Dev. Chem. Eng. Mineral Process., 9(1/2),pp.57-68, 2001. Analysis and Implementation of the DoubleController Scheme Y.-C. Tian*and F. Gao Department of Chemical Engineering, The Hong Kong University of Science and Technology, Clear WaterBay, Kowloon, Hong Kong This paper gives some insight into the double-controller scheme proposed recently. The double-controller scheme has good robustness as well as simultaneous fast setpoint tracking and good load rejection. Investigations show that it enhances the conventionalfeedfonvard control. The stability and robustness of the control scheme are analyzed in thefrequency domain and compared with those of the Smith predictor. To further improve the performance of the double-controller scheme, techniques are proposed to accommodate large uncertainties in process delay time and gain. Two control signals of the double-controller scheme are employedfor extraction of process parameter information and on-line tuning of the control system. The effectiveness of the proposed techniques is veriied through numerical simulations. Introduction Set-point tracking and load rejection are two main objectives of a control system. It is difficult for a conventional control system with a single controller to simultaneously meet these two requirements. To overcome this difficulty, Tian and Gao recently proposed a double-controller scheme [l]. This paper is concerned with analysis and implementation of the double-controller scheme. Figure 1 is the double-controller scheme for processes with dominant time delay. The process is governed by P(s) = G,(s)e* and its model is P'(s) = GS (s) e-d's .The set-point controller (SC) G,,(s) and load controller (LC) GJs) are configured for setpoint tracking and load rejection, respectively. From Figure 1, it follows that: Y(s) G, e-* -...(1) L(s) - H A S ) = 1+ G,, G, e-* With a good process model, Equation (2) becomes: 'Author for correspondence (e-mail: tiany@che.curtin.edu.au). Current address: School of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth, Western Australia 6845. 57 Y.-C. Tian and F. Gao Thus, the load response, H,,is determined p;+q only by GC2and is independent of Gc,.With a T- GdS) good process model, the set-point response, H, is determined only by G,, and is independent of GC2.These are two important features of the double-controller scheme. ed's Thus, the SC and LC can be independently designed and tuned for set-point traclung and Figure I . Double-controller scheme for processes with dominant delay. load rejection, respectively. Suppose the process is governed by a first-order plus delay equation; and a firstorder plus delay model, reasonable for a wide range of processes, is adopted: Ki P ( s ) = C,(s)e-& = K p e-&, P*(s)= Gi(s)e-d's = e-d'r ...(4) Tps+ 1 Tis + 1 The direct synthesis method is used for the SC design. When the desired closedloop response is described by a first-order plus delay expression with unit gain, time constant T, and time delay c f , the resulting SC is a standard PI controller: m' The LC is also designed to be PI type. The controller settings are tuned by modifying Haalmann's formulae [2] through introducing a modification coefficient a 2 1: The selection of a should guarantee sufficiently large gain margin and phase margin. The good robustness, fast set-point trackmg, and good load rejection of the double-controller scheme have been demonstrated in Tian and Gao [l]. Thls paper gives some insight into the novel double-controller scheme. It will be shown that the scheme enhances the conventional feedforward control. The stability and robustness of the scheme are to be investigated through frequency domain analysis. Some techniques will be proposed to accommodate large process uncertainties. conventional control the LC and process are 58 UI L Analysis and Implementation of the Double-ControllerScheme reject load disturbances. On the other hand, all process uncertainties are included in the LC loop, overcoming the process uncertainties is another objective of the LC. The SC loop is also a conventional control loop with the output of the delay-free part of the process model as its feedback signal. The objective of the SC is to provide a fast set-point tracking. On the other hand, the loop does not include load disturbances or any variable affected