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- 1. ISA Transactions 46 (2007) 493–503 www.elsevier.com/locate/isatrans Modiﬁed Smith predictor design for periodic disturbance rejection Han-Qin Zhou a , Qing-Guo Wang b,∗ , Liu Min b a Control Systems Engineering, Jacobs Engineering Group, Inc., Houston, TX 77072, USA b Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore Received 14 September 2005; accepted 5 March 2007 Available online 30 July 2007Abstract In this paper, a modiﬁed Smith predictor control scheme is proposed for periodic disturbance rejection in both stable and unstable processeswith time delay. Without affecting the superior setpoint response of Smith predictor control, the regulation performance under periodic disturbancecan be enhanced signiﬁcantly by the proposed design. Meanwhile, the asymptotical rejection for non-periodic disturbance will also be improved.Internal stability is investigated for a closed-loop control system. The effectiveness of the proposed scheme will be demonstrated by simulationsas well as a test on an experimental thermal system. c 2007, ISA. Published by Elsevier Ltd. All rights reserved.Keywords: Smith predictor; Periodic disturbance; Unstable process; Time delay; PID control1. Introduction in industrial manufacturing processes. Two-degree-of-freedom (2DOF) SP designs have been reported [7,8] for improved In process control, the Smith predictor (SP) [1] is a disturbance attenuation. Essentially, the disturbance is ﬁnallywell-known dead time compensator for stable processes with eliminated due to the integrating action of the controller,large time delays. When the model is accurate, the closed- which is similar to what can be achieved by typical 2DOFloop characteristic equation will be delay free; thus the PI/PID feedback control. Therefore, zero steady-state errorSmith predictor structure has signiﬁcantly facilitated controller is asymptotically achievable only in the face of a constantdesign compared with conventional single-input–single-output disturbance. Nevertheless, with further exploration, it has been(SISO) feedback systems. However, this impressive advantage discovered that the aforementioned 2DOF SP scheme can bewas originally restricted to application on stable processes. enhanced with the capability to handle periodic disturbances,To overcome this limitation, many efforts have been made which exist in many scenarios such as industrial electric motors,to modify the original Smith predictor control scheme. [2] disk drive servomechanisms and power supply systems.presented a modiﬁed Smith predictor for integrator plus dead To counteract periodic disturbances, many control methodstime processes, which can achieve faster setpoint tracking and have been developed basically under two frameworks: feedbackload disturbance rejection. [3] considered the same problem and control with internal model principle (IMP) and activeproposed a more convenient tuning rule. The Smith predictor feedforward cancellation (AFC). Repetitive control [9–12] iscontrol for unstable processes was considered by [4,5] and based on the IMP of the underlying property that a linear[6]. These modiﬁed SP schemes commonly have an inner feedback system has perfect disturbance rejection at somestabilizing loop and employ more controllers. frequency where the controller gain is inﬁnity. Since the Having greatly improved setpoint responses, the Smith additional inner positive feedback loop reduces the stabilitypredictor control structure unfortunately remains deﬁcient in margin, this scheme has to make a tradeoff between systemdisturbance attenuation, which is one of the primary concerns stability and disturbance rejection. AFC [13,14] provides the appealing advantage that feedforward compensation can be ﬂexibly added in an existing control system without affecting its ∗ Corresponding author. Tel.: +65 6516 2282; fax: +65 6779 1103. stability. Yet its application is generally limited to measurable E-mail address: elewqg@nus.edu.sg (Q.-G. Wang). disturbances.0019-0578/$ - see front matter c 2007, ISA. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.isatra.2007.03.007
