Output feedback variable structure control design for singular systems

OUTPUT FEEDBACK VARIABLE STRUCTURE CONTROL DESIGN ...

14th World Congress ofTFAC

C-2a-06-5

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

OUTPUT FEEDBACK VARIABLE STRUCTURE CONTROL DESIGN FOR SINGULAR SYSTEMS

Ma Shuping

Cheng Zhaolin

Depa.rtment of Mathematics, Shandong University, Jinan 250100, F.R. China. E-mrul:[email protected]

Abstract: In this paper, the problem of output feedback variable structure control design for multi variable linear singular systems is discussed. An algorithm of switching surface design and stability analysis of sliding mode motion are given, the output feedback variable structure control design and global asymptotic sta.bility analysis of closed-loop are given. The control design given in this paper assures that dosed-loop tra.jectory converges to zero with arbitrary exponent speed. Copyright © 1999 IFAC

Keywords: variable structure control; output feedback; linear singular systems; ma.trix spectrum analysis; Lyapunov method.

1. INTRODUCTION

in the begining, there are not many representative works, such as Kawaji and Taha (1994), in these papers, the control structure all used the full-order state feedback with observer.

Variable structure control system design is a nonlinear control method with control switch developed in Russia about fourty years ago (Utkin, 1978), because it has robustness to disturbance, in recent ye.;l.rS, it has received considerable attention. There have been many mature works about variable structure control for multivariable linear systems, such as Bondarev, et al.(1985), ELKhazali and Decado (1993, 1995). Two ways were used for control design in these works, which are full-order state feedback via observer and output feedback. Comparing these two ways, the former is easy in theory, but need observer dynamic, which complicates control design in practice. In last decade, many works focused on output feedback. As for the problem of variable structure control for multivariable linear singular systems, since the structure of singular systems is complicated, and the system state trajectory always contains impulsive behaviour, there are more difficulties need to be conquered for singular systems than for linear systems, so at present, the work is

In this paper, the problem of output feedback variable structure control design for multivariable linear singular systems is discussed, there is no report about this question as the author known, and controller using the output feedback accords further with the practical demands.

2. PRELIMINARIES This section gives some results of the linear singular systems

Ex=Ax+Bu, y=Cx

(1)

where E, A E Rnxn, B E Rnxr, E is singular, and the pencil sE - A is regular, i.e. det[sE - A] 1- o. Furthermore, we assume that the system (1)

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contains impulse mode, which is degree det (sE - A)

< rankE.

Lemma 2 (8ee Chen, et al.1996). Matrix [B2 J B2 . .. J/1-1 B 2 ] is of full column rank if and only if JI'-l B2 is of full column rank.

(2)

It is well known, for the system (1), there exist nonsingular matrices P, Q E R:'XTI such that In,

PEQ =

J

Lemma 3 (see Chen, et a1.1996). For any given J :1:2(0-), equation (8) has the solution u(i)(O), i = 0, 1, ... , I-' - 2, if and only if: 1) rank [B2 JB 2 .•. J1-'-1B 2 ] n2; 2) JI'-l B2 is of full column rank.

=

0]

Al

PAQ

=

PH CQ

= [Bi B~ B~r = [Cl C 2 C 3 l

fn~

1

In3

_

X

-

[

XT

T

Xl

2

x~r, X =

Remark. The meaning oflemma 3 lies in: it points out that the no impulse state trajectory and the control of this kind of systems can be described by the following standard state-space linear system:

Qx

(3)

[:1 ]

where Al E Rn,xn" 1nl E R"ixn i is a unit matrix, Bi E Rn;xr, Xi ERn; , i = 1,2,3, J E Rn~xn~ is a nilpotent Jordan matrix with nilpotent index /-" the order of each Jordan block of J is not less than 2. Call the following system xI=A1X1+Bltt,

JX2 = X2 + B 2u, Xg = -Bgu y 001

= [1n,

Xl(O)=

JX2(O-) =

aOl

002

= [0

Xl(t) X2(t)

=

-

+

1t

0-02,

JB

2

(4c)

;=0

2

JP-lE] [ z(t) ]

vet) (10)

2

(11)