by load dsturbances. Also, it does not contain process uncertainties. The system robustness is, therefore, independent of th~sloop includmg the SC. This is also an important feature of the double-controller scheme. The SC loop can be viewed as a block with the transfer function of G,, /(1+ G,,G;). The output u , of this block is fed forward and combined with u2,the output of the LC Gc2,to form the overall control signal, u = u , - u2. This is similar to a feedforward control. The SC loop acts as a feedforward controller. Containing the process model of G;e-d's ,the lower feedforward path of Figure 2 provides an actual set-point, r', for the LC Gc2. A feedforward control is normally introduced to reduce effects of measurable load disturbance on process output. In the double-controller scheme of Figure 2, the feedforward action is for set-point trackmg through designing and tuning the SC Gcl. With this feedforward action, the real set-point signal R cannot be drectly injected into the LC Gc2.Instead, the estimate r' of the process output Y is used as the set-point of Gc2.With a good process model, it follows that r' = Y, indicating that the LC seems to be shut down provided there are no load changes. Thus, the double-controller scheme enhances the conventional feedforward control by providing fast set-point tracking without effect on load rejection. The SC loop produces the feedforward action, while the LC loop rejects load disturbances, compensates model mismatch, and determines the system robustness. A simplified structure of the double-controller scheme can thus be drawn as Figure 3, where the lower part corresponding to a good process L G (s)ed" model serves set-point tracking, and the upper part corresponds to load G *(S) rejection. Due to the special structure R of the control scheme, the set-point tracking and load rejection are separated with a good process model Figure 3. A simplified structure of Figures and can then be designed and tuned 1 and 2 with a good process model. independently. Frequency Domain Analysis Frequency Domain Descriptionfor Load Rejection From Equation (l), the characteristic equation of the LC loop for load rejection is: ...(7) 1 + W(s)= 0, W(s)= Gc2(s)Gp(s)e* where W(s)represents the open-loop transfer function of LC loop. Substituting Equations (4) and (6) into the above equation yields: 59 Y.-C.Tian and F. Gao W ( S )= 2Kp aTis+l 3K;d' s(Tps+ 1) -a .. e-*,a 2 1 The amplitude A and phase angle 4 of W(s)are respectively expressed by: 2Kp /(aT;o)' + 1 , / ( w ) = - K + tg-'(aTio) A ( @ )= I 2 3Kpd w (TPo)*+1 - tg-'(Tpw) - dw ...(9) Consider a unity feedback loop with W(s)in its forward path. Define wg and upas the gain and phase (or critical) crossover fiequencies, respectively. We have: ...(10) M4j= K + &ug),A(cD,) = 1 ; MA= 11A(@,), &up)= --IC Equations (9) and (10) show that an increase in d results in an favorable increase in MA but an unfavorable decrease in MQr.A dominant time delay leads to a large MA but a small (or negative) M@ without introducing a (Len,a = 1). A negative M@ implies that the system is unstable. Thus, the coefficient a provides a simple method to compensate variations in both MA and MQr.A larger d requires a larger a and vice versa. When a increases from a = 1, MA decreases while MQr increases. If a is close to 1, the increase of MQr is dominant and the system stability benefits from this increase of a. However, if a exceeds some critical value, the decrease of MA is dominant and the system performance deteriorates. We recommend taking a to be the value at which M4j I 60". Regarding the system robustness for load rejection, we consider deviations of d, Kp,and Tpfrom their nominal values of d , K ; ,and Ti ,respectively. Denote: A * ( 4 = A ( 0 ) l R p . K ; . 7 , . ~ ~ , ~ , d '' 4*(4= ~ ( w ) I K , = K ; . T , - ~ ~ . d = d ' ...(11) Define the relative amplitude M ( o )= A ( o ) / A* ( w ) and relative phase angle R K u ) = &w) - $(w). M ( w ) and R&w) are measures of relative gain margin and relative phase margin, respectively. Taking into account equations (9) and ( 1 l ) , we have: M ( w )= A(o) K p =- A*(w) K i (T,o)* +1 ...(12) Frequency Domain Propertiesfor Load Rejection Some important properties are directly inferred from Equations (9) through (12). 