- 2. 494 H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 Fig. 1. Modiﬁed Smith predictor control system. In this paper, a modiﬁed SP control system is proposed asan alternative for periodic disturbance control. In this scheme, ˆ G c1 (s)G 0 (s)e−Ls [1 + G c2 (s)G c3 (s)G 0 (s)e− Ls ] ˆa periodic disturbance can be eliminated, provided that the Hr (s) = [1 + (G c1 (s) + G c4 (s))G 0 (s)][1 + G c2 (s)G c3 (s)G 0 (s)e−Ls ] ˆfrequency of the disturbance is detectable. Meanwhile, the G c1 (s)G 0 (s)e−Lssuperior setpoint response of the Smith predictor is retained, = , (1) 1 + (G c1 (s) + G c4 (s))G 0 (s) ˆand the asymptotical rejection of non-periodic disturbance Hd (s) =will also be improved in comparison with existing schemes. ˆMoreover, the internal stability of the proposed modiﬁed SP G 0 (s)e−Ls [1 + (G c1 (s) + G c4 (s))G 0 (s) − G c1 (s)G c3 (s)G 0 (s)e− Ls ] ˆ ˆ .control structure is analyzed, which indicates that the proposed [1 + (G c1 (s) + G c4 (s))G ˆ 0 (s)][1 + G c2 (s)G c3 (s)G 0 (s)e−Ls ]method can be used for both stable and unstable processes. (2) The rest of this paper is organized as follows: the By the merit of the Smith predictor scheme, the denominatormodiﬁed Smith predictor control scheme is proposed in of Hr (s) is now delay free. Thus controller design for setpointSection 2; controller design for some typical delayed processes tracking becomes much easier, that is, G c1 (s) and G c4 (s) canis discussed in Section 3; internal stability is analyzed in be designed for pole placement of the delay-free G 0 as shown inSection 4; the control performance is shown by simulation (1). With respect to both setpoint and disturbance channels, theexamples as well as an experimental test on thermal system in overall system is stabilizable if and only if the process G 0 e−LsSection 5; a ﬁnal conclusion is drawn in Section 6. is stabilizable. For a general unstable process with time delay, the necessity of a stabilizing controller K s (s) = G c2 (s)G c3 (s)2. Proposed scheme is reﬂected by the second part of the denominator of (2). For the choice of K (s), it will be analyzed later, and readers may refer to [15] for more information. The design of G c3 (s) provides The structure of the modiﬁed Smith predictor control system additional control freedom particularly to beat the periodicis depicted in Fig. 1, where G 0 e−Ls is the time-delayed ˆ disturbance, and leads to G c2 (s) = K s (s)/G c3 (s). ˆprocess, G 0 e− Ls is the process model, and d is the periodic Before starting the design procedure, some preliminarydisturbance with period Td . Four controllers are deployed for results are given as follows: the steady state of the output y(t) ofdifferent objectives: G c1 is the primary controller for setpoint a stable system G in response to a periodic input x(t) is periodicresponse; G c2 on the inner loop is for stabilization as well as with the same period as that of the input. For the mathematicalnormal disturbance rejection; G c3 = G c3 (s)e−hs is constructed description of this property, please see the Appendix.speciﬁcally to reject the periodic disturbance d; G c4 is to meet Therefore, it follows from (2) thatthe objective of closed-loop pole placement for desired setpointresponse in case of unstable processes. Note that for stable ˆ G 0 (s)e−Ls [1 + (G c1 (s) + G c4 (s))G 0 (s) − G c1 (s)G c3 (s)G 0 (s)e− Ls ] ˆ ˆprocesses, G c2 and G c4 are not necessary, and that the standard Yd = d. [1 + (G c1 (s) + G c4 (s))G ˆ 0 (s)][1 + G c2 (s)G c3 (s)G 0 (s)e−Ls ]Smith predictor is obtained when G c2 = G c4 = 0 and G c3 = 1 (3)(the switch at position S0 in Fig. 1). Suppose the model used perfectly matches the plant Note that G c3 = G c3 (s)e−hs . Hence Eq. (3) becomes ˆ ˆdynamics, i.e., G 0 = G 0 and e−Ls = e− Ls . Then the ˆ G 0 (s)e−Ls [1 + (G c1 (s) + G c4 (s))G 0 (s) − G c1 (s)G c3 (s)e−hs G 0 (s)e− Ls ] ˆ ˆinput–output transfer function in the proposed control system Yd = d. [1 + (G c1 (s) + G c4 (s))G 0 (s)][1 + G c2 (s)G c3 (s)e−hs G 0 (s)e−Ls ] ˆis (4)Y (s) = Hr (s)r (s) + Hd (s)d(s), In time domain, according to the lemma above, the equationwhere given in Box I results.