= -(Hr Hl)-l Hr

H

= [J B2 J2 B2 ... JI'-I B2l G = [B1 0 ···0], G E R n l X (/1-1)r F = [0 ... 0 frY, FE R(p-1)rxr H1

eAICt-T)H1u(r)dr

;=0

... JP- 2B

(9)

where

L = [Ir 0

~-2

(8)

v

X3(t) = -B3Lz(t) (4b)

In2 OjPo o (4d)

p-1

[:1 ]+ [ ~ ]

[ :\~l ] = [ ;~:2 ]

as the Weierstrass canonical form of system (1). Noticing that the state trajectory of (4a), (4b) is:

e A1t x1(O)

[~1 ~]

(4a)

= Cl:!:l + C2X2 + C 3 xg

0 OJPao,

=

0], L E R rx (I'-I)r

D~[O ~ ~l'

L: Ji B 2u(i)(t) - (L: 6(1)(t)J;+1) /1-2

(X2(0-)

+ L: JS B 2u(i)(0))

DE R(I'-1)rx(p-l)r

(5)

i=O

z(t) = [u"'Ct) ... (u(/1-2)(t)rr vCt) = u(p-1\t)

where u(i)(t) and 6(i)(t) are the ith-order derivative of u(t) and Dirac-6 function, i.e 6(t), respectively. Then, to assure that there is no impulsive behaviour in the system state trajectory, the initial values of the control function and its derivatives must and only must satisfy the following consistent condition:

(12)

where 0101, 0102 are shown as (4d). The above discussion gives us a fine method to discuss the no impulse state trajectory and the control of singular system (1).

1'-2

J ;/;2(0-) = -

E

Ji+1 B 2 u(i)(0).

(6)

3. OUTPUT FEEDBACK VARIABLE

STRUCTURE CONTROL DESIGN

i=O

Denoting the control function set satisfying (6) as U, we have:

This section discusses the output feedback variable structure control design for singular systems

Ex = Ax + Bu, y = CEx

Lemma 1 (see Oben, et a1.1996). For any given JX2(0-), U is nonempty, that is (6) has solution u(i)(O), i = 0,1, ... , /J - 2, if and only if: rank[B2 J B2 ... JP-1 B2l = n2

Ex(O-) =

00

(13)

where E,A E Rnxn, B E Rnxr, C E Rmxn, :x E Rn, u E R r , y E R m are the state vector,

(7)

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the control vector and the output vector respectively. For relating simplicity, we use the Weierstrass cananical form of singular systems as model, i.e. E,A, B, Care Weierstrass canonical form (3). Here, we assume that the autputs are 1inear combinations for dynamic part of the system, i.e. Ex. Obviously, this assumption makes the system contains direct infarmation related to the dynamic part. To realize our designing, we have assumptions for the system as fallows:

14th World Congress ofTFAC

where S E Rrxm is of full row rank matrix, and the choosing of S shoud assure that the switching surface 7 is a (n1 + (~ - 2)r) dimensionallinear subspace. To obtain the sliding mode motion equation on 7, we use the following equation to partition the state vector in (8) again:

z=(z[ z;r, x*::=[xI z[r. Zl

=

[UT (u(1»T .•.

(u(I'-3)rr,

Z2 ::::: u(I'-2} (17)

and rewrite (8)-(9) and (15) as:

Assumption 1. System (13) is impulse controllable. Assumption 2. Matrices JIJ- 1 B2, C~JIJ-lB2' and

[~~ ~C2J B2

] are all of full column rank.

Assumption 3. Matrix [Cl C 2 J B2 ... C2JP-l B2J has full row rank.

Assumption 4. The rank of matrix [Cl C2J B2 ...

where

C2 JP-2 B2J is m - r.

o

Assumption 5. The input, output, and state dimensions of system (13) satisfy the following inequality: (14) nl + n2 S; m + 2r - 1

o All

Assumption 6. System (13) is R-controllable, Robservable.

=

Where parameters A l , J, Ri, C., i shown as (3).

1,2, are

From assumption 1 and the provISIOn about JI'-1 B2 being of full column rank in assumption 2, we can obtain that the no impulse state trajectory and the control af singular systems can completely be described by standard state-space linear systems, so the problem of variable structure control for the system (13) can transform into. the problem of variable structure cantrol for standard state-space linear system, whose state equation is (8)-(12), output equation is y:::::

[Cl - C2J B2 ... - C2JI1- 1 Bd

(

AI:!

