Use of Differential Equations With Variable Coefficients to Describe the Motions of Nonlinear Electromechanical Systems

Roman Voliansky

REVIEW 3 major objections 4 minor 23 references

Use of Differential Equations With Variable Coefficients to Describe the Motions of Nonlinear Electromechanical Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims every nonlinear electromechanical system can be rewritten exactly as a linear-form system with variable coefficients, making standard linear analysis tools applicable.

desk verdict A correct algebraic identity is presented as a reduction to linear systems, but the paper's own equations show it is a tautology and the main claim is unsupported. read the letter →

arxiv 2412.05294 v1 pith:IBGIIDSU submitted 2024-11-22 math.OC

Roman Voliansky This is my paper · ORCID

classification math.OC MSC 34A3493C1093C15

keywords nonlinearelectromechanicalsystemsvariablecoefficientsexactlinearizationpseudo-affinedynamicdifferentialequationstrajectorysynthesisstabilityanalysis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that any nonlinear electromechanical system whose motion is described by the differential equation $d\mathbf{y}/dt = \mathbf{f}(\mathbf{y},\mathbf{u},t)$ can be rewritten exactly as a system of differential equations that has the structural form of a linear system, namely each derivative is a weighted sum of state variables and control inputs, but with coefficients that depend on the state and inputs. The transformation uses only multiplication and division by a linear combination of states and controls, so the original nonlinearity is not approximated away; it is moved into the variable coefficients. If true, the abundance of classical linear analysis and controller-synthesis methods could be applied to strongly nonlinear electromechanical systems without the local-validity limits of Taylor or harmonic linearization. The paper concludes that because the transformed system has a linear-analog form, "analysis and synthesis of motion trajectories can be carried out by any known method."

What carries the argument

The central machinery is a pseudo-affine transformation: the paper rewrites each component of the nonlinearity as the product of a scalar function $K_i$ and a linear combination of states and controls, $f_i = K_i \left(\sum_k a_{ik} y_k + \sum_k c_{ik} u_k\right)$, which is always possible algebraically by defining $K_i$ as the ratio of $f_i$ to that linear combination. The scalar $K_i$ is then absorbed into variable coefficients $b_{ik} = a_{ik} K_i$ and $m_{ik} = c_{ik} K_i$, so the derivative vector becomes a linear function of $\mathbf{y}$ and $\mathbf{u}$ with state-dependent coefficients. This converts the nonlinear system into a system whose form is identical to a linear constant-coefficient system, with the nonlinearity hidden in the varying coefficients.

What would settle it

Take the scalar equation $\dot{y} = y^2$ and choose the denominator $L = y$ (valid for $y \neq 0$), so the transformed system is $\dot{y} = b(y) y$ with $b(y) = y$; a standard linear analysis that freezes $b$ at its initial value predicts exponential growth $y_0 e^{b_0 t}$, whereas the exact solution $y_0/(1 - y_0 t)$ blows up in finite time, so the divergence shows state-dependent coefficients cannot be handled by unmodified linear methods.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is an algebraic identity: starting from $\frac{d\mathbf{y}}{dt} = \mathbf{f}(\mathbf{y},\mathbf{u},t)$, one multiplies and divides each component $f_i$ by a nonzero linear form $L_i = \sum_k a_{ik} y_k + \sum_k c_{ik} u_k$ and defines $K_i = f_i / L_i$. Substituting and absorbing $K_i$ into the constants yields $\frac{dy_i}{dt} = \sum_k b_{ik} y_k + \sum_k m_{ik} u_k$ with $b_{ik} = a_{ik} K_i$ and $m_{ik} = c_{ik} K_i$. This new system (10) is identical to the original (1), not an approximation, yet it has the same structural form as the linear system (4). The paper concludes that because of this structural analogy, analysis and synthesis of motion trajectories for the nonlinear electromechanical system can be carried out by any known method for linear differential equations.

