The paper studies the fault identification problem for linear control systems under the unmatched disturbances. A novel approach to the construction of a sliding mode observer is proposed for systems that do not satisfy common conditions required for fault estimation, in particular matching condition, minimum phase condition, and detectability condition. The suggested approach is based on the reduced order model of the original system. This allows to reduce complexity of sliding mode observer and relax the limitations imposed on the original system.

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[1] H. Alwi and C. Edwards: Fault tolerant control using sliding modes with on-line control allocation. *Automatica*, 44 (2008), 1859–1866, DOI: 10.1016/j.automatica.2007.10.034.

[2] H. Alwi, C. Edwards, and C. Tan: Sliding mode estimation schemes for incipient sensor faults.*Automatica*, 45 (2009), 1679–1685, DOI: 10.1016/j.automatica.2009.02.031.

[3] F. Bejarano, L. Fridman, and A. Pozhyak: Unknown input and state estimation for unobservable systems.*SIAM J. Control and Optimization*, 48 (2009), 1155–1178. DOI: 10.1137/070700322.

[4] F. Bejarano and L. Fridman: High-order sliding mode observer for linear systems with unbounded unknown inputs.*Int. J. Control,* 83 (2010), 1920– 1929, DOI: 10.1080/00207179.2010.501386.

[5] M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki:*Diagnosis and Fault-Tolerant Control.* Berlin: Springer-Verlag, 2006.

[6] A. Brahim, S. Dhahri, F. Hmida, and A. Sellami: Simultaneous actuator and sensor faults reconstruction based on robust sliding mode observer for a class of nonlinear systems.*Asian J. Control, *19 (2017), 362–371, DOI: 10.1002/asjc.1359.

[7] J. Chan, C. Tan, and H. Trinh: Robust fault reconstruction for a class of infinitely unobservable descriptor systems.*Int. J. Systems Science,* (2017), 1–10. DOI: 10.1080/00207721.2017.1280552.

[8] L. Chen, C. Edwards, H. Alwi, and M. Sato: Flight evaluation of a sliding mode online control allocation scheme for fault tolerant control.*Automatica,* 144 (2020), DOI: 10.1016/j.automatica.2020.108829.

[9] M. Defoort, K. Veluvolu, J. Rath, and M. Djemai: Adaptive sensor and actuator fault estimation for a class of uncertain Lipschitz nonlinear systems.*Int. J. Adaptive Control and Signal Processing,* 30 (2016), 271–283, DOI: 10.1002/acs.2556.

[10] S. Ding:*Data-driven Design of Fault Diagnosis and Fault-tolerant Control Systems*. London: Springer-Verlag, 2014.

[11] C. Edwards and S. Spurgeon: On the development of discontinuous observers*. Int. J. Control,* 59 (1994), 1211–1229, DOI: 10.1080/ 00207179408923128.

[12] C. Edwards, S. Spurgeon, and R. Patton: Sliding mode observers for fault detection and isolation.*Automatica,* 36 (2000), 541–553, DOI: 10.1016/S0005-1098(99)00177-6.

[13] C. Edwards, H. Alwi, and C. Tan: Sliding mode methods for fault detection and fault tolerant control with application to aerospace systems.*Int. J. Applied Mathematics and Computer Science*, 22 (2012), 109–124, DOI: 10.2478/v10006-012-0008-7.

[14] V. Filaretov, A. Zuev, A. Zhirabok, and A. Protcenko: Development of fault identification system for electric servo actuators of multilink manipulators using logic-dynamic approach.* J. Control Science and Engineering,* 2017 (2017), 1–8, DOI: 10.1155/2017/8168627.

[15] T. Floquet, C. Edwards, and S. Spurgeon: On sliding mode observers for systems with unknown inputs.*Int. J. Adaptive Control and Signal Processing, *21 (2007), 638–65, DOI: 10.1002/acs.958.

[16] L. Fridman, A. Levant, and J. Davila: Observation of linear systems with unknown inputs via high-order sliding-modes.*Int. J. Systems Science*, 38 (2007), 773–791, DOI: 10.1080/00207720701409538.

[17] L. Fridman, Yu. Shtessel, C. Edwards, and X. Yan: High-order slidingmode observer for state estimation and input reconstruction in nonlinear systems.* Int. J. Robust and Nonlinear Control*, 18 (2008), 399–412, DOI: 10.1002/rnc.1198.

[18] R. Hmidi, A. Brahim, F. Hmida, and A. Sellami: Robust fault tolerant control design for nonlinear systems not satisfying matching and minimum phase conditions.*Int. J. Control, Automation and Systems*, 18 (2020), 1–14, DOI: 10.1007/s12555-019-0516-4.

[19] H. Rios, D. Efimov, J. Davila, T. Raissi, L. Fridman, and A. Zolghadri: Non-minimum phase switched systems: HOSM based fault detection and fault identification via Volterra integral equation.*Int. J. Adaptive Control and Signal Processing*, 28 (2014), 1372–1397, DOI: 10.1002/acs.2448.

[20] I. Samy, I. Postlethwaite, and D. Gu: Survey and application of sensor fault detection and isolation schemes.*Control Engineering Practice*, 19 (2011), 658–674, DOI: 10.1016/j.conengprac.2011.03.002.

[21] C. Tan and C. Edwards: Sliding mode observers for robust detection and reconstruction of actuator and sensor faults.*Int. J. Robust Nonlinear Control*, 13 (2003), 443–463, DOI: 10.1002/rnc.723.

[22] C. Tan and C. Edwards: Robust fault reconstruction using multiple sliding mode observers in cascade: development and design.*Proc. 2009 American Control Conf.*, St. Louis, USA, (2009), DOI: 10.1109/ACC.2009.5160176.

[23] V. Utkin:*Sliding Modes in Control Optimization*, Berlin: Springer, 1992.

[24] X. Wang, C. Tan, and G. Zhou: A novel sliding mode observer for state and fault estimation in systems not satisfying matching and minimum phase conditions.*Automatica*, 79 (2017), 290–295, DOI: 10.1016/ j.automatica.2017.01.027.

[25] X. Yan and C. Edwards: Nonlinear robust fault reconstruction and estimation using a sliding modes observer.*Automatica*, 43 (2007), 1605–1614, DOI: 10.1016/j.automatica.2007.02.008.

[26] J. Yang, F. Zhu, and X. Sun: State estimation and simultaneous unknown input and measurement noise reconstruction based on associated observers.*Int. J. Adaptive Control and Signal Processing*, 27 (2013), 846–858, DOI: 10.1002/acs.2360.

[27] A. Zhirabok: Nonlinear parity relation: A logic-dynamic approach.*Automation and Remote Control,* 69 (2008), 1051-1064, DOI: 10.1134/ S0005117908060155.

[28] A. Zhirabok, A. Shumsky, and S. Pavlov: Diagnosis of linear dynamic systems by the nonparametric method.* Automation and Remote Control,* 78 (2017), 1173–1188, DOI: 10.1134/S0005117917070013.

[29] A. Zhirabok, A. Shumsky, S. Solyanik, and A. Suvorov: Fault detection in nonlinear systems via linear methods.* Int. J. Applied Mathematics and Computer Science,* 27 (2017), 261–272, DOI: 10.1515/amcs-2017-0019.

[30] A. Zhirabok, A. Zuev, and A. Shumsky: Methods of diagnosis in linear systems based on sliding mode observers.*J. Computer and Systems Sciences Int.,* 58 (2019), 898–914, DOI: 10.1134/S1064230719040166.

