Startseite Quadratic dynamic predictive control framework for continuous high-temperature-short-time pasteurization process
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Quadratic dynamic predictive control framework for continuous high-temperature-short-time pasteurization process

  • Marimuthu Indumathy , Sobana Subramani , Rames C. Panda ORCID logo EMAIL logo und Atanu Panda
Veröffentlicht/Copyright: 28. Mai 2025
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International Journal of Food Engineering
Aus der Zeitschrift International Journal of Food Engineering Band 21 Heft 6

Abstract

The present study focuses on developing advanced model-based controllers to enhance pasteurization temperature control in the dairy industry. The analysis and discussions of various multivariable control systems, including the cascade algorithm, model-predictive-control (MPC), and minimum-variance (MV) controller, are presented. A mathematical-model of the plant has been designed using mass and energy-balance equations for the integrated three-stage sections of the system, including the regeneration, heating, and cooling of counter-current gasketed PHE (plate heat exchanger), holding tube, holding tank, boiler, and chiller unit. The plant is modeled as a two-input and single-output system and used for the synthesis of controllers. This paper also uses a model reference (MR) framework for multivariable systems to construct the data-driven quadratic dynamic matrix controller (QDMC). Without building a mathematical model for the process, the data-driven controller is developed using one or more batches of plant data. The efficiency of the controller is evaluated by minimizing the objective function that contains the closed-loop error, and its ability to maintain control despite measurement noise is also examined. The closed-loop results (performances) are compared using the integral absolute error (IAE) performance criteria.

Keywords: milk pasteurization; plate heat exchanger; mathematical model; data-driven QDMC; cascade; minimum variance controller
Corresponding author: Rames C. Panda, Chemical Engineering Department, RajaLakshmi Engg College, Chennai, India, E-mail:

Acknowledgements

The authors thank their institutes for providing facilities to carry out this research.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: The author ‘M.I.’ has carried out data curation, investigation, simulation, observation and writing the rough draft of the paper. The second author ‘S.S.’ has supervised and helped in collecting data. The third author “R.C.P.” has supervised the work, did administration, provided concepts and has corrected the final draft of the paper. The fourth author ‘A.P.’ has contributed through estimation and control related simulations, and drafting for the paper.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state that there is no conflict of interests in this publication.

  6. Research funding: None declared.

  7. Data availability: Not applicable.

Appendix A (Augmented state and parameter estimation using adaptive square root unscented Kalman filter) [48, 50, 51]

Step 1:

Augmented state initialization

(A.1) x ˆ 0 = E x ˆ 0 P 0 = E x 0 − x ˆ 0 x 0 − x ˆ 0 T S 0 = chol P 0

Time updated equations

Step 2:

Computation of sigma points, weights

(A.2) χ i k − 1 = x ˆ k − 1 ;   w i , m = λ n + λ ;   w i , c = λ n + λ + 1 − α 2 + β ;   ∀   i = 0 χ i k − 1 = x ˆ k − 1 + n + λ P k − 1 ;   w i , m = w i , c = λ 2 n + λ ;   ∀   i = 1 , . . . , n χ i k − 1 = x ˆ k − 1 − n + λ P k − 1 ;   w i , m = w i , c = λ 2 n + λ ;   ∀   i = n + 1 , . . . , 2 n

Step 3:

Priori estimation with updation of error co-variance

(A.3) χ i k | k − 1 = f χ i k − 1 , u k − 1 x ˆ k | k − 1 = ∑ i = 0 2 n w i , m χ i k | k − 1 ;   ∀   i = 0 , . . . , 2 n s x k | k − 1 = q r w 1 : 2 n , c χ 1 : 2 n k | k − 1 − x ˆ k | k − 1   Q k − 1 s x k | k − 1 = Cholupdate s x k | k − 1 , w 0 , c χ 0 k | k − 1 − x ˆ k | k − 1

Measurement updated equations:

Step 4:

Determining predicted observation and the covariance matrixes

(A.4) y i k | k − 1 = h χ i k | k − 1 , u k − 1 y ˆ k | k − 1 = ∑ i = 0 2 n w i , m y i k | k − 1 ;   ∀   i = 0 , . . . , 2 n s y k = qr w 1 : 2 n , c y 1 : 2 n k | k − 1 − y ˆ k | k − 1   R k − 1 s y k | k − 1 = Cholupdate s y k , w 0 , c y 0 k | k − 1 − y ˆ k | k − 1 P xy = ∑ i = 0 2 n w i , c χ i k | k − 1 − x ˆ k | k − 1 y i k | k − 1 − y ˆ k | k − 1 T P yy = ∑ i = 0 2 n w i , c y k − y ˆ k | k − 1 y k − y ˆ k | k − 1 T

Step 5:

Deriving filter gain, residual, multi step innovation vector, noise covariances

(A.5) κ k = P x y k s y k s y T k − 1 e ̲ k = y k − y ˆ k | k − 1 ε k = 1 m ∑ i = k − m + 1 k e ̲ i e ̲ T i Q k = κ k ε k κ T k R k = ε k + P y y k

Step 6:

Updation of augmented states and covariance

(A.6) x ˆ k = x ˆ k | k − 1 + κ k e ̲ k P k = P k | k − 1 − κ k P yy k κ T k s k = cholupdate s x k | k − 1 , κ k P xy , − 1

Step 7:

Updation of augmented states and covariance

(A.7) k = k + 1

Appendix B:

Proof: Assuming Lyapunov candidate function as

(B.1) V k = x T k − 1 P − 1 k − 1 x T k − 1

Plunging matrix inversion lemma in 2nd equation of Eq. (A.6), it yields

(B.2) P − 1 k = P − 1 k | k − 1 + P − 1 k | k − 1 P xy k | k − 1 P xy k | k − 1 − P xy T k | k − 1 P − 1 k | k − 1 P xy k | k − 1 − 1 P xy T k | k − 1 P − 1 k | k − 1 = P − 1 k | k − 1 + C T k R − 1 k C k

Thus, 1st equation of Eq. (A.5) reduces to

(B.3) κ k = P k | k − 1 C T k − 1 C k − 1 P k | k − 1 C T k − 1 + R k − 1 = P k C T k R − 1 k

Putting Eq. (A.6) into Eq. (67), it becomes

(B.4) x k = x k − x ˆ k | k − 1 + κ k y k − y ˆ k = x k | k − 1 − κ k y k

Combining Eq. (A.5, B.1, B.4), Lyapunov function reduces to

(B.5) V k = x k | k − 1 κ k e ‾ k − 1 P − 1 k x k | k − 1 κ i e ‾ k = x T k | k − 1 P − 1 k x k | k − 1 − x T k | k − 1 C T k κ T k P − 1 i x k | k − 1 − x T k | k − 1 P − 1 k κ k C k x k | k − 1 + x T k | k − 1 C T k κ T k P − 1 k κ k C i x k | k − 1

Merging Eq. (B.3, B.5), it yields

(B.6) V k = x T k | k − 1 P − 1 k − C T k R − 1 k C k + C T k R − 1 k C k P k C T k R − 1 k C k x k | k − 1 = x T k | k − 1 P − 1 k − C T i R − 1 k − R − 1 k C k P k C T k R − 1 k C k x k | k − 1

However, after simplification, the below terms reduced as

(B.7) R − 1 k − R − 1 k C k P k C T k R − 1 k = R − 1 k − R − 1 k C k P k | k − 1 C T k C i P k | k − 1 C T k + R k − 1 = R − 1 k C i P k | k − 1 C T k + R k − R − 1 k C k P k | k − 1 C T k C k P k | k − 1 C T k + R k − 1 = C k P k | k − 1 C T k + R k − 1 = P yy − 1 k

