[IEEE Proceedings. 1996 IEEE International Workshop on Variable Structure Systems. - VSS'96 - -...

Sliding Mode Time-Delay Systems A Jafari Koshkouei and A S I Zinober

A h L r u c t - In this paper the sliding mode control of time- delay systems is considered Time-delay sliding system stability is studied for the cases of some information about the delay and also lack of information The sliding surface is delay-independent as for the traditional sliding surface and the reaching condition is achieved by applying a conven- tional discontinuous control

Keywords- Sliding mode control time-delay systems stability sliding system

I INTRODUCTION

recent years many methods have been reported for I designing control for time-delay systems [9] [ll] and criteria for the stability of time-delay systems [4] [ lo] have been developed Time-delays may appear in many ways Delays in rrieasurenient of systern variables including phys- ical properties of equipment used in the system or signal transmission Delays in control which arise in rriany cherri- ical processes and radiation problems in physics Time- delay systems are also used to model several different mech- anisms in the dynamics of epidemics Many problems such as incubation periods maturation times age structure sea- sonal or diurnal variations interactions across spatial dis- tances or through complicated paths have been rriodelled by the introduction of time-delay systeriis [2]

The work on the stability of time-delay linear systerns has been reported by many authors and can he found in [3] [4] [8] and [lo] amongst others Work on the stabilization problem for a class of uncertain linear systems with delay on the state has been studied in [9] The proof of stability of closed-loop time-delay systems with discontinuous control is more complicated than for the continuous case

Stability criteria for time-delay systems can be classified into two categories ( i ) there is no information about the delays ie delay-independent criteria (ii) there is some information about the delays ie delay-dependent criteria The delay-independent criteria are strong conditions to test the stability of the system However if the delays are small these criteria may be useful Delay-dependent criteria for a closed-loop system is dependerit upon the kind of contro1 which is applied to the system To prove the stability of both open- and closed-loop time-delay systems an appropriate Lyapunov function can be selected The magnitude of the delay may not necessarily be important when test- ing the system stability The stability may hold for certain sufficient conditions [8]

Sliding mode t irrie-delay systerns have 1)ren developed b y eiiiploying the proI)ortional-integral slidiiig mode (PISM) [ l l ] arid the traditional sliding mode 111 the case of PISM the sliding surface may depend on delays and it is difficult

The authors are with the Applied Mathematics Section School of Mathematics and Statistics The University of Sheffield Sheffield S10 2TN U K emailAJafariOsheffieldacuk AZinobelOsheffieldacuk

1996 IEEE Workshop on Variable Structure Systems 0-7803-3718-296 $500 0 1996 IEEE

to specify its dynamic performance for the traditional sliding mode the sliding surface is independent of delay Here we present a method of designing the sliding surface and appropriate discontinuous control such tha t the stability of the sliding mode and the system is obtained

In this paper the norm of a vector is the Euclidean norm and the norm of a matrix is the maximal singular value of the matrix Both norms are given by 1 1 1 1 and the absolute value of a function or a complex number is 1 I Also A m i n ( + )

Amax() a() and U M ( ) refer to the minimum and maximum eigenvalues and the minimum and maximum singular values of the matrix ( e ) pd pds and upds stand for positive definite positive definite symmetric and unique positive definite symmetric respectively Also p ( ) stands for the spectral radius of the operator ()

11 SLIDING TIME-DELAY SYSTEMS Consider the following uncertain system

i ( t ) = A t ( t ) + Aot(t - T ) + B u ( ~ ) + f ( t ~ t ) z ( t ) = Wl t E [-lo1 (1)

where I E Rn is the state variable A E R B E RxL is full rank U E R is the input control C E Rmx such that CB is nonsingular T is a positive real number and $( t ) is a continuous vector-value initial function with 1 1 $ 1 1 =-go Il4(t)ll Assume that ( A B ) is a completely controllable pair m lt n and the function f(t T t ) E R is a bounded disturbance or uncertain input signal

