Control systems theory
Finite-time stability of linear systems: an approach based on polyhedral Lyapunov functions
In this research we consider the finite-time stability problem for linear systems. Differently from previous papers, the stability analysis is performed with the aid of polyhedral Lyapunov functions rather than with the classical quadratic Lyapunov functions. In this way we are able to manage more realistic constraints on the state variables; indeed, in a way which is naturally compatible with polyhedral functions, we assume that the sets to which the state variables must belong in order to satisfy the finite-time stability requirement are boxes (or more in general polytopes) rather than ellipsoids. The main result, derived by using polyhedral Lyapunov functions, is a sufficient condition for finite-time stability of linear systems, which can also be used in the controller design context. Detailed analysis and design examples are presented to illustrate the advantages of the proposed methodology over existing methods.
Finite-Time Stability of Linear Time-Varying Systems with Jumps: Analysis and Controller Design
This research deals with the finite-time stability problem for continuous-time linear time-varying systems with finite jumps. This class of systems arises in many practical applications and includes, as particular cases, impulsive systems and sampled-data control systems. The paper provides a necessary and sufficient condition for finite-time stability, requiring a test on the state transition matrix of the system under consideration, and a sufficient condition involving two coupled differential/difference linear matrix inequalities. The sufficient condition turns out to be more efficient from the computational point of view. Moreover, it is the starting point for solving the stabilization problem, namely for finding a state feedback controller which finite-time stabilizes the closed loop system. Some examples illustrate the effectiveness of the proposed approach.
On Finite-Time Stability of State Dependent Impulsive Dynamical Systems
This research extends the finite-time stability problem to state dependent impulsive dynamical systems. For this class of hybrid systems, the state jumps when the trajectory reaches a resetting set, which is a subset of the state space. A sufficient condition for finite-time stability of state dependent impulsive dynamical systems is provided. Moreover, S-procedure arguments are exploited to obtain a formulation of this sufficient condition which is numerically tractable by means of Differential Linear Matrix Inequalities (DLMIs). Such a formulation may be in general more conservative, for this reason a procedure which allows to automate its verification, without introduce conservatism, is given both for second order systems, and when the resetting set is ellispoidal. Eventually some examples illustrate the effectiveness of the proposed procedure.
Finite-Time Stability Analysis of Linear Discrete-Time Systems via Polyhedral Lyapunov Functions
In this research we consider the finite-time stability problem for discrete-time linear systems. Differently from previous papers, the stability analysis is performed with the aid of polyhedral Lyapunov functions rather than using the classical quadratic Lyapunov functions. In this way we are able to deal with more realistic constraints on the state variables; indeed, in a way which is naturally compatible with polyhedral functions, we assume that the sets to which the state variables must belong to in order to satisfy the finite-time stability requirement are boxes (or more in general polytopes) rather than ellipsoids. The main result, derived by using polyhedral Lyapunov functions, is a sufficient condition for finite-time stability of discrete-time linear systems, which can also be used in the controller design context. Detailed analysis and design examples are presented to illustrate the advantages of the proposed methodology over existing methods.