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Discrete Variable Structure Integral Controllers
Copyrighl © 19% IFAC 1.1th Triennial Wurld Congre<;.<;. San Frandsco. IJSA
2d-22 1
DISCR ETE VARIA BLE STRU CTUR E INTEG RAL CONT ROLL ERS Claudi o Bonive nto, Mario Sandri , Rober to Zanasi
DE/S, University of Bo/ogna , V.le Risorgimento 2, 40136 Bologna, Italy Fax: +3951 6443073; Phone: +39516 443034 ; Email: [email protected] nioo.it;
Abstra ct. The paper deals with the analysis and the design of a class of robust discrete- time Variable Structu re controllers of integral type. We refer to the Discontinuous Integra l Control (DIC) as the basic techniq ue for reducin g the chatter ing of the system . A modified discretized version of the DIe a.lgorithm is introdu ced and discussed. To co pe with the case of systems having slow time-va rying parame ters, a new adaptIv e cuntrol structu re is proposed and analyzed. The asympt otic stabilit y of the control led system is proved in the case of constan t externa l disturb ances. FInally , some simulat ion f'ef,ults on the velocity control of a. DC motor end the paper. Keywo rds. Robus t control , Sliding-mode control , Integral Control , Discret e-time systems , Adaptiv e algorith ms.
1. INTRO DUCTI ON
While Variabl e-Struc ture Control (VSC) was originally introdu ced in the continu ous-tim e case (Utlein, 1977), a deeper investigation of the discrete case is necessary for a satisfac tory implem entatio n in comput er-cont rolled systems. In this paper we introd uce a discrete -time sliding mode algorith m. The d efinition of t.he discrete -time sliding mode is not uniform in t he literatu re, see for exampl e (Draku nov and Utkin , 1992) and (Yu , 1993). In this paper the expression ... discrete slid ing mode" wi1l be used for referrin g to system trajec tories forced to stay in a. proper neig hborho od of a given sliding manifold. Th. Discont in1l0us Integra l Contro l (DIC) techniq ue, int.roduced in (Nersisian and Zanasi , 1993) and (Zanasi , 1993) and (urther developed in (Buuivent.o et 01., 1994) and (Bonivento and Zanasi , 1994), uses a nonline ar integral action for estimat ing t he externa l disturba.nce and keepin g small the discont inuous term in the control law and the subsequ ent chatter ing effect. The original DIe design was given for the continu ous-tim e case . In this paper, t.he discrctized version of the Dye algorith m is prop-
erly modified in order to taKe into accoun t the uslope" of the control led variable. The so obtaine d discrete -time controller show8 to be very effective in the disturb ance estimat ion and in the chatter ing reduction especially when the system parame ters are known and consta.nt. If this is not the case, a new "adapti ve" control structure which also estimat es the "inertia" of the system is then propooed . This new control .tructu re is analyzed and its asympt otic stabilit y is proved in th e case of constant disturb ances. The paper is organiz ed as follows . In Section 2 a brief review of the DIC technique is present ed togethe r with a first modified discrete version of the DIC control algorithm . In Section 3, t he basic propert ies of this new discrete- time co ntroller arc summa rized. In Section 4, an adaptive version of the proposed control law is introdu ced and anaJyzed. Finally, in Section 5, some simulation results on the velocity control of a DC motor confirm the effectiveness of the proposed control structure.
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2. DISCR ETE VS CONTR OLLER S Let us consider the following uncerta in system
y = .p(t)
+ u(t)
(1 )
where yE R is the state, 1.1 E R the input, and .p(I) an externa l unknow n disturb ance . A large class of MIMO uDcertain systems can be reduced to the parallel of first order systems like (1) by using a proper .tate space transfo rmation (Zanasi, 1993). In the literatu re it is wellknown that when the disturb ance i. bounde d 0.p(1)I < 110) , the slidiog mode conditi on y = 0 can be easily reached and mainta ined by using the simple discontinuous control law 1.1(1) -k'sgny (l) with k' > a o. The conditio n y 0 is reached "exactly" only in the continu ous-tim e case when the switching frequen cy is infinite. On the cont.rary, in the. discrete case when the control law 1.1(1) is actuate d only at the samplin g instants t = n T. (T, is the samplin g period) only the o-boundedness of the output can be obtaine d: Ivl < o. Due to the presence of the switching term k'sgnv( t) the output y is affected by an oscillation whose amplitu de is k' T,. Since k' > .il. o, when a large dist urbance .p(I) acts on the system , the amplitu de of the output oscillation is also large: the system is affected by a large amount of chattering. To redu ce this chatt,ering, in (Nersisian and Zanasi, 1993) and (Zanasi , 1993) a cont.rol structu re was proposed which allows the reduction of k' by introducing in the control law a non linear integral term ~(t) which estimat es the external disturb ance .p(t);
=
=
~(t) { .p(t)
= -ksgny (t) = h.gny( t)
~ (t)
In the first equatio n in (3) , a mixed notatio n has been used: 1i(1) and .p(I) are continu ous-tim e variables, while y(n) and ~(n) are discrete variables which change values only at. samplin g instants t = n T,. An importa nt difference of the discrete control law (3L with respect to the continu ous one, is that estimat ion t/;( n) is now "discontinuous" , that is, at each samplin g time, estimat ion ,j;(n) varies up or down of the quantit y hT•. When the disturb ance is zero (or constan t), the estimat ion ";;(n) switches between the two values ~ and ~ + hT, and produce s on the control led system an effect similar to the one produce d by the switchi ng term k 'goy, that is, an output oscillation, If we want to track a disturb ance ,p with a large derivative (11, > 1~1l, parame ter h must be larger than .il. , (h > .il. , ) , and so the oscillation h T. generat ed by the estimat or will be large loo. This oocillation is the main drawba ck of the discrete control law (3) with respect to the continu ous one. The oscillation generated by the estimat or can be reduced by properly modifying the discrete control law (3), see (Bonivento et al., 1995). By integra ting t.he first equatio n in systcm (3) in the t ime interval [n T" (n + 1) T,], we obtain ("+1) T.
y(n+l) =y(n) -kT.sg ny(n)+ ! .p(I)dl -,j;(n)T . (4) " T.
If we denote with ,,!i tn+ 1) the mean value of the externa l disturb ance .p(I) in the time interval [n T., (n + 1) T.]
