Article pubs.acs.org/IECR
Multiloop Integral Controllability Analysis for Nonlinear MultipleInput Single-Output Processes Yi Zhang,†,‡,¶ Steven Su,*,§ Andrey Savkin,∥ Branko Celler,⊥ and Hung Nguyen§ †
School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu, 611731, China Key Laboratory for NeuroInformation of Ministry of Education, School of Life Science and Technology, and ¶Center for Information in BioMedicine, University of Electronic Science and Technology of China, Chengdu, 610054, China § Centre for Health Technologies, Faculty of Engineering and Information Technology, University of Technology, Sydney, New South Wales 2007, Australia ∥ School of Electrical Engineering and Telecommunications, and ⊥Biomedical Systems Laboratory, University of New South Wales, Sydney, New South Wales 2052, Australia ‡
S Supporting Information *
ABSTRACT: The decentralized integral controllability (DIC) for linear/nonlinear square processes implies the existence of stable decentralized controllers with integral actions capable of achieving offset-free control and detuning any subset of the control loops independently. However, the current version of the DIC cannot be directly applied to nonsquare processes specifically for multiple-input single-output (MISO) processes. This paper presents the new definition and theorem of multiloop integral controllability (MIC) to nonlinear MISO processes, and proposes the sufficient MIC conditions in order for such processes to guarantee decentralized unconditional stability under control loop failure as well as to achieve offset-free tracking performance. Two examples, the quadruple-tank system (model based) and the temperature control system, are modified as two-input single-output (TISO) plants and given to quantitatively interpret the effectiveness of the proposed MIC analysis where the desirable performance of both applications can be obtained.
1. INTRODUCTION Controllability analysis was put forward by R. E. Kalman, who introduced the Kalman’s criterion of state controllability to ensure the output controllability of a linear continuous-time system.1,2 After that, the decentralized control system received much attention as it exhibited several advantages over the centralized system such as flexibility, failure tolerance, simplified design, and tuning.3,4 Aoki et al. proposed the notion of controllability under a decentralized structure for linear, finite dimensional, square systems and the necessary and sufficient conditions for it to be implemented.5 Davison and Wang presented a decentralized fixed model to characterize the models that cannot be altered by using a linear time-invariant decentralized controller.6 Anderson and Moore established the time-varying feedback law to deal with time-invariant systems.7,8 The decentralized controller design was then rapidly applied in chemical control.9 Grosdidier et al. proposed integral controllability to ensure the stability of multiloop control systems for square processes where the number of input and output of systems are equal.10 Furthermore, Skogestad and Morari11,12 proposed decentralized integral controllability (DIC) for square processes as a means to determine whether a multivariable plant can be stabilized by multiloop controllers, whether any subset of © XXXX American Chemical Society
the control loops can be detuned independently (decentralized detunable) or even turned off without endangering closed-loop stability, and whether multiloop controllers can achieve offsetfree control. Following this framework, Lee and Edgar summarized sufficient and necessary conditions for the DIC analysis.13 Su et al. extended the DIC to nonlinear processes by which a steady-state sufficient condition was provided.14 In the process control industry nonsquare processes (indicating that the number of inputs and outputs of a process is unequal) often occur.15−17 For instance, Doukas and Luyben suggested an nonsquare structure for a distillation column that was recognized as the most economical strategy for the separation of toluene, xylene, and benzene. The raised problem was then to control the four impurities with only three manipulated variables, reboil duty, reflux ratio, and side streamflow rate.18 Another example is the Shell control problem published by the company in 1986,19in which a five-input sevenoutput heavy oil fractionator was required. For the purposes of Received: Revised: Accepted: Published: A
May 26, 2017 June 12, 2017 June 14, 2017 June 14, 2017 DOI: 10.1021/acs.iecr.7b02165 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Figure 1. Multiloop Integral Controllability for a MISO system.
