Finite-time Stability Of Linear Systems Research Articles

Robust finite-time stabilization of linear systems with limited state quantization

This paper investigates the robust asymptotic stabilization of a linear time-invariant (LTI) system by static feedback with a static state quantization. It is shown that the controllable LTI system can be stabilized to zero in a finite time by means of nonlinear feedback with a quantizer having a limited (finite) number of values (quantization seeds) even when all parameters of the controller and the quantizer are time-invariant. The control design is based on generalized homogeneity. A homogeneous spherical quantizer is introduced. The static homogeneous feedback is shown to be a local (or global) finite-time stabilizer for any controllable linear system (depending on the system matrix). The tuning rules for both the quantizer and the feedback law are obtained in the form of Linear Matrix Inequalities (LMIs). The closed-loop system is proven to be robust with respect to some bounded matched or vanishing mismatched perturbations. Theoretical results are supported by numerical simulations.

Lower and upper bounds of the solution for the Lyapunov matrix differential equation and an application in input-output finite-time stability of linear systems

In control theory and practical engineering fields, such as the controllability, observability, input–output finite-time stability of the linear systems, it is a significant problem to study the properties of the solution for the Lyapunov matrix differential equation where there are no restrictions on the system matrix. In this paper, by constructing an equivalent form of the Lyapunov matrix differential equation and utilizing some important matrix eigenvalue inequalities, lower and upper bounds for the matrix solution of the Lyapunov matrix differential equation that remove the strict restrictions for the system matrix are proposed. As an application in linear systems, we show that our bounds can be used to discuss the input–output finite-time stability for linear systems. Finally, we give some corresponding numerical examples to illustrate the effectiveness and superiority of our results.

Finite-Time Stabilization of Linear Systems With Unknown Control Direction via Extremum Seeking

In this article, the finite-time stabilization problem is solved for a linear time-varying system with unknown control direction by exploiting a modified version of the classical extremum-seeking algorithm. We propose to use a suitable oscillatory input to modify the system dynamics, at least in an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">average</i> sense, so as to satisfy a differential linear matrix inequality condition, which in turn guarantees that the system’s state remains inside a prescribed time-varying hyperellipsoid in the state space. The finite-time stability (FTS) of the averaged dynamics implies the FTS of the original system, as the distance between the original and the averaged dynamics can be made arbitrarily small by choosing a sufficiently high value of the dithering frequency used by the extremum-seeking algorithm. The main advantage of the proposed approach resides in its capability of dealing with systems with unknown control direction, and/or with a control direction that changes over time. Being FTS a quantitative approach, this article also gives an estimate of the necessary minimum dithering/mixing frequency provided, and the effectiveness of the proposed finite-time stabilization approach is analyzed by means of numerical examples.

Finite-Time Stabilization of Linear Systems With Input Constraints by Event-Triggered Control

Dear editor, This letter designs the event-triggered control (ETC) to achieve finite-time stabilization (FTS) of linear systems with input constraints. The key idea of the established algorithm is that the designed time-varying high-gain is only scheduled on a specified time determined by an event-triggered mechanism. By exploring the properties of the parametric Lyapunov equation, a positive lower bound of inter-event times associated with the designed ETC can be obtained, such that the Zeno phenomenon is avoided. Moreover, semi-global FTS of linear systems with input constraints is also achieved by the designed ETC. Finally, the application of the spacecraft rendezvous control system verifies the effectiveness of the designed ETC.

Finite-time stabilization of linear systems by bounded event-triggered and self-triggered control

This paper is concerned with finite-time stabilization (FTS) of linear systems by bounded event-triggered control (ETC) and self-triggered control (STC). A bounded linear ETC is firstly designed, where the time-varying control gain is only scheduled on a specified time determined by an event-triggered mechanism, such that the FTS of the closed-loop system is achieved and the communication resources are saved. Moreover, the corresponding STC, in which the updates of control law are determined by a self-triggered mechanism that only needs to monitor the previous triggered states, is designed. Specially, by exploring the properties of the parametric Lyapunov equation, the designable minimal inter-event time of the established ETC and STC is obtained, such that the Zeno phenomenon is avoided. In addition, the finite-time semi-global stabilization and the fixed-time (prescribed finite-time) stabilization of linear systems are achieved by bounded ETC and STC. What needs to be emphasized is that the finite-time stabilization in this paper can be called as the practical finite-time stabilization since the state converges to zero exponentially with a fast convergence rate after a designable time. Finally, the established algorithms are used to the design of the spacecraft rendezvous control system and their effectiveness is verified by simulations.

