Anzahl der Treffer: 97
Erstellt: Wed, 31 May 2023 23:08:48 +0200 in 0.0828 sec

Faulwasser, Timm; Ou, Ruchuan; Pan, Guanru; Schmitz, Philipp; Worthmann, Karl
Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again. - In: Annual reviews in control, ISSN 1872-9088, Bd. 55 (2023), S. 92-117

The fundamental lemma by Jan C. Willems and co-workers is deeply rooted in behavioral systems theory and it has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style paper combines recent insights into stochastic and descriptor-system formulations of the lemma to further extend and broaden the formal basis for behavioral theory of stochastic linear systems. We show that series expansions - in particular Polynomial Chaos Expansions (PCE) of L2-random variables, which date back to Norbert Wiener’s seminal work - enable equivalent behavioral characterizations of linear stochastic systems. Specifically, we prove that under mild assumptions the behavior of the dynamics of the L2-random variables is equivalent to the behavior of the dynamics of the series expansion coefficients and that it entails the behavior composed of sampled realization trajectories. We also illustrate the short-comings of the behavior associated to the time-evolution of the statistical moments. The paper culminates in the formulation of the stochastic fundamental lemma for linear time-invariant systems, which in turn enables numerically tractable formulations of data-driven stochastic optimal control combining Hankel matrices in realization data (i.e. in measurements) with PCE concepts.
Kleyman, Viktoria; Schaller, Manuel; Mordmüller, Mario; Wilson, Mitsuru; Brinkmann, Ralf; Worthmann, Karl; Müller, Matthias A.
State and parameter estimation for retinal laser treatment. - In: IEEE transactions on control systems technology, ISSN 1558-0865, Bd. 31 (2023), 3, S. 1366-1378

Adequate therapeutic retinal laser irradiation needs to be adapted to local absorption. This leads to time-consuming treatments as the laser power needs to be successively adjusted to avoid undertreatment and overtreatment caused by too low or too high temperatures. Closed-loop control can overcome this burden by means of temperature measurements. To allow for model predictive control schemes, the current state and the spot-dependent absorption need to be estimated. In this article, we thoroughly compare moving horizon estimator (MHE) and extended Kalman filter (EKF) designs for joint state and parameter estimation. We consider two different scenarios, the estimation of one or two unknown absorption coefficients. For one unknown parameter, both estimators perform very similarly. For two unknown parameters, we found that the MHE benefits from active parameter constraints at the beginning of the estimation, whereas after a settling time, both estimators perform again very similarly as long as the parameters are inside the considered parameter bounds.
Caragea, Andrei; Lee, Dae Gwan; Philipp, Friedrich; Voigtlaender, Felix
A Balian-Low type theorem for Gabor Riesz sequences of arbitrary density. - In: Mathematische Zeitschrift, ISSN 1432-1823, Bd. 303 (2023), 2, 48, S. 1-22
Philipp, Friedrich;
Relatively bounded perturbations of J-non-negative operators. - In: Complex analysis and operator theory, ISSN 1661-8262, Bd. 17 (2023), 1, 14, insges. 30 S.

We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant J-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for J-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators with Lp-potentials. Known bounds on the non-real eigenvalues of such operators are improved.
Lee, Dae Gwan; Philipp, Friedrich; Voigtlaender, Felix
A note on the invertibility of the Gabor frame operator on certain modulation spaces. - In: The journal of Fourier analysis and applications, ISSN 1531-5851, Bd. 29 (2023), 1, 3, S. 1-20

We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $$V_1 = H^1(\mathbb {R}^d)$$, the weighted $$L^2$$-space $$V_2 = L_{1 + |x|}^2(\mathbb {R}^d)$$, and the space $$V_3 = \mathbb {H}^1(\mathbb {R}^d) = V_1 \cap V_2$$consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $$L^2$$spaces. In all cases, we prove that the space of Bessel vectors in $$V_j$$is mapped bijectively onto itself by the Gabor frame operator. As a consequence, if the window function belongs to one of the three spaces, then the canonical dual window also belongs to the same space. In fact, the result not only applies to frames, but also to frame sequences.
Viehweg, Johannes; Worthmann, Karl; Mäder, Patrick
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 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.
Nüske, Feliks; Peitz, Sebastian; Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl
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.
Berger, Thomas; Dennstädt, Dario; Ilchmann, Achim; Worthmann, Karl
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.
Faulwasser, Timm; Maschke, Bernhard; Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl
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.
Grüne, Lars; Philipp, Friedrich; Schaller, Manuel
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.