The sliding surface is defined as s = Ct(t) = 0 The ideal sliding mode exists if there is a finite time t such that

where C E R is the sliding mode matrix Then the virtual equivalent control is given by

c z = o ci=o t gt t

ueq( t ) = -(CB)- (CAz( t ) + CAot( t - T ) + Cf( t T t ) ) and the system in the sliding mode is

i ( t ) = Aeq(t) + A e q t ( t - T ) + Beqf( t 2 7) (2) where

Aeq = ( I - B(CB)- C)A A e q = ( I - B(CB)-C)Ao

and Be = ( I - B(CB)- C)

Assumpt ion Matching condition Assume that f ( t t T ) = B g ( t t r ) Then the system in the sliding mode

is independent of the external input f The matching condition is a suitable condition for which the system in the sliding mode is independent of the external uncertain input

i ( t ) = Aeqz( t ) + Aez(t - T ) ( 3 )

-97-

A Sliding Control where

Ail Aiz A11 A 1 2 Consider the control

u ( t ) = -(CB)- ( C A x ( t ) + CAoz(t - T ) + p sgn s ( t ) ) (4) TATT = [ A21 A22 ] = [ A Aa2 ] where p = p ( t z ( t ) T) = diag(p1 pl p m ) with positive real functions p = p ( t t ( t ) r ) (1 lt i lt m) and sgn s = [ sgn s1 sgn s ~ sgn s 1 with

1 if si gt 0 s g n s = 0 i f s i = O l s i s m -1 if s lt 0

Then bhe system is now given by

i ( t ) = Ax(t) + Ax(t - T) + f ( t l 2 T) -

B ( c ~ ) - p ( t ~ ( t ) T ) sgn s ( 5 )

( 6 )

Hence the sliding dynamics is governed by

S = C f ( t ~ ( t ) T ) - p( t x ( t ) T) sgn s

Subsystem (12) is the system in the sliding mode So the sliding surface is Clz l ( t ) + Czzz(t) = 0 where CTT = [Cl Cz] In the sliding mode q ( t ) = - A z l ( t ) with K = C2-C1 Therefore the reduced order system (12) is converted to

i l ( t ) = (Al l - A 1 2 K ) ~ l ( t ) + ( A i - A l z K ) z l ( t - T) (14)

For any x(0) and any function 4 E C([-TO]R) there exists a function z1 ( t ) satisfying the differential equation (14) almost everywhere [5] In this case for all integers k the function z l ( t ) will be Ck on ((k - l ) ~ CO) ie for all 1 5 j 5 IC the j- th derivative of function z l ( t ) is continu- ously differentiable on ( ( j - l ) ~ CO) With these conditions the solution of (1) is

and for all 0 lt t lt t zl(t) = e ( A i i - A i z K ) t $ l ( ~ ) + e ( A i i - A i a K ) ( t - w )

f ( t x ( t ) T ) d t - p ( t t ( t ) ~ ) ( t - t l ) sgn s (Ai1 - A i amp ) ~ i ( w - T) dw t 2 0 (15)

The reaching sliding mode condition is

S i sgn s lt 0 V i 15 i I m ( 7 ) There exist positive real numbers Q and M such that iizl(t)ii I (Miiw)ii + M2ii$1iir) e ( M 2 + q ) t [51 in the neighbourhood Of = [I2] the We wish to find such that A l l - A12Jlt is stable Con- sider the Riccati equation i-th (1 5 i 5 m) row of (6) by sgn si gives

A l l P + PAllT - P A ~ Z R - ~ A T ~ P = -Q (16)

where Q and R are arbitrary semi-pds and pds matrices receptively I t has a upds matrix solution P For K = R-AT2P the matrix All - AlaK is stable

