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The project aims to develop some contributions to the theory of delayed differential equations (DDE) and switched systems by developing, mainly, necessary and sufficient conditions for stability of equilibrium points and reducing the conservativeness of sufficient conditions given by the use of multiple Lyapunov-Krasovskii functionals.

April, 2022
Dată început proiect
March, 2024
Dată sfârșit proiect

Technical details

UEFISCDI Program: National Research-Development and Innovation Plan 2015 – 2020 (PNCDI III)
Program 1: Development of the national research-development system

Subprogram 1.1: Human Resources
Project Code: PN-III-P1-1.1-PD-2021-0610

Acronym: CONS-TD

Adjacent, the purpose of the project is to elaborate mathematical models for linear and nonlinear systems specific to the aerospace field, namely modeling an airplane wing to ensure flight stability in the presence of command line time-delays and switches between different command laws. Therefore, a secondary goal, but as important as the main goal, is to obtain classical control laws (LQG), as well as from the new generation of artificial intelligence (fuzzy logic and neural networks), in the perspective of counteracting the effects induced by time-delays. The delay, on state or on control (actuator), can lead to instability or poor performance of the control law. However, with a proper choice, the time-delay can be turned into a benefit by playing an important role in attenuating the vibrations of aeroelastic structures like wing with aileron.

Results – Phase I

Phase 1 „Preliminary studies on the stability conditions for linear and nonlinear systems of differential equations with delay and switching and ways to reduce the conservatism of these conditions” includes the following activities: Act 1.1 – State-of-the-art regarding the stability conditions of systems of differential equations with delay and switching; Act 1.2 – Evaluation of the degree of conservatism.

A first representative model considered is that of a linear system with delay, with disturbance

A second model is of a perturbed linear system with delay in control and controllable time-dependent switching (both models can describe the chain of command of the flight of the aircraft)

The switching law σ, is a constant piecewise function

The concepts with which it operates are: input to state stability (ISS), “minimum dwell time d*” stability, where

The Lyapunov Krasovskii functional (LKF) is the mathematical tool frequently used in the study of stability. During Phase I, a stability theorem (ISS) was obtained for system (1) and an ISS stability theorem taken from the literature was analyzed. The demonstration mechanism is laborious and leads to more or less conservative stability conditions.  A first conclusion is that on this LKF path it is unlikely, if not impossible, to obtain non-conservative stability conditions. A suggestion of the difficulty can be seen from the fact that the problem can be put in the form of a multi-objective optimization problem with constraints: minimize

subject to:

A second conclusion is that of a paradigm shift: a method to develop the suggestions coming from some works such as [G. Chesi, P. Colaneri, J. C. Geromel, R. Middleton and R. Shorten, “A Nonconservative LMI Condition for Stability of Switched Systems with Guaranteed Dwell Time]. The method uses homogeneous polynomial Liapunov functions, an idea originating in Hilbert’s famous Problem No. 17. A second way to be developed starts from the convexification of some conditions (W. Xiang et al., Nonconservative lifted convex conditions for stability of discrete-time switched systems under minimum dwell-time constraint, IEEE Transactions on Automatic Control, Volume: 64, Issue: 8, August 2019). 

Results dissemination

1. Critical case of stability for a nonlinear switched system of delay differential equations with applications to a hydraulic servomechanism – D. Enciu, A. Halanay, I. Ursu, 11th International Conference on Pure and Applied Mathematics (ICPAM 2022), 12-22 July 2022, Bratislava, Slovakia


In this paper the complex problem of nonlinear dynamical system with simultaneously critical case for stability, time-delay on state, and an autonomous uncontrolled state dependent switching rule is addressed. A general theorem ensuring simple stability for this type of system is given. This leads to the application of a Malkin-type mathematical apparatus, combined with the use of multiple complete Lyapunov-Krasovskii functionals. Since the characteristic equation of the linear approximation has a null eigenvalue, following some transformations of variables, the initial system consisting of five nonlinear differential equations is decomposed in the canonical form of Malkin approach which consists of a fourth order system and a first order system that contains only nonlinear terms. The equilibrium stability condition of the nonlinear system returns to the fulfillment of the asymptotic stability condition of the linearized equations and the Lyapunov conditions.The mathematical model is applied to the analysis and synthesis of electrohydraulic servomechanism. This real world object is vital for flight safety being an essential equipment in aircraft flight controls. Thus, some thresholds of admissible delay for ensuring the stability of the servomechanism are determined. 

