Proceedings of the 2003 IEEE I n t l m a t i O m l Conference on Robotics &Automation

TaipeiJaiwm, September 14-19. 2003

Sliding Control for Linear Uncertain Systems

C. W. Tao', M.L. Chan', and W.Y. Wang' Department of Electrical Engineering, National I-Lan Institute of Technology, I-Lan, Taiwan

Ernail: cwtao@rnail.ilantech.edu.tw ' Department of Electronic Engineering, h - J e n Catholic University, Taipei, Taiwan.

Abstmct-A new design approach to enhance a termi- nal sliding mode controller for linear systems with mis- matched time-varying uncertainties is presented in this paper. The nonlinear sliding surface is used to have the system states arrive at the equilibrium point in the Rnite time period. The sliding coeiiicient matching condition is extended for the terminal sliding mode control. The un- certain system with the proposed terminal sliding mode controller is shown to he invariant on the sliding surface. The reaching mode of the sliding surface is guaranteed and the close-loop system is stable. Moreover, the u n d e sired chattering is alleviated with the designed terminal sliding mode controller. Simulation results are included to illustrate the effectiveness of the presented terminal sliding mode controller.

I. INTRODUCTION Since the exact mathematical models of the systems

are not always obtainable, the control of the linear sys- tems with time varying uncertainties has been an im- portant research topic in the engineering area [9], [13], [6]. The variable structure with sliding mode is known to have the intrinsic nature of robustness to the uncer- tainties [4]. That is, when the system reaches the sliding mode, the system with variable structure control is in- sensitive to the external disturbances and the variations of the plant parameters [lo]. Thus, the sliding mode control is considered as effective approach for the con- trol of the systems with uncertainties [11], 121, [14], [7].

Recently, a new type of sliding mode control technique called terminal sliding mode control is developed [E]. In- stead of using linear hyperplanes as the sliding surfaces, the terminal sliding mode control adopts nonlinear slid- ing surfaces. The terminal sliding mode controller can have the system states remain on the sliding surface and reach the equilibrium point in a finite time period. In [15], the terminal sliding mode control is sucessfully ap- plied to the uncertain linear systems with uncertainties which match the traditional matching condition. Also, the terminal sliding mode controller is used for the un- certain nonlinear system in [12]. However, the mis- matched time-vaxying uncertain systems with the ter- minal sliding mode control has not yet been studied. Moreover, the undesirable chattering of the (terminal) sliding mode control systems may excite the high fre- quency response of the systems [3], [5].

In this paper, a new terminal sliding mode controller for linear systems with mismatched time varying uncer- tainties is proposed. With the nonlinear sliding surface, the system states approach the equilibrium point in the finite time period. The sliding coefficient matching con- dition in our previous paper [l] is extended for the ter- minal sliding mode control. The uncertain system which

satisfies the extended sliding coefficient matching condi- tion is shown to be invariant on the sliding surface. The reaching mode of the nonlinear sliding surface is guar- anteed and the close-loop system with the proposed ter- minal sliding mode controller is indicated to be stable. Moreover, the undesired chattering associated with the terminal sliding mode control is alleviated for the sys- tem with mismatched uncertainties (no delay or sensor noises are considered). Simulation results are included to illustrate the effectiveness of the proposed terminal sliding mode controller.

The remainder of this paper is organized as follows. The system model considered in this paper and the sliding coefficient matching condition are described in section 11. The reaching mode of the terminal sliding mode controller is discussed in section 111. In section N, the uncertain system with the proposed terminal sliding mode controller is shown to be invariant on the sliding surface. Also, the terminal sliding mode control system is indicated to he stable in section N. In sec- tion V, simulation results of the illustrative examples are presented. Finally, conclusions are provided in section VI.

