Sliding mode controller design for linear systems with mismatched uncertainty

SLIDING MODE CONTROLLER DESIGN FOR LINEAR SYSTEMS ...

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Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

SLIDING MODE CONTROLLER DESIGN FOR LINEAR SYSTEMS WITH MISMA TCHED UNCERTAINTY Shi-Jie Xu Department ofAerospace Engineering and Mechanics Box. 137, Harbin Institute of Technology, 150001 Harbin, China e-mail: [email protected]~edu~cn 1

A9 Rachid Laboratoire des Systernes Automatiques, Universite de Picardie Jules Verne, 7, rue du Moulin Neuf, 80000 Amiens, France. e-mail." [email protected]

Abstract: The present paper deals with sliding mode controller design problem. The linear systems with mismatched uncertainties are considered. A sliding mode controller with

guaranteed cost typed sliding surfaces is proposed, such that the sliding motion is stable and a guaranteed cost bound is provided. Both the sliding motion and the reaching motion are robust against the mismatched uncertainties~ Copyright @ 1999 IFAC Key words: Sliding mode control; variable structure control; uncertain linear system.

1. INTRODUCTION Sliding mode control1er design is an attractive research domain. There is a great number of research results reported (see necarlo et.at (1988); Hung et. at (1993); and the references therein). The sliding mode controller has the advantages that the controller is simple on structure and easy to realize, the sliding motion has invariance against the matched uncertainties. The sliding mode controller design procedure, generally, consists of two steps, Le.. , the reaching motion controller design and the sliding surfaces design. For the reaching motion controller, the inequality approach (Itkis, 1976) and Lyapunoy approach (Grayson, 1967) can ensure the states of system to re.ach the sliding surfaces in limited time, and the reaching law approach (Gao and Cheng, 1988; Gao and Hung, 1993) can guarantee systemts states not on] y to reach the sliding surfaces in limited time but also lo have a specified performance. The sliding surfaces can be designed by using pole~ placement or LQR approach (Dorling and Zinober, 1986) for the systems without uncertainty or with .m atched uncertainty. In fact:. there are some systems with mismatched uncertainties6 In this case, however, the invariance against the perturbation is no longer existed and the robustness of sliding motion canlt be guaranteed, meanwhile the above sliding surface design approaches are no longer available. The specified performance can not be guaranteed if

the reaching law approach is used. Spurgeon and Davies (1993) studied sliding mode control problem for systems with mismatched uncertainties, however, only the nonn-bounded uncertainty was considere(t Kwan (1995) proposed an adaptive sliding surface design approach for linear systems with mismatched uncertainties, a parameter estimating procedure is incorporated, however, the matching condition is once again required for the uncertainty in sliding motion dynamics.. Su et. aI (1996) proposed a sHding surfaces design approach based on Lyapunov's method, which can ensure asymptotic stability for system in sliding motion, however, no uncertainty is considered. The pTesent paper deals with the sliding mode controller design problem~ The linear systems with mismatched uncertainties are considered. A new sliding surface design approach is proposed by using the guaranteed cost control (GCe) technique (petersen and Hollot, 1986), which has the advantage of providing an upper round on a performance index, thus the sliding motion's performance degradation due to uncertainties is known to be less than this bound. The sliding surfaces are obtained by solving a modified Riccati equation. Therefore, there is no

increased difficulty in calculations.. Component typed proposed~ An

sliding mode controllers are then illustrating example is given..

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SLIDING MODE CONTROLLER DESIGN FOR LINEAR SYSTEMS ...

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2. PRELI1\fINARIES Consider the following linear uncertain system x(t) == (A+~) x(t) +(B+~B)u(t),

(1)

system (8) can be rewritten as with a state feedback control law

yet) = (A+~A )y(t) +Bu(t), -1

which minimize the following perfonnance index

-

(10) -1

where,A=TAoT ,B==TBo,~A=TAAT . Using the following partitions

00

J

=

I(xT Qx+u Rn) dt, T

(3)

o

where, Q is a positive semi-definite weighting matrix and R a positive definite weighting matrix.