Load-bearing premise

The load-bearing premise is that a linear denominator $L_i = \sum_k a_{ik} y_k + \sum_k c_{ik} u_k$ can be chosen that never vanishes over the operating range, and that the resulting state-dependent coefficients $b_{ik}, m_{ik}$ can be treated by standard linear analysis methods as though they were constant; the paper states the first as a requirement and asserts the second without proof.

Editorial extensions

If this is right

Any nonlinear electromechanical system of form (1) gets an exact linear-form representation (10), so error from truncating Taylor or harmonic linearization disappears.
The coefficients $b_{ik}$ and $m_{ik}$ are explicitly computable from the original nonlinearity and the chosen constants $a_{ik}, c_{ik}$, which makes the transformation easy to automate.
The paper asserts that because (10) is structurally analogous to the linearized equation (4), classical stability-analysis and trajectory-synthesis methods apply to the nonlinear system.
Since no approximation is introduced, the transformed model inherits the full dynamics of the original system, including any possibility of irregular or chaotic motion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural step the paper leaves implicit: the transformation produces a linear-parameter-varying or gain-scheduled representation, so the LPV control toolbox could be applied, but the paper stops at asserting 'any known method' without naming which.
The denominator non-vanishing condition is a real constraint; for systems with equilibria at the origin, a purely linear denominator may vanish, so practical use likely requires piecewise or region-based selection of $a_{ik}, c_{ik}$, which the paper does not address.
A direct check with $\dot{y} = y^2$ suggests that the 'any known method' claim is too broad: standard linear tools that ignore the state-dependence of coefficients produce qualitatively wrong trajectories, so the paper's practical value depends on adapting methods that explicitly handle state-dependent coefficients.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read the paper. The reader's take holds up: the core manipulation is multiply-and-divide by a linear form, defining K_i = f_i / L, which makes Eq. (10) literally the original nonlinear system in new notation. The paper's own conclusion says the systems are 'totally identical.' So the central claim is not derived; it is asserted.

To be fair, the algebraic identity is correct. The paper clearly notes the denominator must be nonzero over the operating range, and it does cite ref. [21] for the pseudo-affine concept. That is honest engagement. But there is no new mathematical content: this is state-dependent coefficient (SDC) parameterization, already known in the cited literature.

The soft spot is load-bearing and it is the whole point of the paper. Because b_ik and m_ik in Eq. (11) depend on y and U through K_i, Eq. (10) remains nonlinear. The claim that 'analysis and synthesis of motion trajectories can be carried out by any known method' is false in general: constant-coefficient eigenvalue placement, frequency-response, and LTI synthesis methods do not apply to state-dependent coefficient systems merely because the equations are written in linear form. The paper provides no example, simulation, or theorem to support that leap. The branch (12)-(13) is also circular, since it defines K_i using dy_i/dt divided by the same linear form.

In proportion: this is a four-page note in Ukrainian, and the only correct part is an identity. The overclaim is unsupported, and the paper offers no practical method. The reader's assessment is accurate.

Who is this for? Maybe someone looking for a compact illustration of SDC parameterization, but nothing more. It does not deserve a serious referee; a desk reject is appropriate. If the author wants to make the linear-tools claim stick, the paper would need to restrict to slowly varying coefficients or demonstrate that a particular synthesis method works on state-dependent coefficient systems. As it stands, this is a rewrite, not a result.

Referee Report

3 major / 4 minor

Summary. The paper proposes to represent a general nonlinear electromechanical system dy/dt = f(y,u,t) in the form dy_i/dt = sum_k (b_ik y_k + m_ik U_k), where b_ik and m_ik are defined as a_ik K_i and c_ik K_i, with K_i chosen so that the right-hand side is identically equal to f_i. The author claims that because the resulting system (10) has the same form as the linearized system (4), any known linear analysis and synthesis method can be applied to the original nonlinear system. A second branch (12)-(13) is given for the case where f lacks a strict analytic description, using K_i = (dy_i/dt)/L. The paper concludes that the nonlinear system (1) has been 'reduced to a totally identical system' (10) with variable coefficients.

