An improved stability criterion for linear systems with multi-rate sampled data
This paper proposes a stability criterion for linear systems with multi-rate sampled data by constructing novel Lyapunov functionals. The systems are represented as linear systems with input delay. The Lyapunov functionals consist of a quadratic function, an integral term related to the sampling intervals of each sensor and looped-functionals related to the sampling intervals of whole systems. In the view of the Lyapunov stability theorem, the quadratic function and the integral term require the positivity condition, whereas the looped-functionals do not require the positivity condition because the functionals are zeros at the sampling instants. Furthermore, the looped-functionals exploit the information on the sampled state and the state at the next sampling instants. Based on the Lyapunov functionals, this paper derives a stability criterion in terms of linear matrix inequalities. The simulation results show the effectiveness of the proposed stability criterion. This paper considers stability and stabilization problems for linear systems with multi-rate sampled data under aperiodic samplings which are represented as linear systems with an input time delay. To handle the stability problem, the authors construct a Lyapunov functional which consists of a quadratic function, an integral term related to the sampling intervals of each sensor, and looped-functionals related to the sampling intervals of whole systems. The Lyapunov functionals handle the sampling intervals in the view of each sensor and whole systems in the integral term and the looped-functionals, respectively. In the view of the stability theorem, the quadratic function and the integral term require the positivity condition, whereas the looped-functionals do not require the positivity condition because the functionals are zeros at the sampling instants. The looped-functionals utilize the information not only on the interval between the last sampling instant and the present time but also on the interval between the present time and the next sampling instant. Furthermore, the looped-functionals exploit the information on the sampled state. The authors firstly use looped-functionals in the linear systems with multi-rate sampled data. Based on the method of Lyapunov-Krasovskii functionals, this paper derives a stability criterion in terms of linear matrix inequalities. A similar stabilization theorem is used for obtaining the controller gain matrix. The presented results are compared with similar results of the paper: [\textit{M. Moarref} and \textit{L. Rodrigues}, ``Exponential stability and stabilization of linear multi-rate sampled-data systems'', ACC, 158--163 (2013)]. The results of the paper can get the less conservative results than [loc. cit.] because a slack matrix and the relaxation on the positivity constraint in the looped-functionals give the freedom for checking the feasibility of the proposed criterion. The examples given and simulation results show that the authors' results are less conservative but computational complex.
scientific article; zbMATH DE number 7443507