Dynamic delayed controllers for recycling systems with internal delays and complex poles

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1 Received: 9 February 07 Revised: 5 July 07 Accepted: 6 September 07 DOI: 0.00/apj.46 RESEARCH ARTICLE Dynamic delayed controllers for recycling systems with internal delays and complex poles R. J. Vazuez-Guerra J. F. Maruez-Rubio M. A. Hernández-Pérez B. del Muro-Cuéllar J. C. Sánchez-García Escuela Superior de Ingeniería Mecánica y Eléctrica, Unidad Culhuacan, Instituto Politécnico Nacional, México D.F., 04440, México Departamento de Ingeniería Eléctrica, Sección de Mecatrónica, CINVESTAV-IPN, Av. IPN, No. 508, Col. San Pedro Zacatenco, México D.F., 07300, México Correspondence R. J. Vazuez-Guerra, Escuela Superior de Ingeniería Mecánica y Eléctrica, Unidad Culhuacan, Instituto Politécnico Nacional, México D.F., 04440, México. jvg@yahoo.com.mx Abstract Recycling systems are a challenging problem from the control viewpoint due to their detrimental effects. This paper proposes a simple way to stabilize and control a class of high-order recycling systems with time delays, internal instability, and possible complex conjugate poles. The strategy proposes P PI PD PID-like delayed controllers. Contions to guarantee the stability of the controlled closed-loop system are stated. The performance of the proposed delayed controllers is illustrated via numerical simulations of a high-order unstable recycling system and an industrial chemical process. KEYWORDS recycling system, stabilization, time delay, unstable processes INTRODUCTION The purpose of recycling systems is recovering a part of the matter and energy from the output system and reuse it as adtional input. Recycling reduces waste of useful materials and hence increases the efficiency of the overall process. A common example is a plant formed by reactor/separator process, where reactants are recycled back to the reactor., In the literature, specific industrial examples are modeled: the industrial uench column, the reactor-preheater system with energy integration, 3 the reactor/feed-effluent heat exchanger/furnace system, 4 two continuous stirred tank reactor CSTR system with recycle, 5 and so forth. Despite the economic advantages that implies a recycle, in previous years, publications have shown the detrimental effects of recycle. An important conclusion from these stues is that these effects can result from positive or negative feedback, 6 the recycle gain, 7-9 and/or from a delay term presents in the process. 0 Thus, the output behavior of the overall system can be extremely sluggish, oscillating, and even unstable. Therefore, control of recycling systems is a more challenging problem compared to the control of process that receives fresh material and energy streams. Taiwo proposed the idea of recycle compensator; however, any plant model mismatch could have implications on the stability of the closed-loop system. Later, the concept was extended to the design of robust control systems for recycling systems and scussed by Scali and Ferrari. 0 Similar approaches were extended by Lakshminarayanan and Takada 5 and Kwok et al. By another hand, the problem of stabilization and control of delayed unstable processes even without a recycling path is not completely solved. 3-5 Single systems with a delay in the rect input-output path, that is, systems without recycling path, have a delay term only in the numerator transfer function. The problem becomes when a control loop is considered and then the delay term appears in the closed-loop characteristic euation producing an infinite number of roots. Recently, the stabilization and control of delayed systems a single delay in the rect input-output path with only one unstable pole has been stued by Silva. 6 Lee 7 deals with delayed systems containing one unstable pole and n real stable poles and a generalization Asia-Pac J Chem Eng. 08;3:e46. wileyonlinelibrary.com/journal/apj 07 Curtin University and John Wiley &Sons, Ltd. of