[31] A. Zhirabok, A. Zuev, andV. Filaretov: Fault identification in underwater vehicle thrusters via sliding mode observers.*Int. J. Applied Mathematics and Computer Science*, 30 (2020), 679–688, DOI: 10.34768/amcs-2020-0050.

Go to article
[2] H. Alwi, C. Edwards, and C. Tan: Sliding mode estimation schemes for incipient sensor faults.

[3] F. Bejarano, L. Fridman, and A. Pozhyak: Unknown input and state estimation for unobservable systems.

[4] F. Bejarano and L. Fridman: High-order sliding mode observer for linear systems with unbounded unknown inputs.

[5] M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki:

[6] A. Brahim, S. Dhahri, F. Hmida, and A. Sellami: Simultaneous actuator and sensor faults reconstruction based on robust sliding mode observer for a class of nonlinear systems.

[7] J. Chan, C. Tan, and H. Trinh: Robust fault reconstruction for a class of infinitely unobservable descriptor systems.

[8] L. Chen, C. Edwards, H. Alwi, and M. Sato: Flight evaluation of a sliding mode online control allocation scheme for fault tolerant control.

[9] M. Defoort, K. Veluvolu, J. Rath, and M. Djemai: Adaptive sensor and actuator fault estimation for a class of uncertain Lipschitz nonlinear systems.

[10] S. Ding:

[11] C. Edwards and S. Spurgeon: On the development of discontinuous observers

[12] C. Edwards, S. Spurgeon, and R. Patton: Sliding mode observers for fault detection and isolation.

[13] C. Edwards, H. Alwi, and C. Tan: Sliding mode methods for fault detection and fault tolerant control with application to aerospace systems.

[14] V. Filaretov, A. Zuev, A. Zhirabok, and A. Protcenko: Development of fault identification system for electric servo actuators of multilink manipulators using logic-dynamic approach.

[15] T. Floquet, C. Edwards, and S. Spurgeon: On sliding mode observers for systems with unknown inputs.

[16] L. Fridman, A. Levant, and J. Davila: Observation of linear systems with unknown inputs via high-order sliding-modes.

[17] L. Fridman, Yu. Shtessel, C. Edwards, and X. Yan: High-order slidingmode observer for state estimation and input reconstruction in nonlinear systems.

[18] R. Hmidi, A. Brahim, F. Hmida, and A. Sellami: Robust fault tolerant control design for nonlinear systems not satisfying matching and minimum phase conditions.

[19] H. Rios, D. Efimov, J. Davila, T. Raissi, L. Fridman, and A. Zolghadri: Non-minimum phase switched systems: HOSM based fault detection and fault identification via Volterra integral equation.

[20] I. Samy, I. Postlethwaite, and D. Gu: Survey and application of sensor fault detection and isolation schemes.

[21] C. Tan and C. Edwards: Sliding mode observers for robust detection and reconstruction of actuator and sensor faults.

[22] C. Tan and C. Edwards: Robust fault reconstruction using multiple sliding mode observers in cascade: development and design.

[23] V. Utkin:

[24] X. Wang, C. Tan, and G. Zhou: A novel sliding mode observer for state and fault estimation in systems not satisfying matching and minimum phase conditions.

[25] X. Yan and C. Edwards: Nonlinear robust fault reconstruction and estimation using a sliding modes observer.

[26] J. Yang, F. Zhu, and X. Sun: State estimation and simultaneous unknown input and measurement noise reconstruction based on associated observers.

[27] A. Zhirabok: Nonlinear parity relation: A logic-dynamic approach.

[28] A. Zhirabok, A. Shumsky, and S. Pavlov: Diagnosis of linear dynamic systems by the nonparametric method.

[29] A. Zhirabok, A. Shumsky, S. Solyanik, and A. Suvorov: Fault detection in nonlinear systems via linear methods.

[30] A. Zhirabok, A. Zuev, and A. Shumsky: Methods of diagnosis in linear systems based on sliding mode observers.

[31] A. Zhirabok, A. Zuev, andV. Filaretov: Fault identification in underwater vehicle thrusters via sliding mode observers.

Keywords:
optimal control
aerospace applications
nonlinear systems
mechanical/mechatronics applications
robust control

In this paper, model reference output feedback tracking control of an aircraft subject to additive, uncertain, nonlinear disturbances is considered. In order to present the design steps in a clear fashion: first, the aircraft dynamics is temporarily assumed as known with all the states of the system available. Then a feedback linearizing controller minimizing a performance index while only requiring the output measurements of the system is proposed. As the aircraft dynamics is uncertain and only the output is available, the proposed controller makes use of a novel uncertainty estimator. The stability of the closed loop system and global asymptotic tracking of the proposed method are ensured via Lyapunov based arguments, asymptotic convergence of the controller to an optimal controller is also established. Numerical simulations are presented in order to demonstrate the feasibility and performance of the proposed control strategy.

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[31] I. Tanyer, E. Tatlicioglu, and E. Zergeroglu: Neural network based robust control of an aircraft.*Int. J. of Robotics& Automation*, 35(1), (2020), DOI: 10.2316/J.2020.206-0074.

[32] I. Tanyer, E. Tatlicioglu, E. Zergeroglu, M. Deniz, A. Bayrak, and B. Ozdemirel: Robust output tracking control of an unmanned aerial vehicle subject to additive state-dependent disturbance.*IET Control Theory & Applications*, 10(14), (2016), 1612–1619, DOI: 10.1049/iet-cta.2015.1304.

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Go to article
[2] B. Bidikli, E. Tatlicioglu, E. Zergeroglu, and A. Bayrak: An asymptotically stable robust controller formulation for a class of MIMO nonlinear systems with uncertain dynamics.

[3] B. Bidikli, E. Tatlicioglu, A. Bayrak, and E. Zergeroglu: A new robust integral of sign of error feedback controller with adaptive compensation gain. In

[4] B. Bidikli, E. Tatlicioglu, and E. Zergeroglu: A self tuning RISE controller formulation. In

[5] M. Bouchoucha, M. Tadjine, A. Tayebi, P. Mullhaupt, and S. Bouab- dallah: Robust nonlinear pi for attitude stabilization of a four-rotor miniaircraft: From theory to experiment.

[6] A.E. Bryson and Yu-Chi Ho:

[7] Agus Budiyono and Singgih S. Wibowo: Optimal tracking controller design for a small scale helicopter.

[8] Y.N. Chelnokov, I.A. Pankratov, and Y.G. Sapunkov: Optimal reorientation of spacecraft orbit.

[9] W.-H. Chen, D.J. Ballance, P.J. Gawthrop, and J. O’Reilly: A nonlinear disturbance observer for robotic manipulators.

[10] R. Czyba and L. Stajer: Dynamic contraction method approach to digital longitudinal aircraft flight controller design.

[11] Z.T. Dydek, A.M. Annaswamy, and E. Lavretsky: Adaptive control and the NASA X-15-3 flight revisited.

[12] E.N. Johnson and A.J. Calise: Pseudo-control hedging: a new method for adaptive control. In

[13] H.K. Khalil and J.W. Grizzle:

[14] D.E. Kirk:

[15] L.-V. Lai, C.-C. Yang, and C.-J. Wu: Time-optimal control of a hovering quadrotor helicopter.

[16] J. Leitner, A. Calise, and JV.R. Prasad: Analysis of adaptive neural networks for helicopter flight control.

[17] F.L. Lewis, D. Vrabie, and V.L. Syrmos:

[18] W. MacKunis:

[19] W. MacKunis, P.M. Patre, M.K. Kaiser, and W.E. Dixon: Asymptotic tracking for aircraft via robust and adaptive dynamic inversion methods.