Significance. If the claim were true, it would be a remarkable result: every nonlinear system could be analyzed and synthesized with any available linear method. The algebraic manipulation in Eqs. (7)-(9) is correct, but it is a purely definitional rewrite that preserves the nonlinearity inside the coefficients. The paper correctly identifies the limitations of Taylor linearization and harmonic linearization, but it provides no theorem, proof, or numerical demonstration to support the central inference that variable state-dependent coefficients can be treated like constant coefficients. As written, the contribution is a tautology, not a new analysis tool.

major comments (3)

[Materials and results, Eqs. (10)-(11)] The transformation leading to Eq. (10) is algebraically valid but is an identity rewrite. Since K_i in Eq. (8) is defined as f_i divided by the chosen linear form L, substituting it into Eq. (7) reproduces Eq. (1) exactly. The coefficients b_ik and m_ik in Eq. (11) therefore depend on y and U through K_i, so Eq. (10) remains nonlinear. The claim that this system is analogous to Eq. (4) and can be analyzed by 'any known method' for linear systems is unsupported and is false in general; linear methods such as eigenvalue analysis, transfer functions, and LTI synthesis require constant coefficients or specific structural assumptions. No theorem or example is provided to justify this load-bearing step.
[Materials and results, Eqs. (12)-(13)] In the branch for functions without a strict mathematical description, Eq. (13) defines K_i as (dy_i/dt)/L, where L is the same linear denominator used in Eq. (12). Since Eq. (1) already states dy_i/dt = f_i, this substitution is circular: substituting Eq. (13) into Eq. (12) yields the identity dy_i/dt = dy_i/dt. Consequently, the coefficients defined in Eq. (11) carry no new modeling information and cannot serve as a basis for analysis or synthesis unless one already knows the solution dy_i/dt.
[Materials and results, Eq. (7)] The paper requires that the denominator L = sum_i a_ik y_i + sum_i c_ik U_i be nonzero over the entire operating range, but it does not show how to choose the coefficients a_ik and c_ik to satisfy this condition, nor does it prove that such coefficients exist. For any equilibrium point with y = 0 and U = 0, L vanishes identically for any constant coefficients a_ik, c_ik, so the transformation fails exactly at the point most relevant for stability analysis. This is a concrete obstacle to the claimed practical applicability of the method.

minor comments (4)

[Eq. (7)] The same index i is used both for the vector component of f and as the summation index in the numerator and denominator; using a distinct summation index would improve clarity.
[Conclusions] The concluding sentence refers to 'variable coefficients (1)'; this appears to be a typo for Eq. (11), which defines the coefficients.
[General] The paper contains no worked example or numerical simulation, which would be necessary to illustrate or validate the claimed applicability of any known linear method.
[References] The reference list consists mostly of conference papers and omits standard textbooks on nonlinear control and linear parameter-varying systems, making it difficult for a reader to assess the basis for the statement that 'any known method' can be applied.

Circularity Check

2 steps flagged · score 10.0 of 10

The central transformation is an algebraic identity: K_i is defined as f_i divided by an arbitrary nonzero linear form, so Eq. (10) with Eq. (11) reproduces Eq. (1) by construction; the claimed analogy with constant-coefficient Eq. (4) is purely syntactic and does not license linear-analysis methods.

self definitional [Eqs. (7)-(9), (11)]
"Помножимо і розділимо праву частину системи (1) на суму лінійних комбінацій керуючих впливів і координат об’єкта: ... Введемо наступне позначення: K_i = f_i / (Σ a_ik y_i + Σ c_ik U_i) (8) ... та, підставивши (8) у (7), одержимо (9)."