use sufficient MIC condition are given based on the singular perturbation analysis. Two examples, the quadruple-tank system (model based) and the temperature control system (output measurement based), are modified as two-input single-output (TISO) plants and given to quantitatively interpret the effectiveness of the proposed MIC analysis where the decentralized unconditional stability under a partial or complete failure of actuators can be guaranteed and the offset-free tracking can be achieved for both applications when the MIC conditions are satisfied. The remainder of the paper is organized as follows. section 2 introduces the new definition of MIC and the proposed MIC sufficient condition. Section 3 shows the MIC analysis for the quadruple-tank system. Section 4 describes the MIC analysis for the real-time temperature control system. Section 5 concludes the paper. Appendices provide the proof of Theorem 1, and the proof of the Global Asymptotically Stability of the Quadrupletank System, respectively. Supporting Information shows more detailed results for the embedded temperature control system.
controller designs in terms of nonsquare processes, one of widely used strategies was to square the original nonsquare process by adding or discarding the appropriate number of inputs or outputs in order to reshape it into the squared multiple-input multipleoutput (MIMO) system. Once such system is resquared, then the DIC analysis can be applied directly. However, it should be pointed out that nonlinearities almost appear in all process control systems (including the nonsquare process). Therefore, deleting the additional inputs perhaps will decrease the flexibility and reliability of controllers. In addition, adding new outputs may further require extra control efforts. Following this motivation, scholars and researchers began to focus on the nonlinear control for nonsquare MIMO processes.20 Despite such evidence, the literature on Integral Controllability analysis for nonsquare MIMO processes is still limited. This is because handling the multiple equilibriums of the multiple subsets is nontrivial. This study concerns the special case of nonsquare MIMO processes, a multiple-input single-output (MISO) process, and provides the new definition and theorem of multiloop integral controllability (MIC) analysis for MISO processes. It was partially motivated by a human exercise process control where, for safety and reliability purposes, several exercise indexes (e.g., back-carrying load, running speed, and the gradient of the treadmill) were involved in the control of an indicator of human cardiorespiratory responses, heart rate.21−24 Also, in the chemical industry it is desirable if multiple actuators (e.g., feedrate of distributed heaters and fans, pH, and humidity) can be manipulated for one output (the temperature of reactor). In practice, the redundant control inputs of a MISO process were usually discarded in order to simplify it as a single-input singleoutput (SISO) system.25 However, such simplification might also degrade the reliability of system because the discarded inputs might have the potential to facilitate fault accommodation under the control loop failure. In addition, since the actuators in the real engineering process often have physical limitations, the nonsaturation range of the output is always limited. Compared with a nonsquare MISO scheme, discarding the additional inputs of MISO processes will also decrease the amplitude of the maximum open-loop gain. To best of authors’ knowledge, apart from our preliminary study,26 most previous works cannot be directly applied to handle the proposed MIC analysis for nonlinear MISO processes. For instance, the standard robust control analysis/ synthesis approaches, for example, H∞ and μ analyses, cannot be directly applied yet.27−30 In this study, the definition and theorem of MIC for nonlinear MISO processes and an easy-to-
