2.
34 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 Fig. 1. Continuous-time cascaded delay system.the discrete-time sampling period. It should also discrete sample points and one additional point be-be pointed out that while the discretization and tween consecutive discrete sample points.digital control of an analog system with input time The material presented in this paper is organizeddelay can be carried out using the modiﬁed z as follows. In Section 2, we ﬁrst formulate thetransform, the existing methods are mostly devel- standard Smith predictor controller into an aug-oped for the single-input–single-output analog mented system that is subsequently used for digi-system in the frequency domain. The present tal redesign. Section 3 presents the development ofmethod uses a different approach that is capable of the intersampling states for the long and shorthandling much longer time delays relative to the input-delay systems, with the short delay casesampling period and is easily extendable to the shown as a special case of long delay. In Sectionmulti-input–multi-output analog system with in- 4, the augmented system is digitally redesignedput delay in the time domain. detailing the necessary relationships for imple- With the very large installed base of analog con- mentation. For cases where the system states aretrol systems, including Smith predictors, digital not available for measurement, an optimal discreteredesign is a very attractive approach for design- observer for input-delay systems is developed ining controllers for sampled-data systems, that Section 5. To demonstrate the effectiveness of theavoids the problems of direct digital control, yet proposed scheme, an illustrative example withenjoys the beneﬁts of ﬂexibility, reduced cost, ease simulation results for short and long delays is pre-of implementation of complex designs, etc., avail- sented in Section 6. Final observations and conclu-able in today’s digital systems. While numerous sions are presented in Section 7.approaches have been proposed for digital rede-sign, the prediction-based method ͓15͔ that uses anoptimally determined intersample parameter is 2. Augmented reformulated systemvery attractive and is used in this development. Inthe paper, the prediction-based digital redesign Consider the unity feedback time-delay systemmethod that was developed for a delay-free state shown in Fig. 1 below. According to Smith’s for-feedback system ͓15͔ is extended to discretize a mulation, we can redraw Fig. 1 as shown in Fig. 2time-delay cascaded analog Smith predictor for an below for the purpose of designing a controller toinput time-delay plant. be used in the Smith predictor loop in Fig. 3. The In this paper, a classical analog Smith predictor original system in Fig. 1 can be redrawn in asystem is reformulated into an augmented system, Smith predictor formulation as shown in Fig. 3which is then digitally redesigned using the below.prediction-based digital redesign method ͓15͔. In Consider the state-space representation for thethe prediction-based digital redesign scheme, a systems in Fig. 3 to be the following:tuning parameter v is optimally determined suchthat the states in the analog system agree very G 1 ͑ s ͒ e ϪsT d : x c1 ͑ t ͒ ϭA 1 x c1 ͑ t ͒ ϩB 1 u c1 ͑ tϪT d ͒ , ˙closely with the states of the discrete system at ͑1͒ Fig. 2. System used for Smith controller design.
3.
A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 35 Fig. 3. Smith predictor formulation. y c1 ͑ t ͒ ϭC 1 x c1 ͑ t ͒ , ͑2͒ x ec ͑ t ͒ ϭA e x ec ͑ t ͒ ϩB e u c1 ͑ tϪT d ͒ ϩF ec r ͑ t ͒ , ˙ ͑9͒ G 1 ͑ s ͓͒ 1Ϫe ϪsT d ͔ : u c1 ͑ t ͒ ϭϪK ec x ec ͑ t ͒ ϩE ec r ͑ t ͒ , ͑10͒ x c3 ͑ t ͒ ϭA 1 x c3 ͑ t ͒ ϩB 1 ͓ u c1 ͑ t ͒ ˙ ͫ ͬ where Ϫu c1 ͑ tϪT d ͔͒ , ͑3͒ A1 0 0 y c3 ͑ t ͒ ϭC 1 x c3 ͑ t ͒ , ͑4͒ ϪB 2 C 1 A2 ϪB 2 C 1 A eϭ , G 2͑ s ͒ : x c2 ͑ t ͒ ϭA 2 x c2 ͑ t ͒ ϩB 2 u c2 ͑ t ͒ , ˙ ͑5͒ ϪB 1 D 2 C 1 B 1 C 2 A 1 ϪB 1 D 2 C 1 ͫ ͬ ͫ ͬ y c2 ͑ t ͒ ϭC 2 x c2 ͑ t ͒ ϩD 2 u c2 ͑ t ͒ ϭu c1 ͑ t ͒ , ͑6͒ B1 0where B eϭ 0 , F ec ϭ B 2E c , ͑11͒ ϪB 1 B 1D 2E c x c1 ͑ t ͒ R , n1 u c1 ͑ t ͒ R , m1 K ec ϭ ͓ K c1 K c2 K c3 ͔ y c1 ͑ t ͒ Rp1 , x c2 ͑ t ͒ Rn2 , ϭ ͓ D 2C 1 ϪC 2 D 2C 1͔ , ͑12͒ u c2 ͑ t ͒ Rp1 , y c2 ͑ t ͒ Rm1 , E ec ϭD 2 E c . ͑13͒ x c3 ͑ t ͒ Rn1 , u c3 ͑ t ͒ Rm1 , y c3 ͑ t ͒ Rp1 , The system in Eq. ͑9͒ is an input-delay system forand matrices ( A 1 , B 1 , C 1 , A 2 , B 2 , C 2 , and D 2 ) which an exact method exists for developing a dis-are of appropriate dimensions with T d the overall crete model ͓16͔. The input in Eq. ͑10͒ can betime delay. viewed as an available state-feedback analog con- From Fig. 3 and Eq. ͑6͒, we have trol law, for which the corresponding digital con- u c2 ͑ t ͒ ϭe ͑ t ͒ Ϫy c3 ͑ t ͒ trol law is developed in the following sections. ϭE c r ͑ t ͒ Ϫy c1 ͑ t ͒ Ϫy c3 ͑ t ͒ ϭϪC 1 x c1 ͑ t ͒ ϪC 1 x c3 ͑ t ͒ ϩE c r ͑ t ͒ , ͑7͒ 3. Evaluation of the predicted intersampling u c1 ͑ t ͒ ϭC 2 x c2 ͑ t ͒ ϩD 2 u c2 ͑ t ͒ states ϭC 2 x c2 ͑ t ͒ ϪD 2 C 1 x c1 ͑ t ͒ In order to utilize the prediction-based digital ϪD 2 C 1 x c3 ͑ t ͒ ϩD 2 E c r ͑ t ͒ , ͑8͒ redesign scheme ͓15͔, also shown in the Appen- dix, to digitally redesign the control law in Eq.where ͑10͒, it is necessary to evaluate the predicted inter- E c Rp1ϫp1 , r ͑ t ͒ Rp1 . sampling states of the input-delay system as shown in the following sections. Two versions ofFrom Eqs. ͑1͒–͑8͒, the augmented system can be input time delays shown in Figs. 4 and 5 are con-written as sidered in this section. These are, namely, the long
4.
