ISSN 1751-8644 Discretisation Of Continuous-time Dynamic Multi-input .
www.ietdl.orgPublished in IET Control Theory and ApplicationsReceived on 11th August 2010Revised on 28th March 2011doi: 10.1049/iet-cta.2010.0467ISSN 1751-8644Discretisation of continuous-time dynamic multi-inputmulti-output systems with non-uniform delaysZ.M. KassasRadionavigational Laboratory, Wireless Networking & Communications Group, Department of Electrical and ComputerEngineering, The University of Texas at Austin, USAE-mail: zkassas@ieee.orgAbstract: Input and output time delays in continuous-time (CT) dynamic systems impact such systems differently as their effectsare encountered before and after the state dynamics. Given a fixed sampling time, input and output signals in multiple-inputmultiple-output (MIMO) systems may exhibit any combination of the following four cases: no delays, integer-multiple delays,fractional delays and integer-multiple plus fractional delays. A common pitfall in the digital control of delayed systemsliterature is to only consider the system timing diagram to derive the discrete-time (DT) equivalent model; hence, effectively‘lump’ the delays across the system as one total delay. DT equivalent models for systems with input delays are radicallydifferent than those with output delays. Existing discretisation techniques for delayed systems usually consider the delays tobe integer-multiples of the sampling time. This study is intended to serve as a reference for systematically deriving DTequivalent models of MIMO systems exhibiting any combination of the four delay cases. This algorithm is applied towardsdiscretising an MIMO heat exchanger process with non-uniform input and output delays. A significant improvement towardsthe CT response was noted when applying this algorithm as opposed to rounding the delays to the closest integer-multiple ofthe sampling time.1IntroductionTime delays are inevitable in many systems, such asaerospace, chemical, digital, networked control andmanufacturing. Time delays may appear in the systemstates, control inputs or measurements [1]. In aerospacesystems, if the spacecraft is controlled from the Earth,measurements and command signals travel back and forthwith non-negligible time delays [2]. In chemical processes,time delays arise due to piping between different units. Incertain digital systems, computational delays are nonnegligible and must be compensated for [3]. In networkedcontrol systems, where the plants, sensors, controllers andactuators reside on different nodes, time delays may arisebecause of signal routing within the network [4 – 6]. Inmanufacturing facilities, time delays commonly thermocouples, gauges, compensators and convertors do notrespond instantaneously to signal changes. Manipulatedvariables are also delayed from the moment the signals arecomputed to the time they impact the process. Such delaysmay play a detrimental role in process control [7].The control of systems with time delays is generallydifficult in theory and practice [8 – 12]. Time delays oftenput severe restrictions on achievable feedback performance[13 – 15]. Delayed continuous-time (CT) linear systems areinfinite-dimensional. The discrete-time (DT) equivalentsystem is, however, finite-dimensional [16]. This orderreduction makes the discretisation of delayed systems notIET Control Theory Appl., 2011, Vol. 5, Iss. 14, pp. 1637– 1647doi: 10.1049/iet-cta.2010.0467only desirable from a digital implementation perspective,but also from a simplified analysis and synthesisperspective as well. However, such a desirable dimensionalreduction may increase the model mismatch between thedelayed CT system and its DT equivalent. Robustness ofdelayed and perturbed sampled-data systems under digitalredesign has received the attention of many studies [17, 18].Discretisation techniques for delayed CT systems can befound in the literature [19 – 21]. However, these techniquesoften assume the time delay to be an integer-multiple