2. MULTILOOP INTEGRAL CONTROLLABILITY (MIC) As the current version of decentralized integral controllability (DIC)12,31−33 cannot directly be used for the analysis of MISO processes, the new definition of MIC for nonlinear MISO processes is given. As shown in Figure 1, assume the MISO system P can be described by eq 1 with an input vector u ∈ 9 m and a scalar output y ∈ 91: n m ⎧ ⎪ x ̇ = f ̅ (x , u), x ∈ ? ⊂ 9 , u ∈ < ⊂ 9 P⎨ ⎪ 1 ⎩ y = g ̅ (x , u), y ∈ @ ⊂ 9
(1)
It is assumed that the state x(t) is uniquely determined by its initial value x(0) and the input function u(t). On the basis of this assumption, we further state that the process (1) has equilibriums at origin, that is, f ̅(0,0) = 0, and g(0,0) = 0. If the ̅ equilibrium xe is not at origin, a translation can be performed by redefining the state x as x − xe. In the following discussion, it is assumed that such a translation has been done whenever it is applicable. In Figure 1, the block η is a positive scalar, and the blocks C1, C2, ···, Cm are all stable scalar controllers that represent the remaining stable parts (excluding the integrators) of the controller. In the following analysis, it is also assumed that the process (1) is stable (see condition b of Theorem 1). B
DOI: 10.1021/acs.iecr.7b02165 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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2, ···, m. It should be noted if εi(0−) ≠ 0, a simple translation can be performed by redefining the state εi as εi − εi (0−). This transformation, only changing the nominal value of the controller states, does not affect the following stability analysis (see eqs 5 and 6 and their associated discussions for details). The following theorem presents a sufficient condition for MIC for MISO processes. Theorem 1 (Steady-state MIC Conditions for Nonlinear MISO Processes). Consider the closed-loop system in Figure 1. Assume both the reference r0 and the set of detuning factors ki (0 ≤ ki ≤ 1, i = 1, 2, ···, m) are all fixed parameters, and the system has a unique equilibrium. Suppose the general process P̃ and the linear part of the controller Cl are described by eq 2 and 4, respectively, and a translation has been performed so that the equilibrium of the translated system is zero. If the following assumptions are satisfied with respect to the equilibrium (zero), then there exists a positive scalar η0 > 0 such that when 0 < η < η0 the equilibrium is GAS: a. The eq 0 = f(x, ũ) obtained by setting ẋ = 0 in eq 2 implicitly defines a unique C2 function x = h(ũ) for ũ ∈ Rm. b. For any fixed ũ ∈ Rm, the equilibrium x = h(ũ) of the system ẋ = f(x, ũ) is globally asymptotically stable and locally exponentially stable.35 c. When ũ = [k1α k2α ··· kmα]T (0 ≤ ki ≤ 1, i = 1, 2, ···, m), if the steady-state input output function g(h(ũ), ũ) (with respect to the reference input r0) of the general process P̃ satisfies the following conditions:
Definition 1 (Multiloop Integral Controllability for Nonlinear MISO Processes). The nonlinear process P defined in eq 1 is said to be MIC with respect to a given reference r = r0 if the closedloop system depicted in Figure 1 satisfies the following conditions: (1) There exists a multiloop integral controller C, such that the nominal closed-loop system is globally asymptotically stable (GAS) for the equilibrium x = xe0 with respect to the given constant reference r = r0. (2) When each individual loop is detuned independently by a factor ki (0 ≤ ki ≤ 1, i = 1, 2, ···, m), for each set of fixed multiloop gains {k1, k2, ···, km } the closed loop system is GAS for an equilibrium