A System stability without delay information

For this case system (1) is asymptotically stable if all the roots of the characteristic polyqomialof the system (12) p ( s ) = det (s1- A11 +AIamp - (A11 - A1K)er) have negative real parts If all the roots of p ( s ) = 0 lie in the open ~e f t -ha~fcomp~ex plane this condition is equivalent to p ( s ) E + where c+ is the set of all complex numbers with nonnegative real gt 0 there may be infinitely many solutions while for T = 0 there are finitely many solutions So when T gt 0 it is very difficult to find all the infinity roots of p ( s ) = 0 and check if the roots are in the open left-half plane This motivates one to avoid the delay-dependent condition p ( s ) = 0 for stability of the system and use the condition p ( s ) 0 for all s E + to prove the asymptotically stability independent of delay However since p ( s ) is an entire function there are only a finite number of roots of p ( s ) = 0 in any compact set in particular in a vertical strip of the complex plane Furthermore there exists a real number CY such that all the

The function p ( s ) is an analytic function on There- fore according to the Maximum Principle Theorem [l] p ( s )

b sgn si = C i f ( t x ( t ) T ) sgn si - p i ( t z ( t ) T ) (8)

Hence a sufficient condition for the existence of the sliding mode is

Cif ( t x(t) l r)sgn si lt p i ( t ~ ( t ) 7)

and a suficieiit condition is that

IlCf(t z ( t ) ) I 1 lt min pt

111 SYSTEM I N THE S L I D I N G M O D E

(9) l lt lt

The behaviour of the system in the sliding mode is considered in this section The system in the sliding mode is a subsystem of (1) of order n-m Assume T i s an orthogonal riiatrix such that

(10)

for all Note that for

T B = [ i2 ] where Bz is a nonsingular matrix Let z = Tx then

(t ) = TAT^ (t) + T A ~ T ~ ~ ( ~ - T ) + T B u ( ~ ) + T B ~ ( ~ t ( t ) T)

22EIIP NOW ~ZSSUITI~ tT = [ Z I z2] Z I E R- Then

i1( t ) = ~ ~ ~ ~ ~ ( t ) + ~ ~ ~ ~ ~ ( t ) + A l l z l ( t -

(I2) roots of p ( s ) lie to the left of the vertical line z = (Y [7] i 2 ( t ) = A L l Z l ( t ) + A22zz(t) + A 2 1 2 1 (t - T ) + A22z2(t - T )

( 1 3 )

T

+ A l n z z ( t -

+ B2u + Bzg(r( t ) T t )

-98-

(17) For simplicity set A = A11 - Al2K and A0 = A11 - A121lt So from [3] we obtain the following theorem

Proof Since A is a stable matrix p ( ( S I - A)-amp) is an analytic function in the right half-plane and according to the Maximum Principal Theorem [l] it is assumed that it takes its maximum value on the imaginary axis Thus

Since p ( i u I - A)-Ao) is a continuous function there is

a W O such that p ( ( iu0I - A)- amp) = 1 which contradicts 211PAll - A12K)ll l l(-All + A l z K + iu1) - Al l - A1zIlt)ll

II(-A11+ A12K + ~ ~ I ) - ( A I I - A12Kl1~xmin(Q) lt (

(19) Therefore p ( w l - A)- amp) lt 1 Vs E e lt L i n ( Q ) 11(-A11 + A12K + iuI)-Il This complete the proof

( Sufficiency Assume that (18) is satisfied Hence

such a condition free of frequency Its proof is similar to

Theorem 2 Let P be the upds solution of the Riccati [41

equation

A l l P + PAllT - 2PA12R-AT2P = -Q ( 2 0 )

where Q and R are arbitrary semi-pds and pds matrices respectively Then if

11 (A11 - A12K) lt X m i n ( Q ) 2 K = R - A I ~ ~ P

( 2 1 ) the system in the sliding mode is asymptotically stable independent of delay

Proof Let P be the upds matrix solution of the Ric- cati equation (20) Adding and subtracting iwP on the left-hand side of ( 2 0 ) gives

Necessary Assume that the system (14) in the sliding mode is asymptotically stable independent of delay Since A is a stable matrix for all s E det(s1 - A) 0 If s = 0 then p ( (A) - amp) lt 1 and there is nothing to prove

Suppose that s 0 then det I - ( S I - ~ ) - l amp e s T ) 0 ie for all s E C+ 1 is not an eigenvalue of

