properties of multirate digital control system design

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FMll 1:30roceedingsof the 36thConfere nce on Decision & ControlSan Diego, CalifomlaUSA December 1997Some Qualitative Properties of Multirate Digital ControlSystems

Bo Hu and A nthony N.MichelDepartment of Electrical EngineeringUniversity of Notre DameNotre Dame, IN 46556, U.S.A.

AbstractWe consider multirate digital control systems whichconsist of an interconnection of a continuous-time non-linear plant (described by ordinary differential equa-tions), a digital controller (described by ordinary dif-ference equations) which has quantizers (but is other-wise linear), along with the required interface elements(A/D and D/A converters). The input to the digitalcontroller consists of the multirate sampled output ofthe plant.In the present note we show that when quantizernonlinearities are neglected, then under reasonable con-ditions (which exclude the critical cases), the stabilityproperties (in the Lyapunov sense) of the trivial so-lution of the nonlinear multirate digital control sys-tems can be deduced from the stability properties ofthe trivial solution of its linearization. For such sys-tems we also present a result concerning the existenceand construction of stabilizing multirate-output digitalcontrollers.In the present note we also show that the solu-tions of multirate digital feedback control systems withnonlinear plant an d quanizers are uniformly ultimatelybounded if the trivial solution of the corresponding lin-ear systems consisting of the linearization of the plantand with the quantization removed from the digitalcontroller, is asymptotically stable. We also providea result which compares the response of multirate digi-tal control systems with nonlinear plant and quantizersin the controller with the response of the correspond-ing nonlinear multirate digital control systems withoutquantizers in the digital controller.

1. IntroductionA. BackgroundSampled-data control systems, resp., digital controlsystems, are hybrid dynamical systems which usuallyconsist of a continuous-time plant (which can be de-scribed by a set of first order ordinary differential equa-tions) and a digital controller (which can be describedby a set of first order ordinary difference equations).When da ta in such systems are sampled at more thanone rate , such systems are called multirate digital con-

trol systems, resp., multirate sampled-data control sys-tems.

The analysis and synthesis of such systems (espe-cially systems with one sampling rate) have been ofcontinuing interest for several decades (refer, e.g., to[I], [5], [6], [8 ] , [12] and the references cited therein).Although the plant is frequently nonlinear in such sys-tems, for purposes of design and analysis, a lineariza-tion of the plant is usually used in the literature, with-out much justification. Furthermore, most of these re-sults do not take into account quantizer effects in thedigital controller and in the interconnecting elements(A/D and D/A converters). In several works (e.g., [2],[15]-[17]) quantization effects in sampled-data controlsystems with linear plants are studied.In a recent paper [7] we investigated the qualita-tive behavior of single ra te digital control systems withnonlinear continuous-time plants and with digital con-trollers and interface elements (A/D and D /A convert-ers) which include signal quantization. In this work wepresented Lyapunov stability results for systems withnonlinear plants but no quantizers in the controllersand Lagrange stability results for systems with nonlin-ear plants and quantizers in the digital controllers.In the present paper we establish new qualitativeresults for multirate digital control systems with non-linear continuous-time plants and with quantization inthe digital controllers. For the special case when theplant is linear and there is no quantization in the dig-ital controller, the class of systems considered hereinreduces to the class of the systems treated in [6].

B. Multirate Digital Feedback Control SystemsWe consider multirate digital feedback control sys-

tems of the type depicted in Fig. 1. The nonlinearplant is described by equations of the form2 = f ( z ( t ) ) K e (t ), e( t )= E(kTo),t E [kTo,(k+ 1)To) (1.1){ y(t)= Lz( t ) , k = 0 , 1 , 2 , ..

where TO> 0 is the frame period (see, [ 6 ] ) ,and thedigital controller without quantizers is given byU (@ + 1)TO) = Cu(kT0)+Mv ( kT o ) , = 0 , 1 , . .P(kT0) = Nu(kT0) (1.2)I

0-7803-3970-8197 $10.00 0 1997 IEEE 4298

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where x E R", y E I?', U E R", e ,d ,p E Rm,and v(kT0)will be specified in the subsequent discussion.The output of the plant y ( t ) is sampled componen-twise. The ith component of the plant output, gi, isgiven byYi(kT0+pTi ) = LiZ(kT0 +PT i ) , p = 0,. . Ni -