- 3. H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 495 g0 (t) ∗ (t − L) ∗ [1 + (gc1 (t) + gc4 (t)) ∗ g0 (t) − gc1 (t) ∗ gc3 (t) ∗ (t − h) ∗ g0 (t) ∗ 1(t − L)] ∗ d(t) ˆ ˆ yd (t) = [1 + (gc1 (t) + gc4 (t)) ∗ g0 (t)] ∗ [1 + gc2 (t) ∗ gc3 (t) ∗ (t − h) ∗ g0 (t) ∗ 1(t − L)] ˆ g0 (t) ∗ (t − L) ∗ [d(t) + (gc1 (t) + gc4 (t)) ∗ g0 (t) ∗ d(t) − gc1 (t) ∗ gc3 (t) ∗ g0 (t) ∗ d(t − h − L)] ˆ ˆ = . [1 + (gc1 (t) + gc4 (t)) ∗ g0 (t)] ∗ [1 + gc2 (t) ∗ gc3 (t) ∗ (t − h) ∗ g0 (t) ∗ 1(t − L)] ˆ Box I. Apparently, under the periodic disturbance d(t) of period Td , For the ﬁrst-order plus dead time (FOPDT) modeling,the steady state of the system output will unavoidably have anon-zero steady-state error with the same period. K e−Ls G = G 0 e−Ls = , (10) However, if h is adjusted such that Ts + 1 we choose the primary controller in PI formL + h = kTd ,ˆ (5) 1where k is the smallest integer making h ≥ 0, there will be G c1 = K p 1 + . (11) Ti sd(t) = d(t − kTd ) = d(t − h − L). ˆ (6) Theoretically the Smith predictor structure removes the timeThus, the numerator of Box I becomes delay from the closed-loop characteristic equation. The setpoint and disturbance transfer functions are simpliﬁed tog0 (t) ∗ [d(t) + (gc1 (t) + gc4 (t)) ∗ g0 (t) ∗ d(t) G c1 (s)G 0 (s)e−Ls − gc1 (t) ∗ gc3 (t) ∗ g0 (t) ∗ d(t)] Hr (s) = = Hr (s)e−Ls , (12) = g0 (t) ∗ [1 + (gc1 (t) + gc4 (t)) ∗ g0 (t) 1 + G c1 (s)G 0 (s) ˆ Hd (s) − gc1 (t) ∗ gc3 (t) ∗ g0 (t)]d(t). (7) ˆ G 0 (s)e−Ls [1+Gc1 (s)G 0 (s)−G c1 (s)Gc3 (s)G 0 (s)e− Ls ] ˆ ˆIf the controller G c3 (s) is designed as = 1 + G c1 (s)G ˆ 0 (s) 1 + G 0 (s)(G c1 (s) + G c4 (s)) = Hd (s)e−Ls . (13)G c3 (s) = , (8) G 0 (s)G c1 (s) Thus controller G c1 for setpoint tracking can be designed basedthe numerator of the transfer function Hd (7) will be set equal on the delay-free part of the plant transfer function, which isto zero. Consequently, Yd = 0 in Eq. (3), i.e., the effect ofperiodic disturbance will be eliminated completely with the K G0 = . (14)controller (8), if the condition L + h = kTd is satisﬁed. In case Ts + 1of L = kTd , no delay will be needed on the feedback loop, that Letting Ti = T , it follows from (12) that the delay-free portionis, h = 0. of the closed-loop setpoint transfer function can be written as With the prototype of controller G c3 rejecting the periodicdisturbance, the next step is to develop a systematic tuning 1 1 Hr (s) = = , (15)formula for the rest controller settings of G c1 , G c2 , G c4 . (T /K K p )s + 1 λs + 1 where λ > 0 is the adjustable closed-loop design parameter.3. Controller design Typically the value of λ is chosen approximately equal to the3.1. For a stable process time constant of G(s), i.e., λ ≈ T . This is because the Smith predictor control is usually applied to plants whose time delays For stable processes, the inner stabilizing loop is not are greater than their time constants, and to accelerate therequired, i.e., G c2 = G c4 = 0. The control conﬁguration closed-loop response faster than the open-loop normally resultsis reduced to the original Smith predictor structure with an in excessive overshoots. Therefore, a higher value of λ makesadditional controller G c3 . It is well known that most industrial the system response sluggish but more robust, while a lowerprocesses can be adequately approximated by a ﬁrst- or second- value of λ speeds up the system response at the sacriﬁce oforder stable process with dead time as follows: the stability margin. In an ideal case, that is, with a perfect model, the magnitude of the initial control effort is inversely K e−Ls proportional to λ.G = G 0 e−Ls = , (9) (T s + 1)q From Eq. (15), to have the desired closed-loop time constant, the PI setting of G c1 is given aswhere q = 1 or 2. With the model available, the primarycontroller G c1 is then designed as that of a typical Smith Ti = T, (16)predictor, while G c3 is to reject the disturbance. All these 1elements of an auto-tuner will be described in detail in the Kp = . (17) Kfollowing.
