×

Non-smooth optimization for robust control of infinite-dimensional systems. (English) Zbl 1393.90138

Summary: We use a non-smooth trust-region method for \(H_{\infty}\)-control of infinite-dimensional systems. Our method applies in particular to distributed and boundary control of partial differential equations. It is computationally attractive as it avoids the use of system reduction or identification. For illustration the method is applied to control a reaction-convection-diffusion system, a Van de Vusse reactor, and to a cavity flow control problem.

MSC:

90C56 Derivative-free methods and methods using generalized derivatives
49J52 Nonsmooth analysis
93C80 Frequency-response methods in control theory
93C20 Control/observation systems governed by partial differential equations

Software:

QDES
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Apkarian, P; Noll, D, Nonsmooth \(H\)_{∞} synthesis, IEEE Trans. Automat. Control, 51, 71-86, (2006) · Zbl 1366.93148 · doi:10.1109/TAC.2005.860290
[2] Apkarian, P; Noll, D, Nonsmooth optimization for multidisk \(H\)_{∞}-synthesis, Eur. J. Control, 12, 229-244, (2006) · Zbl 1293.93310 · doi:10.3166/ejc.12.229-244
[3] Apkarian, P; Noll, D; Prot, O, A proximity control algorithm to minimize non-smooth and non-convex semi-infinite maximum eigenvalue functions, J. Convex Anal., 16, 641-666, (2009) · Zbl 1182.49012
[4] Apkarian, P; Noll, D; Ravanbod, L, Nonsmooth bundle trust-region algorithm with applications to robust stability, Set-Valued Variational Anal., 24, 115-148, (2016) · Zbl 1334.49092 · doi:10.1007/s11228-015-0352-5
[5] Boyd, S; Balakrishnan, V, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its \(L\)_{∞}-norm, Syst. Control Lett., 15, 1-7, (1990) · Zbl 0704.93014 · doi:10.1016/0167-6911(90)90037-U
[6] Boyd, S; Balakrishnan, V; Kabamba, P, A bisection method for computing the \(H\)_{∞}-norm of a transfer matrix and related problems, Math. Control Signals Syst., 2, 207-219, (1989) · Zbl 0674.93020 · doi:10.1007/BF02551385
[7] Boyd, S., Barratt, C.: Linear controller design. Limits of performance. Prentice Hall, New Jersey (1991) · Zbl 0748.93003
[8] Cullum, J; Donath, W; Wolfe, P, The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices, Math. Programm. Stud., 3, 35-55, (1975) · Zbl 0355.90054 · doi:10.1007/BFb0120698
[9] Curtain, R.F., Zwart, H.J.: An introduction to infinite-dimensional linear system theory, vol. 21. Springer Texts in Applied Mathematics, Berlin (1995) · Zbl 0839.93001
[10] Dao, MN; Gwinner, J; Noll, D; Ovcharova, N, Nonconvex bundle method with application to a delamination problem, Comput. Optim. Appl., 65, 173-203, (2016) · Zbl 1353.90157 · doi:10.1007/s10589-016-9834-0
[11] Dochain, D; Bouaziz, B, Approximation of the dynamical model of fixed bed reactors via a singular perturbation approach, Math. Comput. Simul., 37, 165-172, (1994) · Zbl 0825.93053 · doi:10.1016/0378-4754(94)00005-0
[12] Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations, vol. 194. Springer Graduate Texts in Mathematics, Berlin (2000) · Zbl 0952.47036
[13] Fogler, H.S.: Elements of Chemical Reaction Engineering, Fourth Edition. Prentice Hall International Series in the Physical and Chemical Engineering Sciences (2005) · Zbl 0494.49026
[14] Hare, W; Sagastizábal, C, A redistributed proximal bundle method for nonconvex optimization, SIAM J. Optim., 20, 2442-2473, (2010) · Zbl 1211.90183 · doi:10.1137/090754595