( ( S I - A)-Aoe7 so

p ( ( i d - A ) - A o ) 1 vu 2 0 (19)

and by pre- and post-multiplication by

- 9 9 -

- ( sT ( t ) P B + BT P z ( t ) ) p(t z ( t ) r)sgn( Cz (t) ) (26)

then the matrix -Q + 2cP + PI + r - l P f f e A ~ q P

is a negative definite matrix So if

-Amin(Q) + 2 ~ A m a x ( P ) + P + T-111PAcq112 lt 0 1 P A ~ ~ R - ~ A T ~ P -p(Al l - A 1 2 q [ - (A l l - A I ~ K ) ~ P Q

is pd and V lt 0 implying the stability Note that if then V lt 0 Assume that

then

and the system is stable

I v GLOBAL STABILITY O F T H E SYSTEM

If the ideal sliding mode occurs the system (1) is converted to the reduced order system (14) Otherwise the states lie in a boundary layer of the sliding surface s = 0 and the dynamical motion is no longer governed by a reduced order system ie there is no finite time instant such that after this time instant the states lie on the sliding surface The system is now given by (5) The stability of the sliding system with f = 0 is studied in this section Con- sider system (1) with control ( 4 ) Let c gt 3(Ama(Aeq)) where R() denotes the real part of complex number () Since one eigenvalue of A is 0 E gt 0 Assume that h(s ) = C (SI - (Aeq - euroI))- B is strictly positive real Then matrix A - EI is stable and the Lyapunov equation

(Aeq - E I ) ~ P + P(A - euro1) = -Q (25 )

and

Then V lt 0 if P 2 IlPAII We have proved the stability of the system with a strictly positive real condition However the stability of the system without this condition can be proved by choosing P = CTC and a Q matrix satisfying (25 )

V EXAMPLE Ezample 1 Consider the system

where g ( t z r ) is an external input signal with g(tz)I lt 1 Choose z ( t ) = 4( t ) = [0 -11 for -T lt - - t lt 0 The upd solution of (16) is P = 24142 So K = -24142 and C = [-24142 11 Choose the control

~ ( t ) = [ 24142 -04142 ] ~ ( t ) - where Q is an arbitrary pds matrix has a upd solution P Consider the Lyapunov function

[ 24142 00343 ] z ( t - T) - sgn s ( t )

where s ( t ) = C z ( t ) The sliding surface is s = -24142zl(t) + z z ( t ) = 0 Since the condition (9) is

ing mode is given by 51 = 2 1 - 2 2 - z l ( t - T ) + 04z2(t - T )

where P is the pds solution of the Lyapunov equation Let R = Q = 1 The system in the sliding mode (25 ) and R is an arbitrary pds matrix For simplicity is given by $1 = -1414221 - 00343zl( t - 7) Since

v(t z(t)l z ( t - )) = z T ( t ) P z ( t ) z(e)Rz(e)de satisfied the sliding mode occurs The system in the slid-

- 100 -

( A 1 1 - A l z K P = 01268 lt Xj(Q)2 = 05 the system iS stable independent of delay because Theorem 2 is satisfied The simulation results for T = 04 are shown in Fig 1

constrained controlrdquo IEEE Trans Automat Contr 40 1615- ) I1 1619 1995 [g] M s Mahmoud and N F Al-MuthairillsquoDesign of robust con-

troller for time-delay systemsrdquo IEEE Trans Automat Contr 39 995-999 1994

[lo] T Mori E Noldiis a n d M Kuwahara ldquoA way t o stabilize linear systems with delayed s ta te rdquo Atomntica 19 571-573 1983

[ l l ] K K Shyu a n d J J Yan ldquoRobust stabil i ty of uncertain time- delay systems a n d its stabilization by variable s t ruc ture controlrdquo In t J Control 57 237-246 1993

1121 V I IJtkin Sliding Modes in Control and Oplamt tn t ion

Example 2 Consider the system

1 -1

g

Springer-Verlag Berlin 1992

O A

where g ( t e T ) is an external input signal with 1g(t 2 )I lt 1 So the upd solution of (1G) is P = 24142 Consider the sliding surface and 4 as in Example 1 Select the discontinuous control