(1.3)where Li is the ith row of the output matrixL , and Ti,To are related by Ti = Ni 9 i =1,..., , where Ni is a positive integer. DefineV(kT0)' (kT0)' ( y l ( k T O ) ) T , . * *(Yl(kT0+ (N I -1)T1)lT, * , Yl(kTo))T, . , YI(kT0+ NI - )T1)>TIT,which consists of all the sampled output values in

Finally, we will assume that in ( 1 . 1 ) and (1.2), f EC1[!Rn,!I?fl]i.e., f : 8"+ 8" and f is continuouslydifferentiable), f (0) = 0, and that K , L , C , M , N denote[kTo, ( I c + ) T o ) *

, . . .constant matrices with appropriate dimensions.r ( t )3 0 e ( t ) Y ( t )+-y cUFig. 1 Nonlinear sampled-data control system

2. Stability Analysis of SystemsWithout QuantizersIn this section we assume ideal A/D and D/A con-verters and we assume infinite wordlength digital con-trollers (i.e., there is no quantization in the convert-ers and in the digital controller). Under these as-sumptions, we have e'(kT0)= p(kT0) = Nu(k To ) ,andv(kT0) = d(kT0) = [ ( Y l ( k To ) ) T , - , Y 1 (kT0+ ( N l -l )Tl))T, * , (Yl(kTo))T,* * , (Y'(kT0+W l - )T1))TIT.The multirate digital feedback control system of Fig. 1is now described by the equations

x = f ( z ( t ) )+ B i i ( t ) , i i ( t )= u( k To ) , t E [ W O , k+ 1)To)U ( ( k+ 1)To)= CU(kT0)+Mjj(kTo), k = 0 , 1 , .

(2 .1 ){where B = K N . We note that since f (0) = 0, thetrivial solution ( z T , ~ ~ ) ~(OT, is an equilibriumof the multirate system ( 2 .1 ) .Associated with ( 2 . 1 ) is the system given by

2 = A x ( t )+ B i q t ) , C ( t )= u(kTo) ,t E [kTo,(k+ 1)To)~ ( ( k1)To) Cu(kT0)+ M g ~ ( k T o ) ,= 0 , 1 , .where d,r,(kTo) represents the multirate output of thelinearized system and A denotes the Jacobian off eval-uated at 2 = 0, i.e., A = [g(O)] . The linearsystem of equations (2.2), is called the linearization ofsystem (2 .1 ) .

(2 .2 ){n x n

~

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For s E [O,To],we have for system ( 2 .2 ) ,z(kT0+s) = eA 8 x ( k To )+[J," A td t ] Bu(kTo).Using this equa-tion and (1.3), we obtain tlhe output of the linearizedplant aswhere W1 = [ ( L I ) ~ ,L leA 7 "1 ) T , . L leA ( N1 - l ) T1 7.. L I ) ~ , ( L l e A z I T , .[OT, L ~ [ J + e ~ t d t l . ~ ) ~ ,, L ~ [ J : ~ ~ - ~ ) ~ ~~ t d t ] . ~ ) ~ ,B)T]T. ubstituting (2.3) into (2.2), we abtain for thelinearization of (2 .1 ) the alternative expression

~2 =. , T, (~11~2Atdt1. B I T , . . L ~ [ s ,NI-1)Tl eAtdt]

5 = A z ( t )+B i i ( t ) ,~ ( ( k1)To)= (C +MW2)u(kTo)+M W l z ( k T o ) ,

i i ( t )= ~ ( k T o ) ,t E [ k G , k+ 1)To)k = 0,11,. .

(2.4)Note that the trivial solution = (OT,OT)T isan equilibrium of the single-rate digital control system(2.4).For system (2.4), the following results are wellknown (see, e.g., [4]).Lemma 1. The equilibrium ( z T , ~ ) ~(OT, T) T ofthe linear digital control s,ystem (2.4) is ezponentiallystable in the large if and oiily if the matrix