[20] S. Mishra, T. Rakstad, andW. Zhang: Robust attitude control for quadrotors based on parameter optimization of a nonlinear disturbance observer.

[21] R.M. Murray: Recent research in cooperative control of multivehicle systems.

[22] D. Nodland, H. Zargarzadeh, and S. Jagannathan: Neural networkbased optimal adaptive output feedback control of a helicopter UAV.

[23] A. Phillips and F. Sahin: Optimal control of a twin rotor MIMO system using LQR with integral action. In

[24] Federal Aviation Administration. Federal aviation regulations. part 25: Airworthiness standards: Transport category airplanes, 2002.

[25] R.R. Costa, L. Hsu, A.K. Imai, and P. Kokotovic: Lyapunov-based adaptive control ofMIMOsystems.

[26] A.C. Satici, H. Poonawala, and M.W. Spong: Robust optimal control of quadrotor UAVs.

[27] R.F. Stengel:

[28] V. Stepanyan and A. Kurdila: Asymptotic tracking of uncertain systems with continuous control using adaptive bounding.

[29] B.L. Stevens and F.L. Lewis:

[30] I. Tanyer, E. Tatlicioglu, and E. Zergeroglu: A robust adaptive tracking controller for an aircraft with uncertain dynamical terms. In

[31] I. Tanyer, E. Tatlicioglu, and E. Zergeroglu: Neural network based robust control of an aircraft.

[32] I. Tanyer, E. Tatlicioglu, E. Zergeroglu, M. Deniz, A. Bayrak, and B. Ozdemirel: Robust output tracking control of an unmanned aerial vehicle subject to additive state-dependent disturbance.

[33] G. Tao:

[34] Q. Wang and R.F. Stengel: Robust nonlinear flight control of a highperformance aircraft.

[35] H-N. Wu, M-M. Li, and L. Guo: Finite-horizon approximate optimal guaranteed cost control of uncertain nonlinear systems with application to Mars entry guidance.

[36] Q. Xie, B. Luo, F. Tan, and X. Guan: Optimal control for vertical take-off and landing aircraft non-linear system by online kernel-based dual heuristic programming learning.

Keywords:
non integer order systems
heat transfer equation
finite difference
Caputo operator
positive systems

The paper proposes a new, state space, finite dimensional, fractional order model of a heat transfer in one dimensional body. The time derivative is described by Caputo operator. The second order central difference describes the derivative along the length. The analytical formulae of the model responses are proved. The stability, convergence, and positivity of the model are also discussed. Theoretical results are verified by experiments.

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[22] K. Oprzedkiewicz: Fractional order, discrete model of heat transfer process using time and spatial Grünwald-Letnikov operator.*Bulletin of the Polish Academy of Sciences. Technical Sciences*, 69(1), (2021), 1–10, DOI: 10.24425/bpasts.2021.135843.

[23] K. Oprzedkiewicz, K. Dziedzic, and Ł. Wi˛ eckowski: Non integer order, discrete, state space model of heat transfer process using Grünwald-Letnikov operator.*Bulletin of the Polish Academy of Sciences. Technical Sciences*, 67(5), (2019), 905–914, DOI: 10.24425/bpasts.2019.130873.

[24] K. Oprzedkiewicz and E. Gawin: A non-integer order, state space model for one dimensional heat transfer process.*Archives of Control Sciences*, 26(2), (2016), 261–275, DOI: 10.1515/acsc-2016-0015.

[25] K. Oprzedkiewicz, E. Gawin, and W. Mitkowski: Modeling heat distribution with the use of a non-integer order, state space model.*International Journal of Applied Mathematics and Computer Science*, 26(4), (2016), 749– 756, DOI: 10.1515/amcs-2016-0052.

[26] K. Oprzedkiewicz and W. Mitkowski: A memory-efficient nonintegerorder discrete-time state-space model of a heat transfer process.*International Journal of Applied Mathematics and Computer Science*, 28(4), (2018), 649–659, DOI: 10.2478/amcs-2018-0050.

[27] K. Oprzedkiewicz,W. Mitkowski, E.Gawin, and K. Dziedzic: The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process.*Bulletin of the Polish Academy of Sciences. Technical Sciences,* 66(4), (2018), 501– 507, DOI: 10.24425/124267.

[28] K. Oprzedkiewicz, E. Gawin, and W. Mitkowski: Parameter identification for non integer order, state space models of heat plant. In*MMAR 2016: 21th international conference on Methods and Models in Automation and Robotics: 29 August–01 September 2016, Międzyzdroje, Poland*, pages 184– 188, 2016.

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[32] M. Rozanski: Determinants of two kinds of matrices whose elements involve sine functions.*Open Mathematics,* 17(1), (2019), 1332–1339, DOI: 10.1515/math-2019-0096.

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[2] A. Atangana and D. Baleanu: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer.

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[5] M. Dlugosz and P. Skruch: The application of fractional-order models for thermal process modelling inside buildings.

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[11] A. Kochubei: Fractional-parabolic systems, preprint, arxiv:1009.4996 [math.ap], 2011.

[12] W. Mitkowski: Approximation of fractional diffusion-wave equation.

[13] W. Mitkowski: Finite-dimensional approximations of distributed rc networks.

[14] W. Mitkowski,W. Bauer, and M. Zagorowska: Rc-ladder networks with supercapacitors.

[15] K. Oprzedkiewicz: The discrete-continuous model of heat plant.

[16] K. Oprzedkiewicz: The interval parabolic system.

[17] K. Oprzedkiewicz:Acontrollability problem for a class of uncertain parameters linear dynamic systems.

[18] K. Oprzedkiewicz: An observability problem for a class of uncertainparameter linear dynamic systems.

[19] K. Oprzedkiewicz:Non integer order, state space model of heat transfer process using Caputo-Fabrizio operator.

[20] K. Oprzedkiewicz: Non integer order, state space model of heat transfer process using Atangana-Baleanu operator.

[21] K. Oprzedkiewicz: Positivity problem for the one dimensional heat transfer process.

[22] K. Oprzedkiewicz: Fractional order, discrete model of heat transfer process using time and spatial Grünwald-Letnikov operator.

[23] K. Oprzedkiewicz, K. Dziedzic, and Ł. Wi˛ eckowski: Non integer order, discrete, state space model of heat transfer process using Grünwald-Letnikov operator.

[24] K. Oprzedkiewicz and E. Gawin: A non-integer order, state space model for one dimensional heat transfer process.

[25] K. Oprzedkiewicz, E. Gawin, and W. Mitkowski: Modeling heat distribution with the use of a non-integer order, state space model.

[26] K. Oprzedkiewicz and W. Mitkowski: A memory-efficient nonintegerorder discrete-time state-space model of a heat transfer process.

[27] K. Oprzedkiewicz,W. Mitkowski, E.Gawin, and K. Dziedzic: The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process.

[28] K. Oprzedkiewicz, E. Gawin, and W. Mitkowski: Parameter identification for non integer order, state space models of heat plant. In

[29] P. Ostalczyk:

[30] I. Podlubny:

[31] G. Recktenwald:

[32] M. Rozanski: Determinants of two kinds of matrices whose elements involve sine functions.

[33] N. Al Salti, E. Karimov, and S. Kerbal: Boundary-value problems for fractional heat equation involving caputo-fabrizio derivative.

[34] D. Sierociuk, T. Skovranek, M. Macias, I. Podlubny, I. Petras, A. Dzielinski, and P. Ziubinski: Diffusion process modeling by using fractional-order models.