- 4. 496 H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 It follows from (8), (14), (16) and (17) that where λ is the closed-loop design parameter. A tuning rule is suggested by [5] as follows:G c3 (s) = T s + 1. (18) 1The realizable G c3 is in the ﬁltered form of (18) Kp = , (27) Kλ Ts + 1 1G c3 (s) = , (19) Ti = , (28) αs + 1 KKfwhere the transfer function of G c3 is now bi-proper. The ﬁlter 1parameter α = N is usually speciﬁed with a reasonably T Kf = , (29) Klarge N , such that G c3 will be close enough to the physically where λ is recommended in the range of 1 < λ < 5 forunrealizable one as in (18). robustness. For the second-order modeling Controller K s = G c2 G c3 is essential to reject disturbance as K e−Ls well as to stabilize the second part of the characteristic equationG = G 0 e−Ls = , (20) in (4): (T s + 1)2with the controller G c1 in (11), the delay-free part of the close- s + G c2 (s)G c3 (s)e−hs K e−Ls . (30)loop transfer function for setpoint response becomes According to [3], to satisfy the optimum phase margin criterion G c1 (s)G 0 (s) ω02 of θm = 60◦ , it is computed thatTr (s) = = , (21) 1 + G c1 (s)G 0 (s) s 2 + 2ξ ω0 s + ω0 2 π −2× π K s (s) = K d = 3 . (31)where ω0 and ξ are the natural frequency and damping ratio 2K (L + h)of the desired closed-loop response, respectively. A simple Hence, it follows from (8) thatsolution is suggested by [16] asTi = T, (22) Ti s 2 + K Ti (K f + K p )s + K K f G c3 (s) = . (32) 1 K K p Ti s + 1Kp = 2 . (23) 4ξ K To make G c3 proper, it should be augmented with a ﬁlterThe only user-speciﬁed parameter is thus the damping factor, α = Ti , and turns to Nwhich is normally chosen in the range of [0.5 1]. Similarly, when G c1 is available, G c3 can be obtained with Ti s 2 + K Ti (K f + K p )s + K K f G c3 (s) = . (33)a ﬁlter as αs 2 + K K p Ti s + 1 s 2 + 2ξ ω0 s + ω0 2 With G c3 and K s in hand, G c2 (s) is therefore synthesized asG c3 (s) = . (24) (αs + 1)2 K s (s) K d (αs 2 + K K p Ti s + 1)The ﬁlter parameter α is tuned as 1/N , where 1 is the highest G c2 (s) = = . (34)coefﬁcient of the numerator of G c3 (s) in (24). G c3 (s) Ti s 2 + K Ti (K f + K p )s + K K f3.2. For an integrating process 3.3. For an unstable process In this section, we extend the application of the proposedscheme to integrating processes. As the controller design for Suppose the unstable process is modelled as an FOPDTintegrating/unstable processes is inherently more complicated transfer function:than for a stable one, four controllers, G c1 , G c2 , G c3 , G c4 , are K e−Lsneeded. G = G 0 e−Ls = . (35) For the plant assumed as an integrator with long dead time Ts − 1 Two main controllers G c1 = K p (1 + T1s ) and G c4 = K e−Ls iG = G 0e −Ls = , (25) K f (1 + T f s) are chosen in PI and PD forms respectively. s Similar to the previous design, by setting Ti = T + 2T f andwe choose G c4 = K f on the inner loop as a proportional K K f = 2, a stable closed-loop transfer function for setpointcontroller and the primary controller G c1 = K p (1 + T1s ) still i tracking is obtained as follows:in PI form. Therefore, by setting Ti = 1/(K K f ), a stableclosed-loop transfer function for setpoint tracking is obtained e−Ls e−Ls Hr (s) = = , (36)as follows: (Ti /K K p )s + 1 λs + 1 e−Ls e−Ls where λ is the closed-loop design parameter to be speciﬁed.Hr (s) = = , (26) (1/K K p )s + 1 λs + 1 Letting K K p = 1, this gives Ti = λ. Note that Ti = T + 2T f .
- 5. H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 497Therefore, the controller settings of G c1 and G c4 are given by functions between any two points within the system are stable. A block diagram analysis should be used to derive all of these 1Kp = , (37) transfer functions. However, if the system consists of only K SISO plants, the signal ﬂow graph introduced by [20] is aTi = T + 2T f , (38) systematic and effective tool to represent such a system. A 2 signal ﬂow graph consists of nodes and directed branches. AKf = , (39) node performs the functions of adding all incoming signals and K λ−T the transmission of the sum to all outgoing branches. A branchTf = . (40) connecting two nodes acts as a one-way signal multiplier 2 and the multiplication factor is the transfer function of the The value of λ, the closed-loop time constant, can be corresponding plant. A loop is a closed path that starts at a nodespeciﬁed by the user-desired settling time [6] as and ends at the same node. We say that a signal goes through ts a loop if and only if it passes through all nodes and branchesλ= . (41) in the loop once. If two loops have neither common nodes nor 2.5 branches, then they are said to be non-touching. The loop gain Again a