Definition 1. If there exists a continuous slate feedback control law u*(t) and a positive number J.

such that J ~ Jilt,. then J* is called a guaranteed cost (GC) and u*(t) a GCe law. Lemma 1.. Let xQ be an initial state vector of system (1). The control law (4)

is a GCe law and (5)

is a GC~ if there exists a unique pOSItive semidefinite P such that the following perturbed Riccati equation holds

Yl(t)=(Al1+L\Al1)Yl(t)+(A12+~AliJY2(t)

(I1a)

Y2(t)=(A 21 +h.A21 )y 1(t)+(A 22+L\ A2 2'yz(t)+B 2 u.. (lIb)

Il is obvious that the equation (I1a) is just the sliding motion dynamics~ Choose the sliding surface as

S=Cy=[C 1 Cz]y=O.

(12)

The guaranteed cost sliding surface design problem can be stated as: find a matrix C such that the cost function

ATp + PA _ PBR~lBTp + Q

+ U (P~ ~A~ ~B)= 0

the system (10) can be written as:

J

(6)

=

pT

Qy dt

(13)

ts

~rhere, U(P, LlA. L1.B) is an upper bound of the uncertain matrix/rp, L1A, AB) which is

f(P, /j,A, ~B)

= t1.A Tp +PAA

-PBR- 1.6B T p _p ~BR ~lB T p . (1)

For the proof see Petersen and Hollot (1986). Remark 1. Sufficient conditions for the perturbed Riccati equation (6) to have a unique solution are: the pair (A, B) is stabilizable, (A.

VQ) is detectable~ and

(P, IJ.A, .6.B)~O (for detail see Bemstein and .Haddad, 1990).

-0

is minimized, subjected to constraint (I1a). In (13)~ Q is a positive definite weighting matrix, t is the time for the state rrajeclories to reach the sliding surfaces. Denote 8

Q=

[~~: ~~~ ] T

= Q21'

where, Q12 singular.

•

both

(14) Qll

and Q22 are

000-

3.1 AA has matrixpolytopes typed structure Assume that the uncertain matrix L1A is given by h

AA= ~ aiEi Jail~Si.

3. SLIDING SURFACE DESIGN APPROACH Consider the following linear uncertain system i(t) == (Ao+8A)x(t) +Bou(t))

(8)

where, E i is a known matrix, (Xi. an unknown parameter, Oi a known positive constant. Considering the partition

where, XE Rn, UE R m~ Aa and B o are known matrices, M. an unknown matrix. Denote Bo=[B o1

B 02 ] T.

(9)

No less of generality, assume that the m xm dimensional matrix B 02 is invertible. Using the state transformation y=Tx with

(15)

t

&=[E i11 Ei121, E i21 E i2

(16)

J

one gets (j=1,2)

(17)

where

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SLIDING MODE CONTROLLER DESIGN FOR LINEAR SYSTEMS ...

(18a)

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fr (P~

Lt All ~ Lt A12) is an upper bound of the

uncertain matrix/(P (18b)

-

J

M 11 , L1A 12 ),

-

-

-T

f(p, .&A 1b A A 1z)==Pf'sA 1 1+AA1 ]P

then one can state the following Theorem 1. S=Cy=O with

theorem~

,.."

1

T

1

-'T

-PA A12Q22A12P-PA12Q22l1A12P, (19)

are GC sliding surfaces with a guaranteed cost J=yi(ls)PY1(ts ), if there exists a positive constant E and a semi-positive definite matrix P such that the following perturbed Riccati equation holds

PA+ XTp-P[A12 R -1 ATz-e- t hI]P+ Q =0 (20)

therefore f(p,

6.A 11 , n AIi'

h

=~ C1;[P(Ri-Gi12Q2iAI2P)+(R;.-Gil2QiiAT2P) Tp]

where

r: be a scaling matrix, then from

(21 a)

Let

(21b)

l; [
(21c)

ore gets

h

(21d)

Al~=A11-A12QiiQ21

(2Ie)