11. THE SYSTEM MODEL DESCRiPTloN

The uncertain linear system with the mismatched un- certainties is described in this section. Let the regular form of the state equation of the linear system with the only mismatched time varying uncertainties, AA( t ) , be

where ZI = [ZI ..., z,-,IT, ZZ = [.,-,+I ,..., ..IT, AA(t) E , ~ ( t ) E Em, BZ E RmX", and n - rn 5 m. Also, the constant matrix pair ( A , B ) ,

is assumed to be controllable. The matrix A12 and B are required to have full rank. The AA(t) is a mismatched time-varying uncertainty since it can he easily seen that AA(t) doesnot satisfy the classical matching condition. The mismatched time varying uncertainty AA(t) is fur- ther assumed as in the work of Shen 191 to have the structure

AA( t ) = DF( t )E (3)

0-7803-7736-2/03/$17.00 02003 IEEE 261 1

with the constant matrices D, E, and the uncertain matrix F ( t ) E R'lX'Z satisfying

FT( t )F( t ) 5 I where I is the corresponding identity matrix. From Eq. 3, it can he easily seen that

AA(t) = [AA(t) 01 = DF( t ) [E O] = D F ( t ) f i , (4)

where 0 is a zero matrix with proper dimension, and I? = [E 01. With Z:/' = [#', z$', ..., z $ ! ~ ] ~ , the nonlinear sliding surface of the terminal sliding mode control (as in Man's paper [SI) is designed to he

= c1z,+czz2+c3z:~~=o (5)

where ClrC3 E Rmx(n-m) , and Cz E Rmxm is an in- vertible matrix. Also, q, p are required to be odd inte- gers which satisfy

2 q > P > q Moreover, C,, Cz, C3 are designed to have the following conditions

AI1 - AizCT'Ci = 0,

Ai2CF1C3 = di.g(Vi,Pz, ..., Pn-ml), (6) satisfied. Note that fi, > 0 V i E [ l , n - m]. To make CI, C2, C3 satisfy Eq. 6 , CZ is simply designed to he an identity matrix. Then, the conditions in Eq. 6 become

Aii - AizCi = 0, A12C3 = dia9([fi1,4zr...~fi"--rnl), (7)

with 4; > 0 Vi E [ l , n - m]. Since Alz has full rank and n - m 5 m is assumed, there exists more than one solutions for Cl and C3. For example, the minimum norm solutions,

c1 = A T ~ ( A ~ ~ A T ~ ) - ~ A ~ ~ , C3 = ATz(Ai2ATz)-'diag([4i,4z, -, Pn-m]),

are one of the possible solutions for CI and C3, respec- tively. The sliding coefficient matching condition in [1] is extended as follows for the fuzzy terminal sliding mode control of the uncertain systems.

Definition f; (Extended Sliding Coefficient Matching Condition) I! the uncertainty AA = D F ( t ) E , F ( t ) T F ( t ) 5 I , and E in Eq. 4 satisfy

E = E,C, then the uncertainty AA is said to satisy the extended sliding coefficient matching condition.

For an uncertain system with the state equation in Eq.1, if the mismatched time-varying uncertainty satis- fies the extended sliding coefficient matching condition and B2 has full rank, then the uncertan system is called a extended sliding coefficient matched uncertain system

Definition 2: (A ESCMS system)

(ESCMS) .

Fig. 1. Block diagram of the terminal sliding mode Control system

111. THE REACHING MODE OF T H E TERMINAL SLIDING MODE CONTROL SYSTEM

It is known that the reaching mode of the uncertain system with a terminal sliding mode controller (in Fig- ure 1) is guaranteed if

sTS < 0, S # O

Lemma 1: [9]: If F ( t ) T F ( t ) 5 I, then

2aTF(t )y 5 XTZ + YTY; vx, y E R"

rn Let

Q A I I I Z = AiiZi + AizZz, &a2122 = AziZi + A22Z2,

P Qc3 = gC3diag(~:'-')/~),

Q C , ~ , = C1 + 4 C 3 d i a g ( ~ p - ~ ) / ~ ) . (8 ) P

Then with Lemma 1 applied, the sliding mode reaching condition of a system ESCMS becomes