36 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 Fig. 4. Input-delay system for dϭ1 and v Ͼ ␥ .and short time delays, with the short time delay sponses of the discrete-time and continuous-timeshown to be a special case of the long input time- systems is minimized. While Figs. 4 and 5 onlydelay case. depict the case dϭ1 for simplicity of illustration, the following derivations are for a general delay3.1. Long and short input time-delay systems d. Consider a general input time-delay system with total integer delay d, then the following condi- Consider a sampled-data system with long input tions generally hold:time delay as shown in Fig. 4, in which the inputdelay is greater than the sample period of the dis- dу0, 0р ␥ Ͻ1, 0р v р1,crete system. In Fig. 4, ␥ is the fractional delay Tibeyond the integer multiples of delay in the over- ␥ϭ ⇒T i ϭ ␥ T,all delay time. The tuning parameter v is an inter- Tsample parameter used in the prediction-based T d ϭdTϩT i ϭdTϩ ␥ T,digital redesign scheme, that is determined online,such that the total error between the output re- t v ϭkTϩ v T. ͑14͒ Fig. 5. Input-delay system for dϭ1, and v Ͻ ␥ .
5.
A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 37Depending on the value of the tuning parameter v , wherewhich is optimally determined via digital redesign,two cases need to be considered. G ( v ) ϭe AT v ϭ ͑ e AT ͒ v ,Case 1 In this ﬁrst case, the optimal parameter v is such H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B,that v TуT i ⇒ v у ␥ . This is shown in Fig. 4. Con- dsider a general continuous-time linear system withinput time delay, given by H (dϩ1) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B. (v) x ͑ t ͒ ϭAx ͑ t ͒ ϩBu ͑ tϪT d ͒ . ˙ ͑15͒ Remark 1Evaluating Eq. ͑15͒ at tϭt v ϭkTϩ v T, we get In cases where matrix A is singular, then a gen- eral matrix W x ϭ ͓ G ( ␣ ) ϪI ͔ A Ϫ1 B can be evaluated x ͑ t v ͒ ϭx ͑ kTϩ v T ͒ as ϭe A v T x ͑ kT ͒ ϩ ͵kT tv e A(t v Ϫ) Bu ͑ ϪT d ͒ d. W xϭ ͚ ϱ 1 ͑ A ␣ T ͒ jϪ1 B ␣ T. jϭ1 j! ͑16͒In order to evaluate the integral in Eq. ͑16͒, we Case 2make the following variable substitution: The second case to be considered is for v Ͻ ␥ , as shown in Fig. 5. The result for this case is easily Let ␦ ϭϪdT⇒ϪdTϪT i ϭ ␦ ϪT i , derived from Eq. ͑18͒ and by noting that only the when ϭkT, ␦ ϭkTϪdTϭ ͑ kϪd ͒ T, ˆ set of terms Q 1 exists in this case, with v instead of ␥ in the integration limits. The corresponding when ϭt v ϭkTϩ v T, equation to Eq. ͑19͒ becomes ␦ ϭkTϩ v TϪdTϭ ͑ kϩ v Ϫd ͒ T. ͑17͒ x ͑ t v ͒ ϭx ͑ kTϩ v T ͒Eq. ͑16͒ can then be written as ϭG ( v ) x ͑ kT ͒ ϩH (dϩ1) u ͓͑ kϪdϪ1 ͒ T ͔ , (v) x ͑ t v ͒ ϭx ͑ kTϩ v T ͒ ϭe A v T x ͑ kT ͒ ϩQ 1 ϩQ 2 , ˆ ˆ ͑20͒ ͑18͒where where Q 1 ϭe A(kϩ v Ϫd)T ˆ ͭ͵ (kϪdϩ ␥ )T (kϪd)T e ϪA ␦ Bu ͑ ␦ H (dϩ1) ϭ ͓ G ( v ) ϪI ͔ A Ϫ1 B. (v) ͮ ϪT i ͒ d ␦ , 3.2. Short input time-delay system ͭ͵ The short input time-delay system is a special (kϪdϩ v )T Q 2 ϭe A(kϩ v Ϫd)T ˆ e ϪA ␦ Bu ͑ ␦ case of the long input time-delay system. Hence (kϪdϩ ␥ )T the results for this case are easily derived from the ͮ previous results by setting dϭ0. ϪT i ͒ d ␦ . Case 1 Here again, the optimal parameter v is such that v у ␥ . Substituting dϭ0 in Eq. ͑19͒, we getThis evaluates to x ͑ t v ͒ ϭx ͑ kTϩ v T ͒ x ͑ t v ͒ ϭx ͑ kTϩ v T ͒ ϭG ( v ) x ͑ kT ͒ ϩH ( v ) u ͓͑ kϪd ͒ T ͔ d ϭG ( v ) x ͑ kT ͒ ϩH ( v ) u ͓͑ kT ͔͒ 0 ϩH (dϩ1) u ͓͑ kϪdϪ1 ͒ T ͔ , (v) ͑19͒ ϩH ( v ) u ͓͑ kϪ1 ͒ T ͔ , 1 ͑21͒
6.
