doi: 10.3934/mcrf.2021043
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Convergence of coprime factor perturbations for robust stabilization of Oseen systems

Max Planck Institute for Dynamics of, Complex Technical Systems and OVGU Magdeburg, 39106 Magdeburg, Germany

* Corresponding author: Jan Heiland

Received  September 2019 Revised  February 2021 Early access September 2021

Fund Project: This article was recruited by Andrii Mironchenko

Linearization based controllers for incompressible flows have been proven to work in theory and in simulations. To realize such a controller numerically, the infinite dimensional system has to be linearized and discretized. The unavoidable consistency errors add a small but critical uncertainty to the controller model which will likely make it fail, especially when an observer is involved. Standard robust controller designs can compensate small uncertainties if they can be qualified as a coprime factor perturbation of the plant. We show that for the linearized Navier-Stokes equations, a linearization error can be expressed as a coprime factor perturbation and that this perturbation smoothly depends on the size of the linearization error. In particular, improving the linearization makes the perturbation smaller so that, eventually, standard robust controller will stabilize the system.

Citation: Jan Heiland. Convergence of coprime factor perturbations for robust stabilization of Oseen systems. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021043
References:
[1]

M. Badra, Stabilisation par Feedback et Approximation des Equations de Navier-Stokes, PhD thesis, Université Paul Sabatier, Toulouse, 2006.

[2]

V. Barbu, Stabilization of Navier-Stokes Flows, London: Springer, 2011. doi: 10.1007/978-0-85729-043-4.

[3]

P. Benner and J. Heiland, LQG-Balanced Truncation low-order controller for stabilization of laminar flows, In R. King, editor, Active Flow and Combustion Control 2014, volume 127 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, pages 365–379. Springer, Berlin, (2015). doi: 10.1007/978-3-319-11967-0_22.

[4]

P. Benner and J. Heiland, Robust stabilization of laminar flows in varying flow regimes, IFAC-PapersOnLine, 49 (2016), 31–36. 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations. doi: 10.1016/j.ifacol.2016.07.414.

[5]

P. Benner and J. Heiland, Convergence of approximations to Riccati-based boundary-feedback stabilization of laminar flows, IFAC-PapersOnLine, 50 (2017), 12296–12300. 20th IFAC World Congress. doi: 10.1016/j.ifacol.2017.08.2476.

[6]

P. Benner, J. Heiland and S. W. R. Werner, Robust controller versus numerical model uncertainties for stabilization of Navier-Stokes equations, IFAC-PapersOnLine 52 (2019), 25–29. 3rd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE. doi: 10.1016/j.ifacol.2019.08.005.

[7]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems, Birkhäuser, Basel, Switzerland, 2007. doi: 10.1007/978-0-8176-4581-6.

[8]

R. F. Curtain, Robust stabilizability of normalized coprime factors: The infinite-dimensional case, Internat. J. Control, 51 (1990), 1173-1190.  doi: 10.1080/00207179008934125.

[9]

R. F. Curtain, Model reduction for control design for distributed parameter systems, In R. Smith and M. Demetriou, editors, Research Directions in Distributed Parameter Systems, pages 95–121. SIAM, Philadelphia, PA, (2003). doi: 10.1137/1.9780898717525.ch4.

[10]

R. F. Curtain, A robust LQG-controller design for DPS, Internat. J. Control, 79 (2006), 162-170.  doi: 10.1080/00207170500512985.

[11]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, volume 21 of Texts in Applied Mathematics. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[12]

S. DharmattiJ.-P. Raymond and L. Thevenet, ${H}^\infty$ feedback boundary stabilization of two-dimensional Navier-Stokes equations, SIAM J. Cont. Optim., 49 (2011), 2318-2348.  doi: 10.1137/100782607.

[13]

J. Doyle, Guaranteed margins for LQG regulators, IEEE Trans. Automat. Control, 23 (1978), 756–757. doi: 10.1109/TAC.1978.1101812.

[14]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer, Berlin, Germany, 1986. doi: 10.1007/978-3-642-61623-5.

[15]

B.-Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and colocated observation, SIAM J. Control Optim., 44 (2005), 1598-1613.  doi: 10.1137/040610702.

[16]

X. HeW. Hu and Y. Zhang, Observer-based feedback boundary stabilization of the Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., 339 (2018), 542-566.  doi: 10.1016/j.cma.2018.05.008.