2 of VAZQUEZ-GUERRA ET AL. of this problem to the case when the poles can be complex conjugate has been stued by Hernández. 8 It should be highlighted that the control problem in recycling systems becomes more interesting and complex when time delays in the rect and the recycling paths are involved. In this situation, the transfer function of the overall recycling system has a delay term in the numerator and a second delay term into the characteristic euation, even before closing the control loop. Hence, two fferent delay terms in the characteristic euation are present when the control loop is closed. Unstable recycling systems with time delay in the rect and recycling paths have been addressed by Máruez et al. 9,0 and Bamimore. Máruez et al. 9 propose an observer-prector restricted to unstable first-order system at the rect path. The same approach is used in Máruez 0 in order to generalize the previous result to the case of systems with one unstable pole and several stable poles at the rect path. Also, Babimore has proposed an internal model control IMC-basedPI D for controlling open-loop unstable recycle processes. However, in the last cited work, the system should be perfectly modeled, which represents a sadvantage in a practical application. The main idea of this work is to deal with a more general plant that all previously mentioned; this work considers the control problem of a class of linear high-order unstable recycling systems with time delay in the rect and the recycling paths and considering stable poles that could be real or complex. The proposed controllers have a P PI PD PID-like structure. The key idea in our proposal is introducing a delayed term into the control law. This allows us to establish the contions to guarantee the closed-loop stability. Moreover, the existence contions for the stabilizing controllers are stated in terms of the poles position and the delays magnitude. It is important to note that an observer scheme is not used in this work, which avoids the high sensitivity due to the mismatched model induced from the observer. The paper is organized as follows: First, the problem is stated in Section. Section 3 provides the proposed controllers. Later, the performance of the proposed delayed controllers is shown via simulations in Section 4. Finally, some conclusions are presented. PROBLEM STATEMENT Consider Figure that shows a recycling system, where Us is the process input and Ys the output. G d s and G r s are the transfer functions of the rect and the recycling paths, respectively. The class of recycling systems FIGURE Recycling system stued in this work is characterized by G d s N D s e τ s N e τs, s as + b d s + n s + n G r s N D s e τ s N a e τs, s + b r s + n s + n b where τ,τ 0 are the time delays associated to G d s and G r s. N and N are constants; D s and D s are polynomials on the complex variable s. a, b d, b r R, with a, b d, b r > 0and,. and are the damping relations. n and n are the undamped natural freuencies for i,... and j,....inthecase0< < and 0 < <, the second-order subsystems produce pairs of complex conjugate poles. Finally, G d s is considered unstable, and G r s is Hurwitz stable. The process transfer function of the Figure is given by G T s Ys Us N D se τs D sd s N N e. τ +τ s The presence of the transcendental term in the denominator makes controller design a daunting task. For instance, consider the process transfer function G T s and a trational output feedback control as shown in Figure. Then, the closed-loop transfer function is given by Ys Rs CsN D se τs D sd s+[csn D s N N e τ s ]e. 3 τ s Later, the characteristic uasipolynomial is obtained Es D sd s N N e τ +τ s + CsN D se τ s.

