Simultaneous stabilization of full state information Linear Time‐Invariant systems

Abstract

Single necessary and sufficient stability conditions, are presented, for a common Linear Time-Invariant (LTI) controller that simultaneously stabilizes each of a given family of LTI plants. It is assumed that the strictly proper, lumped and LTI plants have stabilizable realizations and are strongly stabilizable, and that the state-space dimensions of the plants do not change, are even and are double the input-space dimensions of the plants. The stabilizing controller is designed for a nominal plant and its free parameter is fixed assuring stability for all the plants, and assuring a stable controller. The stability conditions for simultaneous stabilization are based on the parity interlacing property, on a matrix that must be unimodular and on an analytical expression of the stabilizing controllers.

Design of simultaneous static output feedback low‐gain H∞ controllers for a collection of time‐delay system

“…Simultaneous control problems are important because they arise frequently in practice, because of plant uncertainty, failure modes, plant variation, plants with several modes of operation, or nonlinear plants linearized at several different equilibria. 1 In the literature, the studies on simultaneous control problems focus on simultaneous stabilization (see, eg, the works of previous studies [2][3][4][5][6][7][8][9][10][11][12][13] ) and simultaneous H ∞ control (see, eg, the works of previous studies [14][15][16][17][18] ) of linear systems. Some other results have been proposed for simultaneous regional pole placement 1 and simultaneous mixed LQR/H ∞ control.…”

Continuous simultaneous stabilization of single‐input nonlinear stochastic systems

“…Therefore, the investigation of simultaneous stabilization problems is quite relevant from a practical point of view, and many important results have been obtained for linear systems: see [21][22][23][24][25] and references therein. For nonlinear systems, the problem is more involved, and there are fewer results in the literature, for example, [26][27][28][29][30].…”

Simultaneous stabilization of a collection of uncertain discrete-time systems with time-varying state-delay via discrete-time sliding mode control

“…Since the work of Saeks and Murray (1982) and Vidyasagar and Viswanadham (1982), the simultaneous stabilization problem has received wide consideration. For linear systems, both linear feedback controllers (Howitt and Luus, 1993; Cao et al., 1999; Savkin and Petersen, 2000; Soliman et al., 2010; Galindo, 2012; Yu and Chi, 2014; Kohan-Sedgh et al., 2016) and nonlinear feedback controllers (Andrade-Da Silva and Edwards, 2009; Emel’yanov et al., 2012)) have been established to address this problem. In the case of nonlinear systems, this problem is more complicated and there are only a few results in the literature.…”