T,
(2)
=
(3)
.p(t) dt
"T.
equatio n (4) can be written as: .il.y(n+ l)
= =
=
!
1
.p(n+ 1) =-
In (2) k << k'. When the control led system is in the sliding mode (li y 0 H k sgny = .p - ~), the switching term k sgny is equal , OD average , to the estimation error e(t) = 1/' (0 - ~. By integra ting the term hsgny, in each instant ,p moves in the direction of reducing the error e(I) , that is, it estimat es the disturb ance .p(t). In (Nersisian and Zanasi, 1993) ""d (Zanasi , 1993) it was shown ,that, if the derivat ive of the disturb ance is bounded (I~I < .il.d, with a proper choi ce of parame ter h (h > .il.d cont.rol structu re (2) ensures the sliding mode on y 0 even if parame ter k is arbitrar ily small (k > 0). In the discrete case, th e reductio n of parame ter k implies a smaller amplitu de of the output oscilJation, that is, a reduction of the chatteri ng. When the discrete version of control law (2) is considered, the dynami c equatio ns of the controlled system bt>come the following:
~(t) = '1'(t) - ksgny( n) - ~(n) { .p(n) .p(n - 1) + hT, sgny(n)
(n+l)T.
-
T
,
+ ksgny( n) =
--
.p(n + 1) - .p(n)
(5)
=
where .il.y(n + 1) y(n + 1) - y(n). The term on the left-han d side of equatio n (5) has the physical meaning of "mean estimat ion error~' in the samplin g interval [n T" (n + 1) T.] and therefore seems to be a suitable function to use in the discrete disturb ance estimat or in the place of functio n sguy. With this modification the controlled system (3) becomes
= .p(I) - ksgllv(n) - ";;(n) { ";; (n) = ~ (n _ I)+k, [.il.~n) +ksgn y(n-l» ) yet)
(6)
where k, is a proper design parame ter. Term .il.y(n)/ T. is the discrete derivative of output y, while the term e(n) = l1y(n)/ T. + ksgny( n - I) is the disturb ance estimatio n error. When e(n) is zero, the estimat or ~(n) is constan t and therefore no additio nal osci1lations are introdu ced in the system .
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=
3. BASIC PROPE RTIES OF THE PROPO SED CONTR OL LAW At the samplin g instants , Hystem (6) becomes
y(n+ I)
{
~(n)
=y(n)-T , [ksgny(n)+~(n+ I)-~(n)~
= ~(n-I)+k. [~~~n) +ksgn y(n-I) j
~y(n) + ksgny( n T,
I)
I)
+ I) = y(n) -
+ e(n + I) T,
(10)
that is, the output dynamics is completely determined by the estimat ion error e(n). From (10) it is clear that a discrete sliding mode on surface y 0 is possible only if the amplitude k of the switching term is greater than t.he amplitu de of disturb ance:
I
k > Je(n)lm ..
(11)
When (11) holds , the discrete sliding mode is reached and the output y i. forced to stay within the region Ivl < (k + lelma.)T,. By eliminating ~(n) from system (9) , we obtain the equatio n
where il ~ (n + I)
= ~ (n + 1) e(z)
~(z) = z
= il~(n + 1)
(12)
~(n)-~(n-I)I T,
<
il
~
the estimat ion error '(00) is also bounde d: e(oo) = ~~T,. When the samplin g time T, ---+ 0 , the estimat ion error e(n) tends to zero, and the sliding mode conditi on (11) is satisfied for an arbitrarily small parameter k. The proposed control structure is:
~(n) .p(n)
I
=
er n + I) + (k, - I) ern)
=
(9)
+ k, e(n)
k1; sgny(n)
= A(w) sin(wt+
d) Ifthe discrete derivati ve ofth. disturbance is bounded
System (9) is now linear with respect to the new state variables e(n) and ~(n), and ~(n) i. the new system input. From (8) we obtain:
y(n
' (00)
=
(8)
= ~(n + I) - ~(n)
= .p(n -
~
where A(w) 2(\ - cos wT.)/( wT,). Note that when T, ~ 0 the estimat ion error tends to zero (liffiwT.-+o~ A(w) 0), that is the estimat ion error is small when the samplin g frequency is much higher than the disturbance frequency: w, » w.
IN
System (7) can be rewritte n as follows eJn + I) { "'(n)
.p(t) = sinwl
(7)
Let e(n) denote the "disturbance estimat ion error":
.(n) =
=
ramp: .,&(t) rot ~ '(00) roT,. The controlled system (7) reaches the discrete sliding mode y = 0 only if k > IroIT,. c) If disturb ance .p(t) is a sinusoi d, the estimat ion error e(oo) is a sinusoid :
e(o)
= -:k sgny(n) + ~(n) = .p(n -
1) + e(o)
= ~~n) +ksgn y(n-I)
(14)
This structure is very simple, easily implementable and very effective in rejecting external disturbances with bounded derivative. More or less it has the same control capabiliti es of the DlC controller (3), but with the following two advanta ges: I) a reduced chatter ing especially when the disturb ance is cODstant; and 2) the amplitu de k(n) of the switching action is adaptive with respect to the · slope" of t he disturb ance.
.,b(n), that i.
z-I
+ k, -
(13)
1
System (12) is stable if and only if the pole of the Z transfer function (13) is inside the unit circle, that is if Ipl 11- k,1< 1 H 0 < k, < 2. [t is evident that the best choice for parameter ke is ke = 1 because in this case the dynamics of system (12) is of deadbe at type . In the following 1 we will consider only the case ke 1. Let e(oo) denote the steady-state value of the disturb ance estimat ion error . In (Bonive nto et al., 1995) it has been shown that for system (12) : a) If disturb ance .p(t) is constan t, the disturb ance estimation error e(oo) is zero .p(!) c ~ '(00) O. In this case, ,p(t) exactly estimat es the disturb ance. b) If disturb ance W(t) is 0 romp , th e estimat ion error c( 00) is constan t and proport ional to the slope of the
=
=
=
=
4. ESTIMATION OF THE SYSTE M "INERTIA" The control structures considered in the previous sections refer to the simple controlled system (1), that is an integrator. A Ia.rge category of systems can be re~ duced to t.his simple structure if their parameters are CODstant and known, see (Zanasi, 1993). To cope with the case of system s with time-varying parameters, let us now consider the followjng system
lil = .p(t) + u(t)
(15)
where J is a parameter (here caUed '-linertia") which is 8upposed to be unknown, and constan t or slowly time-. varying . Let Je (n) denote the "estimation" of the inertia J at the samplin g instant t nT, . The discrete controll ed system is now the following
=
3727
~(n+l) = ~(n)+; [-ksgny(n)+~(n+l)-.p(n)l
=t(n-I) +«n)
tin) -( ) en
1
=
(16)
J.(n - I) .t>y(n)
T,
J are constan t . If the positive parame ter n satisfies the inequality
k
(
+sgn yn-
J
I)
a < 2k
where e(n) is the "disturbance pseudo -estima tion error". The "real" disturb ance estimat ion error e( n) LI;j:
e(n) = J .t>y(n) + ksgny( n _ I) T.
