Value Iteration-Based Cooperative Adaptive Optimal Control for Multi-Player Differential Games With Incomplete Information

This paper presents a novel cooperative value iteration(VI)-based adaptive dynamic programming method for multi-player differential game models with a convergence proof.The players are divided into two groups in the learning process and adapt their policies sequentially.Our method removes the dependence of admissible initial policies,which is one of the main drawbacks of the PI-based frameworks.Furthermore,this algorithm enables the players to adapt their control policies without full knowledge of others’ system parameters or control laws.The efficacy of our method is illustrated by three examples.

Cooperative maneuver decision making for multi-UAV air combat based on incomplete information dynamic game

Cooperative autonomous air combat of multiple unmanned aerial vehicles(UAVs)is one of the main combat modes in future air warfare,which becomes even more complicated with highly changeable situation and uncertain information of the opponents.As such,this paper presents a cooperative decision-making method based on incomplete information dynamic game to generate maneuver strategies for multiple UAVs in air combat.Firstly,a cooperative situation assessment model is presented to measure the overall combat situation.Secondly,an incomplete information dynamic game model is proposed to model the dynamic process of air combat,and a dynamic Bayesian network is designed to infer the tactical intention of the opponent.Then a reinforcement learning framework based on multiagent deep deterministic policy gradient is established to obtain the perfect Bayes-Nash equilibrium solution of the air combat game model.Finally,a series of simulations are conducted to verify the effectiveness of the proposed method,and the simulation results show effective synergies and cooperative tactics.

A rule induction algorithm for incomplete information system

Incompleteness of information about objects may be the greatest obstruct to performing induction learning from examples. In this paper, the concept of limited-non-symmetric similarity relation is used to formulate a new definition of approximation to an incomplete information system. With the new definition of approximation to an object set and the concept of attribute value pair, rough-setsbased methodology for certain rule acquisition in an incomplete information system is developed. The algorithm can deal with incomplete data directly and does not require changing the size of the original incomplete system. Experiments show that the algorithm provides precise and simple certain decision rules and is not affected by the missing values.

Data-Driven Learning Control Algorithms for Unachievable Tracking Problems

For unachievable tracking problems, where the system output cannot precisely track a given reference, achieving the best possible approximation for the reference trajectory becomes the objective. This study aims to investigate solutions using the Ptype learning control scheme. Initially, we demonstrate the necessity of gradient information for achieving the best approximation.Subsequently, we propose an input-output-driven learning gain design to handle the imprecise gradients of a class of uncertain systems. However, it is discovered that the desired performance may not be attainable when faced with incomplete information.To address this issue, an extended iterative learning control scheme is introduced. In this scheme, the tracking errors are modified through output data sampling, which incorporates lowmemory footprints and offers flexibility in learning gain design.The input sequence is shown to converge towards the desired input, resulting in an output that is closest to the given reference in the least square sense. Numerical simulations are provided to validate the theoretical findings.

CONSISTENCY OF MLE OF THE PARAMETER OF EXPONENTIAL LIFETIME DISTRIBUTION FOR RANDOM CENSORING MODEL WITH INCOMPLETE INFORMATION

In this paper, we have discussed a random censoring test with incomplete information, and proved that the maximum likelihood estimator(MLE) of the parameter based on the randomly censored data with incomplete information in the case of the exponential distribution has the strong consistency.

Dynamic Multi-team Antagonistic Games Model with Incomplete Information and Its Application to Multi-UAV

