Stability as the Fundamental Problem of Control Systems

Mansour, Milad Zarif

doi:10.1115/1.3098929

Cited by 5 publications

(6 citation statements)

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Section: Methods Of Moments and Nondimensionalizationmentioning

confidence: 99%

“…The characteristic polynomial of the matrix is shown in eq with n = 7 and λ being the eigenvectors. Then, the Liénord–Chipart criterion is applied, which states that all even Hurwitz determinants D k , k ∈ double-struckN of that matrix as well as all even constants of the characteristic polynomial equation a n have to be positive. , λ n + a 1 λ n − 1 + ··· + a n = 0 For the trivial steady-state, the following seven constants of the characteristic polynomial equation a n and three determinants are obtained: a 1 = 3 ( κ α + κ β ) + 1 > 0 a 2 = 3 ( κ α 2 + κ β 2 ) + 3 ( κ α + κ β ) + 9 κ α κ β > 0 a 3 =...…”

Section: Methods Of Moments and Nondimensionalizationmentioning

confidence: 99%

“…The characteristic polynomial of the matrix is shown in eq with n = 7 and λ being the eigenvectors. Then, the Liénord–Chipart criterion is applied, which states that all even Hurwitz determinants of that matrix as well as all even constants of the characteristic polynomial equation a n have to be positive. , For the trivial steady-state, the following seven constants of the characteristic polynomial equation a n and three determinants are obtained: D 4 and D 6 are unsuitable for print, and thus, omitted here. However, one can see that in case τ α = τ β = τ 0 , the above expressions (including the expressions obtained for D 4 and D 6 ) become identical to the one stated in Farmer et al Thus, it comes at no surprise that the same steady-state conditions are obtained.…”

Section: Methods Of Moments and Nondimensionalizationmentioning

confidence: 99%

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Process Performance and Operational Challenges in Continuous Crystallization: A Study of the Polymorphs of L-Glutamic Acid

Achermann

Košir

Bodák

et al. 2023

Crystal Growth & Design

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The crystallization of the two polymorphs of l-glutamic acid (LGA) is carried out in a continuous crystallization process, and its performance according to different criteria is evaluated. The study aims at identifying suitable operating conditions for producing either αLGA or βLGA with a high polymorphic purity. To this end, we investigate the process both from a theoretical perspective and through experiments using either a single stirred-tank crystallizer or a cascade of two stirred-tank crystallizers in series. In terms of theory, we extend the MSMPR-based steady-state stability analysis of Farmer et al. (AIChE J.20166235053514) by accounting for the possibility of a nonrepresentative withdrawal of the solid phase from the crystallizer. Additionally, the process is simulated using population balance equations, thereby investigating the effect of operating conditions on polymorphic purity, yield, and productivity. Guided by the model-based conclusions, we identified suitable operating conditions and experimentally tested them. The experimental campaign has demonstrated that βLGA could be successfully and continuously produced in both process configurations according to the theory with performance as expected, whereas that was not possible for αLGA. The difference between the two stems from different operational challenges, whose consequence is that steady-state operation is attained in the case of βLGA but not in that of αLGA. In the former case, the needle-like βLGA crystals, which exhibit no agglomeration, tend to be only slightly oversampled; in the latter case, the prismatic αLGA crystals undergo major agglomeration and hence are very difficult to suspend and effectively withdraw from the crystallizer.

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Section: Methods Of Moments and Nondimensionalizationmentioning

confidence: 99%

Section: Methods Of Moments and Nondimensionalizationmentioning

confidence: 99%

“…The characteristic polynomial of the matrix is shown in eq with n = 7 and λ being the eigenvectors. Then, the Liénord–Chipart criterion is applied, which states that all even Hurwitz determinants of that matrix as well as all even constants of the characteristic polynomial equation a n have to be positive. , For the trivial steady-state, the following seven constants of the characteristic polynomial equation a n and three determinants are obtained: D 4 and D 6 are unsuitable for print, and thus, omitted here. However, one can see that in case τ α = τ β = τ 0 , the above expressions (including the expressions obtained for D 4 and D 6 ) become identical to the one stated in Farmer et al Thus, it comes at no surprise that the same steady-state conditions are obtained.…”

Section: Methods Of Moments and Nondimensionalizationmentioning

confidence: 99%

See 1 more Smart Citation

Process Performance and Operational Challenges in Continuous Crystallization: A Study of the Polymorphs of L-Glutamic Acid

Achermann

Košir

Bodák

et al. 2023

Crystal Growth & Design

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“…In this subsection, we propose a robust stability analysis method based on CLM‐criterion and the mapping theorem to reduce the conservativeness.Lemma (CLM‐criterion [1,24–26]) A necessary and sufficient condition for an nth order characteristic polynomial

f false(s false)

to be stable is that the trajectory of

f false(j ω false)

for

0 \leq ω < \infty

(which is referred to as the locus of Leonhard or Mikhailov curve) passes counterclockwise through each of the

n

quadrants in turn in the complex plane . Proof See Sect. 1.3.1 in Ref.…”

Section: Proposed Methods For Reduction Of Conservativenessmentioning

confidence: 99%

“…[20]. In our proposed method, the mapping theorem [1] and the Cremer-Leonhard-Mikhailov criterion (CLM-criterion) [24][25][26] are employed. An update rule of a design parameter is also presented to reduce the conservativeness further.…”

Section: Introductionmentioning

confidence: 99%

Reduction of Conservativeness of Robust PID Control System Design Method: An Approach Using the Mapping Theorem and the Cremer–Leonhard–Mikhailov Criterion

Terui

Matsuda

Nakamura

2023

IEEJ Transactions Elec Engng

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This paper gives a novel method to design less-conservative robust PID control systems. The partial model matching method is a design method for PID control systems. That is a method to design a control system that matches a reference model as closely as possible. This method does not guarantee robust stability of a PID control system because it is assumed that there are no uncertainties on the coefficients of the lower-order terms of the plant. Therefore, in our previous work, a robust design method based on the partial model matching method was proposed. However, the drawback of this conventional method is that the results of stability analysis are conservative. In this paper, we aim to reduce the conservativeness caused by the conventional method. To achieve this goal, we use a robust stability analysis method based on the Cremer-Leonhard-Mikhailov criterion and the mapping theorem. Moreover, we also propose a new method to update the value of a design parameter. A numerical example shown in this paper implies that the conservativeness is reduced by the proposed method.

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Systems theory and human science

Mansour

2002

Annual Reviews in Control

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Stability as the Fundamental Problem of Control Systems

Cited by 5 publications

References 0 publications

Process Performance and Operational Challenges in Continuous Crystallization: A Study of the Polymorphs of L-Glutamic Acid

Process Performance and Operational Challenges in Continuous Crystallization: A Study of the Polymorphs of L-Glutamic Acid

Reduction of Conservativeness of Robust PID Control System Design Method: An Approach Using the Mapping Theorem and the Cremer–Leonhard–Mikhailov Criterion

Systems theory and human science

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