3 VAZQUEZ-GUERRA ET AL. 3of Notice that an alternative expression of the contion 6 can be written as φ< a ζ b nb n n, 7 FIGURE Dynamic delayed Cs controller in closed-loop with a recycling system It is clear that the characteristic euation Es 0 has two transcendental terms e τ +τ s and e τ s.thesetranscendental terms induce an infinite number of poles. Moreover, a characteristic euation with two fferent time delays is more complicated to analyze than a characteristic euation with a single time delay, 3 that is, the trational theory developed for systems with a single delay 6-8 cannot be applied here. Accorngly, recycling systems are challenging problems for control analysis and design.in this paper,p PI PD PID-like delayed output feedback controllers are proposed in order to stabilize the class of recycling system defined above. 3 MAIN RESULTS The general agram of the closed-loop recycling system with the proposed delayed controller Cs is shown in Figure. The closed-loop transfer function is given by 3. Then, inwhatfollowsp PI PD PID-like delayed controllers are presented and the contions to guarantee the stability of the closed-loop system are stated. 3. P PI-like delayed controllers First, let us define the proposed controllers I P-like delayed controller Cs D s e τ s 4 II PI-like delayed controller Cs K P s + α e τ s 5 sd s Theorem. Consider the open-loop recycling system given by and the control scheme shown in Figure. Then there exists a P PI-like delayed controller 4/5 such that the closed-loop system 3 is stable if and only if φ< a b d b r where φ τ + τ. n n, 6 where ζ b b d+b r and nb b d b r. b d b r Proof. Let us consider the P-like delayed controller. Notice that the closed-loop characteristic euation derived from 3 contains two transcendental terms e τ +τ s and e τs. In this way, it is possible to propose that +N N such that substituting the P-like N delayed controller 4 into the characteristic euation associated to 3 yields E P s D sd s+k p e φs 0, or euivalently we have E P s s as + ζ b nb s + nb s + n s + n s + n s + n +K p e φs 0. 8 Then the new characteristic euation 8 has only one delay term. The stability properties of the characteristic euation 8 can be obtained by considering an auxiliary system shown in Figure 3 with σ P; therefore, G CP s e φs D sd s. 9 Thus, the closed-loop characteristic euation of the transfer function Y s R s from Figure 3 is euivalent to the characteristic euation 8. In such case, the controller parameter should be obtained to achieve a stable closed-loop. Therefore, a similar freuency domain analysis used in Hérnandez 8 is considered in order to obtain the stability contions of the closed-loop auxiliary system. Necessity The open-loop representation of the auxiliary system 9 in the freuency domain is given by G CP j FIGURE 3 j aj +ζ b nb j+ n b G CPd G CPr e φj, Auxiliary system G Cσ s 0

4 4of VAZQUEZ-GUERRA ET AL. with G CPd j + n j + n and G CPr j + n j + n. Therefore, the magnitude expression is obtained as M GCP +a 4 + n, ζ + 4 b n M Pd M Pr b b where M Pd 4 + n + 4 n and M Pr 4 + n ζ + 4 n. The phase expression is given as arctan G CP φ π +arctan arctan a arctan n n ζ b nb nb n n. From the Nyuist stability criterion, the closed-loop system shown in Figure 3 is stable iff 0 N + P, where P is the number of poles of the transfer function G CP s on the right half plane and N is the number of counterclockwise rotations to the point, 0 in the Nyuist agram. In this case, P duetog CP s has one unstable pole. Therefore, it is reuired a counterclockwise rotation to the point, 0 to get a stable closed-loop system. Consider the phase expression notice that G CP 0 π. It is clear that in order to obtain a counterclockwise rotation to the point, 0, the phase expression should satisfy G CP > π for 0. Thus, this last contion can be assured by considering d GCP 0 > 0. Since d d GCP d 0 φ + a ζ b nb n producing the stability contion φ< a ζ b nb n n > 0 n. Notice that ζ b +. nb b d b r Sufficiency Consider that the contion φ < a ζ b nb n n is satisfied, with ζ b nb b d + b r. It is clear that d GCP 0 d > 0. Thus, G CP is an increasing function. An adeuate value of the gain produces a counterclockwise rotation to the point, 0 obtaining the closed-loop stability. Remark. Notice that a similar proof can be performed to the case of the PI-like delayed controller by considering +N N K, α P α and an α N +N N sufficiently small to obtain the contion given by P-like delayed controller parameters The parameters N, N, D s and τ in the P-like delayed controller are rectly obtained from