Minimum Number Of Time Steps Research Articles

Generalized min-up/min-down polytopes

Consider a time horizon and a set of N possible states for a given system. The system must be in exactly one state at a time. In this paper, we generalize classical results on min-up/min-down constraints for a 2-state system to an N-state system with N≥3. The minimum-time constraints enforce that if the system switches to state i at time t, then it must remain in state i for a minimum number of time steps. The minimum-time polytope is defined as the convex hull of integer solutions satisfying the minimum-time constraints. A variant of minimum-time constraints is also considered, namely the no-spike constraints. They enforce that if state i is switched on at time t, the system must remain on states j≥i during a minimum time. Symmetrically, they also enforce that if state i is switched off at time t, the system must remain on states j<i during a minimum time. The no-spike polytope is defined as the convex hull of integer solutions satisfying the no-spike constraints. For both the minimum-time polytope and the no-spike polytope, we introduce families of valid inequalities. We prove that these inequalities are facet-defining and lead to a complete description of polynomial size for each polytope.

Reward Value-Based Goal Selection for Agents’ Cooperative Route Learning Without Communication in Reward and Goal Dynamism

This paper proposes a goal selection method to operate agents get maximum reward values per time by noncommunicative learning. In particular, that method aims to enable agents to cooperate along to dynamism of reward values and goal locations. Adaptation against to these dynamisms can enable agents to learn cooperative actions along to changing transportation tasks and changing incomes/rewards because of transporting tasks for heavy/valuable and light/valueless items in a storehouse. Concretely, this paper extends the previous noncommunicative cooperative action learning method (Profit minimizing reinforcement learning with oblivion of memory: PMRL-OM) and sets the two unified conditions combined of the number of time steps and the rewards. One of the unified conditions is calculated the approximated number of time steps if the expected reward values are the same each other for all purposes, and the other is the minimum number of time steps divided by the reward value. The proposed method makes all agents learn to achieve the purposes in the order in which they have the minimum number of the condition values. After that, each agent learns cooperative policy by PMRL-OM as the previous method. This paper analyzes the unified conditions and derives that the condition calculating the approximated time steps can be combined both evaluations with almost same weight unlike the value the other condition, that is, the condition can help the agents to select the appropriate purposes among them with the small difference in terms of the two evaluations. This paper tests empirically the performances of PMRL-OM with the two conditions by comparing with the PMRL-OM in three cases of grid world problems whose goal locations and reward values are changed dynamically. The results of this derive that the unified conditions perform better than PMRL-OM without some conditions in grid world problems. In particular, it is clear that the condition calculating the approximated time step can direct the appropriate goals for the agents.

Heuristics for Spreading Alarm throughout a Network

This paper provides heuristic methods for obtaining a burning number, which is a graph parameter measuring the speed of the spread of alarm, information, or contagion. For discrete time steps, the heuristics determine which nodes (centers, hubs, vertices, users) should be alarmed (in other words, burned) and in which order, when afterwards each alarmed node alarms its neighbors in the network at the next time step. The goal is to minimize the number of discrete time steps (i.e., time) it takes for the alarm to reach the entire network, so that all the nodes in the networks are alarmed. The burning number is the minimum number of time steps (i.e., number of centers in a time sequence alarmed “from outside”) the process must take. Since the problem is NP complete, its solution for larger networks or graphs has to use heuristics. The heuristics proposed here were tested on a wide range of networks. The complexity of the heuristics ranges in correspondence to the quality of their solution, but all the proposed methods provided a significantly better solution than the competing heuristic.

Minimum-Time Consensus-Based Approach for Power System Applications

This paper presents minimum-time consensus-based distributed algorithms for power system applications, such as load shedding and economic dispatch. The proposed algorithms are capable of solving these problems in a minimum number of time steps instead of asymptotically as in most of the existing studies. Moreover, these algorithms are applicable to both undirected and directed communication networks. Simulation results are used to validate the proposed algorithms.

Fire containment in grids of dimension three and higher

We consider a deterministic discrete-time model of fire spread introduced by Hartnell [Firefighter! an application of domination, Presentation, in: 20th Conference on Numerical Mathematics and Computing, University of Manitoba in Winnipeg, Canada, September 1995] and the problem of minimizing the number of burnt vertices when a fixed number of vertices can be defended by firefighters per time step. While only two firefighters per time step are needed in the two-dimensional lattice to contain any outbreak, we prove a conjecture of Wang and Moeller [Fire control on graphs, J. Combin. Math. Combin. Comput. 41 (2002) 19–34] that 2 d - 1 firefighters per time step are needed to contain a fire outbreak starting at a single vertex in the d-dimensional square lattice for d ⩾ 3 ; we also prove that in the d-dimensional lattice, d ⩾ 3 , for each positive integer f there is some outbreak of fire such that f firefighters per time step are insufficient to contain the outbreak. We prove another conjecture of Wang and Moeller that the proportion of elements in the three-dimensional grid P n × P n × P n which can be saved with one firefighter per time step when an outbreak starts at one vertex goes to 0 as n gets large. Finally, we use integer programming to prove results about the minimum number of time steps needed and minimum number of burnt vertices when containing a fire outbreak in the two-dimensional square lattice with two firefighters per time step.