At present, the studies on multi-team antagonistic games(MTAGs) are still in the early stage, because this complicated problem involves not only incompleteness of information and conflict of interests, but also selection of antagonistic targets.Therefore, based on the previous researches, a new framework is proposed in this paper, which is dynamic multi-team antagonistic games with incomplete information(DMTAGII) model.For this model, the corresponding concept of perfect Bayesian Nash equilibrium(PBNE) is established and the existence of PBNE is also proved. Besides, an interactive iteration algorithm is introduced according to the idea of the best response for solving the equilibrium. Then, the scenario of multiple unmanned aerial vehicles(UAVs) against multiple military targets is studied to solve the problems of tactical decision making based on the DMTAGII model. In the process of modeling, the specific expressions of strategy, status and payoff functions of the games are considered, and the strategy is coded to match the structure of genetic algorithm so that the PBNE can be solved by combining the genetic algorithm and the interactive iteration algorithm.Finally, through the simulation the feasibility and effectiveness of the DMTAGII model are verified. Meanwhile, the calculated equilibrium strategies are also found to be realistic, which can provide certain references for improving the autonomous ability of UAV systems.

Multi-criteria decision-making approach with incomplete certain information based on ternary AHP

It is not uncommon in multiple criteria decision-making that the numerical values of alternatives of some criteria are subject to imprecision, uncertainty and indetermination and the information on weights of criteria is incomplete certain. A new multiple criteria decision- making method with incomplete certain information based on ternary AHP is proposed. This improves on Takeda＇s method. In this method, the ternary comparison matrix of the alternatives under each pseudo-criteria is constructed, the eigenvector associated with the maximum eigenvalue of the ternary comparison matrix is attained as to normalize priority vector of the alternatives, then the order of alternatives is obtained by solving two kinds of linear programming problems. Finally, an example is given to show the feasibility and effectiveness of the method.

Remanufacturing Scheme Design for Used Parts Based on Incomplete Information Reconstruction

The different conditions of use of a component result in a variety of damage levels.Therefore,excluding differences in shape and size,used parts show a high degree of uncertainty regarding failure characteristics,quality conditions,and remaining life,which seriously affects the efficiency of a remanufacturing scheme design.Aiming to address this problem,a remanufacturing scheme design method based on the reconstruction of incomplete information of used parts is proposed.First,the remaining life of the reconstructed model is predicted by finite element analysis,and the demand for the next life cycle is determined.Second,the scanned 3D damage point cloud data are registered with the original point cloud data using the integral iterative method to construct a missing point cloud model to achieve the restoration of geometric information.Then,according to reverse engineering and laser cladding remanufacturing,the tool remanufacturing process path can be generated by the tool contact point path section line method.Finally,the proposed method is adopted for turbine blades to evaluate the effectiveness and feasibility of the proposed scheme.This study proposes a remanufacturing scheme design method based on the incomplete reconstruction of used part information to solve the uncertain and highly personalized problems in remanufacturing.

Consistency and Asymptotic Normality of MLE of Parameter Vector in a Randomly Censored GLM with Incomplete Information

The generalized linear model（GLM） based on the observed data with incomplete information in the case of random censorship is defined. Under the given conditions, the existence and uniqueness of the solution on the likelihood equations with respect to the parameter vector β of the model are discussed, and the consistency and asymptotic normality of the maximum likelihood estimator（MLE） βn^-, are proved.

Fuzzy multi-criteria decision-making approach with incomplete information based on evidential reasoning

The weights of criteria are incompletely known and the criteria values are incomplete and uncertain or even default in some fuzzy multi-criteria decision-making problems.For those problems,an approach based on evidential reasoning is proposed,in which the criteria values are integrated on the basis of analytical algorithm of evidential reasoning,and then nonlinear programming models of each alternative are developed with the incomplete information on weights.The genetic algorithm is employed to solve the models,producing the weights and the utility interval of each alternative,and the ranking of the whole set of alternatives can be attained.Finally,an example shows the effectiveness of the method.