By definition (8), K_i is exactly f_i divided by the arbitrarily chosen nonzero linear form L = Σ a_ik y_i + Σ c_ik U_i. Substituting (8) into (7) gives dy_i/dt = K_i L = f_i, i.e., the right-hand side of the original system (1) is recovered identically. The coefficients in (11), b_ik = a_ik K_i and m_ik = c_ik K_i, depend on y and U through K_i, so Eq. (10) is an algebraic rewriting of the original nonlinear system, not a reduction to a constant-coefficient linear system. The appearance of a linear form is manufactured by the definition of K_i; the paper itself calls the result 'тотожної системи' (identical system).
renaming known result [Conclusions]
"Завдяки тому, що система (10) має вид аналогічний (4), аналіз та синтез траєкторій руху розглядаємої системи може здійснюватися будь-яким відомим методом."

This is the load-bearing conclusion: because (10) 'has a form analogous to (4)', the paper asserts that any known method for (4) applies. But the analogy is only syntactic. In (4), B and m are constant matrices evaluated at a working point; in (10), b_ik and m_ik are state- and input-dependent through K_i. The paper neither proves nor cites a theorem that constant-coefficient eigenstructure, frequency-response, or LTI synthesis methods remain valid for such state-dependent coefficients. The nonlinearity is hidden inside the variable coefficients rather than removed, so the claim that any known linear method can be used is an unsupported renaming of the original nonlinear system as a 'linear-form' system.

full rationale

Every load-bearing step of the derivation reduces to its own inputs. Equation (8) defines K_i as f_i divided by the chosen nonzero linear denominator; substituting it back into (7) recovers (1) identically. The paper explicitly calls the result an identical system ('тотожної системи'), so Eq. (10) with Eq. (11) is a tautological rewrite of Eq. (1), not a first-principles reduction to a linear model. The further assertion that methods for constant-coefficient linear system (4) can be applied to (10) is not a consequence of the algebra: the coefficient matrices b_ik and m_ik depend on state and input through K_i, unlike the constant working-point matrices B and m in (4). No theorem, example, or external benchmark is provided to bridge that gap. There is no self-citation chain forcing the result; the circularity is definitional and sits at the center of the paper's claimed contribution. Score 10 because the derivation is equivalent to its input by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces arbitrary coefficients a_ik and c_ik as free parameters but no new physical entities. It relies on the pseudo-affine systems concept from prior work and on an unproved premise that linear tools apply to state-dependent coefficient systems.

free parameters (1)

a_ik, c_ik
Arbitrary coefficients in the denominator of Eq. (7). The author states they are arbitrary and only need to keep the denominator nonzero. The resulting variable coefficients b_ik and m_ik in Eq. (11) inherit this arbitrariness.

assumptions (4)

domain assumption The denominator L_i = sum_k a_ik y_k + sum_k c_ik U_k in Eq. (7) can be chosen nonzero over the entire operating domain.
The transformation divides by this sum. The author says the arbitrary coefficients must be selected so the denominator is not zero, but existence over all relevant states and controls is assumed, not proved.
ad hoc to paper Known linear analysis and synthesis methods remain valid for systems whose coefficients depend on the state and control, i.e. Eq. (10) can be treated like Eq. (4).
This is the load-bearing premise that makes the conclusion useful. It is stated in the Conclusions without proof and is not a standard result for state-dependent coefficient systems.
domain assumption The concept of pseudo-affine dynamical systems from ref. [21] provides a valid basis for the transformation.
The paper imports this concept from a previous publication rather than deriving it; the present note relies on it for the form of Eqs. (7)-(11).
domain assumption For systems without a strict description of f_i, the unknown derivative dy_i/dt can be used inside K_i in Eq. (13).
This presumes the derivative is available or measurable, and it makes the proposed equation implicit in the unknown derivative, which the paper does not discuss.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Use of Differential Equations With Variable Coefficients to Describe the Motions of Nonlinear Electromechanical Systems." pith.science (2026). https://pith.science/paper/IBGIIDSU