Ql;=QI1-Q12Q2iQ21

(21f)

-Gi12Q2iAT2P)-r-tp]~o

~ [
~ ~ a.;[PCRi-Gi12Q~A i2P )+(Ri-G il2QiiAT2P )TP], J

choosing ri=f§! yields

-

Proof; Using state transformation

V=Q2~Q21Yl+Y2;

,...."

f(P~ ~A'1, ~ A 12) h '}

(22)

2

-1

T

~ (:;1 EO i (Ri- G il2Q22 A 12P )

and (21e), equation (1 la) can be re\Vritten as

T

)

-

-

,....,

1

J;::

J (YIQ11Yl + v T...

T

Q22v )dt.

T

-1

T

== £ t;1 Ui (Ri R i+PA 12Q22G i12

(23b)

L\ '-

)

G

i12

Q-1AT P 22

12

1Gi12Q21AI2P)+£ -l hp2

-PA12Qi~GT12Rr-R

Using (2lt), (13) can be rewritten as 00

s=2

(23 a)

where ttAll=flAII-AA12QilQ2I -

T

+E- 1 hp 2 h

Yl=(Atl+6.A 11 )Yl +(A12+L\A 1:0v ,

-1

(Ri- G i12Q22A lZP

(24)

=U

-

-

(P:o ~Al1' AA 1Z),

substituting the above fJ (p) L1A 111 L1 A12) into (27) and using the notations (21a)-(2Ic) yields (20). Comparing (22) with (25) yields

tg

- Q2~A{;,Pyl=Q2~Q21Yl+Y2

According to the lemma 1 ,

v=-

Q2~Al~PYl

(25)

is a GCe law and

i.e.. ~

this is just the sliding surfaces S =Cy =0 with

T

(26)

J=Yl (ts)Py}(t.s)

is a GC; where~ P is the solution of the perturbed Riccati equation

(28)

The proof is complete.

3.2 M satiifles a generalized matching condition

Q-JAf2P +Qtl

PAt'1+A;ip-PA 12

(27)

Assume that the uncertain matrix AA. is given by (29)

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SLIDING MODE CONTROLLER DESIGN FOR LINEAR SYSTEMS ...

where, Eo and F 0 are known matrices, D an unknown matrix. One gets

l1A=TE~FoT-l=EDF, -1

where,. E=TEo. F=FoT

~:J F=[Fl

~

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hence f(P)

(30)

Denote

dA11 , &A l 2J -1 T T T T P) , =(L-F2Q2JA12P) D E 1P+PE.D(L-F 2Q-1A 22 12 T

choosing T;= t?-l yields (31)

FzJ.

rep) ~Al1' ~Alz} ~PEIE1p+E-l(L-F2Q2iAI2P)TDTD(L-F2Q~AI2P)

~PEIETp+E-l(LTL-PA12Qi~F~-L Tp2QilAf2P

(32)

1 TIT ..1 +PA12Qi2F2F2Q22A12P)= U (P, ~Al1' AA 12),

Theorem 2 .. S=Cy=O with (33)

are GC sliding surfaces with a guaranteed cost J==yiCls)PYl (ts), if there exists a posilivc constant E and a semi-positive definite matrix P satisfying the following perturbed Riccati equation

Using the same procedure as that in the proof of theorem 1 , one gets (38)

The proof is complete. 4. REACIDNG MOTION CONTROLLER DESIGN

P A + ATp-Pf A 12 R -1 Ai2-f.E 1Ei"lP+ where

Q =0, (34)

A=AI;-E-IA12Q2~F~L

(35a)

Consider the system (10), choose the following nominal reaching law: (39)

S=-Ks-esign(S), (35b)

S=[St S2 ...