the recycling system model characterized in. Consider the 0, then the phase in the Nyuist agram G CP 0 πafter this, the phase increases up to a maximum and later it decreases until there is an intersection with the negative real axis, that is, G CP c π. The phase crossover freuency c is given by c ζ c b nb φ c + arctan arctan a c nb ζ c n ζ c n arctan arctan 0, c c n n 3 where c > 0. Then the stability contion M GCP < can be used in order to assure one counterclockwise rotation to the point, 0. Therefore, the set of stabilizing control parameter,, becomes with T n b n n and a T < < M P, 4 M P c + a 4 c + n c ζ + 4 b n M Pd M Pr b b with M Pd M Pr 4 c + n c 4 c + n c ζ + 4 n ζ + 4 n. and 3.3 PI-like delayed controller parameters Following a similar procedure, the parameters N, N, D s and τ in the PI-like delayed controller are obtained from the recycling system model. Then an α sufficiently

5 VAZQUEZ-GUERRA ET AL. small should be proposed such that the following euation has solution at least for two freuencies c > 0 c ζ c b α nb φ c +arctan arctan arctan a c ζ c n ζ c n arctan arctan c c n n c nb 0, 5 We define c > 0and c > 0 as the first two positive phase crossover freuencies of 5. Notice that 5 is derived from the phase of the open-loop auxiliary system G CPI, inthiscasewehave G CPI c π. Then the range of is given by M PIg M PI < < M PI 6 cg +a 4 cg+ n cg ζ + 4 b n M PId M PIr b b + α cg with M PId M PIr,. 4 cg + n cg ζ + 4 n, 4 cg + n cg ζ + 4 n and g Remark. In the case when the recycling system does not satisfy the stabilizing contion 6 of the P PI-like delayed controllers, in the next paragraphs, thepd PID-like delayed controllers are proposed. 3.4 PD PID-like delayed controllers Let us define the proposed controllers III PD-like delayed controller K p s + Cs e τ s 7 D s IV PID-like delayed controller K p s + s + α Cs e τ s 8 sd s Theorem. Consider the open-loop recycling system given by and the control scheme shown in Figure. 5of Then there exists a PD PID-like delayed controller such that the closed-loop system 3 is stable if and only if φ< a b d b r + a + b d + b r + where φ τ + τ. n n ζ ζ +, n n Proof. The proof of this Theorem can be consulted in the Appenx A. 3.5 PD-like delayed controller parameters 9 The parameters N, N, D s and τ in the PD-like delayed controller 7 are obtained from the recycling system model. Considering that can be chosen accorng to Euation A and that the phase G CPD in the Nyuist agram starts with G CPD 0 π. Thus, after this, the phase increases up to maximum and later it decreases; therefore, there is an intersection with the negative real axis with G CPD c π. The phase crossover freuency c is given as φ c +arctan ζ c arctan c c +arctan a n n ζ b arctan c ζ c arctan c n n c nb c nb 0, 0 with c > 0. Then, in order to obtain the counterclockwise rotation to the point, 0, M GCPD < is reuired. Then, the set of stabilizing gain is given by a T with T n b n n and < < M, c +a 4 c + n c ζ + 4 b n M d M r b b M + c

6 6of with M d M r 4 c + n c 4 c + n c ζ + 4 n. 3.6 PID-like delayed controller parameters ζ + 4 n and Following a similar procedure, N, N, D s and τ in the PID-like delayed controller are obtained from the recycling system model. Considering that is selected from A and α is sufficiently small, such that the following euationissolved c c φ c + arctan + arctan a α c ζ c b nb arctan c nb ζ c arctan c n n ζ c arctan c n n 0. for c > 0and c > 0 being the first two phase crossover freuency. Thus, the expression is derived from G CPID c π.therangeof is given by M < < M where cg +a 4 cg + n cg b M g with M d 3 ζ b + 4 n b M d M r cg + α cg 4 cg + n cg ζ + 4 n, M r 4 cg + n cg ζ + 4 n and g,. Since the proposed controllers are PI PID-like delayed controllers, the properties of step tracking references and rejecting step sturbances are fulfilled in the closed-loop system. This can be easily verified by using the proposed controller and applying the final value theorem. Remark 3. Consider the obtained stability contion 6 in Theorem and assuming that there are not complex conjugated poles, that is,, >. Then, the