:1]

(15)

We choose the switching surface ([xI z'Y : o-(y) = O} of problem P2 as u(y) Sy. So the switching surface is

=

'lr

= ([xI zT' : u(y) ;::: By

:=

O}

(16)

= [0, : 0

f31 :::::

Ir

[In1 +(p-2)r

f32 = [0.· ·0

Ir 1[

Ir

r

0] [

o

;;~02

]

~O!02 ]

cf = [Cl

- C2J Bz ... - C 2 JJl-2 B 2] cg = -C2 JP-l B2 (21)

fh ER!, A12 E (31 E Rn,+(Jl-2)r. Noticing that the sliding mode motian on 1r satisfies

All

E

R("1+(p-2)r)x(n 1+(p-2)r),

R(tl1 +(p-2)r)Xr,

o-(y(t»

:=

so on the premise that (18) and (20) yields: Z2

Denoting the former as PI, the latter as P 2 • Obviausly, solving P 2 can obtain PI. For this, we begin to. solve problem Pz.

3.1 Switching surface design and stability for sliding mode motion

=

x+ =

By(t)

== 0

(22)

scg has full rank, from

~ -(Scg)-lSCfz'"

(23)

(All - A12(SC~)-1 SCP)x'"

y = (Irn - cg(SC~)-l)S)Cfx* Veq

=

-(SC~)-lSCf(Al1x"

+ A 12 zz ).

(24)

(25) (26)

In which, (24) is called the sliding mode motion equa.tion on switching surface 7, (25) is the autput equation of sliding mode matian, (26) is the equivalent controL The first step in the following is choosing S such tha.t the caefficient matrix All - A 12 (SCg)-lSCP of the sliding mode equation is stable matrix. For this we intraduce the

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(cgy,

rank of Cp, C~ and the null-space of (CfY, from assumptions 2, 4 and (21), yields:

following discussion.

Lemma 4. If assumptions 2, 6 hold, then linear system [Cf, All, An] is state controllable and observable.

rankCr

==

m - r, rankGg

==

r , rank[Cr C~]

== m (34)

Lemma 5 (see Kimura,1975) . Assume that linear system is [C, A, Bl, in which A, B, C are constant matrices, A E Rnxn, B E Rnxr, C E RmXfI. If the system is state controllable and observable, the input, output, and state dimensions satisfy Davison-Kimura condition: n 5 m+r-l, then for n arbitrary given numbers, a1,"' , an (complex numbers apear as conjugate pairs), and a small enough positive number ~, there exists feedback matrix K E K"xm such that the eigenvalues of dosed-loop system via output feedback are all located in e-domain of a, i = 1,2"", n.

where M is shown as (31), Ml E Rrxm, M E R(m-r)xm are all full row rank matrices. More further, there are:

The computation for feedback matrix K lemma 5 can see Kimura(1975).

Lemma 8. For any given positive number Q, there must exist P E Rrx(tn-r) such that the eigenvalues Ai, i == 1,2, ... , nl + (p.- 2)r of .All - A 12 P(\ satisfy: 1) Ai are distinct real; 2) Ai < -a.

Lemma 6. Matrix [

lTI

Lemma 9. If K is shown as (29), SE Rrxrn , then S is a solution to (27) if and only if S can be shown as: S=LK (35)

where L is an arbitrary r x r full rank square matrix.

Notice that K satisfies

(28)

The proofs of Lemma 6-9 are easy, in lemma 8, the computation of matrix P can see Kimura(1975).

namely that K can not arbitrarily be choosed. Analyse equatio~ (28), we know that the general solution to (28) can be written as:

We conclude the above disscusion as theorem.

Theorem 1. The following algorithm gives the design of switching surface, and assures that the sliding mode motion trajectory x"(t) converges to zero with exponent speed:

(29)

where

(30) is the generalized inverse of c~, P E Rrx(m-r) is an arbitrary parameter matrix, M E R(m-r)x m is of full row rarlk , and M satisfies

MCg=O

where a- is an arbitrary given positive number, mo is positive real, and depends on a and initial values of the state trajectory, is independent oft 2: O.

(31).