[17]

L. S. Hou and S. S. Ravindran, A penalized Neumann control approach for solving an optimal Dirichlet control problem for the Navier–Stokes equations, SIAM J. Cont. Optim., 36 (1998), 1795–1814. doi: 10.1137/S0363012996304870.

[18]

K. Ito, Strong convergence and convergence rates of approximationg solutions for algebraic Riccati equations in Hilbert spaces, In Distributed Parameter Systems, Proc. 3rd Int. Conf., Vorau/Austria, (1987), 153–166. doi: 10.1007/BFb0041988.

[19]

V. Maz'ya and J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Rational Mech. Anal., 194 (2009), 669-712.  doi: 10.1007/s00205-008-0171-z.

[20]

D. C. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Lecture Notes in Control and Information Sciences, volume 138. Springer, 1990. doi: 10.1007/BFb0043199.

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Berlin, Germany, 2012. doi: 10.1007/978-3-642-10455-8.

[22]

P. A. Nguyen and J.-P. Raymond, Boundary stabilization of the Navier–Stokes equations in the case of mixed boundary conditions, SIAM J. Cont. Optim., 53 (2015), 3006–3039. doi: 10.1137/13091364X.

[23]

M. Orlt and A.-M. Sändig, Regularity of viscous Navier-Stokes flows in nonsmooth domains, In Boundary Value Problems and Integral Equations in Nonsmooth Domains, New York, NY: Marcel Dekker, 167 (1995), 185–201.

[24]

L. Paunonen and D. Phan, Reduced order controller design for robust output regulation, IEEE Trans. Automat. Control, 65 (2020), 2480–2493. doi: 10.1109/TAC.2019.2930185.

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations., volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York etc., 1983. doi: 10.1007/978-1-4612-5561-1.

[26]

S. S. Ravindran, Stabilization of Navier-Stokes equations by boundary feedback, International Journal of Numerical Analysis and Modeling, 4 (2007), 608–624. Available from: http://www.math.ualberta.ca/ijnam/Volume-4-2007/No-3-07/2007-03-17.pdf.

[27]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier–Stokes equations, SIAM J. Cont. Optim., 45 (2006), 790–828. doi: 10.1137/050628726.

[28]

J.-P. Raymond, A family of stabilization problems for the Oseen equations, In K. Kunisch, J. Sprekels, G. Leugering, and F. Tröltzsch, editors, Control of Coupled Partial Differential Equations, pages 269–291, Basel, (2007). Birkhäuser Basel. doi: 10.1007/978-3-7643-7721-2_12.

[29]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations, J. Math. Pures Appl., 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.

[30]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, Switzerland, 2005. doi: 10.1007/3-7643-7397-0.

[31]

D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.

[32]

M. Schäfer and S. Turek, Benchmark computations of laminar flow around a cylinder. (With support by F. Durst, E. Krause and R. Rannacher), In E. Hirschel, editor, Flow Simulation with High-Performance Computers II. DFG priority research program results 1993-1995, number 52 in Notes Numer. Fluid Mech., pages 547–566. Vieweg, Wiesbaden, (1996). doi: 10.1007/978-3-322-89849-4_39.

[33]

E. D. Sontag, Mathematical Control Theory. Deterministic Finite Dimensional Systems, Springer, New York, NY, 1998. doi: 10.1007/978-1-4612-0577-7.

[34] O. Staffans, Well-Posed Linear Systems, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511543197.
[35]

R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, volume 66., Philadelphia, PA: SIAM, 2 edition, 1995. doi: 10.1137/1.9781611970050.

[36]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[37]

M. Tucsnak and G. Weiss, Survey paper: Well-posed systems-The LTI case and beyond, Automatica J. IFAC, 50 (2014), 1757-1779.  doi: 10.1016/j.automatica.2014.04.016.

[38]

G. Weiss, Representation of shift-invariant operators on ${L}^ 2$ by ${H}^{\infty}$ transfer functions: An elementary proof, a generalization to ${L}^ p$, and a counterexample for ${L}^{\infty}$, Math. Control Signals Syst., 4 (1991), 193-203.  doi: 10.1007/BF02551266.

show all references

References:
[1]

M. Badra, Stabilisation par Feedback et Approximation des Equations de Navier-Stokes, PhD thesis, Université Paul Sabatier, Toulouse, 2006.

