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u=x2+kTx,,. (2) where g is in R”. k is in R”-' and can be selected by pole placement so ... are in the range space of b (Drazenovic, 1%9; El-Ghezawi et al., 1983).
Auromaticu, Vol. 31, No. 2, pp. 303-307, 1995 Copyright Q 1995 Elsetier Science Ltd Printed in Great Britain. All rights reserved ooos-1098/95 $9.50 + 0.00

Peqpmon

Brief Paper

Sliding Mode Control of Linear Systems with Mismatched Uncertainties* CHI-MAN Key Words-Linear

systems; adaptive control; variable structure systems.

robust to matched uncertainties. It is certainly true that many systems can be classified under this category. For example, the rigid robot is one of them. However, there are even more systems which unfortunately are affected by mismatched uncertainties and do not enjoy the nice matching condition. It appears that there does not exist any explicit way of handling mismatched uncertainties in the literature since the work of Drazenovic (1%9). In this paper we present some results to explicitly tackle the mismatched uncertainties. We shall describe two schemes to handle mismatched uncertainties. Although the design procedures are quite similar, the assumptions and stability results are quite different. Our new dynamical approach of sliding variable formulation introduces on-line estimation capability to the sliding variable so that, when the system is in sliding mode, an adaptive reference signal is realized which can explicitly deal with mismatched uncertainties. Note that when the system is in sliding mode, the system behavior is completely determined by the reduced-order system. Hence we can allow (lb) to be highly nonlinear and time-varying. This is a result of the well-known model reduction capability of sliding control. The stability analysis and design are modular and intuitively simple. In Section 3, we shall apply our new method to two examples to illustrate the design procedure as well as its effectiveness. Finally, conclusions will be drawn and comments will be made in Section 4.

Abstract-It is well-known that sliding mode control is robust to matched uncertainties that lie in the range space of the input matrix. However many systems are affected by mismatched uncertainties and yet do not enjoy the matching conditions. It is the purpose of this paper to present a new dynamical approach of sliding variable formulation which, when the system is in sliding mode, can explicidy deal with mismatched uncertainties. Our first step is to treat certain states as inputs to a reduced-order system and use adaptive techniques to design fictitious controllers for these inputs, which can then tackle the mismatched uncertainties. The second step is to use sliding control to realize the adaptive fictitious controllers. The design is systematic, modular and intuitively simple. 1. Introducdon Consider the regulation single-input system:

of

KWANt

the

following

controllable

*I = (All + ~~I)XI + (Arz + u,z)xz xz = (A*, + AA2,)x, + (AZ2 + A&,)x,

+ (b + Ab)u,

(la) b > 0, (lb)

where u, xz, AZ2 are in R' x,, A,,, A$ are in R"-' A,, is in R(n-‘)x(n-‘). A(.) denotes uncertainties in (.). Tracking and MIMO systems can be similarly treated. Usually the sliding variable is defined as (for example, Drazenovic, 1969; Utkin, 1978, El-Ghezawi er al., 1983; Bailey and Arapostathis, 1987; Yeung and Chen, 1988; Slotine and Li, 1991)

2. Method We shall describe two schemes to handle mismatched uncertainties. One is with a reference model and the other is without a reference model. Although they are similar in design procedures, they require different assumptions. The types of stability are also different. 2.1. Tackling mismatched uncertainties without using a

u=grx or without loss of generality u=x2+kTx,,

(2)

where g is in R”. k is in R”-’ and can be selected by pole placement so that (A,, - A12kT) is asymptotically stable and has some desirable convergence rate. This approach is completely robust to those uncertainties in (lb) because they are in the range space of b (Drazenovic, 1%9; El-Ghezawi et al., 1983). However, since AA,, and AA,* are not in the range space of b, the method is not robust to these so-called mismatched uncertainties (Drazenovic, 1969). This can be considered as one of the major limitations of sliding mode control. Sliding mode control (or variable structure systems) has been widely applied to various systems since its introduction about three decades ago. As pointed out by Drazenovic (1969), a salient feature of this control is that it is completely

reference model. Step 1. Treating q as the input to (la), we find a reference

signal xzd by using an adaptive technique counteract the mismatched uncertainties. The following assumptions are needed: Assumption 1.