1) Deviations of d from its nominal value of d' do not affect the gain margin for load rejection. However, they affect the phase margin of the control system. 2) RQ(o) is depends on d - d . 3 ) Positive deviations of d fiom its nominal value, i.e. d > d , result in decreases of phase margin. Without uncertainties in Kpand Tp the accessible deviation of the d 60 Analysis and Implementation of the Double-Controller Scheme is bounded by d - d* < M@'og. With the recommended MO = 60" in nominal case, this upper bound of d becomes d - d' < nI3 up. 4) In contrast to 3) above, negative deviations of d lead to increases of phase margin, implying that the lower bound for variations of d is d 2 0. 5) Deviations of Kp from its nominal value of K; have no effect on phase margin for load rejection. By contrast, they affect the gain margin of the system. 6) RA(0)is dependent on KJ Ki . 7) Positive deviations of K, from its nominal value, i.e., Kp == K ; , result in decreases of gain margin. Without uncertainties in T, and d, the accessible deviation of the 1/ A ( o , ) . 8) In contrast to 7) above, negative deviations of K, lead to increases of phase margin, suggesting that the lower bound for variations of K, except Kp 2 0. 9) Deviations of T, affects both gain and phase margins for load rejection. They K, is bounded by K, / Ki C MA = affect RA( w) and R&w) through a factor of (Tim)' +1 (TPw)*+ 1 and an addtive term of tg-I (Ti@)- tg-'(T,o) ,respectively. The effects are not si@icant unless a very large variation in T, appears. For the same percentage of the relative variations in K,, T, and d, the variation of Tp gives the minimum effect on system performance. 10) Both RA(w) and R 4 w ) are independent of the coefficient a of Equation (6). Frequency Domain Analysisfor Set-Point Tracking Correspondmg to Equation (2), the characteristic equation is expressed by: [l + W,(s)][l + W(S)]= 0, W,(s) = GCl(s)G;(s),W(S)= Gc2(s)Gp(s)e-& ...( 13) The above equation implies Equation (7), whch is the characteristic equation of the LC loop for load rejection and has already been analyzed, and ...(14) 1 + w,(s)= 0, W, ( s ) = G,, (s)G;( s ) WAS) is the open-loop transfer function of the SC loop. Because neither Gc,(s)nor G;(s) contains any uncertainty, the system robustness is independent of the SC. Considering Equations (4)and (9,we have W,.(s) = 1/ T,s , implying that MA = Q) and M O = 90"for the SC loop. Thus, the SC loop is always stable for VTe> 0. Analysisfor the Smith Predictor The characteristic equation of the Smith predictor is: 1 + W,(S)= 0, W , ( S )= G,(s)G;(s)+ GC(s)[Gp(s)e-& -G;(s)~-~*'] ...(15) The controller Gc(s)is designed to be the same as the SC G,,(s), both giving the same set-point tracking with a good process model. Thus, W,(s)reads: 61 Y.-C. Tim and F. Gao . ..(16) The amplitude A(w)and phase angle Kw) of W,(s)can thus be derived as A ( @ ) = - ~I [ l + z ( w ) c o s P ( w ) - c o s ( d * w ) + ~~(w)sin/?(o)-sin(d*o)~ Tew r -- - && = I) tg-' 2 z(w) = z(w) sinP(w) - sin(dL) 1 +z(w)cos/?(w) - cos(d*w) s / pP(o) K; l+(T,w)* = dw + tg-'(T,w)- ...(17) tg-'(T;w) Thus, RA(w), R&w), MA, and MOcan be easily obtained from the above equation. In the double-controller scheme, RA(w) is independent of variations in d; and R d w ) is not affected by Kp uncertainty. These properties, however, do not hold in the Smith predictor. Moreover, negative deviations of d increase the closed-loop M @ in the double-controller scheme, while in the Smith predictor, both positive and negative deviations of d may lead to decreases of M O (and MA as well). Now consider K,,deviations. Without variations of d and T,, Equation (17) reads: 1 A ( @ ) = -d[l + 6 cos(d 'w)! T-W + [ssin(d 'w)! ..(18) An important result can be obtained: the Smith predictor is stable for' ,K 2Ki if there are no uncertainties in d and T,. In fact, Kp < 2K; implies 6 < 1 and consequently 1 + Gsin(d*w) r < tg-' < - . From Equation (18) it 2 1+ 6 c o s ( d h ) 2 follows that Kw) > - r f o r Vw. The closed-loop system is, thus, stable for Vw. As in double-controller scheme, deviations of T, affect both RA(w) and R d w ) in the Smith predictor. This will be analyzed in detail below through an example. 