Technical Notes and Correspondence
_______________________________ A State-Dependent Boundary Layer Design for SlidingMode Control
Min-Shin Chen, Yean-Ren Hwang, and Masayoshi Tomizuka
Abstract—The use of a boundary layer in sliding-mode control has been
a common technique to reduce chattering of the control signal. However, different choices of the boundary layer width lead to conflicting effects: a large/small boundary layer width can more/less effectively alleviate the chattering phenomenon, but leads to less/more accurate control results. This note proposes online adjusting the width of the boundary layer based on the state norm for an uncertain linear system. The proposed state-de-pendent boundary layer design can effectively eliminate chattering while at the same time ensuring almost perfect control accuracy.
Index Terms—Boundary layer control, chattering, control accuracy,
sliding mode control, variable structure system.
I. INTRODUCTION
Sliding-mode control is known to be robust against parameter un-certainties and external disturbances [1]–[3]. However, for the sliding surface to be attractive, a switching function must be used in the con-trol law, which causes chattering of the concon-trol signals. In order to re-duce chattering, one can introre-duce a boundary layer [4], [5] around the sliding surface. Inside the boundary layer, the discontinuous switching function is interpolated by a continuous function to avoid discontinuity of the control signals. The width of the boundary layer is normally con-stant, and the larger the boundary layer width, the smoother the control signal. Even though the boundary layer design alleviates the chattering phenomenon, it no longer drives the system state to the origin, but to a small residual set around the origin. The size of the residual set is de-termined by the width of the boundary layer: the larger the width of the boundary layer, the larger the size of the residual set. As a consequence, there exists a design conflict between requirements on the smoothness of control signals and on the control accuracy. For smoothness of the control signals, a large boundary layer width is preferred, but for better control accuracy, a small boundary layer width is preferred.
Instead of using a constant width, one can also use a time-varying boundary layer width. In one example [3], the width can be the filtered output of the reference trajectory. In other examples [6], [7], the width is scheduled to decay exponentially starting from some initial value. The decaying-width design is attractive since it ensures the exponential convergence of the system state to zero. The simulation results in [8] also demonstrate that the decaying-width design is effective in maintaining stability in the face of high-frequency unmodeled dynamics. However, in the decaying-width design, the chattering phenomenon inevitably shows up when the width has decayed practically to zero. Another drawback of the decaying width design, which uses a simple “open-loop” tuning of the boundary
Manuscript received July 24, 2001; revised April 3, 2002. Recommended by Associate Editor P. Tomei.
M.-S. Chen is with the Department of Mechanical Engineering, Na-tional Taiwan University, Taipei 106, Taiwan, Republic of China (e-mail: [email protected]).
Y.-R. Hwang is with the Department of Mechanical Engineering, National Central University, 310 Chung-Li, Taiwan (e-mail: [email protected]).
M. Tomizuka is with the Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740 USA (e-mail: [email protected]).
Digital Object Identifier 10.1109/TAC.2002.803534.
layer width, is its lack of vigilance to the change of system conditions such as the sudden injection of a disturbance.
A more reasonable approach to the boundary layer design is to schedule the width based on the system state; forming a “closed-loop” tuning of the boundary layer width. In this note, it is proposed that the width of the boundary layer be proportional to the modulus of the system state when controlling an uncertain linear system. The new design can control the system state to almost zero with no chattering in the control signals. It is interesting to note the work in [9], where they use a first-order filter to smooth the control signal in a fuzzy sliding mode control design. The filter bandwidth is scheduled to be propor-tional to the angle between the state vector and the normal vector of the sliding surface. Their design, lacking rigorous mathematical proof, is in spirit similar to the state-dependent boundary layer width design in this note. There are other various approaches proposed to alleviate the chattering problem. Prefiltering of the control signal is one approach. This can be done either by direct insertion of a low-pass filter before the plant [10], or by treating the time derivative of the control input as the design input [11], or as used in the disturbance estimator design in [12]. Another approach is the observer-based sliding mode control [13], which can ease the chattering problem due to unmodeled dynamics by constructing a high-frequency bypass loop. The most common approach is to shape the phase portrait near the sliding surface such that the state velocity vector is almost parallel to the sliding surface. See for example [14]–[16], among which the boundary layer control is recognized to be the most simple and widely used approach.
