Optimal input design for uncertain max‐plus linear systems

Abstract

Summary This paper investigates the optimal input design for uncertain max‐plus linear systems whose parameters are only known to belong to certain intervals. The exact interval input is introduced to minimize the regulating range of the input that ensures that the system can output at any specified value in the desired interval regardless of how the parameters vary in certain ranges. The optimal input design for uncertain max‐plus linear systems is formulated as an optimal control problem that finds the maximum element of the set of exact interval inputs with respect to the order of max‐plus interval vectors. This paper also considers the optimal approximate interval input of an uncertain max‐plus linear system that has no exact interval input. A polynomial algorithm is developed to find the optimal interval input and optimal approximate interval input of an uncertain max‐plus linear system. A data transmission system with parameter perturbations is employed as an example to demonstrate how the proposed results work in practical applications.

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

“…The optimal solution of problem (20) can determine the optimal allocation scheme such that the completion time of overall task is the earliest. For example, in the distributed system with task precedence graph shown in Figure 2, it can be seen that T 7 is the last subtask…”

“…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.…”

Joint source and relay precoder design for energy‐efficient MIMO‐cognitive relay networks

“…where P s_max and P r_max are the maximum transmit powers at the SU-Tx and SU-R, respectively; I s_max and I r_max denote the tolerable interference threshold at the PU-Rx due to the SU-Tx and SU-R, respectively; R min is the minimum rate requirement of the secondary system. In problem (P1), constraints (10) and (11) are the transmit power constraints. Interference constraints (12) and (13) ensure that the interference from the SU is below the tolerable interference thresholds at the PU.…”

“…Moreover, a relay node improves the coverage and performance of the SU without utilising extra spectrum resources [8]. On the other hand, the multi-input multi-output (MIMO) technique improves the capacity of wireless networks [9,10]. Particularly, the MIMO system in cognitive relay networks (CRNs) allows the SU to transmit multiple data streams by means of spatial multiplexing and may adjust radiation pattern to reduce the interference at the PU [11].…”

A Note on Global Optimization for Max-Plus Linear Systems

“…Max-plus linear systems can describe some nonlinear time-evolution systems with synchronization but no concurrency, such as flexible manufacturing systems, flow shop scheduling, traffic managements, communication networks (see, e.g., [1][2][3]). Many progresses have been made in control and optimization of max-plus linear systems (see, e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]).…”