sTS = sT(C1il + c2zz + c3z:I') = ST(Cl (Q~nlz + A4Z) + CZQAZIZZ

+Cz&U + Q C ~ ( Q A I I I ~ + A A Z ) ) = ST(Qc,~,Q~il lz + CZQAZIZZ

f C z B z ~ + Qc,c , (DF( t )EZ)) = s ~ ( Q ~ , ~ , Q ~ ~ ~ ~ ~ + CZQAZIZZ

+CzBzu + Qc,c,(DF(t)@Z)) 5 ST(Qc,c,Q~iil* + CZQA~IZZ + CZBZU

+~/z(Qc,c~DD~Q&,s + E:E,s)), (9)

Also, it can he easily seen that

DF(t )EZ = D F ( t ) E i ,

and the ESCMS is defined to satisfy the extended sliding coefficient matching condition. For S # 0 and z; # 0, V i = 1,2, ..., n - m, the terminal sliding mode controller T S M C is designed to have output U ,

u=ueg+ug ,

where

ueg = ~(CZ~Z)- '(QC,C,QAIIIZ + CZQAZIZZ + ~ / ~ ( Q ~ , ~ , D D ~ & & , , s + E , T E ~ S ) ) , (10)

U6 = -(CzBz)-'kS; k > 0, (11)

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and ,fu-P)/P

fI-P)/P =(n-m)

diag(&P)lP) = ". 1 Thus, the reaching condition of the sliding surface is

sTS 2 S T ( Q ~ , ~ , Q A l l l z + CZQAZIZZ + CZBZU + ~ I ~ ( Q G c , D D ~ Q & c ~ S + E Z E J ) )

< O ; S # 0 . (12)

It can be found that if S # 0 and there exists one zi = 0, i E {1,2, ..., n - m},the singular points will occurred (since q - p < 0 in Eq.10). In this case, the output of the terminal sliding mode controller T S M C is designed as in [8] ,

5 -kSTS

U = B;'(-AziZi - AZZZZ - yZz); y > 0, (13)

to have the system dynamics away from the singular points. Thus, the discussion in this section can be sum- marized in Theorem 1.

Theorem I: The reaching mode of the sliding surface of a linear uncertain system E S C M S (see section 11) with the output U of a terminal sliding mode controller T S M C defined as

U = - ( C Z B 2 ) - 1 ( Q ~ 1 ~ 3 Q ~ 1 1 1 2 + CZQAZIZZ + I I ~ ( Q ~ , ~ $ D ~ Q & , , S + E Z E J ) ) - (CzBz)-'kS;

w h e n S # 0 , Vzi#O, i = 1 , 2 ,..., n - m , a n d

U = B;'(-AzIZI - AzzZz - yZz); y > 0,

is guaranteed. w whenS#O, thereis z i = O , i = { 1 , 2 ,..., n-m},

Iv. THE INVARIANT CHARACTERISTIC OF THE TERMINAL SLIDING MODE CONTROL SYSTEM

In this section, the invariant characteristic of the un- certain system E S C M S with the terminal sliding mode controller T S M C is derived in the following theorem.

Theorem 2: A linear uncertain system E S C M S (see section 11) with the terminal sliding mode controller T S M C is invariant with respect t o time varying un- certainties on the sliding surface. rn Proof For the uncertain system E S C M S in the regular form,

Z(t) =A[Ei[:]+[A%(t)]Z(t) +Bu(t) ,

the mismatched time varying uncertainty,

AAZ = AAZ,

and AA = D F ( t ) k

where F T ( t ) F ( t ) 2 I , and 1 is an identity matrix with corresponding dimensions. Since the extended sliding coefficient matching condition is satisfied,

E = E,C and AAZ = AA,? = DF(t )E,Ci .

When the states of the terminal sliding mode control system are on the sliding surface,

s = cz = 0,

thus, AAZ = DF(t)E,CZ = 0.