to explicitly

and IV&II < IIAlzll, i.e. led < 1. Assumption 2. The elements in 8, and ez are bounded. Remark 1. The first assumption says that the parametric uncertainties M,, and AAl* are in the range space of A,*. In other words, matching conditions are required for the reduced-order system. It also implies that the uncertainties are linearly parametrizable. This will be useful for deriving an adaptive law to suppress the effects of these uncertainties. The second one requires some knowledge on the bounds of the unknown parameters. We shall need this information when we derive the sliding controller. Using Assumption 1, equation (la) can be rewritten as

*Received 31 August 1993; revised 28 January 1994; received in final form 25 May 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. K&era under the direction of Editor Ruth Curtain. Corresponding author Dr C.-M. Kwan. Tel. 1 817 794 5933; E-mail [email protected]. t SSC Laboratory, 604 Causley Ave., #108, Arlington, TX 76010, U.S.A.

i, = A,,x,

+ A,2(~2 + We),

(3)

where W is a known function of x1 and xz, and 6 = [ef&]‘. 303

Brief Papers Treating xz as the input controller of the form

to equation

x2,, = -k’x,

(3). we find a fictitious

- We.

of parameter errors (Assumption 2) and p is a known scalar. Then the sliding condition (Bailey and Arapostathis, 1987) is satisfied, i.e.

(4) (T&S -7

where ,k is chosen to stabilize the linear part of equation and 0 IS the parameter estimate of 0. Defining A’;, =A,, and substituting

e=e-e

-A,>k’

(4) into (3) gives i, =A;,x,

+AlzWi%

(5)

Note that A;, should be asymptotically stable by suitably choosing k. Consider the following Lyapunov function candidate 1’ = $x:Px,

+ $‘r-

‘6.

(6)

where P and I‘ are symmetric and positive definite matrices. Differentiating (6) along the trajectory of (5) yields ti = fx:(PA;,

+ (A;‘I)‘P)x,

+ 6 ‘WTA:$x,

+ @r

‘8.

Selecting PA;, + (A;,)‘P

= -Q,

/VI.

(11)

(3) Hence cr will reach zero in finite time, i.e. x, converges to X~ in finite time. xZd will then make xi -+O and r? is bounded following the arguments in Step 1. From (4) it can be easily seen that x2 = ( -kTx, - &x,)/(1 + 8,). Thus we have x2-+0 as x, +O. 2.2. Tackling mismatched uncertainties using a reference model. It should be noted that Assumption 1 is quite stringent. It requires those mismatched uncertainties to lie in the range space of A,*. To alleviate this problem we introduce a reference model which may provide some extra freedoms in designing the adaptive reference signal. Here we assume the following nominal model: i Inr =&xi,,,

+ &,r,

(12)

where A,, is chosen to be asymptotically stable and has the same dimension as A,, xlm also has the same dimension as xi. The following assumptions are required: Assumption 3. Matrices 0, 4 of appropriate dimensions exist and satisfy the following two simultaneous equations:

Q >O (A,, +MA,,)+&,e=&

and using the adaptation

law

(A,z + AA,,)4

6 = -I‘WTA:;Px,

(7)

will give k’ = -+x:Qx,

xi =&xi

of the reference

signal

x2,,. Define

+ B,,(Yx,

by (4). Differentiating

(8) and using

@ = C#J’ -

(1)

(14)

(9)

f(x) = (1 + &)](&,

+ uz,)x,

where

Q satisfies

and Li. 1991)

- h(x) sgn (u)]/(l

the nominal values of Note that there is one avoid this, we can use the 1. Parameter projection then be used to ensure chosen as

h(x) = (P - 1) If(x)1 + W(x)

+

VI? rl >(I

where --f(x)1 < F(x)

of states

I p’

6 ; < P.

depending

(16)

equation Q=QT>O.

(17)

(10) Using the adaptation

can be used where (-) denotes (.). sgn (.) is the Signum function. singularity in (lo), i.e. f3* = ~1. To condition l&l < 1 in Assumption (Sastry and Bodson. 1989) can It?,/ < 1. The switching gain h(x) is

(15)

+ eTPB,@x,,

+ Tr(QT&),

the following

PA,+A;P=-Q,

+ 4,)

+ &.

+ :Tr(WTY).