is Schur stable (i.e., all the eigenvalues of H are withinthe unit circle of the complex plane).Lemma 2. Assume that H has at least one eigen-value outside the unit cixcle. Then the equilibrium( x T ,u ~ ) ~( O T , O T ) T of the linear digital control sys-tem (2.4) is unstable.When the eigenvalues of H satisfy either Lemma1 or Lemma 2, they are said to be noncritical. Other-wise, they are said to be critical eigenvalues. (For thesedefinitions and the usual definitions and results of theLyapunov stability of an equilibrium and the bounded-ness of solutions (Lagrange stability), the reader maywant to refer, e.g., to [13]and [14].)Theorem 1. The equilibrium ( x T , ~ T ) T (OT,OT)Tof the nonlinear multirate digital control system (2 .1 )is uniformly asymptotically stable if the equilibrium( x T ,u ~ ) ~( O T , O T ) T of tbe linear digital control sys-tem (2.4) is exponentially stable, or equivalently, if thematrix H given in (2.5) is Schur stable.Proof: Since f : Rn -+ 3Ifl is continuously differ-entiable with f (0) = 0, we can represent f as f ( x )=Az +g(z), where A = [%.(O)] n x n and g E Cl[%", n]._ . _satisfies the condition liml,,o = 0, where 11 * 11denotes the Euclidean norm on W. The first equationof (2.1) now assumes the form x = A z ( t )+ g ( z ( t ) )+B i i ( t ) , G ( t ) = zc(kT0) for t E [kTo,(k+ 1)To). Its



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now in a pogtion to show that the trivial sblution ofthe system (2.1) is uniformly asymptotically stable ifthe trivial solution of system (2.4) s exponentially sta-ble. Since H is assumed to be Schur stable, there ex-ists a positive definite symmetric matrix P such thatH T P H - P = -In+,, where In+ss the ( n+s ) x ( n+s )identity matrix. Choosing V(w) = w T P w , wherew E !IFs, nd letting w ( k ) = [~~(kTo),u~(kTo)]~~(2.10) can be written as

u ( k + 1) = Hw(k)+ O ( k ) . (2.11)The first forward difference of V evaluated along thesolutions of (2.11) yields

V ( w ( k+ 1)) - V(w(k))= wT(k)[HTPH- P]w(k)+2RT(k)PHw(k)+ R(k)PR(k) = -l/w(k)112+2RT(lc)PHw(k) + l ( k ) *PO ( k ) -llw(k)112+2llQ(k)ll . IlPHll * Il4k)l l + I l ~ ( ~ > I l 2P11. (2.12)Before proceeding further, we need the following inter-mediate result.

Proposition 1. For any given Y > 0, there exists a6 = S(v) > 0, such that(k+l)TOeTIln(k>II5 JkT0 o l l A l l Il9(4t))ldt (2.13)+llMll IlWdkT0)ll I l l4k) I I

Awhenever Ilw(k)\l < 6 , for any k E Z+ = ( 0 , 1 , 2 , . . }.The proof of the above proposition is omitted dueto the space limitations. If we now choose a vo > 0such that 2vollPHII + vO2llPll < 1, then there exists aS(YO) > 0 such that

V(W(k+ 1)) - V ( w ( k ) )I llW(k>1l2+2YOllPHll * llW(~>1l2 .0211Pll - l lW(k)1l2= - U - 2vollPHII - vo211PII) - llw(k>l12

(2.14)

whenever Ilw(k)II < ~ ( v o ) .t follows from the definitionof V(w) nd (2.14) that Xmi"(P)llw(k + 1)112 I v(w(k+1)) I ( w ( k ) )5 Anax(P)llW(k)112, where Xmin(P) andA,,(P) denote the smallest an d largest eigenvaluesofP , respectively. Let d ( ~ ) ~ S ( Y O ) .f Ilw(k0)ll ll vollw(k>>ll- (eTollAllJ1+ TO2e2TollAllll~112 u o ) ~ ~ w ( ~ ) ~ ~where in the second-last line, inequality (2.13) wasused. Therefore, z(t ) converges to the origin simul-taneously with w ( k ) = [ z T ( k T o ) , T(kTo)lTwheneverlIw(ko)ll < d . Noting that d is independent of ko, weconclude that the trivial solution of system (2.1)s uni-formly asymptotically stable if H , given by (2.5), isSchur stable. 0To accommodate hybrid systems, we have modifiedthe usual definition of asymptotic stability in Theo-rem 1. Specifically, the equilibrium w = ( z ~ , u ~ )(OT, T )T of the hybrid system (2.1) is taken to be uni-formly asymptotically stable if the equilibrium w = 0of the corresponding discrete-time system (2.10) is uni-formly asymptotically stable and in addition Ilz(t))Icllw(k)II for all t E [kTo,(k+ )To), k 2 ko,where c isa constant independent of ko.Theorem 2. Assume that H given in (2.5) has atleast one eigenvalue outside the unit circle in the com-plex plane. Then the equilibrium w = ( z ~ , u ~ )( OT , OT)T of the nonlinear sampled-data control sys-