State variables

u ( t ) = [ 24142 -04142 ] c ( t ) -t 0 5 10 [ 0 -05172 ] z ( t - 7) - sgn s ( t ) t

Sliding function

where s ( t ) = Cz( t ) The condition (9) is satisfied O 5 so the sliding rnode occurs The systern in the sliding mode is given by il = e1 - x2 + 0222(t - T ) Let R = Q = 1 The system in the sliding mode is given by i 1 = -1 4142~1+ 04828sl(t - 7) On the other hand

(I ( A l l - AlzK P = 07464 gt Xin(amp)2 = 05 1 I1 So the condition (21) is not satisfied But the system is stable independent of delay The simulation results for r = 04 are shown in Fig 2

VI CONCLUSIONS Time-delay systems appear in many practical problems

The sliding mode on a specified surface is achieved if the state converges to the siirface Two kinds of sliding surface can be designed (i) when the sliding surface is independent of the delays or (ii) the sliding surface depends on the delays In the second case the delays should be constant otherwise the sliding surface is not a simple hyperplane In this paper the stability of the sliding mode control of a system with a delay on the state has been considered

REFERENCES 111 L V Ahlfors Complez Analysis McGraw-Hill Inc New York

1979 [2] SBusenberg a n d K Cooke Vertically Transmitted Diseases

Models and Dynamics Springer-Verlag 1992 [3] J Chen a n d H A La tchman ldquoFrequency sweeping tests for

stability independent of delayrdquo IEEE Trans Automat Contr

J Chen D Xu a n d B Shafai ldquoOn sufficient conditions for stb- bility independent of delayrdquo IEEE Trans Automat Contr 40

R F Cur ta in a n d H J Zwart A n Introdrrction to Inlinrte- Dimensional Linear Sys tems Theory Springer-Verlag Berlin 1995 C M Dorling a n d A S I Zinober ldquoTwo approaches t o hyperplane design in multivariable variable s t ruc ture control systemsrdquo Int J Control 44 65 - 82 1986 J Hale Theory of FTrnctional Differential Eqtrations Springer- Verlag New York 1977 A Hmamed A Benzaouiaand H Bensalah ldquoRegulator problem for linear continuous-time delay systems with nonsymmetrical

40 1640-1645 1995 141

1675-1 6780 1995 [5]

[6]

(71

[8]

-1 10

time External disturbance

- 0

-0 5 1-l 0 5 10

time

Control input 2

-0 a 5

0 005 0 1 015

Fig 1 Responses of Example 1

Stale Variables 0 5

0

X l

Control input

2-

h e Sliding function

O 5-

I ldquo 1 5 10

time Extemal disturbance

- 0

lime

-2 - 0 5 10

lime Equivalent mntrol

U 10

time Phase plane

-20

lsquopi 0 5

1 0 0 1 0 2 0 3

Xl

Fig 2 Responses of Examplr 2

- 101 -

A Sliding Control where

Ail Aiz A11 A 1 2 Consider the control

u ( t ) = -(CB)- ( C A x ( t ) + CAoz(t - T ) + p sgn s ( t ) ) (4) TATT = [ A21 A22 ] = [ A Aa2 ] where p = p ( t z ( t ) T) = diag(p1 pl p m ) with positive real functions p = p ( t t ( t ) r ) (1 lt i lt m) and sgn s = [ sgn s1 sgn s ~ sgn s 1 with

1 if si gt 0 s g n s = 0 i f s i = O l s i s m -1 if s lt 0

Then bhe system is now given by

i ( t ) = Ax(t) + Ax(t - T) + f ( t l 2 T) -

B ( c ~ ) - p ( t ~ ( t ) T ) sgn s ( 5 )

( 6 )