Keywords:
Navier–Stokes equations
hydrodynamic
approximations
mathematical models
incompressible melt

The article presents "-approximation of hydrodynamics equations’ stationary model along with the proof of a theorem about existence of a hydrodynamics equations’ strongly generalized solution. It was proved by a theorem on the existence of uniqueness of the hydrodynamics equations’ temperature model’s solution, taking into account energy dissipation. There was implemented the Galerkin method to study the Navier–Stokes equations, which provides the study of the boundary value problems correctness for an incompressible viscous flow both numerically and analytically. Approximations of stationary and non-stationary models of the hydrodynamics equations were constructed by a system of Cauchy–Kovalevsky equations with a small parameter ". There was developed an algorithm for numerical modelling of the Navier– Stokes equations by the finite difference method.

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Go to article
[2] M.R. Malik, T.A. Zang, and M.Y. Hussaini:Aspectral collocation method for the Navier–Stokes equations.

[3] P.M. Gresho: Incompressible fluid dynamics: some fundamental formulation issues.

[4] R. Lakshminarayana, K. Dadzie, R. Ocone, M. Borg, and J. Reese: Recasting Navier–Stokes equations.

[5] S.Sh. Kazhikenova, S.N. Shaltakov, D. Belomestny, and G.S. Shai- hova: Finite difference method implementation for numerical integration hydrodynamic equations melts.

[6] O.A. Ladijenskaya:

[7] Z.R. Safarova: On a finding the coefficient of one nonlinear wave equation in the mixed problem.

[8] A. Abramov and L.F. Yukhno: Solving some problems for systems of linear ordinary differential equations with redundant conditions.

[9] K. Yasumasa and T. Takahico: Finite-element method for three-dimensional incompressible viscous flow using simultaneous relaxation of velocity and Bernoulli function. 1st report flow in a lid-driven cubic cavity at Re = 5000.

[10] H. Itsuro, Î. Hideki, T. Yuji, and N. Tetsuji: Numerical analysis of a flow in a three-dimensional cubic cavity.

[11] X. Yan, L. Wei, Y. Lei, X. Xue, Y.Wang, G. Zhao, J. Li, and X. Qingyan: Numerical simulation of Meso-Micro structure in Ni-based superalloy during liquid metal cooling.

[12] T.A. Barannyk, A.F. Barannyk, and I.I. Yuryk: Exact Solutions of the nonliear equation.

[13] S. Tleugabulov, D. Ryzhonkov, N. Aytbayev, G. Koishina, and G. Sul- tamurat: The reduction smelting of metal-containing industrial wastes.

[14] S.L. Skorokhodov and N.P. Kuzmina: Analytical-numerical method for solving an Orr–Sommerfeld-type problem for analysis of instability of ocean currents.

[15] N.B. Iskakova, A.T. Assanova, and E.A. Bakirova: Numerical method for the solution of linear boundary-value problem for integrodifferential equations based on spline approximations.

[16] S.Sh. Kazhikenova, M.I. Ramazanov, and A.A. Khairkulova: epsilon- Approximation of the temperatures model of inhomogeneous melts with allowance for energy dissipation.

[17] J.A. Iskenderova and Sh. Smagulov: The Cauchy problem for the equations of a viscous heat-conducting gas with degenerate density.

[18] A.M. Molchanov:

[19] Y. Achdou and J.-L. Guermond: Convergence Analysis of a finite element projection / Lagrange-Galerkin method for the incompressible Navier–Stokes equations.

[20] M.P. de Carvalho, V.L. Scalon, and A. Padilha: Analysis of CBS numerical algorithm execution to flow simulation using the finite element method.

[21] G. Muratova, T. Martynova, E. Andreeva, V. Bavin, and Z-Q. Wang: Numerical solution of the Navier–Stokes equations using multigrid methods with HSS-based and STS-based smoother.

[22] M. Rosenfeld and M. Israeli: Numerical solution of incompressible flows by a marching multigrid nonlinear method.

5
Synchronization of FitzHugh-Nagumo reaction-diffusion systems via one-dimensional linear control law

Keywords:
FitzHugh-Nagumo
synchronization
uni-dimensional control
linear control
reaction-diffusion system
neuronal networks
Lyapunov’s second method

The Fitzhugh-Nagumo model (FN model), which is successfully employed in modeling the function of the so-called membrane potential, exhibits various formations in neuronal networks and rich complex dynamics. This work deals with the problem of control and synchronization of the FN reaction-diffusion model. The proposed control law in this study is designed to be uni-dimensional and linear law for the purpose of reducing the cost of implementation. In order to analytically prove this assertion, Lyapunov’s second method is utilized and illustrated numerically in one- and/or two-spatial dimensions.

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[3] B. Ambrosio, M.A. Aziz-Alaoui, and V.L.E. Phan: Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type.*Discrete & Continuous Dynamical Systems*, 23(9), (2018), 3787–3797, DOI: 10.3934/dcdsb.2018077.

[4] B. Ambrosio, M. A. Aziz-Alaoui, and V.L.E. Phan: Large time behaviour and synchronization of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type.*IMA Journal of Applied Mathematics, *84(2), (2019), 416–443, DOI: 10.1093/imamat/hxy064.

[5] M. Aqil, K.-S. Hong, and M.-Y. Jeong: Synchronization of coupled chaotic FitzHugh–Nagumo systems.*Communications in Nonlinear Science and Numerical Simulation*, 17(4), (2012), 1615–1627, DOI: 10.1016/j.cnsns. 2011.09.028.

[6] S. Bendoukha, S. Abdelmalek, and M. Kirane: The global existence and asymptotic stability of solutions for a reaction–diffusion system.*Nonlinear Analysis: Real World Applications*. 53, (2020), 103052, DOI: 10.1016/j.nonrwa.2019.103052.

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[16] F. Mesdoui, N. Shawagfeh, and A. Ouannas: Global synchronization of fractional-order and integer-order N component reaction diffusion systems: Application to biochemical models.*Mathematical Methods in the Applied Sciences*, 44(1), (2021), 1003–1012, DOI: 10.1002/mma.6807.

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[22] A. Ouannas, Z. Odibat, N. Shawagfeh, A. Alsaedi, and B. Ahmad: Universal chaos synchronization control laws for general quadratic discrete systems.*Applied Mathematical Modelling*, 45 (2017), 636–641, DOI: 10.1016/j.apm.2017.01.012.

[23] A. Ouannas, Z. Odibat, and N. Shawagfeh: A new Q–S synchronization results for discrete chaotic systems.*Differential Equations and Dynamical Systems*, 27(4), (2019), 413–422, DOI: 10.1007/s12591-016-0278-x.

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[33] K.-N. Wu, T. Tian, and L. Wang: Synchronization for a class of coupled linear partial differential systems via boundary control.*Journal of the Franklin Institute*, 353(16), (2016), 4062–4073, DOI: 10.1016/ j.jfranklin.2016.07.019.

Go to article
[2] B. Ambrosio and M.A. Aziz-Alaoui: Synchronization and control of coupled reaction–diffusion systems of the FitzHugh–Nagumo type.

[3] B. Ambrosio, M.A. Aziz-Alaoui, and V.L.E. Phan: Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type.

[4] B. Ambrosio, M. A. Aziz-Alaoui, and V.L.E. Phan: Large time behaviour and synchronization of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type.

[5] M. Aqil, K.-S. Hong, and M.-Y. Jeong: Synchronization of coupled chaotic FitzHugh–Nagumo systems.