stabilizing controller K s (s) = G c2 (s)G c3 (s) will be for a loop is the product of all transfer functions of the plants inused in the inner loop, and this is shown in the second part of the loop. The Mason formula is very important for signal ﬂowthe denominator of Eq. (4): analysis, and it makes use of the system determinant ∆ deﬁned by1 + G c2 (s)G c3 (s)e−hs G 0 (s)e−Ls . (42)For the FOPDT unstable case, the method of [17] would be ∆=1− F1i + F2 j − F3k + · · ·, (47) i j kadequate by suggesting a proportional controller based on anoptimum phase margin: where F1i are the loop gains, F2 j are the products of two non- touching loop gains, F3k are products of three non-touching 1 T loop gains, and so on.K s (s) = K d = . (43) K L +h If an LTI interconnected system consists of n plants and eachHence, it follows from (8) that of them is of SISO type, then such a system is here called a system with scalar signals, because all the signals in theG c3 (s) system are scalar. Let n plants be, respectively, described by K K f Ti T f s 2 +(K K f Ti +K K p Ti +Ti T )s+K K p −Ti transfer functions gi (s), i = 1, 2, . . . , n, and let pi (s) be the = K K p Ti s+K K p . (44) characteristic polynomial of gi (s) for each i. Deﬁne nTo make G c3 physically realizable, it should be augmented with pc (s) = ∆ pi (s). (48)a ﬁlter, and it becomes i=1G c3 (s) K K f Ti T f s 2 +(K K f Ti +K K p Ti +Ti T )s+K K p −Ti Theorem 1. A linear time-invariant interconnected system with = αs 2 +K K p Ti s+K K p , (45) scalar signals is internally stable if and only if all the roots of pc (s) in Eq. (48) are in the open left half of the complex plane. KK T Twhere α = f i f N . To apply this theorem, we ﬁnd that the system in Fig. 1 is With G c3 and K s in hand, G c2 (s) is computed as comprised of seven subsystems: K s (s)G c2 (s) = G c1 (s), G c2 (s), G c3 (s), G c4 (s) G 0 (s), ˆ G c3 (s) ˆ e− Ls and G 0 e−Ls . K d (αs 2 +K K p Ti s+K K p ) = K K f Ti T f s 2 +(K K T +K K T +T T )s+K K −T . (46) Let G 0 (s) = a(s)/b(s), G 0 (s) = a(s)/b(s), G c1 (s) = ˆ ˆ ˆ f i p i i p i c(s)/d(s), G c2 (s) = e(s)/ f (s), G c3 (s) = g(s)/ h(s) andThe controller design procedure is thus completed for both G c4 (s) = i(s)/j (s) be coprime polynomial fractions.stable and unstable processes with time delay. Their pi , respectively, are4. Internal stability p1 = d(s), p2 = f (s), p3 = h(s), p4 = j (s), ˆ p5 = b(s), p6 = 1 and p7 = b(s). Since internal stability is the prerequisite for any controlsystem, we will be using the internal stability criterion for The system has ﬁve loops:SISO systems developed by [18] to check whether the proposed L 1 : −G c1 G 0 , ˆcontrol structure is internally stable or not. It is pointed out by [19] that a linear time-invariant (LTI) L 2 : −G c4 G 0 , ˆsystem is internally stable if and only if all of its transfer L 3 : −G c2 G 0 e−Ls G c3 ,
- 6. 498 H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503L 4 : −G c1 G 0 e−Ls G c3 , Suppose a periodic disturbance ˆL 5 : G c1 G 0 e− Ls G c3 , ˆ −1, if − T < t < 0, D(t) = (50) 1, if 0 < t < T,where L 1 and L 2 are non-touching. It follows that the systemdeterminant ∆ is with the frequency of 2π rad/s is injected at the moment of t = 30. The amplitudes of the setpoint and disturbance are both∆ = 1 − L1 − L2 − L3 − L4 − L5 + L1 ∗ L3 + L2 ∗ L3 unity. From (16), (17) and (19), the PI controller G c1 is given = 1 + (G c1 + G c4 )G 0 + GG c4 + G c2 G 0 e−Ls G c3 ˆ ˆ as 1 + 5.414s and controller G c3 = (5.414s+1)e 1 −0.95s , where h is αs+1 ˆ + G c1 G 0 e−Ls G c3 − G c1 G 0 e− Ls G c3 ˆ set as 0.95 to satisfy condition (5), and α = 5.414/N . Then the frequency of the disturbance changes to π rad/s after t = 80. + G c1 G 0 G c2 G 0 e−Ls G c3 (s) + G c2 G 0 (s)e−Ls G c3 GG c4 ˆ ˆ Once the change is detected, say, at t = 110, h is adjusted to = [1 + (G c1 + G c4 )G 0 ][1 + G c2 G c3 G 0 e−Ls ] ˆ 1.95 to meet (5) again. The simulation result is shown in Fig. 2. ˆ + G c1 G c3 (G 0 e−Ls − G 0 e− Ls ). ˆ It can be seen from Plot A that the disturbance obviously ˆIn case of perfect model match, G(s) = G(s), there is causes a periodic steady-state output error in the control scheme [5], where no controller is designed speciﬁcally as the∆ = [1 + (G c1 (s) + G c4 (s))G 0 (s)] ˆ countermeasure for periodic disturbance. × [1 + G c2 (s)G c3 (s)G 0 (s)e−Ls ]. In Plots B and C, the disturbance effect can be reduced by activating and tuning G c3 adaptively. In Plot B, it is set N = 20, Hence, we have and the disturbance can be attenuated considerably, while in 7 Plot C, with N = 200, it can almost eliminate the disturbancepc (s) = ∆(s) Pi (s) effect. i=1 Remark 1. Note that the ﬁlter parameter