38 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 Fig. 6. Digitally redesigned Smith predictor for dϭ1.where The objective now is to digitally redesign this analog input using the prediction-based digital re- H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B, 0 design technique ͓15͔, such that the analog and H ( v ) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B. discrete states match very closely even at inter- 1 sample points. For this development, we assumeCase 2 that the continuous-time controller u c1 ( t ) in Eq. As stated previously, for this case, v Ͻ ␥ . Setting ͑23͒ is approximated by a piecewise-constantdϭ0 in Eq. ͑20͒, we get discrete-time controller u d1 ( kT ) , yet to be deter- mined. Also, the choice of the tuning parameter v , x ͑ t v ͒ ϭx ͑ kTϩ v T ͒ used in all the subsequent equations, is determined ϭG ( v ) x ͑ kT ͒ ϩH ( v ) u ͓͑ kϪ1 ͒ T ͔ , ͑22͒ by minimizing the following performance index, 1 in which t f is the ﬁnite time of interest,where H ( v ) ϭ ͓ G ( v ) ϪI ͔ A Ϫ1 B. ͵ ͉y 1 tf J͑ v ͒ϭ c1 ͑ t ͒ Ϫy d1 ͑ t ͒ ͉ dt, ͑25͒4. Prediction-based digital redesign 0 The augmented equations for the Smith predic-tor presented previously in Eqs. ͑9͒ and ͑10͒ are where y c1 ( t ) and y d1 ( t ) are shown in Figs. 3 and x ec ͑ t ͒ ϭA e x ec ͑ t ͒ ϩB e u c1 ͑ tϪT d ͒ ϩF ec r ͑ t ͒ , ˙ 6, respectively. We again consider two cases de- ͑23͒ pending on the relative values of the parameters v and ␥ as shown in Figs. 4 and 5. u c1 ͑ t ͒ ϭϪK ec x ec ͑ t ͒ ϩE ec r ͑ t ͒ , ͑24͒ Case 1where u c1 ( t ) in Eq. ͑24͒ is the control input before In this case, v у ␥ as shown in Fig. 4. From Eq.the input time delay of the plant G 1 ( s ) and r ( t ) is ͑19͒, we can write the equivalent discrete expres-a constant setpoint. sion for x ec ( t ) in Eq. ͑23͒ as
7.
A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 39x ec ͑ kTϩ v T ͒ ϭG ( v ) x ec ͑ kT ͒ ϩH ( v ) u c1 ͓͑ kϪd ͒ T ͔ e ed which can be written as ϩH e(dϩ1) u c1 ͓͑ kϪdϪ1 ͒ T ͔ (v) u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ ed u d1 ͓͑ kϪd ͒ T ͔ ϪJ e(dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔ ϩ ͵ kT tv e A(t v Ϫ) F ec r ͑ ͒ d, ͑26͒ ϩE ed r ͑ kT ͒ , ͑31͒which evaluates to wherex ec ͑ kTϩ v T ͒ ϭG ( v ) x ec ͑ kT ͒ ϩH ( v ) u c1 ͓͑ kϪd ͒ T ͔ e ed K ed ϭK ec G ( v ) ϭ ͓ K d1 e K d2 K d3 ͔ , ϩH e(dϩ1) u c1 ͓͑ kϪdϪ1 ͒ T ͔ (v) J ed ϭK ec H ( v ) , ed ϩE ( v ) r ͑ kT ͒ , e ͑27͒ J e(dϩ1) ϭK ec H e(dϩ1) , (v)where E ed ϭ ͓ E ec ϪK ec E ( v ) ͔ . e G ( v ) ϭe A e v T , e For the special case when dϭ0, Eq. ͑31͒ becomes H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B e , ed e e u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ e1 u d1 ͓͑ kϪ1 ͒ T ͔ H e(dϩ1) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B e , (v) ϩE ed r ͑ kT ͒ , ͑32͒ e e e where E ( v ) ϭ ͓ G ( v ) ϪI ͔ A Ϫ1 F ec . e e e H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B e ,The discrete model of Eq. ͑23͒ is given ͓16͔ as e0 e e x ed ͑ kTϩT ͒ ϭG e x ed ͑ kT ͒ ϩH ed u d1 ͓͑ kϪd ͒ T ͔ H ( v ) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B e , e1 e e e ϩH e(dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔ K ed ϭ ͓ IϩK ec H ( v ) ͔ Ϫ1 K ec G ( v ) e0 e ϩE e r ͑ kT ͒ , ͑28͒ ϭ ͓ K d1 K d2 K d3 ͔ ,where J e1 ϭ ͓ IϩK ec H ( v ) ͔ Ϫ1 K ec H ( v ) , e0 e1 G e ϭe A e T , E ed ϭ ͓ IϩK ec H ( v ) ͔ Ϫ1 ͓ E ec ϪK ec E ( v ) ͔ . e0 e H ed ϭ ͓ G (1Ϫ ␥ ) ϪI ͔ A Ϫ1 B e , e e To illustrate the procedure and for use as part of the subsequent simulation, let dϭ1. Then we get H e(dϩ1) ϭ ͓ G e ϪG (1Ϫ ␥ ) ͔ A Ϫ1 B e , e e from Eqs. ͑28͒ and ͑31͒ that E e ϭ ͓ G e ϪI ͔ A Ϫ1 F ec . e x ed ͑ kTϩT ͒ ϭG e x ed ͑ kT ͒ ϩH e1 u d1 ͓͑ kϪ1 ͒ T ͔Using the prediction-based digital control law ϩH e2 u d1 ͓͑ kϪ2 ͒ T ͔ ϩE e r ͑ kT ͒ ,͑A10͒ in the Appendix, we can substitute Eq. ͑27͒ ͑33͒into the following continuous-time control law,which is equivalent to Eq. ͑24͒: where u d1 ͑ kT ͒ ϭϪK ec x ec ͑ t v ͒ ϩE ec r ͑ t v ͒ . ͑29͒ G e ϭe A e T ,As a result, we get H e1 ϭ ͓ G (1Ϫ ␥ ) ϪI ͔ A Ϫ1 B e , e e u d1 ͑ kT ͒ ϭϪK ec ͕ G ( v ) x ed ͑ kT ͒ ϩH ( v ) u d1 ͓͑ k e ed H e2 ϭ ͓ G e ϪG (1Ϫ ␥ ) ͔ A Ϫ1 B e , e e Ϫd ͒ T ͔ ϩH e(dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔ (v) E e ϭ ͓ G e ϪI ͔ A Ϫ1 F ec . e ϩE ( v ) r ͑ kT ͒ ͖ ϩE ec r ͑ kT ͒ , e ͑30͒ For the control law, we get
8.