x̃e (not necessarily x̃e = xe0). Remark 1. In the above definition, the reference is a constant value r0. For any other interested (constant) reference value, all the above statements should be valid for a new equilibrium with respect to a new reference. Different with DIC, for MIC, the equilibrium of the MISO system under control not only is determined by the reference input r = r0 but also the ratio of detuning factors ki (0 ≤ ki ≤ 1, i = 1,2,···m) may influence the equilibrium. We will discuss how the equilibrium can be calculated later. In the following, the initial reference r0 is assumed that r0 = 0. In Figure 1, we assume the state equation of the general process P̃ (inclusive of original process P and m stable scalar controllers C1, C2, ···, and Cm) is modeled as below (with the same assumptions for eq 1 of process P) ⎧ = f (x , u)̃ ⎪ ẋ P :̃ ⎨ ⎪ ⎩ y = g (x , u)̃
m
(2)
α ·g (h(u), ̃ u)̃ > 0
(c.1)
and
⎡ ε1̇ ⎤ ⎡1⎤ ⎡1⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ε2̇ ⎥ ⎢ 1 ⎥ = − ηy ⎢ 1 ⎥ = η e ⎢ ⎥ ⎢⋮⎥ ⎢⋮⎥ ⎢⋮⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ εṁ ⎥⎦ ⎣1⎦ ⎣1⎦
α ·g (h(u), ̃ u)̃ > ρ ·α 2 m
(when α ≠ 0, and
(c.2)
i=1
diag{[k1 , k 2 , ⋯km]}ε
(3)
Equation 3 indicates the input multiplicity of MIC. On the basis of MIC where the output of the system converges to its reference, the equilibrium of both inputs and states may converge to different equilibriums if the set of detuning factors [k1, k2, ···, km ] is different. To cope with input multiplicity, we swap the bank of integrators with a single integrator (see Figure 1). This means, we use only one scalar variable (ξ) to take the role of all integral states (εi, i = 1, 2, ···, m) in eq 3 for the multiloop integral controller. Considering the controller with zero initial conditions (i.e., εi(0−) = 0, i = 1, 2, ···, m), eq 3 is then simplified as follows
k1
∂g (h(u), ∂g (h(u), ̃ u)̃ ̃ u)̃ + k2 + ··· + km ∂u1̃ ∂u 2̃ ∂g (h(u), ̃ u)̃ >0 ∂um̃
m
(when α ≠ 0, and
∑ ki ≠ 0) i=1
(c′.1)
and ⎛ ∂g (h(u), ∂g (h(u), ̃ u)̃ ̃ u)̃ + k2 + ⎜k 1 ∂u 2̃ ∂u1̃ ⎝
ηe = − ηy ⎡ k1 ⎤ ⎢ ⎥ ⎢ k2 ⎥ ⎢ ⎥ξ ⎢⋮⎥ ⎢⎣ k ⎥⎦ m
∑ ki ≠ 0)
in a neighborhood of α = 0, where ρ is a positive real number. c′. When ũ = [k1α k2α ··· kmα]T, if the steady-state input output function g(h(ũ), ũ) (with respect to its equilibrium point) of the general process P̃ satisfies the following requirements:
34
⎧ξ ̇ = ⎪ ⎪ ⎪ Cl : ⎨ ⎪ũ = ⎪ ⎪ ⎩
∑ ki ≠ 0) i=1
Considering that r0 = 0, the state equation for the linear integral controller is expressed as ⎧ ⎪ ⎪ ⎪ε ̇ = Cl : ⎨ ⎪ ⎪ ⎪ ⎩ũ =
(when α ≠ 0, and
··· + km m with ∑i = 1 ki
(4)
∂g (h(u), ̃ u)̃ ⎞ ⎟>ρ>0 ∂um̃ ⎠
(c′.2)
≠ 0 in a neighborhood of α = 0 (excluding α
= 0), where ρ is a positive real number. If for any set of detuning factors ki (0 ≤ ki ≤ 1, i = 1, 2, ···, m), the above conditions are satisfied with its equilibriums, then the
where ξ is a scalar, which can be used to represent every εi, (i = 1, 2, ···, m) when εi has zero initial condition, that is, εi(0−) = 0, i = 1, C
DOI: 10.1021/acs.iecr.7b02165 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research nonlinear MISO process is MIC with respect to the reference input r0. Proof 1. The Proof of the above theorem (similar to that of Theorem 1 in ref 14) is based on the singular perturbation theory.11 By using the singular perturbation theory, the overall dynamic system can be reduced as the integrator dynamics feedback with the steady-state input−output mapping of the process. For details, see Appendix A. ■ Note 1. When the steady-state input-state equation is available, condition a of Theorem 1 (the uniqueness of a C2 function x = h(ũ) of a steady-state function) is easily checked. The GAS of the open-loop process ẋ = f(x, ũ) (the fast time scale system) (the first part of condition b) can be