23 research outputs found
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.Comment: 8 pages, 2 tables, to be published in the proceedings of the 22nd
World Congress of the International Federation of Automatic Control (2023
How scaling of the disturbance set affects robust positively invariant sets for linear systems
This paper presents new results on robust positively invariant (RPI) sets for
linear discrete-time systems with additive disturbances. In particular, we
study how RPI sets change with scaling of the disturbance set. More precisely,
we show that many properties of RPI sets crucially depend on a unique scaling
factor which determines the transition from nonempty to empty RPI sets. We
characterize this critical scaling factor, present an efficient algorithm for
its computation, and analyze it for a number of examples from the literature
Implicit predictors in regularized data-driven predictive control
We introduce the notion of implicit predictors, which characterize the
input-(state)-output prediction behavior underlying a predictive control
scheme, even if it is not explicitly enforced as an equality constraint (as in
traditional model or subspace predictive control). To demonstrate this concept,
we derive and analyze implicit predictors for some basic data-driven predictive
control (DPC) schemes, which offers a new perspective on this popular approach
that may form the basis for modified DPC schemes and further theoretical
insights.Comment: This paper is a reprint of a contribution to the IEEE Control Systems
Letters. 6 pages, 2 figure
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.Comment: 7 pages, 2 figures, 1 tabl
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.Comment: 8 pages, 2 figures, to be published in the proceedings of the 22nd
World Congress of the International Federation of Automatic Control (2023
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.Comment: 7 pages, 2 figure, submitted to 61st IEE Conference on Decision and
Control 202
On the maximal controller gain in linear MPC
The paper addresses the computation of Lipschitz constants for model predictive control (MPC) laws. Such Lipschitz constants are useful to assess the inherent robustness of nominal MPC for disturbed systems. It is shown that a Lipschitz constant can be computed by identifying the maximal controller gain of the MPC. Clearly, given the explicit description of the MPC, this gain can be easily identified. The computation of the explicit MPC may, however, be numerically demanding. The goal of the paper thus is to overestimate the maximal controller gain without using the explicit control law