40 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ e1 u d1 ͓͑ kϪ1 ͒ T ͔ which can be written as ϪJ e2 u d1 ͓͑ kϪ2 ͒ T ͔ ϩE ed r ͑ kT ͒ , u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ e(dϩ1) u d1 ͓͑ kϪd ͑34͒ Ϫ1 ͒ T ͔ ϩE ed r ͑ kT ͒ , ͑37͒where where G ( v ) ϭe A e v T , e K ed ϭK ec G ( v ) ϭ ͓ K d1 e K d2 K d3 ͔ , H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B e , J e(dϩ1) ϭK ec H e(dϩ1) , (v) e1 e e H ( v ) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B e , E ed ϭ ͓ E ec ϪK ec E ( v ) ͔ . e e2 e e e Again, to illustrate the procedure and for later E ( v ) ϭ ͓ G ( v ) ϪI ͔ A Ϫ1 F ec , e e e simulation, let dϭ1. Then from Eqs. ͑28͒ and ͑37͒ we have K ed ϭK ec G ( v ) ϭ ͓ K d1 e K d2 K d3 ͔ , x ed ͑ kTϩT ͒ ϭG e x ed ͑ kT ͒ ϩH e1 u d1 ͓͑ kϪ1 ͒ T ͔ J e1 ϭK ec H ( v ) , e1 ϩH e2 u d1 ͓͑ kϪ2 ͒ T ͔ ϩE e r ͑ kT ͒ , J e2 ϭK ec H ( v ) , e2 ͑38͒ E ed ϭ ͓ E ec ϪK ec E ( v ) ͔ . e whereEq. ͑33͒ can be rewritten as G e ϭe A e T , ͫ x d1 ͑ kTϩT ͒ ͬͫ G 11 0 0 x d2 ͑ kTϩT ͒ ϭ G 21 G 22 G 23 x d3 ͑ kTϩT ͒ G 31 G 32 G 33 ͬͫ ͬx d1 ͑ kT ͒ x d2 ͑ kT ͒ x d3 ͑ kT ͒ H e1 ϭ ͓ G (1Ϫ ␥ ) ϪI ͔ A Ϫ1 B e , e e e H e2 ϭ ͓ G e ϪG (1Ϫ ␥ ) ͔ A Ϫ1 B e , e ͫ ͬ E e ϭ ͓ G e ϪI ͔ A Ϫ1 F ec . e H e11 ϩ H e12 u d1 ͑ kTϪT ͒ For the control law H e13 u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ e2 u d1 ͓͑ kϪ2 ͒ T ͔ ͫ ͬ H e21 ϩ H e22 u d1 ͑ kTϪ2T ͒ H e23 where ϩE ed r ͑ kT ͒ , ͑39͒ ϩ 0 ͫ ͬ E e2 r ͑ kT ͒ . E e3 ͑35͒ G ( v ) ϭe A e v T , e H ( v ) ϭ ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B e , e2 eFrom Eqs. ͑34͒ and ͑35͒, we can develop the K ed ϭK ec G ( v ) ϭ ͓ K d1 e K d2 K d3 ͔ ,simulation diagram shown in Fig. 6.Case 2 J e2 ϭK ec H ( v ) , e2 In this case, v Ͻ ␥ as shown in Fig. 5. Substitut- E ed ϭ ͓ E ec ϪK ec E ( v ) ͔ .ing from Eq. ͑20͒ into Eq. ͑29͒, we get the digital econtrol law as Remark 2 Due to the speciﬁc structures of the system ma-u d1 ͑ kT ͒ ϭϪK ec ͕ G ( v ) x ed ͑ kT ͒ ϩH e(dϩ1) u d1 ͓͑ k (v) e trix A e , the input vector B e in Eq. ͑11͒ and the ϪdϪ1 ͒ T ͔ ϩE ( v ) r ͑ kT ͒ ͖ ϩE ec r ͑ kT ͒ , virtual feedback gain K ec in Eq. ͑12͒, all digital e gains J ep ͓for pϭd, ( dϩ1 ) ] in Eqs. ͑31͒ and ͑37͒ ͑36͒ are zero. To verify this, consider
9.