Substituting of (29), (27) into (24) yields: :1:" = (All - A 12 P(\)X'"

is full rank square ma-

Lemma 7. Linear system [(\, All, A12J is state cont rollable and observable and satisfies DavisonKimura condition, in which C1, All, A12 are shown as (33) , (21).

(27)

= (C~)+ + PM

]

trix, in which M, Ml are shown as (34).

Obviously, the coefficient matrix of sliding mode motion equation (24), i.e. All - A12(SCg)-lSCP, can be looked as the coefficient matrix of closedloop of system [CP,All,Ad via output feedback, where the output feedback matrix is

K

~l

Algorithm. 1) For arbitrary given positive number Cl', according to follows:

(32)

< Qi - 1, i = 1,2, . .. ,n1 + (J.l < -(Q + 1)

a'+1

in which

al

2)r - 1

(37)

we assign nl + (~ - 2)r real numbers ai, and by lemma 8, using the Kimura algorithm to find matrix P E K"x(m-r) such that the eigenvalues >'1, >'2, ... , An1 +(p-2)r of All - A 12 PC1 satisfy:

Notice t hat the asymptotic stability of sliding mode motion is determined by the eigenvalues distribution of matrix All - A 12 PG\. In the following we first study state controllability and observability of the system [<\, All, Ad. Observing the

I),i - a-d

1

< "2' i = 1,2,· .. ,n1 + (I-' - 2)r. (38)

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2) According to (30), (31), find (C~)+ and M , from (29), find K. 3)According to (35), find 5, particularly can let S=K.

Next, use Lyapunov method to discuss global asymptotic stability for the closed-loop system

3.2 Output feedback variable structure control design and global asymptotic stability of closed-loop

(46) of system (43),(44) via the output feedback (39), and define the domain of coefficients Fo and F l · We introduce quadratic Lyapunov function:

In this section, we discuss the design of output feedback variable structure controller. For generality, the output feedback variable structure control law takes the form v

= FoY + FI . sgnu,

Vu(y):f:. 0

take

(39)

Fo =

where sgnu = [sgnuI sgnu2 ... sgnurr

=

[T0] [ Ser scg I

x" ] Z2

.

Ko = kolr, ko ~ 1 +

(40)

u;. is the ith vector of u(y) = Sy, i = 1,2, .. " r, Fo E Rrxm, FI E R rxr are the feedback gain matrix waiting decided. Obviously, the key to control design is to define the domain of Fo and FI. For this, we introduce coordinate transformation as follows:

[ uq]

{30 -(SC~)-l(A22n + Ko)S

FI = -(5Cn- l

P analysing

(48) (49)

p2

IA11

(50)

= IIAl2nll + IIA2lnll

(51)

V:

17= q1" q + u.,. iT = qT(AUnq + A12 n O") +u1" «A21n + B2nFoCln)q +(A22n + B2nFOC2n)0") + UT B2nFt . sgnu $; -(I).llllqI12 - pllqllllO"II + k olluif2) - .80110"11 = -~I).dllqIl2 _ (v'IAllllqll_ -P-lluID 2

(41)

4

2

y"j"I;j

p2

where T E RC n l+(JL-2)r)x(n 1 +(1'-2)r) satisfies

lAd )ll ull 2- {3ollull 3 $ -(4 1 ).11I1qW + Ilu11 2) - .801 lull < 2 'v'lIull + IIqll2 i- 0

-(ko -

(42)

0 (52)

Using Lyapunov theorem, we know immediatly that closed-loop system (46) is globally asymptotically stable. Above results are concluded as theorem, that is

It is easy to know that in the coordinate transfor-

mation (41), the system (18) and the output (20) transform as:

[ iTq]

=

[Alln A 2ln

A12n] [ q ] A 22n U

+[

0 ] B 2n

Theorem 2. Assume that the open-loop system is shown as (18)-(21), the output feedback variable structure control is shown as (39), coefficients Fo, FI are shown as (48)-(51). Then the closedloop system is globally asymptotically stable.