The remainder of this note is organized as follows. Section II re-views the switching sliding mode control for linear uncertain systems. Section III examines the constant-width and decaying-width boundary layer controls. Section IV introduces the new state-dependent boundary layer control. Finally, Section V gives conclusions. Notice that this note offers a “unified” stability analysis for various boundary layer controls, and this analysis is more concise than those previously proposed. In order not to obscure the development of the control designs, all the proofs of lemmas and theorems are placed in the Appendix.
II. SWITCHINGSLIDINGMODECONTROL
Consider a switching sliding mode control design for a linear system with “matching” uncertainties [17]
_x = Ax + B(u + 1Ex + d); x(0) = x0 (1) wherex 2 Rnis the system state,u is a scalar control input, A 2 Rn2nandB 2 Rnare nominal system matrices satisfying the
con-trollability condition [18], uncertainty1E is possibly time-varying, andd an unknown disturbance. The system uncertainties are bounded by two known constants:
k1Ek E kdk D: (2)
Note that one can always perform a state transformation such that the controllable pair(A; B) is in the controller canonical form [18]
A = 0 1 1 1 1 0 1 1 1 1 1 1 a1 1 1 an B = 0 1 0 1 : (3) 0018-9286/02$17.00 © 2002 IEEE
= x(n01)
1 + cn01x(n02)1 + 1 1 1 + c1x1+ c0 t 0 x1d
(6) where the coefficientscis are chosen such that the differential equation (6) is stable (has only left-half plane characteristic roots). The purpose of adding an integral term in (6) is for the special case when the system dimensionn = 1. Note from (3) and (5) that CB = 1.
The differential equations (4) and (6) can be cast into a state-space form _z = F z + Gs; wherez = t 0 x1d x1 1 1 xn01 2 Rn (7)
and matricesF and G are in controller canonical form:
F = 0 1 1 1 1 0 1 1 1 1 1 1 0c0 1 1 0cn01 2 Rn2n G = 0 1 0 1 2 Rn: (8) Since the differential equation (6) is stable by the choice of the coeffi-cientsci’s, the matrixF in (7) is stable.
Several results regarding (7) that will be repeatedly used in later sec-tions are listed below. Firstly, given the stable matrixF in (8), there exist positive constantsm and such that
eF (t0 ) me0 (t0 ) 8 t (9)
where is treated as a control design parameter since its value is deter-mined by the choices ofc0is in (6). Second, given any positive constant 0Re[i(F )] > 0 for all i, where Re[i(F )] denotes the real part
of eigenvalues ofF , there exists a positive definite matrix P 2 Rn2n satisfying the following Lyapunov inequality:
(0F 0I)TP +P (0F 0I)0; 0Re[
i(F )]>0 8 i:
(10) Finally, from linear system theory [18], the state in (7) satisfies
z(t) = eF tz(0) + t 0 e
F (t0 )Gs( ) d : (11)
Stage II—Design of the Control Input: The stable state space equa-tion (7) suggests that if the sliding variable s can be driven to zero by some control design, the statez will decay to zero. Therefore, one chooses the following “switching” sliding mode control to drives to zero:
u = 0s 0 c0x10 CAx 0 (x)f0(s) (12)
exponentially.
III. CONSTANT/DECAYING-WIDTHBOUNDARYLAYERCONTROL
In practical implementation of the switching control (12), the imper-fect switching of the discontinuous functionf0(s) = sgn(s) causes
the control signal to chatter [3]. Such chattering may bring damage to the actuator or excite high-frequency unmodeled dynamics. To alle-viate the problem, a boundary layer around the sliding surfaces = 0 is suggested to smooth the control signal. The result is the so-called boundary layer control
u = 0s 0 c0x10 CAx 0 (x)f1(s) (14) where the discontinuous functionf0(s) = sgn(s) in the switching
control (12) is now replaced by a continuous function f1(s) =jsj + s
0e0t; > 0; 0> 0 (15)
in which0e0tis the width of the boundary layer, which decays ex-ponentially to zero when 6= 0, and remains constant when = 0.
