3 | D. Bankmann
**On linear-quadratic control theory of implicit difference equations** Master's Thesis, Technische Universität Berlin, 2016. [abstract] [bibtex]
In this master's thesis we adapt recent results on optimal control of continuous-time linear differential-algebraic equations to the discrete-time case of implicit difference equations. First, we adapt equivalent characterizations of solvability of the so-called Kalman- Yakubovich-Popov inequality for differential-algebraic equations to the case of implicit difference equations. That is, we relate the solvability of a certain matrix inequality to the positivity of the Popov function on the unit circle. An essential difference between the continuous-time and the discrete-time linear-quadratic optimal control problem is due to different structures occurring during the analysis in the form of even or palindromic matrix pencils, respectively. Therefore, with the help of certain structured Kronecker canonical forms, we adapt characterizations of inertia of even matrix pencils to palindromic matrix pencils. To this end, we first introduce a suitable notion of inertia for palindromic matrix pencils. These results are used – analogously to the continuous-time case – to characterize solvability of Lur'e equations equivalently by the existence of certain deflating subspaces of the palindromic matrix pencil. Then we use these findings to describe feasibility and the structure of solutions of the linear-quadratic control problem with infinite time horizon. Finally, these results are illustrated by means of an example. @PHDTHESIS{Ban16, title = {On linear-quadratic control theory of implicit difference equations}, year = {2016}, month = {August}, address = {{Berlin}}, url = {https://depositonce.tu-berlin.de/handle/11303/5838}, copyright = {https://creativecommons.org/licenses/by/4.0/}, language = {English}, school = {Technische Universit{\"a}t Berlin}, type = {Master's {{Thesis}}}, author = {Bankmann, D.}, } |

2 | D. Bankmann, V. Mehrmann, Y. Nesterov, and P. Van Dooren
**Computation of the analytic center of the solution set of the linear matrix inequality arising in continuous- and discrete-time passivity analysis**
*arXiv:1904.08202 [math]*, 2019. [abstract] [bibtex]
In this paper formulas are derived for the analytic center of the solution set of linear matrix inequalities (LMIs) defining passive transfer functions. The algebraic Riccati equations that are usually associated with such systems are related to boundary points of the convex set defined by the solution set of the LMI. It is shown that the analytic center is described by closely related matrix equations, and their properties are analyzed for continuous- and discrete-time systems. Numerical methods are derived to solve these equations via steepest ascent and Newton-like methods. It is also shown that the analytic center has nice robustness properties when it is used to represent passive systems. The results are illustrated by numerical examples. @ARTICLE{BanMNV19, title = {Computation of the analytic center of the solution set of the linear matrix inequality arising in continuous- and discrete-time passivity analysis}, year = {2019}, month = {April}, url = {http://arxiv.org/abs/1904.08202}, urldate = {2019-04-18}, archiveprefix = {arXiv}, eprint = {1904.08202}, eprinttype = {arxiv}, journal = {arXiv:1904.08202 [math]}, primaryclass = {math}, author = {Bankmann, D. and Mehrmann, V. and Nesterov, Y. and Van Dooren, P.}, } |

1 | D. Bankmann and M. Voigt
**On linear-quadratic optimal control of implicit difference equations**
*IMA J Math Control Info*, 2018. [abstract] [preprint] [bibtex]
In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur'e equation and a corresponding Kalman–Yakubovich–Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur'e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context. @ARTICLE{BanV18, title = {On linear-quadratic optimal control of implicit difference equations}, year = {2018}, month = {February}, doi = {10.1093/imamci/dny007}, journal = {IMA J Math Control Info}, language = {en}, preprint = {https://arxiv.org/abs/1703.01217}, pubdate = {2018-02-28}, author = {Bankmann, D. and Voigt, M.}, } |