Q=Ql~ +E- 1L T L L=F t -

where,

sign (S)=-[sign(SI) sign(S~4.4 sign (Sn,)]T,

F2Q;~Q21'

(3Sd)

A:

and Ql~ have the same forms as (21e) and (21t) respectively . I

Proof: Using the same procedure as that in the proof of theorem 1 , one knows that

Ei

are positive

S==CY=C(A+AA )y(t) +C2 B zu(t). Denote

(40)

K=diag(k i ) and e=diag(Ei), k j constants~ From S=CY, one gets

c=

(36)

is a GCe law and with lhc GC

where) (37)

where P is the solution of the perturbed Riccati equation (27). Using (23b), (32) and (35d) yields -

Sm]T,

(35c)

-1

AAll=ElD(Fl-F2Q22Q21)~EtDL~

[~l c = [~l

Ch c2i

2

and

Vi

and

1

(CzB0- =

[J.

are the i-th raw of C., C 2 and

(C2B 2 )-1, respectively.

4.1 AA has matrix polytopes typed structure

Assume that the uncertain matrix AA satisfies then one gets

let Tbe a scaling matrix, then from

(15)~

(41)

[fErp-rlD(L-F2Qi~Ai2P)]T[rEip-r-lD(L -1

-F2 Q2iAI2P)]

~O

one gets

where Gi=TEiT . Theorem 3. The reaching conditions are satisfied if the following control law is used:

(L-F202}AT2P)TDTEIp+PEID(L-F2Q21AI2P)

~= -viKS-Viesign(S)-ViCAy -Vi z~

if

~~

(i=l,

2,.~.)m)

(42a)

where"

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SLIDING MODE CONTROLLER DESIGN FOR LINEAR SYSTEMS ...

14th World Congress of IFAC

h

-0.9571 -O~4584 2.0427

z=( ~ Sjlc}Gj y(l)lsign(Sl)

A==

h

~.~ ') OjlcmGj y(t)lsign(Sm) ]T,

1=1

-1.4945

00347]

~O:5529

1.0213

[ -1.7000 0.2200

2.2400 0.0203 -0.5300 -0.2600 0.5500 0.5452

(42b)

Proof: According to (42a) , one gets u(t)=-(C 2Bz)-1 [KS+e sign(S )+CAy+zJ..

1.1519

'

B=[ ~ ~]

(43)

substituting (43) into (40) yields

0.5

-1.0

1.. 55

1

Assume that

S==C&A y(t) - KS-.ssign(S)- z,

2

using (41) and (42b), onc gets the i-th component of

AA=> CtiEi, falf~O.lS; rclll~O.15 f;;1

Sas h

Si= ')
(44)

When Si>O, onc has h

Sj= ')

ftj

-kisi-£i-

J=1

on the other hand) when Si
,t=;;1

!~~0]'

2

000

calculation one gets 2.5089 -4.6436 ] p= [ -4.6436 10.5326 ' the sliding surfaces are S=Cy=O Viith

~ 3}CiGj y(t)1

:$ -~Si-Ci
Si= )'

,E= [~

choose £=2, Q=diag{l, 1, 1, O.2}, after a simple

h

ajCjGj yet)

1 0 1 0 100 0 E I =[ 0 0 1 0 o 0 0 0

b

CtjCiGj yet) -kisi+Ei

+ ~

-11.8418

8+4318

OjlCiGj y(t)t

J=1

c= [

~ ..kisi+ei>O.

13.8071

-21~4916

1

0 ]

0 0.2

'

the sliding mode controller is given by

The proof is complete. 4.2 L1A satisfies a generalized matching condition

Assume that LlA satisfies (29). Theorem 4. The reaching conditions ace satisfied if the following control law is used: Ui=-ViKS- viesign(S) - viCAy -

if

~;t:{},

2

ViZ,

(i=l, 2,... ,m),

where,

(45)

where

2

z=[ ~ OjlCl G j y(t)lsign(Sl)

~ OjlczGj y(t)lsign(sz)]T,

with

z=[( ±royTpTpy+ ro-1c lEETc})sign(Sl) ...