Finite-time stability analysis and stabilization by bounded linear time-varying feedback

This paper studies finite-time stability analysis and finite-time stabilization of linear systems by bounded linear time-varying feedback. On the one hand, (both local and global) finite-time stability of general nonlinear time-varying systems is investigated by the comparison principle and the notion of finite-time escaping functions. Some Lyapunov-like stability theorems are established. On the other hand, finite-time and prescribed-time stabilization of linear systems by bounded linear time-varying feedback are revisited based on the proposed finite-time stability theorems. Two classes of new time-varying high-gain functions are proposed to reduce the regulation time. Some connections of the proposed results to existing results on Lyapunov-inequality based finite-time stability analysis and nonlinear feedback based finite-time stabilization are revealed. The effectiveness of the proposed methods is illustrated by a numerical example.

Finite-time stabilization of linear systems by bounded linear time-varying feedback

Time-varying features are generally considered to be detrimental to the analysis and design of control systems. This paper establishes methods to design bounded linear time-varying (LTV) controllers such that the control performance of a linear time-invariant (LTI) system can be improved, that is, the finite-time stability of the closed-loop system can be obtained. Specifically, for an LTI control system, by using the solution to a parametric Lyapunov equation (PLE), a bounded LTV controller containing a suitable time-varying parameter is designed. By fully exploiting properties of the solution to the PLE, it is shown that the closed-loop system is finite-time stable. Both state feedback and observer based output feedback, in which both the observer gain and the state feedback gain are time-varying, are considered. As a consequence, the finite-time semi-global stabilization and the fixed-time (prescribed finite-time) stabilization problems for linear systems by bounded controls are solved. The established method is utilized to the design of the spacecraft rendezvous control system and its effectiveness is verified by simulations.

Finite-Time Stabilizability, Detectability, and Dynamic Output Feedback Finite-Time Stabilization of Linear Systems

In this note, we deal with linear systems and we extend to the finite-time setting the concepts of stabilizability and detectability. It will be shown that, similarly to what happens in the classical Lyapunov framework, even in the finite-time context, stabilizability and detectability play a role into the existence of stabilizing dynamical controllers. We prove that a dynamic output feedback controller, which finite-time stabilizes the overall closed-loop system, exists if and only if the open-loop system is finite-time detectable and stabilizable plus a further linear matrix inequality coupling condition. We also show that, in the finite-time context, the equivalence between stabilizability via output feedback and stabilizability via observer-based controllers is no longer true.

$L_p$-Stabilization of Integrator Chains Subject to Input Saturation Using Lyapunov-Based Homogeneous Design

Consider the $n$th integrator $\dot x=J_nx+\sigma(u)e_n$, where $x\in\mathbb{R}^n$, $u\in\mathbb{R}$, $J_n$ is the $n$th Jordan block and $e_n=(0 \cdots 0 1)^T\in\mathbb{R}^n$. We provide easily implementable state feedback laws $u=k(x)$ which not only render the closed-loop system globally asymptotically stable but also are finite-gain $L_p$-stabilizing with arbitrarily small gain, as in [A. Saberi, P. Hou, and A. Stoorvogel, IEEE Trans. Automat. Control, 45 (2000), pp. 1042--1052]. These $L_p$-stabilizing state feedbacks are built from homogeneous feedbacks appearing in finite-time stabilization of linear systems. We also provide additional $L_\infty$-stabilization results for the case of both internal and external disturbances of the $n$th integrator, namely, for the perturbed system $\dot x=J_nx+e_n\sigma (k(x)+d)+D$, where $d\in\mathbb{R}$ and $D\in\mathbb{R}^n$.

