Strengthened Circle and Popov Criteria for the stability analysis of feedback systems with ReLU neural networks
Strengthened Circle and Popov Criteria for the stability analysis of feedback systems with ReLU neural networks
This letter considers the stability analysis of a Lurie system with a static repeated ReLU (rectified linear unit) nonlinearity. Properties of the ReLU function are leveraged to derive new tailored quadratic constraints (QCs) which are satisfied by the repeated ReLU. These QCs are used to strengthen the Circle and Popov Criteria for this specialised Lurie system. It is shown that the criteria can be cast as a set of linear matrix inequalities (LMIs) with less restrictive conditions on the matrix variables. Many systems involving a neural network (NN) with ReLU activations are important instances of this specialised Lurie system; for example, a continuous time recurrent neural network (RNN) or the interconnection of a linear system with a feedforward NN. Numerical examples show the strengthened criteria strike an appealing balance between reduced conservatism and complexity, compared to existing criteria.
LMI, Lyapunov methods, Robust control, Stability of nonlinear systems, neural networks (NNs), stability of nonlinear systems, neural networks, LMIs, robust control
2635-2640
Richardson, Carl
3406b6af-f00d-410b-8051-a0ecc27baba5
Turner, Matthew
6befa01e-0045-4806-9c91-a107c53acba0
Gunn, Stephen
306af9b3-a7fa-4381-baf9-5d6a6ec89868
19 June 2023
Richardson, Carl
3406b6af-f00d-410b-8051-a0ecc27baba5
Turner, Matthew
6befa01e-0045-4806-9c91-a107c53acba0
Gunn, Stephen
306af9b3-a7fa-4381-baf9-5d6a6ec89868
Richardson, Carl, Turner, Matthew and Gunn, Stephen
(2023)
Strengthened Circle and Popov Criteria for the stability analysis of feedback systems with ReLU neural networks.
IEEE Control Systems Letters, 7, .
(doi:10.1109/LCSYS.2023.3287494).
Abstract
This letter considers the stability analysis of a Lurie system with a static repeated ReLU (rectified linear unit) nonlinearity. Properties of the ReLU function are leveraged to derive new tailored quadratic constraints (QCs) which are satisfied by the repeated ReLU. These QCs are used to strengthen the Circle and Popov Criteria for this specialised Lurie system. It is shown that the criteria can be cast as a set of linear matrix inequalities (LMIs) with less restrictive conditions on the matrix variables. Many systems involving a neural network (NN) with ReLU activations are important instances of this specialised Lurie system; for example, a continuous time recurrent neural network (RNN) or the interconnection of a linear system with a feedforward NN. Numerical examples show the strengthened criteria strike an appealing balance between reduced conservatism and complexity, compared to existing criteria.
Text
Strengthened_Circle_Popov
- Accepted Manuscript
More information
Accepted/In Press date: 19 June 2023
e-pub ahead of print date: 19 June 2023
Published date: 19 June 2023
Additional Information:
Funding Information:
This work was supported in part by the Defence Science and Technology Laboratory (DSTL) and in part by the U.K. Research and Innovation (UKRI) Centre of Machine Intelligence for Nano-Electronic Devices and Systems under Grant EP/S024298/1.
Publisher Copyright:
© 2017 IEEE.
Keywords:
LMI, Lyapunov methods, Robust control, Stability of nonlinear systems, neural networks (NNs), stability of nonlinear systems, neural networks, LMIs, robust control
Identifiers
Local EPrints ID: 478495
URI: http://eprints.soton.ac.uk/id/eprint/478495
ISSN: 2475-1456
PURE UUID: 305a544d-08d6-4d9e-9e52-0b6012bc4c01
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Date deposited: 04 Jul 2023 17:38
Last modified: 17 Mar 2024 04:09
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Contributors
Author:
Carl Richardson
Author:
Matthew Turner
Author:
Stephen Gunn
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