checked by finding a Lyapnouve function (refer to the example in ref 14). In engineering practice, the stability of the open-loop asymptotical stability is often possible to be verified experimentally without the need of an explicit model. The LES of the open-loop process (the second part of condition b) can be verified by checking the asymptotical stability of its linearized model around the equilibrium state. Condition c is a sector condition with parameters ki (0 ≤ ki ≤ 1, i = 1, 2, ···, m). If an explicit steady state model in the operation range cannot be obtained, condition c.1 is not easily verified (see an illustrative example in section 3). We also provide condition c′.1, a sufficient condition of condition c, which can be checked experimentally as this qualitative type condition does not need an explicit model to verify (see an illustrative example in section 4 for details). The following discussion highlights the major issues raised in the conditions of Theorem 1: 1. The MIC conditions in Theorem 1 are involved in the input−output function of the process only in the steadystate. Models of the steady-state behavior of processes are generally more accurate and readily available than are those of the transient.3 Thus, the proposed steady state sufficient condition is easy to use. For a linear MISO process (G(s)1×m), assume its steady-state gain is G(0) = [g1, g2, ···, gm]. If all of the items of its steady-state gain are nonzero (i.e., gi ≠ 0), then it can be verified that all the conditions of Theorem 1 can be satisfied by properly selecting the sign of each control loop. 2. Similar with the sufficient conditions for DIC of nonlinear square processes (see condition (iii) of Theorem 1 in our previous study14), both conditions c and c′ contain condition c.2 and condition c′.2, respectively, which ensure LES of the reduced model (slow time scale) around its equilibrium. Actually, these conditions are rather mild. For example, it can be checked that the following steady-state input−output function, g1 (h(ũ), ũ) = ũ31 + ũ1 + 5ũ52 + 2ũ2, satisfies LES conditions (both c.2 and c′.2) for any of equilibrium. For a reference with the form r = a3 + a + 5b 5 + 2b (which will shift the input−output equilibrium), the equilibrium translated form of g1 (h(ũ), 3 ũ) can be written as g1̅ (h(u), ̅ u)̅ = (u̅1 + a) + (u1̅ + a) + 5 5(u̅2 + b) + 2(u̅2 + b) − r. As c′.2 is a sufficient condition of c.2, we only show that c.2 is valid. This can be confirmed by
∂g1̅ (h(u ̅ ), u ̅ ) ∂u1̅
= 3(u1̅ + a)2 + 1 > 0, and
∂g2̅ (h(u ̅ ), u ̅ ) = 25(u 2̅ + b)4 + 2 > 0 ∂u 2̅
It should be emphasized that although the function g2 (h(ũ), ũ) = ũ31 + 5ũ52 does not satisfy LES conditions ( c.2 and c′.2) around the equilibriums which are in the planes of ũ1 = 0 and ũ2 = 0 (inflection points), it satisfies these conditions around any other equilibriums. To see this, we select a reference with the form r = a3 + 5b5 (where a ≠ 0 and b ≠ 0) to shift the input−output equilibrium; the equilibrium translated form of g2 (h(ũ), ũ) can be written 3 5 as g2̅ (h(u), ̅ u)̅ = (u̅1 + a) + 5(u̅2 + b) − r. As c′.2 is a sufficient condition of c.2, we only show that c′.2 is valid. This can be confirmed by (considering that a ≠ 0, b ≠ 0, and u̅ is near the neighborhood of input−output equilibrium zero), ∂g1̅ (h(u ̅ ), u ̅ ) ∂u1̅
= 3(u1̅ + a)2 ≈ 3a 2 > 0, and
∂g2̅ (h(u ̅ ), u ̅ ) = 25(u 2̅ + b)4 ≈ 25b4 > 0 ∂u 2̅
3. As condition c′ is a sufficient condition of c, certain steadystate functions satisfy condition c but may not satisfy condition c′. See the following examples for details. g3(h(u), ̃ u)̃ = u1̃ 3 + and
15 2 u1̃ + u1̃ + u 2̃ 3 + 3u 2̃ 2
g4 (h(u), ̃ u)̃ = u1̃ 3 + u1̃ 2u 2̃ 3 + u1̃ + u 2̃ 3 + 3u 2̃
Considering that ũ = [k1α k2α ]T, we have ⎡ ⎛ α ·g3(h(u), ̃ u)̃ = α 2⎢k1⎜k1α + ⎢⎣ ⎝