following stability contion can easily be derived φ< a b d b r c VAZQUEZ-GUERRA ET AL. c, 4 where c > 0andc > 0 are the real poles position of the rect and recycle paths, respectively. Therefore, the stability contion 4 intends to provide a simpler expression than the one presented in 6, when, >, considering now the associated real poles position. Similarly, the contion 9 in Theorem for the case, >, becomes φ< a b d b r c c + a b b d r c c 4 SIMULATION RESULTS 5 In this section, the performance of the proposed delayed controllers is evaluated through a high-order academic recycling system and a reactor-separator-recycle process inclung time delays at rect and recycling paths. Example. Consider the high-order recycling system of Figure with G d s N D s e τ s 0 s 0.3s + s + 8s + 0s + 6s + 3 e 0.4s, 6a G r s N D s e τs 5 s + 5s + s + 0 e 0.5s. 6b The parameters of recycling system 6 are φ 0.9, a 0.3, ζ b.068, nb 3.63, ζ d , nd 4.47, ζ d 0.83, nd , ζ r 0.36 and nr Since 0.9 φ< a b d b r n n.578 the contion 6 of the Theorem is fulfilled. It follows that the recycling system 6 can be stabilized by a P PI-like delayed controller. For the P-like delayed controller, from 3, 0.869, and the range of stabilizing values is given by 4, that is, < < Specifically, it is selected From 4, thecorresponngp-like delayed controller is

7 VAZQUEZ-GUERRA ET AL. Cs D s e τs 805 s + 5s + s + 0 e 0.5s 7 The output response of the recycling system 6 in the closed-loop with the P-like delayed controller is shown in Figure 4. When a PI-like delayed controller is considered, the parameter α is chosen as α From 5, and 0.89 are obtained. Then is bounded by < < 088 from 6. The parameter 8300 is chosen. Then, from 5, the corresponngpi-like delayed controller is given by Cs s + α 835s e τs 8350 sd s ss + 5s + s + 0 e 0.5s. 8 To improve the tracking properties of the system together with an adeuate overshoot response and set time reduction, it has been proposed in the literature a two degree of freedom control implementation, 4 also known as PI-setpoint weighting tuning. The proposed PI-like delayed controller given by 8 can be implemented as yt FIGURE 4 controller Us RsG ff s YsG c s, Time s Closed-loop output response with P-like delayed with G ff s D η + α s e τ s, 7of G c s + α D s e τs, where η is a parameter that should be select from 0 < η <. For the simulations η 0.0 is chosen. The closed-loop performance of the output signal is shown in Figure 5. An unitary step reference and a step sturbance of magnitude 0. acting at 500 s are considered. Dashed line shows the performance for the recycling system under delay uncertainty of rect path to +75%. On the other hand, the stability contion 9 of the Theorem 0.9 φ< a b d b r n n + a + b + d b + ζ ζ + 4.9, r n n m m is also satisfied. Recycling system 6 can also be stabilized by a PD PID-like delayed controller. When a PD-like delayed controller is considered, the range of < and is taking from A, <.5 is chosen. Considering from 0, it is obtained is determined as < < from, where it is selected 3000 for the simulation experiment. From 7, the corresponngpd-like delayed controller is Cs K p s + D s e τ s 300s s + 5s + s + 0 e 0.5s. 9 Figure 6 illustrates the evolution of the closed-loop output response under the mentioned controller design yt 0.6 yt Nominal recycling system Delay Uncertainly of rect path 75% Time s Time s FIGURE 5 controller Closed-loop output response with PI-like delayed FIGURE 6 controller Closed-loop output response with PD-like delayed