(17)
tben the overaJJ controlled system is asympt otically stable in a neighborbood of the equilibrium point e( n) = 0 and J.(n) == J. Proof. From (18) it follows that J,(n-l J ) e(n)+k \ 1- J,(n-I J )l sgny(n -l) J.(n-2 ) J,(n-2 ) .::..:::-'.-cJ--'-e(n-I)-k 1J sgny(n -2)
.t>e(n)
The "pseud o-estim ation error" i( n) differs from real one
e(n) because it uses the inertia J, (n - I), estimat ed at the previous inslant t == (n - I )T" instead of using the real inertia J . The errors e(n) and i(n) are related by
the equatio n
e(n) - ksgny( n - I) e(n) - ksgny( n - I)
=
J.(n - I) J
(18)
When J, --I J, i (n) --I e(n). From (16) and (17) one obtains the dynami cs of the estimat ion error e(n) : e(7I+I) +
[J,(~- I)
I] e(n)
Note that when J, = J the equation (19) reduces to the equation (12). The linear part of system (19) is stable only if
1J,(~-I) -11 < I
By using the sliding mode condition sgny( n-2) sgny( n1) == -1, the updatin g funct,ion (21) transforms as follows J,(n) = J,(n-I ) +tle(n ) sgny(n -I) =J.(n -l)+2k J,(n-I) 2+;, (n-2)] +
[1
-> J,(n-I )
< 2J
If we introduce the "inertia estimat ion error'" J~ (n) J,(n) - J as new state space variable substitu ting J,(n), equations (19) and (23) are transform ed as follows
=
J,(n):
Inequalities (20) clearly show that when J is not known
2
(21)
where .t>e(n) = «n)-i( o-I), .!>sgny (n-l) = sgny(n 1)- sgny(n -2) and <> is a proper design parame ter. In (21) the term [.t>sgny(n- I)]/2 has the function to allow the updatin g of inertia J,(n) only when the controlled system is in the neighborhood of the
~'f(~: ~:(~:~I[J:;:)~;::(~~~~2)1
- a 1+
J.(n-2) J
(24)
e(n-I) sgny(n -I)
Let us cont.~ider the case when the externa l disturb ance
tIt) is constant: tl~(n+l) == O. If we introduce the new
state space variables
e(n) J,(n) x\(n) =-J ' x,(n) =--, x3(n)
system, let us consider t,he followin g updatin g function
= J,(n _ ll+atl e(n)tls gny(n -I)
J
k
it's better to "undere stimate" the initiaJ va)ue of J to e
J,(n)
J.(n-I )
e(n)+.t >t(n+l ) J.(n-I ) +k J sgny(n -l)
.(n+l) =-
(20)
ensure the st.ability of the controlled system . Due to the presence of the last term in the right-hand side of eq. (19), the estimat ion error e(n) is not null even when the disturbance .p(n + I) is constan t (.t>~(n + I) == 0). To ensure in (16) both the asympt oti,' estimation of the real value J and the global stabilit.y of the controlled
(23)
J,(n-2 ) ] + [ J . (n-I) J e(n)J e(n-l) sgny(n -I)
=
-I)] (19) = .t>t(n+ I)- k [J,(n 1.T sgny(n -I)
(22)
.J
and the auxiliary constan t {3 comes
J,(n-I ) J
= k/J , system (24) be-
x\(o+l ) = %3 [Bsgny (n-I)-% '] %,(n+l ) = (I - {3a)z,- {3ax3 +<>{(I +x,)z3 [{3sgn y(n-I) -xd - (I + X3)Z.) .gny(n ) x3(n+l )
1
(25)
=z,
For the sake of brevity, in the right-h and side of equations (25) the state variables %t(n) , %,(n) and x3(n)
have been denoted as
Xl, %2
and
X3
respectively. System
(25) can also be represented in the compact form ,,( n+
3728
.....
1) = F(z(n) , ni, where x(n) = [z,(n) r,(n) z3(n)JT and F(x(n), n) is a proper non lineM function of the state r(n). One can directly verify that z, = 0, z, = 0 and X3 = 0 is an equilibr ium point for system (25). Equivalently e(n) = 0 and J,(n) = 0 i. an equilibrium point for system (24). The stabilit y of the discrete nonlinear system (25) in a neighborhood of the "zero" equilibrium point can be proved by using the first Lyapunov theo-
",
3r-----~----~~=-------------, 2
of--- 1---- -----
rem. The Ja.cobian matrix of the system is A(r)
.,
.....
= of(x(n ), n) ox(n) ~~------~~------.~,------~O------~2
all :::.....z3
a'3 = psgny (n-I)- x, a" =-0- [{I + X,)X3 + (1 + X3)J sgny(n) = (1 - pa) + a"'3 [(Jsgn y(n-I) -rd agny(n) = +a(1 + "") [(Jsgny (n-I) - "',J sgny(n) -{3a-o" " sgny(n)
0" a"
The characteristic equatio n Iz /- A(O)I A(x) in the equilibrium point x=O is
= 0 of matrix
Fig. 1. The global behavior of the root locus when per
o
>
equilibrium point ern) = 'Po, J,(n) = 0 is asymptotically stable is ". is "sufficiently small". When inertia J i. slOWly time-varying, system (24) does not guarantee the asymptotic estimation of J(t), but still it shows good Utracking" performances.
