Zero dynamics and funnel control of general linear differential-algebraic systems

Berger, Thomas

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 371-403.

We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the zero dynamics and tracking control. We introduce and characterize the concept of autonomous zero dynamics as an important system theoretic tool for the analysis of differential-algebraic systems. We use the autonomous zero dynamics and $(E,A,B)$-invariant subspaces to derive the so called zero dynamics form – which decouples the zero dynamics of the system – and exploit it for the characterization of system invertibility and asymptotic stability of the zero dynamics. A refinement of the zero dynamics form is then used to show that the funnel controller (that is a static nonlinear output error feedback) achieves – for a special class of right-invertible systems with asymptotically stable zero dynamics – tracking of a reference signal by the output signal within a pre-specified performance funnel. It is shown that the results can be applied to a class of passive electrical networks.

Classification : 15A22, 15A21, 34A09, 34A30, 93D15
Mots clés : Differential-algebraic systems, zero dynamics, invariant subspaces, system inversion, funnel control, relative degree

R.A. Adams, Sobolev Spaces, Number 65 in Pure Appl. Math. Academic Press, New York, London (1975). | MR | Zbl

B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis – A Modern Systems Theory Approach. Prentice-Hall, Englewood Cliffs, NJ (1973).

A. Banaszuk, M. Kociȩcki and K. Maciej Przyłuski, Implicit linear discrete-time systems. Math. Control Signals Syst. 3 (1990) 271–297. | DOI | MR | Zbl

T. Berger, On differential-algebraic control systems. Ph.D. thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Ilmenau, Germany (2014).

T. Berger, Zero Dynamics and Stabilization for Linear DAEs. Progress in Differential-Algebraic Equations. In Differential-Algebraic Equations Forum, edited by Sebastian Schöps, Andreas Bartel, Michael Günther, E. Jan W. ter Maten and Peter C. Müller. Springer-Verlag, Berlin-Heidelberg (2014) 21–45. | MR

T. Berger and S. Trenn, The quasi-Kronecker form for matrix pencils. SIAM J. Matrix Anal. Appl. 33 (2012) 336–368. | DOI | MR | Zbl

T. Berger and S. Trenn, Addition to “The quasi-Kronecker form for matrix pencils”. SIAM J. Matrix Anal. Appl. 34 (2013) 94–101. | DOI | MR | Zbl

T. Berger and T. Reis, Zero dynamics and funnel control for linear electrical circuits. J. Franklin Inst. 351 (2014) 5099–5132. | DOI | MR | Zbl

T. Berger and T. Reis, Controllability of Linear Differential-Algebraic Systems – A Survey. Surveys in Differential-Algebraic Equations I. In Differential-Algebraic Equations Forum, edited by A. Ilchmann and T. Reis. Springer-Verlag, Berlin-Heidelberg (2013) 1–61. | MR | Zbl

T. Berger, A. Ilchmann and T. Reis, Normal Forms, High-gain, and Funnel Control for Linear Differential-Algebraic Systems. Control and Optimization with Differential-Algebraic Constraints. Vol. 23 of Advances in Design and Control, edited by L.T. Biegler, S.L. Campbell and Volker Mehrmann. SIAM, Philadelphia (2012) 127–164. | MR

T. Berger, A. Ilchmann and T. Reis, Zero dynamics and funnel control of linear differential-algebraic systems. Math. Control Signals Syst. 24 (2012) 219–263. | DOI | MR | Zbl

T. Berger, A. Ilchmann and S. Trenn, The quasi-Weierstraß form for regular matrix pencils. Linear Algebra Appl. 436 (2012) 4052–4069. | DOI | MR | Zbl

T. Berger, A. Ilchmann and T. Reis, Funnel Control for Nonlinear Functional Differential-Algebraic Systems. In Proc. of the MTNS 2014. Groningen, The Netherlands (2014) 46–53.

T. Berger, A. Ilchmann and F. Wirth, Zero dynamics and stabilization for analytic linear systems. Acta Applicandae Mathematicae 138 (2015) 17–57. | DOI | MR | Zbl

K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, Amsterdam (1989). | MR | Zbl

J. Dwight Aplevich, Minimal representations of implicit linear systems. Automatica 21 (1985) 259–269. | DOI | MR | Zbl

C.I. Byrnes and A. Isidori, A frequency domain philosophy for nonlinear systems, with application to stabilization and to adaptive control. In Proc. of 23rd IEEE Conf. Decis. Control 1 (1984) 1569–1573.

B. Dziurla and R.W. Newcomb, Nonregular Semistate Systems: Examples and Input-Output Pairing. IEEE Press, New York (1987).