V

(43) (44) where

All n = [

In the following, observing the possibility for that the state trajectory of the closed-loop system reaches the switching surface in finite time, and realizes sliding mode motion on it forced by the control (39), we have:

1

).1 ),nl+(/I-2)I"

A2ln

= T- I A 12 (SCn- l = 5C?(All - A12(SC~)-ISC?)T

A22n

= SCP A I2(SCg}-I

A12n

B 2n =

Theorem 3. The closed-loop system related in theroem 2 must reach the switching surface in finite time, and realize sliding mode motion on it.

scg

C ln

= (Im -

C2n

= C~(SC~)-l.

cg(SC~)-lS)C~T

Now, we pay attention back to system (13), and observe question Pt. Noticing that control u(t) is

(45)

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OUTPUT FEEDBACK VARIABLE STRUCTURE CONTROL DESIGN ...

the solution of the differential equation (8) satisfying initial condition (9), so

z(t) = compute

u(t)

e Dt

+ z(O)

1t

e Dt ,

we have

=

u(O)

eD(t-T) Fv( r)dr

(53)

2

+ tu(J)(O) + t2! U(2)(0)

+ ... +

t",-2

(I' - 2)!

+ Cl-' ~ 2)1

lot

U C",-2)(O)

(t - r)I-'-2 . v(r)d7(54)

where vCr) is shown as (39),(48)-(51) Analyse asymptotic properties for the state trajectory x(t) = [xHt) x~(t) x;{tW of system (13) forced by u(t). Noticing (10),(11),(17), and noticing that x'" et) reaches the switching surface 11" in finite time, and realizes stable sliding mode motion on it with the exponent speed (36), then

where ex is shown as (36), m* is positive real number, and depends on the initial value Ex(O-) and a, is independent of t ~ O. Concluding above results, we have:

Theorem 4: If system (13) satisfies assumptions 16, then the system state trajectory drived by output feedback control law (54) converges to zero with assigned exponent speed. Example. Assume that singular system is shown as (13), in which

1 0 E =

0

0 0

n, o

B~ Ex(O-)

=(0

~l' A=[~ ~

000

c=[~ ~ ~], 1

0

r.

by calculating, above system satisfies assumptions 1-6, and take P = 3, then

S = K = [-1 3), Take Ko

= 26, Fo = [29 - 87], /30 = 1, FJ

then obtain control law:

+ f; v(r)dr

u

= -1

v

= [29 - 87)y - sgnO',

0'

= [-1 3Jy.

'10'

t= 0

= -1

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4. CONCLUSIONS This paper has discussed the problem of output feedback variable structure control design for multivariable linear singular systems. Given: 1. the design of switching surface and stability analysis of sliding mode motion; 2. the output feedback variable structure control design and global asymptotic stability analysis of closed-loop system. The design of control given in this paper guarantees that the closed-loop state trajectory converges to zero with arbitrary assigned exponent speed. An importa.nt difference between singular systems and linear systems is: to assure that the state trajectory of singular systems has no impulsive behaviour, the initial values of the control and up to the 1-'-2th-order derivative must satisfy consistent condition (6). So the control design for singular systems, which we can freely design is the p- lthorder derivative of the control, but not the control inputs, that is what the discussion is based on in this paper.

REFERENCES Bondarev, A.G, S. A. Bondarev, N. E. Kosbyrera and V.1. Utkin(1985). Sliding modes in systems with asymptotic state observers. Automatica, 21, 6-1l. Dai, L. (1989). Singular Control Systems. Springer-Ver lag. Davision, E.J. and S.H. Wang (1975). On pole assignment in linear multivariable systems using output feedback. IEEE 'Dans. Autom. Control, AC-20, 516-518. EL-Khazali, R. and R. Decarlo (1993). Output feedback variable structure control using dynamic conpensation for linear systems. Proceedings of American Control Conf., San Francisco, 954-958. EL-Khazali, R. and R. Decarlo (1995). Output feedback variable structure control design. Automatica, 31, 805-816. Kawaji, S. and E.Z. Taha(1994). Hyperplane design in variable structure of descriptor systems. Proceeding of the 33rd IEEE Conf. Decision and Control, Florida, USA. Kimura, H. (1975). Pole assignment by gain output feedback. IEEE 'Dans. Autom. Control,

AC-20, 509-516. Utkin, V.I.(1978). Sliding Modesand Their Application in Variable Structure Systems. MlR, Moscow. Chen,Y., S. Ma and Z. Cheng(1996). Singular optimal control problem of linear singular systems with linear-quadratic cost. Proceedings of Chinese Control Conference, Qingdao, 5-9.

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