[6] S. Bendoukha, S. Abdelmalek, and M. Kirane: The global existence and asymptotic stability of solutions for a reaction–diffusion system.

[7] X.R. Chen and C.X. Liu: Chaos synchronization of fractional order unified chaotic system via nonlinear control.

[8] D. Eroglu, J.S.W. Lamb, and Y. Pereira: Synchronisation of chaos and its applications.

[9] R. Fitzhugh: Thresholds and Plateaus in the Hodgkin-Huxley Nerve Equations.

[10] P.Garcia, A.Acosta, and H. Leiva: Synchronization conditions for masterslave reaction diffusion systems

[11] A.L. Hodgkin and A.F. Huxley: A quantitative description of membrane current and its application to conduction and excitation in nerve.

[12] T. Kapitaniak: Continuous control and synchronization in chaotic systems.

[13] A.C.J. Luo:

[14] D. Mansouri, S. Bendoukha, S. Abdelmalek, and A. Youkana: On the complete synchronization of a time-fractional reaction–diffusion system with the Newton–Leipnik nonlinearity.

[15] F. Mesdoui, A. Ouannas, N. Shawagfeh, G. Grassi, and V.-T. Pham: Synchronization Methods for the Degn-Harrison Reaction-Diffusion Systems.

[16] F. Mesdoui, N. Shawagfeh, and A. Ouannas: Global synchronization of fractional-order and integer-order N component reaction diffusion systems: Application to biochemical models.

[17] J. Nagumo, S. Arimoto, and S. Yoshizawa: An active pulse transmission line simulating nerve axon.

[18] L.H. Nguyen and K.-S. Hong: Synchronization of coupled chaotic FitzHugh–Nagumo neurons via Lyapunov functions.

[19] Z.M. Odibat: Adaptive feedback control and synchronization of nonidentical chaotic fractional order systems.

[20] Z.M. Odibat, N. Corson, M.A. Aziz-Alaoui, and C. Bertelle: Synchronization of chaotic fractional-order systems via linear control.

[21] A. Ouannas, M. Abdelli, Z. Odibat, X. Wang, V.-T. Pham, G. Grassi, and A. Alsaedi:

[22] A. Ouannas, Z. Odibat, N. Shawagfeh, A. Alsaedi, and B. Ahmad: Universal chaos synchronization control laws for general quadratic discrete systems.

[23] A. Ouannas, Z. Odibat, and N. Shawagfeh: A new Q–S synchronization results for discrete chaotic systems.

[24] N. Parekh, V.R. Kumar, and B.D. Kulkarni: Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system. Pramana –

[25] L.M. Pecora and T.L. Carroll: Synchronization in chaotic systems.

[26] M. Srivastava, S.P. Ansari, S.K. Agrawal, S. Das, and A.Y.T. Le- ung: Anti-synchronization between identical and non-identical fractionalorder chaotic systems using active control method.

[27] J. Wang, T. Zhang, and B. Deng: Synchronization of FitzHugh–Nagumo neurons in external electrical stimulation via nonlinear control.

[28] J. Wang, Z. Zhang, and H. Li: Synchronization of FitzHugh–Nagumo systems in EES via H1 variable universe adaptive fuzzy control.

[29] L. Wang and H. Zhao: Synchronized stability in a reaction–diffusion neural network model.

[30] J. Wei and M. Winter: Standingwaves in the FitzHugh-Nagumo system and a problem in combinatorial geometry.

[31] X. Wei, J.Wang, and B. Deng: Introducing internal model to robust output synchronization of FitzHugh–Nagumo neurons in external electrical stimulation.

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Keywords:
continuous interval systems
Kharitonov polynomials
Routh approximation
modelling
SISO systems
MIMO systems

In recent, modeling practical systems as interval systems is gaining more attention of control researchers due to various advantages of interval systems. This research work presents a new approach for reducing the high-order continuous interval system (HOCIS) utilizing improved Gamma approximation. The denominator polynomial of reduced-order continuous interval model (ROCIM) is obtained using modified Routh table, while the numerator polynomial is derived using Gamma parameters. The distinctive features of this approach are: (i) It always generates a stable model for stable HOCIS in contrast to other recent existing techniques; (ii) It always produces interval models for interval systems in contrast to other relevant methods, and, (iii) The proposed technique can be applied to any system in opposite to some existing techniques which are applicable to second-order and third-order systems only. The accuracy and effectiveness of the proposed method are demonstrated by considering test cases of single-inputsingle- output (SISO) and multi-input-multi-output (MIMO) continuous interval systems. The robust stability analysis for ROCIM is also presented to support the effectiveness of proposed technique.

Go to article
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[27] A. Abdelhak and M. Rachik: Model reduction problem of linear discrete systems: Admissibles initial states.*Archives of Control Sciences,* 29(1), (2019), 41–55, DOI: 10.24425/acs.2019.127522.

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[37] M.S. Kumar, N.V. Anand, and R.S. Rao: Impulse energy approximation of higher-order interval systems using Kharitonov’s polynomials.*Transactions of the Institute of Measurement and Control, *38(10), (2016), 1225–1235, DOI: 10.1177/0142331215583326.

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[40] M. Sharma, A. Sachan and D. Kumar: Order reduction of higher order interval systems by stability preservation approach. In*2014 International Conference on Power, Control and Embedded Systems* (ICPCES), (2014), 1–6.

[41] G. Sastry and P.M. Rao: A new method for modelling of large scale interval systems. IETE Journal of Research, 49(6), (2003), 423–430, DOI: 10.1080/03772063.2003.11416366.

Go to article
[2] A. Gupta, R. Saini, and M. Sharma: Modelling of hybrid energy system— part i: Problem formulation and model development.

[3] S. Singh, V. Singh, and V. Singh: Analytic hierarchy process based approximation of high-order continuous systems using tlbo algorithm.

[4] J. Hu, Y. Yang, M. Jia, Y. Guan, C. Fu, and S. Liao: Research on harmonic torque reduction strategy for integrated electric drive system in pure electric vehicle.

[5] K. Takahashi, N. Jargalsaikhan, S. Rangarajan, A. M. Hemeida, H. Takahashi and T. Senjyu: Output control of three-axis pmsg wind turbine considering torsional vibration using h infinity control.

[6] V. Singh, D.P.S. Chauhan, S.P. Singh, and T. Prakash: On time moments and markov parameters of continuous interval systems.

[7] B. Pariyar and R.Wagle: Mathematical modeling of isolated wind-dieselsolar photo voltaic hybrid power system for load frequency control. arXiv preprint arXiv:2004.05616, (2020).

[8] N. Karkar, K. Benmhammed, and A. Bartil: Parameter estimation of planar robot manipulator using interval arithmetic approach.

[9] F.P.G. Marquez: A new method for maintenance management employing principal component analysis.

[10] F.P.G. Marquez: An approach to remote condition monitoring systems management.

[11] D. Li, S. Zhang, andY. Xiao: Interval optimization-based optimal design of distributed energy resource systems under uncertainties.

[12] A.K. Choudhary and S.K. Nagar: Order reduction in z-domain for interval system using an arithmetic operator.

[13] A.K. Choudhary and S.K. Nagar: Order reduction techniques via routh approximation: a critical survey.

[14] V.P. Singh and D. Chandra: Model reduction of discrete interval system using dominant poles retention and direct series expansion method. In

[15] V. Singh and D. Chandra: Reduction of discrete interval system using clustering of poles with Padé approximation: a computer-aided approach.

[16] Y. Dolgin and E. Zeheb: On Routh-Pade model reduction of interval systems.