[2]

V. Barbu, Stabilization of Navier-Stokes Flows, London: Springer, 2011. doi: 10.1007/978-0-85729-043-4.

[3]

P. Benner and J. Heiland, LQG-Balanced Truncation low-order controller for stabilization of laminar flows, In R. King, editor, Active Flow and Combustion Control 2014, volume 127 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, pages 365–379. Springer, Berlin, (2015). doi: 10.1007/978-3-319-11967-0_22.

[4]

P. Benner and J. Heiland, Robust stabilization of laminar flows in varying flow regimes, IFAC-PapersOnLine, 49 (2016), 31–36. 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations. doi: 10.1016/j.ifacol.2016.07.414.

[5]

P. Benner and J. Heiland, Convergence of approximations to Riccati-based boundary-feedback stabilization of laminar flows, IFAC-PapersOnLine, 50 (2017), 12296–12300. 20th IFAC World Congress. doi: 10.1016/j.ifacol.2017.08.2476.

[6]

P. Benner, J. Heiland and S. W. R. Werner, Robust controller versus numerical model uncertainties for stabilization of Navier-Stokes equations, IFAC-PapersOnLine 52 (2019), 25–29. 3rd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE. doi: 10.1016/j.ifacol.2019.08.005.

[7]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems, Birkhäuser, Basel, Switzerland, 2007. doi: 10.1007/978-0-8176-4581-6.

[8]

R. F. Curtain, Robust stabilizability of normalized coprime factors: The infinite-dimensional case, Internat. J. Control, 51 (1990), 1173-1190.  doi: 10.1080/00207179008934125.

[9]

R. F. Curtain, Model reduction for control design for distributed parameter systems, In R. Smith and M. Demetriou, editors, Research Directions in Distributed Parameter Systems, pages 95–121. SIAM, Philadelphia, PA, (2003). doi: 10.1137/1.9780898717525.ch4.

[10]

R. F. Curtain, A robust LQG-controller design for DPS, Internat. J. Control, 79 (2006), 162-170.  doi: 10.1080/00207170500512985.

[11]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, volume 21 of Texts in Applied Mathematics. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[12]

S. DharmattiJ.-P. Raymond and L. Thevenet, ${H}^\infty$ feedback boundary stabilization of two-dimensional Navier-Stokes equations, SIAM J. Cont. Optim., 49 (2011), 2318-2348.  doi: 10.1137/100782607.

[13]

J. Doyle, Guaranteed margins for LQG regulators, IEEE Trans. Automat. Control, 23 (1978), 756–757. doi: 10.1109/TAC.1978.1101812.

[14]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer, Berlin, Germany, 1986. doi: 10.1007/978-3-642-61623-5.

[15]

B.-Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and colocated observation, SIAM J. Control Optim., 44 (2005), 1598-1613.  doi: 10.1137/040610702.

[16]

X. HeW. Hu and Y. Zhang, Observer-based feedback boundary stabilization of the Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., 339 (2018), 542-566.  doi: 10.1016/j.cma.2018.05.008.

[17]

L. S. Hou and S. S. Ravindran, A penalized Neumann control approach for solving an optimal Dirichlet control problem for the Navier–Stokes equations, SIAM J. Cont. Optim., 36 (1998), 1795–1814. doi: 10.1137/S0363012996304870.

[18]

K. Ito, Strong convergence and convergence rates of approximationg solutions for algebraic Riccati equations in Hilbert spaces, In Distributed Parameter Systems, Proc. 3rd Int. Conf., Vorau/Austria, (1987), 153–166. doi: 10.1007/BFb0041988.

[19]

V. Maz'ya and J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Rational Mech. Anal., 194 (2009), 669-712.  doi: 10.1007/s00205-008-0171-z.

[20]

D. C. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Lecture Notes in Control and Information Sciences, volume 138. Springer, 1990. doi: 10.1007/BFb0043199.

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Berlin, Germany, 2012. doi: 10.1007/978-3-642-10455-8.

[22]

P. A. Nguyen and J.-P. Raymond, Boundary stabilization of the Navier–Stokes equations in the case of mixed boundary conditions, SIAM J. Cont. Optim., 53 (2015), 3006–3039. doi: 10.1137/13091364X.