6 cos(dw) > o for ~ wn u . s, - An Illustrative Example System Description and Stability Bounds Consider a fust-order plus delay process with K; = 1, T,' = 1 and d = 5. Let T,= T,' 1. With our design procedure, we have the following controller settings: r2 = K,, = T.,= 1; a = 2.17, Kc2= 0.2893, = 2.17 a = 2.17 gives a 60" M#, as recommended previously; the corresponding wg FZ 0.1347. The resulting MA = 2.3529 (i.e. 7.4322&), and the correspondmg wpz 0.38. 62 Analysis and Implementation of the Double-Controller Scheme Without process uncertainties, Figure 4 shows responses of the double-controller scheme to a unit step change in set-point at t = 0 and a negative unit step change in load at t =75. The Bode diagram of W(s)is depicted in Figure 5 . The maximum accessible deviation of d is d - d 8 = M a w g = ld(3o.J = 7.77 if there are no uncertainties in Kpand T,. So, the double-controller scheme is stable over 0 Id < 7.77+d* = 12.77. By contrast, computation shows that the accessible delay deviations for the Smith predictor are bounded by about 3.38 < d c 6.53. Without uncertainties in Tpand d, the maximum accessible deviation of the K, for the double-controller scheme is K, / K i = MA = 11 A(@,) = 2.3529. The accessible Kpis then bounded by 0 I K, < 2.3529Ki = 2.3529. By contrast, simulation reveals that the accessible deviations of K, for the Smith predictor are bounded by 0 I K, < 2.135. As proved previously, the Smith predxtor is stable for Kp€2 K i = 2. I 0 ' 50 nm t 1W 150 -&no 50 10" -F I 0 to.' 10' 1W T i1 Figure 4. Responses of the doublecontroller scheme with a good process model. 150 -360 loJ -- - - -1- - - 1I -'I - I I - - -1- - 10' FnoaY Figure 5. Bode diagram of W(s) in nominal case case: A'(o) (upper) and qS(o) (lower). Process Uncertainties in the Double-Controller Scheme Time delay d. Figure 6 shows R 4 o ) with deviations of d from its nominal value of 5 . Corresponding to the nine lines in the figure from top to bottom, the deviations of d are -40% to 40% by 10% increments. RA(o)is not plotted here since it is independent of deviations of d. It is clearly seen from Figure 6 that positive deviations of d lead to decreases of R$((o) and consequently decreases of the closed-loop phase margin. By contrast, negative deviations of d result in increases of R K o ) and consequently increases of the closed-loop phase margin. Process gain Kp. Figure 7 shows RA(o)with Kp deviations. Corresponding to the nine lines in the figure from bottom to top, the deviations of K, are from -40% to 40% by 10% increments. R 4 o ) is independent of d deviations and is thus not plotted here. Figure 7 clearly shows that positive deviations of Kp yield increases of RA(o)and consequently deterioration of the system stability. By contrast, negative deviations of K, lead to decreases of RA( o)and consequently increases of the system stability. 63 Y.- C.Tim and F. Gao \Y 10.' 10' 10' Fw-w __10.' 1oo 10' FrsOvy Figure 6. RA(o)of the double-controller Figure 7. R#o) of the double-controller scheme with Kp deviations rangingfrom scheme with d deviations rangingfiom -40% to 40% by 10% increments @om -40% to 40% by 10% increments lFom bottom to top line). top to bottom lines). I 2, -2M' 1 6 1OD 10' FnquKy Figure 8. RA(0) and RNw) of the doublecontroller scheme with Tpdeviations rangingfrom -40% to 40% by 10% increments (bothporn top to bottom). F&n* Figure 9. RA(w) and RHm) of the double-controller scheme with 10% variations in process time delay d, gain K, and time constant T,. Time constant TP.Its deviations affect both M(w) and RKw), as shown in Figure 8 where deviations of Tpare from -40% to 40% by 10% increments and M ( o )and R ~ o are ) both from top to bottom. It is seen from Figure 8 that positive deviations of Tpresult in unfavorable decreases of RKw) but favorable decreases of M(w) and vice versa. The effects of deviations of Tpon system stability are thus not sigmfkant. Figure 9 shows M ( m ) and RKw) for 10% deviations in d, K,, and T,, respectively. It is clear that deviations of T, have the weakest effect on system stability. Hence, emphases should be placed on d and K, for improving system robustness. Uncertainties in the Smittr Predictor Ws(s)in Equations (17) and (18) describes the stability and robustness of the Smith predictor. In the example under consideration, Ws*(s)= lls in nominal case. The Bode diagram of Ws*(s) is simple: the corresponding A*(@)= l/w and qj'(w) is a horizontal line at -90'. The