of thesampling time. Some approximations have been proposedfor systems with delays that are non-integer multiples of thesampling time, that is, systems with fractional delays.Among them are to round the delays to the closest integermultiple of the sampling time, the use of Taylor-seriesexpansions and the use of the Runge –Kutta (RK) methodinvolving polynomial interpolations [22 – 24]. Nevertheless,these approximations introduce model mismatches, whichprevent the DT model response from reaching the actual CTresponse at the sampling instants.Transfer functions (TFs) of systems with fractional timedelays have been thoroughly analysed through the modifiedz-transform [25, 26]. TFs characterise the input – outputbehaviour of a system, while the state information issuppressed. Single-input single-output (SISO) CT TFs withtime delay(s) have the structure G(s) ¼ e 2lsH(s), whereH(s) is the delay-free TF and l is the time delay. Thisstructure shows that the DT equivalent TF, G(z), will be thesame regardless if such delay(s) correspond to the input or1637& The Institution of Engineering and Technology 2011
www.ietdl.orgthe output signal [19]. This is not the case, however, fordelayed state-space (SS) models. SS models with fractionaltime delays have not been analysed to the same extent asTFs. State variables often have physical interpretations;hence, it is desirable to obtain a DT equivalent SSrepresentation. Discretisation of SISO linear time-invariant(LTI) systems with only a fractional input delay has beenconsidered in [19, 20, 27]. Additionally, discretisation ofsystems with ‘inner’ time delays has been considered in[16, 28], whereas discretisation of systems with internal andinput delays via finite-impulse response (FIR) filteringapproximation has been addressed in [29].In [30], an algorithm for discretising CT multiple-inputmultiple-output (MIMO) LTI systems with non-uniforminput and/or output fractional time delays was presented.However, that algorithm assumed that all input and/oroutput signals were delayed by fractional values. This is abit restrictive, since given an arbitrary CT MIMO systemand a fixed sampling time, T, input and output channelsmay exhibit any combination of the following cases: nodelays, integer-multiple delays, fractional delays andinteger-multiple plus fractional delays. This paper extendsthe algorithm of [30] to the general case of discretising CTMIMO LTI systems, where input and/or output signalsexhibit any combination of the four delay cases.Particularly, this algorithm will handle the 28 different casesan MIMO system may experience.Input and output time delays impact SS models differentlyas their effects are encountered before and after the statedynamics. As such, discretisation of systems with suchdelays must compensate for these delays by consideringtheir separate and combined effects on the system states andoutputs. Unfortunately, in digital control of delayed systemsliterature, some authors do not pay attention to such fact,and they commit the common pitfall of considering onlythe system timing diagram to derive the DT equivalentmodel. So, they effectively ‘lump’ the delay across thesystem as one ‘total’ delay (e.g. see [31, 32]). As will bediscussed in this paper, DT equivalent models for systemswith input delays are radically different than those withoutput delays. This paper is intended to serve as a referencefor systematically deriving DT equivalent models of MIMOsystems exhibiting any combination of input and outputdelays. In particular, this paper considers the systemin Fig. 1, which depicts a delayed CT MIMO LTI system(A, B, C, D) in the prototypical sampled-data systemconfiguration, where the delayed inputs and outputs of theCT system are interfaced through a digital to analogue (D/A) converter via zero-order hold (ZOH) and an analogue todigital (A/D) converter via periodic sampling, respectively.The objective is to find the DT equivalent system, (Ad , Bd ,Cd , Dd), such