,3 +(P" -1)z'
-po [sgny(n - l) sgny(n )-IJ z+ +p" sgny(n -l) sgny{n) = 0
If we impose the SM condition sgny(n -l)sgny (n) =-1 we obtain the time-constant equation
P(z)
=,3 + (P" _ l)z' + 2paz -
pa
=0
(26)
5. SIMULATION RESULTS Let us consider to the block scheme of Fig. 2 repr_n t-
ing the velocity control of a DC motor where J is the
The roots of equation (26) are functions of the produc t per. By using the contour locus method, equation (26) can be transfonned. as follows
1+ pa(,
+ I - v'2)(z + 1 + ,,12)
r--'
(I-z ')J,(n -I) e(n) ~(n) 1 T, 1- Z-1 -ksgny (n-I)
=0
I
z'(z - I) The corresponding root locus when (In varies from 0 to +00 is reported in Fig. 1. Hy using t.he Jury's test one
r(n)
can easily find th.t the system is .table for 0 < per < 0.5 , that is when 0 < a < J / (2k) . From Fig. I it is evident that for 0 < pa < 0.5 the three roots of equation (26) are inside the unit circle and so for the first
Lyapunov theorem the nonlinear discrete system (25) is asympt.otically stable. in the equilibrium point z O.
=
The stated Result is proved. When disturb ance ,p(t) and inertia J are constan t and unknown , inequality (22) is a sufficient condition for
the asympt otic .. timatio n of the real values: ;j(n) .... W and Jc(n) ~ J. Since inertia J is supposed unknown,
in the synthesis of the parameter Q we have to use the lower-bound Jmin of the inertia J , that is Q < Jm;n/{2 k). Note that Result I can also be extended: when the disturbance "/.I(tL instead of being constant,
is a ramp (t;.;j
= 'Po) , it
y(n)
.p,(t u(n)
~(n) - k agny( n) f----. Ho{s)
",(t)
"'(n) T.
1 b+Ja
r6 u{t)
I--
Fig. 2. Block scheme used in .imulat ion. inertia of the motor, b the linear friction, u(t) the input torque, W,(t) an external disturbance, r(n) the reference signal, y(n) = r(n) -",{n) the tracking error and Ho{s) the zero order hold. The aim of the controller i. to keep the tracking error y(n) as small as poesible even when
the inertia J of the motor is time-varying. ]n Fig. 3 are reported the simulation results obtained when inertia J varies as follows
can be proved that the new
J(t)
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= 0.04 + 0.01 sgn[sin(15 III
and when the disturb ance is constan t: r(n) = 10 and 1/I(t) = -7.5. In the upper part of Fig. 3 the controlled
vious simu1ation when the reference signal r(n) and the
disturb ance ",.(t) are time-varying:
11
r(n)
= IO cos(10nT,},
"'T(t)
w(n)
The obtaine d simulat ion resulta show the good performance of the proposed control structure even when the
v
disturbance is time-varying.
9
15
= -7sin(8 t)
)l/~
psUilde(n
o
0.051 Je(D)
O.Olll
o
6. CONCLUSIONS The Discontinuous Integral Control technique has been
sr
V
discretized and suitable modifications have been pro-
I
V-
(secoodl)
Fig. 3. Simula tion results when r(n) and J is time-varying.
1.2
= 10, 1/I(t) = -7.5
variabl ew(n) is reported togethe r with the reference signal r(n) , while in the middle and low parts of the figure are shown the estimat ed disturb ance .);(n) and the estimated inertia JeCn) . The used parameters are: T, = 0.005 s, k 0.8, '" 0.005, b 0.002 and J.(O) 0.001. The estimat ion .);(n) and the input u(n) have been saturated to the maxim um available torque [-15 , 15]Nm. Note that in steady- state conditions the estimat es ";;(n) and J,(n) tend asympt.otically to the real values 1/I(t) e J , despite of the fact that a small oscillation is present on th e controlled variable "'(t). In Fig. 4 are reported
=
=
=
=
11
wen)
· 11
15
posed and analyzed . The first introdu ced modification improves the discrete-time estimation of the external disturb ance "'(t) . The second modification allows also the estimation of the unknown "inertia" J(t) of the system. The asympt otic stabilit y of the proposed control structure has been proved. The obtained simulation results support the theoretical results. Extensions to higher dimensional system s is under construction. 7. REFER ENCES Utkin, V.1. (1997) . Variable Structu re System s with Sliding Modes. IEEE Trans. A ulom.I ic Control. Vol. 22. pp. 212-222. Drakunov, S.V., V.1. Utkin (1992), Sliding Mode Control in dynami c systems . Int. J. Cont. Vol. 55, no. 4. Vu, X.H. (1993). Discrete Variable Structu re Control System . 1nl. J. Systems Science. Vol. 24. no. 2. pp. 373-386 . Baida, S.V. (1993). Unit Sliding Mode Contro l in Continuous and Discrete-time Systems. 1nl. J. Control. Vol. 57. no. 5. Nersisian, A., R . Zanasi (1993). A Modified Variable Structu re Contro l Algorit hm for Stabiliz ation of Uncertain Dynam ical Systems. 1nl. J. of Robust .nd Nonlineor Contml. Vol. 3. pp. 199-209 . Fridma n, L. , A. Levant (1994) . Higher Order Sliding Modes as a Natural Phenom enon in Control The-
·15 O.M! b---------------------------------~ Je(n)
0.001
'=.C=---- - . -- - o
---~ -=-__"i' (secoodi) 1.2
Fig. 4. Simula tion resulta obtaine d when J, r(n) and "' (t) are time-varying. th e results obtained in the same conditions of the pre-
ory. IEEE VSLT'94 , Benevenlo, Italy. Zanasi, R. (1993). Sliding Mode Using Discontinuou. Control Algorit hms of Integra l Type. lnt. Journal of Conlrol. Special Issue on Sliding Mode Control. Vol. 57 . no. 5. pp. 1079-1099. Bonivento, C ., A. Nersisian , A. Tonielli, R . Zanasi (1994), A Cascade Structu re For Robust Control Design . IEEE Trons. on A ut. Conlr. Vol. 39. no. 4. Bonivento, C., R. Zanasi (1994). Discontinuous Integral Control. IEEE VSLT'94, Benevento, Italy. Bonivenlo, C., M. Sandri, R . Zanasi (1995). Discrete Low-C hattering Variable Structu re Controllers. ECC'95, Rama, Italy.