@misc{pith2026241205294,
  author       = {Pith},
  title        = {Pith review of: Use of Differential Equations With Variable Coefficients to Describe the Motions of Nonlinear Electromechanical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IBGIIDSU}},
  note         = {Machine review of arXiv:2412.05294}
}

read the original abstract

Due to the processes that occur during the functioning of modern electromechanical systems, these systems can be considered complex nonlinear dynamic systems from the point of view of the theory of dynamic systems. The movement of such systems is completely determined by external influences acting on the EMS, their parameters, and initial operating conditions. The above-mentioned factors complicate the study of electromechanical systems and, in the general case, make it impossible to use classical methods of analyzing the dynamics of the EMS since the latter neglect the features of nonlinear systems and describe their dynamics using ordinary linear differential equations with constant coefficients. At the same time, many methods and approaches have been developed in control theory for analyzing stability and synthesizing motion trajectories based on linear differential equations. Therefore, an important task arises to create mathematical models of nonlinear EMS that consider the peculiarities of their motion but have a form similar to linear models.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[21]

ХПІ": зб. науч. пр. Темат. в ип. : Механ іко-технологічні системи та комплекси. – Харків: НТУ

Садовой О. В. Лінеаризація зворотними зв’язками рівнянь динаміки узагальненого електромеханічного об’єкта з диференціальним рівнянням спостережуваності // Вісник Нац. техн. ун-ту "ХПІ": зб. науч. пр. Темат. в ип. : Механ іко-технологічні системи та комплекси. – Харків: НТУ "ХПІ". – 2014. – № 60 (1102). – С. 52-57. (на рос.)

work page 2014
[1]

Синтез оптимально ї системи керування з іраціональною активаційною функцією // Вісник НТУ «ХПІ» «Проблеми автоматизованого електроприводу Теорія та практика)»

Садовой O.В. Синтез оптимально ї системи керування з іраціональною активаційною функцією // Вісник НТУ «ХПІ» «Проблеми автоматизованого електроприводу Теорія та практика)». – Харків: НТУ «ХПИ». – 2010.– Вип. 28. – С.49 – 51 (на рос.)

work page 2010
[2]

Синтез оптимально ї системи керування з нел інійною активаційною функцією

Садовой О.В. Синтез оптимально ї системи керування з нел інійною активаційною функцією. Електротехнічні і комп'ютерні системи. Київ: Техніка. 2014, № 15 (91), С. 69 – 71 (на рос.)

work page 2014
[3]

Математич ні моделі усунення помпажа в відцентровому компресорі // Вісник КДУ ім

Цaбенко М.В. Математич ні моделі усунення помпажа в відцентровому компресорі // Вісник КДУ ім. М. Остоградського. - 2010. - Випуск 4(63). - С. 167 - 169. (на рос.)

work page 2010
[4]

Система оптимального керування натягом прокатуваної полос и стану холодної прокатки // З бірка наукових прац ь Донбаського державного техн ічного університету

Назарова Е.С. Система оптимального керування натягом прокатуваної полос и стану холодної прокатки // З бірка наукових прац ь Донбаського державного техн ічного університету. –2011. – Вип. 34. – С.122-130 (на рос.)

work page 2011
[5]

Система керування слідкуючим електроприводом із ковзним режимом 2- го порядка, Вісник Кременчуцького державного університету імені Михайла Остроградського

Волянський Р.С. Система керування слідкуючим електроприводом із ковзним режимом 2- го порядка, Вісник Кременчуцького державного університету імені Михайла Остроградського. – Кременчук: КДУ, 2010. – Вип. 4/2010 (63) частина 3. – с.11-14 (на рос.)

work page 2010
[6]

In: IEEE Int ernational Conference on Modern Electrical and Energy Systems (MEES), Kremenchuk, Ukraine, pp

