Robust controller design with novel sliding surface

S.K. Park and H.K. Ahn

Abstract: A novel sliding surface is proposed by defining a virtual state. This sliding surface has the nominal dynamics of an original system and makes it possible to use the sliding-mode control technique with various types of controller. Its design is based on an augmented system whose dynamics have a higher order than that of the original system. The reaching phase is eliminated by using an initial virtual state that makes the initial sliding function equal to zero.

1 Introduction

Eliding-mode control (SMC) is a popular robust control method which has many good points and applications[l]. However, it has a reaching phase problem and an input chattering prohlem[Z, 31. As well a, these two problems, SMC is very conservative in use with other controller design methods because the state trajectory of the SMC system is determined by sliding-mode dynamics which cannot have the same order dynamics of the original system. In this paper, a novel sliding surface is proposed to overcome the conservatism and eliminate the reaching phase. It can have the desired dynamics controlled by various control strategies. It is based on the controllable canonical form of the nominal system and by using a newly defined virtual state. There are already some existing virtual states [4], however, their definitions are different from that of the new virtual state in this paper and they have been used only to attenuate the high-frequency component that would otherwise appear in the dynamics of the SMC. The control objective of this paper is lo make the controlled system have the desired nominal dynamics in spite of uncertainties. Optimal control is combined with SMC and the novel sliding mode can have the same dynamics as the nominal optimal control system. By using an initial virtual stale which makes the initial sliding function equal to zero, the reaching phase is removed.

2 Problem statement

Consider the ntb-order system described by

X(t) = (A + AA)x(t) + (B + AB)u(t) + Df(t) (I)

where x E R", U t R, f E R' and the hounded uncertainties AA, AB and the disturbance matrix D satisfy the following matching condition:

rank([B :AA :AB :D]) = rankB (2)

0 IEE, 1999 IEE Pmceedings online no. 19990436 ~01:10.1n49/ip-cta:1999n436 Paper first received 30th June 1998, and in revised farm 16th Fehruaiy 1999 The authors are with the Department of Electrical Engineering, Changwon National University, Changwon, Kyungnain 641-773, Korea E-mail: skpark@sarim.chan~won.ac.kr

With this condition the uncertainties and disturbance can be expressed as

AAx(t) = BAAIx(t)

ABu(t) = BABlu(t) (3)

Df(t) = BDIf(t)

The existing sliding-mode surfaces have the following form [2]:

s(x) + C ( , _ I J X ( , , - I J ( f ) + " ' f C I X l ( t ) = C X ( t ) (4)

where C=[cl ' . .e,,] and c l , . . .,e,, are given so that sliding-mode dynamics can be stable. The sliding surface has (n - l)th-order dynamics which are not the same as the nth-order dynamics of the original system. The reach- ing phase exists when the initial x(t) is not on the sliding surface, i.e. the initial s(x) is not zero.

The following condition guarantees the sliding mode [I]:

,i(x) is calculated as follows: s(x)S(x) < 0 ( 5 )

.E(x) = CX = C(A + AA)x(t) + C(B + AB)u(t) + CDf(t)) (6)

There exist positive constants pI, p2, p, satisfying the following condition because the uncertainties and distur- bance are bounded:

CAAx(t) + PI lIx(t)ll

CABdf) < P21dt)I (7)

CDf(t) + p3

The following input guarantees the condition of expr. 5 :

u(t) = -(CB)-'(cAX(t)) - (CB)-l(pJlx(t)ll + pzlu(t)l

+ p3)sgn(s) (8)

The problem to be solved is -to overcome the conservatism of the SMC by using a novel sliding surface which has the same dynamics of the nominal original system controlled by a nominal controller -to eliminate the reaching phase.

The control objective is to make the state follow the desired nominal trajectories in spite of uncertainties.