3730
2d-22 1
DISCR ETE VARIA BLE STRU CTUR E INTEG RAL CONT ROLL ERS Claudi o Bonive nto, Mario Sandri , Rober to Zanasi
DE/S, University of Bo/ogna , V.le Risorgimento 2, 40136 Bologna, Italy Fax: +3951 6443073; Phone: +39516 443034 ; Email: [email protected] nioo.it;
Abstra ct. The paper deals with the analysis and the design of a class of robust discrete- time Variable Structu re controllers of integral type. We refer to the Discontinuous Integra l Control (DIC) as the basic techniq ue for reducin g the chatter ing of the system . A modified discretized version of the DIe a.lgorithm is introdu ced and discussed. To co pe with the case of systems having slow time-va rying parame ters, a new adaptIv e cuntrol structu re is proposed and analyzed. The asympt otic stabilit y of the control led system is proved in the case of constan t externa l disturb ances. FInally , some simulat ion f'ef,ults on the velocity control of a. DC motor end the paper. Keywo rds. Robus t control , Sliding-mode control , Integral Control , Discret e-time systems , Adaptiv e algorith ms.
1. INTRO DUCTI ON
While Variabl e-Struc ture Control (VSC) was originally introdu ced in the continu ous-tim e case (Utlein, 1977), a deeper investigation of the discrete case is necessary for a satisfac tory implem entatio n in comput er-cont rolled systems. In this paper we introd uce a discrete -time sliding mode algorith m. The d efinition of t.he discrete -time sliding mode is not uniform in t he literatu re, see for exampl e (Draku nov and Utkin , 1992) and (Yu , 1993). In this paper the expression ... discrete slid ing mode" wi1l be used for referrin g to system trajec tories forced to stay in a. proper neig hborho od of a given sliding manifold. Th. Discont in1l0us Integra l Contro l (DIC) techniq ue, int.roduced in (Nersisian and Zanasi , 1993) and (Zanasi , 1993) and (urther developed in (Buuivent.o et 01., 1994) and (Bonivento and Zanasi , 1994), uses a nonline ar integral action for estimat ing t he externa l disturba.nce and keepin g small the discont inuous term in the control law and the subsequ ent chatter ing effect. The original DIe design was given for the continu ous-tim e case . In this paper, t.he discrctized version of the Dye algorith m is prop-
erly modified in order to taKe into accoun t the uslope" of the control led variable. The so obtaine d discrete -time controller show8 to be very effective in the disturb ance estimat ion and in the chatter ing reduction especially when the system parame ters are known and consta.nt. If this is not the case, a new "adapti ve" control structure which also estimat es the "inertia" of the system is then propooed . This new control .tructu re is analyzed and its asympt otic stabilit y is proved in th e case of constant disturb ances. The paper is organiz ed as follows . In Section 2 a brief review of the DIC technique is present ed togethe r with a first modified discrete version of the DIC control algorithm . In Section 3, t he basic propert ies of this new discrete- time co ntroller arc summa rized. In Section 4, an adaptive version of the proposed control law is introdu ced and anaJyzed. Finally, in Section 5, some simulation results on the velocity control of a DC motor confirm the effectiveness of the proposed control structure.
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2. DISCR ETE VS CONTR OLLER S Let us consider the following uncerta in system
y = .p(t)
+ u(t)
(1 )
where yE R is the state, 1.1 E R the input, and .p(I) an externa l unknow n disturb ance . A large class of MIMO uDcertain systems can be reduced to the parallel of first order systems like (1) by using a proper .tate space transfo rmation (Zanasi, 1993). In the literatu re it is wellknown that when the disturb ance i. bounde d 0.p(1)I < 110) , the slidiog mode conditi on y = 0 can be easily reached and mainta ined by using the simple discontinuous control law 1.1(1) -k'sgny (l) with k' > a o. The conditio n y 0 is reached "exactly" only in the continu ous-tim e case when the switching frequen cy is infinite. On the cont.rary, in the. discrete case when the control law 1.1(1) is actuate d only at the samplin g instants t = n T. (T, is the samplin g period) only the o-boundedness of the output can be obtaine d: Ivl < o. Due to the presence of the switching term k'sgnv( t) the output y is affected by an oscillation whose amplitu de is k' T,. Since k' > .il. o, when a large dist urbance .p(I) acts on the system , the amplitu de of the output oscillation is also large: the system is affected by a large amount of chattering. To redu ce this chatt,ering, in (Nersisian and Zanasi, 1993) and (Zanasi , 1993) a cont.rol structu re was proposed which allows the reduction of k' by introducing in the control law a non linear integral term ~(t) which estimat es the external disturb ance .p(t);
=
=
~(t) { .p(t)
= -ksgny (t) = h.gny( t)
~ (t)
In the first equatio n in (3) , a mixed notatio n has been used: 1i(1) and .p(I) are continu ous-tim e variables, while y(n) and ~(n) are discrete variables which change values only at. samplin g instants t = n T,. An importa nt difference of the discrete control law (3L with respect to the continu ous one, is that estimat ion t/;( n) is now "discontinuous" , that is, at each samplin g time, estimat ion ,j;(n) varies up or down of the quantit y hT•. When the disturb ance is zero (or constan t), the estimat ion ";;(n) switches between the two values ~ and ~ + hT, and produce s on the control led system an effect similar to the one produce d by the switchi ng term k 'goy, that is, an output oscillation, If we want to track a disturb ance ,p with a large derivative (11, > 1~1l, parame ter h must be larger than .il. , (h > .il. , ) , and so the oscillation h T. generat ed by the estimat or will be large loo. This oocillation is the main drawba ck of the discrete control law (3) with respect to the continu ous one. The oscillation generated by the estimat or can be reduced by properly modifying the discrete control law (3), see (Bonivento et al., 1995). By integra ting t.he first equatio n in systcm (3) in the t ime interval [n T" (n + 1) T,], we obtain ("+1) T.
y(n+l) =y(n) -kT.sg ny(n)+ ! .p(I)dl -,j;(n)T . (4) " T.
If we denote with ,,!i tn+ 1) the mean value of the externa l disturb ance .p(I) in the time interval [n T., (n + 1) T.]