Sokhina, Y Active Suspension Control System. In: IEEE Int ernational Conference on Modern Electrical and Energy Systems (MEES), Kremenchuk, Ukraine, pp. 10-13 (2019)

work page 2019
[7]

Волянський Р. С. Синтез керуючого впливу для електромеханічних об’єктів з невідомими параметрами // Вісник НТУ «ХПІ». — Харьков: ХПИ. — 2015. — № 12 (1121). — С. 60— 63 (на рос.)

work page 2015

Show all 23 references

[8]

Pranolo А, Parallel Mathematical Models of Dynamic Objects, Int. J. Adv. Intell. Informs. 4(2) (2018) 120–131

2018
[9]

Anti-swing Control System for the One Class of Underactuated Dynamic Objects,

I. Shramko, "Anti-swing Control System for the One Class of Underactuated Dynamic Objects," 2020 IEEE Problems of Automated Electrodrive. Theory and Practice (PAEP), Kremenchuk, Ukraine, 2020, pp. 1-4

2020
[10]

Chaotic Time-variant Dynamical System,

Kluev, O., "Chaotic Time-variant Dynamical System," 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Eng ineering (TCSET), 2020, pp. 606-609

2020
[11]

Chua's circuits interval synchronization,

R. Voliansky, "Chua's circuits interval synchronization," 2017 4th International Scientific - Practical Conference Problems of Infocommunications. Science and Technology (PIC S&T), Kharkov, Ukraine, 2017, pp. 439-443

2017
[12]

Sliding Mode Control for DC Generator with Uncertain Load,

O. Sinkevych, "Sliding Mode Control for DC Generator with Uncertain Load," 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), Lviv-Slavske, Ukraine, 2020, pp. 313-316

2020
[13]

Voliansky, R. (2017). Transformation of the generalized chaotic system into canonical form. International Journal of Advances in Intelligent Informatics, 3(3), 117–124

2017
[14]

The transformation of linear dynamical object's equation to Brunovsky canonical form,

Sadovoi A. V., "The transformation of linear dynamical object's equation to Brunovsky canonical form," 2017 IEEE 4th International Conference Actual Problems of Unmanned Aerial Vehicles Developments (APUAVD), Kiev, Ukraine, 2017, pp. 196-199

2017
[15]

Садовой А. В. Дослідження частотн их характеристик динам ічних ланок із похідним дробових порядк ів / Р. С. Вол янский, // Збірка наукових праць Д ДТУ (техн ічні науки). Тематичний випуск „Проблеми автоматизованого електроприводу. Теорія і практика”. –
[16]

Construction of Parallel Piecewise -Linear Interval Models for Nonlinear Dynamical Objects,

O. Sinkevych, "Construction of Parallel Piecewise -Linear Interval Models for Nonlinear Dynamical Objects," 2019 9th International Conference on Advanced Computer Information Technologies (ACIT), 2019, pp. 97-100

2019
[17]

Sliding Mode Interval Controll er for the Mobile Robot,

Y. Sokhina, "Sliding Mode Interval Controll er for the Mobile Robot," 2019 XIth International Scientific and Practical Conference on Electronics and Information Technologies (ELIT), Lviv, Ukraine, 2019, pp. 76-81

2019
[18]

Interval model of the piezoelectric drive,

N. Volianska, "Interval model of the piezoelectric drive," 2018 14th International Conference on Advanced Trends in Radioelecrtronics, Telecommunications and Computer Engineering (TCSET), Lviv-Slavske, Ukraine, 2018, pp. 1-6

2018
[19]

Root Methods for Dynamic Analysis of the One Class Chaotic Systems,

O. Sinkevych, "Root Methods for Dynamic Analysis of the One Class Chaotic Systems," 2019 IEEE 14th International Con ference on Computer Sciences and Information Technologies (CSIT), 2019, pp. 117-121

2019
[20]

