Optimal control of axial dispersion tubular reactors with recycle: Addressing state‐delay through transport PDEs
Open accessAbstract
The optimal control of an axial tubular reactor with a recycle stream is addressed as a key type of setting for distributed parameter systems in chemical engineering. The intrinsic time delay from the recycle process, thus far overlooked in relevant literature, is modelled as a transport partial differential equation (PDE), resulting in a system of coupled parabolic and hyperbolic PDEs. Utilizing Danckwerts boundary conditions, the reactor is boundary‐controlled with the control input at the inlet. A continuous‐time optimal linear quadratic regulator is developed to stabilize the infinite‐dimensional system, employing a late lumping approach in order to preserve the properties of the original infinite dimensional system in controller design. The full‐state feedback regulator is designed by solving the Operator Riccati Equation (ORE), leveraging the system's Riesz‐spectral properties. To address practical limitations of full‐state feedback, a Luenberger observer is also proposed, enabling state reconstruction from boundary measurements. Numerical simulations are conducted to evaluate the proposed control strategies. The results demonstrate that the full‐state feedback regulator effectively stabilizes the system. A comparison is made between two configurations where different numbers of eigenmodes were selected to design the controller. The observer‐based regulator also stabilizes the system successfully using merely output measurements, effectively overcoming the challenge of limited state access.
AI snapshot
Written from the paper’s full text.
TL;DR
The paper develops an optimal control framework for an axial dispersion tubular reactor with recycle, modeling the recycle-induced state delay as a transport PDE coupled to the reactor’s parabolic PDE. It designs full-state and observer-based LQ regulators using a late-lumping, Riesz-spectral approach and demonstrates stabilization in numerical simulations.
Key findings
- The recycle delay is represented as a first-order hyperbolic transport PDE, yielding a coupled parabolic–hyperbolic PDE system with Danckwerts boundary conditions; the non-self-adjoint generator is Riesz-spectral, allowing a bi-orthogonal eigenbasis for controller design.
- A full-state LQ regulator, obtained by solving the operator Riccati equation and truncating to matrix Riccati equations, stabilizes the unstable linearized system; using \(N=7\) eigenmodes performs better than \(N=3\), stabilizing quicker with lower cost function values.
- A Luenberger observer-based output feedback regulator stabilizes the system using only outlet concentration measurements, though it is slightly more sluggish than full-state feedback.
- Sensitivity analysis shows stabilization when the actual delay is less than the assumed \(\tau=80\) s, but significant deviation as actual delay increases beyond a threshold near the recycle delay; recycle ratio near unity makes behavior well-mixed and dilutes control action, while lower Peclet number dampens oscillatory behavior.
Methods
- First-principles modeling of an axial dispersion tubular reactor with recycle, linearized around steady state; the delay is converted into a transport PDE, giving a boundary-controlled infinite-dimensional LTI system with control at the inlet and output at the outlet.
- Late-lumping LQR design: the operator Riccati equation is solved using the system’s Riesz-spectral properties and bi-orthogonal eigenfunctions, then truncated to \(N\) dominant modes to compute the feedback gain; an observer is designed via pole placement.
- Numerical evaluation uses finite-difference spatial discretization with 100 equidistributed points and RK45 time integration; parameters include \(D=2\times10^{-5}\,\mathrm{m^2/s}\), \(v=0.01\,\mathrm{m/s}\), \(k_r=-1.5\,\mathrm{s^{-1}}\), \(\tau=80\,\mathrm{s}\), \(R=0.3\), with \(Q=0.05\) and \(\Re=50\), comparing \(N=3\) and \(N=7\) controllers.
Limitations
- Stated: observer gain is constrained by output noise and plant–model mismatches; the sensitivity analysis is deliberately limited in scope; future work is needed for non-isothermal reactors, MPC, measurement noise, plant–model mismatches, and eigenmode selection.
- Evident: the study is purely numerical/simulation-based with no experimental validation; the model is linearized around steady state and assumes constant physical properties; no datasets were generated or analyzed.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
1 of 11 cited sources checked · 0 supported · 0 flagged · accuracy not yet measured
- Premisecitation
Luenberger-based observers are well-suited for infinite-dimensional systems when the system parameters are perfectly known.