N will have a = [1 + (G c1 (s) + G c4 (s))G 0 (s)] ˆ substantial effect on performance of the proposed design, i.e., a larger N makes G c3 more ideal to its improper prototype in Eq. × [1 + G c2 (s)G c3 (s)G 0 (s)e−Ls ] (18) and therefore more effective in eliminating the periodic ·d(s) · f (s) · h(s) · j (s) · 1 · b(s) · b(s) ˆ disturbance. However, in both cases, the setpoint response is = [d(s)b(s) j (s) + c(s)a(s) j (s) + a(s)d(s)i(s)] ˆ ˆ ˆ never affected by G c3 . × [ f (s)h(s)b(s) + e(s)g(s)a(s)e−Ls ]. In Plot D, we can verify that if the incoming disturbance is non-periodic, e.g., a step one of −0.5 magnitude injected atThe polynomial, d(s)b(s) j (s) + c(s)a(s) j (s) + a(s)d(s)i(s), ˆ ˆ ˆ t = 80, the system still can reject it effectively with a betterreﬂects the stabilization of delay-free G 0 (s) by the controllers performance over the existing scheme.G c1 (s) and G c4 (s), which is always realizable by pole Remark 2. For Plot D, the parameters of G c1 and G c3 remainplacement. Thus the overall closed-loop system is stabilizable the same as for Plot A. Nevertheless, G c3 can be deactivatedif and only if the delayed process G(s) = G 0 (s)e−Ls is by leaving the switch at position S0 (G c3 = 1) such that thestabilizable, which would be the task to ﬁnd the proper resultant step disturbance rejection will be the same as thatcontrollers G c2 (s) and G c3 (s) to ensure all the roots of of [5]. f (s)h(s)b(s) + e(s)g(s)a(s)e−Ls located in the open left halfplane. Example 2. Consider the ﬁrst-order integrating process with It is observed from above analysis that unlike the original time delay:Smith predictor conﬁguration for stable processes, all of e−5sthe existing modiﬁed Smith predictor schemes dealing with G(s) = . (51)unstable processes are essentially with a delay-dependent sclosed-loop characteristic equation. The periodic disturbance in (50) comes in at t = 40. Choosing the closed-loop design parameter λ = 2, the controller settings5. Simulations of G c1 and G c4 are K p = 0.5, Ti = 1 and K f = 1. Since L = kTd , no time delay is needed in G c3 . Thus from (34), In this section, simulation examples are given in different s 2 + 1.5s + 0.5scenarios to demonstrate the control performance of the G c3 (s) = G c3 (s) = . (52) 0.5s + 0.5proposed method, and an experimental system is used to test To make the controller G c3 (s) practically realizable, it shouldthe effectiveness for practical applications. The results are be augmented with a ﬁlter α = 1/N such thatcompared with those of existing methods [21,5]. s 2 + 1.5s + 0.5Example 1. Consider a stable FOPDT process with long time G c3 (s) = . (53) αs 2 + 0.5s + 0.5delay: The stabilizing controller K s is calculated as K d = 0.105 by e−16.05 (31).G(s) = . (49) Hence, with K d and G c3 , 5.414s + 1
- 7. H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 499 Fig. 2. Simulation results of Example 1. Fig. 3. Simulation results of Example 2. 0.105(αs 2 + 0.5s + 0.5) respectively. The attenuation of a step disturbance d = −0.2G c2 (s) = . (54) coming at t = 40 is depicted in Plot D. s 2 + 1.5s + 1 Plot A shows the control performance of normal design Example 3. Consider the unstable FOPDT process[5] under periodic disturbance. In Plots B and C of Fig. 3, 4e−2sthe control effects are shown for N = 20 and N = 200, G(s) = . (55) 4s − 1
- 8. 500 H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 Fig. 4. Simulation results of Example 3.Specify the settling time ts = 5. Then from Eqs. (37)–(40), the Examples 1–3 in Plots A, B, C, respectively. The setpointcontroller settings of G c1 and G c4 are K p = 0.25, Ti = 2, command of a rectangle wave with the period T = 200 isK f = 0.5 and T f = −1. Since L = kTd , no time delay needs shown by the dotted line, while the actual plant output is shownto be added in G c3 , which is thus given as by the solid line. It can be seen that, in all cases, the outputs can follow the command quite well in a periodic manner. However, 4s 2 + 4s + 1G c3 (s) = G c3 (s) = , (56) the length of the process dead time imposes a substantial effect αs 2 + 2s + 1 on the fast-tracking ability.where α = 4/N . Example 4. Finally, an experimental thermal system shown in The stabilizing controller K s is calculated as K d = 0.354 Fig. 7 is set up to examine the proposed scheme, which consistsfrom (43). of two parts: (1) a thermal chamber set made by National With K d and G c3 , it gives Instruments (NI); (2) a personal computer equipped with data 0.354(αs 2 + 2s + 1) acquisition cards linked to the NI LabVIEW software. TheG c2 (s) = . (57) control input u(t) scaled in the range [0 1] is to manipulate 4s 2 + 4s + 1 the voltage