Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach

Abstract

V. CONCLUSIONThis note has developed a sliding-mode controller which requires only output information for a class of uncertain linear systems. The controller comprises both linear and nonlinear components and is static in nature, i.e., no compensation/observation is included. The novelty of the approach is in the rationale and method used to synthesize the linear control component. The reachability condition is not required to be satisfied globally. Instead, sliding is only expected to take place within a subset of the state-space containing the origin referred to as the sliding patch. This region is shown to be rendered invariant by the control law. The linear static output feedback control component is synthesized using an LMI optimization. The resulting LMI formulation can be solved easily by standard commercially available software. The efficacy of the approach has been demonstrated on a numerical example taken from the sliding-modeliterature. Control, vol. 3, pp. 115-132, 1993. [11] C. M. Kwan, "On variable structure output feedback controllers," IEEE Trans. Automat. Contr., vol. 41, pp. 1691-1693, Nov. 1996 [12] J. J. E. Slotine, J. K. Hedrick, and E. A. Misawa, "On sliding observers for nonlinear systems," ASME: J. Dyn. Syst., Meas. Control, vol. 109, pp. 245-252, Sept. 1987. [13] S. K. Spurgeon and R. Davies, "A nonlinear control strategy for robust sliding-mode performance in the presence of unmatched uncertainty," Int. J. Control, vol. 57, no. 5, pp. 1107Control, vol. 57, no. 5, pp. -1123Control, vol. 57, no. 5, pp. , 1993 Abstract-This note deals with the problem of local stabilization of linear discrete-time systems subject to control saturation. A linear matrix inequalitie-based framework is proposed in order to compute a saturating state feedback that stabilizes the system with respect to a given set of admissible initial states and, in addition, guarantees some dynamical performances when the system operates in the zone of linear behavior (i.e., when the controls are not saturated). REFERENCES

Tracking control of leader-follower multi-agent systems subject to actuator saturation

“…and P (ε), ε ∈ (0, 1] is the positive definite solution of the Riccati equation (4). We refer to the control law (9) as lowand-high-gain feedback laws and ε as the low-gain parameter, since in view of Lemma 1, the norm of the feedback gain matrix B T P (ε) can be made arbitrarily small by choosing a sufficiently small ε, and ρ > 0 is called a high-gain parameter for it can possibly take a high value.…”

“…Obviously, if the adjacent graphḠ is connected, the tracking problem can be solved if and only if system (18) is simultaneously asymptotically stable for i = 1, 2, · · · , N, where P (ε), ε ∈ (0, 1] is the positive definite solution of the Riccati equation (4).…”

“…The eigenvalues of H are λ 1 = 2.6180, λ 2 = 0.3820, λ 3 = 0.3820, and λ 4 = 2.6180. To construct the low-and-high-gain feedback laws (9), we choose Q(ε) = εI 2 and solve the algebraic Ricatti equation (4) to obtain the following equation [9] P (ε) =…”

Oracle based approach to admissible invariant sets

“…An AIS can be obtained using different approaches [10], [11], [12], [13], [14], [15], [16]. where P is a symmetric definite positive matrix and ρ + ∈ ℝ , can be obtained using a linear difference inclusion (LDI) of the system (2) around the origin [17].…”

A novel design method for nonlinear excitation control system with saturating input

“…In the last years, the problem of stabilization of linear system subject to control saturating input has received the attention of many authors [8]- [12]. In this context, the present paper proposes the application of the LMI for the development of a systematic design methodology of multi-objective control to guarantee the asymptotic stability not of the whole state space but only of a given set 0 χ of admissible initial states, which can be viewed as the zone of operation of the system.…”