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Observers for differential-algebraic systems with Lipschitz or monotone nonlinearities. (English) Zbl 1453.93094
Reis, Timo (ed.) et al., Progress in differential-algebraic equations II. Proceedings of the 9th workshop on descriptor systems, Paderborn, Germany, March 17–20, 2019. Cham: Springer. Differ.-Algebr. Equ. Forum, 257-289 (2020).
Summary: We study state estimation for nonlinear differential-algebraic systems, where the nonlinearity satisfies a Lipschitz condition or a generalized monotonicity condition or a combination of these. The presented observer design unifies earlier approaches and extends the standard Luenberger type observer design. The design parameters of the observer can be obtained from the solution of a linear matrix inequality restricted to a subspace determined by the Wong sequences. Some illustrative examples and a comparative discussion are given.
For the entire collection see [Zbl 1445.34004].
MSC:
93B53 Observers
93C10 Nonlinear systems in control theory
93C23 Control/observation systems governed by functional-differential equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
Software:
PENLAB; YALMIP
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References:
[1] Åslund, J., Frisk, E.: An observer for non-linear differential-algebraic systems. Automatica 42(6), 959-965 (2006) · Zbl 1117.93309
[2] Berger, T.: On differential-algebraic control systems. Ph.D. Thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau (2014)
[3] Berger, T.: Controlled invariance for nonlinear differential-algebraic systems. Automatica 64, 226-233 (2016) · Zbl 1328.93125
[4] Berger, T.: The zero dynamics form for nonlinear differential-algebraic systems. IEEE Trans. Autom. Control 62(8), 4131-4137 (2017) · Zbl 1373.93069
[5] Berger, T.: On observers for nonlinear differential-algebraic systems. IEEE Trans. Autom. Control 64(5), 2150-2157 (2019) · Zbl 07082445
[6] Berger, T., Reis, T.: Observers and dynamic controllers for linear differential-algebraic systems. SIAM J. Control Optim. 55(6), 3564-3591 (2017) · Zbl 1376.93081
[7] Berger, T., Reis, T.: ODE observers for DAE systems. IMA J. Math. Control Inf. 36, 1375-1393 (2019)
[8] Berger, T., Trenn, S.: The quasi-Kronecker form for matrix pencils. SIAM J. Matrix Anal. Appl. 33(2), 336-368 (2012) · Zbl 1253.15021
[9] Berger, T., Trenn, S.: Addition to “The quasi-Kronecker form for matrix pencils”. SIAM J. Matrix Anal. Appl. 34(1), 94-101 (2013) · Zbl 1270.15008
[10] Berger, T., Ilchmann, A., Trenn, S.: The quasi-Weierstraß form for regular matrix pencils. Linear Algebra Appl. 436(10), 4052-4069 (2012) · Zbl 1244.15008
[11] Boutayeb, M., Darouach, M., Rafaralahy, H.: Generalized state-space observers for chaotic synchronization and secure communication. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 49(3), 345-349 (2002) · Zbl 1368.94087
[12] Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, Amsterdam (1989) · Zbl 0699.65057
[13] Campbell, S.L.: Singular Systems of Differential Equations II. Pitman, New York (1982) · Zbl 0482.34008
[14] Corradini, M.L., Cristofaro, A., Pettinari, S.: Design of robust fault detection filters for linear descriptor systems using sliding-mode observers. IFAC Proc. Vol. 45(13), 778-783 (2012)
[15] Darouach, M., Boutat-Baddas, L.: Observers for a class of nonlinear singular systems. IEEE Trans. Autom. Control 53(11), 2627-2633 (2008) · Zbl 1367.93088
[16] Eich-Soellner, E., Führer, C.: Numerical Methods in Multibody Dynamics. Teubner, Stuttgart (1998) · Zbl 0899.70001
[17] Fiala, J., Kočvara, M., Stingl, M.: PENLAB: a MATLAB solver for nonlinear semidefinite optimization (2013). https://arxiv.org/abs/1311.5240
[18] Gantmacher, F.R.: The Theory of Matrices (Vol. I & II). Chelsea, New York (1959) · Zbl 0085.01001