E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics. Teubner, Stuttgart (1998). | MR | Zbl

D. Estévez Schwarz and C. Tischendorf, Structural analysis for electric circuits and consequences for MNA. Int. J. Circuit Theory Appl. 28 (2000) 131–162. | DOI | Zbl

F.R. Gantmacher, The Theory of Matrices. In vol. I & II. Chelsea, New York (1959). | MR

A.H.W. (Ton) Geerts, Invariant subspaces and invertibility properties for singular systems: the general case. Linear Algebra Appl. 183 (1993) 61–88. | DOI | MR | Zbl

C.-W. Ho, A.E. Ruehli and P.A. Brennan, The modified nodal approach to network analysis. IEEE Trans. Circuits Syst. 22 (1975) 504–509. | DOI

A. Ilchmann and E.P. Ryan, High-gain control without identification: a survey. GAMM Mitt. 31 (2008) 115–125. | DOI | MR | Zbl

A. Ilchmann and E.P. Ryan, Performance funnels and tracking control. Int. J. Control 82 (2009) 1828–1840. | DOI | MR | Zbl

A. Ilchmann, E.P. Ryan and C.J. Sangwin, Tracking with prescribed transient behaviour. ESAIM: COCV 7 (2002) 471–493. | Numdam | MR | Zbl

A. Ilchmann, E.P. Ryan and P. Townsend, Tracking with prescribed transient behavior for nonlinear systems of known relative degree. SIAM J. Control Optim. 46 (2007) 210–230. | DOI | MR | Zbl

A. Isidori, Nonlinear Control Systems. Commun. Control Eng. Series, 3rd edition. Springer-Verlag, Berlin (1995). | MR | Zbl

T. Kailath, Linear Systems. Prentice-Hall, Englewood Cliffs, NJ (1980). | MR | Zbl

A. Kumar and P. Daoutidis, Control of Nonlinear Differential Algebraic Equation Systems with Applications to Chemical Processes. Vol. 397 of Chapman Hall/CRC Res. Notes Math. Chapman and Hall, Boca Raton, FL (1999). | MR | Zbl

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich, Switzerland (2006). | MR | Zbl

R. Lamour, R. März and C. Tischendorf, Differential Algebraic Equations: A Projector Based Analysis. Vol. 1 of Differ. Algebr. Eq. Forum. Springer-Verlag, Heidelberg-Berlin (2013). | MR | Zbl

M. Malabre, Generalized linear systems: geometric and structural approaches. Linear Algebra Appl. 122 (1989) 591–621. | DOI | MR | Zbl

M. Mueller, Normal form for linear systems with respect to its vector relative degree. Linear Algebra Appl. 430 (2009) 1292–1312. | DOI | MR | Zbl

K. Özçaldiran, A geometric characterization of the reachable and controllable subspaces of descriptor systems. IEEE Proc. Circuits, Systems and Signal Processing 5 (1986) 37–48. | MR | Zbl

J.W. Polderman and J.C. Willems, Introduction to Mathematical Systems Theory. A Behavioral Approach. Springer-Verlag, New York (1998). | MR | Zbl

T. Reis, Circuit synthesis of passive descriptor systems – a modified nodal approach. Int. J. Circ. Theor. Appl. 38 (2010) 44–68. | DOI | Zbl

W. Respondek, Right and Left Invertibility of Nonlinear Control Systems. In Nonlinear Controllability and Optimal Control, edited by H.J. Sussmann. Marcel Dekker, New York (1990) 133–177. | MR | Zbl

R. Riaza, Differential-Algebraic Systems, Analytical Aspects and Circuit Applications. World Scientific Publishing, Basel (2008). | MR | Zbl

H.H. Rosenbrock, Structural properties of linear dynamical systems. Int. J. Control 20 (1974) 191–202. | DOI | MR | Zbl

P. Sannuti and A. Saberi, Special coordinate basis for multivariable linear system – finite and infinite zero structure, squaring down and decoupling. Int. J. Control 45 (1987) 1655–1704. | DOI | MR | Zbl

L.M. Silverman, Inversion of multivariable linear systems. IEEE Trans. Auto. Control 14 (1969) 270–276. | DOI | MR

L.M. Silverman and H.J. Payne, Input-output structure of linear systems with application to the decoupling problem. SIAM J. Control 9 (1971) 199–233. | DOI | MR | Zbl

C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Number 41 in Appl. Math. Sci. Springer-Verlag (1982). | MR | Zbl

S. Trenn, Solution Concepts for Linear DAEs: A Survey. Surveys in Differential-Algebraic Equations I. In Differ. Algebr. Eq. Forum, edited by A. Ilchmann and T. Reis. Springer-Verlag, Berlin-Heidelberg (2013) 137–172. | MR | Zbl

H.L. Trentelman, A.A. Stoorvogel and M.L.J. Hautus, Control Theory for Linear Systems. Commun. Control Engineering. Springer-Verlag, London (2001). | MR | Zbl

L.M. Wedepohl and L. Jackson, Modified nodal analysis: an essential addition to electrical circuit theory and analysis. Eng. Sci. Educ. J. 11 (2002) 84–92. | DOI

H. Zhao and D. Chen, Stable Inversion for Exact and Stable Tracking Controllers: A Flight Control Example. In Proc. of 4th IEEE Conference on Control Applications (1995) 482–487.