8 8of When a PID-like delayed controller is considered, the range of is taking from A and.6 is chosen. Considering 0.65 and α 0. into, and are obtained. Finally, from 3, we obtain < < The parameter 000 is chosen. Then, from 8, the corresponngpid-like delayed controller is given by Cs K p s + s + α sd s e τ s 00 s s e 0.5s. ss + 5s + s + 0 Considering the two degree of freedom control implementation, 4 the proposed PID-like delayed controller given by 30 can be implemented as with 30 Us RsG ff s YsG c s 3 G ff s η s + η s + α e τs, 3 sd s G c s s + s + α e τs, 33 sd s where η and η are parameters that should be select from 0 <η < and0<η <. For the simulations η η 0. are chosen. Figure 7 illustrates the performance nominal case of the recycling system 6 in the closed-loop with the PID-like delayed controller given by 30. In the same figure, two dashed lines show the output response obtained under unstable pole uncertainty. Also, an unitary step reference and a step sturbance of magnitude 0.3 acting at 00 s are considered. In order to illustrate the closed-loop robustness with respect to parametric model uncertainties provided by VAZQUEZ-GUERRA ET AL. the proposed controller, consider the rect path of the recycling system under an adtive bounded uncertainty, given by N G d 34 s ās + b d s + n s + n where the adtive uncertainty is defined as ā a + Δ a. Therefore, with a 0.3 andδ a 0, the nominal case is referred. Then, considering a 0.3and Δ a [ 0.3, 0.40], a deviation on the nominal parameter is considered, that is, a parametric uncertainty is consideredontheparameterā. Under these assumptions, a proposed PID-like delayed controller given by 30 is designed to the whole recycling system in the nominal case, and then a freuency domain analysis based on the Nyuist stability is applied to the closed-loop system for fferent values of the parametric uncertainty. In this way, Figure 8 shows the Nyuist agram trajectory when fferent values of the parametric uncertainty. It is possible to see in Figure 8 that the closed-loop stability is preserved until an uncertainty of the 34%, since the closed-loop stability is satisfied, that is, an anticlockwise trajectory to the point. Example. Let us consider the reactor-separatorrecycle process modeled in Beuette 5 inclung time delays at rect and recycling paths. Thus, a state space representation of the open-loop linear model is given by ẋ Āx + A xt τ + A xt τ + But y C xt τ, yt Δ 0 nominal case 0.6 Δ Δ Tiempo s FIGURE 7 controller Closed-loop output response with PID-like delayed Imaginary Axis Nyuist Diagram Δ 0 nominal case 0.5 Δ Δ Real Axis FIGURE 8 Nyuist agram for fferent values of uncertainty in a

9 VAZQUEZ-GUERRA ET AL. yt Time FIGURE 9 Output response of the system 35 in closed-loop with P-like delayed controller with rect and recycling paths Figure are obtained as G d s 9of s s e 0.5s, 35a G r s e 0.8s. 35b The process 35 is a particular case of the contion 4 of the Remark 3, where there exists an unstable pole, a real stable pole in the rect path and a gain in the recycling path, as well as delays on both paths. Thus, the contion 4 of the Remark 3 is used, and it is fulfilled. It follows that the recycling system is stabilizable by P-like delayed controller. Computing x [x d x r ] T, Ā F V λ F V UA + ΔH C As k ps 0 Vρc p ρc p 0 F V λ F, V k s [ ] 00 0 ΔH k s UA Ā, Ā ρc p, B Vρc p C As k ps where k s k 0 exp E a RT s and k ps k s E a RT s. The parameters values are given in Table. The state x d is the reactor temperature T, and x r is the component concentration C A. The manipulated input is the jacket reactor temperature T j. Also, it is assumed that τ is the time delay due to concentration and τ is the time delay caused by transport of material. The output matrix is C [ 0 ], since concentration second state is the measured signal. The models of.3649 and using the range.506 < < is chosen. From 4, the corresponng P-like delayed controller can be written as Cs D s e τ s e 0.8s. In this example, there is a negative gain in the rect path; therefore, in order to obtain a positive output TABLE Constant parameters for Example Operating volume V 500 ft 3 Operating flowrate F 000 ft 3 hr Reactor ameter D r 7.5ft Overall heat-transfer coefficient U Btu hrft R Heat transfer area through reactor wall A ft Preexponential factor k hr Activation energy E a 3400 Btu lbmol Ideal gas constant R.987 Btu lbmol R Heat of reaction ΔH Btu lbmolpo Density of coolant ρ 53.5 lb ft 3 Heat capacity of coolant c p Btu lb R Operating concentration C As lbmol ft 3 Operating temperature T s R Direct path time delay τ 0.5 hr Recycling path time delay τ 0.8hr Recirculation coefficient λ 0.