(~yTFTFy+ro-lCmEETC:)sign(Sm)]T

v= [ (46)

w is a positive constant Proof: Noting that 1 T T T -1 T T cjEDFys 4'CJ)y F D DFy+oo CjEE C i'

0

-6.4516 ]

-1

342258

£2=0.. 02;

' k 1 =1; k 2=1; £1=0.02;

0)=02=0.15;

The simulation results are shown by Fig.l-Fig4.

and using a similar procedure as that in the proof of theorem 3, one gets ~=~-kiSi-EiO

si==~-kiSi+Ei>O, for Si
The proof is complete.

-4

L....-_----''-----_---'-_~--'_ _- - - > - - _ - - s

o

5. NU:MERICAL EXA1\.1PLE

123 time (s)

4

5

Fig.1 Response cases of states Y

Example 1. Consider the system (10), where

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SLIDING MODE CONTROLLER DESIGN FOR LINEAR SYSTEMS ...

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15....---....,.........---,...------------.

6. CONCLUSIONS In this paper, the sliding mode controller design for

10t,:\(2 ...\ \_~

linear system with mismatched uncertainties is investigated. A new sliding surface design approach is proposed by llSing the GCe technique, which has

I",:.. or,

s?:>:~.

5

.,

o o

the advantage of providing a GC. The reaching motion control problem is also studied, a component •.. :;:::....,

controller is proposed for MIMO systems by means .......:... '~,

': "': :.:': :~'~'~~" ..' '.'

123

_.~

of reaching law approach. The controller has the

.

4

advantage of ensuring specified reaching law. Both the sliding motion and the reaching motion are robust against the mismatched uncertainties.

5

time (s) Fig.2 Convergent cases of reaching motion

REFERENCES

10

XlO-3

5

o -51.....-..-----....0..------------....0 2.5 3 3.5 4 time (s) Fig.3 Chattering on sliding surface SI

0.005

o -0.005 -0.0 1

--I--_--.&~_---'

L.---

2

2.5

3

3.5

4

time (s) Fig.4 Chattering on sliding surface

S2,

Fig.l shows the response cases of state trajectories with time. Fig.2 shows the convergent cases of S. The time of reaching sliding surface, for Si is about 2.7sec. and for 82 is about 2.28sec.. Fig.3 and Fig.4 show the chattering cases on sliding surfaces Sl and S2 respectively. The chattering is very small.

Bernstein,. D~S. and W.M. Haddad (1990). Robust stability and perlonnance analysis for state space systems via quadratic Lyapunov bound) SIAM J. Matrix Anal. Appl., 11, 239-271 . DeCarlo, R.A., S.H. Zak and G.P. Matthews (1988) Variable structure ontrol of nonlinear multivariable systems: a tutorial, Proceedings of IEEE, 76, 212-232~ Darling, C.M4 and A4S.I. Zinober (1986). Two approaches to hyperplane design in multivariable variable structure control systems, Int. J. Control, 44, 65-82~ Gao, W.B. and M. Cheng (1988). Quality of variable structure control systes, Control & Decision, 4, 1.. 7. (in Chinese)~ Gao, W.B. and I.C. Hung (1993)~ Variable stricture control of nonlinear systems: a new approach, IEEE Trans. Industrial Electronics.. 40, 45-55. Grayson, L.P. (1967). The status of synthesis using Lyapunovts method, Automatica, 3, 91-121. Hung, J.Y., W.B. Gao and I.C. Hung (1993). Variable structure control: A survey, IEEE Trans. Industrial Electronics, 40, 2-22. ltkis, U. (1976). Control Systems of Variable Structure, John Wiley & Sons, New York. Kwan, Chi-Man (1995). Sliding mode control of linear systems with mismatched uncertainties, Automatica, 3 t, 303-307.

Peterscn, I.R. and C.V. Hollot (1986)~ A Riccati equation approach to the stabilization of uncertain systems, Automatica, 22, 397-411. Spurgeon, S.K. and R. Davies (1993). A nonlinear control strategy for robust sliding mode performance in the presence of unmatched uncertainty, Int. J. Control, 57, 1107-1123. Su, Wu-Chung, S.Y. DTakunov and U~Ozguner (1996)~ Constructing discontinuity surfaces fro variable structure systems: a Lyapunov approach, Automatica, 32, 925-928.

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