Lemma 2: If the boundary layer control (14) is applied to (1), then js(t)j 0e0t+ js(0)je0t; 0= 0
00 1: (16)
If chosen to be positive, it is a decaying-width boundary layer design. In this case, exponential stability of the controlled system is guaranteed by the control (14).
Theorem 2: If the decaying-width boundary layer control (14) ( > 0) is applied to the uncertain system (1), the system state x will con-verge to zero exponentially.
If is chosen to be zero, it is a constant-width boundary layer de-sign. In this case, the so-called practical stability is guaranteed in the sense that given any small neighborhood of the origin, there always exist control design parameters in (14) such that the state will eventu-ally be trapped in the specified neighborhood.
Theorem 3: If the constant-width boundary layer control (14) ( = 0) is applied to the uncertain system (1), the system state x will asymp-totically approach a residual set around the origin, with the size of the residual set proportional to0=(00 1); 0 > 1.
The following simulation examples demonstrate the effects of using different values of0and in the boundary layer control (14).
Example 1 Constant Boundary Layer Width: Consider a distur-bance rejection problem for (1) with
A = 0 10 0 B = 01
the disturbanced(t) = sin(t), system uncertainty 1E = 0, and xT(0) = [5; 02]. The constant-width boundary layer control (14) is
applied to the system with = 0, 0 = 1:5; = 2; E = 0; D = 1
andc0 = 1; c1 = 2; c2 = 1 in (6). Two computer simulations are
Fig. 1. Time history of state normkx(t)k.
and the other with a large width0 = 0:1. Fig. 1 plots kx(t)k versus
t for t 2 [10; 20], which clearly shows that the control with a smaller boundary layer width (the upper plot) results in better control accuracy, while the control with a larger width (the lower plot) results in much worse control accuracy. On the other hand, one can see from Fig. 2 that the use of a smaller boundary layer width (the upper plot) causes severe chattering in the control signal during the transient (sayt 2 [2; 8]), while the use of a larger width (the lower plot) can effectively alleviate the chattering phenomenon.
Example 2 Decaying Boundary Layer Width: The same system as in Example 1 is simulated by the control (14) with = 0:1, 0= 0:1, and
all other design parameters the same as in Example 1. In this case, the boundary layer width decays exponentially because > 0. At t = 50 second, the system is subject to an impulsive type disturbance, which brings the state toxT(50+) = [3; 4]. When the control drives the state close to the sliding surface again, chattering shows up as shown in the lower plot of Fig. 3 (t 2 [40; 60]), and this is because the boundary layer width has decayed practically to zero. This example reflects that the decaying-width design lacks the ability to respond to the change of system conditions.
IV. STATE-DEPENDENTBOUNDARYLAYERCONTROL
As is demonstrated by the first simulation example in Section III, the constant-width boundary layer design can reduce chattering of the control signals, but it decreases the control accuracy. The compromise between the smoothness of control signals and the accuracy of con-trol results is dictated by the choice of the boundary layer width [0in (15)]. A solution to this design conflict is revealed by a careful exam-ination of Figs. 1 and 2, which shows that for the control with a small boundary layer width, chattering occurs only during the transient stage when the system state is far from the origin. When the state becomes closer and closer to the origin, the chattering phenomenon gradually disappears even though the boundary layer width is very small. This observation suggests that whenkxk is large, one should use a large boundary layer width to avoid chattering, and whenkxk is small, use a small boundary layer width to achieve good control accuracy. There-fore, this note proposes that the boundary layer width be proportional to the modulus of the system state. Such a design automatically ad-justs the boundary layer width based on the system condition, and will be more capable of dealing with unexpected situations such as the one seen in the simulation Example 2.
The aforementioned reasoning leads to the following state-depen-dent boundary layer control
u = 0s 0 c0x10 CAx 0 (x)f2(s) + 12GTP z + 01GTP ez
(17)
Fig. 2. Time history of control signalu(t).
