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IEEE Conference on Decision and Control (CDC)

Asymptotic Stabilization over Encrypted Data with Limited Controller Capacity and Time-varying Quantizer

Junsoo Kim, Moritz Schulze Darup, Henrik Sandberg, Karl Henrik Johansson

Published Dec 6, 2022• 1 citation
Publisher

Abstract

We consider a problem of implementing dynamic controllers over encrypted data for asymptotic stabilization of closed-loop systems. Though a time-varying quantizer is used and it can be infinitesimally fine with time, a major issue is that the underlying space for encrypted messages is unavoidably finite and the controller receives a limited amount of quantized data. To resolve this issue, the proposed method takes advantage of the state matrix consisting of integers, which enables the controller to generate only lower bits of the same output without computing the upper bits. Whenever a portion of the upper bits of output has converged, the computation scope can be moved further lower, receiving only lower bits of the measurement. The quantization is scheduled and the size of the message space is predetermined from the convergence rate, so that the feedback input is restored from the outcome of the lower bits, no matter how fine quantization is performed in the end. As a consequence, asymptotic stabilization can be achieved by encrypted operation, despite the limited controller capacity.

DOI
10.1109/cdc51059.2022.9993215

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We consider a problem of implementing dynamic controllers over encrypted data for asymptotic stabilization of closed-loop systems. Though a time-varying quantizer is used and it can be infinitesimally fine with time, a major issue is that the underlying space for encrypted messages is unavoidably finite and the controller receives a limited amount of quantized data. To resolve this issue, the proposed method takes advantage of the state matrix consisting of integers, which enables the controller to generate only lower bits of the same output without computing the upper bits. Whenever a portion of the upper bits of output has converged, the computation scope can be moved further lower, receiving only lower bits of the measurement. The quantization is scheduled and the size of the message space is predetermined from the convergence rate, so that the feedback input is restored from the outcome of the lower bits, no matter how fine quantization is performed in the end. As a consequence, asymptotic stabilization can be achieved by encrypted operation, despite the limited controller capacity.
This paper proposes a novel event-driven encrypted control framework for linear networked control systems (NCSs), which relies on two modified uniform quantization policies, the Paillier cryptosystem, and an event-triggered strategy. Due to the fact that only integers can work in the Pailler cryptosystem, both the real-valued control gain and system state need to be first quantized before encryption. This is dramatically different from the existing quantized control methods, where only the quantization of a single value, e.g., the control input or the system state, is considered. To handle this issue, static and dynamic quantization policies are presented, which achieve the desired integer conversions and guarantee asymptotic convergence of the quantized system state to the equilibrium. Then, the quantized system state is encrypted and sent to the controller when the triggering condition, specified by a state-based event-triggered strategy, is satisfied. By doing so, not only the security and confidentiality of data transmitted over the communication network are protected, but also the ciphertext expansion phenomenon can be relieved. Additionally, by tactfully designing the quantization sensitivities and triggering error, the proposed event-driven encrypted control framework ensures the asymptotic stability of the overall closed-loop system. Finally, a simulation example of the secure motion control for an inverted pendulum cart system is presented to evaluate the effectiveness of the theoretical results.
Encrypted control has been extensively studied to ensure the confidentiality of system states and control inputs for networked control systems. This paper presents a computationally efficient encrypted control framework for networked systems enabled by quantum communication. A quantum channel between sensors and actuators is used to generate identical secret keys, whose security is further enhanced through quantum key distribution. These keys enable lightweight encryption and decryption while preserving confidentiality and control accuracy. We develop a novel encryption-decryption architecture for state-feedback control of linear systems based on quantum keys, and characterize the impact of quantum state errors on closed-loop stability. In particular, we establish the existence of a critical threshold on intrinsic quantum noise below which stability is guaranteed. In contrast to classical encrypted control schemes, which may collapse under a single key-bit error, the proposed quantum encrypted control exhibits strong robustness to key imperfections. We further adopt quantization techniques to address the scenarios with limited communication bits in practical situations, and implement privacy protection for quantum keys based on a stochastic quantizer. These results demonstrate that integrating quantum technologies into control systems in a nontrivial and principled manner, even at their current level of maturity, can yield substantial performance gains in reducing computational complexity and improving resilience to key errors while ensuring security against multiple eavesdropping sources.
In practical applications, quantum systems are inevitably subject to significant uncertainties, including unknown initial states, imprecise physical parameters, and unmodeled environmental noise, all of which pose major challenges to robust quantum feedback control. This paper proposes a feedback stabilization strategy based on a reduced quantum filter that achieves robustness against time-varying Hamiltonian perturbations and additional dissipative effects, without requiring prior knowledge of the initial state or exact system parameters. The proposed filter estimates only O(N) real variables corresponding to the diagonal elements of the system density matrix in a quantum non-demolition basis in contrast to the O(N^2) variables required by a full stochastic master equation, where N is the Hilbert space dimension. This dimensionality reduction substantially simplifies real-time computation and feedback implementation while preserving both convergence and robustness guarantees. Rigorous analysis further establishes global exponential stability of the target subspace. The results provide a scalable framework for robust and efficient measurement-based feedback control applicable to high-dimensional perturbed open quantum systems.
The design of encrypted controllers, which perform control operation directly over encrypted signals via homomorphic cryptosystems, should consider both security of the control data and performance of the controller. Considering the use of Learning With Errors (LWE) based cryptosystem, in this paper, we present a design procedure for encrypted linear dynamic controllers. Providing a guideline for choosing the parameters of the cryptosystem as well as quantization, the procedure guarantees both the desired level of security and performance. Receiving the encrypted signals of the plant input as well as the output, the proposed controller is able to perform the dynamic operation over encrypted data for infinite time horizon, without the use of decryption or reset of the state. Thanks to additively and multiplicatively homomorphic property of the LWE-based cryptosystem, information of both control signals and gain matrices is protected by encryption.
Due to the dynamic arrival of data packets and time-varying channel states, secure transmission opportunities will be wasted when there is no confidential message to transmit, and data packet transmission can be delayed while waiting for the next available secure transmission opportunity. In order to seize every precious secure transmission opportunity and reduce the data packet waiting time, we propose a secure transmission protocol in which the under-utilized secure transmission opportunities can be exploited to transmit key packets, and these key packets will encrypt the confidential messages in the subsequent transmissions. Following the above-mentioned principle, we first apply this protocol to a single-input single-output network, and optimally allocate the secure transmission opportunities for the key and data transmissions, such that the secrecy rate is maximized under the constraints of the minimum energy efficiency and queue stability requirements. In addition, the packet delay for the proposed protocol is investigated using a generating function approach and a closed-form expression of the packet delay is derived. Then, we extend our work to the multiple-input single-output network and utilize the key queue as well as the artificial noise to protect the confidential messages. Similarly, we also optimally allocate the transmission opportunities for the key and data transmissions to maximize the secrecy rate and study the data packet delay performance. Since all secrecy transmission opportunities are utilized in the proposed protocol, the full secrecy rate is achieved. Numerical results are demonstrated to verify the performance superiority of the proposed protocols when compared with the other key encrypted schemes in terms of the data packet delay and the secrecy rate.