+ e[PB,Yx,

+ Tr(Y%J)

x=[X:,X?,e’]‘.

function

+ $Tr(aT@)

ti = -:e:Qei

+ WI’WTPA IZ ‘-x I

of the form (Slotine

xLd = $4x,

where P is symmetric and positive definite and Tr(.) denotes the trace of (.). Differentiating (15) along (14) gives

+ (AX + A.~ZZ)XZ+ Abn]

+(k’+B:)](A,,+M,,)x,+(A,,+;\A,,)x,J

n = I; ‘[-,7(x)

4

For regulation problems, r is usually chosen to be zero. From now on, we set r = 0. Choose the following Lyapunov candidate (Narendra and Annaswamy, 1989) V = &:Pe,

F(x) is a known

+ @x,,),

v=&e

cl = xl ~ xlnl

where

If(x)

- @xi).

fictitious input, 4 and 0 are in the equivalent to saying that matching the reduced-order system. input to (la), we find a reference adaptive approach to counteract uncertainties. and using (13) yields

the

(8)

ti = f(x) + (1 + &)hu.

A controller

(13b)

where (7 = x2 - x2,,.

where xZd is given and (4) yields

+ (A,2 + u,z)(xz

Hence by treating x2 as a range space of x2. This is conditions are needed for Step 1. Treating x2 as the signal xZd by using an explicitly the mismatched Subtracting (12) from (la) c, =&c,

Srep 2. Realization sliding variable as

= &.

Assumption 4. The elements in 0, 4 are bounded. Remark 3. Substituting (13a) and (13b) into (la) yields

5 0.

Hence x, E Lzn L,, 8 E L,. It then follows from (5) that i, E L,. By using Barbalat’s Lemma in adaptive control theory (Sastry and Bodson, 1989; Slotine and Li, 1991), it can be easily shown that xi goes to zero and b is bounded. Remark 2. It should be noted that, as can be seen from the Lyapunov function in (6) the closed-loop system is globally stable.

(13a)

on the bounds

and substituting

laws Y = -BKPe,x:

(18)

6 = -BLPe,x?,,

(19)

(18) and (19) into (16) yields i, = -feTQe]

% 0.

(20)

In getting (20), a relationship in trace operation has been used, i.e. Tr(xyT) = yTx where x, y are vectors. Similar to the analysis of Section 2.1, we can easily show that ei converges to zero. Remark 4. Although the system is globally stable in ei, Y and (D, it is not globally stable in +I - 4 (Narendra and Annaswamy, 1989). This is because the global stability of

Brief Papers @ = 4-i - 4-i does not imply that 4 - 4 is globally stable. This is the main difference between the two adaptive schemes in this section. Step 2. Realization of adaptive reference signal xZd We define the sliding variable as a=xz-x~,

Following the procedure in Section 2.1, we can easily derive a sliding mode control law to drive v to zero in finite time. Simulation studies. Case 1: conventioal sliding mode control. (la) No mismatched uncertainties. Drazenovic (1969) defined the sliding variable as V = 7x, + xz + xj.

(21)

(26)

where xM is given in (14). The procedure in Step 2 of Section 2.1 can be carried out to drive (Tto zero in finite time. When the system is in sliding mode, the adaptive reference X~ is realized and hence e, approaches zero. Hence x1 will be zero because the regulation point r is assumed to be zero. Consequently, X~ and xz will eventually go to zero.

When there are no parameter variations in (22a), (22b), the system behavior in sliding mode is completely determined by

3. Examples

The trajectories of the above ideal system are plotted in Fig. 1 for comparison. (lb) With mismatched uncertainties. When there are parameter variations in (22a) and (22b), the system behavior in sliding mode is distorted by the uncertainties even though it is still stable. The trajectories in this case are also plotted in Fig. 1. Case 2. New sliding mode control with mismatched uncertainties. We choose the adaption gain matrix in (7) as l? = diag(0.05, 0.005, 0.0002). n in the controller (10) is chosen to be 140. The initial conditions for x1, x2, x3 are 10, 0, 0, respectively. The state trajectories are plotted and superimposed on those trajectories in Cases (la) and (lb) as shown in Fig. 1. It can be seen that our method has a quite close response as the ideal case (la) with all the mismatched uncertainties in place. This clearly demonstrates the robustness improvement over conventional sliding mode control. For completeness, we also plotted the estimated parameters, sliding variable (+ and control u in Fig. 1. Note that Case (2) and Case (la) are not exactly the same due to the fact that the parameters have not converged to their true values. To guarantee the convergence of parameter estimates, one would require the reference trajectory to be persistently exciting, which is clearly not possible in the regulation case since the set point is always zero. 3.2. Example 2. Simple pendulum with motor dynamics. Consider a more realistic example in the paper of Barmish et al. (1983). The effect of the motor was not included in their example. Here we add the motor dynamics in the nonlinear pendulum model shown below:

