# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021043
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## 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

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 & Related Fields, doi: 10.3934/mcrf.2021043
##### References:
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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.  Google Scholar [10] R. F. Curtain, A robust LQG-controller design for DPS, Internat. J. Control, 79 (2006), 162-170.  doi: 10.1080/00207170500512985.  Google Scholar [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.  Google Scholar [12] S. Dharmatti, J.-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.  Google Scholar [13] J. Doyle, Guaranteed margins for LQG regulators, IEEE Trans. Automat. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Berlin, Germany, 2012. doi: 10.1007/978-3-642-10455-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, Switzerland, 2005. doi: 10.1007/3-7643-7397-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] E. D. Sontag, Mathematical Control Theory. Deterministic Finite Dimensional Systems, Springer, New York, NY, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar [34] O. Staffans, Well-Posed Linear Systems, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511543197.  Google Scholar [35] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, volume 66., Philadelphia, PA: SIAM, 2 edition, 1995. doi: 10.1137/1.9781611970050.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] M. Badra, Stabilisation par Feedback et Approximation des Equations de Navier-Stokes, PhD thesis, Université Paul Sabatier, Toulouse, 2006. Google Scholar [2] V. Barbu, Stabilization of Navier-Stokes Flows, London: Springer, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] R. F. Curtain, A robust LQG-controller design for DPS, Internat. J. Control, 79 (2006), 162-170.  doi: 10.1080/00207170500512985.  Google Scholar [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.  Google Scholar [12] S. Dharmatti, J.-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.  Google Scholar [13] J. Doyle, Guaranteed margins for LQG regulators, IEEE Trans. Automat. Control, 23 (1978), 756–757. doi: 10.1109/TAC.1978.1101812.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] X. He, W. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, Berlin, Germany, 2012. doi: 10.1007/978-3-642-10455-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, Switzerland, 2005. doi: 10.1007/3-7643-7397-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] E. D. Sontag, Mathematical Control Theory. Deterministic Finite Dimensional Systems, Springer, New York, NY, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar [34] O. Staffans, Well-Posed Linear Systems, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511543197.  Google Scholar [35] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, volume 66., Philadelphia, PA: SIAM, 2 edition, 1995. doi: 10.1137/1.9781611970050.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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