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tem (2.1) is unstable.Proof: Omitted due to space limitations. 0For single-rate digital control systems considered in[ 7 ] ,we have W1 = L, and W2 = 0. Clearly then, theresults of the present paper constitute generalizationsto the results given in [ 7 ] .3. Stabilization b y Multirate-OutputControllers

In this section, we will present a result concerningthe existence and construction of multirate-output con-trollers which stabilize nonlinear systems of the typeconsidered herein. The nonlinear plant may not bestable at the outset.Definition 1. [8] The frame period TO s said to benonpathological if whenever X is an eigenvalue of A , nopoint X + .27rk/To is an eigenvalue of A when k is anonzero integer.Definition 2. [6 ] (Observability Index Vector)Consider an observable pair ( A , L ) where A EWXn and L E eXnnd express L as L =[LT ,- . TIT. A set of 1 integers (121,. - e , nr) is saidto be an observability index v ector (abbreviated OIV)of the pair ( A , L ) if Ci=lni= n and the vectorsare linearly independent. Here ni = 0 means that thevectors of the form LiA" do not appear in the aboveseries.Note that an observable pair has at least one OIV,e.g., the set of Kronecker invariants [9] of the pair( A T ,L T ) ,and may have more OIV's if 1 2 2. An algo-rithm to determine OIV's was presented in Luenberger[ll].Also, it is known that any OIV is preserved undersimilarity transformations.In the following, we use the triplet ( A ,K , L ) to de-note the linearized system 5 = A z ( t )+ K e ( t ) , y ( t )=L z ( t ) corresponding to the nonlinear plant (1.1).Theorem 3. If the linear system ( A , K , L ) s a con-trollable and observable triplet, where A , K , L are thesame as in Section 11, and if (ml , . ,ml ) , the OIVof the augmented system ([ A ] , [L,oI) satisfyNi 2 m,,i = l , . . - , Z , then for any nonpathologicalTo> 0 , there exists a multirate digital controller of theform (1.2) which stabilizes the nonlinear plant ( l . l ) ,i.e., makes the nonlinear multirate digital control sys-tem (2 .1) uniformly exponentially stable.Proof. This theorem is a direct consequence of thelinear system result given in [6 ] and Theorem 1. Inthe present case, we let N = I, E PXg,he s x sidentity matrix. Consider the linearized system. Byz(kTo+s) = eA uz( k To )+[J," A td t ] Bu(kTo),we have

L1,. L I A n l - l ;L a , . L2 A n2 - 1 ; .. Ll , . . LrAnJ-1

~ ( ( k 1)To) = A~(lcT0) Bu(kT0) (3.1)where = eA To ,B= [J eA td t ] K. For a nonpatho-logical frame period TO> 0, we know that a,B is con-trollable [lo] and thus there exists a stable feedback

gain matrix F such tha t -B F is stable (its eigenval-ues can be arbitrarilly assigned).By [ 6 ] , it follaws that under the present as-sumptions, [6',4 as full column rank, where 6' =[ ( L ~- AT0 ) T , ( L ~-A (To-TI1 " ,.. , L l e v A T 1 ) , .

= (WleATO)T,& = [ ( ~ 1 [ ~ , - ~ 0 e A t d t ]BIT, ~ 1J,- '~O-~" e ~ t d t ]B I T , . , ~ 1 e ~ t d t ]

. . Lr[&* eAtdt] * B ) T ] T .For any given statetransition matrix C in the second equation of (2.2), wesolve - M [ C , G ] = [.F,C] o obtain one solution, e.g.,M = -[~,C]([6',G:IT[6',G])-1[d,G]T.ow for theabove choice of F, , M , system (2.2) is uniformly ex-ponentially stable (see [ S I ) . In fact, in the present case,the multirate digital controller can be reduced to theequivalent form,