supplied to a 12 V light with 20 W halogen bulb, Again, the periodic disturbance in (50) with the frequency while the periodic disturbance d(t) is simulated by the coolingof 2π rad/s is injected at t = 40. In Plot A of Fig. 4, the effect of a 12 V fan. The system output y(t) is the chamberdisturbance effect is signiﬁcant with no countermeasure. Then temperature.in Plots B and C, the performance of the proposed control is The ﬁrst-order modeling of this experimental system givesshown with N = 20 and 200, respectively. And the rejection ofa step disturbance d = −0.2 loaded at t = 40 is presented by 11.16Plot D. ˆ G(s) = e−9.8s . (58) 15.46s + 1Remark 3. Since the choice of the ﬁlter parameter α is Here the main controller setting is obtained according to thean important factor in the proposed design, we repeat allthree examples in the presence of process noise to test the model-based predictive PI (PIP) control [21] as follows:noise sensitivity of the controller G c3 . It is observed in 1 1Fig. 5 that, with a large ﬁlter parameter, our proposed design G c1 (s) = 1+ . 11.16 11.56smildly ampliﬁes the effect of process noise than the existingscheme [5] does. Correspondingly,Remark 4. The periodic command tracking performance is 11.56s + 1exhibited in Fig. 6, where all three examples are repeated: G c3 (s) = . 1.156s + 1
- 9. H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 501 Fig. 5. Setpoint tracking in the presence of process noise (top: Example 1; middle: Example 2; bottom: Example 3). Fig. 6. Command tracking for periodic trajectories (dotted line: setpoint; solid line: output).At the beginning of the test, the control system follows the is activated at t = 500 s to reject it. The test has been repeatedstructure of PIP control, i.e., the switch is at position S0 and three times and the veriﬁed result is given in Fig. 8. It can betherefore G c3 = 1. A disturbance is injected shortly and then observed that the disturbance effect, alternatively output steady-becomes periodic at t = 200 s with Td = 50 s. Once this state error, can be reduced substantially.periodic perturbation is detected, the controller Remark 5. It can be seen that the curves for u and y are non- 11.56s + 1 −40.2 smooth. This is due to the unavoidable measurement noisesG c3 (s) = e in actual practice. Furthermore, robustness issues have be to 1.156s + 1
- 10. 502 H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 modiﬁed SP scheme is proposed with the capability of rejecting periodic disturbances in both stable and unstable delayed processes, provided that the frequency of the disturbance is detectable. Meanwhile, the attenuation of a non-periodic disturbance is also enhanced, and the setpoint response remains the same as those best achievable by existing methods which yet cannot exercise effective control for periodic disturbances. In addition, the internal stability of the proposed control system is analyzed explicitly. Simulations and experiment results sustained the effectiveness of the proposed method. Acknowledgements The authors appreciate the valuable comments provided by anonymous reviewers, which helped to improve the quality of Fig. 7. Experiment setup of a thermal chamber. this manuscript. The ﬁrst author is also grateful to Mr. Aleck Wilbar, the control systems department administrator of Jacobsbe considered in practical applications. The design of Smith Engineering Group, Inc. (Houston Ofﬁce), for proofreading thispredictor control in model mismatch was resolved by [22].For delay uncertainties, [23] has proposed a sophisticated manuscript from the viewpoint of an experienced engineeringimplementation scheme with real-time online computation of practitioner.varying process time delay. These methods can be combinedwith the proposed control for robust/adaptive application with Appendixrespect to parametric uncertainties.6. Conclusions With the standard convolution operation ∗, the response of a stable system Y = G 2 G 1 X can be written as y(t) = g2 (t) ∗ The Smith predictor is an effective dead time compensator g1 (t) ∗ x(t), where g1 (t) and g2 (t) are impulse responses of G 1for stable processes with large delays. Modiﬁed SP schemes and G 2 , respectively. For simplicity, denote g2 (t) ∗ g1 (t) ∗ x(t)have extended its applications to unstable processes control. by (g2 ∗ g1 ∗ x)(t). For a signal f (t) with a periodic steady state,However, the potential of a 2DOF modiﬁed SP has not been let its steady state be f s (t), and the corresponding Laplacefully explored regarding disturbance rejection. In this paper, a transform be F s (s) = L{ f s (t)}. Fig. 8. Test results obtained for periodic disturbance rejection.