The uncertain system E S C M S on the sliding surface becomes

and the terminal sliding mode control system is invari- ant to the mismatched time-varying uncertainty on the

In the next theorem, the stability and convergence characteristics of the uncertain system ESCMS with the proposed terminal sliding mode controller T S M C are indicated.

Theorem 3: A linear uncertain system E S C M S with the terminal sliding mode controller T S M C is stable and the system states in the sliding mode converges to

rn Proof: As in section 11, the sliding aiirface for the proposed terminal sliding mode controller is designed to satisfy the conditions in Eq. 7. Also, the uncertain system E S C M S is shown to be invariant on the sliding surface. Then with the same approach in the proof of Theorem 2 in Man's paper [SI, the uncertain system E S C M S with the terminal sliding mode controller T S M C is stable and the system states in the sliding mode converges to

rn

sliding surface. rn

equilibrium point in the finite time period.

equilibrium point in the finite time period.

v. SIMULATION RESULTS

Let the uncertain linear system be described by the Eq. 2 with

A = [ : 0 1 :'I, B = [H i],

1 1 -1 -2 -3

and

AA(t) = O.OGsin(O.lt) 0 O.OGsin(O.lt) . (14)

That is, All = 0, AIZ = [l 01, Azl = [0 - 1IT, and

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From the equation Eq. 14, it is easy to see that AA( t ) is a mismatched time varying uncertainty, and the mis- matched uncertainty A.4(t) can he represented as

A A ( t ) = D F ( t ) E

= 1 *O.OGsin(O.lt)* [ 1 0 11 (15)

where F ( t ) T F ( t ) = (0.06,~in(O.lt))~ 5 1.

Let CZ of the sliding coefficient matrix C be designed as an iden titymatrix. With the @I = 1 in Eq.7, the

minimum norm solution of 6 3 = [ ,!j] is adopted. Since

All = 0 and the minimum norm solution of C1 is a trivial solution (CI = 0), the other solution Cl =

is utilized. Thus,

0 1 0 1 C = [ I 0 1 01

The parameters p , q in the sliding function are defined to be odd integers,

p = 7 , q = 5 ,

to satisfy the condition 2q > p > q. It can be seen

that CZ = [i y ] is an invertiblematrix. .4lso, the conditions in Eq.7 is satisfied. Moreover,

E = 1 1 0 1 O ] = E , C = [ O l ] * C ,

the extended sliding coefficient matching condition is satisfied. Then, the control action u(t) is designed as in Eq.10 with k = 1 for S # 0 and z; # 0, V i = 1,2, ...,, n - m. If S # 0 and there exists zi = 0, i E {1,2, ..., n - m}, t,he control action u( t ) is designed as in Eq.13 with y = 1. Simulation results are provided for the terminal sliding niode control system with initial conditions

z(0) = [0.15 ~ 0.15 0.1IT.

The performance for the states of the uncertain system ESCMS with the proposed terminal sliding mode con- troller TSMC is shown in Figure 2. Figure 2 indicates that the terminal sliding control system is stable. With only the system uncertainties (no delay or sensor noises) considered in this example, the control actions of the terminal sliding mode con trollerTSMC are presented in Figures 3 and 4 to illustrate that the chattering is alleviated.

VI. CONCI.USIONS

A terminal sliding mode controller is proposed for the linear systems with mismatched time varying uncertain- ties in this papcr. The sliding coefficient matching con- dition is extended for the terminal sliding mode COIL- trol. The reaching mode of the uncertain system with

-0 - I I 1 1 I LI 1

"-

Fig. 2. States of the uncertain sy8tem:zI (solid h e ) , zz (dashed line) , z3 (dash-dot line)

--I

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the proposed terminal sliding mode controller is gnarm- teed. The system is shown to be invariant and stable on the sliding surface when the extended sliding coefficient matching condition is matched. Also, the chattering around the sliding surface for the terminal sliding mode control is alleviated. Simulation results are included to illustrate the effectiveness of the proposed terminal slid- ing mode controller.

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