arXiv subjects

Moritz Schulze Darup

Publications and source records attributed to Moritz Schulze Darup.

At least 37 records · Page 2Linked to original sources

arXiv2024

On the design of stabilizing FIR controllers

Recently, it has been observed that finite impulse response controllers are an excellent basis for encrypted control, where privacy-preserving controller evaluations via special cryptosystems are the main focus. Beneficial properties of FIR filters are also well-known from digital signal processing, which makes them preferable over infinite impulse response filters in many applications. Their appeal extends to feedback control, offering design flexibility grounded solely on output measurements. However, designing FIR controllers is challenging, which motivates this work. To address the design challenge, we initially show that FIR controller designs for linear systems can equivalently be stated as static or dynamic output feedback problems. After focusing on the existence of stabilizing FIR controllers for a given plant, we tailor two common design approaches for output feedback to the case of FIR controllers. Unfortunately, it will turn out that the FIR characteristics add further restrictions to the LMI-based approaches. Hence, we finally turn to designs building on non-convex optimization, which provide satisfactory results for a selection of benchmark systems.

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arXiv2024

Privacy-preserving gradient-based fair federated learning

Federated learning (FL) schemes allow multiple participants to collaboratively train neural networks without the need to directly share the underlying data.However, in early schemes, all participants eventually obtain the same model. Moreover, the aggregation is typically carried out by a third party, who obtains combined gradients or weights, which may reveal the model. These downsides underscore the demand for fair and privacy-preserving FL schemes. Here, collaborative fairness asks for individual model quality depending on the individual data contribution. Privacy is demanded with respect to any kind of data outsourced to the third party. Now, there already exist some approaches aiming for either fair or privacy-preserving FL and a few works even address both features. In our paper, we build upon these seminal works and present a novel, fair and privacy-preserving FL scheme. Our approach, which mainly relies on homomorphic encryption, stands out for exclusively using local gradients. This increases the usability in comparison to state-of-the-art approaches and thereby opens the door to applications in control.

cs.LG↗

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↗

arXiv2024

A code-driven tutorial on encrypted control: From pioneering realizations to modern implementations

The growing interconnectivity in control systems due to robust wireless communication and cloud usage paves the way for exciting new opportunities such as data-driven control and service-based decision-making. At the same time, connected systems are susceptible to cyberattacks and data leakages. Against this background, encrypted control aims to increase the security and safety of cyber-physical systems. A central goal is to ensure confidentiality of process data during networked controller evaluations, which is enabled by, e.g., homomorphic encryption. However, the integration of advanced cryptographic systems renders the design of encrypted controllers an interdisciplinary challenge. This code-driven tutorial paper aims to facilitate the access to encrypted control by providing exemplary realizations based on popular homomorphic cryptosystems. In particular, we discuss the encrypted implementation of state feedback and PI controllers using the Paillier, GSW, and CKKS cryptosystem.

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arXiv2024

Towards a unifying framework for data-driven predictive control with quadratic regularization

Data-driven predictive control (DPC) has recently gained popularity as an alternative to model predictive control (MPC). Amidst the surge in proposed DPC frameworks, upon closer inspection, many of these frameworks are more closely related (or perhaps even equivalent) to each other than it may first appear. We argue for a more formal characterization of these relationships so that results can be freely transferred from one framework to another, rather than being uniquely attributed to a particular framework. We demonstrate this idea by examining the connection between $γ$-DDPC and the original DeePC formulation.

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arXiv2024

Privacy Analysis of Affine Transformations in Cloud-based MPC: Vulnerability to Side-knowledge

Search for the optimizer in computationally demanding model predictive control (MPC) setups can be facilitated by Cloud as a service provider in cyber-physical systems. This advantage introduces the risk that Cloud can obtain unauthorized access to the privacy-sensitive parameters of the system and cost function. To solve this issue, i.e., preventing Cloud from accessing the parameters while benefiting from Cloud computation, random affine transformations provide an exact yet light weight in computation solution. This research deals with analyzing privacy preserving properties of these transformations when they are adopted for MPC problems. We consider two common strategies for outsourcing the optimization required in MPC problems, namely separate and dense forms, and establish that random affine transformations utilized in these forms are vulnerable to side-knowledge from Cloud. Specifically, we prove that the privacy guarantees of these methods and their extensions for separate form are undermined when a mild side-knowledge about the problem in terms of structure of MPC cost function is available. In addition, while we prove that outsourcing the MPC problem in the dense form inherently leads to some degree of privacy for the system and cost function parameters, we also establish that affine transformations applied to this form are nevertheless prone to be undermined by a Cloud with mild side-knowledge. Numerical simulations confirm our results.

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arXiv2023

On the security of randomly transformed quadratic programs for privacy-preserving cloud-based control

Control related data, such as system states and inputs or controller specifications, is often sensitive. Meanwhile, the increasing connectivity of cloud-based or networked control results in vast amounts of such data, which poses a privacy threat, especially when evaluation on external platforms is considered. In this context, a cipher based on a random affine transformation gained attention, which is supposed to enable privacy-preserving evaluations of quadratic programs (QPs) with little computational overhead compared to other methods. This paper deals with the security of such randomly transformed QPs in the context of model predictive control (MPC). In particular, we show how to construct attacks against this cipher and thereby underpin concerns regarding its security in a practical setting. To this end, we exploit invariants under the transformations and common specifications of MPC-related QPs. Our numerical examples then illustrate that these two ingredients suffice to extract information from ciphertexts.