closed-loop M@ = 90'. In the double-controller, the system stability benefits fiom negative deviations of the d; and M(o) is independent of these deviations. But the Smith predictor is sensitive to both positive and negative deviations of d; and these deviations affect both M ( o ) and R K o ) . Figure 10 depicts M ( o ) and R K o ) of the Smith predictor 64 Analysis and Implementation of the Double-Controller Scheme -1woL \ ; -1wo \ Figure 11. RA(o)and R&m) of the Smith predictor with Kpdeviations of 11 0% and deviations rangingfiom -40% to 40% by I0 % increments. with d deviations ranging fiom -20% to 20% by 10% increments. Clearly, both positive and negative deviations of d lead to increases of U ( w ) and simultaneous , resulting in deterioration of the system stability. decreases of R ~ u ) both Deviations of Kp lead to variations of RA(w) and RKw) and consequently decreases of the closed-loop gain and phase margins, as shown in Figure 11. It has been proved previously that the Smith predictor is stable for K, < 2 Ki if there are no uncertainties in d and T,. Once deviations of K, exceed loo%, the resulting #(w) of W,(s)will cross the critical value of -K Figure 11 contains plots for a 110% deviation of Kp. A sharp decrease of K w ) is clearly seen. Note that here 40)= R 4 w ) - 762. The effects of deviations of T, on system stability and robusmess are shown in Figure 11. It is seen fiom this figure that the Smith predictor is stable for small variations in Tpsince the resulting 4 m ) of W,(s)remains to be larger than -w. A -50% deviation of T,, however, leads to a cross of -win Kw), as shown in Figure 12. With the same 10% deviations of d, Kp and T,, Figure 13 shows a performance comparison for the Smith predictor. The effects of uncertainty in d on system stability and robustness are clearly the strongest among d, K, and Tp Thus, a special attention should be paid to uncertainties of d to improve the Smith predictor. Comparisons of the Double-Controller Scheme and the Smith Predictor Figures 5 through 9 correspond to the double-controller scheme and Figures 10 through 13 correspond to the Smith predictor. Comparisons of these figures reveal that effects of process uncertainties on system stability and robustness of the Smith predictor are stronger than those of the double-controller scheme. In other words, the 65 Y.-C. Tian and F. Cao lo.' td to' -F lW, 1 I -100 10" d 1D( 1 -F Figure 13. Performance comparisons for the Smith predictor with 10% variations in d, Kp and T,- double-controller scheme is more robust than the Smith predictor. Comparisons of the control schemes in time domain can also be found in Tian and Gao [ 11. Implementation of the Double-Controller Scheme For processes with dominant time delay, the double-controller scheme of Figure 1 outperforms the Smith predictor in the presence of large process uncertainties. Some techniques are proposed here to further improve the system performance. Many industrial processes contain large process uncertainties. A typical example is the injection velocity control of thermoplastic injection molding. It is a typical cyclic process with varying process gain and time constant over a wide range. A detailed analysis of the process dynamics is given in [4]. Among time delay, gain, and time constant, time constant has the weakest d u e n c e on system performance for both the double-controller scheme and the Smith predictor. Thus, this section addresses uncertainties of the process delay and gain. Negative deviations of process delay result in increases of phase margin for the double-controller scheme. Thus, a value close to the maximum possible value of d may be selected as its nominal value, b. "his selection, however, may lead to a sacrifice in set-point tracking performance. To compensate dominant and variable time delay, an effective technique has been proposed recently by Tian and Gao [ 5 ] . It continuously tunes the delay parameter in the process model by minimizing a performance index: the average magnitude difference function. The same techmque can be used in the double-controller scheme. With regard to the process gain, if the process is governed by a first-order plus delay equation and the SC and LC are PI type, the double-controller scheme gives: ...(19) s-+o limu,(s)=RIKi = L + R ( ~K I ; - 11K,)= L + (I/ ~ iimU,(s) s+o .