Studies have shown that fractional calculus can describe and characterize a practical system satisfactorily. Therefore, the stabilization of fractional-order systems is of great significance. The asymptotic stabilization problem of delayed linear fractional-order systems (DLFS) subject to state and control constraints is studied in this article. Firstly, the existence conditions for feedback controllers of DLFS subject to both state and control constraints are given. Furthermore, a sufficient condition for invariance of polyhedron set is established by using invariant set theory. A new Lyapunov function is constructed on the basis of the constraints, and some sufficient conditions for the asymptotic stability of DLFS are obtained. Then, the feedback controller and the corresponding solution algorithms are given to ensure the asymptotic stability under state and control input constraints. The proposed solution algorithm transforms the asymptotic stabilization problem into a linear/nonlinear programming (LP/NP) problem which is easy to solve from the perspective of computation. Finally, three numerical examples are offered to illustrate the effectiveness of the proposed method.

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This paper deals with the Constrained Regulation Problem (CRP) for linear continuous-times fractional-order systems. The aim is to find the existence conditions of linear feedback control law for CRP of fractional-order systems and to provide numerical solving method by means of positively invariant sets. Under two different types of the initial state constraints, the algebraic condition guaranteeing the existence of linear feedback control law for CRP is obtained. Necessary and sufficient conditions for the polyhedral set to be a positive invariant set of linear fractional-order systems are presented, an optimization model and corresponding algorithm for solving linear state feedback control law are proposed based on the positive invariance of polyhedral sets. The proposed model and algorithm transform the fractional-order CRP problem into a linear programming problem which can readily solved from the computational point of view. Numerical examples illustrate the proposed results and show the effectiveness of our approach.

Constrained regulation problem (CRP) for continuous-time stochastic systems is investigated in this article. New existence conditions of linear feedback control law for continuous-time stochastic systems under constraints are proposed. The computation method for solving constrained regulation problem of stochastic systems considered in this article is also presented. Continuous-time stochastic linear systems and stochastic nonlinear systems are focused on, respectively. First, the condition of polyhedral invariance for stochastic systems is established by using the theory of positive invariant set and the principle of comparison. Second, the asymptotic stability conditions in the sense of expectation for two types of stochastic systems are established. Finally, finding the linear feedback controller model and corresponding algorithm of constrained regulation problem for two types of stochastic systems are also proposed by using the obtained condition. The presented model of the stochastic constrained regulation problem in this article is formulated as a linear programming problem, which can be easily implemented from a computational point of view. Our approach establishes a connection between the stochastic constrained regulation problem and positively invariant set theory, as well as provides the possibility of using optimization methodology to find the solution of stochastic constrained regulation problem, which differs from other methods. Numerical examples illustrate the proposed method.

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