2. New strategy for the safety and comfort of the passengers and aircraft crew during atmospheric turbulence – D. Enciu, I. Ursu, G. Tecuceanu, 7th European Conference on Structural Control (EACS 2022), 10-13 July 2022, Warsaw, Poland


An airplane trip can be psychological terrifying for any traveler. If, during the flight, the airplane meets a turbulent air front, then the scenario is perfect for a Hollywood movie, and the panic among passengers increases proportionally with the severity of the turbulence. In this paper, a new approach of the turbulence mitigation methodology is proposed based on a solid background using an active control vibration. The experimental model is represented by a realistic, elastic airplane wing model controlled by an electric linear servoactuator. The mathematical model is completed by numerical simulations and experiments in the subsonic wind tunnel upgraded with a turbulence generator. The qualification of an emergent technology of this type will have double impact: for the passengers – safety and mental comfort increasing given by the significant reduction of the dynamic effects produced by the turbulent field; for the airplane – weight optimization based on the loads control generated by the atmospheric turbulence.

3. A critical case for stability in a model of an electrohydraulic servomechanism – D. Enciu, A. Halanay, I. Ursu,  29th Conference on Applied and Industrial Mathematics (CAIM 2022), 25-29 August 2022, Chisinau, Republica Moldova


In this paper, the conditions required for the stability of a nonlinear system with time-delay and switching are studied. The starting point of the theory is based on a real-world mathematical model of an electrohydraulic servomechanism located in ailerons flight controls of the Romanian IAR 99 Hawk jet training airplane. For this model, a general theorem on the equilibrium stability in a critical case for a switched nonlinear system of delay differential equations is stated. The framework uses multiple complete Lyapunov-Krasovskii functionals. The characteristic equation has one zero root which claims the use of a special approach given by a Lyapunov-Malkin theorem. Therefore, some transformations are made to write the linearized system in a canonical form where the stability Lyapunov theorem of the linear approximation can be applied. The study of the stability of equilibria relies on two conditions: a Lyapunov condition and an asymptotic stability condition. The transformation of the nonlinear system into the specific form of the Lyapunov-Malkin theorem and the verification of the two conditions mentioned above requires a double perspective – analytical developments and numerical simulations – since the mathematical models are too complex to be approached only analytically. Accordingly, an important result is calculated, regarding an admissible delay threshold in preserving stability of the electrohydraulic servomechanism as vital system for the safety of the aircraft. Some considerations regarding the conservatism and the non-necessity of sufficient conditions conclude the work.

4. Lyapunov-Malkin type approach of equilibrium stability in a critical case applied to a switched model of a servomechanism with state delay –  D. Enciu, A. Halanay, A. Toader, I. Ursu, accepted for publication in International Journal of Control


A general theorem on equilibrium stability in a critical case is applied to switched strong nonlinear differential equations with time delay on state, characterizing electrohydraulic servomechanisms dynamics. Basically, its proof involves the use of the Lyapunov-Malkin approach to stability and multiple complete Lyapunov-Krasovskii functionals. The fulfilment of the equilibrium stability condition of the nonlinear system returns to the asymptotic stability condition of the linearized equations and to the so-called Lyapunov conditions for the latter. The transformation of the nonlinear system into the canonical form specific to the Lyapunov-Malkin theorem, and the verification of the two conditions mentioned above require analytical developments doubled by numerical simulations, since the mathematical models are too complex to be approached only analytically. As a consequence, an important result is obtained, for the first time, regarding the thresholds of admissible delay in preserving the stability of a real world object, vital for the safety of the aircraft.

Additional information can be obtained at I.N.C.A.S. Bucharest
Research & development

Dr. math. Daniela Enciu
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