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Prof. Dr. Achim Ilchmann

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Behrndt, Jussi; Luger, Annemarie; Trunk, Carsten
On the negative squares of a class of self-adjoint extensions in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 34 S., 386 KB). . - (Preprint. - M10,16)

A description of all exit space extensions with finitely many negative squares of a symmetric operator of defect one is given via Krein's formula. As one of the main results an exact characterization of the number of negative squares in terms of a fixed canonical extension and the behaviour of a function t (that determines the exit space extension in Krein's formula) at zero and at infinity is obtained. To this end the class of matrix valued D k n×n -functions is introduced and, in particular, the properties of the inverse of a certain D k 2×2 -function which is closely connected with the spectral properties of the exit space extensions with finitely many negative squares is investigated in detail. Among the main tools here are the analytic characterization of the degree of non-positivity of generalized poles of matrix valued generalized Nevanlinna functions and some extensions of recent factorization results.
Berger, Thomas; Ilchmann, Achim; Reis, Timo
Zero dynamics and funnel control of linear differential-algebraic systems. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 35 S., 674 KB). . - (Preprint. - M10,15)

We study the class of linear differential-algebraic m-input m-output systems which have a transfer function with proper inverse. A sufficient condition for the transfer function to have proper inverse it that the system has 'strict and non-positive relative degree'. We present two main results: First, a so called 'zero dynamics form' is derived: this form is - within the class of system equivalence - a simple ("almost normal") form of the DAE; it is a counterpart to the well-known Byrnes-Isidori form for ODE systems with strictly proper transfer function. The 'zero dynamics form' is exploited to characterize structural properties such as asymptotically stable zero dynamics, minimum phase, and high-gain stabilizability. The zero dynamics are characterized by (A,E,B)-invariant subspaces. Secondly, it is shown that the 'funnel controller' (that is a static nonlinear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.
Kuzhel, Sergii; Trunk, Carsten
On a class of J-self-adjoint operators with empty resolvent set. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 29 S., 257,3 KB). . - (Preprint. - M10,11)
Philipp, Friedrich; Strauss, Vladimir A.; Trunk, Carsten;
Local spectral theory for normal operators in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 18 S., 172 KB). . - (Preprint. - M10,09)

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of nite order, then the normal operator possesses a local spectral function dened for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space.
Berger, Thomas; Ilchmann, Achim
On stability of time-varying linear differential-algebraic equations. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 25 S., 284,9 KB). . - (Preprint. - M10,12)

We develop a stability theory for time-varying linear differential algebraic equations (DAEs). Standard stability concepts for ODEs are formulated for DAEs and characterized. Lyapunovs direct method is derived as well as the converse of the stability theorems. Stronger results are achieved for DAEs which are transferable into standard canonical form; in this case the existence of the generalized transition matrix is exploited.
Trunk, Carsten;
Spectral theory for second order systems and indefinite Sturm-Liouville problems. - Getr. Zählung. Ilmenau : Techn. Univ., Habil.-Schr., 2010

Berger, Thomas; Ilchmann, Achim
Time-varying linear DAEs transferable into standard canonical form. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 19 S., 270,1 KB). . - (Preprint. - M10,07)
Hackl, Christoph; Hopfe, Norman; Ilchmann, Achim; Müller, Markus; Trenn, Stephan
Funnel control for systems with relative degree two. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 36 S., 523,7 KB). . - (Preprint. - M10,06)

Tracking of reference signals yref(&hahog;) by the output y(&hahog;) of linear (as well as a considerably large class of nonlinear) single-input, single-output system is considered. The system is assumed to have strict relative degree two with ("weak") stable zero dynamics. The control objective is tracking of the error e = y - yref and its derivative e&hahog; within two prespecified performance funnels, resp. This is achieved by the so called 'funnel controller': u(t) = -k0(t)2e(t) - k1(t)e&hahog;(t), where the simple proportional error feedback has gain functions k0 and k1 designed in such a way to preclude contactof e and e&hahog; with the funnel boundaries, resp. The funnel controller also ensures boundedness of all signals. --- We also show that the same funnel controller is (i) applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control.
Berger, Thomas; Ilchmann, Achim
Zero dynamics of time-varying linear systems. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 23 S., 297,2 KB). . - (Preprint. - M10,05)

The Byrnes-Isidori form with respect to the relative degree is studied for time-varying linear multi-input, multi-output systems. It is clarified in which sense this form is a normal form. (A,B)-invarianttime-varying subspaces are defined and the maximal(A,B)-invariant time-varying subspace included in the kernel of C is characterized. This is exploited to characterize the zero dynamics of the system. Finally, a high-gain derivative output feedback controller is introduced for the class of systems with higher relative degree and stable zero dynamics. All results are also new for time-invariant linear systems.
Azizov, Tomas Ya.; Trunk, Carsten
On domains of PT symmetric operators related to -y"(x) + (-1) n x 2n y(x). - In: Journal of physics. Mathematical and theoretical. - Bristol : IOP Publ., 1968- , ISSN: 1751-8121 , ZDB-ID: 1363010-6, ISSN 1751-8121, Bd. 43.2010, 17, 175303, insges. 13 S.