Multicast in hypercube multiprocessors

An efficient interprocessor communication mechanism is essential to the performance of hypercube multiprocessors. All existing hypercube multiprocessors basically support one-to-one interprocessor communication only. However, multicast (one-to-many) communication is highly demanded in executing many data parallel algorithms. A multicast algorithm should attempt to inform each destination in a minimum number of time steps while generating a least amount of traffic. In this paper, we first propose a graph theoretical model, the Optimal Multicast Tree (OMT), for interprocessor communication in distributed-memory multiprocessors. The problem of finding an OMT is conjectured to be NP-hard even for hypercube multiprocessors. A heuristic Greedy multicast algorithm which guarantees a minimized message delivery time is proposed. Simulation results show that the performance of the Greedy algorithm is very close to the optimal solution. Routing of multicast messages is done in a distributed manner. The hardware design of a VLSI router which supports all types of communications is briefly discussed.

A VLSI router design for hypercube multiprocessors

An efficient interprocessor communication mechanism is essential to the performance of hypercube multiprocessors. All existing hypercube multiprocessors only support one-to-one (unicast) communication. Multi-destination communication (multicast), which is highly demanded in many application areas, is not supported. Moreover, a store-and-forward routing scheme is usually used at each intermediate node, which further increases the communication delay. In this paper we present the hardware design of a VLSI router which is versatile in the sense that it supports unicast communication as well as multicast and broadcast communications. The router can handle messages with variable length and arbitrary number of destinations. For any type of communication, each destination will receive the source message in the minimum number of time steps. The total amount traffic created for message routing is minimum in the case of unicast and broadcast, and is very close to the optimal solution for multicast. Message passing is done in a distributed manner. The router handles message routing in a relay approach in which the routing decision is made immediately after the destination addresses are received. Architectural description of the router design is presented. A VLSI prototype design of the Message Handling Unit, an essential part of the router, is detailed.

Note on 'Time-optimal control strategy for saturating linear discrete systems’

The control policy taking a linear system from a given state to the origin in a minimum number of time steps is not unique. Such flexibility also offers an opportunity to minimize the control effort in a straightforward manner.

線形離散時間システムの最短時間定値制御

In this paper, the settling problem of the output of linear, time-invariant, discrete-time systems is discussed, i.e. the problem is to find a control sequence which transfers and settles the output to a given constant reference value with the minimum time (in the sense of the minimum number of time steps). The solution to the posed problem can be obtained in the form of an extension of the results for the problem of the zero reference value case. Also, the internal stability and the full time settling property of the constant-value control system are considered. It is shown that the transmission zero of the system play some important roles on the synthesis and the evaluating performance of such control systems. Finally a simple numerical example is presented to illustrate the above results.

Output dead-beat control—a geometric approach

The problem of forcing the output of a multivariate sampled-data system to zero in a minimum number of time steps is discussed. The solution to the posed problem is given as a linear state feedback. The zeros of the system are contained among the poles of the closed-loop system. The performance of the closed-loop system will be unsatisfactory if any of these zeros are located close to or outside the unit circle. In this case suboptimal dead-beat controllers are derived.

Multivariable dead-beat control

The problem of forcing the state of a linear, multivariable, sampled-data system to zero in a minimum number of time steps is discussed. The solution to the posed problem is given as linear state feedbacks. Numerical aspects of the algorithms for computing these feedbacks are presented. The methods are successfully used to control the temperature profile of a diffusion process in the laboratory.

Multivariate control theory in linear sequential machines

This paper presents a unified survey of some modern multivariate control theory aspects and techniques applied to linear sequential machines over a Galois field GF(p), Utilizing the concepts of controllability and observability for a Mealy-type machine the canonical decomposition problem, the state minimization problem, the identification problem, the transformation to canonical form problem, and the controllability/observability problem of combined machines are studied. Then the problem of designing a linear feedback controller for driving any state of a linear machine to the zero state after a minimum number of time steps A(clock periods), as well as the dual problem of designing a time-optimal state reconstructor for the same machine are solved. Finally, the problems of inverting a linear sequential machine and decoupling its inputs and outputs by using state feedforward and feedback are examined. Several examples illustrate the theoretical results.