that the two systems are equivalent, and theresponse from the DT system matches that of its CTcounterpart at the sampling instants.This paper is organised as follows. Section 2 derives thealgorithms for discretising delayed CT systems byconsidering the cases of input, output and simultaneousinput and output delays. Section 3 presents the applicationof the proposed discretisation algorithm to a particularexample. Concluding remarks are discussed in Section 4.2 Discretisation algorithms for systems withinput and output time delaysThis section derives algorithms for discretising CT LTIsystems with input and/or output time delays. Thediscretisation algorithms assume ZOH of the input signal,which implies that the input signal takes piecewise constantvalues within T, that is, {u(t) ¼ u(kT), kT t , (k 1)T}.The input signals, u(t) [ Rr , are assumed to be delayed byu [ Rr , whereas the output signals, y(t) [ Rm , areassumed to be delayed by f [ Rm . Without loss ofgenerality, the input (output) signals will be assumed to beordered such that the first R1 inputs (M1 outputs) are nondelayed; the second R2 inputs (M2 outputs) are delayed byinteger-multiples of the sampling time; the third R3 inputs(M3 outputs) are delayed by fractions of the sampling timeand the fourth R4 inputs (M4 outputs) are delayed byinteger-multiples plus fractions of the sampling time. Asignal with non-uniform delay s(t q) [ Rp will beinterpretedass(t q)W[s1 (t q1 ) s2 (t q2 ) . . .sp (t qp )]T . In this paper, the following notation isadopted. Given a matrix P, the vector pj is the jth columnvector in P, the vector pTi as the ith row vector in P, and thescalar pi,j as the (i, j)th element in P, where (i, j) [ N.The next subsections will derive the discretisationalgorithms for systems with input delays, outputs delaysand simultaneous input and output delays.2.1Systems with input delaysConsider a CT LTI system (A, B, C, D) with non-uniforminput delaysẋ(t) Ax(t) Bu(t u)(1)y(t) Cx(t) Du(t u)(2)where A [ Rn n , B [ Rn r , C [ Rm n and D [ Rm r . Ifthe inputs are not ordered as discussed previously, thenordering can be achieved by applying the permutationmatrix Q [ Rr r to the system (1) and (2) to obtain thesystem with ordered inputs (A, B̃, C, D̃), whereTheB̃ BQ 1 , D̃ DQ 1 and ũ(t ũ) Qu(t u).delay vectors will be decomposed asuj 0, l T,j(1 p )T , (l pj )T ,jjj [ J R1 W{1, 2, . . . , r1 }j [ J R2 W{r1 1, r1 2, . . . , r2 }j [ J R3 W{r2 1, r2 2, . . . , r3 }j [ J R4 W{r3 1, r3 2, . . . , r4 }where{lj 2, 3, . . . ; j [ J R4 }and{pj [ (0, 1),j [ J R3 , j [ J R4 }. The general solution for the statedynamics in (1) isFig. 1 CT MIMO LTI system with non-uniform input and outputdelays in the prototypical sampled-data system configuration andits DT equivalent1638& The Institution of Engineering and Technology 2011 teA(t t) Bu(t u) dtx(t) eA(t t0 ) x(t0 ) (3)t0IET Control Theory Appl., 2011, Vol. 5, Iss. 14, pp. 1637–1647doi: 10.1049/iet-cta.2010.0467
www.ietdl.orgLetting t0 ¼ kT and t ¼ (k 1)T in (3), and using the changeof variables s ¼ (k 1)T 2 t, yieldsx[(k 1)T] F(T)x(kT ) r T j 10(4)The integrals in (4) evaluate to cj (T )uj (kT ), c (T )u [(k l )T ], jjjIj cj (pj T )uj (kT ) lj (pj T )uj [(k 1)T ], c (p T )u [(k l 1)T] j l j(p Tj)u [(k j l )T ],j jjj j jTR2 ,r1 1 jR2 ,j jR2 ,j,1eAs bj uj [(k 1)T uj s] dsexpressed compactly asj [ J R4 tF(t) W e ,cj (t) WFedt bj ,lj (t) W F(t)cj (T t)0In evaluating {Ij , j [ J R1 , j [ J R2 } we used the ZOHassumption. In evaluating {Ij , j [ J R3 , j [ J R4 } we usedFig. 