10 0 of VAZQUEZ-GUERRA ET AL. response, a negative input reference is considered. Figure 9 shows the stable output response of the system 35 in closed-loop with the P-like delayed controller, subject to an input step-type of. 5 CONCLUSIONS This paper proposes P PI PD PID-like delayed controllers and presents necessary and sufficient contions to guarantee the closed-loop stabilization of high-order unstable recycling systems considering time delays at the rect and recycling paths, and adtionally possible complex conjugate stable poles. Also, it is included a procedure to obtain the parameters of the delayed controllers using the domain freuency approach. When the proposed PI PID-like delayed controllers are used, the properties of step tracking references and rejecting step sturbances are fulfilled in the closed-loop system. Numerical simulations show the adeuate closed-loop performance with the proposed controllers. ORCID R. J. Vazuez-Guerra M. A. Hernández-Pérez REFERENCES. Luyben WL, Tyreus BD, Luyben ML, eds. Plantwide Process Control. New York: McGraw-Hill; Wu KL, Yu CC. Reactor/separator processes with recycle-. Candate control structure for operability. Comput Chem Eng. 996;0: Madhukar GM, Goraa DB, Agrawal P, Lakshminarayanan S. Feedback control of processes with recycle: a control loop performance perspective. Chem Eng Res Design. 005;83A: Chodavarapu SK, Zheng A. Control system design for recycle systems. J Process Contr. 00;: Lakshminarayanan S, Takada H. Empirical modelling and control of processes with recycle: some insights via case stues. Chem Eng Sci. 00;56: Morud J, Skogestad S. Dynamic behavior of integrated plants. J Process Contr. 996;6: Luyben WL. Dynamics and control of recycle systems. Simple open-loop and closed-loop systems. Ind Eng Chem Res. 993;3: Luyben WL. Dynamics and control of recycle systems. Comparison of alternative process designs. Ind Eng Chem Res. 993;3: Luyben WL. Temperature control of autorefrigerated reactor. J Process Contr. 999;9: Scali C, Ferrari F. Performance of control systems based on recycle compensators in integrated plants. J Process Control. 999;9: Taiwo O. The Dynamics and Control of Plants with Recycle Control Systems Centre Report 65, UMIST. Manchester, England; Kwok KE, Chong-Ping M, Dumont GA. Seasonal model based control of processes with recycle dynamics. Ind Eng Chem Res. 00;40: Lee J, Lee Y, Yang DR, Edgar TF. Half order plus time delay HOPTD models to tune PI controllers. AIChE Journal. 07;63: Liu T, Gao F. Enhanced IMC design of load sturbance rejection for integrating and unstable processes with slow dynamics. ISA Trans. 0;50: Yin C, Gao J, Sun Q. Enhanced PID controllers design based on mofied smith prector control for unstable process with time delay. Math Prob Eng. 04;04. Article ID 5460, 7 pages. Etor: Hongli Dong. 6. Silva GJ, Datta A, Bhattacharyya SP, eds. Pid Controllers for Time-Delay Systems. Boston: Birkhuser; Lee SC, Wang QG, Xiang C. Stabilization of all-pole unstable delay processes by simple controllers. J Process Control. 00;0: Hernández MA, del Muro B, Velasco M. PID for the stabilization of high-order unstable delayed systems with possible complex conjugate poles. Asia-Pacific J Chemical Eng. 05;0: Máruez JF, del Muro B, Velasco M, Cortés D, Sename O. Control of delayed recycling systems with unstable first order forward loop. J Process Control. 0;: Máruez JF, Vazuez-Guerra RJ, del Muro-Cuellar B, Sename O. Stabilization and control of delayed recycling high order systems with one unstable pole at the rect path. Asian J Contr. 