Defining of Lyapunov Functions for the Generalized Nonlinear Object,

O. Sadovoi, "Defining of Lyapunov Functions for the Generalized Nonlinear Object," 2018 IEEE 5th International Conference on Methods and Systems of Navigation and Motion Control (MSNMC), Kiev, Ukraine, 2018, pp. 222-228

2018
[22]

Electromechanical System Motion Control in Direct and Inverse Time,

N. Krasnoshapka, O. Statsenko, "Electromechanical System Motion Control in Direct and Inverse Time," 2022 IEEE 3rd KhPI Week on Advanced Technology (KhPIWeek), Kharkiv, Ukraine, 2022, pp. 1-6

2022
[2007]

– С. 82–85. (на рос.)

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [21]

ХПІ": зб. науч. пр. Темат. в ип. : Механ іко-технологічні системи та комплекси. – Харків: НТУ

Садовой О. В. Лінеаризація зворотними зв’язками рівнянь динаміки узагальненого електромеханічного об’єкта з диференціальним рівнянням спостережуваності // Вісник Нац. техн. ун-ту "ХПІ": зб. науч. пр. Темат. в ип. : Механ іко-технологічні системи та комплекси. – Харків: НТУ "ХПІ". – 2014. – № 60 (1102). – С. 52-57. (на рос.)

work page 2014

[2] [1]

Синтез оптимально ї системи керування з іраціональною активаційною функцією // Вісник НТУ «ХПІ» «Проблеми автоматизованого електроприводу Теорія та практика)»

Садовой O.В. Синтез оптимально ї системи керування з іраціональною активаційною функцією // Вісник НТУ «ХПІ» «Проблеми автоматизованого електроприводу Теорія та практика)». – Харків: НТУ «ХПИ». – 2010.– Вип. 28. – С.49 – 51 (на рос.)

work page 2010

[3] [2]

Синтез оптимально ї системи керування з нел інійною активаційною функцією

Садовой О.В. Синтез оптимально ї системи керування з нел інійною активаційною функцією. Електротехнічні і комп'ютерні системи. Київ: Техніка. 2014, № 15 (91), С. 69 – 71 (на рос.)

work page 2014

[4] [3]

Математич ні моделі усунення помпажа в відцентровому компресорі // Вісник КДУ ім

Цaбенко М.В. Математич ні моделі усунення помпажа в відцентровому компресорі // Вісник КДУ ім. М. Остоградського. - 2010. - Випуск 4(63). - С. 167 - 169. (на рос.)

work page 2010

[5] [4]

Система оптимального керування натягом прокатуваної полос и стану холодної прокатки // З бірка наукових прац ь Донбаського державного техн ічного університету

Назарова Е.С. Система оптимального керування натягом прокатуваної полос и стану холодної прокатки // З бірка наукових прац ь Донбаського державного техн ічного університету. –2011. – Вип. 34. – С.122-130 (на рос.)

work page 2011

[6] [5]

Система керування слідкуючим електроприводом із ковзним режимом 2- го порядка, Вісник Кременчуцького державного університету імені Михайла Остроградського

Волянський Р.С. Система керування слідкуючим електроприводом із ковзним режимом 2- го порядка, Вісник Кременчуцького державного університету імені Михайла Остроградського. – Кременчук: КДУ, 2010. – Вип. 4/2010 (63) частина 3. – с.11-14 (на рос.)

work page 2010

[7] [6]

In: IEEE Int ernational Conference on Modern Electrical and Energy Systems (MEES), Kremenchuk, Ukraine, pp

Sokhina, Y Active Suspension Control System. In: IEEE Int ernational Conference on Modern Electrical and Energy Systems (MEES), Kremenchuk, Ukraine, pp. 10-13 (2019)

work page 2019

[8] [7]

Волянський Р. С. Синтез керуючого впливу для електромеханічних об’єктів з невідомими параметрами // Вісник НТУ «ХПІ». — Харьков: ХПИ. — 2015. — № 12 (1121). — С. 60— 63 (на рос.)

work page 2015