( L ~ ~ - A T O ) T ,L~~-A("O-TI )T , re-^")^]^

BIT,. L,[s , -~oA'tdt1 . B I T , (LIIJo- (To-*) Atdt]

u(kT0)= -Fz(kTo) , (k2 1) (3.2)regardless of the choice of the state transition matrixC. For a detailed discussion concerning the results forlinear systems, refer to [ 6 ] . Now, by (3.1), (3.2) andTheorem 1, we obtain Theorem 3. 0

4. Analysis of !Systems wit h Quantiza tionNonlinearitiesIn the implementatj on of digital controllers, quanti-zation is unavoidable. In [3], [15] and [16], detailsconcerning the inclusion of quantizers into digital con-trollers are discussedl. In the present section, we studyqualitative properties of the nonlinear multirate digi-ta l control system of Fig.1 with fixed-point quantiza-tion included in both the A/D converter and the digitalcontroller. Fixed-point quantization can be character-ized by the relation &(e) = 8 + q(8) where lq(8)l < efor all 8 E 92 and t:, the quantization level, dependson the desired preci,sion. From Fig. 1, we obtain, as-suming that r = O,v(kTo) = Q(Q(kT0)) = Q(kT0)q i ( i j ( k To ) ) ,u ( ( k+ 1:ITo) = Q(Cu(kTo)+Mv(kTo))=Cu(kT0)+Mv(kTo ) -472 (Cu(kTo)+Mv(kTo) ) ,(kT0)=Q ( N ~ ( k 2 ' 0 ) ) N ~ ( l c T 0 ) qS(NU(kTo)) ,k= 0 , 1 , 2 , * * *where the q1, 42 and q3 should be interpreted as vec-tors whose components contain quantization terms. Bya slight abuse of notatton, we will henceforth write q1 (k )in place of q1(Q(kTo:l), Z(k) in place of q2(Cu(kTo)+M v ( k T o ) )and so forth. It is easily verified that thereexist positive constants Ji which are independent of Esuch that Ilqi(k)II 5 i e , i = 1,2 ,3 , k = O , l , . .. In thepresence of quantizer nonlinearities, we can no longerexpect that the system of Fig. 1will have a uniformlyasymptotically stable equilibrium at the origin; in fact,there may not even be an equilibrium at the origin. Inview of this, we will investigate the (ultimate) bound-edness of the solutioin of the system of Fig. 1, includingthe dependence of thle bounds on the quantization size.Similarly as in Section 11, we can show that the4301


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system of Fig. 1can be represented by the equationsk ( t ) = A z ( t )+ g ( z ( t ) )+Bu(kT0)+K q 3 ( k ) ,

t E W O , k+WO)u ( ( k+ 1)To)= MWlZ(kT0)+ (C MWz)u(kTo)+MW3(kTO)+Mql(k) + q2(k)* (4.1)Letting w ( k )= [ ~ ~ ( k ) , u ~ ( k ) ] ~ ,e have the equivalentrepresentation of (4.1), valid at t = kT0, k = 0,1,. .-,given by w ( k ) = Hw ( k )+ n ( k ) , where H is defined by(2.5) and where

Theorem 4. For system (4.1), if the matrix H definedin (2.5) is Schur stable, then there exist 0 > 0, do > 0,and a constant ro > 0 such that

whenever E 5 0 and Ilw(k)ll I ,.

U

Remark. From the above theorem we can concludethat the bound of the solution of (4.1) can be made ar-bitrarily small by choosing the quantization level suf-ficiently small, provided that the initial state of thesystem is close enough to the origin. Indeed, replacing60 by 2 n Theorem 4, we obtain from (4.3) and thelast statement in the above proof that2 ko ;