*. K, = Ki lim s-0 66 1 1- [u2(s) - L]/u, (s) j -,11~ , )s+o~ i m(s)~ , u2(s)--L - [u* (s) s-0 u, (3) - L] ...(20) ...(21) Analysis and Implementation of the Double-ControllerScheme Without considering the process load disturbances, this equation is simplified to This suggests that the ratio uJu, reflects the degree of the gain mismatch. The process gain and consequently the controller settings can be self-tuned based on equation (25). A recursive relation for gain tuning is: ...(23) where the superscript (n) denote the recursive step, the parameter O< h I 1 is used to adjust the self-tuning rate. This self-tuning method is especially suitable for cyclic processes in which a cycle-to-cycle learning of K, is easy to be implemented. Reconsider the example of the last section. Assume that Kp deviates to 1.2 from its nominal value of 1, and T, deviates to 0.8 from its nominal value of 1. A unit step change in set-point is introduced at t = 0, and a white noise load with zero mean and variance of 0.3 is also introduced, as shown in Figure 14.The sampling period is 0.05. Without self-tuning of K,,, Figure 15 shows the closed-loop response in the first cycle. From Figure 15, U J U , = 0.2 at steady state. Thus, g = 0.25 when h = 1. Therefore, Kp is closer to 1.25 compared to the original estimate of 1. With this estimation, the uncertainty of Kp is reduced from 16.67% to 4.17%. The controller settings are recomputed based on the updated Kp' The closed-loop response is shown in Figure 16 for the second cycle. Clearly, the set-point tracking is improved and satisfactory. 2 R 2 1 -1' . - I I 10 0 20 50 40 30 line t Figure 14. A step change in set-point and a white noise load. . 1.5 -0s 10 m ' rm1 30 40 Figure 15. Closed-loop response to excitations of Figure 14 in thefirst cycle without self-tuning of Kp 1.5 4.5; 10 20 nmI 30 40 50 Figure 16. Closed-loop response to excitations of Figure I 4 in the second cycle with self-tuning of K,. 67 Y.-C. Tian and F. Gao Conclusions Some insight into the novel double-controller scheme has been given. It has been shown that the scheme enhances the conventional feedforward control. The system stability and robustness have been analyzed in the frequency domain. Some important results have been derived from this analysis. Without process uncertainties, the Smith predictor is a good choice for processes with (dominant) time delay. But in the presence of large process uncertainties, the double-controller scheme outperforms the Smith predictor. Some techniques have also been proposed to accommodate large process uncertainties. It has been found that the process gain information can be extracted from two control signals of the double-controller scheme. This nature has been employed to develop a simple self-tuning algorithm. The effectiveness of this self-tuning method has been demonstrated through an example. Nomenclature Amplitude Process time delay G Delay-free part of P G,, Set-point controller (SC) Load controller (LC) Closed-loop load response Closed-loop set-point response T., PI settings of G,, T2 PI settings of Gc2 Process Gain Load disturbance Gain margin A d Greek Letters a Coefficient w Frequency in radian wg wp M@ P R RA Phasemargin Plant transfer function Set-point Relative amplitude (AIA') Relative phase angle ( 4 - 4') Process time constant Desired time constant Overall control signal, u = u, - u2 outputs of sc Output of LC Process output . I Gain crossover frequency,A( wZ) = 1 Phase crossover frequency, 4 ( w p )= -n Superscripts * Estimate or nominal values References 1. Tian, Y.C. and Gao, F., Double~ontrollascheme for separating load rejection from set-point backing, IChemE Part A: Chem Eng Res Des, 76,1998. 2. Haalman, A. 1965. Adjusting controllers for deadtime processes, Conrrol Eng., 12,71-73. 3. Smith,O.J.M.1957. Closer control of loops with dead time, Chem Eng Prog, 53(2), 21 17-219. 4. Tian, Y.C. and Gao, F. 1999. Analysis and control of ram velocity of thermoplastic injection molding, Ind Eng Chem Res, 38(9), 3396-3406. 5. Tian, Y.-C. and Gao, F. 1998. Compensation of Dominant and Variable Delay in Process Systems, Ind Eng Chem Res. 37(3), 982-986. Received: 26 October 1999;Accepted afier revision: 27 May 2000. 68

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