[17] S.F. Yang: Comments on “On Routh-Pade model reduction of interval systems”.

[18] Y. Dolgin: Author’s reply [to comments on ‘On Routh-Pade model reduction of interval systems’

[19] B. Bandyopadhyay, O. Ismail, and R. Gorez: Routh-Pade approximation for interval systems.

[20] Y.V. Hote, A.N. Jha, and J.R. Gupta: Reduced order modelling for some class of interval systems.

[21] B. Bandyopadhyay, A. Upadhye, and O. Ismail: /spl gamma/-/spl delta/routh approximation for interval systems.

[22] J. Bokam, V. Singh, and S. Raw: Comments on large scale interval system modelling using routh approximants.

[23] G. Sastry, G.R. Rao, and P.M. Rao: Large scale interval system modelling using Routh approximants.

[24] M.S. Kumar and G. Begum: Model order reduction of linear time interval system using stability equation method and a soft computing technique.

[25] S.R. Potturu and R. Prasad: Qualitative analysis of stable reduced order models for interval systems using mixed methods.

[26] N. Vijaya Anand, M. Siva Kumar, and R. Srinivasa Rao: A novel reduced order modeling of interval system using soft computing optimization approach.

[27] A. Abdelhak and M. Rachik: Model reduction problem of linear discrete systems: Admissibles initial states.

[28] M. Buslowicz: Robust stability of a class of uncertain fractional order linear systems with pure delay.

[29] S.R. Potturu and R. Prasad: Model order reduction of LTI interval systems using differentiation method based on Kharitonov’s theorem.

[30] E.-H. Dulf: Simplified fractional order controller design algorithm.

[31] Y. Menasria, H. Bouras, and N. Debbache: An interval observer design for uncertain nonlinear systems based on the ts fuzzy model.

[32] A. Khan, W. Xie, L. Zhang, and Ihsanullah: Interval state estimation for linear time-varying (LTV) discrete-time systems subject to component faults and uncertainties.

[33] N. Akram, M. Alam, R. Hussain, A. Ali, S. Muhammad, R. Malik, and A.U. Haq: Passivity preserving model order reduction using the reduce norm method.

[34] K. Kumar Deveerasetty and S. Nagar: Model order reduction of interval systems using an arithmetic operation.

[35] K.K. Deveerasetty,Y. Zhou, S. Kamal, and S.K.Nagar: Computation of impulse-response gramian for interval systems.

[36] P. Dewangan, V. Singh, and S. Sinha: Improved approximation for SISO and MIMO continuous interval systems ensuring stability.

[37] M.S. Kumar, N.V. Anand, and R.S. Rao: Impulse energy approximation of higher-order interval systems using Kharitonov’s polynomials.

[38] S.K. Mangipudi and G. Begum: A new biased model order reduction for higher order interval systems.

[39] V.L. Kharitonov: The asymptotic stability of the equilibrium state of a family of systems of linear differential equations.

[40] M. Sharma, A. Sachan and D. Kumar: Order reduction of higher order interval systems by stability preservation approach. In

[41] G. Sastry and P.M. Rao: A new method for modelling of large scale interval systems. IETE Journal of Research, 49(6), (2003), 423–430, DOI: 10.1080/03772063.2003.11416366.

Keywords:
planning
conformant planning
conditional planning
parallel planning
uncertainty
linear programming
computational complexity

Classical planning in Artificial Intelligence is a computationally expensive problem of finding a sequence of actions that transforms a given initial state of the problem to a desired goal situation. Lack of information about the initial state leads to conditional and conformant planning that is more difficult than classical one. A parallel plan is the plan in which some actions can be executed in parallel, usually leading to decrease of the plan execution time but increase of the difficulty of finding the plan. This paper is focused on three planning problems which are computationally difficult: conditional, conformant and parallel conformant. To avoid these difficulties a set of transformations to Linear Programming Problem (LPP), illustrated by examples, is proposed. The results show that solving LPP corresponding to the planning problem can be computationally easier than solving the planning problem by exploring the problem state space. The cost is that not always the LPP solution can be interpreted directly as a plan.

Go to article
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Go to article
[2] Ch. Backstrom: Computational Aspects of Reordering Plans.

[3] Ch. Baral, V. Kreinovich, and R. Trejo: Computational complexity of planning and approximate planning in the presence of incompleteness.

[4] R. Bartak: Constraint satisfaction techniques in planning and scheduling: An introduction.

[5] A. Bhattacharya and P. Vasant: Soft-sensing of level of satisfaction in TOC product-mix decision heuristic using robust fuzzy-LP,

[6] J. Blythe: An Overview of Planning Under Uncertainty.

[7] T. Bylander: The Computational Complexity of Propositional STRIPS Planning.

[8] T. Bylander: A Linear Programming Heuristic for Optimal Planning. In

[9] L.G. Chaczijan: A polynomial algorithm for linear programming.

[10] E.R. Dougherty and Ch.R. Giardina:

[11] I. Elamvazuthi, P. Vasant, and T. Ganesan: Fuzzy Linear Programming using Modified Logistic Membership Function,

[12] A. Galuszka: On transformation of STRIPS planning to linear programming.

[13] A. Galuszka, W. Ilewicz, and A. Olczyk: On Translation of Conformant Action Planning to Linear Programming.

[14] A. Galuszka, T. Grzejszczak, J. Smieja, A. Olczyk, and J. Kocerka: On parallel conformant planning as an optimization problem.

[15] M. Ghallab et al.:

[16] A. Grastien and E. Scala: Sampling Strategies for Conformant Planning.

[17] A. Grastien and E. Scala: CPCES: A planning framework to solve conformant planning problems through a counterexample guided refinement.

[18] D. Hoeller, G. Behnke, P. Bercher, S. Biundo, H. Fiorino, D. Pellier, and R. Alford: HDDL: An extension to PDDL for expressing hierarchical planning problems.

[19] J. Koehler and K. Schuster:

[20] R. van der. Krogt: Modification strategies for SAT-based plan adaptation.

[21] M.D. Madronero, D. Peidro, and P. Vasant: Vendor selection problem by using an interactive fuzzy multi-objective approach with modified s-curve membership functions.

[22] A. Nareyek, C. Freuder, R. Fourer, E. Giunchiglia, R.P. Goldman, H. Kautz, J. Rintanen, and A. Tate: Constraitns and AI Planning.

[23] N.J. Nilson:

[24] E.P.D. Pednault: ADL and the state-transition model of action.

[25] D. Peidro and P. Vasant: Transportation planning with modified scurve membership functions using an interactive fuzzy multi-objective approach,

[26] F. Pommerening, G. Roger, M. Helmert, H. Cambazard, L.M. Rousseau, and D. Salvagnin: Lagrangian decomposition for classical planning.

[27] T. Rosa, S. Jimenez, R. Fuentetaja, and D. Barrajo: Scaling up heuristic planning with relational decision trees.

[28] S.J. Russell and P. Norvig:

[29] J. Seipp, T. Keller, and M. Helmert: Saturated post-hoc optimization for classical planning.

[30] D.E. Smith and D.S. Weld: Conformant Graphplan. Proc. 15th National Conf. on AI, (1998).

[31] D.S. Weld: Recent Advantages in AI Planning.

[32] D.S. Weld, C.R. Anderson, and D.E. Smith: Extending graphplan to handle uncertainty & sensing actions.

[33] X. Zhang, A. Grastien, and E. Scala: Computing superior counterexamples for conformant planning.