242 IKE Pw.-Conlrol Theory Appl., Yo/. 146, No. 3. Moy 1999

4 SMC with new sliding surface I 'arious types of sliding surface have been proposed 1 nclnding a time-varying sliding-mode surface [ 5 ] . These Iiliding surfaces cannot have the dynamics of the original kystem controlled by a different type of controller. This lnakes the conventional SMC very conservative in comhin- og with the other types of control strategy. To overcome his conservatism completely, a new SMC with a novel

1;liding surface that has the dynamics of the nominal ixiginal system is proposed. The novel sliding surface is liesigned based on an augmented system which has a h t u a l state. The virtual state is defined from the control- lable canonical form of the nominal system. i The following controllable canonical form is obtained p m eqn. 1 by a state transformation z=Px. The para- peter uncertainties and disturbances can be expressed as Lumped uncertainty h ( f ) because they satisfy the matching p i t i o n of eqn. 2

i(f) = A&) + B,u(t) + B,h(t) i I r o I 0 . . . 0 1 roi

(9)

0 0 l . . . i ' where A, = 1 : , : : :

i -a1 -az ... !

. . . I

nominal system for eqn. 9 is

i o ( 0 = A,zo(t) + Bc~o(zo, 0 (10) I ;\where uo(zo, t ) is a nominal regulating control input and ldifferentiable. A virtual state z.. proposed in this paper, is idefined as a derivative of zOn and its dynamic is

I l ! 1 zov( t ) = -anzov(t) - %1zon(f) - ' ' ' - %zd - alzoz

+ 2 t ) (11)

~

ikhis is the differential form of the last equation (eqn. 10). lljrom eqn. 11 the following novel virtual state z, is defined by replacing nominal state z, with state z (it does not 'include any uncertainty):

(12) i iwhere U&, t) and U&, t) are obtained from U,,(Z., t ) and iio(z0, t ) 'respectively' by replacing nominal state zo with original state z. Any uncertainty must not be considered in 1:his procedure. With the novel virtual state, the augmented !system is constructed as follows:

i X(t) = (A + AA)x(f) + (B + AB)u(f) + Df(t)

1 i"@) = -a&) - a,_,z,(t) ' ' ' - C(ZZj(t) - a&) + U,(z,t)

(13)

1 where u(t) is the SMC input which guarantees the sliding lmode on the sliding surface. For this augmented system, the novel sliding surface is defined as

I (14) ~ - u,(z,t) = 0 ~

"The state z can be expressed as x at any time by the ~ transformation P. It is noted that the nominal control input ~ is used to define the noble sliding surface.

i [EE Pmc.-Contml Theow Appl., Yol. 146, No. 3, May 1999

The following initial virtual state makes the initial value

zv(to) -a.Z,(fo) - ' ' - ~ i ~ i ( t o ) + u,(to) (15) This eliminates the reaching phase completely. Now the following theorem is obtained. Theorem 1: If the states of the eqn. 13 system are on the novel sliding surface of eqn. 14, then the states of eqn. 1 have the dynamics of the nominal system of eqn. 1. Proo$ Ifzl , zz,-.z,, zy are on the novel sliding surface then the following equation is satisfied by eqn. 14:

of s(z, 2") equal to zero:

z,(t) + a,z,(t) + %I+l)(t) ' ' ' + E l Z 1 ( t ) - u,(z,t) = 0

By differentiating

(17) is obtained.

i , ( f )+a, i , ( f )+a.-~z," '+Elz~(t) - U & f ) = 0 (18)

is obtained by the following relations in eqn. 9:

z2 = i,, . . . ,z, = 2(n-l) (19)

These relations are guaranteed under the matching condi- tion. According to eqn. 12, zv has the following dynamic:

ZJt) = -E,z"(t) - CI.-lZ,(f) - ' ' ' - E223 - "I22 + U,(z,t)

(20)

2" = zn (21)

(22)

Comparing eqns. 18 and 20,

is obtained. Now the following is obtained from eqn. 16:

in(() = -a,z,(t) ' ' ' - a,z,(t) - a,z,(t) + u,(z,t)

Eqns. 19 and 22 are the same as the nominal controllable canonical system (eqn. 10). It can be transformed to the nominal system of eqn. 1 by the state transformation P. Therefore the states of eqn. 1 on the novel sliding-mode surface (eqn. 14) have the same dynamics as those of the nominal system of eqn. 1.