[19] Gao, Z., Ho, D.W.: State/noise estimator for descriptor systems with application to sensor fault diagnosis. IEEE Trans. Signal Proc. 54(4), 1316-1326 (2006) · Zbl 1373.94594
[20] Gupta, M.K., Tomar, N.K., Darouach, M.: Unknown inputs observer design for descriptor systems with monotone nonlinearities. Int. J. Robust Nonlinear Control 28(17), 5481-5494 (2018) · Zbl 1408.93077
[21] Gutú, O., Jaramillo, J.A.: Global homeomorphisms and covering projections on metric spaces. Math. Ann. 338, 75-95 (2007) · Zbl 1115.58006
[22] Ha, Q., Trinh, H.: State and input simultaneous estimation for a class of nonlinear systems. Automatica 40, 1779-1785 (2004) · Zbl 1088.93004
[23] Kumar, A., Daoutidis, P.: Control of Nonlinear Differential Algebraic Equation Systems with Applications to Chemical Processes. Chapman and Hall/CRC Research Notes in Mathematics, vol. 397. Chapman and Hall, Boca Raton (1999) · Zbl 0916.93001
[24] Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich (2006) · Zbl 1095.34004
[25] Lamour, R., März, R., Tischendorf, C.: Differential Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum, vol. 1. Springer, Heidelberg (2013) · Zbl 1276.65045
[26] Liu, H.Y., Duan, Z.S.: Unknown input observer design for systems with monotone non-linearities. IET Control Theory Appl. 6(12), 1941-1947 (2012)
[27] Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design, pp. 284-289 (2004)
[28] Logemann, H., Ryan, E.P.: Ordinary Differential Equations. Springer, London (2014) · Zbl 1300.34001
[29] Lu, G., Ho, D.W.: Full-order and reduced-order observers for Lipschitz descriptor systems: the unified LMI approach. IEEE Trans. Circuits Syst. Express Briefs 53(7), 563-567 (2006)
[30] Luenberger, D.G.: Observing the state of a linear system. IEEE Trans. Mil. Electron. MIL-8, 74-80 (1964)
[31] Luenberger, D.G.: An introduction to observers. IEEE Trans. Autom. Control 16(6), 596-602 (1971)
[32] Luenberger, D.G.: Nonlinear descriptor systems. J. Econ. Dyn. Control 1, 219-242 (1979)
[33] Luenberger, D.G., Arbel, A.: Singular dynamic Leontief systems. Econometrica 45, 991-995 (1977) · Zbl 0368.90029
[34] Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory. A Behavioral Approach. Springer, New York (1998) · Zbl 0940.93002
[35] Riaza, R.: Double SIB points in differential-algebraic systems. IEEE Trans. Autom. Control 48(9), 1625-1629 (2003) · Zbl 1364.34052
[36] Riaza, R.: Differential-Algebraic Systems. Analytical Aspects and Circuit Applications. World Scientific, Basel (2008) · Zbl 1184.34004
[37] Valcher, M.E., Willems, J.C.: Observer synthesis in the behavioral approach. IEEE Trans. Autom. Control 44(12), 2297-2307 (1999) · Zbl 1136.93340
[38] VanAntwerp, J.G., Braatz, R.D.: A tutorial on linear and bilinear matrix inequalities. J. Process Control 10(4), 363-385 (2000)
[39] Wong, K.T.: The eigenvalue problem λTx + Sx. J. Diff. Equ. 16, 270-280 (1974) · Zbl 0327.15015
[40] Yang, C., Zhang, Q., Chai, T.: Nonlinear observers for a class of nonlinear descriptor systems. Optim. Control Appl. Methods 34(3), 348-363 (2012) · Zbl 1270.93023
[41] Yang, C., Kong, Q., Zhang, Q.: Observer design for a class of nonlinear descriptor systems. J. Franklin Inst. 350(5), 1284-1297 (2013) · Zbl 1293.93122
[42] Yeu, T.K., Kawaji, S.: Sliding mode observer based fault detection and isolation in descriptor systems. In: Proceedings of American Control Conference 2002, pp. 4543-4548. Anchorage (2002)
[43] Zhang, J., Swain, A.K., Nguang, S.K.: Simultaneous estimation of actuator and sensor faults for descriptor systems. In: Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLABⓇ, Advances in Industrial Control, pp. 165-197. Springer, Berlin (2016)
[44] Zheng, G., Boutat, D., Wang, H.: A nonlinear Luenberger-like observer for nonlinear singular systems. Automatica 86, 11-17 (2017) · Zbl 1375.93028
[45] Zulfiqar, A. · Zbl 1452.93006
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