Hence the sliding dynamics is governed by

S = C f ( t ~ ( t ) T ) - p( t x ( t ) T) sgn s

Subsystem (12) is the system in the sliding mode So the sliding surface is Clz l ( t ) + Czzz(t) = 0 where CTT = [Cl Cz] In the sliding mode q ( t ) = - A z l ( t ) with K = C2-C1 Therefore the reduced order system (12) is converted to

i l ( t ) = (Al l - A 1 2 K ) ~ l ( t ) + ( A i - A l z K ) z l ( t - T) (14)

For any x(0) and any function 4 E C([-TO]R) there exists a function z1 ( t ) satisfying the differential equation (14) almost everywhere [5] In this case for all integers k the function z l ( t ) will be Ck on ((k - l ) ~ CO) ie for all 1 5 j 5 IC the j- th derivative of function z l ( t ) is continu- ously differentiable on ( ( j - l ) ~ CO) With these conditions the solution of (1) is

and for all 0 lt t lt t zl(t) = e ( A i i - A i z K ) t $ l ( ~ ) + e ( A i i - A i a K ) ( t - w )

f ( t x ( t ) T ) d t - p ( t t ( t ) ~ ) ( t - t l ) sgn s (Ai1 - A i amp ) ~ i ( w - T) dw t 2 0 (15)

The reaching sliding mode condition is

S i sgn s lt 0 V i 15 i I m ( 7 ) There exist positive real numbers Q and M such that iizl(t)ii I (Miiw)ii + M2ii$1iir) e ( M 2 + q ) t [51 in the neighbourhood Of = [I2] the We wish to find such that A l l - A12Jlt is stable Con- sider the Riccati equation i-th (1 5 i 5 m) row of (6) by sgn si gives

A l l P + PAllT - P A ~ Z R - ~ A T ~ P = -Q (16)

where Q and R are arbitrary semi-pds and pds matrices receptively I t has a upds matrix solution P For K = R-AT2P the matrix All - AlaK is stable

A System stability without delay information

For this case system (1) is asymptotically stable if all the roots of the characteristic polyqomialof the system (12) p ( s ) = det (s1- A11 +AIamp - (A11 - A1K)er) have negative real parts If all the roots of p ( s ) = 0 lie in the open ~e f t -ha~fcomp~ex plane this condition is equivalent to p ( s ) E + where c+ is the set of all complex numbers with nonnegative real gt 0 there may be infinitely many solutions while for T = 0 there are finitely many solutions So when T gt 0 it is very difficult to find all the infinity roots of p ( s ) = 0 and check if the roots are in the open left-half plane This motivates one to avoid the delay-dependent condition p ( s ) = 0 for stability of the system and use the condition p ( s ) 0 for all s E + to prove the asymptotically stability independent of delay However since p ( s ) is an entire function there are only a finite number of roots of p ( s ) = 0 in any compact set in particular in a vertical strip of the complex plane Furthermore there exists a real number CY such that all the

The function p ( s ) is an analytic function on There- fore according to the Maximum Principle Theorem [l] p ( s )

b sgn si = C i f ( t x ( t ) T ) sgn si - p i ( t z ( t ) T ) (8)

Hence a sufficient condition for the existence of the sliding mode is

Cif ( t x(t) l r)sgn si lt p i ( t ~ ( t ) 7)

and a suficieiit condition is that

IlCf(t z ( t ) ) I 1 lt min pt

111 SYSTEM I N THE S L I D I N G M O D E

(9) l lt lt

The behaviour of the system in the sliding mode is considered in this section The system in the sliding mode is a subsystem of (1) of order n-m Assume T i s an orthogonal riiatrix such that

(10)

for all Note that for

T B = [ i2 ] where Bz is a nonsingular matrix Let z = Tx then

(t ) = TAT^ (t) + T A ~ T ~ ~ ( ~ - T ) + T B u ( ~ ) + T B ~ ( ~ t ( t ) T)

22EIIP NOW ~ZSSUITI~ tT = [ Z I z2] Z I E R- Then

i1( t ) = ~ ~ ~ ~ ~ ( t ) + ~ ~ ~ ~ ~ ( t ) + A l l z l ( t -

(I2) roots of p ( s ) lie to the left of the vertical line z = (Y [7] i 2 ( t ) = A L l Z l ( t ) + A22zz(t) + A 2 1 2 1 (t - T ) + A22z2(t - T )

