University of Groningen A geometric approach to differential-algebraic systems Megawati, Noorma

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University of Groningen

A geometric approach to differential-algebraic systems

Megawati, Noorma

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Megawati, N. (2017). A geometric approach to differential-algebraic systems: from bisimulation to control by interconnection. University of Groningen.

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[1] J.D. Aplevich. Time-domain input-output representations of linear systems. Automatica, 17:509–522, 1981.

[2] J.D. Aplevich. Minimal representations of implicit linear systems. Automatica, 21:259–269, 1985.

[3] V. Armentano. The pencil (sE − A) and controllability - observability for generalized linear systems: A geometric approach. SIAM Journal of Control and Optimization, 24(4):616–638, 1986.

[4] U. Bas¸er and J.M. Schumacher. The equivalence structure of descriptor representations of systems with possibly inconsistent initial conditions. Linear Algebra and Its Applications, 318:53–77, 2000.

[5] G. Basile and G. Marro. Controlled and conditioned invariant subspaces in linear system theory. Journal of Optimization Theory and Applications, 3(5):306–315, 1969.

[6] G. Basile and G. Marro. Controlled and Conditioned Invariants in Linear System Theory. Prentice-Hall, Englewood Cliffs, NJ, 1992.

[7] T. Berger. Controlled invariance for nonlinear differential-algebraic systems. Automatica, 64:226–233, 2016.

[8] T. Berger. Disturbance decoupling by behavioral feedback for linear differential-algebraic systems. Automatica, 80:272–283, 2017.

[9] T. Berger, A. Ilchmann, and S. Trenn. The quasi-Weierstrass form for regular matrix pencils. Linear Algebra and its Applications, 436(10):4052–4069, 2012.

(3)

100 BIBLIOGRAPHY

[10] T. Berger and T. Reis. Controllability of linear differential-algebraic systems— a survey. In A. Ilchmann and T. Reis, editors, Surveys in Differential-Algebraic Equations I, pages 1–61. Springer, Berlin, Heidelberg, 2013.

[11] T. Berger, T. Reis, and S. Trenn. Observability of linear differential-algebraic systems: A survey. In A. Ilchmann and T. Reis, editors, Surveys in Differential-Algebraic Equations IV, pages 161–219. Springer, Cham, 2017.

[12] P. Bernhard. On singular implicit linear dynamical systems. SIAM Journal of Control and Optimization, 20(5):612–633, 1982.

[13] S.L. Campbell. Singular Systems of Differential Equations I. Pitman Advanced Publishing Program, London, 1980.

[14] E. M. Clarke, Jr., O. Grumberg, and D.A. Peled. Model Checking. MIT Press, Cambridge, MA, USA, 1999.

[15] D. Cobb. On the solutions of linear differential equations with singular coefﬁcients. Journal of Differential Equations, 46(3):310 – 323, 1982. [16] L. Dai. Singular Control Systems. Springer-Verlag New York, Inc., Secaucus,

NJ, USA, 1989.

[17] J. Dieudonn´e. Sur la r´eduction canonique des couples des matrices. Bull. Soc. Math. Fr., 74:130–146, 1946.

[18] A. R. F. Everts and M. K. C¸amlıbel. The disturbance decoupling problem for continuous piecewise afﬁne systems. In Proc. 53th_{IEEE Conference on Decision}

and Control, pages 6365–6370, 2014.

[19] A. R. F. Everts and M. K. C¸amlıbel. When is a linear complementarity system disturbance decoupled? In M. N. Belur, M. K C¸amlıbel, P Rapisarda, and J.M.A. Scherpen, editors, Mathematical Control Theory II: Behavioral Systems and Robust Control, pages 243–255. Springer, Cham, 2015.

[20] A.R.F. Everts. A geometric approach to multi-modal and multi-agent systems: From disturbance decoupling to consensus. PhD thesis, Rijksuniversiteit Gronin-gen, 2016.