Ilz(t)II 5 ( ( e d a +OJ~~~KI~)~~O(Iwhenever E 5 EO and Ilw(k0)ll 5 *e for some ko E Z+and t 2 koT0. Clearly, these bounds are all linear in E ,In our final result we establish an estimate of thenorm of the difference between the response of nonlin-ear multirate digital control systems without quantiz-ers, given by (2.1),and nonlinear digital feedback con-trol systems with quantizers, given by (4.1). For the

llw(k)ll 5 9 6

the quantization level. 0

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present purpose, we rewrite (2.8) and (2.9) as k ( k ) =H 3 ( k ) + fi(k), where 3 ( k ) = [ZT(kTo),iiT(kTo)]T- . -,z(k) = w ( k ) : ( I C ) , we obtain

z(k + 1) = Hz(lc)+ [ SkTok + l )T o eA((k+l)To-t)[g(z)- (2 )+ ~ q 3 ( 1 c ) l c i tM Q l IC) + Q2 (k) (4.7)Suppose g ( - ) has the property tha t

lim,,o,*+o b = 0. Following a similar proce-dure as in Section 11, we can establish the next result.We will omit the details of the proof.Theorem 5. Assume that H defined in (2.5) is Schurstable and lim,,o,%,o j e 0. Then there ex-ist 0 > 0, 60 > 0, and a constant TO > 0, such tha t

References[l] M. Araki and K. Yamamoto, MultivariableMultirate Sampled-data Systems: State-space Descrip-tion, Transfer Characteristics, and Nyquist Criterion,IEEE Trans. Automat. Contr., vol. 31 , pp. 145-154,1986.[2] J. E. Bertram, The effect of Quantization inSampled Feedback Systems, Dans. AIEE Appl. Ind.,part 2, vol. 77 , pp. 177-181, Sept., 1988.[3] J. A. Farrell and A. N. Michel, Estimates ofAsymptotic Trajectory Bounds in Digital Implemen-tations of Linear Feedback Control Systems, IEEERuns. Automat. Contr., vol. 34, No. 12 , pp 1319-1324,1989.[4] B.A. Francis and T.T. Georgiou, Stability The-ory for Linear Time-invariant Plants with Periodic Dig-ital Controllers, IEEE Dans. Automat. Contr., vol.[5] G. F. Franklin and J. D. owell, Digital Controlof Dynamic Systems, Addison-Wesley, 1980.[6] T. Hagiwara and M. Araki, Design of a Sta-ble State Feedback Controller Based on the MultirateSampling of the Plant Output , IEEE %ns. Automat.Contr., vol. 33 , No. 9 , pp. 812-819, 1988.

33 , NO. 9 , pp 820-832, 1988.*

(71 L. Hou, A.N. INichel and H. Ye, Some Quali-tative Properties of Sampled-Data Control Systems,IEEE Runs. Automat. Contr., to appear.[8] H. Ito, H. Ohmori and A. Sano, Stability Analy-sis of Multirate Sampled-data Control Systems, IM AJournal of Mathematical Control & Information, vol.[9] R.E. Kalman, Kronecker Invariants and Feed-back, Proc. Conf. Ordinay Differential Equations,Math. Research Center, Naval Research Labs., Wash-ington, DC, pp 459-471, 1971.[ lo] R.E. Kalman, Y. C. Ho and K. S. Narendra,Controllability of linear dynamical systems, Contri-butions to DifferentialEquations, vol. 1, No. 2, pp. 189-213, 1962.[l l] D. G . Luenberger, Canonical Forms for the Lin-ear Multivariable Systems, IEEE lkans. Automat.Contr., vol. 12 , pp. 290-293, 1967.[12] D. G. Meyer, A Parameterization of Stabil-ity Controllers for Multirate Sampled-data Systems,IEEE lkans. Automat. Contr., vol. 35 , pp. 223-235,1990.[13] A.N. Michel and K. Wang, Qualitative Theory ofDynamical Systems, ]Marcel Dekker, New York, 1995.[14] R. K. Miller and A. N. Michel, Ordinary Differ-ential Equations, Academic Press, New York, 1982.[15] R. K. Miller, ,4. N. Michel and J. A. Farrell,Quantizer Effects on Steady-State Error Specifica-tions of Digital Feedback Control Systems, IEEERuns. Automat. Coiatr., vol. 34 , No. 6 , pp. 651-654,1989.[16] R. K. Miller, Ivl. S. Mousa and A. N. Michel,Quantization and Overflow Effects in Digital Im-plementations of Linjear Dynamic Controllers, IEEETrans. Automat. Contr., vol. 33, NO. 7, pp. 698-704,1988.[17] J. E. Slaughter, Quantization Errors in DigitalControl Systems, IE:EE Runs. Automat. Contr., vol.

11 , pp. 341-354, 1994.

9, pp. 70-74, 1964.

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