T,
(2)
=
(3)
.p(t) dt
"T.
equatio n (4) can be written as: .il.y(n+ l)
= =
=
!
1
.p(n+ 1) =-
In (2) k << k'. When the control led system is in the sliding mode (li y 0 H k sgny = .p - ~), the switching term k sgny is equal , OD average , to the estimation error e(t) = 1/' (0 - ~. By integra ting the term hsgny, in each instant ,p moves in the direction of reducing the error e(I) , that is, it estimat es the disturb ance .p(t). In (Nersisian and Zanasi, 1993) ""d (Zanasi , 1993) it was shown ,that, if the derivat ive of the disturb ance is bounded (I~I < .il.d, with a proper choi ce of parame ter h (h > .il.d cont.rol structu re (2) ensures the sliding mode on y 0 even if parame ter k is arbitrar ily small (k > 0). In the discrete case, th e reductio n of parame ter k implies a smaller amplitu de of the output oscilJation, that is, a reduction of the chatteri ng. When the discrete version of control law (2) is considered, the dynami c equatio ns of the controlled system bt>come the following:
~(t) = '1'(t) - ksgny( n) - ~(n) { .p(n) .p(n - 1) + hT, sgny(n)
(n+l)T.
-
T
,
+ ksgny( n) =
--
.p(n + 1) - .p(n)
(5)
=
where .il.y(n + 1) y(n + 1) - y(n). The term on the left-han d side of equatio n (5) has the physical meaning of "mean estimat ion error~' in the samplin g interval [n T" (n + 1) T.] and therefore seems to be a suitable function to use in the discrete disturb ance estimat or in the place of functio n sguy. With this modification the controlled system (3) becomes
= .p(I) - ksgllv(n) - ";;(n) { ";; (n) = ~ (n _ I)+k, [.il.~n) +ksgn y(n-l» ) yet)
(6)
where k, is a proper design parame ter. Term .il.y(n)/ T. is the discrete derivative of output y, while the term e(n) = l1y(n)/ T. + ksgny( n - I) is the disturb ance estimatio n error. When e(n) is zero, the estimat or ~(n) is constan t and therefore no additio nal osci1lations are introdu ced in the system .
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=
3. BASIC PROPE RTIES OF THE PROPO SED CONTR OL LAW At the samplin g instants , Hystem (6) becomes
y(n+ I)
{
~(n)
=y(n)-T , [ksgny(n)+~(n+ I)-~(n)~
= ~(n-I)+k. [~~~n) +ksgn y(n-I) j
~y(n) + ksgny( n T,
I)
I)
+ I) = y(n) -
+ e(n + I) T,
(10)
that is, the output dynamics is completely determined by the estimat ion error e(n). From (10) it is clear that a discrete sliding mode on surface y 0 is possible only if the amplitude k of the switching term is greater than t.he amplitu de of disturb ance:
I
k > Je(n)lm ..
(11)
When (11) holds , the discrete sliding mode is reached and the output y i. forced to stay within the region Ivl < (k + lelma.)T,. By eliminating ~(n) from system (9) , we obtain the equatio n
where il ~ (n + I)
= ~ (n + 1) e(z)
~(z) = z
= il~(n + 1)
(12)
~(n)-~(n-I)I T,
<
il
~
the estimat ion error '(00) is also bounde d: e(oo) = ~~T,. When the samplin g time T, ---+ 0 , the estimat ion error e(n) tends to zero, and the sliding mode conditi on (11) is satisfied for an arbitrarily small parameter k. The proposed control structure is:
~(n) .p(n)
I
=
er n + I) + (k, - I) ern)
=
(9)
+ k, e(n)
k1; sgny(n)
= A(w) sin(wt+
d) Ifthe discrete derivati ve ofth. disturbance is bounded
System (9) is now linear with respect to the new state variables e(n) and ~(n), and ~(n) i. the new system input. From (8) we obtain:
y(n
' (00)
=
(8)
= ~(n + I) - ~(n)
= .p(n -
~
where A(w) 2(\ - cos wT.)/( wT,). Note that when T, ~ 0 the estimat ion error tends to zero (liffiwT.-+o~ A(w) 0), that is the estimat ion error is small when the samplin g frequency is much higher than the disturbance frequency: w, » w.
IN
System (7) can be rewritte n as follows eJn + I) { "'(n)
.p(t) = sinwl
(7)
Let e(n) denote the "disturbance estimat ion error":
.(n) =
=
ramp: .,&(t) rot ~ '(00) roT,. The controlled system (7) reaches the discrete sliding mode y = 0 only if k > IroIT,. c) If disturb ance .p(t) is a sinusoi d, the estimat ion error e(oo) is a sinusoid :
e(o)
= -:k sgny(n) + ~(n) = .p(n -
1) + e(o)
= ~~n) +ksgn y(n-I)
(14)
This structure is very simple, easily implementable and very effective in rejecting external disturbances with bounded derivative. More or less it has the same control capabiliti es of the DlC controller (3), but with the following two advanta ges: I) a reduced chatter ing especially when the disturb ance is cODstant; and 2) the amplitu de k(n) of the switching action is adaptive with respect to the · slope" of t he disturb ance.
.,b(n), that i.
z-I
+ k, -
(13)
1
System (12) is stable if and only if the pole of the Z transfer function (13) is inside the unit circle, that is if Ipl 11- k,1< 1 H 0 < k, < 2. [t is evident that the best choice for parameter ke is ke = 1 because in this case the dynamics of system (12) is of deadbe at type . In the following 1 we will consider only the case ke 1. Let e(oo) denote the steady-state value of the disturb ance estimat ion error . In (Bonive nto et al., 1995) it has been shown that for system (12) : a) If disturb ance .p(t) is constan t, the disturb ance estimation error e(oo) is zero .p(!) c ~ '(00) O. In this case, ,p(t) exactly estimat es the disturb ance. b) If disturb ance W(t) is 0 romp , th e estimat ion error c( 00) is constan t and proport ional to the slope of the
=
=
=
=
4. ESTIMATION OF THE SYSTE M "INERTIA" The control structures considered in the previous sections refer to the simple controlled system (1), that is an integrator. A Ia.rge category of systems can be re~ duced to t.his simple structure if their parameters are CODstant and known, see (Zanasi, 1993). To cope with the case of system s with time-varying parameters, let us now consider the followjng system
lil = .p(t) + u(t)
(15)
where J is a parameter (here caUed '-linertia") which is 8upposed to be unknown, and constan t or slowly time-. varying . Let Je (n) denote the "estimation" of the inertia J at the samplin g instant t nT, . The discrete controll ed system is now the following
=
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~(n+l) = ~(n)+; [-ksgny(n)+~(n+l)-.p(n)l
=t(n-I) +«n)
tin) -( ) en
1
=
(16)
J.(n - I) .t>y(n)
T,
J are constan t . If the positive parame ter n satisfies the inequality
k
(
+sgn yn-
J
I)
a < 2k
where e(n) is the "disturbance pseudo -estima tion error". The "real" disturb ance estimat ion error e( n) LI;j:
e(n) = J .t>y(n) + ksgny( n _ I) T.