06;85:-3.. Bamimore A, Taiwo F. Recycle compensator facilitates rapid parameterization of proportional integral/derivative PI/D controllers for open-loop unstable recycle processes. Ind Eng Chem Res. 05;54: Atay FM. Complex Time-Delay Systems: Theory and Applications. Berlin: Springer; Gu K, Niculescu SI, Chenc J. On stability crossing curves for general systems with two delay. J Math Anal Appl. 005;3: Astrom KJ, Hagglund T, eds. Pid Controllers, Theory, Design and Tuning. New York: International Society for Measurement and Control; Beuette BW. Process Control: Modeling, Design and Simulation. The University of Michigan: Prentice Hall; 003. ISBN: , , 769 pages. Howtocitethisarticle: Vazuez-Guerra RJ, Maruez-Rubio JF, Hernandez-Perez MA, del Muro-Cuellar B, Sánchez-García JC. Dynamic delayed controllers for recycling systems with internal delays and complex poles. Asia-Pac J Chem Eng. 08;3:e46. APPENDIX A: PROOF OF THEOREM. Let us consider the PD-like delayed controller. As shown previously, the closed-loop characteristic euation derived

11 VAZQUEZ-GUERRA ET AL. of from 3 has two fferent transcendental terms. Therefore, it is possible to obtain a characteristic euation with a single time delay by considering and + N N. Under these assumptions and substituting the PD-like delayed controller 7 into the characteristic euation 3, it yields E PD s D sd s+k p s + e φs 0, or euivalently we have E PD s s as + ζ b nb s + nb s + n s + n s + n s + n +K p s + e φs 0, A3 where ζ b b d+b r b d b r and nb b d b r. The stability contion of the characteristic euation A3 is obtained from the auxiliary system shown in Figure 3 with σ PD;thus, G CPD s s + e φs. A4 D sd s It is easy to see that the closed-loop characteristic euation of the auxiliary system shown in Figure 3 and the characteristic euation A3 are similar. In such case, thefreecontrolparametersare and. Therefore, the stability properties of the closed-loop auxiliary system can be obtained by considering a freuency domain analysis similar to the used approach in Hernández. 8 Necessity The open-loop representation of the auxiliary system A4 in the freuency domain is given by G CPD j j + e φj e φj, j aj +ζ b nb j+ n b G CPDd G CPDr A5 with G CPDd j + n j + n and G CPDr j + n j + n. The magnitude of the auxiliary system A4 is expressed as The phase expression of the auxiliary system A4 is obtained as G CPD φ π + arctan a + arctan arctan n n ζ b arctan arctan nb nb. A8 n n In order to assure the counterclockwise rotation to the point, 0 in the Nyuist agram, the magnitude M GCPD should be a decreasing function of and the phase G CPD an increasing function around 0. First, the decreasing property can be assured by considering d d M G CPD < 0, producing the stability 0 contion < a + ζ ζ b + ζ +. n b n n Thus, the magnitude M GCPD decreases monotonically. Then the phase expression should satisfy G CPD > π for 0. It can be assured by considering d GCPD 0 > 0. d Since d GCP d 0 φ + a + ζ b nb Thus, the stability contion φ< a ζ b nb + a + ζ b + n b n n n n > 0. ζ ζ + n n + M GCPD + a 4 + n, A6 ζ + 4 b n M PDd M PDr b b with M PDd M PDr 4 + n + 4 n and 4 + n ζ + 4 n. is obtained, where ζ b + and ζ b n b b d b r nb is possible to choose within the range b d + b r.it

12 of VAZQUEZ-GUERRA ET AL. φ a + ζ b + nb < a + ζ b + n b n + n < ζ + n Sufficiency Consider that the contion φ< a ζ b nb a + ζ b + n b n n + ζ ζ + n n. n A is satisfied, with ζ b + and ζ b +, n b b d b r nb b d b r from d GCPD 0 > 0and d M G CPD < 0. d d 0 Thus, the phase, G CP, is an increasing function for freuencies 0, and the magnitude M GCPD is a decreasing function of considering the expression A. The closed-loop stability can be obtained with a gain that produces a counterclockwise rotation to the point, 0. A similar proof as in the PD-like delayed controller can be made to the PID-like delayed controller by considering, + N, a sufficiently small α and following N the development presented above to obtain the contion given by 9.