¡ sT K0 sign … s†
¡
2
s ; . . . ; j sm j Š
‡
‰j j
2† 1 … t† j …
j
‡
~ … 2† … t† †
3
6
7
6
7
6
7
4
j
~ m2† … t† j
5
…
¡
sT … K ¡ » 2 Im † s ¡ sT K0 sign … s†
‡
s Im sign … s† … » 1 k x
¡
T
…
1†
k ‡
l†
T
s ‰ K0 ¡ … » 1 ¯ x 1 …
Now let s ˆ w1 ; x is proved.
…
…
1 1 x ; # … x ; 0† † …
for all the possible choose parameters s ! 0 is the ideal closed-loop system is omitted here. 4.
†
†
‡
1†
…
3: 3†
l † Im Š sign … s†
< 0
&
f
…
1†
…
x 1 ; # … x 1 ; 0† † …
†
…
†
‡
t
1… †
†
1 … t† †
‡
2L k Pk . This means that (4.1) always has a bounded solution. …
1350
X.-Y . L u and S. K. Spurgeon
Essentially, this analysis is based on the fact that the stability of linear systems is a global property. However, the separability described above for linear systems using sliding mode control does not apply to non-linear systems. This is mainly due to the so-called peaking phenomenon. It is noted that the peaking phenomenon appears also for linear systems, where it is called overshoot. Separability for more general non-linear systems is more complicated. It can be shown that the separability does not hold for the following non-linear a ne system in a regular form x_ 1 …
†
x_ 2
†
…
ˆ
f
…
1†
ˆ
f
…
2†
…
x1 ;x2
†
…
x1 ;x2
†
…
…
†
…
†
…
† † ‡
x_
1†
…
†
…
…
…
1†
s_ ˆ
¡
…
…
†
…
1†
†
†
® … s; t†
…
4: 3
…
4: 4†
†
G… x 1 ; # … x 1 ; 0† ; t† …
ˆ
†
…
†
is uniformly asymptotically stable, one can conclude that the dynamics (4.3) is uniformly asymptotically stable. However, consider the following counter example. Consider the regulation of the following non-linear system Example 1:
x_ 1
ˆ
¡
¬ x1
‡
…
x21 ‡ x22 † sin t ‡ 1 x43
x_ 2
ˆ
¡
¬ x2
‡
…
x21 ‡ x22 † cot t ‡ 2 x43
x_ 3
ˆ
x3 … 0: 04 ‡ x3 †
4
‡
¡
¬ x1
0 such that (A-1) For k w1 k Filippov (1964) ; …
A-2†
c1 , F1 satis® es the Condition B in F2 … t; w1 ; w2 † ¡ F2 … t; 0; w2 † < k w1 k
sup sup t 0
k
wi k ci
1
If in (A.1), F2 is locally Lipschitz with respect to w1 at w1 ˆ 0 and uniformly in t, (A-2) is satis® ed. Remark 3:
Suppose
Theorem 2:
(1) (A-1) and (A-2) hold for c1
ˆ
¯ 0 > 0; c2
ˆ
¯ 2 > 0.
(2) For any " 1 > 0 su ciently small there exists a feedback control u ˆ u² … w1 ; t† such that for the closed-loop system w_ 1 ()
ˆ
F1 … t; w1 ; u² … w1 ; t† †
…
A: 4†
F1 satis® es Condition B in Filippov … 1964† and there exists a continuously di erentiable function V 1: R N¯ 0 … 0† ! R satisfying ‡
¬ 1 … k w1 k †
V_ 1 j
…
6:4†
V 1 … t; w1 † ˆ
@V 1 @t ¡
8 k
w1 k
‡
@V 1 @W1
9
¬ 2 … k w1 k †
> > > > > > > > > > =
F1 … t; w1 ; u² … w1 ; t† †
¬ 3 … k w1 k †
"1 > 0;
8 …
t; w1 †
2
R‡
N¯ 0 … 0† …
()
> > > > > > > > > > ;
A: 5†
where ¬ i … : † (i ˆ 1; 2; 3) are class- K functions and 1 "1 < ¬ 2 … ¬ 1 … ¯0 † † . ¡
(3) The system … A:3† is locally uniformly asymptotically stable for w2 … 0† 2 N¯ 2 … w02 † . Then for any " > 0, there exists 0 < ¯ 1 < ¯ 0 and a control u ˆ u² … w1 ; t† such that the closed loop system … A : 1† is uniformly ultimately bounded by " for any … w1 … 0† ; w2 … 0† † 2 N¯ 1 N¯ 2 . Proof:
See Lu and Spurgeon (1997 b).
References
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