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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.

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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.

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arXiv2023

Convex reformulations for a special class of nonlinear MPC problems

We show how the solution to NMPC problems for a special type of input-affine discrete-time systems can be obtained by reformulating the underlying non-convex optimal control problem in terms of a finite number of convex subproblems. The reformulation is facilitated by exact (input-state) linearization, which is shown to provide beneficial properties for the treated class of systems. We characterize possible types of the resulting convex subproblems and illustrate our approach with three numerical examples.

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arXiv2023

Cryptanalysis of Random Affine Transformations for Encrypted Control

Cloud-based and distributed computations are of growing interest in modern control systems. However, these technologies require performing computations on not necessarily trustworthy platforms and, thus, put the confidentiality of sensitive control-related data at risk. Encrypted control has dealt with this issue by utilizing modern cryptosystems with homomorphic properties, which allow a secure evaluation at the cost of an increased computation or communication effort (among others). Recently, a cipher based on a random affine transformation gained attention in the encrypted control community. Its appeal stems from the possibility to construct security providing homomorphisms that do not suffer from the restrictions of ``conventional'' approaches. This paper provides a cryptanalysis of random affine transformations in the context of encrypted control. To this end, a deterministic and probabilistic variant of the cipher over real numbers are analyzed in a generalized setup, where we use cryptographic definitions for security and attacker models. It is shown that the deterministic cipher breaks under a known-plaintext attack, and unavoidably leaks information of the closed-loop, which opens another angle of attack. For the probabilistic variant, statistical indistinguishability of ciphertexts can be achieved, which makes successful attacks unlikely. We complete our analysis by investigating a floating point realization of the probabilistic random affine transformation cipher, which unfortunately suggests the impracticality of the scheme if a security guarantee is needed.

cs.CR↗

arXiv2022

Encrypted extremum seeking for privacy-preserving PID tuning as-a-Service

Wireless communication offers many benefits for control such as substantially reduced deployment costs, higher flexibility, as well as easier data access. It is thus not surprising that smart and wireless sensors and actuators are increasingly used in industry. With these enhanced possibilities, exciting new technologies such as Control-as-a-Service arise, where (for example) controller design or tuning based on input-output-data can be outsourced to a cloud or mobile device. This implies, however, that sensitive plant information may become available to service providers or, possibly, attackers. Against this background, we focus on privacy-preserving optimal PID tuning as-a-Service here. In particular, we combine homomorphic encryption with extremum seeking in order to provide a purely data-driven and confidential tuning algorithm. The encrypted realization requires several adaptions of established extremum seekers. These encompass relative parameter updates, stochastic gradient approximations, and a normalized objective function. As a result, and as illustrated by various numerical examples, the proposed encrypted extremum seeker is able to tune PID controllers for a wide variety of plants without being too conservative.

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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.

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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.

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arXiv2021

Novel convex decomposition of piecewise affine functions

In this paper, we present a novel approach to decompose a given piecewise affine (PWA) function into two convex PWA functions. Convex decompositions are useful to speed up or distribute evaluations of PWA functions. Different approaches to construct a convex decomposition have already been published. However, either the two resulting convex functions have very high or very different complexities, which is often undesirable, or the decomposition procedure is inapplicable even for simple cases. Our novel methodology significantly reduces these drawbacks in order to extend the applicability of convex decompositions.

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arXiv2021

Teaching MPC: Which Way to the Promised Land?

Since the earliest conceptualizations by Lee and Markus, and Propoi in the 1960s, Model Predictive Control (MPC) has become a major success story of systems and control with respect to industrial impact and with respect to continued and wide-spread research interest. The field has evolved from conceptually simple linear-quadratic (convex) settings in discrete and continuous time to nonlinear and distributed settings including hybrid, stochastic, and infinite-dimensional systems. Put differently, essentially the entire spectrum of dynamic systems can be considered in the MPC framework with respect to both -- system theoretic analysis and tailored numerics. Moreover, recent developments in machine learning also leverage MPC concepts and learning-based and data-driven MPC have become highly active research areas. However, this evident and continued success renders it increasingly complex to live up to industrial expectations while enabling graduate students for state-of-the-art research in teaching MPC. Hence, this position paper attempts to trigger a discussion on teaching MPC. To lay the basis for a fruitful debate, we subsequently investigate the prospect of covering MPC in undergraduate courses; we comment on teaching textbooks; and we discuss the increasing complexity of research-oriented graduate teaching of~MPC.

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arXiv2019

On closed-loop dynamics of ADMM-based MPC

This paper studies the closed-loop dynamics of linear systems under approximate model predictive control (MPC). More precisely, we consider MPC implementations based on a finite number of ADMM iterations per time-step. We first show that the closed-loop dynamics can be described based on a nonlinear augmented model. We then characterize an invariant set around the augmented origin, where the dynamics become linear. Finally, we investigate the performance of the approximate MPC for various choices of the ADMM parameters based on a comprehensive numerical benchmark.

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