Introduction
Production–regeneration systems describe processes that must alternate between a productive phase and a recovery phase. During production, the system generates an output (for instance filtered water), but this phase also progressively degrades an internal state (such as fouling or resistance). The regeneration phase does the opposite: it restores the system’s internal condition, but does not directly produce useful output.
Our goal is to show how the Control-Toolbox ecosystem, in particular the OptimalControl.jl package, can be used in the context of membrane filtration processes where fouling accumulates during filtration and must be periodically removed through regeneration (e.g., backwashing). This system is shown in the following figure.
A package dedicated to production-regeneration systems, specialized for such applications, is available and denoted Filtration.jl.
Description of membrane filtration system
The goal is to briefly describe the modeling of a membrane filtration process. The permeate flux $J~(\mathrm m. \mathrm s^{-1})$ corresponds to the permeate volume $v~(\mathrm m^3)$ flow per membrane area $A~(\mathrm m^2)$ and is related to the total resistance $R~(\mathrm m^{-1})$ of the membrane according to Darcy's law
\[ J = \frac{\dot v}{A} = \frac{\Delta p}{\mu R}\]
where $\Delta p~(\mathrm{Pa})$ corresponds to the trans-membrane pressure and $\mu~(\mathrm{Pa}.\mathrm s)$ is the permeate viscosity. The total resistance of the membrane corresponds to the sum of the intrinsic resistance $R_0~(\mathrm m^{-1})$ and the residual cake resistance $R_c~(\mathrm m^{-1})$ using the "resistances-in-series" concept
\[ R = R_0 + R_c. \]
The pump consumes energy $e~(\mathrm W.\mathrm h)$ at a rate proportional to the hydraulic power $P_h~(\mathrm W)$, which is the product of the permeate flow $J$ and the trans-membrane pressure $\Delta p$
\[ \dot e = \frac{J\Delta p}{\eta},\]
where $\eta$ is the efficiency parameter of the pump.
The dynamics of the resistance $R_c$ involve complex phenomena, but a simple and effective formulation is given by the following:
- In filtration phases (when $u = +1$), this resistance variation is proportional to the flux $J$ and the total suspended solids concentration $C~(\mathrm{Kg}.\mathrm m^{-3})$
\[ \dot R_c = J \beta C \quad \text{when} \quad u = +1,\]
where $\beta~(\mathrm m.\mathrm{Kg}^{-1})$ is a parameter that describes the resistance of the cake layer.
- In backwash phases (when $u = -1$), this resistance variation is proportional to the flux $J$ and the resistance cake layer $R_c$
\[ \dot R_c = J \omega R_c \quad \text{when} \quad u = -1,\]
where $\omega~(\mathrm m^{-1})$ models the detachment resistance of fouling.
Problem statement
Let us assume that the flux $J$ is constant during filtration ($J = J_f > 0$) and backwash ($J = -J_b < 0$) phases. The goal is to minimize the total power used to produce a targeted permeate volume $v_f$ at a free final time $t_f$. Using equations given previously, the dynamics of the cost and the state variables are given in filtration and backwash modes by
\[\mathrm{Filtration} : \left\{ \begin{array}{rl} \dot e & = \frac{J_f^2 \mu}{\eta} (R_0 + R_c), \\[0.5em] \dot R_c & = J_f \beta C, \\[0.5em] \dot v & = J_f A, \end{array} \right. \quad \text{and} \quad \mathrm{Backwash} : \left\{ \begin{array}{rl} \dot e & = \frac{J_b^2 \mu}{\eta} (R_0 + R_c), \\[0.5em] \dot R_c & = -J_b \omega R_c, \\[0.5em] \dot v & = - J_b A. \end{array} \right. \]
Let $u \in [-1, 1]$ be the control variable, where $u = +1$ denotes filtration mode and $u = -1$ denotes backwash mode. We consider the following optimal control problem:
\[\text{(OCP)} \quad \left\{ \begin {array}{ll} \displaystyle \min_{x,y,t_f} \int_{t_0}^{t_f} \frac{\mu(R_0 + R_c(t))}{2\eta}\big(J_f^2 + J_b^2 + u(t)(J_f^2 - J_b^2)\big) \, \mathrm dt, & \\[1em] \displaystyle \mathrm{s.t.} \ \dot R_c(t) = J_f\beta C - J_b \omega R_c(t) + u(t)\big(J_f \beta C + J_b \omega R_c(t)\big), & t \in [t_0, t_f] \ \mathrm{a.e.}, \\[1em] \displaystyle \phantom{\mathrm{s.t.} \ } \dot v(t) = A\big((J_f - J_b) + u(t)(J_f + J_b)\big), \, & t \in [t_0, t_f] \ \mathrm{a.e.}, \\[1em] \phantom{\mathrm{s.t.} \ } u(t) \in [-1, 1], & t \in [t_0, t_f], \\[1em] \phantom{\mathrm{s.t.} \ } R_c(t_0) = R_{c0}, \quad v(t_0) = v_0, \quad v(t_f) = v_f, \end{array} \right.\]
where $t_0 \in \mathbb R$, $R_{c0} > 0$, $v_0 > 0$, and $v_f > 0$ are given.
