Parameterizing echo state networks for multi-step time series prediction. - In: Neurocomputing, ISSN 1872-8286, Bd. 522 (2023), S. 214-228
Prediction of multi-dimensional time-series data, which may represent such diverse phenomena as climate changes or financial markets, remains a challenging task in view of inherent nonlinearities and non-periodic behavior In contrast to other recurrent neural networks, echo state networks (ESNs) are attractive for (online) learning due to lower requirements w.r.t.training data and computational power. However, the randomly-generated reservoir renders the choice of suitable hyper-parameters as an open research topic. We systematically derive and exemplarily demonstrate design guidelines for the hyper-parameter optimization of ESNs. For the evaluation, we focus on the prediction of chaotic time series, an especially challenging problem in machine learning. Our findings demonstrate the power of a hyper-parameter-tuned ESN when auto-regressively predicting time series over several hundred steps. We found that ESNs’ performance improved by 85.1%-99.8% over an already wisely chosen default parameter initialization. In addition, the fluctuation range is considerably reduced such that significantly worse performance becomes very unlikely across random reservoir seeds. Moreover, we report individual findings per hyper-parameter partly contradicting common knowledge to further, help researchers when training new models.
https://doi.org/10.1016/j.neucom.2022.11.044
Finite-data error bounds for Koopman-based prediction and control. - In: Journal of nonlinear science, ISSN 1432-1467, Bd. 33 (2023), 1, 14, S. 1-34
The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein-Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.
https://doi.org/10.1007/s00332-022-09862-1
Towards Funnel MPC for nonlinear systems with relative degree two. - In: Extended abstracts presented at the 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022, (2022), S. 656-659
Funnel MPC, a novel Model Predictive Control (MPC) scheme, allows guaranteed output tracking of smooth reference signals with prescribed error bounds for nonlinear multi-input multi-output systems. To this end, the stage cost resembles the high-gain idea of funnel control. Without imposing additional output constraints or terminal conditions, the Funnel MPC scheme is initially and recursively feasible for systems with relative degree one and stable internal dynamics. Using an additional funnel for the derivative as a penalty term in the stage cost, these results can be also extended to single-input single-output systems with relative degree two.
https://doi.org/10.15495/EPub_UBT_00006809
A note on efficient and reliable prediction-based control in the Koopman framework. - In: Extended abstracts presented at the 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022, (2022), S. 584-587
Extended Dynamic Mode Decomposition, embedded in the Koopman framework, is a widely-applied technique to predict the evolution of an observable along the flow of a dynamical (control) system. However, despite its popularity, the error analysis for control systems is still fragmentary. Here, we provide a complete and rigorous analysis of the approximation error for control systems. To this end, the approximation error is split up according to its two sources of error: the finite dictionary size (projection) and the finite amount of i.i.d. data used to generate the surrogate model (estimation). Then, invoking - among others - finite-elements techniques and the Chebyshev inequality, probabilistic error bounds are derived. Finally, we demonstrate the applicability of the novel error bounds in optimal control with state and control constraints.
https://doi.org/10.15495/EPub_UBT_00006809
Port-Hamiltonian system nodes. - In: Extended abstracts presented at the 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022, (2022), S. 441-444
We present a framework to formulate infinite dimensional port-Hamiltonian systems by means of system nodes, which provide a very general and powerful setting for unbounded input and output operators that appear, e.g., in the context of boundary control or observation. One novelty of our approach is that we allow for unbounded and not necessarily coercive Hamiltonian energies. To this end, we construct finite energy spaces to define the port-Hamiltonian dynamics and give an application in case of multiplication operator Hamiltonians where the Hamiltonian density does not need to be positive or bounded. In order to model systems involving differential operators on these finite energy spaces, we show that if the total mass w.r.t. the Hamiltonian density (and its inverse) is finite, one can define a unique weak derivative.
https://doi.org/10.15495/EPub_UBT_00006809
Verbund: 05M2018 - KONSENS : Schlussbericht : Förderzeitraum: 01.01.2018-30.06.2021. - Ilmenau : Technische Universität Ilmenau. - 1 Online-Ressource (16 Seiten, 450,82 KB)Förderkennzeichen BMBF 05M18SIA
https://edocs.tib.eu/files/e01fb23/1870887948.pdf
Extended abstracts presented at the 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022 : held 12-16 September 2022 in Bayreuth, Germany. - Bayreuth : Universität Bayreuth, 2022. - 1 Online-Ressource
Foreword: After more than two years of limited social and scientific interactions due to the Covid-19 pandemic, it was a pleasure to welcome more than 300 participants in person and about 60 online participants at MTNS 2022 in Bayreuth. Submissions to MTNS 2022 were possible as extended abstracts and full papers. The accepted full papers that were presented at the conference are published in IFAC PapersOnline https://www.sciencedirect.com/journal/ifacpapersonline/vol/55/issue/30. In this volume you find the extended abstracts that were presented at the conference. Further, you also find the titles of the plenary and semi-plenary talks as well as their abstracts resp. links to the corresponding full papers. We hope you enjoy these abstracts and to see you in person at MTNS in the future. The Editors M. H. Baumann, L. Grüne, B. Jacob, and K. Worthmann
https://doi.org/10.15495/EPub_UBT_00006809
Funnel model predictive control for nonlinear systems with relative degree one. - In: SIAM journal on control and optimization, ISSN 1095-7138, Bd. 60 (2022), 6, S. 3358-3383
We show that Funnel MPC, a novel model predictive control (MPC) scheme, allows tracking of smooth reference signals with prescribed performance for nonlinear multi-input multioutput systems of relative degree one with stable internal dynamics. The optimal control problem solved in each iteration of funnel MPC resembles the basic idea of penalty methods used in optimization. To this end, we present a new stage cost design to mimic the high-gain idea of (adaptive) funnel control. We rigorously show initial and recursive feasibility of funnel MPC without imposing terminal conditions or other requirements like a sufficiently long prediction horizon.
https://doi.org/10.1137/21M1431655
Optimal control of port-Hamiltonian descriptor systems with minimal energy supply. - In: SIAM journal on control and optimization, ISSN 1095-7138, Bd. 60 (2022), 4, S. 2132-2158
We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. Further, we derive a input-state turnpike toward a subspace for optimal control of port-Hamiltonian ordinary differential equations with a feed-through term and a turnpike property for the corresponding adjoint states toward zero. In an appendix we characterize the class of dissipative Hamiltonian matrices and pencils.
https://doi.org/10.1137/21M1427723
Strict dissipativity for generalized linear-quadratic problems in infinite dimensions. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 55 (2022), 30, S. 311-316
We analyze strict dissipativity of generalized linear quadratic optimal control problems on Hilbert spaces. Here, the term “generalized” refers to cost functions containing both quadratic and linear terms. We characterize strict pre-dissipativity with a quadratic storage function via coercivity of a particular Lyapunov-like quadratic form. Further, we show that under an additional algebraic assumption, strict pre-dissipativity can be strengthened to strict dissipativity. Last, we relate the obtained characterizations of dissipativity with exponential detectability.
https://doi.org/10.1016/j.ifacol.2022.11.071