2 and noted that in the interval s [ [0, pj T ), the inputtakes a piecewise constant value of uj (kT), whereas in theinterval s [ [ pj T, T ) it takes a piecewise constant value ofuj [(k– 1)T ]. In order to eliminate past inputs, we introducethe state variablesjR3 ,j (kT ) W uj [(k 1)T],j [ J R3jRz ,j,s (kT ) W uj [(k lj s 1)T ]0Cd,I C FR2jR2 ,j,ljSR4jTR4 ,r3 1T···,j [ J R2,j [ J R4TjTR4 ,r4T (6)j [ J R1j [ J R3otherwise2.2DR3ER4FR4 ,Dd,I DR100 0Systems with output delaysConsider a CT LTI system with output delaysẋ(t) Ax(t) Bu(t)(7)y(t f) Cx(t) Du(t)(8)If the outputs are not ordered as discussed previously, thenordering can be achieved by applying the permutationmatrix R [ Rm m to the system in (7) and (8) to obtain thesystem with ordered outputs, (A, B, Ĉ, D̂), whereĈ RC, D̂ RD and ŷ(t f̂) Ry(t f).The delay vectors will be decomposed asfi 0, h T,i(1 q )T , (h qi )T ,iFinally, dropping the dependency of F on T for simplicity ofnotation and defining the augmented DT state xI W [xT jT ]T ,the DT equivalent SS model (Ad,I , Bd,I , Cd,I , Dd,I) can be0Explicit expressions for the matrix blocks of the DTequivalent SS model are given in Appendix (Section 7.1).(5)where {z ; 2, j [ J R2 }, {z ; 4, j [ J R4 } and s ¼ 1, . . . , lj .Next, to discretise the output (2) we note that uj (kT),uj (kT uj ) uj [(k 1)T ], u [(k l )T ],jj···SR3SR20AtjTR3 0 J00 R2 Ad,I 000 0000JR4 0 VR30CR1 000 ER2 Bd,I 0I0 0where we define F(t), cj (t) and lj (t) asAtjR2 ,j,2jTR2 ,r2TjR3 jr2 1 jr2 2 · · · jr3 jR4 ,j jR4 ,j,1 jR4 ,j,2 · · · jR4 ,j,lj j [ J R1j [ J R2j [ J R3···ii [ I M1 W{1, 2, . . . , m1 }i [ I M2 W{m1 1, . . . , m2 }i [ I M3 W{m2 1, . . . , m3 }i [ I M4 W{m3 1, . . . , m4 }where {hi [ N, i [ I M2 }, {hi 2, 3, . . . ; i [ I M4 } and{qj [ (0, 1), i [ I M3 , i [ I M4 }. Discretising the stateequation in (7) and the output equation in (8), we obtainx[(k 1)T] F(T )x(kT ) C(T )u(kT )C(t) W c1 (t) c2 (t) . . cr (t) Tci x(kT) dTi u(kT ), cT x[(k h )T ] dT u[(k h )T ],iiiiyi (kT) TT ci x[(k 1 qi )T ] di u[(k 1 qi )T ], Tci x[(k hi qi )T ] dTi u[(k hi qi )T ],Fig. 2 Effect of input fractional time delayIET Control Theory Appl., 2011, Vol. 5, Iss. 14, pp. 1637– 1647doi: 10.1049/iet-cta.2010.0467i [ I M1i [ I M2i [ I M3i [ I M4In order to capture the effect of the delayed signal1639& The Institution of Engineering and Technology 2011
www.ietdl.orgyi [(k hi)T ], we introduce hi state variables such thathMz ,i,s (kT ) W yi [(k hi s)T ](9)where {z ; 2, i [ I M2 } and s ¼ 1, 2, . . . , hi . The termx[(k 2 1 qi)T ] can be evaluated by starting with thesolution of the state dynamics in (3) with u ; 0, lettingt0 ¼ (k 2 1)T and t ¼ kT 2 (1 2 qi)T, noting thatu(t) ¼ u[(k 2 1)T ] for [(k 2 1)T t , (k 2 1 qi)T , kT]for any qi [ (0,1), and finally using the change of variables ¼ [(k 2 1 qi)T 2 t] to arrive atx[(k 1 qi )T ] F(qi T )x[(k 1)T ] C(qi T )u[(k 1)T ]To evaluate u[(k 2 1 qi)T ], we note that under ZOH,u[(k 2 1 qi)T ] ¼ u[(k 2 1)T ], for any qi [ (0, 1). Inorder to capture the effect of the delayed signalyi [(k hi 2 qi)T ], we introduce hi state variables, whichwill be identical to the ones defined in (9) with{z ; 4, i [ I M4 }. In order to evaluate hM4 ,i,1 (kT ), we startwith yi[(k hi 2 qi)T ] and proceed to find thatyi [(k hi 1)T ] cTi x[(k 1 qi )T ] dTi u[(k 1 qi )T ]Finally, dropping the dependency of (F, C) on T, definingthe augmented DT state xO W [ xT hT ]T , the DTequivalent SS model (Ad,O , Bd,O , Cd,O , Dd,O) can beexpressed as h W hTM2 ,m1 1 · · · hTM2 ,m2 hTM3 hTM4 ,m3 1 · · · hTM4 ,m4hM2 ,i W hM2 ,i,1 hM2 ,i,2 · · · hM2 ,i,hihM3 W hm2 1 hm2 2 · · · hm3Tyi (kT ) cTi x(kT ) dTR1 ,i uR1 (kT ) dTR2 ,i jR2 ,R2 ,1 (kT ) dTR3 ,i jR3 (kT ) dTR4 ,i jR4 ,R4 ,1 (kT ),dTR1 ,i di,1···di,r1 ,···dTR3 ,i di,r2 1di,r3 ,dTR2 ,i di,r1 1i [ I M1···dTR4 ,i di,r3 1di,r2···jR2 ,R2 ,1 (kT ) W jR2 ,r1 1,1 (kT ) · · · jR2 ,r2 ,1 (kT )TjR4 ,R4 ,1 (kT ) W jR4 ,r3 1,1 (kT ) · · · jR4 ,r4 ,1 (kT )Tdi,r4Second, for each {yi , i [ I M2 }, we introduce hi statevariables, as defined in (9). To calculate hM2 ,i,1 [(k 1)T ],we consider the effect of each of the delayed inputs. First,we note that each {uj , j [ J R2 } corresponds to the statevariable jR2 ,j,1 (kT ) that was introduced in (6). Second, wenote that under ZOH, uj [(k 2 1 pj )T ] ¼ uj [(k 2 1)T ] forany pj [ (0, 1) and for all {uj , j [ J R3 }. Similarly, wenote that under ZOH, uj [(k 2 lj pj)T ] ¼ uj [(k 2 lj)T ] forany pj [ (0, 1) and any lj ¼ 2, 