Abstract
This paper points to sufficient conditions for practical stability of linear systems with time delay in state. Particularly, in control system engineering practice, despite the contribution to the contemporary control theory and system thinking, the problems of practical stability are not developed in details. Taking into account that the system can be stable in a classic way, but it can also possess inappropriate quality of dynamic behavior, and because of that, it is not applicable. For engineers it is crucial to take the system into consideration in relation to permitted states in phase space which are defined for such a problem, Dihovicni et al. (2006). Although, there are some papers covering practical stability problems, the lack of exploring it by using fundamental matrix and matrix measure was observed. Our main idea is to present definitions and conditions for practical stability, applying matrix measure approach. From a practical view point, it is crucial to find intervalson which the system is stable, and to know the function of initial state, the "prehistory" of system motion. The practical stability for a class of a distributed parameter system is also presented. The system is described in state space and a unique theory for such a problem is developed where a fundamental matrix of system and matrix measure is used. Using an efficient approach based on matrix measure and system fundamental matrix, the theorems for practical stability of distributed parameter systems are developed, and superiority of our results is illustrated with a numerical example.
Highlights
This paper contradicts the majority of theories dealing with this topic by developing a unique theory in state space using a fundamental matrix and matrix measure approach combined with conditions of singular values of matrix A0 and A1
Definition 1: The distributed parameter system described by equation (32) that satisfies the initial condition (33) is stable on finite time interval in relation to [ξ(t,z), β, T, Z] if and only if:
Theorem4: The distributed parameter system described by equation (32) that satisfies the internal condition (33) is stable on finite time interval in relation to [ξ(t,z), β, T, Z] if and only if the following condition is satisfied: b e 1 a 2n_Ai$t$z
Summary
During the process of analysis and synthesis of control systems, the fundamental problem is stability, Wang (1992). The most often case for consideration of control systems, is behaving on an infinite interval which in real cases is only of academic importance in spite of practical stability which has significance in real life, Nedic et al (2006). Considering engineering practice needs, it is important to explore the time intervals where the system is stable. This paper contradicts the majority of theories dealing with this topic by developing a unique theory in state space using a fundamental matrix and matrix measure approach combined with conditions of singular values of matrix A0 and A1. Where the system described by equation (1) is presented in a free working state, x(t) is the state vector, and A0 and A1 are constants of the system matrix of appropriate dimension, and τ is time delay. The system behavior described by equation (1) is defined on time interval, J ϭ {t0, t0 ϩ T}, Nenadic et al (1997), and T can be a positive real number
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