Combining Eq. (72, B.6, B.7), the Lyapunov function can be derived as

(B.8) V k = x T k | k − 1 F T k η ′ k P − 1 k | k − 1 − C T k P yy − 1 k C k η ′ k F k x k | k − 1

Since, V(k) would be a decreasing sequence if there exists a positive scalar (1>ρ>0) such that

(B.9) V k ≤ 1 − ρ V k − 1

Eq. (B.9) can be extended into the below LMI

(B.10) x T k | k − 1 F T k η ′ k P − 1 k | k − 1 − C T k P yy − 1 k C k η ′ k F k − 1 − ρ P − 1 k − 1 x k | k − 1 ≤ 0

To obey Eq. (B.10), below necessary condition is introduced

(B.11) μ l F T k η ′ k C T k P yy − 1 C k η ′ k F k + μ l 1 − ρ P − 1 k − 1 ≥ μ u F T k η ′ k P − 1 k | k − 1 η ′ k F k

To corroborate Eq. (B.11), below inequalities are introduced

(B.12) μ u F 2 k μ u η ′ k 2 μ u P − 1 k | k − 1 ≥ μ u F T k η ′ k P − 1 k | k − 1 η ′ k F k μ l F 2 k μ l η ′ k 2 μ l C 2 k μ l P yy − 1 ≤ μ l F T k η ′ k C T k P yy − 1 C k η ′ k F k

Eq. (B.11) again transforms as

(B.13) μ l F 2 k μ l η ′ k 2 μ l C 2 k μ l P y y − 1 + 1 − ρ μ l P − 1 k − 1 ≥ μ u F 2 k μ u ς 2 k μ u P − 1 k | k − 1

Thus, Eq. (B.13) yields as

(B.14) μ l η ′ k 2 η ′ k F 2 k η ′ k C 2 k μ u F 2 k μ u P − 1 k | k − 1 μ u P yy − 1 + 1 − ρ μ l P − 1 k − 1 μ u F 2 k μ u P − 1 k | k − 1 ≥ μ u η ′ k 2 i . e .   μ l η ′ k 2 μ l C 2 k μ l P k | k − 1 μ u P yy − 1 + 1 − ρ μ l P − 1 k − 1 μ l P k | k − 1 μ u F 2 k ≥ μ u η ′ k 2

Eq. (B.14) shows that the Lyapunov function (as defined in Eq. (B.1)) is a monotonically decreasing function. To bring forward, below criteria must be met.

(B.15) lim K → ∞ x T k = 0

Thus, Eq. (B.15) shows that error due to plant-model mismatch tends to zero asymptotically.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/ijfe-2025-0038).

Received: 2024-11-14
Accepted: 2025-05-09
Published Online: 2025-05-28

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Articles
  3. Interaction of water chestnut starch with oleic acid and linolenic acid: a molecular dynamic simulation approach to assess the impact on physicochemical, rheological, and thermal properties
  4. Preparation and identification of ADH-promoting peptides from the meat of sturgeon
  5. Quadratic dynamic predictive control framework for continuous high-temperature-short-time pasteurization process
  6. Moisture migration and rheological properties of dual-frequency ultrasound combined vacuum drying on Pangasius bocourti surimi
  7. The antioxidant activity of grapes after different drying treatments
https://doi.org/10.1515/ijfe-2025-0038
Schlagwörter für diesen Artikel
milk pasteurization; plate heat exchanger; mathematical model; data-driven QDMC; cascade; minimum variance controller

Artikel in diesem Heft

  1. Frontmatter
  2. Articles
  3. Interaction of water chestnut starch with oleic acid and linolenic acid: a molecular dynamic simulation approach to assess the impact on physicochemical, rheological, and thermal properties
  4. Preparation and identification of ADH-promoting peptides from the meat of sturgeon
  5. Quadratic dynamic predictive control framework for continuous high-temperature-short-time pasteurization process
  6. Moisture migration and rheological properties of dual-frequency ultrasound combined vacuum drying on Pangasius bocourti surimi
  7. The antioxidant activity of grapes after different drying treatments
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