( 1 3 )

T

+ A l n z z ( t -

+ B2u + Bzg(r( t ) T t )

-98-










equation









p ( ( i d - A ) - A o ) 1 vu 2 0 (19)


- 9 9 -






then




(Aeq - E I ) ~ P + P(A - euro1) = -Q (25 )

and





[ 24142 00343 ] z ( t - T) - sgn s ( t )





- 100 -








1 -1

g


O A


State variables

u ( t ) = [ 24142 -04142 ] c ( t ) -t 0 5 10 [ 0 -05172 ] z ( t - 7) - sgn s ( t ) t

Sliding function











40 1640-1645 1995 141

1675-1 6780 1995 [5]

[6]

(71

[8]

-1 10


- 0

-0 5 1-l 0 5 10

time

Control input 2

-0 a 5

0 005 0 1 015


Stale Variables 0 5

0

X l

Control input

2-


O 5-

I ldquo 1 5 10


- 0

lime

-2 - 0 5 10


U 10

time Phase plane

-20

lsquopi 0 5

1 0 0 1 0 2 0 3

Xl


- 101 -










equation









p ( ( i d - A ) - A o ) 1 vu 2 0 (19)


- 9 9 -






then




(Aeq - E I ) ~ P + P(A - euro1) = -Q (25 )

and





[ 24142 00343 ] z ( t - T) - sgn s ( t )





- 100 -








1 -1

g


O A


State variables

u ( t ) = [ 24142 -04142 ] c ( t ) -t 0 5 10 [ 0 -05172 ] z ( t - 7) - sgn s ( t ) t

Sliding function











40 1640-1645 1995 141

1675-1 6780 1995 [5]

[6]

(71

[8]

-1 10


- 0

-0 5 1-l 0 5 10

time

Control input 2

-0 a 5

0 005 0 1 015


Stale Variables 0 5

0

X l

Control input

2-


O 5-

I ldquo 1 5 10


- 0

lime

-2 - 0 5 10


U 10

time Phase plane

-20

lsquopi 0 5

1 0 0 1 0 2 0 3

Xl


- 101 -






then




(Aeq - E I ) ~ P + P(A - euro1) = -Q (25 )

and





[ 24142 00343 ] z ( t - T) - sgn s ( t )





- 100 -








1 -1

g


O A


State variables

u ( t ) = [ 24142 -04142 ] c ( t ) -t 0 5 10 [ 0 -05172 ] z ( t - 7) - sgn s ( t ) t

Sliding function











40 1640-1645 1995 141

1675-1 6780 1995 [5]

[6]

(71

[8]

-1 10


- 0

-0 5 1-l 0 5 10

time

Control input 2

-0 a 5

0 005 0 1 015


Stale Variables 0 5

0

X l

Control input

2-


O 5-

I ldquo 1 5 10


- 0

lime

-2 - 0 5 10


U 10

time Phase plane

-20

lsquopi 0 5

1 0 0 1 0 2 0 3

Xl


- 101 -








1 -1

g


O A


State variables

u ( t ) = [ 24142 -04142 ] c ( t ) -t 0 5 10 [ 0 -05172 ] z ( t - 7) - sgn s ( t ) t

Sliding function











40 1640-1645 1995 141

1675-1 6780 1995 [5]

[6]

(71

[8]

-1 10


- 0

-0 5 1-l 0 5 10

time

Control input 2

-0 a 5

0 005 0 1 015


Stale Variables 0 5

0

X l

Control input

2-


O 5-

I ldquo 1 5 10


- 0

lime

-2 - 0 5 10


U 10

time Phase plane

-20

lsquopi 0 5

1 0 0 1 0 2 0 3

Xl


- 101 -

[IEEE Proceedings. 1996 IEEE International Workshop on Variable Structure Systems. - VSS'96 - -...

Documents

Transcript of [IEEE Proceedings. 1996 IEEE International Workshop on Variable Structure Systems. - VSS'96 - -...