Notes on Recent Achievements in Proving Stability using KeYmaeraX

KeYmaeraX is a Hoare-style theorem prover for hybrid systems. A hybrid system can be seen as an aggregation of both discrete and continuous variables, whose values can change abruptly or continuously, respectively. KeYmaeraX supports only variables having the primitive type bool or real. Due to the mixture of discrete and continuous system elements, one promising application area for KeYmaeraX are closed-loop control systems. A closed-loop control system consists of a plant and a controller. While the plant is basically an aggregation of continuous variables whose values change over time accordingly to physical laws, the controller can be seen as an algorithm formulated in a classical programming language. In this paper, we review some recent extensions of the proof calculus applied by KeYmaeraX that make formal proofs on the stability of dynamic systems more feasible. Based on an example, we first introduce to the topic and prove asymptotic stability of a given system in a hand-written mathematical style. This approach is then compared with a formal encoding of the problem and a formal proof established in KeYmaeraX. We also discuss open problems such as the formalization of asymptotic stability.

Dual Sequence Controller with Delayed Signal Cancellation in the Rotating Reference Frame

Manuscript submitted to the Twenty-second IEEE Workshop on Control and Modeling for Power Electronics (COMPEL 2021).<div>Abstract: Dual-sequence current controllers of voltage source converters (VSCs) feature two separate rotating reference frames (RRFs), commonly named dq frames, and rely on techniques that isolate the positive and negative sequences of three-phase measurements. One of these techniques is the delayed signal cancellation (DSC). It is performed in the stationary reference frame (SRF), also known as αβ frame. The DSC combines old values of one axis with new values of the other axis of the SRF. The results are, then, transformed into the RRFs for use in the current controller. This filtering process introduces an extra layer of complexity for dual-sequence current controllers, which could otherwise operate solely in the RRFs. This paper introduces a frequency adaptive DSC method that operates directly in the RRF. Moreover, an averaging of two of the proposed DSC filters with contiguous integer delays is employed for reducing discretization errors caused by grid frequency excursions. A formal proof of the equivalence between the αβ and dq DSC methods is presented. Furthermore, computer simulations of a case study support the interpretation of the results.</div>

Dual Sequence Controller with Delayed Signal Cancellation in the Rotating Reference Frame

Manuscript submitted to the Twenty-second IEEE Workshop on Control and Modeling for Power Electronics (COMPEL 2021).<div>Abstract: Dual-sequence current controllers of voltage source converters (VSCs) feature two separate rotating reference frames (RRFs), commonly named dq frames, and rely on techniques that isolate the positive and negative sequences of three-phase measurements. One of these techniques is the delayed signal cancellation (DSC). It is performed in the stationary reference frame (SRF), also known as αβ frame. The DSC combines old values of one axis with new values of the other axis of the SRF. The results are, then, transformed into the RRFs for use in the current controller. This filtering process introduces an extra layer of complexity for dual-sequence current controllers, which could otherwise operate solely in the RRFs. This paper introduces a frequency adaptive DSC method that operates directly in the RRF. Moreover, an averaging of two of the proposed DSC filters with contiguous integer delays is employed for reducing discretization errors caused by grid frequency excursions. A formal proof of the equivalence between the αβ and dq DSC methods is presented. Furthermore, computer simulations of a case study support the interpretation of the results.</div>

Ownership and Immutability in Coq

<p>A significant issue in modern programming languages is unsafe aliasing. Modern type systems have attempted to address this in two prominent ways; immutability and ownership, and often a combination of the two [4][17]. The goal of this thesis is to formalise Immutability and Ownership using the Coq Proof Assistant, a formal proof management system [13]. We encode three type systems using Coq; Featherweight Immutable Java, Featherweight Generic Java and Featherweight Ownership Generic Java, and prove them sound. We describe the challenges presented in encoding immutability, ownership and type systems in general in Coq.</p>

Ownership and Immutability in Coq

<p>A significant issue in modern programming languages is unsafe aliasing. Modern type systems have attempted to address this in two prominent ways; immutability and ownership, and often a combination of the two [4][17]. The goal of this thesis is to formalise Immutability and Ownership using the Coq Proof Assistant, a formal proof management system [13]. We encode three type systems using Coq; Featherweight Immutable Java, Featherweight Generic Java and Featherweight Ownership Generic Java, and prove them sound. We describe the challenges presented in encoding immutability, ownership and type systems in general in Coq.</p>

Optimality of a Linear Decision Rule in Discrete Time AK Model

Surprisingly, formal proof on the optimality of a linear decision rule in the discrete time AK model with a CRRA utility function has not been established in the growth literature while that in the continuous time counterpart is well-established. This note fills such a gap: I provide a formal proof that consumption being linearly related to investment is a sufficient and necessary condition for Pareto optimality in the discrete time AK model.

Conformal Chebyshev chaotic map-based remote user password authentication protocol using smart card

With the rapid advancement and growth of computer networks, there have been greater and greater demands for remote user password authentication protocols. In current ages, smartcard-based authentication protocol has formed the standard with their incredibly insubstantial, user-friendly equipment and low-cost apps. In this study, we proposed an effective robust authentication protocol using the conformable chaotic map, where a conformable calculus is a branch of newly appearing fractional calculus. It has a magnificent property, because it formulates using a controller term. We shall also offer formal proof of smooth execution of the proposed authenticated protocol. Our new protocol is more secure as compared to several comparable protocols.

Formal proof of the requirement of MESP1 and MESP2 in mesoderm specification and their transcriptional control via specific enhancers in mice

ABSTRACT MESP1 and MESP2 are transcriptional factors involved in mesoderm specification, somite boundary formation and somite polarity regulation. However, Mesp quadruple mutant zebrafish displayed only abnormal somite polarity without mesoderm specification defects. In order to re-evaluate Mesp1/Mesp2 mutants in mice, Mesp1 and Mesp2 single knockouts (KOs), and a Mesp1/Mesp2 double KO were established using genome-editing techniques without introducing selection markers commonly used before. The Mesp1/Mesp2 double KO embryos exhibited markedly severe mesoderm formation defects that were similar to the previously reported Mesp1/Mesp2 double KO embryos, indicating species differences in the function of MESP family proteins. However, the Mesp1 KO did not display any phenotype, including heart formation defects, which have been reported previously. We noted upregulation of Mesp2 in the Mesp1 KO embryos, suggesting that MESP2 rescues the loss of MESP1 in mesoderm specification. We also found that Mesp1 and Mesp2 expression in the early mesoderm is regulated by the cooperation of two independent enhancers containing T-box- and TCF/Lef-binding sites. Deletion of both enhancers caused the downregulation of both genes, resulting in heart formation defects. This study suggests dose-dependent roles of MESP1 and MESP2 in early mesoderm formation.