“Luenberger-based observers are well-suited for infinite-dimensional systems when the system parameters are perfectly known.[27]”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites [27] source
Section 3.3 says Luenberger-based observers are appropriate when system parameters are well known, and the conclusions specify perfect knowledge of those parameters. But Section 2 discusses extension to infinite-dimensional systems for asymptotic, exponential, and interval observers, not Luenberger-based observers; it describes the latter as suitable for less complex linear systems. Page numbers are not available in our copy. (LLM, conf=0.87)
Experimental — accuracy not yet measured
- Findingown data
The full-state feedback regulator effectively stabilizes the system.
“The results demonstrate that the full-state feedback regulator effectively stabilizes the system.”
Quote in the paperSubstantial claim - Findingreasoning
The system results in a non-self-adjoint operator.
“The system results in a non-self-adjoint operator.”
- Also stated in: § 2.3 | Adjoint operator · plain
“Given that A is not self-adjoint (i.e., A ≠ A Ã ), their combined eigenmodes may still form a bi-orthonormal basis, typical of a Riesz-spectral operator.[16]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [16] source
- Also stated in: § 2.3 | Adjoint operator · plain
- Findingreasoning
For the linearized system to have an unstable steady state, the reaction coefficient k_r must be negative.
“It has been observed that for the linearized system to have an unstable steady state, the reaction coefficient, k r , must be negative.”
Quote in the paperNot a substantial claim- Rests on an assumption: A negative reaction coefficient can arise in certain reaction mechanisms such as autocatalytic reactions, enzyme-catalyzed reactions, or reactions involving inhibitory effects.
- Findingnone given
As long as the actual time delay of a system is less than the assumed delay used in the controller design, the controller is still able to stabilize the system within a finite time horizon, although transient deviations from the desired behaviour are observed.
“It can be seen that as long as the actual time delay of a system is less than the assumed delay used in the controller design, the controller is still able to stabilize the system within a finite time horizon, although transient deviations from the desired behaviour are observed.”
Quote in the paperSubstantial claim
- Findingreasoning
The open-loop linearized system is inherently unstable.
“Additionally, the parameters are deliberately selected to introduce instability into the system, emphasizing the proposed control strategy's ability to stabilize an inherently unstable system.”
Quote in the paperSubstantial claim- Rests on an assumption: A negative reaction coefficient can arise in certain reaction mechanisms such as autocatalytic reactions, enzyme-catalyzed reactions, or reactions involving inhibitory effects.
- Findingnone given
The proposed modelling and control strategy effectively stabilizes the system over a broad range of parameter sets, including variations in the imposed time delay, Peclet number, and recycle ratio.
“the proposed modelling and control strategy is able to effectively stabilize the system over a broad range of parameter sets, including but not limited to variations in the imposed time delay, Peclet number, and recycle ratio, ensuring the practicality of the proposed framework across different system configurations.”
Quote in the paperSubstantial claim - Findingown data
Both optimal feedback gains (computed with N = 3 and N = 7 eigenmodes) successfully stabilize the system within a finite time horizon.
“Both optimal feedback gains are able to successfully stabilize the system within finite time horizon.”
Quote in the paperSubstantial claim - Findingreasoning
A decrease in the mass transfer Peclet number causes the eigenvalues of the system generator to shift closer to the real axis of the complex plane.
“a decrease in the Peclet number causes the eigenvalues of the system generator to shift closer to the real axis of the complex plane.”
Quote in the paperNot a substantial claim - Findingnone given
As the recycle ratio approaches unity, the open-loop system exhibits behaviour similar to that of a well-mixed reactor, with concentration profiles flattening.
“as the recycle ratio approaches unity, the open-loop system exhibits behaviour similar to that of a well-mixed reactor, with concentration profiles flattening.”
Quote in the paperNot a substantial claim - Findingreasoning
As the recycle ratio approaches unity, it becomes more challenging to affect the system dynamics with the control input because the controller's action becomes diluted at the reactor inlet due to mixing with the recycle stream.