1. Consider the following uncertain system which was used by Drazenovic (1969): 3.1. Example

& = (ai, + Aaii)x, + (aiz + Aa&& = (azi + Aa&

+ (ai3 + ha&s

(22a)

+ (a22 + A.azz)xz+ (a23 + Aa+

(22b)

4 = (a3, + Aa3,)x, + (a32 + Aas2)x2 + (a33 + Aa33)x3 +f + u. (22c) The following parameters

are used:

a ,, = -0.03

aI = 0.01

a2, = -0.05

a22 = -0.15

a3, = -0.09

a3* = 0.03

aI = 0.01 a23 = 0.05 a33 = -0.17

Aa,, =O.ll + 0.044sin (3.14t)

ha,, = 0.01 +O.O04cos (3.14r)

Aa,, = 0.55 + 0.22 sin (3.14t)

Au22= 0.05 + 0.02 cos (3.14t)

Aua,,= 0.5 sin (3.14t)

AaS, = 0

Aa,, = 0.008 + 0.002 sin (6.28t) Aa

= 0.04 + 0.01 sin (6.28~)

Aas3 = 0 f = 5 sin (3.142). Unlike the example in Drazenovic (1%9), it should be noted that there are many parameter variations in (22a) and (22b) which are not in the range space of control a. We can rewrite (22a) and (22b) as xi = -0.03 [:Ix2 [ -0.05

0.01 -0.15 +

x, x2

x, = -7x, - x2.

i, =x2 sin(x,)+x3

+02x,+

03x3),

(23)

where

e2 = 1 + 0.4 cos

(4.14t)

0, = 0.8 + 0.2 sin (6.28t).

The similarity between (3) and (23) can be easily seen. Hence we can readily apply the method in Section 2.1 to tackle these mismatched uncertainties in (23). Sfep 1. Treating x3 as an input to (22a), (22b), design an adaptive reference signal xgd to counteract the mismatched uncertainties in (22a), (22b). Following the procedures in Section 2.1, we design the following fictitious controller: X3d= -7x, -x2 - B,x, - 4,x2 - 6,x,. Choosing Q = I, we get -3.3 2.5

1’

(r =x3 + 7x, + x2 + B,x, + B,x, + &x3.

(25)

(27b)

(motor dynamics)

(27~)

X 3d=-klx1-k2x2+dsin(xl)+q^cos(xl) a" =

-&(P2xI

-&(P,xl

+P3x2)

+ P3Xd

(284

sin (x1)

(28b)

cm (Xl),

(28~)

where g,, g, are positive gains. p,, p2, and p3 are the elements of P in the Lyapunov function

V=$xTPx+@

Using adaptation laws (7) and Lyapunov function (6) it can be easily shown that x1, x2 will converge to zero. Step 2. Realization of xjd Our sliding variable is defined as

(pendulum equation)

where q represents a bounded unknown disturbance to the pendulum. a > 0 is a constant unknown parameter in the pendulum. d(t) denotes bounded unknown disturbances. It is quite clear that both the disturbance term q cos (x1) and uncertain term a sin (x1) in (27b) are not in the range space of a. Fortunately they are in the range space of x3. Hence our method can be applied. Owing to page limitations, we do not show all the details of derivations. Following the procedure described in Section 2 or the example in Section 3.1, we get the following fictitious controller:

4 =

(24)

(27a)

- q cos(x,)

x3 = -x3 + u + d(t),

(x3+,9,x,

0, = 11 + 4.4 sin (3.14t)

i, = -0.4x, - 0.2x,

x2 = -a

I[ I 1

;';;

[.

i, = -0.1x,

,“;,“:

P= [

I

q,[gd,“,1’[;]

(‘) = (0) - (‘).

Using (28) and (29) it can be easily verified that (28) can globally stabilize the reduced-order system (27a) and (27b).

