On Finite Time Instability of Continuous Time Delay Systems Dragutin Lj. Debeljkovic
Ivan M. Buzurovic
Department of Control Engineering School of Mechanical Engineering, University of Belgrade Belgrade, Serbia
[email protected]
Medical Physics and Biophysics Division Harvard Medical School Boston, MA, USA
[email protected]
Nebojsa J. Dimitrijevic
Milan A. Misic
School of High Applied Professional Education Vranje, Serbia
[email protected]
School of Technical Science, University of Pristina Kosovska Mitrovica, Serbia
[email protected]
Abstract— Finite time instability for linear continuous timedelay systems was investigated in this paper. The novel Lyapunov-like functions were used in the analysis. The functions do not need to fulfill the following conditions: being positive definite on the whole state space domain and possessing negative derivatives along the system trajectories. These functions were previously used for the development of both the delay-dependent and delay-independent sufficient conditions for the investigation of the finite time stability of control systems. However, the reported conditions cannot be used for the precise calculation of the instant when the system trajectory leaves the prescribed boundaries. In this paper, a novel concept of finite time instability was introduced to solve this problem. Numerical examples were used to additionally clarify the procedure. Keywords— Non-Lyapunov instability, time delay system, sufficient conditions
I.
INTRODUCTION
has allowed the formulation of less conservative conditions that can guarantee both FTS and finite-time stabilization of the linear continuous time systems. Many valuable results have been obtained for this type of stability, such as the ones reported in [4-11]. Time delay and parameter uncertainty are commonly encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, and long transmission lines. It has been shown that the existence of delay and uncertainty is the source of instability and poor performance of control systems. Similar to the systems without delay, there is a need to investigate FTS for a class of time-delay systems. There are few results on FTS of time-delay systems. Some early results on FTS of time-delay systems can be found in [12–18]. The results of these investigations are conservative because they use boundedness proprieties of the system response, i.e., of the solution of system models. Recently, based on the linear matrix inequality (LMI) theory, some results have been obtained for FTS for some particular classes of time-delay systems [19–22].
The concept of Lyapunov asymptotic stability is widely known in the control community. However, in some cases, Lyapunov asymptotic stability approach is not sufficient in the practical applications. Sometimes large values of state variables are not practically acceptable, for instance in the cases where saturation is present. In these cases, it is of particular significance to consider the behavior of dynamical systems only over a finite time interval. For this purpose, the concept of finite-time stability (FTS) can be used. For a system, it is said to be FTS once a time interval is fixed if its state does not exceed some bounds during this finite time interval.
In this article, a novel delay dependent condition for the finite-time instability of the linear continuous time-delay systems has been presented. To solve the problem of FTS, we used the Lyapunov-like method. The sufficient condition is expressed in the form of algebraic inequality. The computation of the proposed conditions was presented throughout the numerical example.
This concept stability dates back to the 1950s [1-3]. Since then, the researchers’ interest has moved toward the classical Lyapunov stability due to the lack of operative test conditions for FTS. Recently, the concept of FTS has been revisited in the prospective of the linear matrix inequality theory, which
The following notations has been used throughout the article. Superscript “T” stands for matrix transposition. ℜn denotes the n-dimensional Euclidean space and ℜ n× m is the set of all real matrices having dimension (n × m). F > 0
c 978-1-4799-4315-9/14/$31.00 2014 IEEE
II.
1416
PRELIMINARIES AND PROBLEM FORMULATION
means that F is real symmetric and positive definite and F > G means that the matrix (F - G) is positive definite. μ (F) and μ2 (F), where μ2(F) = ½ λmax (F + FT) are the matrix measure and the second matrix measure of matrix F , respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. Consider the following linear system with time delay:
x ( t ) = A0 x ( t ) + A1 ( x ( t − τ ) )
(1)
with a known vector valued function of the initial conditions:
x ( t ) = ϕ ( t ) , t ∈ [ − τ, 0 ]
(2)
Λ max = λ max
n× n
In this study, the finite-time stability of the class of systems (1) has been investigated. Before continuing further, the following definition of finite-time stability for the timedelay system (1) is introduced.
Definition 1. Time-delay system (1) satisfying the given initial condition (2) is said to be finite-time stable (FTS) with respect to {α , β ,T } if : x ( t ) x ( t ) < β , ∀t ∈ [0, T ] (3) T
t∈[ −τ ,0]
Definition 2 Time-delay system (1) satisfying the given initial condition (2) is said to be finite-time unstable (FTUS) with respect to {α , β , T } , α < β , if there exist real positive
δ , δ ∈ ]0, α [ ,
number
(
∗
∗
) (
+ A0 + A1T + A1
and
time
instant
)
t , t = t : ∃ t > t0 ∈ ℑ that if: sup ijT ( t ) ij ( t ) < α
t∈[ −τ ,0]
(4)
))
(
)
Theorem 2. Time-delayed system (1) with the function of initial conditions (2) is finite time stable with respect to α , β , T if there exist a positive scalar Λ max such that the
{
}
following condition holds:
(1 + τ ) eΛ max ⋅t
t0 ∈ ℑ for
where:
which the next condition is fulfilled:
R = A0 + Q ( 0)
(21)
Ξ=R +R T
(
ψ = λ max Q ( 0 ) QT ( 0 )
(22) 2 μ2 ( R )τ
) e 2μ ( R )− 1
(23)
μ2 ( R ) being matrix measure of matrix R and Q ( 0 ) is any solution of the following nonlinear transcendental matrix equation: e
A0 + Q ( 0 )τ
Q ( 0 ) = A1
(24)
[24].
Theorem 5. The time-delayed system (1) with the function of initial conditions (2), possessing the following properties:
σ { A0 } ∈ \ ∧ σ { A1} ∈ \
( )
∃!λ i A0 ∈ &
+
(25)
is finite time stable with respect to {α , β , T } if the following condition is satisfied:
1418
β , ∀t ∈ ℑ α
R = A0 + Q ( 0 )
ψ = λ max Q ( 0 ) QT ( 0 )
where:
℘1,2 =