“This also influences the controller performance as it becomes more challenging to affect the system dynamics with the control input as the controller's action becomes diluted at the reactor inlet due to mixing with the recycle stream.”
Quote in the paperNot a substantial claim - Findingown data
As the actual delay increases, the response of the system starts to deviate significantly from the desired behaviour, especially after a threshold close to the actual recycle delay of the system.
“However, as the actual delay increases, the response of the system starts to deviate significantly from the desired behaviour, especially after a certain threshold close to the actual recycle delay of the system.”
Quote in the paperSubstantial claim - Findingreasoning
The proposed observer design mechanism achieves system stabilization with reasonable performance.
“Despite these challenges, the proposed observer design mechanism achieves system stabilization with reasonable performance.”
Quote in the paperSubstantial claim- Rests on an assumption: In practice, the observer gain is constrained by factors such as noise in the system output and plant-model mismatches.
- Interpretationreasoning
The full-state feedback regulator is designed by solving the Operator Riccati Equation (ORE), leveraging the system's Riesz-spectral properties.
“The full-state feedback regulator is designed by solving the Operator Riccati Equation (ORE), leveraging the system's Riesz-spectral properties.”
Quote in the paperSubstantial claim - Interpretationnone given
A Luenberger observer is proposed, enabling state reconstruction from boundary measurements.
“To address practical limitations of full-state feedback, a Luenberger observer is also proposed, enabling state reconstruction from boundary measurements.”
Quote in the paperSubstantial claim - Interpretationreasoning
The control feedback is derived by solving an ORE in order to implement a late lumping approach.
“The control feedback is derived by solving an ORE in order to implement a late lumping approach.”
Quote in the paperSubstantial claim - Interpretationreasoning
A continuous-time optimal linear quadratic regulator is developed to stabilize the infinite-dimensional system, employing a late lumping approach to preserve the properties of the original infinite-dimensional system in controller design.
“A continuous-time optimal linear quadratic regulator is developed to stabilize the infinite-dimensional system, employing a late lumping approach in order to preserve the properties of the original infinite dimensional system in controller design.”
Quote in the paperSubstantial claim - Premisecitation
The diffusion term was not addressed in Qi et al., leaving a gap in the literature regarding diffusion-convection-reaction systems with a recycle stream imposing state delay.
“However, the diffusion term was not addressed, leaving a gap in the literature regarding diffusion-convection-reaction systems with a recycle stream imposing state delay.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [23] source
- Premisecitation
The dynamics of the reactant concentration can be described by the PDE given in Equation (1), belonging to the class of second order parabolic PDEs commonly used to characterize diffusion-convection-reaction systems.
“By applying first-principle modelling through relevant mass balance relations on an infinitesimally small section of the reactor, the dynamics of the reactant concentration can be described by the PDE given in Equation ( 1), belonging to the class of second order parabolic PDEs commonly used to characterize diffusion-convection-reaction systems [25] in chemical engineering.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [25] source
- The authors state this as an assumption: The system properties are constant against changes in temperature and pressure.
- Premisecitation
In Khatibi et al.'s work, the recycle is assumed to be instantaneous.
“Even in Khatibi et al.'s work, the recycle is assumed to be instantaneous [14]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] source
- Premisereasoning
Because the operator P is infinite-dimensional, a numerical solution of the operator Riccati equation is impractical.
“Due to the infinite-dimensional nature of P, a numerical solution is impractical.”
Quote in the paperSubstantial claim - Interpretationreasoning
A negative reaction coefficient causes a decline in the reaction rate as the reactant accumulates, leading to further reactant accumulation and thus driving the system away from its steady state.
“This instability can be qualitatively understood as follows: a negative reaction coefficient causes a decline in the reaction rate as the reactant accumulates, leading to further reactant accumulation and thus, driving the system away from its steady state.”
Quote in the paperSubstantial claim- Rests on an assumption: A negative reaction coefficient can arise in certain reaction mechanisms such as autocatalytic reactions, enzyme-catalyzed reactions, or reactions involving inhibitory effects.