306

Brief Papers

e

20

48

68

L___

80

a,68]

;

48

68

89

4il

60

88

time

8.66,

e2

8064, 8.62, 1

1.2, u

1

8.8. 1

I/f,

a

time 28

48

68

88

I

9

time 29

Fig. 1. Simulation results of our new sliding mode controller. Case (la): conventional sliding mode control without mismatched uncertainties. Case (lb): conventional sliding mode control with mismatched uncertainties. Case 2: new sliding mode control with mismatched uncertainties.

To realize as

this fictitious

signal,

the sliding

variable

is defined

fl = x3 - xx,,. The sliding mode controller can be easily results in Section 2. We used the following simulation: k, = 25

kz = 10

g, =s

g2 = 25

p, = 1.5 a=11

p2 =0.02 q=4

derived using the parameters in the p3 = 0.0052

d(t) = 5 sin (I).

The results are shown in Fig. 2. We also used conventional sliding control in this problem where the sliding variable is defined as c =x3 + k,x, + kzxz x1 found by using conventional sliding control is also plotted in Fig. 2. It is clear that conventional sliding control failed to drive x1 to zero

because of the presence of mismatched uncertain disturbances in (27b). The importance of this example is two-fold. Firstly, our method can be extended to nonlinear systems even though a more general theory is still under development. Secondly, this is a more practical example than Example 1. Our method can guarantee stability and robustness to both the pendulum and motor systems. All simulations were done by using SIMNON PC/3.1.

4. Conclusions The main contribution of this note is to resolve one of the major problems of sliding mode control: how to counteract mismatched uncertainties explicitly? The reasoning behind our new method is the belief that if we want to increase robustness of systems with mismatched uncertainties we must

Brief Papers

307

new sliding control conventional sliding control

-i.s~*,

time

0

g

2

4

6

0

2

4

6

8

10

3*:li ,,,,,

18

B

2

4

6

0

10

0I-8 2

4

6

8

18

18

5

Fig. 2. Our new sliding

mode controller

provide some dynamics in the sliding variable so that when the system is in sliding mode, these dynamics will work against the mismatched uncertainties. Our dynamical sliding variable formulation is an adaptive one. If we compare the sliding variables defined in (2) and (8) [or between (25) and (26)], one will see that our new approach is a simple and natural extension of conventional sliding variable definition. An adaptation law estimates the unknown mismatched parameters on-line and provides information for a fictitious adaptive controller which consequently reduces the effects of mismatched uncertainties. The fictitious controller is robustly realized by the sliding mode technique. Furthermore, by virtue of the so-called model reduction capability of sliding mode control, we can allow equation (lb) to be highly nonlinear and time-varying. Although we cannot handle all kinds of mismatched uncertainties, we believe that this is still a significant step towards a more general theory of handling mismatched uncertainties. Acknowledgments-The author would like to thank the anonymous reviewers and one associate editor for their many comments and suggestions which helped in improving this paper.

AUTO 31-2-J

applied

to the pendulum

plus motor

system.

References Bailey, E. and A. Arapostathis (1987). Simple sliding mode control scheme applied to robot manipulators. Inr. J. Contr., 42,1197-1209. Barmish, B. R., M. Corless and G. Leitmann (1983). A new class of stabilizing controllers for uncertain dynamical systems. SZAM J. Contr. Optimization, 21,246-2X Drazenovic, B. (1969). The invariance conditions in variable structure systems. Automatica, $287-297. El-Ghezawi, 0. M. E., A. S. I. Zinober and S. A. Billings (1983). Analysis and design of variable structure systems using a geometric approach. Znt. J. Conzr., 38,657-671. Narendra, K. S. and A. M. Annaswamy (1989). Stable Adaptive Systems. Prentice-Hall, New Jersey. Sastry, S. and M. Bodson (1989). Adaptive Control: Stability, Convergence and Robustness. Prentice-Hall, New Jersey. Slotine, J. S. and W. Li (1991). Applied Nonlinear Control. Prentice-Hall, New Jersey. Utkin, V. I. (1978). Sliding Modes and Their Application in Variable Structure Systems. MIR, Moscow. Yeung, K. S. and Y. P. Chen (1988). A new controller design for manipulators using the theory of variable structure systems. IEEE Trans. Autom. Contr., AC-33,200-206.