In this paper,we start by the research of the existence of Lyapunov homogeneous function for a class of homogeneous fractional Systems, then we shall prove that local and global behaviors are the same. The uniform Mittag-Leffler stability of homogeneous fractional time-varying systems is studied. A numerical example is given to illustrate the efficiency of the obtained results.

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[1] V. Andrieu, L. Praly, and A. Astolfi: Homogeneous approximation, recursive observer design, and output feedback. *SIAM Journal on Control and Optimization*, 47(4), (2008), 1814–1850, DOI: 10.1137/060675861.

[2] A. Bacciotti and L. Rosier: Liapunov Functions and Stability in Control Theory.*Lecture Notes in Control and Inform. Sci*, 267 (2001), DOI: 10.1007/b139028.

[3] K. Diethelm:*The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type*. Series on Complexity, Nonlinearity and Chaos, Springer, Heidelberg, 2010.

[4] M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, and R. Castro-Linares: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems.*Commun. Nonlinear Sci. Numer. Simul*., 22(1-3) (2015), 650–659, DOI: 10.1016/j.cnsns. 2014.10.008.

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[7] Y. Li, Y. Chen, and I. Podlubny: Stability of fractional-order nonlinear dynamic system: Lyapunov direct method and generalized Mittag- Leffler stability.*Comput. Math. Appl,* 59(5) (2010), 1810–1821, DOI: 10.1016/j.camwa.2009.08.019.

[8] Y. Li, Y. Chen, and I. Podlubny: Mittag-Leffler stability of fractional order nonlinear dynamic systems.*Automatica,* 45 (2009), 1965–1969, DOI: 10.1140/epjst/e2011-01379-1.

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[10] T. Menard, E. Moulay, and W. Perruquetti: Homogeneous approximations and local observer design.*ESAIM: Control, Optimization and Calculus of Variations*, 19 (2013), 906–929, DOI: 10.1051/cocv/2012038.

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[12] I. Podlubny:*Fractional Differential Equations.* Mathematics in Sciences and Engineering. Academic Press, San Diego, 1999.

[13] H. Rios, D. Efmov, L. Fridman, J. Moreno, and W. Perruquetti: Homogeneity based uniform stability analysis for time-varying systems.*IEEE Transactions on automatic control*, 61(3), (2016), 725–734, DOI: 10.1109/TAC.2015.2446371.

[14] R. Rosier: Homogeneous Lyapunov function for homogeneous continuous vector field.*System & Control Letters*, 19 (1992), 467–473, DOI: 10.1016/0167-6911(92)90078-7.

[15] H.T. Tuan and H. Trinh: Stability of fractional-order nonlinear systems by Lyapunov direct method.*IET Control Theory Appl,* 12 (2018), DOI: 10.1049/ict-cta.2018.5233.

[16] F. Zhang, C. Li, and Y.Q. Chen: Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int. J. Differ. Equ., (2011), 1–12, DOI: 10.1155/2011/635165.

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[2] A. Bacciotti and L. Rosier: Liapunov Functions and Stability in Control Theory.

[3] K. Diethelm:

[4] M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, and R. Castro-Linares: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems.

[5] H. Hermes: Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. In:

[6] H. Hermes: Nilpotent and high-order approximations of vector field systems.

[7] Y. Li, Y. Chen, and I. Podlubny: Stability of fractional-order nonlinear dynamic system: Lyapunov direct method and generalized Mittag- Leffler stability.

[8] Y. Li, Y. Chen, and I. Podlubny: Mittag-Leffler stability of fractional order nonlinear dynamic systems.

[9] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo:

[10] T. Menard, E. Moulay, and W. Perruquetti: Homogeneous approximations and local observer design.

[11] K.B. Oldham and J. Spanier:

[12] I. Podlubny:

[13] H. Rios, D. Efmov, L. Fridman, J. Moreno, and W. Perruquetti: Homogeneity based uniform stability analysis for time-varying systems.

[14] R. Rosier: Homogeneous Lyapunov function for homogeneous continuous vector field.

[15] H.T. Tuan and H. Trinh: Stability of fractional-order nonlinear systems by Lyapunov direct method.

[16] F. Zhang, C. Li, and Y.Q. Chen: Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int. J. Differ. Equ., (2011), 1–12, DOI: 10.1155/2011/635165.

Keywords:
null-controllability
mobile control
nonlinear constraints
triangular wave
rectangular wave
Green’s function approach
heuristic control
lack of exact controllability

The Green’s function approach is applied for studying the exact and approximate nullcontrollability of a finite rod in finite time by means of a source moving along the rod with controllable trajectory. The intensity of the source remains constant. Applying the recently developed Green’s function approach, the analysis of the exact null-controllability is reduced to an infinite system of nonlinear constraints with respect to the control function. A sufficient condition for the approximate null-controllability of the rod is obtained. Since the exact solution of the system of constraints is a long-standing open problem, some heuristic solutions are used instead. The efficiency of these solutions is shown on particular cases of approximate controllability.

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[15] V.A. Kubyshkin and V.I. Finyagina:*Moving control of systems with distributed parameters* (in Russian). Moscow: SINTEG, 2005.

[16] Sh.Kh. Arakelyan and As.Zh. Khurshudyan: The Bubnov-Galerkin procedure for solving mobile control problems for systems with distributed parameters. Mechanics.*PNAS Armenia*, 68(3), (2015), 54–75.

[17] A.G. Butkovskiy: Some problems of control of the distributed-parameter systems.*Automation and Remote Control*, 72 (2011), 1237–1241, DOI: 10.1134/S0005117911060105.

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[19] A.S. Avetisyan and As.Zh. Khurshudyan: Green’s function approach in approximate controllability of nonlinear physical processes.*Modern Physics Letters A*, 32 1730015, (2017), DOI: 10.1142/S0217732317300154.

[20] As.Zh. Khurshudyan: Resolving controls for the exact and approximate controllabilities of the viscous Burgers’ equation: the Green’s function approach.*International Journal of Modern Physics C,* 29(6), 1850045, (2018), DOI: 10.1142/S0129183118500456.

[21] A.S. Avetisyan and As.Zh. Khurshudyan: Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green’s function approach.*ZAMM*, 98(11), (2018), 1992–2009, DOI: 10.1002/zamm.201800122.

[22] As.Zh. Khurshudyan: Exact and approximate controllability conditions for the micro-swimmers deflection governed by electric field on a plane: The Green’s function approach.*Archives of Control Sciences*, 28(3), (2018), 335–347. DOI: 10.24425/acs.2018.124706.

[23] J. Klamka and As.Zh. Khurshudyan: Averaged controllability of heat equation in unbounded domains with uncertain geometry and location of controls: The Green’s function approach. Archives of Control Sciences, 29(4), (2019), 573–584, DOI: 10.24425/acs.2018.124706.

[24] J. Klamka, A.S. Avetisyan and As.Zh. Khurshudyan: Exact and approximate distributed controllability of the KdV and Boussinesq equations: The Green’s function approach. Archives of Control Sciences, 30(1), (2020), 177–193, DOI: 10.24425/acs.2020.132591.

[25] J. Klamka and As.Zh. Khurshudyan: Approximate controllability of second order infinite dimensional systems. Archives of Control Sciences, 31(1), (2021), 165–184, DOI: 10.24425/acs.2021.136885.

[28] A.G. Butkovskii and L.M. Pustyl’nikov: Characteristics of Distributed- Parameter Systems. Kluwer Academic Publishers, 1993.