[21] L.R. Fletcher and A. Aasaraai. On disturbance decoupling in descriptor systems. SIAM Journal of Control and Optimization, 27(6):1319–1332, 1989. [22] T. Geerts. Invariant subspaces and invertibillity properties for singular sys-tems: The general case. Linear Algebra and Its Applications, 183:61–88, 1993.

(4)

[23] T. Geerts. Solvability conditions, consistency, and weak consistency for linear differential-algebraic equations and time-invariant singular systems:the general case. Linear Algebra and its Applications, 181:111 – 130, 1993. [24] J. Grimm. Realization and canonicity for implicit systems. SIAM Journal of

Control and Optimization, 26:1331–1347, 1988.

[25] M.L.J. Hautus and L.M. Silverman. System structure and singular control. Linear Algebra and Its Applications, 50:369–402, 1983.

[26] W.P.M.H. Heemels, J.M. Schumacher, and S. Weiland. Linear complementarity systems. SIAM Journal of Applied Mathematics, 60(4):1234–1269, 2000. [27] H. Hermanns. Interactive Markov Chains and The Quest for Quantiﬁed Quality.

Springer-Verlag, Berlin, 2002.

[28] A.A. Julius and M.N. Belur. Behavioral control in the presence of disturbance. In Proc. 44th_{IEEE Conference on Decision and Control and The European Control}

Conference, pages 155–160, Seville, Spain, December 12-15, 2005.

[29] A.A. Julius, J.C. Willems, M.N. Belur, and H.L. Trentelman. The canonical controllers and regular interconnection. Systems & Control Letters, 54(8):787– 797, 2005.

[30] N. Karcanias and G. Hayton. Geometric theory and feedback invariants of generalized linear systems:a matrix pencil approach. Journal of Circuits, systems, and Signal processing, 8(3):375–397, 1989.

[31] C. Kazantzidou and L. Ntogramatzidis. On the computation of the funda-mental subspaces for descriptor systems. International Journal of Control, 89:1481–1494, 2016.

[32] M. Kuijper and J.M. Schumacher. Minimality of descriptor representations inder external equivalence. Automatica, 27(6):985–995, 1991.

[33] P. Kunkel and V. Mehrmann. Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society, Zurich, Switzerland, 2006.

[34] F.L. Lewis. A survey of linear singular systems. Journal of Circuits, Systems, and Signal Processing, 5(1):3–36, 1986.

[35] F.L. Lewis. A tutorial on the geometric analysis of linear time-invariant implicit systems. Automatica, 28(1):119–137, 1992.

[36] D.G. Luenberger. Time-invariant descriptor systems. Automatica, 14:473–480, 1978.

(5)

102 BIBLIOGRAPHY

[37] M. Malabre. Generalized linear systems: geometric and structural approaches. Linear Algebra and its Applications, 122:591–621, 1989.

[38] D.S. Malik, J.N. Mordeson, and M.K. Sen. Fundamentals of Abstract Algebra. McGraw-Hill Education, London, UK, 1997.

[39] N.Y. Megawati and A.J. van der Schaft. Bisimulation equivalence of DAE systems. In Proc. 7th_{European Congress of Mathematics, Berlin, Germany, July}

18-22, 2016.

[40] N.Y. Megawati and A.J. van der Schaft. Bisimulation equivalence of differential-algebraic systems. International Journal of Control, pages 1–12, 2016. In press.

[41] N.Y. Megawati and A.J. van der Schaft. Equivalence of regular matrix pencil DAE systems by bisimulation. In Proc. 44th _{European Control Conference,}

pages 1093–1098, Aalborg, Denmark, June 29- July 1, 2016.

[42] N.Y. Megawati and A.J. van der Schaft. Abstraction and control by intercon-nection of linear systems: A geometric approach. Systems & Control Letters, 105:27–33, 2017.