- Premisereasoning
The presence of at least one eigenvalue with a positive real part indicates that the open-loop linearized system is exponentially unstable.
“Quantitative confirmation of this behaviour can be achieved through eigenvalue analysis, where the presence of at least one eigenvalue with a positive real part indicates the fact that the open-loop linearized system is exponentially unstable.”
Quote in the paperNot a substantial claim - Interpretationreasoning
The open-loop system can be described in the standard state-space form of an infinite-dimensional linear time-invariant (LTI) system as _x = Ax.
“With all state variables now expressed explicitly at a specific time instance t-in contrast to the previous representation where states at t were directly involved with states at t À τ ð Þ-the open-loop system can be described in the standard state-space form of an infinitedimensional linear time-invariant (LTI) system as _ x ¼ Ax.”
Quote in the paperSubstantial claim - Interpretationreasoning
Including the time delay in the controller design is important to ensure the stability of the system within an optimal framework.
“This shows the importance of including the time delay in the controller design to ensure the stability of the system within an optimal framework.”
Quote in the paperSubstantial claim - Premisenone given
The LQR problem is solved by finding the unique positive semi-definite operator Π, which satisfies the ORE presented in Equation (14).
“The LQR problem is solved by finding the unique positive semi-definite operator Π, which satisfies the ORE presented in Equation ( 14).”
Quote in the paperSubstantial claim - Interpretationreasoning
Greater diffusion relative to convection dampens the oscillatory behaviour of the system that is originally imposed as a result of the delayed recycle stream.
“This implies that greater diffusion relative to convection dampens the oscillatory behaviour of the system that is originally imposed as the result of the delayed recycle stream.”
Quote in the paperNot a substantial claim
- Classified as probably not a substantial claim. 11
- May combine two claims. 11
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Findingown dataAttempts to analytically solve the characteristic equation det Λ = 0 have failed; therefore, it is solved numerically using the parameters in Table 1.
“Attempts to analytically solve the characteristic equation det Λ À Á ¼ 0 have failed; therefore, it is solved numerically using the parameters in Table 1.”
Quote in the paperNot a substantial claimMay combine two claims.
FindingreasoningThe eigenvalues of A match the ones of its adjoint, confirming that A and A* form a bi-orthogonal basis according to Equation (12).
“Following the same procedure for A Ã shows that the eigenvalues of A match the ones of its adjoint, confirming that A and A Ã form a bi-orthogonal basis according to Equation (12):”
Quote in the paperNot a substantial claimMay combine two claims.
FindingreasoningThe observer-based controller successfully stabilizes the system within a finite time horizon using only output measurements instead of full state information, although its performance is slightly more sluggish than the full-state feedback regulator.
“While the performance of the observer-based controller is slightly more sluggish compared to that of the full-state feedback regulator, it successfully stabilizes the system within a finite time horizon using only output measurements instead of full state information.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningThe eigenvalues of both control systems are identical after the seventh mode, because the real parts of these eigenvalues are already sufficiently negative.
“Note that the eigenvalues of both control systems are identical after the seventh mode, as the real parts of these eigenvalues are already sufficiently negative.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown dataThe controller designed with more eigenmodes (N = 7) performs better than the N = 3 controller, stabilizing the system quicker with lower cost function values in general.
“However, the case where more eigenmodes are considered in the controller design shows better performance, as the higher dimensional controller is able to stabilize the system quicker with lower cost function values in general.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
PremisecitationMismatches between the dynamical properties of the original DPS and the approximate lumped parameter model can occur, negatively affecting the performance of the designed regulator.
“While this method enables the use of standard regulator design techniques, mismatches between the dynamical properties of the original DPS and the approximate lumped parameter model can occur, negatively affecting the performance of the designed regulator.[4]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] source
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationEarly lumping approximates the infinite-dimensional system with a finite-dimensional model.
“One is early lumping, which approximates the infinitedimensional system with a finite-dimensional model.[2,3]”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
Interpretationnone givenThe intrinsic time delay from the recycle process, overlooked in relevant literature, is modelled as a transport PDE, resulting in a system of coupled parabolic and hyperbolic PDEs.