A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 41 A eB eϭ ͫ A1 ϪB 2 C 1 0 A2 ϪB 1 D 2 C 1 B 1 C 2 A 1 ϪB 1 D 2 C 1 0 ϪB 2 C 1 ͬ Let the discrete model of Eqs. ͑40͒ and ͑41͒ be given by x c1 ͑ kTϩT ͒ ϭG 1 x c1 ͑ kT ͒ ϩH d u d1 ͓͑ kϪd ͒ T ͔ ϫ B1 ͫ ͬͫ ͬ 0 ϭ ϪB 1 A 1B 1 0 ϪA 1 B 1 , ϩH (dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔ , y c1 ͑ kT ͒ ϭC 1 x c1 ͑ kT ͒ , ͑42͒ ͑43͒ A 2 B e ϭA e ͑ A e B e ͒ ϭ e 1 0ͫ ͬ A 2B 1 ϪA 2 B 1 1 , where ␥ is as shown in Figs. 4 and 5, for the long and short time-delay cases. Also, T d ϭdTϩ ␥ T,then Ti ␥ϭ ,H ( ␣ ) ϭ ͓ G ( ␣ ) ϪI ͔ A Ϫ1 B e e e e T ϱ ␣T G 1 ϭe A 1 T , ϭ͚ ͑ ␣ TA e ͒ jϪ1 B e ͫ ͬ j! jϭ1 H d ϭ ͓ G (1Ϫ ␥ ) ϪI ͔ A Ϫ1 B 1 , 1 1 ϱ ␣T H (dϩ1) ϭ ͓ G 1 ϪG (1Ϫ ␥ ) ͔ A Ϫ1 B 1 . ͫ ͬ ͚ jϭ1 j! ͑ ␣ TA 1 ͒ jϪ1 B 1 H (␣) 1 1 e1 The desired digital observer is given by ϭ 0 ϭ 0 , ϱ ␣T ϪH ( ␣ ) e1 x o ͑ kTϩT ͒ ϭG 1 x o ͑ kT ͒ ϩH d u d1 ͓͑ kϪd ͒ T ͔ ˆ ˆ Ϫ͚ ͑ ␣ TA 1 ͒ jϪ1 B 1 jϭ1 j! ϩH (dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔ K ec H ( ␣ ) ϭ ͓ D 2 C 1 e ϪC 2 D 2C 1͔ H (␣) 0 e1 ͫ ͬ ϪH ( ␣ ) e1 ϭ0, where ϪK o e y ͑ kT ͒ , e y ͑ kT ͒ ϭC 1 ͓ x o ͑ kT ͒ Ϫx c1 ͑ kT ͔͒ ˆ ͑44͒which implies ϭy o ͑ kT ͒ Ϫy c1 ͑ kT ͒ , ˆ ͑45͒ J e ϭK ec H ( ␣ ) ϭ ͓ IϩK ec H ( ␣ ) ͔ Ϫ1 K ec H ( ␣ ) ϭ0. e eo e y o ͑ kT ͒ ϭC 1 x o ͑ kT ͒ . ˆ ˆ ͑46͒5. Digital state observer design From Eqs. ͑44͒ and ͑42͒, In order to implement the system in Fig. 6, the x o ͑ kTϩT ͒ Ϫx c1 ͑ kTϩT ͒ ˆdiscrete state x d1 ( kT ) must be available for mea-surement. In some practical cases, this state may ϭG 1 ͓ x o ͑ kT ͒ Ϫx c1 ͑ kT ͔͒ ϪK o C 1 ͓ x o ͑ kT ͒ ˆ ˆnot be accessible and an observer will be requiredto estimate it. Ϫx c1 ͑ kT ͔͒ . ͑47͒ Consider the following state-space representa- Lettion of an input time-delay system, e ͑ kT ͒ ϭx o ͑ kT ͒ Ϫx c1 ͑ kT ͒ , ˆ ͑48͒ x c1 ͑ t ͒ ϭA 1 x c1 ͑ t ͒ ϩB 1 u c1 ͑ tϪT d ͒ , ˙ ͑40͒ then from Eq. ͑47͒, we get y c1 ͑ t ͒ ϭC 1 x c1 ͑ t ͒ , ͑41͒ e ͑ kTϩT ͒ ϭ ͓ G 1 ϪK o C 1 ͔ e ͑ kT ͒ . ͑49͒where x c1 ( t ) Rn1 , u c1 ( t ) Rm1 , y c1 ( t ) Rp1and ( A 1 , B 1 , and C 1 ) are known constant matri- We now need to ﬁnd the optimal K o in Eq. ͑49͒ces of appropriate dimensions. such that
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42 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 lim ͕ e ͑ kT ͒ ͖ →0. ͑50͒ which the PID controller has been pre-designed to k→ϱ meet the following control speciﬁcations ͓14,18͔: velocity error constant, K v ϭ20, crossover fre-This optimal K o can be designed from duality. quency, c ϭ5 rad/s, damping ratio, ϭ0.7. LetConsider the following delay-free system: the transfer functions for the plant and controller x ͑ kTϩT ͒ ϭGx ͑ kT ͒ ϩHu ͑ kT ͒ , ͑51͒ in Fig. 2 be given as u ͑ kT ͒ ϭϪK d x ͑ kT ͒ . ͑52͒ 6000 G 1͑ s ͒ ϭ , ͑61͒ ͑ s 2 ϩ32.44sϩ20͒͑ sϩ30͒Then, for a free ﬁnal-state closed-loop system withperformance index given by s 2 ϩ10.42sϩ20 Ki K ds G 2͑ s ͒ ϭ ϭK p ϩ ϩ , 1 1 NϪ1 s ͑ sϩ10͒ s ͑ sϩ ␣ ͒Jϭ x T ͑ NT ͒ Px ͑ NT ͒ ϩ 2 2 ͚ kϭ1 ͓ x T ͑ kT ͒ Qx ͑ kT ͒ ͑62͒ where the PID controller parameters are K p ϩu T ͑ kT ͒ Ru ͑ kT ͔͒ , ͑53͒ ϭ0.842, K i ϭ2, K d ϭ0.158, and ␣ ϭ10. The state-space models for G 1 ( s ) and G 2 ( s ) are ͫ ͬ ͫͬwhere Qу0, RϾ0, and PϾ0, with ( G,H ) con-trollable and ( G,Q ) observable, the steady-state Ϫ62.44 Ϫ993.20 Ϫ600gain is given by the solution to the Riccati equa- 1tions ͓17͔ as A 1ϭ 1 0 0 , B 1ϭ 0 , 0 1 0 0 Ϫ1 PϭG ͓ PϪ PH ͑ H PHϩR ͒ T T H P ͔ GϩQ, T ͑54͒ C 1 ϭ ͓ 0 0 6000͔ , D 1 ϭ0, ͑63͒ ͫ ͬ Ϫ1 K d ϭ ͑ H PHϩR ͒ T T ͑55͒ ͫͬ H PG, Ϫ10 0 1giving the closed-loop system as A 2ϭ , B 2ϭ 0 , 1 0 x ͑ kTϩT ͒ ϭ ͓ GϪHK d ͔ x ͑ kT ͒ . ͑56͒ C 2 ϭ ͓ 0.42 20͔ , D 2 ϭ1. ͑64͒The dual system of Eqs. ͑51͒ and ͑52͒ is given by The bandwidth of the delay-free system with the x ͑ kTϩT ͒ ϭG T x ͑ kT ͒ ϩC T u ͑ kT ͒ , 1 ͑57͒ above G 1 ( s ) , G 2 ( s ) , is b ϭ9.56 rads/s. The sampling period T can be approximately evaluated u ͑ kT ͒ ϭϪK T x ͑ kT ͒ . d ͑58͒ ͓16͔ as TХ / ( 3 – 10) b ϭ0.033– 0.11 s. In Ref.From Eqs. ͑54͒ and ͑55͒, we get the desired gain ͓14͔, the sampling period was chosen as 0.035 andas 0.07 s. In this paper, simulation runs are shown for sampling periods 0.035 and 0.1 s, with E c ϭ1 and K o ϭ ͓͑ C 1 PC T ϩR ͒ Ϫ1 C 1 PG T ͔ T 1 1 ͑59͒ r ( t ) being a step input applied at time tϭ0. It should be noted that the true sampling period ofwith the solution for P given by the following this system is closer to 0.035 s and that the simu-Riccati equation: lations for 0.1 s are only shown to illustrate the PϭG 1 ͓ PϪ PC T ͑ C 1 PC T ϩR ͒ Ϫ1 C 1 P ͔ G T ϩQ, tolerance of the system to inaccurate sampling pe- 1 1 1 ͑60͒ riod selection. In general, in selecting a suitable sampling period with respect to the dead time, awhere G T in Eq. ͑54͒ equals G 1 in Eq. ͑59͒ and H bisection search method is suggested to ﬁnd anin Eq. ͑54͒ equals C T in Eq. ͑59͒. 1 appropriate sampling period, so that a reasonable tradeoff between the closed-loop response and the6. Illustrative example stability of the closed-loop response can be achieved. Consider the unity output feedback continuous- From the development, we have for the case intime delay control system shown in Fig. 2, in Eq. ͑9͒ with Tϭ0.1 s and T d ϭ0.17 s, that