- 11. H.-Q. Zhou et al. / ISA Transactions 46 (2007) 493–503 503Lemma 1. Under a periodic input x(t), the output steady state [15] Bonnet Catherine, Partington JonathanR. Bezout factors and l 1 -optimalof the system Y = G 2 G 1 X with stable G 2 and G 1 satisﬁes controllers for delay systems using a two-parameter compensator scheme. IEEE Trans Automat Control 1999;44(8):1512–21.(g2 ∗ g1 ∗ x)s (t) = [g2 ∗ (g1 ∗ x)s ]s (t), [16] Hang CC, Wang QG, Cao LS. Self-tuning Smith predictors for processes with long dead time. Internat J Adapt Control Signal Process 1995;9(3):and for any τ , 255–70. [17] DePaor AM, O’Malley Mark. Control of Ziegler–Nichols type fory s (t − τ ) (g2 ∗ g1 ∗ x)s (t − τ ) unstable process with time delay. Internat J Control 1989;49(4):1273–84. = {g2 (t) ∗ [g1 (t) ∗ x(t − τ )]s }s . [18] Wang QG, Lee TH, He JB. Internal stability of interconnected systems. IEEE Trans Automat Control 1999;44(3):593–6.Equivalently, in the s-domain, there holds [19] Doyle JC, Francis BA, Tannenbaum AR. Feedback control theory. New York (USA): Macmillan; 1992.(G 2 G 1 X )s = [G 2 (G 1 X )s ]s . [20] Mason SJ. Feedback theory-further properties of singal ﬂow graphs. In: Proc. IRE. July 1956. For the proof of this lemma, please refer to [24]. [21] Hagglund T. A predictive PI controller for processes with long dead times. IEEE Control Syst 1992;12(1):57–60. [22] Wang QG, Bi Q, Zhang Y. Redesign of Smith predictor systems forReferences performance enhancement. ISA Trans 2000;39:79–92. [23] Nortcliffe A, Love J. Varing time delay Smith predictor process controller. [1] Smith OJ. A controller to overcome dead time. ISA J 1959;6(2):28–33. ISA Trans 2004;43:61–71. [2] Astrom KJ, Hang CC, Lim BC. A new Smith predictor for controlling [24] Cartwright M. Fourier methods for mathematicians, scientists and a process with an integrator and long dead time. IEEE Trans Automat engineers. New York: Ellis Horwood; 1990. Control 1994;39(2):343–5. [3] Matausek MR, Micic AD. A modiﬁed Smith predictor for controlling a process with an integrator and long dead-time. IEEE Trans Automat Han-Qin Zhou received his B.Eng. (2001) in Control 1996;41(8):1199–203. Automation, M.Eng. (2003) and Ph.D. (2006), both [4] Majhi S, Atherton DP. Modiﬁed Smith predictor and controller for in Electrical Engineering, from Shanghai Jiao Tong processes with time delay. IEE Proc -Control Theory Appl 1999;146(5): University, National University of Singapore and the University of Houston, respectively. Since 2006, he has 359–66. been with the Control Systems Department of Jacobs [5] Majhi S, Atherton DP. Obtaining controller parameters for a new Smith Engineering Group, Inc. His research interests include predictor using autotuning. Automatica 2000;36:1651–8. PID tuning, disturbance rejection, digital design, robust [6] Kaya I. Obtaining controller parameters for a new PI-PD Smith predictor multivariable control, and their industrial applications. using autotuning. J Proc Control 2003;13(5):465–72. He is a member of the Instrumentation, Systems, and [7] Zhang WD, Sun YX, Xu XM. Two degree-of-freedom Smith predictor for Automation Society. processes with time delay. Automatica 1998;34(10):1279–82. [8] Lu X, Yang YS, Wang QG, Zheng WX. A double two-degree-of-freedom Qing-Guo Wang received his B.Eng. in Chemical control scheme for improved control of unstable delay processes. J Proc Engineering in 1982, M.Eng. in 1984 and Ph.D. Control 2005;15:605–14. in 1987, both in Industrial Automation, all from [9] Hara S, Yamamoto Y, Omata T, Nakano M. Repetitive control system: Zhejiang University of the PRC. He held an Alexander- A new type servo system for periodic exogenous signals. IEEE Trans von-Humboldt Research Fellowship of Germany with Automat Control 1988;33(7):659–68. Duisburg University and Kassel University from 1990 to 1992. Since 1992 he has been with the Department[10] Manayathara TJ, Tsao TC, Bentsman J, Ross D. Rejection of unknown of Electrical Engineering of National University of periodic load disturbances in continuous steel casting process using Singapore. His research interests are mainly in systems learning repetitive control approach. IEEE Trans Control Syst Technol theory, robust, adaptive and multivariable control and 1996;4(1):72–7. optimization, with emphasis on their applications in process, chemical and[11] Moon JH, Lee MN, Chung MJ. Repetitive control for the track-following environmental industries. servo system of an optical disk drive. IEEE Trans Control Syst Technol 1998;6(5):663–70. Liu Min received his B.Eng. degree in Electrical Engi-[12] Steinbuch M. Repetitive control for systems with uncertain period-time. neering in 1999 and his M.Eng. degree in Industrial Au- Automatica 2002;38:2103–9. tomation in 2002, both from Tianjin University of the[13] Bodson M, Sacks A, Khosla P. Harmonic generation in adaptive PRC. He is currently a Ph.D. candidate in the Depart- feedforward cancellation schemes. IEEE Trans Automat Control 1994; ment of Electrical and Computer Engineering, National 39(9):1939–44. University of Singapore. His present research interests are process identiﬁcation multivariable control systems[14] Wang QG, Zhang Y, Huang XG. Virtual feedforward control for and PID tuning. asymptotic rejection of periodic disturbance. IEEE Trans Indus Electron 2002;49(3):566–73.

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