3, . . ., and for all{uj , j [ J R4 }. Hence, we may expresshM2 ,i,1 [(k 1)T ] cTi x(kT ) dTR1 ,i uR1 (kT ) dTR3 ,i jR3 dTR2 ,i jR2 ,R2 ,1 dTR4 ,i jR4 ,R4 ,1Twhere i [ I M2 . Third, for each {yi , i [ I M3 }, the DTequivalent output equation is given by, i [ I M2Tyi (kT) cTi x[(k 1 qi )T ] dTi u[(k 1 qi )T u]ThM4 ,i W hM4 ,i,1 hM4 ,i,2 · · · hM4 ,i,hi , i [ I M4 F0 0 0C L Y M HM 2 0 0 M Ad,O 2 , Bd,O 2 LM3 0 0 0 YM3 YM4LM4 0 0 HM4 Cd,O diag[CM1 , GM2 , I, GM4 ], DTd,O DTM1 0 0 0Explicit expressions for the matrix blocks of the DTequivalent SS model are given in Appendix (Section 7.2).2.3was shown in Section 2.1. Next, we consider thediscretisation of the output (11). First, for each{yi , i [ I M1 }, we note that this is identical to discretisingthe output equation with input delays only. Consequently,the DT equivalent is nothing butTo evaluate x[(k 2 1 qi)T ], we use the change of variabless ¼ kT 2 (1 2 qi)T 2 t in (3) to obtainx[(k 1 qi )T ] F(qi T )x[(k 1)T] 4 IRs,j,is 1 j[J Rs qi TIRs,j,i WeAs bj uj [(k 1 qi )T uj s] ds0The integrals IR1,j,i and IR2,j,i can be evaluated with the aid ofFig. 3 by noting that in the interval s [ [0, qiT ], the inputsignal takes a piecewise constant value of uj [(k 2 1 2 lj )T ],so we obtainSystems with input and output delaysConsider a CT LTI system with input and output delaysẋ(t) Ax(t) Bu(t u)(10)y(t f) Cx(t) Du(t u)(11)If the inputs and outputs are not ordered as discussedpreviously, then applying the permutation matrices (Q, R) D), C, to (10) and (11) yields the ordered system (A, B, ) RC, D BQ 1 , C RDQ 1 , u (t uwhereB ) Ry(t f). The system states areQu(t u) and y(t faffected by the input but not the output delays.Consequently, the state (10) DT equivalent is derived as1640& The Institution of Engineering and Technology 2011(12) IRs,j,i cj (qi T)uj [(k 1)T ],cj (qi T)uj [(k 1 lj )T ],s 1s 2The integrals IR3,j,i and IR4, j,i can be evaluated with the aid ofFigs. 4 and 5, where Fig. 4 depicts the case where0 , pj qi , 1, whereas Fig. 5 depicts the case where1 pj qi , 2. For the former case, we note that in theinterval s [ [0, qiT), the input signal takes a piecewiseconstant value of uj [(k 2 1 2 lj )T ]. However, in the lattercase, the input takes a piecewise constant value ofIET Control Theory Appl., 2011, Vol. 5, Iss. 14, pp. 1637–1647doi: 10.1049/iet-cta.2010.0467
www.ietdl.orgj [ J R1 , and uj [(k lR2 1 qi )T ] uj [(k lR2 1)T ],for j [ J R2 and any qi [ (0, 1).Moreover uj [(k 1 lj pj qi )T ] uj [(k 1 lj )T ], 0 , qi pj , 1uj [(k lj )T ],1 qi pj , 2where {lj 1, j [ J R3 } and j [ J R4 . Therefore we canexpress the R3 and R4 inputs compactly asFig. 3 Effect of input and output delay on uR1 and uR2uRz [(k lRz pRz 1 qi )T ] VRz ,i uRz [(k lj )T ] WRz ,i uRz [(k 1 lj )T ]VRz ,i diag[vRz ,i,11 , vRz ,i,22 , . . . , vRz ,i,rz rz ] 0, 0 , qi pj , 1vRz ,i,ff 1, 1 qi pj , 2Fig. 4 Effect of input and output time delays on uR3 and uR4 with0 , pj qi , 1wheneverz ¼ 3,wherez ¼ {3, 4},lRz 1f [ F Rz W{1, 2, . . . , Rz } and WRz ,i I VRz ,i . Combiningx[(k 2 1 qi)T ] and the different inputs we obtain yi(kT),which upon advancing one sample yields the signal thatwill be augmented to the state vectoryi [(k 1)T ] cTi {F(qi T )x(kT ) mR1 ,1,iFig. 5 Effect of input and output time delays on uR3 and uR4 with1 pj qi , 2uj [(k 2 lj)T ] in the interval s [ [0, ( pj qi 2 1)T ) and thenswitches to the constant value of uj [(k 2 1 2 lj )T ] in theinterval s [ [( pj qi 2 1)T, qiT ). ThereforeIRz,j,i cj (qi T )uj [(k 1 lj )T ],0 , qi pj , 1; c [(q p 1)T ]u [(k l )T ]jijjj p 1)T]c[(1 pj )T ] F[(qijj 1 qi pj , 2 uj [(k 1 lj )T ], dRz,j,i uj [(k lj )T ] gRz,j,i uj [(k 1 lj )T ] dRz,j,i gRz,j,i0,cj [(qi pj 1)T ],0 , qi pj , 11 qi pj , 2 0 , qi pj , 1 cj (qi T ),F[(q p 1)T] ij c [(1 pj )T ], 1 qi pj , 2jwhere z ¼ {3, 4} and lj ¼ 1 whenever z ¼ 3.Next, we will evaluate the inputs in (12). First, we knowthat under ZOH uj [(k 2 1 qi)T ] ¼ uj [(k 2 1)T ], forIET Control Theory Appl., 2011, Vol. 5, Iss. 14, pp. 