[15] Jacobson, CA; Nett, CN, Linear state-space systems in infinite-dimensional space: the role and characterization of joint stabilizability/detectability, IEEE Trans. Autom. Control, 33, 541-549, (1988) · Zbl 0645.93025 · doi:10.1109/9.1249
[16] Kato, T.: Perturbation theory for linear operators; 2nd ed. Series Grundlehren Mathematische Wissenschaftliche. Springer, Berlin (1976) · Zbl 0342.47009
[17] Kiwiel, C, An aggregate subgradient method for nonsmooth convex minimization, Math. Programm., 27, 320-341, (1983) · Zbl 0525.90074 · doi:10.1007/BF02591907
[18] Kiwiel, KC, A linearization algorithm for computing control systems subject to singular value inequalities, IEEE Trans. Automat. Control, AC-31, 595-602, (1986) · Zbl 0622.49015 · doi:10.1109/TAC.1986.1104345
[19] Mayne, D; Polak, E, Algorithms for the design of control systems subject to singular value inequalities, Math. Programming Stud., 18, 112-134, (1982) · Zbl 0494.49026 · doi:10.1007/BFb0120976
[20] Mayne, D; Polak, E; Sangiovanni, A, Computer aided design via optimization, Automatica, 18, 147-154, (1982) · Zbl 0478.93021 · doi:10.1016/0005-1098(82)90104-2
[21] Noll, D; Prot, O; Rondepierre, A, A proximity control algorithm to minimize non-smooth and nonconvex functions, Pac. J. Optim., 4, 571-604, (2008) · Zbl 1162.49020
[22] Polak, E; Wardi, Y, A nondifferential optimization algorithm for the design of control systems subject to singular value inequalities over the frequency range, Automatica, 18, 267-283, (1982) · Zbl 0522.49020 · doi:10.1016/0005-1098(82)90087-5
[23] Polak, E; Salcudean, S, On the design of linear multivariable feedback systems via constrained nondifferentiable optimization in \(H\)_{∞} space, IEEE Trans. Autom. Control, AC-34, 268-276, (1989) · Zbl 0669.93058 · doi:10.1109/9.16416
[24] Ravanbod, L., Noll, D., Apkarian, P.: Computing the structured distance to instability. In: Proceedings of the SIAM Conference on Control and Applications, pp 423-430, Paris (2015) · Zbl 0645.93025
[25] Ruszczynski, A.: Nonlinear optimization. Princeton University Press, Princeton (2007) · Zbl 1108.90001
[26] Sagastizábal, C, Composite proximal bundle method, Math. Progr., 140, 189-233, (2013) · Zbl 1273.90163 · doi:10.1007/s10107-012-0600-5
[27] Salamon, D, Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach, Trans. Amer. Math. Soc., 300, 383-431, (1987) · Zbl 0623.93040
[28] Sano, H, \(H\)_{∞}-control of a parallel-flow heat exchange process, IFAC-PapersOnLIne, 48-25, 050-055, (2015) · doi:10.1016/j.ifacol.2015.11.058
[29] Schramm, H; Zowe, J, A version of the bundle idea for minimizing a nonsmooth function: conceptual ideas, convergence analysis, numerical results, SIAM J. Opt., 2, 121-152, (1992) · Zbl 0761.90090 · doi:10.1137/0802008
[30] Spingarn, JE, Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc., 264, 77-89, (1981) · Zbl 0465.26008 · doi:10.1090/S0002-9947-1981-0597868-8
[31] Vusse, JG, Plug-flow type reactor versus tank reactor, Chem. Eng. Sci., 19, 994-998, (1964) · doi:10.1016/0009-2509(64)85109-5
[32] Yan, P; Debiasi, M; Yuan, X; Little, J; Özbay, H; Samimy, M, Experimental study of linear closed-loop control of subsonic cavity flow, AIAA J., 44, 929-938, (2006) · doi:10.2514/1.14873
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.