“The intrinsic time delay from the recycle process, thus far overlooked in relevant literature, is modelled as a transport partial differential equation (PDE), resulting in a system of coupled parabolic and hyperbolic PDEs.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningBy utilizing the bi-orthogonal theorem, given that the generator is Riesz-spectral, a full-state feedback optimal LQ regulator is developed, followed by an output feedback regulator.
“However, by utilizing the bi-orthogonal theorem, given that the generator is Riesz-spectral, a full-state feedback optimal LQ regulator is developed, followed by an output feedback regulator.”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationQi et al. addressed the challenge of state delay imposed by a recycle stream in a system modelled by interconnected first-order hyperbolic partial integro differential equations (PIDEs), introducing a transport PDE to account for the in-domain recycle delay.
“In another attempt, Qi et al. addressed the challenge of state delay imposed by a recycle stream in a system modelled by interconnected first-order hyperbolic partial integro differential equations (PIDEs), introducing a transport PDE to account for the in-domain recycle delay.[23]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [23] source
Probably not a substantial claim: may not pass S2 (checkable).
Interpretationnone givenThe assumption of instantaneous recycle does not resonate well with reality.
“a simplifying assumption that does not resonate well with reality.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationreasoningThe model comprises a second-order parabolic PDE to capture the diffusion-convection-reaction nature of the reactor, coupled with a first-order hyperbolic PDE to account for the delay.
“The model comprises a second-order parabolic PDE to capture the diffusion-convection-reaction nature of the reactor, coupled with a first-order hyperbolic PDE to account for the delay.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The system properties are constant against changes in temperature and pressure.
- The authors state this as an assumption: The system output is the deviation of the reactant concentration from the steady-state measured at the reactor outlet.
- The authors state this as an assumption: Transport lags in connecting lines are negligible.
Classified as probably not a substantial claim. May combine two claims.
PremisereasoningThe solution to the ORE is unique for any set of functions in the domain of operator A, so one can arbitrarily set x = φ_m and y = φ_n, the eigenfunctions of A.
“Given that the solution to the ORE is unique for any set of functions in the domain of operator A, we can arbitrarily set x ¼ ϕ m and y ¼ ϕ n , that is, the eigenfunctions of A.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone givenDanckwerts boundary conditions are particularly suited for the class of axial tubular reactors.
“Particularly suited for the class of axial tubular reactors, Danckwerts boundary conditions are considered.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
InterpretationreasoningThe delay is modelled as a pure transport process, as if the first state is being transported from the reactor outlet to the inlet, experiencing a delay of τ time units while in the recycle stream.
“The delay is thus modelled as a pure transport process, as if the first state x 1 ζ, t ð Þ is being transported from the reactor outlet to the inlet, experiencing a delay of τ time units while in the recycle stream.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationPlacing the poles of an observer-based controller to the left of those of the full-state feedback controller is common practice in the design of observer-based controllers for infinite-dimensional systems.
“This practice is common in the design of observer-based controllers for infinite-dimensional systems.[15]”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [15] source
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationAccounting for deviations from perfect mixing and piston flow and assuming negligible transport lags in connecting lines, the Danckwerts boundary conditions have become an inseparable part of modelling axial tubular reactors in the field of chemical engineering process control and dynamics.
“Accounting for deviations from perfect mixing and piston flow and assuming negligible transport lags in connecting lines, [26] the Danckwerts boundary conditions have become an inseparable part of modelling axial tubular reactors in the field of chemical engineering process control and dynamics.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [26] source
- The authors state this as an assumption: Transport lags in connecting lines are negligible.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Interpretationnone givenIn the absence of uncertainty in the system model, the observer gain can theoretically be designed so that the state estimation error converges to zero very fast compared to the full-state feedback regulator dynamics.
“In the absence of uncertainty in the system model, the observer gain can theoretically be designed so that the state estimation error converges to zero very fast compared to full-state feedback regulator dynamics.”