Fig. 3. Control with decaying width = 0:1e .
whereP is as in (10), G as in (8), z the state in (7), 1= 1=(001) >
0, 0= 0=(00 1) > 0, ez = z=kzkpand f2(s) =jsj + s
1kzkp+ 0; kzkp 1
=pzTP z (18)
in which1 > 0 and 1 0 > 0. Since 0 0, the boundary layer
width in (18) is approximately proportional to the state normkzkp. The reason for adding this small0in the boundary layer width is to prevent f2(1) from degenerating into the discontinuous sign(1) function when
the statez has decayed practically to zero. Note that there are two extra feedback terms (the last two terms) in the new control (17), which serve to ensure that the inequality (19) in Lemma 3 holds.
Lemma 3: If the new boundary layer control (17) is applied to the uncertain system (1), the sliding variables will be bounded by, for all t > 0
js(t)j 1kz(t)kp+ 0+ js(0)je0t
1= 1=(00 1); 0= 0=(00 1): (19) Lemma 4 (Bellman–Gronwall’s Lemma) [18]: If a continuous func-tionf(t) 0 satisfies
f(t) b(t) + t
t k( )f( ) d 8 t t0
whereb(t) 0 and k(t) 0 are continuous and nonnegative for all t t0, then
f(t) b(t) + t
t b( )k( ) exp t
k(v) dv d 8 t t0:
Substituting the result of Lemma 3 into (11), and using Bellman–Gronwall’s Lemma, one can establish the following stability result for the state-dependent boundary layer control.
Fig. 4. State-dependent boundary layer control with = 0:001 and =
0:1.
Fig. 5. Control with state-dependent width.
Theorem 4: Consider the uncertain system (1) and the state-depen-dent boundary layer control (17). If the control design parameters are chosen to satisfy
1 > 0; 1 0> 0; 0> 1
0Re[i(F )] > 0; > 1
00 1m p (20)
where and m are as in (9), pis the maximum singular value ofP in (10),F is given by (8), then the system state x will asymptotically approach a residual set around the origin, with the size of the residual set proportional to0.
The following example verifies the new boundary layer design. Example 3 State-Dependent Boundary Layer Width: Consider the same problem as in Example 1. A computer simulation with the new state-dependent boundary layer control (17) is performed with1 =
0:1; 0 = 0:001, and all other design parameters the same as in
Ex-ample 1. The upper plot in Fig. 4 showskx(t)k versus time t for t 2 [10; 20]. As can be seen that the state-dependent boundary layer con-trol achieves the same concon-trol accuracy as the previous concon-trol (14) with a small constant boundary layer width (the upper plot in Fig. 1). Furthermore, the lower plot in Fig. 4 shows that the proposed state-de-pendent boundary layer control (17) successfully avoids chattering of the control signal.
In the second simulation, att = 50 s, similar to the experiment done in Example 2, the system state is suddenly transferred toxT(50+) = [3; 4] by an impulse disturbance, and the result for t 2 [40; 60] is shown in Fig. 5. It is seen from the lower plot that the proposed new de-sign successfully avoids the chattering problem in the decaying-width design in Fig. 3, and this is because the new design can respond imme-diately to the change of system conditions.
should be modified toh2(s) = s=(1kzkp+ 0) if jsj 1kzkp+ 0
andh2(s) = sgn(s=(1kzkp+ 0)), otherwise. APPENDIX
Proof for Lemma 2: Define a region0in the extended state space by
0= xv=1 vx 2Rn+1: jsj 0e0t; 0= 0=(00 1) :
Check the time derivative of(s2 0 20e02t) along the closed-loop trajectory (1) and (14)
d
dt(s20 20e02t) 02s2+ 220e02t
00jsj E kxk + D jsj + jsj
0e0t 0 10 :
Since the term in the rightmost parenthesis is greater than zero forjsj > 0e0t(orxv =2 0), one has, using <
d
dt(s20 20e02t) 02(s20 20e02t) 8 xv =2 0: (A1) Integrating the inequality suggests that0is attractive since
s2(t) 0 2
0e02t (s2(0) 0 20)e02t (A2)
wheret is any time before the extended system state xv enters the re-gion0. One can further deduce from (A2) thats2(t) 02e02t+ s2(0)e02t (
0e0t+ js(0)je0t)2. Taking the square root of the inequality proves that (16) holds for any time beforexv enters the re-gion0.