From theorem 1 and SMC theory the following result is obtained. Theorem 2: If SMC input u(t) is designed to force the states of the system onto the sliding surface of eqn. 14 then the states x(t) follow the trajectories of the nominal system of eqn. 1. Theproof: is obvious from theorem 1 and SMC theory.

To force the states onto the sliding surface of eqn. 14, a SMC input that satisfies the following condition has to be derived

~n(z,z")~"(zJ") < 0 (23) 8, (2, z,) is calculated as follows:

auo(z t ) . i,(Z,Z") = 2, + C,PX - L P X

az

= -unz, - C ~ P X + U,(z,t) + CzP((A + AA)x(t) + (B + AB)u(t) + Df(t))

LE.1 243

There exist positive constants PI, P,, P 3 satisfying the following condition because the uncertainties and distur- bance are bounded:

C,PAAx(t) < Ilx(t)ll

C2PABu(t) < Pzl4t)l (25)

CzPDf(t) < P 3

The following input guarantees thc condition of eqn. 23.

u(t) = - (C2PB)-'(-a,z, - CnPx + k,,(z,f) + C2PAx(t))

- ( c 2 P W 1 ( f l 1 Ilx(t)ll + P2Iu(t)l + B3)sgn(s,,)

(26)

Note that the nominal control input U&, f ) can be any type of control input and this makes it possible that the SMC is used with the various types of controllers. This means that the conservatism of the SMC is removed.

4 Robust optimal control using the Novel SMC

A robust optimal controller which makes the states follow the optimal trajectories in spite of parameter uncertainties is designed. Consider the systcm of eqn. 1. The perfor- mance index for the nominal system is

m

J = J, i (XZQX, + rui)dt

uE(x0) = --BTSx,(t) = -K x d 0

(27)

The optimal control input for the nominal system is [6]

(28) 1 r

where S is the solution of the Riccati equation

(29) 1

-SA - ATS - Q + -SBBTS = 0

ko'(x) is calculated as follows (the uncertainties are not considered in this calculation):

r

kz(X) = -K(Ax + Bu:(t)) = -K(Ax - BKX) = L x (30)

where L = - K(A - BK). According to eqn. 12, the virtual state zv is defined as

i,(f)= -a,z2( t )" ' -a ,_ ,z"( t ) -a ,z , ( f )+Lx(t) (31)

The augmented system is constructed as follows:

x(t) = (A + AA)x(t) + (B + AB)u(f) + Df(t) (32)

z " = - a , z , ' " - a , ~ , z , - a , , z , , + L x

For this system the proposed novel sliding surface is given by

s , , = z , ,+ a , z , + ~ ( * ~ ~ . . . a , , z , , + K ~ (33 )

S, is calculated as,

S, = &(t) + C,Pi(t) + Ki(t) = -a,,z,,(t) - C,Px(t) + Lx(t)

+ (CiP + K)((A + AA)X(t) + (B + AB)u(t) + Df(t)) (34)

244

There cxist positive constants yI, y 2 ' , y 3 satisfying the following condition becausc the uncertainties and distur- bance are bounded:

(CiP + K)AAx(t) 4 YI Ilx(t)ll

( c i P + K)ABu(t) < 72ldOl (35)

( C , P + K)Df(t) < y ,

The following input guarantees the condition sn(z, z,)S,,(z, 2") < 0:

u(t) = - ( ( C I P + K ) B ) - ' ( - ~ , z , - C o P x + L x + ( C , P + KIWt))

- ((cIp + K)B)-'(./, Ilx(t)li + y21u(t)l + y3)sgn(s,,)

(36)