Quote in the paperNot a substantial claimMay combine two claims.
Study design and conductreasoningThe control law is derived directly in the infinite-dimensional space (a late-lumping approach), so the infinite-dimensional characteristics of the system are preserved in the control design.
“The control law is derived directly in the infinite-dimensional space, fully capturing the continuous nature of the original system.”
- Also stated in: § 5 | CONCLUSION · plain
“a late lumping approach is employed, ensuring that the infinite-dimensional characteristics of the system are preserved in the control design.”
Quote in the paperSubstantial claim- Also stated in: § 5 | CONCLUSION · plain
Probably not a substantial claim: may not pass S4 (based).
Interpretationnone givenThe optimal full-state feedback gain calculated from the truncated matrix Riccati equation via Equation (20) ensures closed-loop stability.
“The optimal full-state feedback gain is then calculated using Equation ( 20), ensuring closed-loop stability.”
Quote in the paperNot a substantial claim
The authors state this as an assumption.
The system properties are constant against changes in temperature and pressure.
“It is worth noting that the system properties are assumed to be constant against changes in temperature and pressure.”
The authors state this as an assumption.
The system output is the deviation of the reactant concentration from the steady-state measured at the reactor outlet.
“The system output is assumed to be the deviation of the reactant concentration from the steadystate measured at the reactor outlet”
The authors state this as an assumption.
Transport lags in connecting lines are negligible.
“assuming negligible transport lags in connecting lines, [26]”
The paper relies on this without support.
The presence of recycle is common in industrial reactor designs.
“Although the presence of recycle is common in industrial reactor designs”
The paper relies on this without support.
A negative reaction coefficient can arise in certain reaction mechanisms such as autocatalytic reactions, enzyme-catalyzed reactions, or reactions involving inhibitory effects.
“Although rare, this scenario can arise in certain reaction mechanisms where the reaction rate decreases as the reactant concentration increases, such as autocatalytic reactions, enzyme-catalyzed reactions, or reactions involving inhibitory effects.”
The paper relies on this without support.
The linearized model adequately represents the reactor dynamics near the steady state.
“Considering the reaction term in general can be nonlinear, the model is further linearized around its steadystate”
The authors state this as an assumption.
The infinite-dimensional system is boundary controlled, with the control input acting on the reactor inlet.
“The infinite-dimensional system is assumed to be boundary controlled, with the control input acting on the reactor inlet.”
The paper relies on this without support.
The time delay introduced by the recycle process has not been adequately addressed in the literature, despite being a common and intrinsic feature of such systems.
“The notion of time delay introduced by the recycle process has not been adequately addressed in the literature despite being a common and intrinsic feature of such systems; introducing a rare example of state-delay in this field.”
The paper relies on this without support.
Full access to the system's states is not feasible in realistic applications.
“However, this assumption is not feasible in realistic applications.”
The paper relies on this without support.
In practice, the observer gain is constrained by factors such as noise in the system output and plant-model mismatches.
“In practice, however, the observer gain is constrained by factors such as noise in the system output and plantmodel mismatches.”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Reaction term changes sign in the PDE reformulation
MajorEquation (2) uses Àk r c for the reaction term, but Equation (4), presented as a reformulation of Equation (2), uses þ k r for the same state. The two equations therefore specify different reactor dynamics.