Input–output finite-time stabilization of linear systems with finite-time boundedness

The paper presents linear system Input–Output Finite-Time Stabilization (IO-FTS) method under Finite-Time Boundedness (FTB) constraint. A state feedback controller is designed, via Linear Matrix Inequalities (LMIs), to guarantee the system both IO-FTS and FTB. The proposed methods are applied to the guidance design of a class of terminal guidance systems to suppress disturbances with IO-FTS method and FTB constraints simultaneously satisfied. The simulation results illustrate the effectiveness of the proposed methods.

Finite-Time Stabilization of Linear Systems Over Networks with Bounded Packet Dropout

Finite-Time Stabilization of Linear Systems Over Networks with Bounded Packet Dropout

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

In the recent paper “Input-output finite-time stabilization of linear systems,” (F. Amato ) a sufficient condition for input-output finite-time stability (IO-FTS), when the inputs of the system are <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> signals, has been provided; such condition requires the existence of a feasible solution to an optimization problem involving a certain differential linear matrix inequality (DLMI). Roughly speaking, a system is said to be input-output finite-time stable if, given a class of norm bounded input signals over a specified time interval of length <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> , the outputs of the system do not exceed an assigned threshold during such time interval. IO-FTS constraints permit to specify quantitative bounds on the controlled variables to be fulfilled during the transient response. In this context, this paper presents several novel contributions. First, by using an approach based on the reachability Gramian theory, we show that the main theorem of F. Amato is actually also a necessary condition for IO-FTS; at the same time we provide an alternative-still necessary and sufficient-condition for IO-FTS, in this case based on the existence of a suitable solution to a differential Lyapunov equality (DLE). We show that the last condition is computationally more efficient; however, the formulation via DLMI allows to solve the problem of the IO finite-time stabilization via output feedback. The effectiveness and computational issues of the two approaches for the analysis and the synthesis, respectively, are discussed in two examples; in particular, our methodology is used in the second example to minimize the maximum displacement and velocity of a building subject to an earthquake of given magnitude.

Sufficient Conditions for Robust Input-Output Finite-Time Stability of Linear Systems in Presence of Uncertainties

AbstractThe concept of Bounded-Input Bounded-Output (BIBO) stability arises when one wants to focus on the study of the the input-output behavior of a dynamical system, as opposed to the classical Lyapunov stability. 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. IO-FTS has been already investigated in previous papers, whereas this is the first work dealing with the input-output behavior of an uncertain dynamical system in the FTS framework. Two sufficient conditions are given, concerning the class of 2 and ∞ input signals, for the analysis of robust IO-FTS. The applicability of the results is illustrated by means of a numerical examples.

Finite-time stability of linear systems: an approach based on polyhedral Lyapunov functions

In this study, the authors consider the finite-time stability (FTS) problem for linear systems. Differently from previous studies, the authors assume that the sets to which the state variables must belong in order to satisfy the FTS requirement are boxes (or more in general polytopes) rather than ellipsoids. To deal with these more realistic constraints on the state variables the stability analysis is performed with the aid of polyhedral Lyapunov functions rather than with the classical quadratic Lyapunov functions. The main result, derived by using polyhedral Lyapunov functions, is a sufficient condition for FTS of linear systems. Detailed analysis and design examples are presented to illustrate the advantages of the proposed methodology over existing methods.

Finite-time stabilization of linear systems via impulsive control

This paper introduces a new concept of finite-time stability of linear time-invariant impulsive systems. The sufficient condition of finite-time stability of the system via impulsive control at fixed times and variable times is obtained, respectively. Such sufficient conditions of finite-time stabilization are given in terms of matrix inequalities. A numerical example is presented to illustrate the efficiency of the proposed results.

Finite-time stability and stabilization of time-delay systems

Finite-time stability and stabilization of retarded-type functional differential equations are developed. First, a theoretical result on finite-time stability inspired by the theory of differential equations, using Lyapunov functionals, is given. As it may appear not easily usable in practice, we show how to obtain finite-time stabilization of linear systems with delays in the input by using an extension of Artstein’s model reduction to nonlinear feedback. With this approach, we give an explicit finite-time controller for scalar linear systems and for the chain of integrators with delays in the input.