Threat sequencing of multiple UCAVs with incomplete information based on game theory

The threat sequencing of multiple unmanned combat air vehicles(UCAVs) is a multi-attribute decision-making(MADM)problem. In the threat sequencing process of multiple UCAVs,due to the strong confrontation and high dynamics of the air combat environment, the weight coefficients of the threat indicators are usually time-varying. Moreover, the air combat data is difficult to be obtained accurately. In this study, a threat sequencing method of multiple UCAVs is proposed based on game theory by considering the incomplete information. Firstly, a zero-sum game model of decision maker( D) and nature(N)with fuzzy payoffs is established to obtain the uncertain parameters which are the weight coefficient parameters of the threat indicators and the interval parameters of the threat matrix. Then,the established zero-sum game with fuzzy payoffs is transformed into a zero-sum game with crisp payoffs(matrix game) to solve. Moreover, a decision rule is addressed for the threat sequencing problem of multiple UCAVs based on the obtained uncertain parameters. Finally, numerical simulation results are presented to show the effectiveness of the proposed approach.

Strategy Selection for Moving Target Defense in Incomplete Information Game

As a core component of the network,web applications have become one of the preferred targets for attackers because the static configuration of web applications simplifies the exploitation of vulnerabilities by attackers.Although the moving target defense(MTD)has been proposed to increase the attack difficulty for the attackers,there is no solo approach can cope with different attacks;in addition,it is impossible to implement all these approaches simultaneously due to the resource limitation.Thus,the selection of an optimal defense strategy based on MTD has become the focus of research.In general,the confrontation of two players in the security domain is viewed as a stochastic game,and the reward matrices are known to both players.However,in a real security confrontation,this scenario represents an incomplete information game.Each player can only observe the actions performed by the opponent,and the observed actions are not completely accurate.To accurately describe the attacker’s reward function to reach the Nash equilibrium,this work simulated and updated the strategy selection distribution of the attacker by observing and investigating the strategy selection history of the attacker.Next,the possible rewards of the attacker in each confrontation via the observation matrix were corrected.On this basis,the Nash-Q learning algorithm with reward quantification was proposed to select the optimal strategy.Moreover,the performances of the Minimax-Q learning algorithm and Naive-Q learning algorithm were compared and analyzed in the MTD environment.Finally,the experimental results showed that the strategy selection algorithm can enable defenders to select a more reasonable defensive strategy and achieve the maximum possible reward.

Multi-criteria classification approach with polynomial aggregation function and incomplete certain information

The relationship between the importance of criterion and the criterion aggregation function is discussed, criterion＇s weight and combinational weights between some criteria are defined, and a multi-criteria classification method with incomplete certain information and polynomial aggregation function is proposed. First, linear programming is constructed by classification to reference alternative set （assignment examples） and incomplete certain information on criterion＇s weights. Then the coefficient of the polynomial aggregation function and thresholds of categories are gained by solving the linear programming. And the consistency index of alternatives is obtained, the classification of the alternatives is achieved. The certain criteria＇s values of categories and uncertain criteria＇s values of categories are discussed in the method. Finally, an example shows the feasibility and availability of this method.

Fusion of multiagent preference orderings with information on agent's importance being incomplete certain

The problem of fusing multiagent preference orderings, with information on agent＇s importance being incomplete certain with respect to a set of possible courses of action, is described. The approach is developed for dealing with the fusion problem described in the following sections and requires that each agent provides a preference ordering over the different alternatives completely independent of the other agents, and the information on agent＇s importance is incomplete certain. In this approach, the ternary comparison matrix of the alternatives is constructed, the eigenvector associated with the maximum eigenvalue of the ternary comparison matrix is attained so as to normalize priority vector of the alternatives. The interval number of the alternatives is then obtained by solving two sorts of linear programming problems. By comparing the interval numbers of the alternatives, the ranking of alternatives can be generated. Finally, some examples are given to show the feasibility and effectiveness of the method.

Information Measures of Roughness of Knowledge and Rough Sets for Incomplete Information Systems

In this paper we address information measures of roughness of knowledge and rough sets for incomplete information systems. The definition of rough entropy of knowledge and its important properties are given. In particular, the relationship between rough entropy of knowledge and the Hartley measure of uncertainty is established. We show that rough entropy of know1edge decreases monotonously as granularity of information become smaller. This gives an information interpretation for roughness of knowledge. Based on rough entropy of knowledge and roughness of rough set. a definition of rough entropy of rough set is proposed, and we show that rough entropy of rough set decreases monotonousIy as granularity of information become smaller. This gives more accurate measure for roughness of rough set.