[43] N.Y. Megawati and A.J. van der Schaft. Disturbance decoupling problem for linear systems with complementarity switching. In preparation, 2017. [44] R. Milner. Communication and Concurrency. Prentice-Hall, Inc., Upper Saddle

River, NJ, USA, 1989.

[45] W. K. Nicholson. Introduction to Abstract Algebra. Wiley Publishing, 4th edition, 2012.

[46] N. Otsuka. Disturbance decoupling with quadratic stability for switched linear systems. Systems & Control Letters, 59(6):349–352, 2010.

[47] K. ¨Ozc¸aldıran. A geometric characterization of the reachable and the control-lable subspaces of descriptor systems. Circuits, Systems and Signal Processing, 5(1):37–48, 1986.

[48] G. J. Pappas. Bisimilar linear systems. Automatica, 39(12):2035–2047, 2003. [49] G. J. Pappas, G. Lafferriere, and S. Sastry. Hierarchically consistent control

systems. IEEE Transactions on Automatic Control, 45(6):1144–1160, 2000. [50] K.M. Przyłuski and A. Sosnowski. Remarks on the theory of implicit linear

(6)

[51] G. Reissig, A. Weber, and M. Rungger. Feedback reﬁnement relations for the synthesis of symbolic controllers. IEEE Transactions on Automatic Control, 62(4):1781–1796, 2017.

[52] P. Tabuada. Controller synthesis for bisimulation equivalence. Systems & Control Letters, 57(6):443 – 452, 2008.

[53] P. Tabuada. Veriﬁcation and Control of Hybrid Systems: A Symbolic Approach. Springer, 1st edition, 2009.

[54] P. Tabuada and G.J. Pappas. Bisimilar control afﬁne systems. Systems & Control Letters, 52(1):49–58, 2004.

[55] S. Trenn. Solution concepts for linear DAEs: A survey. In A. Ilchmann and T. Reis, editors, Surveys in Differential-Algebraic Equations I, pages 137–172. Springer, Berlin, Heidelberg, 2013.

[56] H.L. Trentelman, A.A. Stoorvogel, and M.L.J. Hautus. Control theory for linear systems. Springer-Verlag, London, 2001.

[57] A.J. van der Schaft. Achievable behavior of general systems. System & Control Letters, 49:141–149, 2003.

[58] A.J. van der Schaft. Equivalence of dynamical systems by bisimulation. IEEE Transactions on Automatic Control, 49(12):2160–2172, 2004.

[59] A.J. van der Schaft. Equivalence of hybrid dynamical system. In Proc. of 16th

International Symposium on Mathematical Theory of Networks and Systems, 2004.

[60] A.J. van der Schaft and J.M. Schumacher. Complementarity modelling of hybrid systems. IEEE Transactions on Automatic Control, 43(4):483–490, 1998.

[61] A.J. van der Schaft and J.M. Schumacher. An Introduction to Hybrid Dynamical Systems. Springer-Verlag, 2000.

[62] H. Vinjamoor. Achievability of linear systems up to bisimulation. PhD thesis, Rijksuniversiteit Groningen, 2011.

[63] H. Vinjamoor and A.J. van der Schaft. The achievable dynamics via control by interconnection. IEEE Transactions on Automatic Control, 56(5):1110–1117, 2011.

[64] J.C. Willems. On interconnections, control, and feedback. IEEE Transactions on Automatic Control, 42(3):326–339, 1997.

(7)

104 BIBLIOGRAPHY

[65] K.T. Wong. The eigenvalue problem λT x + Sx. Journal of Differential Equa-tions, 16(2):270–280, 1974.

[66] W.M. Wonham. Linear Multivariable Control: A geometric approach. Springer-Verlag, New York, 3rd edition, 1985.

[67] E. Yurtseven, W.P.M.H. Heemels, and M.K. C¸amlıbel. Disturbance decoupling of switched linear systems. Systems & Control Letters, 61:69–78, 2012.