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44 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47Fig. 7. Continuous-time and discrete-time system outputs Fig. 8. Continuous-time and discrete-time control laws forfor delays, dϭ0, 1, and 2. delays, dϭ0, 1, and 2. the system was potentially unstable at the inaccu-Similarly, the digitally redesigned control gains rate sampling period, Tϭ0.1 s, for dϾ5, whilefor the digital control law in Eq. ͑34͒ with T stable at the correct sampling period, Tϭ0.1 s, dϭ1, v ϭ0.95, and ␥ ϭ0.7 are given by ϭ0.035 s. In general, the system also shows good K ed ϭ ͓ 4.6 308.8 4966.8 Ϫ1.2 robustness to sampling period selection, even in the case where the chosen sampling period is Ϫ15.3 4.6 308.8 4966.8͔ , longer than the sampling period chosen in Ref. ͓14͔. J e1 ϭϪ2.8623e Ϫ16, A representative case for the observer perfor- J e2 ϭϪ2.3065e Ϫ14, mance is shown in Figs. 10 and 11 for dϭ1, with Qϭ103 I and RϭI. E ed ϭ0.8514. ͑69͒ 7. ConclusionSimulating the example as shown in Fig. 6, usingthe calculated coefﬁcients above, with initial con- This paper has proposed a digital redesignditions x c1 ( 0 ) ϭ ͓ 0 0 0 ͔ T and x o( 0 ) ˆ scheme for the analog Smith predictor, that forcesϭ ͓ 0.001 0.0001 0.00001͔ T , we get the resultsshown in Figs. 7–11 below. The plots in Figs. 7and 8 show the plant outputs and control laws fortwo sampling periods, Tϭ0.035 s and Tϭ0.1 s,with integer time delays of dϭ0,1, and 2. Fig. 9shows outputs and control laws for cases dϭ5 and10. For each case, the optimally determined inter-sample parameter v is shown with the correspond-ing error between the continuous-time anddiscrete-time systems. As expected, larger errorsin J ( v ) occur as the delay d increases. As theplots demonstrate, the proposed system providesgood performance even for delays up to ten sam-pling periods, when the sampling period is suit-ably selected. Not surprisingly, as the delay in-creases the need for more careful sampling periodselection becomes crucial. This point was seen Fig. 9. Continuous-time and discrete-time outputs and con-during the simulation, where it was observed that trol laws for delays, dϭ5 and 10.
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A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 45 Acknowledgments This work was supported in part by the US Army Research Ofﬁce, under Grant No. DAAD 19-02-1-0321 and the National Science Council of the Republic of China, under Contract No. NSC- 91-2213-E006-050. Appendix: Development of the prediction- based digital redesign method Consider a linear controllable continuous-time system described by ͓15͔Fig. 10. Observer-based continuous-time and discrete-time x c ͑ t ͒ ϭAx c ͑ t ͒ ϩBu c ͑ t ͒ , x c ͑ 0 ͒ ϭx 0 , ͑A1͒ ˙system outputs for delay, dϭ1. where x c ( t ) R n , u c ( t ) R m , and A and B are constant matrices of appropriate dimensions. Let the continuous-time state-feedback controller be u c ͑ t ͒ ϭϪK c x c ͑ t ͒ ϩE c r ͑ t ͒ , ͑A2͒state matching between the continuous-time anddiscrete-time system states, even at intersample where K c R mϫn and E c R mϫm have been de-points. The proposed method reformulates a tradi- signed to satisfy some speciﬁed goals, and r ( t )tional analog Smith predictor into an augmented R m is a piecewise-constant reference input vec-system, which is then digitally redesigned using tor. The controlled system isthe predicted intersampling states. As evidencedby the simulation plots, the proposed system is x c ͑ t ͒ ϭA c x c ͑ t ͒ ϩBE c r ͑ t ͒ , x c ͑ 0 ͒ ϭx 0 , ˙capable of dealing with long delays, several times ͑A3͒greater than the sampling period. The method also where A c ϭAϪBK c . Let the state equation of ademonstrates good robustness in relation to the corresponding hybrid model bediscrete-time sampling period selection for moder-ate time delays. x d ͑ t ͒ ϭAx d ͑ t ͒ ϩBu d ͑ t ͒ , x d ͑ 0 ͒ ϭx 0 , ͑A4͒ ˙ where u d ( t ) R m is a piecewise-constant input vector, satisfying u d ͑ t ͒ ϭu d ͑ kT ͒ for kTрtϽ ͑ kϩ1 ͒ T and TϾ0 is the sampling period. Let the discrete- time state-feedback controller be u d ͑ kT ͒ ϭϪK d x d ͑ kT ͒ ϩE d r * ͑ kT ͒ , ͑A5͒ where K d R mϫn is a digital state-feedback gain, E d R mϫm is a digital feedforward gain, and r * ( kT ) R m is a piecewise-constant reference in- put vector to be determined in terms of r ( t ) for tracking purposes. The digitally controlled closed- loop system thus becomes x d ͑ t ͒ ϭAx d ͑ t ͒ ϩB ͓ ϪK d x d ͑ kT ͒ ˙ ϩE d r * ͑ kT ͔͒ , x d ͑ 0 ͒ ϭx 0 ͑A6͒Fig. 11. Observer-based continuous-time and discrete-timecontrol laws for delay, dϭ1. for kTрtϽ ( kϩ1 ) T.