Generalized binary discernibility matrix for attribute reduction in incomplete information systems

To extract and express the knowledge hidden in information systems, discernibility matrix and its extensions were introduced and applied successfully in many real life applications. Binary discernibility matrix, as a representative approach, has many interesting superior properties and has been rapidly developed to find intuitive and easy to understand knowledge. However, at present, the binary discernibility matrix is mainly adopted in the complete information system. It is a challenging topic how to achieve the attribute reduction by using binary discernibility matrix in incomplete information system. A form of generalized binary discernibility matrix is further developed for a number of representative extended rough set models that deal with incomplete information systems. Some useful properties and criteria are introduced for judging the attribute core and attribute relative reduction. Thereafter, a new algorithm is formulated which supports attribute core and attribute relative reduction based on the generalized binary discernibility matrix. This algorithm is not only suitable for consistent information systems but also inconsistent information systems. The feasibility of the proposed methods was demonstrated by worked examples and experimental analysis.

Target-tracked prioritization to surveille ballistic missiles

The missile-tracked priority assessment for the early warning system monitoring multi-missile very well, is the first task to defend them and useful to perform optimally the sensor-to-missile assignment. The problem of the missile-tracked priority assessment is of incomplete information, of multi-attribute and dynamic. To solve the dificult problem, the index system, which includes six classification indices, is established by means of reducing the target＇s primary information which the early warning system focuses on. The lack of some attributes values, which is caused by the incomplete information, is handled by the approach： first classifying each attribute as unknown one or known one, and then subdividing the latter, last using the expectation and the information entropy if the attribute is known but uncertain. With a view of reality, nine qualitative evaluation criteria are given. Based on them each index is quantified by the eigenvector method. And then based on the improved technique for order preference by similarity to ideal solution （TOPSIS）, the targets-tracked priorities are assessed and sorted by an evaluation function from three aspects： 1） the quantity of the available information, 2） the affirmative or accurate degree of the available information, 3） the classification or trait of the available information.

Method for multi-attribute group decision-making based on the compromise weights

Multi-attribute group decision-making problems are considered where information on both attribute weights and value scores of consequences is incomplete.In group decision analysis,if preference information about alternatives is provided by participants,it should be verified whether there exist compromise weights that can support all the preference relations.The different compromise weight vectors may differ for the ranking of the alternatives.In the case that compromise weights exist,the method is proposed to find out all the compromise weight vectors in order to rank the alternatives.Based on the new feasible domain of attribute weights determined by all the compromise weight vectors and the incomplete information on value scores of consequences,dominance relations between alternatives are checked by a nonlinear goal programming model which can be transformed into a linear one by adopting a transformation.The checked dominance relations uniformly hold for all compromise weight vectors and the incomplete information on value scores of consequences.A final ranking of the alternatives can be obtained by aggregating these dominance relations.

Rough set extensions in incomplete information systems

All eight possible extended rough set models in incomplete information systems are proposed.By analyzing existing extended models and technical meth-ods of rough set theory,the strategy of model extension is found to be suitable for processing incomplete information systems instead of filling possible values for missing attributes.After analyzing the definitions of existing extended models,a new general extended model is proposed.The new model is a generalization of indiscernibility relations,tolerance relations and non-symmetric similarity relations.Finally,suggestions for further study of rough set theory in incomplete informa-tion systems are put forward.

Extended Rough Set Model Based on Prior Probability and Valued Tolerance Relation

Based on equivalence relation,the classical rough set theory is unable to deal with incomplete information systems.In this case,an extended rough set model based on valued tolerance relation and prior probability obtained from incomplete information systems is firstly founded.As a part of the model,the corresponding discernibility matrix and an attribute reduction of incomplete information system are then proposed.Finally,the extended rough set model and the proposed attribute reduction algorithm are verified under an incomplete information system.