(17)
tben the overaJJ controlled system is asympt otically stable in a neighborbood of the equilibrium point e( n) = 0 and J.(n) == J. Proof. From (18) it follows that J,(n-l J ) e(n)+k \ 1- J,(n-I J )l sgny(n -l) J.(n-2 ) J,(n-2 ) .::..:::-'.-cJ--'-e(n-I)-k 1J sgny(n -2)
.t>e(n)
The "pseud o-estim ation error" i( n) differs from real one
e(n) because it uses the inertia J, (n - I), estimat ed at the previous inslant t == (n - I )T" instead of using the real inertia J . The errors e(n) and i(n) are related by
the equatio n
e(n) - ksgny( n - I) e(n) - ksgny( n - I)
=
J.(n - I) J
(18)
When J, --I J, i (n) --I e(n). From (16) and (17) one obtains the dynami cs of the estimat ion error e(n) : e(7I+I) +
[J,(~- I)
I] e(n)
Note that when J, = J the equation (19) reduces to the equation (12). The linear part of system (19) is stable only if
1J,(~-I) -11 < I
By using the sliding mode condition sgny( n-2) sgny( n1) == -1, the updatin g funct,ion (21) transforms as follows J,(n) = J,(n-I ) +tle(n ) sgny(n -I) =J.(n -l)+2k J,(n-I) 2+;, (n-2)] +
[1
-> J,(n-I )
< 2J
If we introduce the "inertia estimat ion error'" J~ (n) J,(n) - J as new state space variable substitu ting J,(n), equations (19) and (23) are transform ed as follows
=
J,(n):
Inequalities (20) clearly show that when J is not known
2
(21)
where .t>e(n) = «n)-i( o-I), .!>sgny (n-l) = sgny(n 1)- sgny(n -2) and <> is a proper design parame ter. In (21) the term [.t>sgny(n- I)]/2 has the function to allow the updatin g of inertia J,(n) only when the controlled system is in the neighborhood of the
~'f(~: ~:(~:~I[J:;:)~;::(~~~~2)1
- a 1+
J.(n-2) J
(24)
e(n-I) sgny(n -I)
Let us cont.~ider the case when the externa l disturb ance
tIt) is constant: tl~(n+l) == O. If we introduce the new
state space variables
e(n) J,(n) x\(n) =-J ' x,(n) =--, x3(n)
system, let us consider t,he followin g updatin g function
= J,(n _ ll+atl e(n)tls gny(n -I)
J
k
it's better to "undere stimate" the initiaJ va)ue of J to e
J,(n)
J.(n-I )
e(n)+.t >t(n+l ) J.(n-I ) +k J sgny(n -l)
.(n+l) =-
(20)
ensure the st.ability of the controlled system . Due to the presence of the last term in the right-hand side of eq. (19), the estimat ion error e(n) is not null even when the disturbance .p(n + I) is constan t (.t>~(n + I) == 0). To ensure in (16) both the asympt oti,' estimation of the real value J and the global stabilit.y of the controlled
(23)
J,(n-2 ) ] + [ J . (n-I) J e(n)J e(n-l) sgny(n -I)
=
-I)] (19) = .t>t(n+ I)- k [J,(n 1.T sgny(n -I)
(22)
.J
and the auxiliary constan t {3 comes
J,(n-I ) J
= k/J , system (24) be-
x\(o+l ) = %3 [Bsgny (n-I)-% '] %,(n+l ) = (I - {3a)z,- {3ax3 +<>{(I +x,)z3 [{3sgn y(n-I) -xd - (I + X3)Z.) .gny(n ) x3(n+l )
1
(25)
=z,
For the sake of brevity, in the right-h and side of equations (25) the state variables %t(n) , %,(n) and x3(n)
have been denoted as
Xl, %2
and
X3
respectively. System
(25) can also be represented in the compact form ,,( n+
3728
.....
1) = F(z(n) , ni, where x(n) = [z,(n) r,(n) z3(n)JT and F(x(n), n) is a proper non lineM function of the state r(n). One can directly verify that z, = 0, z, = 0 and X3 = 0 is an equilibr ium point for system (25). Equivalently e(n) = 0 and J,(n) = 0 i. an equilibrium point for system (24). The stabilit y of the discrete nonlinear system (25) in a neighborhood of the "zero" equilibrium point can be proved by using the first Lyapunov theo-
",
3r-----~----~~=-------------, 2
of--- 1---- -----
rem. The Ja.cobian matrix of the system is A(r)
.,
.....
= of(x(n ), n) ox(n) ~~------~~------.~,------~O------~2
all :::.....z3
a'3 = psgny (n-I)- x, a" =-0- [{I + X,)X3 + (1 + X3)J sgny(n) = (1 - pa) + a"'3 [(Jsgn y(n-I) -rd agny(n) = +a(1 + "") [(Jsgny (n-I) - "',J sgny(n) -{3a-o" " sgny(n)
0" a"
The characteristic equatio n Iz /- A(O)I A(x) in the equilibrium point x=O is
= 0 of matrix
Fig. 1. The global behavior of the root locus when per
o
>
equilibrium point ern) = 'Po, J,(n) = 0 is asymptotically stable is ". is "sufficiently small". When inertia J i. slOWly time-varying, system (24) does not guarantee the asymptotic estimation of J(t), but still it shows good Utracking" performances.