1637– 1647doi: 10.1049/iet-cta.2010.0467···mR1 ,r1 ,i uR (kT )1 mR2 ,r1 1,i···mR2 ,r2 ,i jR2 ,R2 ,1 (kT ) pR3 ,r2 1,i···pR3 ,r3 ,i uR (kT )3 rR3 ,r2 1,i···rR3 ,r3 ,i jR (kT )3 pR4 ,r3 1,i···pR4 ,r4 ,i jR ,R ,2 (kT )4 4 rR4 ,r3 1,i···rR4 ,r4 ,i jR ,R ,1 (kT )}4 4wherejR4 ,R4 ,2 (kT ) jR4 ,r3 1,2 (kT ) · · · jR4 ,r4 ,2 (kT )mR1 ,j,i cj (qi T ) dR1 ,j,i ,j [ J R1mR2 ,j,i cj (qi T ) dR2 ,j,i ,j [ J R2TpR3 ,j,i dR3 ,j,i dTR3 ,i vR3 ,i,f ,j [ J R3 , f [ F R3rR3 ,j,i gR3 ,j,i dTR3 ,i wR3 ,i,f ,j [ J R3 , f [ F R3pR4 ,j,i dR4 ,j,i dTR4 ,i vR4 ,i,f ,j [ J R4 , f [ F R4rR4 ,j,i gR4 ,j,i dTR4 ,i wR4 ,i,f ,j [ J R4 , f [ F R4Fourth, for each {yi , i [ I M4 }, we introduce hi states asdefined in (9) with {z 4, i [ I M4 }. To evaluatehM4 ,i,1 (kT ) yi [(k hi 1)T ], it is easy to establish thatyi [(k hi 1)T ] cTi x[(k 1 qi )T ] dTi u[(k 1 qi )T u]The term x[(k 2 1 qi)T ] and the contributions fromdifferent inputs have been evaluated when computing (12).1641& The Institution of Engineering and Technology 2011
IO W [ xT jT hT ]T , the equivalent SS model (Ad,IO ,Bd,IO , Cd,IO , Dd,IO) can be expressed as Ad,IO Ad,IO,11 Ad,IO,12,Ad,IO,21 Ad,IO,22 Bd,IO Bd,IO,11Bd,IO,21Cd,IO Cd,IO,11 Cd,IO,12Ad,IO,11 ; Ad,I , Ad,IO,12 ; 0, Bd,IO,11 ; Bd,I LM2 LM2 JM2 NM2 Ad,IO,12 LM3 LM3 JM3 NM3 LM4 LM4 JM4 NM4 HM 2 0 0 Ad,IO,22 0 0 0 0 0 HM 4 PM2 ,R1 000 Bd,IO,21 PM3 ,R1 0 PM3 ,R3 0 PM4 ,R1 0 PM4 ,R3 0 CM1 FM1 ,R2 DM1 ,R3 FM1 ,R4 0 Cd,IO,11 00000 0 0 G M 0Cd,IO,12 2I 00 00 ,0 0 GM4000000 Dd,IO DM1 ,R1 0 0 0 000 0 0 0 0 0 00 0 0Explicit expressions for the matrix blocks of the DTequivalent SS model are given in Appendix (Section 7.3).3ExampleThis section illustrates the application of the algorithmderived in Section 2 into discretising a fourth-order heatexchanger process with four inputs and four outputs, whereeach of the delays happen to be one of the four differentdelay cases. The system under study consists of two sets ofsingle shell heat exchangers filled with water, placed inparallel and cooled by a liquid saturated refrigerant flowingFig. 6 Heat exchanger process with non-uniform input and output delaysInlet temperatures T1in , . . . , T4in are the system inputs, while vapour flows F1 , . . . , F4 are the outputs1642& The Institution of Engineering and Technology 2011IET Control Theory Appl., 2011, Vol. 5, Iss. 14, pp. 1637–1647doi: 10.1049/iet-cta.2010.0467
www.ietdl.orgthrough a coil system, as it is illustrated in Fig. 6. Thesaturated vapour generated in the coil system is separatedfrom the liquid phase in the stages S1 and S2, both ofneglected volumes. This vapour, withdrawn in S1 and S2,reduces the refrigerant mass flow rate along the coolingsystem, and only the saturated liquid portion is used forcooling purposes. Table 1 provides the fluid properties andequipment dimensions. The temperature of the refrigerantremains constant at Tc as the liquid is saturated, and theenergy exchanged with water is used to vapourise a smallportion of the refrigerant fluid. The energy balance aroundeach heat exchanger is given byM 1 CpM2 CpdT1 (ṁ1 ṁ3 )Cp [T2 (t) T1 (t)] Q1 (t)dtdT2 ṁ1 Cp T1 in (t u1 ) ṁ3 Cp T3 in (t u3 )dt (ṁ1 ṁ3 )Cp T2 (t) Q2 (t)M 3 CpM4 CpdT3 (ṁ2 ṁ4 )Cp [T4 (t) T3 (t)] Q3 (t)dtdT4 ṁ2 Cp T2 in (t u2 ) ṁ4 Cp T4 in (t u4 )dt (ṁ2 ṁ4 )Cp T4 (t) Q4 (t)where Qs , s ¼ 1, . . . , 4, is the heat exchanged in Es , which isestimated using the overall surface heat transfer coefficientand the refrigerant temperature Tc , such thatQs(t) ¼ hA[Ts(t) 2 Tc], s ¼ 1, . . . , 4. The input time delaysuj in the inlet water temperatures are due to thetransportation time, such that uj Vj r/ṁj , j {1, 2, 4}.This implies that the inputs are delayed by the vectorTu 0.5 2 0 1.5 . The measured output flow Fi isexpressed as Fi(t fi) ¼ [Qi(t) Qi 2(t)]/hlv , i ¼ 1, . . . , 4,where hlv is the refrigerant heat of vapourisation.TThese outputs are delayed by f 2.4 0 0.6 4 ,which are due to fluid property compensators. Theindividual heat exchanger energy balances can be