Go to article
[2] S.A. Avdonin and S.A. Ivanov:

[3] A. Fursikov and O.Yu. Imanuvilov:

[4] E. Zuazua:

[5] R. Glowinski, J.-L. Lions and J. He:

[6] A.S. Avetisyan and As.Zh. Khurshudyan:

[7] S. Micu and E. Zuazua: On the lack of null-controllability of the heat equation on the half-line.

[8] S. Micu and E. Zuazua:

[9] V. Barbu: Exact null internal controllability for the heat equation on unbounded convex domain.

[10] As.Zh. Khurshudyan: (2019), Distributed controllability of heat equation in un-bounded domains: The Green’s function approach.

[11] S. Ivanov and L. Pandolfi: Heat equation with memory: Lack of controllability to rest.

[12] A. Halanay and L. Pandolfi: Approximate controllability and lack of controllability to zero of the heat equation with memory.

[13] B.S. Yilbas:

[14] A.G. Butkovskiy and L.M. Pustylnikov:

[15] V.A. Kubyshkin and V.I. Finyagina:

[16] Sh.Kh. Arakelyan and As.Zh. Khurshudyan: The Bubnov-Galerkin procedure for solving mobile control problems for systems with distributed parameters. Mechanics.

[17] A.G. Butkovskiy: Some problems of control of the distributed-parameter systems.

[18] A.S. Avetisyan and As.Zh. Khurshudyan: Green’s function approach in approximate controllability problems.

[19] A.S. Avetisyan and As.Zh. Khurshudyan: Green’s function approach in approximate controllability of nonlinear physical processes.

[20] As.Zh. Khurshudyan: Resolving controls for the exact and approximate controllabilities of the viscous Burgers’ equation: the Green’s function approach.

[21] A.S. Avetisyan and As.Zh. Khurshudyan: Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green’s function approach.

[22] As.Zh. Khurshudyan: Exact and approximate controllability conditions for the micro-swimmers deflection governed by electric field on a plane: The Green’s function approach.

[23] J. Klamka and As.Zh. Khurshudyan: Averaged controllability of heat equation in unbounded domains with uncertain geometry and location of controls: The Green’s function approach. Archives of Control Sciences, 29(4), (2019), 573–584, DOI: 10.24425/acs.2018.124706.

[24] J. Klamka, A.S. Avetisyan and As.Zh. Khurshudyan: Exact and approximate distributed controllability of the KdV and Boussinesq equations: The Green’s function approach. Archives of Control Sciences, 30(1), (2020), 177–193, DOI: 10.24425/acs.2020.132591.

[25] J. Klamka and As.Zh. Khurshudyan: Approximate controllability of second order infinite dimensional systems. Archives of Control Sciences, 31(1), (2021), 165–184, DOI: 10.24425/acs.2021.136885.

[26] As.Zh. Khurshudyan: Heuristic determination of resolving controls for exact and approximate controllability of nonlinear dynamic systems. Mathematical Problems in Engineering, (2018), Article ID 9496371, DOI: 10.1155/2018/9496371.

[27] H. Hossain and As.Zh. Khurshudyan: Heuristic control of nonlinear power systems: Application to the infinite bus problem. Archives of Control Sciences, 29(2), (2019), 279–288, DOI: 10.24425/acs.2019.129382. [28] A.G. Butkovskii and L.M. Pustyl’nikov: Characteristics of Distributed- Parameter Systems. Kluwer Academic Publishers, 1993.

We devise a tool-supported framework for achieving power-efficiency of data-flowhardware circuits. Our approach relies on formal control techniques, where the goal is to compute a strategy that can be used to drive a given model so that it satisfies a set of control objectives. More specifically, we give an algorithm that derives abstract behavioral models directly in a symbolic form from original designs described at Register-transfer Level using a Hardware Description Language, and for formulating suitable scheduling constraints and power-efficiency objectives. We show how a resulting strategy can be translated into a piece of synchronous circuit that, when paired with the original design, ensures the aforementioned objectives. We illustrate and validate our approach experimentally using various hardware designs and objectives.

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[1] P. Babighian, L. Benini, and E. Macii: A scalable algorithm for RTL insertion of gated clocks based onODCscomputation. *IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,* 24(1), (2005), 29–42, DOI: 10.1109/TCAD.2004.839489.

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[13] M. Özbaltan:*Achieving Power Efficiency in Hardware Circuits with Symbolic Discrete Control. PhD thesis,* University of Liverpool, 2020.

[14] M. Özbaltan and N. Berthier: Exercising symbolic discrete control for designing low-power hardware circuits: an application to clock-gating.*IFAC-PapersOnLine*, 51(7), (2018), 120–126, DOI: 10.1016/j.ifacol.2018.06.289.

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Go to article
[2] R. Bellman: Dynamic programming and stochastic control processes.

[3] L. Benini, P. Siegel, and G. De Micheli: Saving power by synthesizing gated clocks for sequential circuits.

[4] R. Bhutada and Y. Manoli: Complex clock gating with integrated clock gating logic cell. In

[5] J. Billon:

[6] E.M. Clarke, E.A. Emerson, and A.P. Sistla: Automatic verification of finite-state concurrent systems using temporal logic specifications.

[7] E. Dumitrescu, A. Girault, H. Marchand, and E. Rutten: Multicriteria optimal reconfiguration of fault-tolerant real-time tasks.

[8] K. Gilles: The semantics of a simple language for parallel programming.

[9] E.A. Lee and T.M. Parks: Dataflow process networks.

[10] H. Marchand and M.L. Borgne: On the optimal control of polynomial dynamical systems over z/pz. In

[11] H. Marchand, P. Bournai, M.L. Borgne, and P.L. Guernic: Synthesis of discrete-event controllers based on the signal environment.

[12] S. Miremadi, B. Lennartson, and K. Akesson: A BDD-based approach for modeling plant and supervisor by extended finite automata.

[13] M. Özbaltan:

[14] M. Özbaltan and N. Berthier: Exercising symbolic discrete control for designing low-power hardware circuits: an application to clock-gating.

[15] M. Özbaltan and N. Berthier: A case for symbolic limited optimal discrete control: Energy management in reactive data-flow circuits.

[16] M. Pedram and Q.Wu: Design considerations for battery-powered electronics. In

[17] N. Raghavan, V. Akella, and S. Bakshi: Automatic insertion of gated clocks at register transfer level. In

[18] P. Ramadge and W. Wonham: The control of discrete event systems.

[19] S. Tripakis, R. Limaye, K. Ravindran, G. Wang, H. Andrade, and A. Ghosal: Tokens vs. signals: On conformance between formal models of dataflow and hardware.

The time delay element present in the PI controller brings dead-time compensation capability and shows improved performance for dead-time processes. However, design of robust time delayed PI controller needs much responsiveness for uncertainty in dead-time processes. Hence in this paper, robustness of time delayed PI controller has been analyzed for First Order plus Dead-Time (FOPDT) process model. The process having dead-time greater than three times of time constant is very sensitive to dead-time variation. A first order filter is introduced to ensure robustness. Furthermore, integral time constant of time delayed PI controller is modified to attain better regulatory performance for the lag-dominant processes. The FOPDT process models are classified into dead-time/lag dominated on the basis of dead-time to time constant ratio. A unified tuning method is developed for processes with a number of dead-time to time constant ratio. Several simulation examples and experimental evaluation are exhibited to show the efficiency of the proposed unified tuning technique. The applicability to the process models other than FOPDT such as high-order, integrating, right half plane zero systems are also demonstrated via simulation examples.

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[2] A. O’Dwyer:

[3] A.R. Pathiran and J. Prakash: Design and implementation of a modelbased PI-like control scheme in a reset configuration for stable single-loop systems.

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