Improved Results on Finite Time Stability of Time Delay Systems: Jensen's Inequality-based Approach

DRAGUTIN LJ. DEBELJKOVIĆ, Megatrend University, Original scientific paper Faculty of Civil Aviation, Belgrade UDC: 100.404,7 IVAN M. BUZUROVIĆ Medical Physics and DOI: 10.5937/tehnika1801076D Biophysics Division, Harvard Medical School Boston, MA, USA GORAN V. SIMEUNOVIĆ University of Belgrade, Faculty of Mechanical Engineering, Innovation Centre, Belgrade DARKO M. RADOJEVIĆ, University of Belgrade, Faculty of Mechanical Engineering, Belgrade


INTRODUCTION
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 time interval.
This concept stability dates back to the 1950s [1][2][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 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][5][6][7][8][9][10][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][13][14][15][16][17][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][20][21][22]. In this article, a novel delay dependent condition for the finite-time stability 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.

PRELIMINARIES AND PROBLEM FORMULATION
The following notations has been used throughout the article. Superscript "T" stands for matrix transposition. n denotes the n-dimensional Euclidean space and nm  is the set of all real matrices having dimension (n  m). F > 0 means that F is real symmetric and positive definite and F > G means that the matrix (F -G) is positive definite.  (F) and  2 (F), where  2 (F) = ½  max (F + F T ) are the matrix measures of matrix F , respectively. Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.
Consider the following linear system with time delay: with a known vector valued function of the initial conditions: The initial condition,   t φ is a continuous and differentiable vector-valued function of In this study, the finite-time stability of the class of systems (1) has been investigated. Definition 1. Time-delay system (1) satisfying the given initial condition (2) is said to be finite-time In the following part, some existing results on delay dependent stability conditions are presented. These stability conditions were used for comparison against the results derived in this study. Theorem 1. The time-delayed system (1) with the function of initial conditions (2) is finite time stable with respect to {α,β,T} if there exists a positive scalar  such that the following condition holds: where: where: with  being symmetric matrix with all eigenvalues defined over the set of real numbers, [18].
Theorem 3. Time-delayed system (1) with the function of initial conditions (2) is finite time stable 3  exist as well as positive definite symmetric matrices P and Q such that the following conditions holds, [21]:

MAIN RESULT
In this section, the Lyapunov-like approach was used in order to find sufficient delay dependent conditions of finite time stability for the time delayed systems.
In the following part, a lemma necessary for construction of the system aggregation function is presented.
It was observed that the novel result presented here is based on the result given in [23].

Lemma 2. Let a scalar aggregation function
y y (12) where vector   t y is defined in the following manner: matrix which is continuous and differentiable over time interval   0, satisfying the following differential matrix equation: The further part of the proof is straightforward if In that sense, let us look at this expression after the derivation on the variable  has been performed: The previous relation can be further derived as: (23) so, by virtue of (23), the expression can be derived as: or: After rearrangement, the previous equation can be expressed as follows: (29) By virtue of (14), one can get: (30) and: (31) as well as: (32) and finally:  (14) and if the following condition is satisfied:  (12) , one can find: (45) Based on the known inequality 1 , and with the particular choice of I  , one can get: (46) Using the Jensen's integral inequality, as in Lemma 1, the following inequalities are valid: (47) Introducing the general solution of (14), given with: and by substituting (48) into (47), the following expression is obtained: (50) and : and finally: (56) Based on the crucial assumption of Theorem 4, in connection with definiteness of matrix Q (v) over prescribed time interval and using the assumption given in (36), follows: what directly leads to: Taking into account (44) and (56), it follows: Finally, condition (37) and the above inequality imply: what was to be proven. Q.E.D.
Remark 1. It can be noticed that the sufficient condition (37) entirely depends on the existence of the solution of (42). This requires that nonlinear algebraic matrix equation (42) Consequently, all conditions given (36) are satisfied.
In order to verify the finite time stability of system, given (61), the system operation is simulated under the following conditions: It is noticeable that:   , using various methods presented in Theorem 1 [17], Theorem 2 [18], Theorem 3 [21] , Theorem 4 [23] and Theorem 4 (this study) .
By simulating system (61), the actual values of parameter T , 1.945 a T  are estimated from the norm of state vector. The results are presented in Table 1. Furthermore, corresponding values of the parameter  (Theorem 4, [23]) are given in Table 1.
It is calculated that Theorem 4 (this study) gives noticeably better results than all other obtained using the previous Theorems. However, unlike Theorem 3 which uses LMI, Theorem 4 [23] is based on algebraic inequalities only.
That inequality can be solved without using appropriate optimization methods.

CONCLUSION
This paper extends some of the basic results in the area of the non-Lyapunov stability to the linear continuous invariant time-delay systems.
Under certain assumptions, the new sufficient, delay-dependent criteria for the finite time stability has been presented.
The derived result is based on algebraic inequalities only, which can be solved without using appropriate optimization methods. It has been shown that, under some circumstances, the conditions derived in this study leads to significant improvement in the finite time stability analysis, particularly in comparison with results given in [23], [24] 6. ACKNOWLEDGMENT This work has been supported by the Ministary of Science and Technological Department of Serbia under the Project ON 174 001.