Systems theory of partial differential-algebraic equations
Final Report Abstract
We have considered (control) systems govered by differential-algebraic equations. Starting with finite- dimensional systems, we have introduced a novel index concept, namely the “the input index”. This allows for single-input single output systems to systematically characterize the set of consistent initial values and the regularity requirements on the control. An application of this to linear-quadratic Optimal Control Problems (OCPs) allows to properly construct terminal conditions in Model Predictive Control (MPC) for state- and input-constrained systems. Moreover, it also enables to determine suitable lengths of the optimisation horizon such that MPC without stabilising terminal conditions renders the origin asymptotically stable with respect to the resulting closed-loop system. Subsequently, we have addressed the two main drawbacks of the proposed approach. Using the so-called Feedback-Equivalence-Form (FEF) or unimodular transformation and, thus, the staircase form, allows to extend the deduced framework to multi-input multi-output systems and to avoid numerical disadvantages of the Quasi- Weierstraß-Form (QWF). Secondly, we have revisited our results from a truly differential-algebraic perspective. We have de- veloped an approach to the OCP zero-state terminal conditions by using so-called “Lur’e equations”. This gives - in contrast to many other approaches - equivalent criteria for optimality. Then, in a follow-up step – based on a suitable, newly derived characterisation of output stabilisability – MPC for differential-algebraic systems has been treated. This is of particular interest since it enables to verify the assumed sufficient conditions without employing a canonical form beforehand. Thirdly, stability concepts were suitably generalised to the infinite-dimensional setting such that the respective optimal control task can be tackled based on the developed control methodology. In addition, we studied port-Hamiltonian systems. Again, we contributed by first deriving general system-theoretic concepts, e.g. on existence, uniqueness, and stability, and second by analysing an intrinsically motivated optimal control problem, i.e. state transition with minimal energy. The latter leads to a singular OCP, which can, however, due to the port-Hamiltonian structure be properly solved. Here, we started to extend our considerations to infinite-dimensional systems towards the very end of the project, which opens up interesting, future research directions.
Publications
- Constant-coefficient differential-algebraic operators and the Kronecker form. Linear Algebra and its Applications, 552, 29-41, 2018
M. Puche, F.L. Schwenninger, T. Reis
(See online at https://doi.org/10.1016/j.laa.2018.04.005) - Model predictive control for linear differential-algebraic equations. IFAC-PapersOnLine 51(20):98-103, 2018
A. Ilchmann, J. Witschel, and K. Worthmann
(See online at https://doi.org/10.1016/j.ifacol.2018.10.181) - Linear-quadratic optimal control of differential-algebraic systems: the infinite time horizon problem with zero terminal state. SIAM Journal on Control and Optimization, 57(3), 1567-1596, 2019
T. Reis and M. Voigt
(See online at https://doi.org/10.1137/18M1189609) - Optimal control of differential-algebraic equations from an ordinary differential equation perspective. Optimal Control Applications and Methods 40(2):351-366, 2019
A. Ilchmann, L. Leben, J. Witschel, and K. Worthmann
(See online at https://doi.org/10.1002/oca.2481) - Invariance of the essential spectra of operator pencils. R.E. Curto, W. Helton, H. Lin, X. Tang, R. Yang, G. Yu (eds.): Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology, 203-219, Springer, 2020
H. Gernandt, N. Moalla, F. Philipp, W. Selmi, C. Trunk
(See online at https://doi.org/10.1007/978-3-030-43380-2_10) - A linear relations approach to port-Hamiltonian differentialalgebraic equations. SIAM Journal on Matrix Analysis and Applications 42 (2), 1011-1044, 2021
H. Gernandt, F.E. Haller, and T. Reis
(See online at https://doi.org/10.1137/20M1371166) - Control of port-Hamiltonian systems with minimal energy supply. European Journal of Control, 62, 33-40, 2021
M. Schaller, F. Philipp, T. Faulwasser, K. Worthmann, and B. Maschke
(See online at https://doi.org/10.1016/j.ejcon.2021.06.017) - Minimizing the energy supply of infinite-dimensional linear port-Hamiltonian systems. IFAC-PapersOnLine, 54, 155-160, 2021
F. Philipp, M. Schaller, T. Faulwasser, B. Maschke, and K. Worthmann
(See online at https://doi.org/10.1016/j.ifacol.2021.11.071) - Model predictive control for singular differentialalgebraic equations. International Journal of Control, 2021
A. Ilchmann, J. Witschel, and K. Worthmann
(See online at https://doi.org/10.1080/00207179.2021.1900604) - The spectrum and the Weyr characteristics of operator pencils and linear relations. SJM-NAOS 1:73–89, 2021
H. Gernandt and C. Trunk
(See online at https://doi.org/10.48550/arXiv.2106.08726)