15 ⎞ 1 k1 ⎟ + 2 ⎠ 16 ⎤ ⎛ ⎞ 1 + (3k 2 + k 23α 2)⎥ > ⎜ k1 + 3k 2⎟α 2 , ⎠ ⎥⎦ ⎝ 16
∂g3(h(u), ̃ u)̃ ∂u1̃
= 3u1̃ 2 +
2
15 u1̃ + 1
⎛ = ⎜ 3 u1̃ + ⎝
2 5⎞ 1 ⎟ − , 4⎠ 4
α ·g4 (h(u), ̃ u)̃ = α 2(k1 + 3k 2 + k13α 2 + k12k 23α 4 + k13α 2) > (k1 + 3k 2)α 2 , and ∂g4 (h(u), ̃ u)̃ ∂u1̃
= 3u1̃ 2 + 2u1̃ u 2̃ 3 + 1
That means both g3(h(ũ), ũ) and g4(h(ũ), ũ) satisfy condition c but not c′ (c′.2 actually). 4. Although some functions cannot satisfy condition c or c′ at all equilibriums, it can satisfy these conditions at certain equilibriums. For example, g5 (h(ũ), ũ) = ũ21 + ũ22 cannot meet condition c and c′ around zero. However, for a new equilibrium [ũ10, ũ20]T = [k1 α0, k2 α0]T (0 < c0 < α0 < ∞) and u ̃ ∈ 0. We ̇ have V(ξ) = −ξg(h(ũ), ũ) < 0 for ξ ≠ 0. Furthermore, as ∂g ∂g ∂g k1 ∂u ̃ + k 2 ∂u ̃ + ··· + km ∂u ̃ > ρ > 0 in a neighborhood 1
x4̇ = −
A2
m
2
∂g k 2 ∂u ̃ 2
(12c)
γ2
(11)
1
1 − γ2 a3 2g ( x3 + x3e) + ( u 2 + u 2e) A3 A3
As xe is the equilibrium corresponding to ue, the corresponding items of the right-hand side of the above eq (12) should be zeros when x̅ = u̅ = 0, that is, γ1 a a u1e = 1 2gx1e − 3 2gx3e A1 A1 A1 (13a)
It is easy to check that both V(ξ) and V̇ (ξ) will satisfy the requirements for GAS and LES given that condition c.2 is satisfied. Now, we prove that condition c′ is sufficient to ensure condition c. As the steady state input−output function g(h(ũ), ũ) is a function of ũ and ũ = [k1ξ k2ξ ··· kmξ]T, the derivative of g ( h ( ũ ) , ũ ) w i t h ξ c a n b e w r i t t e n a s ∂g ∂g ∂g ∂ (g (h(u), ̃ u)) ̃ = k1 ∂u ̃ + k 2 ∂u ̃ + ··· + km ∂u ̃ . Consider ∂ξ ∂g that g(0,0) = 0, and k1 ∂u ̃ 1
x3̇ = −
x 2̇ = −
a1 2g 2A1 h
x1̅ 2 +
2g 2 a3 2g a x1̅ x3̅ − m 3 x3 A1 2 l A3 2 h ̅
2g 2g 2 a a a 2 2g 2 x2 + 4 x 2x4 − m 4 x4 A2 2 h ̅ A2 2 l ̅ ̅ A4 2 h ̅ (16)
It is easy to see that V̇ (x)̅ ≤ 0 if m is big enough, and V̇ (x)̅ = 0 if and only if x̅ = 0. Thus, model (12) is GAS around the equilibrium xe in the region of interest.
(12b) J
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b02165. Detail results in figures and tables for close-loop characteristics of the embedded temperature control system, rising set-points at 40−80 deg, falling setpoints at 30−70 deg; heater controller rising and falling steps; fan controller rising and falling steps, and both controllers rising and falling steps (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +61 2 9514 7603. Fax: +61 2 9514 2435. ORCID
Yi Zhang: 0000-0001-5263-5823 Steven Su: 0000-0002-5720-8852 Branko Celler: 0000-0003-3790-2895 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Fundamental Research Funds for the Central Universities, China (Grant No. ZYGX2015J118), National Natural Science Foundation of China (Grant No. 51675087, Grant No. 61522105), and China Postdoctoral S c i en c e F o u n d a t i o n f u n d e d p r o je c t ( G r a n t N o . 2017M612950). The authors are thankful for the support from the Faculty of Aeronautics and Astronautics, University of Electronic Science and Technology of China, the Centre for Health Technologies, University of Technology Sydney, the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, and Biomedical Systems Laboratory, University of New South Wales, Sydney.
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REFERENCES
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DOI: 10.1021/acs.iecr.7b02165 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research (37) Ma, M.; Chen, H.; Findeisen, R.; Allgower, F. Nonlinear receding horizon control of quadruple tank system and real time implementation. Int. J. Innovat. Comput., Inform., Control 2012, 8, 10B.
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DOI: 10.1021/acs.iecr.7b02165 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
