Optimization approach to the constrained regulation problem for linear continuous-time fractional-order systems

Si, Xindong; Yang, Hongli

doi:10.1515/ijnsns-2019-0267

Cited by 4 publications

(5 citation statements)

References 31 publications

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“…The problem of estimating the maximum robust invariant set for discrete time nonlinear regenerative systems in an optimal control framework is considered in [22]. The study of polyhedral sets positive invariance by optimization is investigated in [23,24]. The ellipsoid and Lorenz cone invariance condition in a nonconvex optimization form for continuous time systems is given in [25], and the existence problem of the solution is discussed in conjunction with KKT theorem.…”

Section: Of 17mentioning

confidence: 99%

“…As long as the initial state and the subsequent trajectory of the system are always in a certain positive invariant set, the state quantity of the system can be guaranteed to remain in the positive invariant set. Due to its good properties, positive invariant sets play an important role in the study of system stability analysis and feedback controller design [1][2][3][4][5]. Lyapunov stability theory provides theoretical support to the study the stability of dynamic systems.…”

Section: Introductionmentioning

confidence: 99%

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Characterization of Positive Invariance of Quadratic Convex Sets for Discrete Systems Using Optimization Approaches

Lei¹,

Yang²,

Ivanov³

2023

Preprint

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Positive invariant set is an important concept of dynamic systems. The purpose of this paper is to study the sufficient and necessary conditions that the set of ellipsoids and the Lorenz cone are positive invariant sets of discrete time systems. By means of nonlinear programming and induced norm, the problem of positive invariance is formulated as an optimization problem, and the equivalent dual form optimization is also presented using the dual optimization method. Our results provide more alternative methods for determining the positive invariance of quadratic form convex sets from the point of view of optimization and dual optimization. The effectiveness of this method is demonstrated by numerical examples.

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Section: Of 17mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Characterization of Positive Invariance of Quadratic Convex Sets for Discrete Systems Using Optimization Approaches

Lei¹,

Yang²,

Ivanov³

2023

Preprint

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“…Descriptor systems can be considered a powerful modeling tool since they can describe processes governed by differential and algebraic equations [1]. They play an important role in the field of system control theory because of their extensive practical background, such as in chemistry, robotics, and circuit systems [2][3][4]. CSRP is a basic problem in control systems in that there are always hard constraints on states and control inputs in practical engineering problems.…”

Section: Introductionmentioning

confidence: 99%

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

Yang,

Si,

Ivanov

2024

Fractal Fract

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This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method.

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“…As long as the initial state and the subsequent trajectory of the system are always in a particular positively invariant set, the state quantity of the system can be guaranteed to remain in the positively invariant set. Due to their good properties, positively invariant sets play an important role in the study of system stability analysis and feedback controller design [1][2][3][4][5]. Lyapunov's stability theory provides theoretical support for the study of the stability of dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

“…The problem of estimating the maximum robust invariant set for discrete-time nonlinear regenerative systems in an optimal control framework is considered in [22]. The study of polyhedral sets' positive invariances via optimization is investigated in [23,24]. In [25], Nagumo's theorem is used to study the sufficient and necessary positive invariance conditions of the ellipsoid and Lorenz cone for continuous-time systems using optimization techniques.…”

Section: Introductionmentioning

confidence: 99%

Characterization of Positive Invariance of Quadratic Convex Sets for Discrete-Time Systems Using Optimization Approaches

2023

Self Cite

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A positively invariant set is an important concept in dynamical systems. The study of positively invariant set conditions for discrete-time systems is one interesting topic in both theoretical studies and practical applications research. Different methods for characterizing the invariance of different types of sets have been established. For example, the ellipsoidal and the Lorenz cone, which are quadratic convex sets, have different properties from a polyhedral set. This paper presents an optimization method and a dual optimization method to characterize the positive invariance of the ellipsoidal and the Lorenz cone. The proposed methods are applicable to both linear and nonlinear discrete-time systems. Using nonlinear programming and an induced norm, the positive invariance condition problems are transformed into optimization problems, and the dual optimization method is also used to give equivalent dual forms. Fewer results on the positive invariance condition of Lorenz cones can be found than for the other type of set; this paper fulfills the results of this problem. In addition, the proposed methods in this paper provide more options for checking the positive invariance of quadratic convex sets from the perspective of optimization and dual optimization. The effectiveness of this method is demonstrated by numerical examples.

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Optimization approach to the constrained regulation problem for linear continuous-time fractional-order systems

Cited by 4 publications

References 31 publications

Characterization of Positive Invariance of Quadratic Convex Sets for Discrete Systems Using Optimization Approaches

Characterization of Positive Invariance of Quadratic Convex Sets for Discrete Systems Using Optimization Approaches

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

Characterization of Positive Invariance of Quadratic Convex Sets for Discrete-Time Systems Using Optimization Approaches

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