,3 +(P" -1)z'
-po [sgny(n - l) sgny(n )-IJ z+ +p" sgny(n -l) sgny{n) = 0
If we impose the SM condition sgny(n -l)sgny (n) =-1 we obtain the time-constant equation
P(z)
=,3 + (P" _ l)z' + 2paz -
pa
=0
(26)
5. SIMULATION RESULTS Let us consider to the block scheme of Fig. 2 repr_n t-
ing the velocity control of a DC motor where J is the
The roots of equation (26) are functions of the produc t per. By using the contour locus method, equation (26) can be transfonned. as follows
1+ pa(,
+ I - v'2)(z + 1 + ,,12)
r--'
(I-z ')J,(n -I) e(n) ~(n) 1 T, 1- Z-1 -ksgny (n-I)
=0
I
z'(z - I) The corresponding root locus when (In varies from 0 to +00 is reported in Fig. 1. Hy using t.he Jury's test one
r(n)
can easily find th.t the system is .table for 0 < per < 0.5 , that is when 0 < a < J / (2k) . From Fig. I it is evident that for 0 < pa < 0.5 the three roots of equation (26) are inside the unit circle and so for the first
Lyapunov theorem the nonlinear discrete system (25) is asympt.otically stable. in the equilibrium point z O.
=
The stated Result is proved. When disturb ance ,p(t) and inertia J are constan t and unknown , inequality (22) is a sufficient condition for
the asympt otic .. timatio n of the real values: ;j(n) .... W and Jc(n) ~ J. Since inertia J is supposed unknown,
in the synthesis of the parameter Q we have to use the lower-bound Jmin of the inertia J , that is Q < Jm;n/{2 k). Note that Result I can also be extended: when the disturbance "/.I(tL instead of being constant,
is a ramp (t;.;j
= 'Po) , it
y(n)
.p,(t u(n)
~(n) - k agny( n) f----. Ho{s)
",(t)
"'(n) T.
1 b+Ja
r6 u{t)
I--
Fig. 2. Block scheme used in .imulat ion. inertia of the motor, b the linear friction, u(t) the input torque, W,(t) an external disturbance, r(n) the reference signal, y(n) = r(n) -",{n) the tracking error and Ho{s) the zero order hold. The aim of the controller i. to keep the tracking error y(n) as small as poesible even when
the inertia J of the motor is time-varying. ]n Fig. 3 are reported the simulation results obtained when inertia J varies as follows
can be proved that the new
J(t)
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= 0.04 + 0.01 sgn[sin(15 III
and when the disturb ance is constan t: r(n) = 10 and 1/I(t) = -7.5. In the upper part of Fig. 3 the controlled
vious simu1ation when the reference signal r(n) and the
disturb ance ",.(t) are time-varying:
11
r(n)
= IO cos(10nT,},
"'T(t)
w(n)
The obtaine d simulat ion resulta show the good performance of the proposed control structure even when the
v
disturbance is time-varying.
9
15
= -7sin(8 t)
)l/~
psUilde(n
o
0.051 Je(D)
O.Olll
o
6. CONCLUSIONS The Discontinuous Integral Control technique has been
sr
V
discretized and suitable modifications have been pro-
I
V-
(secoodl)
Fig. 3. Simula tion results when r(n) and J is time-varying.
1.2
= 10, 1/I(t) = -7.5
variabl ew(n) is reported togethe r with the reference signal r(n) , while in the middle and low parts of the figure are shown the estimat ed disturb ance .);(n) and the estimated inertia JeCn) . The used parameters are: T, = 0.005 s, k 0.8, '" 0.005, b 0.002 and J.(O) 0.001. The estimat ion .);(n) and the input u(n) have been saturated to the maxim um available torque [-15 , 15]Nm. Note that in steady- state conditions the estimat es ";;(n) and J,(n) tend asympt.otically to the real values 1/I(t) e J , despite of the fact that a small oscillation is present on th e controlled variable "'(t). In Fig. 4 are reported
=
=
=
=
11
wen)
· 11
15
posed and analyzed . The first introdu ced modification improves the discrete-time estimation of the external disturb ance "'(t) . The second modification allows also the estimation of the unknown "inertia" J(t) of the system. The asympt otic stabilit y of the proposed control structure has been proved. The obtained simulation results support the theoretical results. Extensions to higher dimensional system s is under construction. 7. REFER ENCES Utkin, V.1. (1997) . Variable Structu re System s with Sliding Modes. IEEE Trans. A ulom.I ic Control. Vol. 22. pp. 212-222. Drakunov, S.V., V.1. Utkin (1992), Sliding Mode Control in dynami c systems . Int. J. Cont. Vol. 55, no. 4. Vu, X.H. (1993). Discrete Variable Structu re Control System . 1nl. J. Systems Science. Vol. 24. no. 2. pp. 373-386 . Baida, S.V. (1993). Unit Sliding Mode Contro l in Continuous and Discrete-time Systems. 1nl. J. Control. Vol. 57. no. 5. Nersisian, A., R . Zanasi (1993). A Modified Variable Structu re Contro l Algorit hm for Stabiliz ation of Uncertain Dynam ical Systems. 1nl. J. of Robust .nd Nonlineor Contml. Vol. 3. pp. 199-209 . Fridma n, L. , A. Levant (1994) . Higher Order Sliding Modes as a Natural Phenom enon in Control The-
·15 O.M! b---------------------------------~ Je(n)
0.001
'=.C=---- - . -- - o
---~ -=-__"i' (secoodi) 1.2
Fig. 4. Simula tion resulta obtaine d when J, r(n) and "' (t) are time-varying. th e results obtained in the same conditions of the pre-
ory. IEEE VSLT'94 , Benevenlo, Italy. Zanasi, R. (1993). Sliding Mode Using Discontinuou. Control Algorit hms of Integra l Type. lnt. Journal of Conlrol. Special Issue on Sliding Mode Control. Vol. 57 . no. 5. pp. 1079-1099. Bonivento, C ., A. Nersisian , A. Tonielli, R . Zanasi (1994), A Cascade Structu re For Robust Control Design . IEEE Trons. on A ut. Conlr. Vol. 39. no. 4. Bonivento, C., R. Zanasi (1994). Discontinuous Integral Control. IEEE VSLT'94, Benevento, Italy. Bonivenlo, C., M. Sandri, R . Zanasi (1995). Discrete Low-C hattering Variable Structu re Controllers. ECC'95, Rama, Italy.
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