5 Numerical examples and simulation results

Consider the second-order system

where lAall < 3, f ( t ) < 0.5. The parametric uncertainty Aal,f(f) are assumed to be constant 2 and 0.4, respectively, in the simulation. The pesformance index is

The optimal gain is K = [2.2845 0.13921. The transforma- tion matrix to controllable canonical form is

p = [ l l .'I The calculation results needed to coiistruct the SMC are

= [ 1.2845 4.13921

(GIP + K)B = 5.4237, L = -K(A - BK) = [7.8214 0.61581

(CIP+K)AAx(t) < 16.27111x1l

(C,P + K)Df(t) < 2.71 18 = y 3

(yI = 16.2711)

From eqn. 3 1, the virtual state is defined as z , = 9.82 I 5xl - 3 . 3 8 4 3 ~ ~ -32, The noble sliding surface is constructed as s,=z,+ 1 . 2 8 4 5 ~ ~ +4.1392x2. The following SMC input guarantees the condition s,S, < 0:

(-32, - 8 . 5 3 7 0 ~ ~ + 1 1 . 6 6 2 7 ~ ~ ) 1

5.4237 u(t) = - -

I 5.4237

(16.27111~~1 +2.7118)sgn(sn) -~

The siinulatioii results are shown in the following Figures. Fig. 1 shows that the reaching phase is eliminated. Optimal trajectories ofn l and x2 without parameter unccrtainties are presented in Fig. 2. Fig. 3 shows the state trajectories ofx , and x2 controlled by optimal controller with uncertainties. They are not optimal trajectories any more. The state trajectories of xI and x2 contsollcd by the new SMC with

IEE Pnic.-Conlrol TIieo,y Appl., Yo/, 146, NCI. 3, May 1999

0 0.5 1.0 1.5 2.0 2.5 3.0

/ ::::w , , , -0.6

0 0.5 1.0 1.5 2.0 2.5 3.0 Y2

ig. 1 Phase ImjeetoFy of novel slidinb. mode

0 1 2 3 4 5 time, s

-1.0' ' ' ' ' ' ' ' ' ' '

ig. 2 Optimal trajectories sfs, and x, without uncertainties

3.0 F

0 1 2 3 4 5 time. s

ig. 3 State trujectories qf x, and x1 controlled by optimal controller 'th uncertainties

irameter uncertainties are shown in Fig. 4. Fig. 4 shows hat the proposed sliding mode has the nominal dynamics F the optimal control system Fig. 2. This means that the ajectories of the system controlled by the new SMC illow the nominal optimal trajectories in spite of para- leter uncertainties. Fig. 5 shows the trajectory of zv Fig. 6 for the control input. Fig. 7 is the value of switching 1 function s,,(z, 2")

~

, IEE Proc.-Contml Theory Appl, Yal. 146, No. 3, May 1999

-1.0 0 1 2 3 4 5

lime, s

Fig. 4 State trajectorim of x, and xi contmlled by new SMC with aneerlainlle.~

I l l I1 I I I

i

1 2 3 4 5 -201 ' ' " ' ' ' ' ' 1

00 time, 6

Fig. 6 SMC input u(t)

6 Conclusions

A novel design method of sliding surface has been proposed. With this sliding surface a new SMC that makes the states of the system follow the nominal trajec- tory controlled by a nominal controller can be designed. Any type of controller which is differentiable can be a nominal controller. It has been shown that a robust optimal controller with the novel sliding mode is designed using

245

0.08 o'lol

246

optimal controller as a nominal one. The reaching phase is easily eliminated by an initial virtual state chosen appro- priately. This work opens up the attractive area that various types of controller can be combined with the SMC.

7 References

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survey,' IEEE Tmni.. 1993, IE-40, (I), 2-22 UTKIN, VI.: 'Sliding modes and their application in variable stmciure swtems' (Morcow~ Mir Pohliahnvs 107111

2 ~~~~~~~ ., . ~~~

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