expressedin terms of deviation variables to define the following LTITable 1system (1 nA )100 t1 t1 (1 n)A 000 t2 A (1 nB )1 00 tt33 (1 nB ) 000 t4 0 0 0 0kk3m 0 m 0 1 0 0 t2 0 m 0 m t2 B 0 0 0 0 , C m 0 m 0 k2k40 m 0 m00t4t4D [0]where nA W (hA/Cp (ṁ1 ṁ3 )), nB W (hA/Cp (ṁ2 ṁ4 )),t1 W (M1 /(ṁ1 ṁ3 )), t2 W (M2 /(ṁ1 ṁ3 )),t3 W (M3 /(ṁ2 ṁ4 )), t4 W(M4 /(ṁ4 ṁ3 )), k1 W(ṁ1 /(ṁ1 ṁ3 )), k2 W(ṁ2 / (ṁ2 ṁ4 )), k3 W (ṁ3 / (ṁ1 ṁ3 )), k4 W (ṁ4 / ṁ2 ṁ4 ) and m W (hA/hlv ). This system was discretised with asampling time of T ¼ 1 s using the algorithm outlined inSection 2 and was discretised while rounding the input andoutput delays to the closest integer-multiples of T.The systems were driven by step inputs of amplitudes {5,25, 5, 25} that were applied at time instants {1, 10, 1, 10}seconds, respectively. The LabVIEW control design andsimulation module was used to define the system dynamicswith the appropriate input and output delays and simulatethe CT system and its DT equivalents. Fig. 7 shows theblock diagram of the simulated systems. Fig. 8 illustrates theoutput responses of the three systems over 40 s. We can notethe response mismatch of the DT system that rounds thedelay to the nearest integer-multiple of the sampling time.None of its response points lie on the CT system responseline. However, the response obtained by the DT system thatcompensates for the delays matches exactly the CT systemresponse at the precise sampling instants.Table 2 quantifies the detrimental effect of rounding theinput and output delays. The average relative percentageerror along the simulation time is used to determine theaccuracy of the approximation relative to the magnitude ofthe actual response, defined by1i Fluid properties and equipment dimensions for thei 1, . . . , 4,N 40processCphlvTs(0)Tjin(0)TcṁjMshAV1V2V4rN1 yci (kT) ydi (kT ) 100%,N k 1yci (kT )4.217 kJ/kg K850 kJ/kg408C408C408C1 kg/s50 kg8 kJ/kg0.5 1023 m32 1023 m31.5 1023 m31000 kg/m3water specific heatrefrigerant heat of vapourisationinitial temperature in Esinitial water inlet temperature jrefrigerant temperaturewater mass flow jmass of water in Esoverall surface heat transferinlet water pipe volume 1inlet water pipe volume 2inlet water pipe volume 4water densityIET Control Theory Appl., 2011, Vol. 5, Iss. 14, pp. 1637– 1647doi: 10.1049/iet-cta.2010.0467Here, yci and ydi represent the CT and DT responses for outputi. It can be noted from Table 2 that the average errors for theZOH method that rounds the delays are significant comparedto the ZOH method that compensates for the delays byincorporating them into the DT equivalent model.4Conclusions and future workThis paper presented an algorithm for systematicallydiscretising CT systems with non-uniform input and/oroutput time delays, where the delays could be anycombination of the four cases: no delays, integer-multiple1643& The Institution of Engineering and Technology 2011
www.ietdl.orgFig. 7 Simulation of delayed CT SS and its DT equivalentFig. 8 Response of CT system and its DT equivalents: rounding the delays and compensating for the delaysTable 2Relative percentage errors along the simulation timeZOH method11 , %12 , %13 , %14 , %delay compensationdelay rounding0.019.10.015.60.021.60.015.8delays, fractional delays and integer-multiple plus fractionaldelays. In particular, this algorithm is capable of handlingall 28 possible cases of CT LTI systems with input and/oroutput delays. This algorithm was applied to discretise afourth-order MIMO heat exchanger process with four inputsand four outputs, where each input and output signalexperienced one of the four different delay cases. Moreover,this MIMO system was discretised by rounding the inputand output delays to the closest integer-multiple of thesampling time. The two resulting DT models weresimulated and compared against the CT model. It was notedthat the response from the DT system obtained through theproposed algorithm matched its CT counterpart at thesampling instants. However, the