[23]

M. Orlt and A.-M. Sändig, Regularity of viscous Navier-Stokes flows in nonsmooth domains, In Boundary Value Problems and Integral Equations in Nonsmooth Domains, New York, NY: Marcel Dekker, 167 (1995), 185–201.

[24]

L. Paunonen and D. Phan, Reduced order controller design for robust output regulation, IEEE Trans. Automat. Control, 65 (2020), 2480–2493. doi: 10.1109/TAC.2019.2930185.

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations., volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York etc., 1983. doi: 10.1007/978-1-4612-5561-1.

[26]

S. S. Ravindran, Stabilization of Navier-Stokes equations by boundary feedback, International Journal of Numerical Analysis and Modeling, 4 (2007), 608–624. Available from: http://www.math.ualberta.ca/ijnam/Volume-4-2007/No-3-07/2007-03-17.pdf.

[27]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier–Stokes equations, SIAM J. Cont. Optim., 45 (2006), 790–828. doi: 10.1137/050628726.

[28]

J.-P. Raymond, A family of stabilization problems for the Oseen equations, In K. Kunisch, J. Sprekels, G. Leugering, and F. Tröltzsch, editors, Control of Coupled Partial Differential Equations, pages 269–291, Basel, (2007). Birkhäuser Basel. doi: 10.1007/978-3-7643-7721-2_12.

[29]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations, J. Math. Pures Appl., 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.

[30]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, Switzerland, 2005. doi: 10.1007/3-7643-7397-0.

[31]

D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.

[32]

M. Schäfer and S. Turek, Benchmark computations of laminar flow around a cylinder. (With support by F. Durst, E. Krause and R. Rannacher), In E. Hirschel, editor, Flow Simulation with High-Performance Computers II. DFG priority research program results 1993-1995, number 52 in Notes Numer. Fluid Mech., pages 547–566. Vieweg, Wiesbaden, (1996). doi: 10.1007/978-3-322-89849-4_39.

[33]

E. D. Sontag, Mathematical Control Theory. Deterministic Finite Dimensional Systems, Springer, New York, NY, 1998. doi: 10.1007/978-1-4612-0577-7.

[34] O. Staffans, Well-Posed Linear Systems, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511543197.
[35]

R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, volume 66., Philadelphia, PA: SIAM, 2 edition, 1995. doi: 10.1137/1.9781611970050.

[36]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[37]

M. Tucsnak and G. Weiss, Survey paper: Well-posed systems-The LTI case and beyond, Automatica J. IFAC, 50 (2014), 1757-1779.  doi: 10.1016/j.automatica.2014.04.016.

[38]

G. Weiss, Representation of shift-invariant operators on ${L}^ 2$ by ${H}^{\infty}$ transfer functions: An elementary proof, a generalization to ${L}^ p$, and a counterexample for ${L}^{\infty}$, Math. Control Signals Syst., 4 (1991), 193-203.  doi: 10.1007/BF02551266.

Figure 1.  Computational domain of the cylinder wake
[1]

N. U. Ahmed. Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach. Evolution Equations and Control Theory, 2012, 1 (2) : 235-250. doi: 10.3934/eect.2012.1.235

[2]

Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations and Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072

[3]

Didier Georges. Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems. Networks and Heterogeneous Media, 2009, 4 (2) : 267-285. doi: 10.3934/nhm.2009.4.267

[4]

Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control and Related Fields, 2022, 12 (1) : 17-47. doi: 10.3934/mcrf.2021001

[5]

Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control and Related Fields, 2022, 12 (1) : 245-273. doi: 10.3934/mcrf.2021021

[6]

Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004

[7]

Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control and Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019

[8]

Marius Tucsnak. Preface to the special issue on control of infinite dimensional systems. Mathematical Control and Related Fields, 2019, 9 (4) : i-ii. doi: 10.3934/mcrf.2019042

[9]

Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303

[10]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control and Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029

[11]

Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations and Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207

[12]

Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149

[13]

Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control and Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83

[14]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066

[15]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77

[16]

Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems and Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053

[17]

James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445

[18]

Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063

[19]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control and Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[20]

Dongyun Wang. Sliding mode observer based control for T-S fuzzy descriptor systems. Mathematical Foundations of Computing, 2022, 5 (1) : 17-32. doi: 10.3934/mfc.2021017

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (309)
  • HTML views (299)
  • Cited by (0)

Other articles
by authors

[Back to Top]