Francesco Amato

Technical Communique: Finite-time control of linear systems subject to parametric uncertainties and disturbances

In this paper we consider finite-time control problems for linear systems subject to time-varying parametric uncertainties and to exogenous constant disturbances. The main result provided is a sufficient condition for robust finite-time stabilization via state feedback. It can be applied to problems with both non-zero initial conditions and unknown constant disturbances. This condition is then reduced to a feasibility problem involving linear matrix inequalities (LMIs). A detailed example is presented to illustrate the proposed methodology.

Technical communique: Finite-time stabilization via dynamic output feedback

In this paper the finite-time stabilization of continuous-time linear systems is considered; this problem has been previously solved in the state feedback case. In this work the assumption that the state is available for feedback is removed and the output feedback problem is investigated. The main result provided is a sufficient condition for the design of a dynamic output feedback controller which makes the closed loop system finite-time stable. Such sufficient condition is given in terms of an LMI optimization problem; this gives the opportunity of fitting the finite-time control problem in the general framework of the LMI approach to the multi-objective synthesis. In this context an example illustrates the design of a controller which guarantees, at the same time, finite-time stability together with some pole placement requirements.

Technical communique: Finite-time stability of linear time-varying systems with jumps

This paper 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. Some examples illustrate the effectiveness of the proposed approach.

Finite-Time Stability of Linear Time-Varying Systems: Analysis and Controller Design

The note deals with the finite-time analysis and design problems for continuous-time, time-varying linear systems. Necessary and sufficient conditions and a sufficient condition for finite-time stability are devised. Moreover, sufficient conditions for the solvability of both the state and the output feedback problems are stated. Such results require the feasibility of optimization problems involving Differential Linear Matrix Inequalities. Some numerical examples illustrate the effectiveness of the proposed approach.

Brief paper: Finite-time control of discrete-time linear systems: Analysis and design conditions

In this paper we deal with some finite-time control problems for discrete-time, time-varying linear systems. First we provide necessary and sufficient conditions for finite-time stability; these conditions require either the computation of the state transition matrix of the system or the solution of a certain difference Lyapunov inequality. Then we address the design problem. The proposed conditions allow us to find output feedback controllers which stabilize the closed loop system in the finite-time sense; all these conditions can be expressed in terms of LMIs and therefore are numerically tractable, as shown in the example included in the paper.

Technical communique: On the region of attraction of nonlinear quadratic systems

Quadratic systems play an important role in the modelling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems, it is of mandatory importance not only to determine whether the origin of the state space is locally asymptotically stable but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the region of attraction of the equilibrium. The proposed algorithm requires the solution of a suitable feasibility problem involving linear matrix inequalities constraints. An example illustrates the effectiveness of the proposed procedure by exploiting a population interaction model of three species.

Technical communique: Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems

Sufficient conditions for the finite-time stability (FTS) of impulsive dynamical linear systems (IDLSs) have been provided in Amato, Ambrosino, Cosentino, and De Tommasi (2011). In this brief note we show that, when time-dependent IDLSs are dealt with, the sufficient condition for FTS derived in Amato, Ambrosino, Cosentino et al. (2011) is also necessary. Moreover, an alternative necessary and sufficient condition for FTS is proposed, which is based on the solution of a coupled differential-difference Lyapunov equation. This condition turns out to be more efficient from the computational point of view. The proposed approach is illustrated through some examples.

Input–Output Finite-Time Stability of Linear Systems: Necessary and Sufficient Conditions

When only the input-output behavior of a dynamical system is of concern, usually Bounded-Input Bounded-Output (BIBO) stability is studied, for which several results exist in literature. The present paper investigates the analogous concept in the framework of Finite Time Stability (FTS), namely the Input-Output FTS. A system is said to be IO finite time stable if, assigned a bounded input class and some boundaries in the output signal space, the output never exceeds such boundaries over a prespecified (finite) interval of time. FTS has been already investigated in several papers in terms of state boundedness, whereas this is the first work dealing with the characterization of the input-output behavior. Sufficient conditions are given, concerning the class of L2 and L∞ input signals, for the analysis of IO-FTS and for the design of a static state feedback controller, guaranteeing IO-FTS of the closed loop system. Finally, the applicability of the results is illustrated by means of two numerical examples.

Finite-Time Stability and Control

Part I: Linear Systems.- Finite-time Stability Analysis of Continuous-Time Linear Systems.- Controller Design for the Finite-Time Stabilization of Continuous-Time Linear Systems.- Robustness Issues.- Finite-time Stability of Discrete-Time Linear Systems.- Finite-time Stability Analysis via PQLFs.- Part II: Hybrid Systems.- Finite-time Stability of Impulsive Dynamical Linear Systems.- Controller Design for the Finite-time Stability of Impulsive Dynamical Linear Systems.- Robustness Issues for Impulsive Dynamical Linear Systems.

Technical communique: Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions

Quadratic systems play an important role in the modeling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is of mandatory importance not only to determine whether the origin of the state space is locally asymptotically stable, but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the region of attraction of the equilibrium. The proposed approach is based on polyhedral Lyapunov functions, rather than on the classical quadratic Lyapunov functions. An example shows that our methodology may return less conservative results than those obtainable with previous approaches.