Global optimization for max-plus linear systems and applications in distributed systems

Tao, Yuegang; Wang, Cailu

doi:10.1016/j.automatica.2020.109104

Cited by 8 publications

(22 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Max-plus algebra [1][2][3][4] has a nice algebraic structure and is effectively used to model, analyze, control and optimize some nonlinear time-evolution systems with synchronization but no concurrency (see, e.g., [5][6][7][8][9][10][11]). These nonlinear systems can be described by a max-plus linear time-invariant model, which is called the max-plus linear system.…”

Section: Introductionmentioning

confidence: 99%

Ordered Structures of Polynomials over Max-Plus Algebra

2021

Self Cite

View full text Add to dashboard Cite

The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element.

show abstract

Section: Introductionmentioning

confidence: 99%

Ordered Structures of Polynomials over Max-Plus Algebra

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…Much work has been done in modeling, analysis, control, and optimization of max-plus-linear systems, including reachability analysis MPL systems ( [13], [14]). Further discussion of the MPL system presented the necessary and sufficient conditions for the existence and uniqueness of globally optimal solutions MPL systems ( [15], [16]).…”

Section: Introductionmentioning

confidence: 99%

Modeling Predictive Tracking Control for Max-Plus Linear Systems in Manufacturing

Aulia

Widowati

Tjahjana

et al. 2020

JFMA

View full text Add to dashboard Cite

Discrete event systems, also known as DES, are class of system that can be applied to systems having an event that occurred instantaneously and may change the state. It can also be said that a discrete event system occurs under certain conditions for a certain period because of the network that describes the process flow or sequence of events. Discrete event systems belong to class of nonlinear systems in classical algebra. Based on this situation, it is necessary to do some treatments, one of which is linearization process. In the other hand, a Max-Plus Linear system is known as a system that produces linear models. This system is a development of a discrete event system that contains synchronization when it is modeled in Max-Plus Algebra. This paper discusses the production system model in manufacturing industries where the model pays the attention into the process flow or sequence of events at each time step. In particular, Model Predictive Control (MPC) is a popular control design method used in many fields including manufacturing systems. MPC for Max-Plus-Linear Systems is used here as the approach that can be used to model the optimal input and output sequences of discrete event systems. The main advantage of MPC is its ability to provide certain constraints on the input and output control signals. While deciding the optimal control value, a cost criterion is minimized by determining the optimal time in the production system that modeled as a Max-Plus Linear (MPL) system. A numerical experiment is performed in the end of this paper for tracking control purposes of a production system. The results were good that is the controlled system showed a good performance.

show abstract

“…This paper will further address the global optimisation problem considered in ref. [26]. The global optimisation problem involves some operations between infinity elements and real numbers, which are not clearly defined in ref.…”

mentioning

confidence: 99%

“…The global optimisation problem involves some operations between infinity elements and real numbers, which are not clearly defined in ref. [26]. This paper will explain these operations definitely, and simplify the formula of the greatest lower bound and the discriminant of the existence of globally optimal solutions given in ref.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Locally and globally optimal solutions of global optimisation for max‐plus linear systems

Wang

Tao

2021

IET Control Theory & Appl

Self Cite

View full text Add to dashboard Cite

This paper considers the locally and globally optimal solutions of the global optimisation problem whose objective function is a max-plus vector-valued function and constraint function is a real affine function. The formulas about global optimisation in the existing work are further explained and simplified. The necessary and sufficient conditions for the existence and uniqueness of locally optimal solutions of a single objective are established, which are then used to deduce these of globally optimal solutions. Furthermore, the general formulas of locally and globally optimal solutions are presented, respectively. The local optimisation is then applied to solve the load distribution problem of distributed systems with no globally optimal solution, and the optimal allocation scheme is proposed to complete the overall task at the earliest time. The proposed method is constructive, and the obtained results are illustrated by the numerical examples. INTRODUCTIONMax-plus linear systems can describe some nonlinear timeevolution systems with synchronisation but no concurrency, such as manufacturing systems, flow shop scheduling, traffic managements, communication networks (see, e.g. refs.[1-4]).Many researches have been made on control and optimisation of max-plus linear systems (see, e.g. refs. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]).Global optimisation is to find the global minimiser of a function or a set of functions over a given set, which has been a basic tool in all areas of engineering, medicine, economics, and so on (see, e.g. refs. [23, 24]). The global optimisation of maxplus linear systems has been considered in refs. [25] and[26], whose objective function is a max-plus vector-valued function and constraint function is a real affine function. This paper will further address the global optimisation problem considered in ref. [26]. The global optimisation problem involves some operations between infinity elements and real numbers, which are not clearly defined in ref. [26]. This paper will explain these operations definitely, and simplify the formula of the greatest lower bound and the discriminant of the existence of globally optimal solutions given in ref. [26]. In addition, the zero coefficient in the constraint function is neglected when it discusses the uniqueness of globally optimal solutions (see refs. [26] and [27]), and a sufficient condition for the uniqueness is presented.

show abstract

Global optimization for max-plus linear systems and applications in distributed systems

Cited by 8 publications

References 22 publications

Ordered Structures of Polynomials over Max-Plus Algebra

Ordered Structures of Polynomials over Max-Plus Algebra

Modeling Predictive Tracking Control for Max-Plus Linear Systems in Manufacturing

Locally and globally optimal solutions of global optimisation for max‐plus linear systems

Contact Info

Product

Resources

About