Eulerian and Lagrangian methods for stability analysis. An interval approach

  • Mario Martinez Guerrero

Student thesis: Phd


Stability analysis is one of the most important areas of research in engineering nowadays. The goal of stability analysis is to determine a working region in which a system's output remains bounded for a given input or perturbation. Since the previous century, the problem of verifying if a system is stable has been addressed using Lyapunov-based methods. However if the system is nonlinear, time variant or uncertain; obtaining a stable region becomes a challenging problem and no generalised or reliable methods have been developed. This thesis proposes some novel concepts to study the stability of nonlinear time variant systems with uncertainties in a set membership context. Two new concepts of G-Stability related to the stability of time variant systems, and T-Stability which refers to the asymptotic stability of time variant nonlinear systems, are introduced. These new approaches have been developed by extending the existing theory of nonlinear stability analysis towards time variant nonlinear systems and combining it alongside interval methods for the validation of reliable and robust results for the stability problem. The main principle is to establish a safe time variant region in the state space, namely a time variant tube, where the system is attracted and cannot escape for all time. Furthermore, this methodology is illustrated on the trajectory following problem for a sailboat system. In modern control systems, Lyapunov theory has been used to develop different control strategies, such as Sliding Mode Control (SMC). In this thesis, using the new stability concepts, a tube based control strategy was defined. In general, this control strategy is based on time variant tubes and the ability of the system to become attracted by the tube, reach this region of attraction in a finite time, and remain there for all time. This control strategy was implemented for the problem of trajectory tracking of a nonholonomic robotic system. Viability theory analyses the state evolution of a dynamical system such that its states remain inside for a determined time. In this thesis, some novel concepts and methods for viability theory are proposed for the analysis of nonlinear systems with additive bounded uncertainties based on interval methods. Furthermore, an extension of these concepts towards time variant systems is also proposed. The methodology and concepts defined previously, are illustrated and validated on the car on the hill problem with additive bounded uncertainties.
Date of Award1 Aug 2022
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorZhengtao Ding (Supervisor) & Alexandru Stancu (Supervisor)


  • Control systems
  • Time variant systems
  • Set Mambership
  • Nonlinear systems
  • Interval Methods
  • Stable systems
  • Stability Analysis

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