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46 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 A zero-order-hold device is used for Eq. ͑A5͒. Thus from Eqs. ͑A8͒ and ͑A9͒ it follows that toThe digital redesign problem is to ﬁnd digital con- obtain the state x c ( t v ) ϭx d ( t v ) , under the assump-troller gains ( K d ,E d ) in Eq. ͑A5͒ from the analog tion of x c ( kT ) ϭx d ( kT ) , it is necessary to havegains ( K c ,E c ) in Eq. ͑A2͒, so that the closed-loop u d ( kT ) ϭu c ( t v ) . This leads to the followingstate x d ( t ) in Eq. ͑A6͒ can closely match the prediction-based digital controller:closed-loop state in Eq. ͑A3͒ at all the samplinginstants, for a given r ( t ) ϵr ( kT ) , kϭ0,1,2,... . u d ͑ kT ͒ ϭu c ͑ t v ͒ The state x c ( t ) in Eq. ͑A1͒, at tϭt v ϭKTϩ v T ϭϪK c x c ͑ t v ͒ ϩE c r ͑ t v ͒for 0р v Ͻ1, is found to be ϭϪK c x d ͑ t v ͒ ϩE c r ͑ t v ͒ , ͑A10͒ x c ͑ t v ͒ ϭexp͓ A ͑ t v ϪkT ͔͒ x c ͑ kT ͒ ϩ ͵ kT tv exp͓ A ͑ t v where the future state x d ( t v ) ͑denoted as the pre- dicted state͒ needs to be predicted based on the Ϫ ͔͒ Bu c ͑ ͒ d . ͑A7͒ available causal signals x d ( kT ) and u d ( kT ) . Substituting the predicted state x d ( t v ) in Eq.Let u c ( t v ) be a piecewise-constant input. Then Eq. ͑A9͒ into Eq. ͑A10͒ and then solving it for u d ( kT )͑A7͒ reduces to results in x c ͑ t v ͒ Ϸexp͑ A v T ͒ x c ͑ kT ͒ ϩ ͵ kT kTϩ v T exp͓ A ͑ kT u d ͑ kT ͒ ϭ ͑ I m ϩK c H ( v ) ͒ Ϫ1 ͓ ϪK c G ( v ) x d ͑ kT ͒ ϩE c r ͑ t v ͔͒ . ͑A11͒ ϩ v TϪ ͔͒ Bd u c ͑ t v ͒ Consequently, the desired predicted digital con- ϭG (v) x c ͑ kT ͒ ϩH (v) u c͑ t v ͒ , ͑A8͒ troller ͑A5͒ is found, from Eq. ͑A11͒, to be u d ͑ kT ͒ ϭϪK ( v ) x d ͑ kT ͒ ϩE ( v ) r * ͑ kT ͒ , d dwhere ͑A12͒ G (v) ϭexp͓ A ͑ t v ϪKT ͔͒ where, for tracking purposes, r * ( kT ) ϭr ( kT ϭexp͑ A v T ͒ ϭ ͓ exp͑ AT ͔͒ vϭ ͑G͒ v, ϩ v T ) , and K ( v ) ϭ ͑ I m ϩK c H ( v ) ͒ Ϫ1 K c G ( v ) , ͵ tv d H (v)ϭ exp͓ A ͑ t v Ϫ ͔͒ Bd kT E ( v ) ϭ ͑ I m ϩK c H ( v ) ͒ Ϫ1 E c . d ϭ͵ vT In particular, if v ϭ1 then the prerequisite exp͑ A ͒ Bd x c ( kT ) ϭx d ( kT ) is ensured. Thus, for any k 0 ϭ0,1,2,..., the controller is given by ϭ ͓ G ( v ) ϪI n ͔ A Ϫ1 B. u d ͑ kT ͒ ϭϪK d x d ͑ kT ͒ ϩE d r * ͑ kT ͒ , ͑A13͒ Ϫ1Here, it must be noted that ͓ G ϪI n ͔ A (v) is a whereshorthand notation, which is well deﬁned as canbe veriﬁed by cancellation of A Ϫ1 in the series K d ϭ ͑ I m ϩK c H ͒ Ϫ1 K c G,expansion of the term ͓ G ( v ) ϪI n ͔ . This convenientnotation for an otherwise long series is used E d ϭ ͑ I m ϩK c H ͒ Ϫ1 E c ,throughout this appendix. Also, the state x d ( t ) ofEq. ͑A4͒, at tϭt v ϭkTϩ v T for 0р v р1, is ob- r * ͑ kT ͒ ϭr ͑ kTϩT ͒ ,tained as in which x d ͑ t v ͒ ϭexp͓ A ͑ t v ϪkT ͔͒ x d ͑ kT ͒ ϩ ͵ kT tv exp͓ A ͑ t v Gϭexp͑ AT ͒ and Hϭ ͑ GϪI n ͒ A Ϫ1 B. In selecting a suitable sampling period for the Ϫ ͔͒ Bd u d ͑ kT ͒ digital redesign method, a bisection searching method is suggested to ﬁnd an appropriate long ϭG ( v ) x d ͑ kT ͒ ϩH ( v ) u d ͑ kT ͒ . ͑A9͒ sampling period, so that the reasonable tradeoff
15.