Once the system statexvhas entered0, which is invariant due to the definition of0and (A1), it will remain in0forever. Ifxv(t) 2
0, one has, by definition,js(t)j 0e0t 0e0t+ js(0)je0t.
Hence, (16) holds both before and afterxventers0.
Proof for Theorem 2: Substituting the result of Lemma 2 with > 0 into (11), one can derive that
kz(t)k m kz(0)k 0 0 0 0 js 0 0j e0 t
+ mjs(0)j 0 e0t+ m0
0 e0t: Hence,z(t) decays to zero exponentially, and so does x(t) following the definitions ofz(t) in (7) and s(t) in (6).
Proof for Theorem 3: Substituting the result of Lemma 2 with = 0 into (11), one can derive, using (9) and kGk = 1, that
kz(t)k m kz(0)k0 00 js 0 0j e0 t+ mjs(0)j 0 e0t+ m 0:
Therefore
lim
t!1kz(t)k m 0: (B1)
By definitions of s in (5) and z in (7), xn = s 0 Cz, where
the previous inequality with kxk kzk + jxnj, one obtains
kxk jsj + (1 + kCk)kzk. Finally, using (16) and (B1), one concludes that limt!1kxk [1 + (1 + kCk)m= ] 1 0, where 0 = 0=(00 1), and , 0and0are control design parameters in the control law.
Proof for Lemma 3: The proof is very much similar to that of Lemma 2; hence only the first few steps are shown as follows. Define a regionzin the extended state space by
z= xv=1 vx 2 Rn+1: jsj 1kzkp+ 0;
1= 1
00 1; 0=
0
00 1 :
Check the time derivative of the following quantity along the trajectory (1) and (17): d dt[s20 (1kzkp+ 0)2] = 2s _s 0 2 1dkzk 2 p dt + 10 1 kzkp dkzk2 p dt 02[s20 ( 1kzkp+ 0)2] 0 20jsj E kxk + D 1 jsj + jsj 1kzkp+ 0 0 10 02[s20 (1kzkp+ 0)2] 8 xv =2 z
where one has purposely added two terms2(10kzk+ 20) and used
(2), (10) and (14) to derive the first inequality. The remainder of this proof follows exactly the same argument as in Lemma 2, and is omitted here.
Proof for Theorem 4: Substituting (19) into (11), and using (9), kGk = 1 and kzkpppkzk, one can derive
kz(t)kp pme0 tkz(0)k + t
0 pme 0 (t0 )
1(1kz( )kp+ 0+ js(0)je0 ) d : (C1)
Letq(t) = e t(1kz(t)kp+ 0+ js(0)je0t). The following can be deduced from (C1):
q(t) 1m pkz(0)k + 0e t+ js(0)je0(0 )t
+ t
0 1m pq( ) d :
Applying the Bellman–Gronwall Lemma to the aforementioned inequality yields q(t) 1m pkz(0)ke m p t+ 0e t + js(0)je( 0)t+ 1mpp0 0 1mpp e t0 e mp t + 1m p pjs(0)j 0 1mpp0 e ( 0)t0 e mp t :
Recalling the definition ofq(t), one can deduce that kzkpsatisfies kz(t)kp m pkz(0)ke0( 0 m p )t + m p p0 0 1mpp 1 0 e 0( 0 mp )t + m p pjs(0)j ( 0 1mpp0 ) e 0t0 e0( 0 mp )t :
Since > 1mppby (20), one obtains lim t!1kz(t)kp 00; 0 = mpp 0 1mpp 1 00 1 :
Finally, following the same procedure as in the proof of Theorem 2, one can show that
lim
t!1kx(t)k 10; 1=
1
00 1+ 1 + kCkpp + 1 0:
Therefore, one concludes thatx asymptotically approaches a residual set around the origin, with the size of the residual set proportional to 0.
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