“_ c ζ, t ð Þ¼D∂ ζζ c ζ, t ð ÞÀv∂ ζ c ζ,t ð ÞÀk r c ζ,t ð Þð2Þ”
“∂ t x 1 ζ, t ð Þ x 2 ζ, t ð Þ ! ¼ D∂ ζζ À v∂ ζ þ k r 0 0 1 τ ∂ ζ 2 4 3 5 x 1 ζ, t ð Þ x 2 ζ, t ð Þ !ð4Þ”
Cites
SciPy 1.0: fundamental algorithms for scientific computing in Python
Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., et al. (2020) · Nature Methods · doi:10.1038/s41592-019-0686-2 · Cited by 38,430 · see citing works
Levenspiel, O. (1999) · Industrial & Engineering Chemistry Research · doi:10.1021/ie990488g · Cited by 10,011 · see citing works
A family of embedded Runge-Kutta formulae
Dormand, J. R., & Prince, P. J. (1980) · Journal of Computational and Applied Mathematics · doi:10.1016/0771-050x(80)90013-3 · Cited by 3,634 · see citing works
Continuous flow systems. Distribution of residence times
Danckwerts, P. V. (1995) · Chemical Engineering Science · doi:10.1016/0009-2509(96)81811-2 · Cited by 1,274 · see citing works
The robust control of a servomechanism problem for linear time-invariant multivariable systems
Davison, E. (1976) · IEEE Transactions on Automatic Control · doi:10.1109/tac.1976.1101137 · Cited by 1,271 · see citing works
The Linear Multivariable Regulator Problem
Francis, B. A. (1977) · SIAM Journal on Control and Optimization · doi:10.1137/0315033 · Cited by 1,092 · see citing works
Delay Compensation for Nonlinear, Adaptive, and PDE Systems
Krstić, M. (2009) · Systems & control · doi:10.1007/978-0-8176-4877-0 · Cited by 1,064 · see citing works
Krstić, M., & Smyshlyaev, A. (2008) · Systems & Control Letters · doi:10.1016/j.sysconle.2008.02.005 · Cited by 796 · see citing works
Review and classification of recent observers applied in chemical process systems
Mohd Ali, J., Ha Hoang, N., Hussain, M. A., & Dochain, D. (2015) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2015.01.019 · Cited by 239 · see citing works
Some practical Runge-Kutta formulas
Shampine, L. F. (1986) · Mathematics of Computation · doi:10.1090/s0025-5718-1986-0815836-3 · Cited by 198 · see citing works
Robust control of parabolic PDE systems
Christofides, P. D. (1998) · Chemical Engineering Science · doi:10.1016/s0009-2509(98)00091-8 · Cited by 187 · see citing works
The bifurcation behavior of tubular reactors
Jensen, K. F., & Ray, W. H. (1982) · Chemical Engineering Science · doi:10.1016/0009-2509(82)80155-3 · Cited by 180 · see citing works
Introduction to Infinite-Dimensional Systems Theory: A State-Space Approach
Curtain, R., & Zwart, H. (2020) · Texts in Applied Mathematics · doi:10.1007/978-1-0716-0590-5 · Cited by 150 · see citing works
Rijnsdorp, J. E. (1982) · Automatica · doi:10.1016/0005-1098(82)90084-x · Cited by 143 · see citing works
Predictive control of parabolic PDEs with state and control constraints
Dubljevic, S. S., El‐Farra, N. H., Mhaskar, P., & Christofides, P. D. (2006) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.1097 · Cited by 135 · see citing works
Self-citationLQ control design of a class of hyperbolic PDE systems: Application to fixed-bed reactor
Aksikas, I., Fuxman, A. M., Forbes, J. F., et al. (2009) · Automatica · doi:10.1016/j.automatica.2009.02.017 · Cited by 127 · see citing works
Controller Design for Distributed Parameter Systems
Morris, K. A. (2020) · Communications and Control Engineering · doi:10.1007/978-3-030-34949-3 · Cited by 43 · see citing works
LQ-boundary control of a diffusion-convection-reaction system
Mohammadi, L., Aksikas, I., Dubljevic, S. S., et al. (2011) · International Journal of Control · doi:10.1080/00207179.2011.642308 · Cited by 31 · see citing works
Self-citationXu, X., & Dubljevic, S. S. (2016) · European Journal of Control · doi:10.1016/j.ejcon.2016.02.002 · Cited by 31 · see citing works
Self-citationOptimal control of an advection-dominated catalytic fixed-bed reactor with catalyst deactivation
Aksikas, I., Mohammadi, L., Forbes, J. F., et al. (2013) · Journal of Process Control · doi:10.1016/j.jprocont.2013.09.016 · Cited by 21 · see citing works
Self-citationInfinite-dimensional LQ optimal control of a dimethyl ether (DME) catalytic distillation column