A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 47between the closed-loop response ͓i.e., matching ͓18͔ Chen, C. F. and Shieh, L. S., An algebraic method forof the states x c ( kT ) in Eq. ͑A8͒ and x d ( kT ) in Eq. control systems design. Int. J. Control 11, 717–739 ͑1970͒.͑A9͔͒ and the stability of the closed-loop systemcan be achieved.References Alex C. Dunn received B.S., ͓1͔ Astrom, K. J. and Hagglund, T., PID Controllers: M.S., and Ph.D. degrees in Theory, Design and Tuning. Instrument Society of electrical engineering from the America, Research Triangle Park, NC, 1995. University of Sierra Leone ͑Si- erra Leone͒, 1976, the Univer- ͓2͔ Tan, K. K., Wang, Q. G., and Hang, C. C., Advances sity of Aston ͑UK͒, 1982, and in PID Control. Springer-Verlag, London, 1999. the University of Houston ͓3͔ Morari, M. and Zaﬁrriou, E., Robust Process Control. ͑USA͒, 2003, respectively. Prentice-Hall, Englewood Cliffs, NJ, 1989. Alex has an extensive industry ͓4͔ Marshall, J. E., Gorecki, H., and Walton, K., Time background in control systems Delay Systems: Stability and Performance Criteria and information technology, having worked for companies With Applications, 1st ed. Ellis Horwood, New York, such as Shell Oil, Honeywell, 1992. Inc., and Setpoint Inc./Aspen ͓5͔ Laughlin, D. L., Rivera, D. E., and Morari, M., Smith Tech. His research interests include multivariable control of industrial predictor design for robust performance. Int. J. Control plants, intelligent controls via soft computing techniques, and digital 46, 477–504 ͑1987͒. control of input time-delay and constrained nonlinear systems. ͓6͔ Astrom, K. J., Hang, C. C., and Lim, B. C., A new Smith predictor for controlling a process with an inte- grator and long dead-time. IEEE Trans. Autom. Con- Leang-San Shieh received his trol 39, 343–345 ͑1994͒. B.S. degree from the National ͓7͔ Hagglund, T., A predictive PI controller for processes Taiwan University, Taiwan in with long dead times. IEEE Control Syst. Mag. 12, 1958, and his M.S. and Ph.D. degrees from the University of 57– 60 ͑1992͒. Houston, Houston, Texas, in ͓8͔ Huang, J. J. and DeBra, D. B., Automatic Smith- 1968 and 1970, respectively, predictor tuning using optimal parameter mismatch. all in electrical engineering. IEEE Trans. Control Syst. Technol. 10, 447– 459 He is a professor in the Depart- ͑2002͒. ment of Electrical and Com- ͓9͔ Tan, K. K., Lee, T. H., and Leu, F. M., Predictive PI puter Engineering and the di- rector of the Computer and versus Smith control for dead-time compensation. ISA Systems Engineering. He was Trans. 40, 17–29 ͑2000͒. the recipient of more than ten͓10͔ Fliess, M., Marquez, R., and Mounier, H., An exten- College Outstanding Teacher Awards, the 1973 and 1997 College sion of predictive control, PID regulation and Smith Teaching Excellence Awards, and the 1988 College Senior Faculty predictors to some linear delay systems. Int. J. Control Research Excellence Award from the Cullen College of Engineering, 75, 728 –743 ͑2002͒. University of Houston, and the 1976 University Teaching Excellence Award and the 2002 El Paso Faculty Achievement Award from the͓11͔ Vrecko, D., Vrancic, D., Juricic, D., and Strmcnik, S., University of Houston. He has published more than two hundred ar- A new modiﬁed Smith predictor: The concept, design ticles in various referred scientiﬁc journals. His ﬁelds of interest are and tuning. ISA Trans. 40, 111–121 ͑2001͒. digital control, optimal control, self-tuning control, and hybrid control͓12͔ Zhang, W. and Xu, X., Analytical design and analysis of uncertain systems. of mismatched Smith predictor. ISA Trans. 40, 133– 138 ͑2001͒.͓13͔ Smith, O. J. M., Closer control of loops with dead Shu-Mei Guo received the time. Chem. Eng. Prog. 53, 217–219 ͑1957͒. M.S. degree in Department of͓14͔ Guo, S. M., Wang, W., and Shieh, L. S., Discretization Computer and Information of two degree-of-freedom controller and system with Science from the New Jersey state, input and output delays. IEE Proc.: Control Institute of Technology, USA in 1987. She received the Theory Appl. 147, 87–96 ͑2000͒. Ph.D. degree in Computer and͓15͔ Guo, S. M., Shieh, L. S., Chen, G., and Lin, C. F., Systems Engineering from the Effective chaotic orbit tracker: A prediction-based University of Houston, USA in digital redesign approach. IEEE Trans. Circuits Syst., May 2000. Since June 2000, I: Fundam. Theory Appl. 47, 1557–1570 ͑2000͒. she has been an assistant pro-͓16͔ Astrom, K. J. and Wittenmark, B., Computer Con- fessor in the Department of Computer System and Infor- trolled Systems. Prentice-Hall, Upper Saddle River, mation Engineering, National NJ, 1997. Cheng-Kung University, Taiwan. Her research interests include vari-͓17͔ Goodwin, G. C., Graebe, S. F., and Salgado, M. E., ous applications on evolutionary programming, chaos systems, Kal- Control System Design. Prentice-Hall, Upper Saddle man ﬁltering, fuzzy methodology, sampled-data systems, image pro- River, NJ, 2001. cessing, and computer and systems engineering.
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