Main theoretical results
Based on the theoretical developments presented in Dutto et al., 2026, all possible optimal solution structures are characterized by the following result:
Under standard regularity assumptions, and denoting
- $\sigma_-$ a regeneration arc associated to $u = -1$,
- $\sigma_+$ a production arc associated to $u = +1$,
- $\sigma_s$ a singular arc associated to $u = u_s$,
the structure of an optimal solution can only be one of the following:
\[\sigma_+, \ \sigma_-\sigma_+, \ \sigma_s\sigma_+, \ \sigma_-\sigma_s\sigma_+ ~ \text{or} ~ \sigma_+\sigma_s\sigma_+.\]
Here, $u_s$ denotes a singular control such that $\dot x = 0$. See Dutto et al., 2026 for a detailed explanation and proof.
References
- Filtration.jl
- OptimalControl.jl
- Rémy Dutto, Jérôme Harmand, Alain Rapaport (2026). Optimal control synthesis for a class of production-regeneration systems -Application to membrane filtration.
- F. Aichouche, N. Kalboussi, A. Rapaport, J. Harmand (2020). Modeling and optimal control for production-regeneration systems - preliminary results -, 2020 European Control Conference (ECC)
- B. Benyahia, A. Charfi, N. Benamar, M. Heran, A. Grasmick, B. Cherki, J. Harmand (2013). A simple model of anaerobic membrane bioreactor for control design: coupling the “AM2b” model with a simple membrane fouling dynamics, World Congress on Anerobic Digestion: Recovering (bio) Ressources for the World
- F. Ellouze, Y. Kammoun, N. Kalboussi, A. Rapaport, J. Harmand, S. Nasr, N. Ben Amar (2023) Optimal control of backwash scheduling for pumping energy saving: Application to the treatment of urban wastewater, Journal of Water Process Engineering
- N. Kalboussi, A. Rapaport, T. Bayen, N. Ben Amar, F. Ellouze, J. Harmand (2017) Optimal control of a membrane filtration system, IFAC-PapersOnLine
- N. Kalboussi, J. Harmand, A. Rapaport, T. Bayen, F. Ellouze, N. Ben Amar (2018) Optimal control of physical backwash strategy - towards the enhancement of membrane filtration process performance, Journal of Membrane Science
Reproducibility
You can download the exact environment used to build this documentation:
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Julia Version 1.12.6
Commit 15346901f00 (2026-04-09 19:20 UTC)
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[89763e89] Libtiff_jll v4.7.2+0
[38a345b3] Libuuid_jll v2.42.0+0
[d00139f3] METIS_jll v5.1.3+0
[856f044c] MKL_jll v2025.2.0+0
[d7ed1dd3] MUMPS_seq_jll v500.800.200+0
[c8ffd9c3] MbedTLS_jll v2.28.1010+0
[e7412a2a] Ogg_jll v1.3.6+0
[656ef2d0] OpenBLAS32_jll v0.3.30+0
[9bd350c2] OpenSSH_jll v10.3.1+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.57.0+0
⌅ [30392449] Pixman_jll v0.44.2+0
[c0090381] Qt6Base_jll v6.10.2+1
[629bc702] Qt6Declarative_jll v6.10.2+1
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[319450e9] SPRAL_jll v2025.9.18+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
⌅ [02c8fc9c] XML2_jll v2.13.9+0
[ffd25f8a] XZ_jll v5.8.3+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[a51aa0fd] Xorg_libXi_jll v1.8.3+0
[d1454406] Xorg_libXinerama_jll v1.1.7+0
[ec84b674] Xorg_libXrandr_jll v1.5.6+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[a65dc6b1] Xorg_libpciaccess_jll v0.18.1+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.2.0+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[35661453] Xorg_xkbcomp_jll v1.4.7+0
[33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[3161d3a3] Zstd_jll v1.5.7+1
[b792d7bf] cminpack_jll v1.3.12+0
[35ca27e7] eudev_jll v3.2.14+0
[214eeab7] fzf_jll v0.61.1+0
[a4ae2306] libaom_jll v3.13.3+0
[0ac62f75] libass_jll v0.17.4+0
[1183f4f0] libdecor_jll v0.2.2+0
[8e53e030] libdrm_jll v2.4.125+1
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
[b53b4c65] libpng_jll v1.6.57+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.0.0+1
⌅ [1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.12.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.12.1
[de0858da] Printf v1.11.0
[9abbd945] Profile v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.12.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.3.0+1
[deac9b47] LibCURL_jll v8.15.0+0
[e37daf67] LibGit2_jll v1.9.0+0
[29816b5a] LibSSH2_jll v1.11.3+1
[14a3606d] MozillaCACerts_jll v2025.11.4
[4536629a] OpenBLAS_jll v0.3.29+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.4+0
[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
[83775a58] Zlib_jll v1.3.1+2
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.64.0+1
[3f19e933] p7zip_jll v17.7.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`Partners and fundings