response obtained from theDT system obtained through rounding the delays exhibitedsignificant mismatches with the CT response.1644& The Institution of Engineering and Technology 2011Future work would study whether the stability of the DTsystem is endangered by the additional system modes,especially if the chosen sampling time is large. Since thealgorithm presented in this paper derived explicit closedform expressions for the DT equivalent system matrices,this assessment should not be troublesome. In addition, it isimportant to analyse the spectrum of the discretised systemagainst its CT counterpart, and study how the additionalpoles are distributed with respect to the poles of the originalCT system.While the algorithm considered in this paper assumed afixed sampling rate and ZOH of the inputs, extending suchformalism to the multi-rate context apart or combined withextensions to other hold techniques, such as first-order hold(FOH) is a potential future research direction. Moreover,while the derived DT equivalent system was exact, it wouldbe instructive to compare it against a DT equivalent systemderived through the Padé approximation method. Suchcomparison would
input and output time delays This section derives algorithms for discretising CT LTI systems with input and/or output time delays. The discretisation algorithms assume ZOH of the input signal, which implies that the input signal takes piecewise constant values within T, that is, {u(t) ¼ u(kT), kT t , (k 1)T}. The input signals, u(t) [ Rr .
TSI 147kW GTI DSG Wheelrims / Tyres (continued) Full size steel spare wheel with original equipment tyres X X X X - - - Space-saver spare wheel - - - - X X X Toolkit & jack X X X X X X X Dimensions / Capacities Length (mm) 4053 4053 4053 4053 4053 4053 4067 Width (mm) 1751 1751 1751 175
Centers for Medicare & Medicaid Services U.S. Department of Health and Human Services CMS-1751-P Mail Stop C4-26-05 7500 Security Boulevard Baltimore, MD 21244-1850 Attention: CMS-1751-P, RIN 0938-AU42 Subject: CY 2022 Payment Policies under the
NAME OF CA 90822, United StatesJOURNAL World Journal of Gastroenterology ISSN ISSN 1007-9327 (print) ISSN 2219-2840 (online) LAUNCH DATE October 1, 1995 FREQUENCY Weekly EDITORS-IN-CHIEF Damian Garcia-Olmo, MD, PhD, Doctor, Profes-sor, Surgeon, Department of Surgery, Universidad Autonoma de Madrid; Department of General Sur-
Civil Engineering and Environmental Systems ISSN 10286608 0.512 Q2 United Kingdom Construction Innovation ISSN 14770857, 14714175 0.508 Q2 United Kingdom Zhongguo Gonglu Xuebao/China Journal of Highway and Transport ISSN 10017372 0.502 Q2 China . Department of Civil Engineering International Journal of Offshore and Polar Engineering ISSN 10535381 0.486 Q2 United States Advances in Structural .
Int J Ayu Pharm Chem (IJAPC) e- ISSN- 2350-0204 10/01/2020 . 6 “Role of Vanaspati Swaras in Pharmaeutics of Rasaushadhi in various Disaeases Raview Through Rasashastra Text” Dr. Komal Motghare, Shrikant Solarke, Vaibhav Yeokar, IJAAM P-ISSN- 2395-3985 E- ISSN-2348-0173 Aug/2015
P-ISSN 2086-6038 E-ISSN 2580-9717 KEMENTERIAN PENDIDIKAN DAN KEBUDAYAAN Volume 9, Nomor 2, Edisi Oktober 2018 Teks Berita Republika Terkait Perceraian Ahok dan Veronica Tan Ali Kusno Dinamika Kepribadian Tokoh Nayla dalam Kumpulan Cerpen Saia Elva Yusanti
ISSN: 1992-8645 www.jatit.org E-ISSN: 1817-3195 155 In this work, the system identification methods Neural Network time series: Neural Network based on Nonlinear Auto-Regressive External (Exogenous) Input (NARX) and Neural Network based on Nonlinear Auto-Regressive (NAR) used
Annual Report 2014-2015 “ get it right, and we’ll see work which empowers and connects, work which is unique, authentic and life-affirming, work which at its best is genuinely transfor-mational ” (Nick Capaldi, Chief Exec, Arts Council of Wales, March 2015, Introduction to ‘Person-Centred Creativity’ publication, Valley and Vale Community Arts) One of the key aims and proven .
