arXiv subjects

Dieter Teichrib

Publications and source records attributed to Dieter Teichrib.

9 recordsLinked to original sources

arXiv2025

A mixed-integer framework for analyzing neural network-based controllers for piecewise affine systems with bounded disturbances

We present a method for representing the closed-loop dynamics of piecewise affine (PWA) systems with bounded additive disturbances and neural network-based controllers as mixed-integer (MI) linear constraints. We show that such representations enable the computation of robustly positively invariant (RPI) sets for the specified system class by solving MI linear programs. These RPI sets can subsequently be used to certify stability and constraint satisfaction. Furthermore, the approach allows to handle non-linear systems based on suitable PWA approximations and corresponding error bounds, which can be interpreted as the bounded disturbances from above.

eess.SY↗

arXiv2024

Reachability analysis for piecewise affine systems with neural network-based controllers

Neural networks (NN) have been successfully applied to approximate various types of complex control laws, resulting in low-complexity NN-based controllers that are fast to evaluate. However, when approximating control laws using NN, performance and stability guarantees of the original controller may not be preserved. Recently, it has been shown that it is possible to provide such guarantees for linear systems with NN-based controllers by analyzing the approximation error with respect to a stabilizing base-line controller or by computing reachable sets of the closed-loop system. The latter has the advantage of not requiring a base-line controller. In this paper, we show that similar ideas can be used to analyze the closed-loop behavior of piecewise affine (PWA) systems with an NN-based controller. Our approach builds on computing over-approximations of reachable sets using mixed-integer linear programming, which allows to certify that the closed-loop system converges to a small set containing the origin while satisfying input and state constraints. We also show how to modify a given NN-based controller to ensure asymptotic stability for the controlled PWA system.

eess.SY↗

arXiv2024

Piecewise regression via mixed-integer programming for MPC

Piecewise regression is a versatile approach used in various disciplines to approximate complex functions from limited, potentially noisy data points. In control, piecewise regression is, e.g., used to approximate the optimal control law of model predictive control (MPC), the optimal value function, or unknown system dynamics. Neural networks are a common choice to solve the piecewise regression problem. However, due to their nonlinear structure, training is often based on gradient-based methods, which may fail to find a global optimum or even a solution that leads to a small approximation error. To overcome this problem and to find a global optimal solution, methods based on mixed-integer programming (MIP) can be used. However, the known MIP-based methods are either limited to a special class of functions, e.g., convex piecewise affine functions, or they lead to complex approximations in terms of the number of regions of the piecewise defined function. Both complicate a usage in the framework of control. We propose a new MIP-based method that is not restricted to a particular class of piecewise defined functions and leads to functions that are fast to evaluate and can be used within an optimization problem, making them well suited for use in control.

math.OC↗

arXiv2023

Efficient computation of Lipschitz constants for MPC with symmetries

Lipschitz constants for linear MPC are useful for certifying inherent robustness against unmodeled disturbances or robustness for neural network-based approximations of the control law. In both cases, knowing the minimum Lipschitz constant leads to less conservative certifications. Computing this minimum Lipschitz constant is trivial given the explicit MPC. However, the computation of the explicit MPC may be intractable for complex systems. The paper discusses a method for efficiently computing the minimum Lipschitz constant without using the explicit control law. The proposed method simplifies a recently presented mixed-integer linear program (MILP) that computes the minimum Lipschitz constant. The simplification is obtained by exploiting saturation and symmetries of the control law and irrelevant constraints of the optimal control problem.

math.OC↗

arXiv2023

Error bounds for maxout neural network approximations of model predictive control

Neural network (NN) approximations of model predictive control (MPC) are a versatile approach if the online solution of the underlying optimal control problem (OCP) is too demanding and if an exact computation of the explicit MPC law is intractable. The drawback of such approximations is that they typically do not preserve stability and performance guarantees of the original MPC. However, such guarantees can be recovered if the maximum error with respect to the optimal control law and the Lipschitz constant of that error are known. We show in this work how to compute both values exactly when the control law is approximated by a maxout NN. We build upon related results for ReLU NN approximations and derive mixed-integer (MI) linear constraints that allow a computation of the output and the local gain of a maxout NN by solving an MI feasibility problem. Furthermore, we show theoretically and experimentally that maxout NN exist for which the maximum error is zero.

eess.SY↗

arXiv2023

A deterministic view on explicit data-driven (M)PC

We show that the explicit realization of data-driven predictive control (DPC) for linear deterministic systems is more tractable than previously thought. To this end, we compare the optimal control problems (OCP) corresponding to deterministic DPC and classical model predictive control (MPC), specify its close relation, and systematically eliminate ambiguity inherent in DPC. As a central result, we find that the explicit solutions to these types of DPC and MPC are of exactly the same complexity. We illustrate our results with two numerical examples highlighting features of our approach.

eess.SY↗

arXiv2022

An asymmetric non-uniform 3-dB directional coupler with 300-nm bandwidth

Here, we demonstrate a new type of 3-dB coupler that has an ultra-broadband operational range starting from 1300 nm to 1600 nm with low fabrication sensitivity. Overall device size is 800 micron including in/out S-bend waveguides. The coupler is an asymmetric non-uniform directional coupler that consists of two tapered waveguides. One of the coupler arms is shifted by 130 micron in the propagation direction, which reduces the phase mismatch between the odd and even mode of the coupled system resulting in a better wavelength-flattened response compared to a not shifted coupler. Moreover, compared to a long adiabatic coupler, we achieved a similar wavelength response at 16 times smaller device length. The couplers were fabricated using the silicon nitride platform of Lionix International. We also experimentally demonstrated an optical switch that is made by using two of these couplers in a Mach Zehnder interferometer configuration. According to experimental results, this optical switch exhibits -10 dB of extinction ratio over a 100 nm wavelength range (1500-1600 nm). Our results indicate that the asymmetric non-uniform directional coupler holds great promise for various applications including optical imaging, telecommunications, and reconfigurable photonic processors where compact, fabrication-tolerant and wavelength-insensitive couplers are essential.

physics.optics↗

arXiv2022

Tailored max-out networks for learning convex PWQ functions

Convex piecewise quadratic (PWQ) functions frequently appear in control and elsewhere. For instance, it is well-known that the optimal value function (OVF) as well as Q-functions for linear MPC are convex PWQ functions. Now, in learning-based control, these functions are often represented with the help of artificial neural networks (NN). In this context, a recurring question is how to choose the topology of the NN in terms of depth, width, and activations in order to enable efficient learning. An elegant answer to that question could be a topology that, in principle, allows to exactly describe the function to be learned. Such solutions are already available for related problems. In fact, suitable topologies are known for piecewise affine (PWA) functions that can, for example, reflect the optimal control law in linear MPC. Following this direction, we show in this paper that convex PWQ functions can be exactly described by max-out-NN with only one hidden layer and two neurons.

eess.SY↗

arXiv2021

Tailored neural networks for learning optimal value functions in MPC

Learning-based predictive control is a promising alternative to optimization-based MPC. However, efficiently learning the optimal control policy, the optimal value function, or the Q-function requires suitable function approximators. Often, artificial neural networks (ANN) are considered but choosing a suitable topology is also non-trivial. Against this background, it has recently been shown that tailored ANN allow, in principle, to exactly describe the optimal control policy in linear MPC by exploiting its piecewise affine structure. In this paper, we provide a similar result for representing the optimal value function and the Q-function that are both known to be piecewise quadratic for linear MPC.

eess.SY↗