Alizadeh Moghadam, A., Aksikas, I., Dubljevic, S., & Forbes, J. F. (2012) · Journal of Process Control · doi:10.1016/j.jprocont.2012.06.018 · Cited by 20 · see citing works
Self-citationModel predictive control of a non-isothermal axial dispersion tubular reactor with recycle
Khatibi, S., Ozorio Cassol, G., & Dubljevic, S. (2021) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2020.107159 · Cited by 18 · see citing works
Self-citationHeat exchanger system boundary regulation
Cassol, G. O., Ni, D., & Dubljevic, S. S. (2019) · AIChE Journal · doi:10.1002/aic.16623 · Cited by 17 · see citing works
Self-citationQi, J., Dubljevic, S., & Kong, W. (2021) · Automatica · doi:10.1016/j.automatica.2021.109565 · Cited by 17 · see citing works
Self-citationDiscrete-time modeling and output regulation of gas pipeline networks
Xie, J., & Dubljevic, S. S. (2020) · Journal of Process Control · doi:10.1016/j.jprocont.2020.12.002 · Cited by 15 · see citing works
Self-citationZhang, L., Xie, J., & Dubljevic, S. S. (2023) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2023.108381 · Cited by 10 · see citing works
Self-citationDynamic modeling and model predictive control of a continuous pulp digester
Zhang, L., Xie, J., & Dubljevic, S. S. (2021) · AIChE Journal · doi:10.1002/aic.17534 · Cited by 9 · see citing works
Self-citationDiscrete Output Regulator Design for a Mono-tubular Reactor with Recycle
Cassol, G. O., & Dubljevic, S. S. (2019) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.23919/acc.2019.8815250 · Cited by 6 · see citing works
Self-citationSpectral-based optimal output-feedback boundary control of a cracking catalytic reactor PDE model
Aksikas, I. (2024) · International Journal of Control · doi:10.1080/00207179.2024.2302057 · Cited by 5 · see citing works
Morris, K. (2020) · Communications and control engineering/Communications and control engineering series · doi:10.1007/978-3-030-34949-3_6 · Cited by 1 · see citing works
Flagged first, then by citations
1 of the 31 references couldn’t be matched to a known paper.
Research works printed but not linked: 1
- Christofides P. D., 2012, Nonlinear and Robust Control of PDE Systems; Systems & Control: Foundations & Applications
Cited by
Sorted by citations
(2025) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.25333
Moadeli, B., & Dubljevic, S. S. (2025) · European Control Conference (ECC) · doi:10.23919/ecc65951.2025.11186872
“The full modeling rationale—including boundary condition selection, linearization procedure, and physical assumptions—follows our prior work [24], to which the reader is referred for further detail.”
Self-citationCelebrating open access articles at CJCE : The 20 top‐read articles of 2025
Soares, J. B. P. (2025) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.70157
Moadeli, B., & Dubljevic, S. S. (2025) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.70189
Self-citation
- Authors
- Moadeli, B., Cassol, G. O., & Dubljevic, S.
- Affiliations
- University of Alberta (CA)
- Corresponding author
- Moadeli, Behrad
- Venue
- The Canadian Journal of Chemical Engineering
- Publisher
- Wiley
- Published
- 2025-02-06
- Volume
- 103
- Issue
- 8
- Pages
- 3751-3766
- ISSN
- 0008-4034 (ISSN-L)
- 1939-019X
- OpenAlex
- W4407202974
- Cited by
- 4
- References
- 31
- Citations by year
- 2025: 4
- FWCI
- 1.57
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- The Canadian Journal of Chemical Engineering (publishedVersion, cc-by-nc-nd)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Stability and Controllability of Differential Equations
- Physical Sciences › Engineering › Control and Systems Engineering › Advanced Control Systems Optimization
- Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
- Keywords
- optimal control, tubular reactor, recycle streams, axial dispersion, state delay, Danckwerts boundary conditions, boundary control, linear quadratic regulator, operator Riccati equation, Luenberger observer, full state feedback, output feedback, distributed parameter systems
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-11.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-11.
Cited sources are checked by gpt-6-sol.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
