Stability Verification of Neural Network Controllers Using Mixed-Integer Programming

We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline pol...

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Published in:	IEEE transactions on automatic control Vol. 68; no. 12; pp. 1 - 16
Main Authors:	Schwan, Roland, Jones, Colin N., Kuhn, Daniel
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.12.2023 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Approximation Approximation error Artificial neural networks Closed loops Computing costs Integer programming Linear programming Mathematical analysis Mixed integer Neural networks Numerical stability Parameterization Policies Power converters Power electronics Predictive control Programming Quadratic programming Stability analysis Symmetric matrices Verification
ISSN:	0018-9286, 1558-2523
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Abstract	We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline policy, which is known to be stable but expensive to evaluate. We provide sufficient conditions for the closed-loop stability of the candidate policy in terms of the worst-case approximation error with respect to the baseline policy, and we show that these conditions can be checked by solving a Mixed-Integer Quadratic Program (MIQP). Additionally, we demonstrate that an outer and inner approximation of the stability region of the candidate policy can be computed by solving an MILP. The proposed framework is sufficiently general to accommodate a broad range of candidate policies including ReLU Neural Networks (NNs), optimal solution maps of parametric quadratic programs, and Model Predictive Control (MPC) policies. We also present an open-source toolbox in Python based on the proposed framework, which allows for the easy verification of custom NN architectures and MPC formulations. We showcase the flexibility and reliability of our framework in the context of a DC-DC power converter case study and investigate its computational complexity.
AbstractList	We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline policy, which is known to be stable but expensive to evaluate. We provide sufficient conditions for the closed-loop stability of the candidate policy in terms of the worst-case approximation error with respect to the baseline policy, and we show that these conditions can be checked by solving a Mixed-Integer Quadratic Program (MIQP). Additionally, we demonstrate that an outer and inner approximation of the stability region of the candidate policy can be computed by solving an MILP. The proposed framework is sufficiently general to accommodate a broad range of candidate policies including ReLU Neural Networks (NNs), optimal solution maps of parametric quadratic programs, and Model Predictive Control (MPC) policies. We also present an open-source toolbox in Python based on the proposed framework, which allows for the easy verification of custom NN architectures and MPC formulations. We showcase the flexibility and reliability of our framework in the context of a DC-DC power converter case study and investigate its computational complexity. In this article, we propose a framework for the stability verification of mixed-integer linear programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline policy, which is known to be stable but expensive to evaluate. We provide sufficient conditions for the closed-loop stability of the candidate policy in terms of the worst-case approximation error with respect to the baseline policy, and we show that these conditions can be checked by solving a mixed-integer quadratic program. In addition, we demonstrate that an outer and inner approximation of the stability region of the candidate policy can be computed by solving an MILP. The proposed framework is sufficiently general to accommodate a broad range of candidate policies including ReLU neural networks (NNs), optimal solution maps of parametric quadratic programs, and model predictive control (MPC) policies. We also present an open-source toolbox in Python based on the proposed framework, which allows for the easy verification of custom NN architectures and MPC formulations. We showcase the flexibility and reliability of our framework in the context of a dc–dc power converter case study and investigate its computational complexity.
Author	Schwan, Roland Jones, Colin N. Kuhn, Daniel
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Snippet	We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a... In this article, we propose a framework for the stability verification of mixed-integer linear programming (MILP) representable control policies. This...
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SubjectTerms	Approximation Approximation error Artificial neural networks Closed loops Computing costs Integer programming Linear programming Mathematical analysis Mixed integer Neural networks Numerical stability Parameterization Policies Power converters Power electronics Predictive control Programming Quadratic programming Stability analysis Symmetric matrices Verification
Title	Stability Verification of Neural Network Controllers Using Mixed-Integer Programming
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