Differential Game Theory Research Articles

Distributed optimal integrated tracking control for separate kinematic and dynamic uncertain non‐holonomic mobile mechanical multi‐agent systems

This study addres,ses a distributed optimal integrated tracking control method with disturbance rejection for separate kinematic and dynamic uncertain non-holonomic mobile mechanical multi-agent ( N M 3 ) systems. Initially, based on the graph theory, the overall tracking systems of agents are defined and the distributed optimal tracking problem of separate kinematics and dynamics is transformed into an equivalent distributed optimal regulation problem of the integrated affine system. Then, neural network (NN)-based adaptive dynamic programming and cooperative differential game theory is utilised for control, in which only one NN is required for each agent. The NN weight-tuning law and the online algorithm is developed to approximate the value function, and synchronously update both optimal control and worst disturbance laws in only one iterative loop. The tracking errors and function approximation errors are proven to be uniformly ultimately bounded using Lyapunov theory. Finally, as applications of the proposed method, control of the wheeled mobile multi-robot system is discussed. The effectiveness of the method is demonstrated by the results of the comparative numerical simulation.

A Comprehensive Differential Game Theoretic Solution to a Game of Two Cars

In this paper, a pursuit-evasion game involving two non-holonomic agents is examined using the theory of differential games. It is assumed that the two players move on the Euclidean plane with fixed but different speeds and they each have a lower bound on their achievable turn radii. Both players steer at each instant by choosing their turn radii value and directions of turn. By formulating the game as a game of kind, we characterize the regions of initial conditions that lead to capture as well as the regions that lead to evasion, when both the players play optimally. The game is then formulated as a game of degree to obtain time-optimal paths for the pursuer and evader inside a capture region. Besides, all possible scenarios are considered for both players that differ in speed ratios and maneuverability constraints. Solutions are provided for those cases using appropriate simulation parameters, which aid in understanding the characteristics of the game of two cars under a wide range of constraints.

The program iteration method in a game problem of guidance

A variant of the program iteration method for solving a game problem of guidance to a target set under state constraints is considered. We study a procedure for the construction of a positional absorption set corresponding to N. N.Krasovskii and A. I. Subbotin’s alternative theorem, which underlies the modern theory of differential games. Important results on the alternative solvability of differential games for systems with distributed parameters and aftereffect belong to Yu. S.Osipov. These results are an essential complement to the ideas related to the alternative for dynamic problems of infinite-dimensional nature. The solution method from the present paper is intended for the “finite-dimensional” case of a differential game of approach–evasion.

Tracking Control for Electromechanical Systems Using Robust Discrete Time H∞

In this work we propose a robust controller to do tracking using the sub-optimal H∞technique with the approach of differential game theory. The problem is solved in two steps using the Block Control technique. The controller is designed in discrete time and it is synthesized for electromechanical systems which are modeled by means of the Euler-Lagrange formulation. Making use of the discrete Hamilton-Jacobi-Isaacs equation and the discrete Riccati equation the control law is derived. The control law is then applied to a continuous-time 6-DOF bipedal robot model in order to track the walking pattern references for each link. The system along with the control law is simulated, where the system is subjected to a disturbance that emulates the action of a group of external unknown bounded forces over the links of the bipedal robot. The simulation results are shown displaying robustness against the disturbance, torques required from the motors are plot and since the control input was optimized the values lie within a reasonable bound. Furthermore, this work is compared to a similar approach that uses H∞ technique in continuous time.

Pursuit-evasion games: a tractable framework for antijamming games in aerial attacks

We solve a communication problem between a UAV and a set of receivers, in the presence of a jamming UAV, using differential game theory tools. We propose a new approach in which this kind of games can be approximated as pursuit-evasion games. The problem is posed in terms of optimizing capacity, and it is solved in two ways: firstly, a surrogate function approach is used to approximate it as a pursuit-evasion game; secondly, the game is solved without that approximation. In both cases, Isaacs equations are used to find the solution. Finally, both approaches are compared in terms of relative distance and complexity.

Altruistic behavior in a nonantagonistic positional differential game

We consider a two-person nonantagonistic positional differential game (NPDG) whose dynamics is described by an ordinary nonlinear vector differential equation. Constraints on values of players' controls are geometric. Final time of the game is fixed. Payoff functionals of both players are terminal. The formalization of positional strategies in an NPDG is based on the formalization and results of the general theory of antagonistic positional differential games (APDGs) (see monographs by N.N. Krasovskii and A.I. Subbotin [3, 4]). Additionally, in the present paper we assume that each player, together with the usual, normal (nor), type of behavior aimed at maximizing his own functional, can use other behavior types introduced in [2, 5]. In particular, these may be altruistic (alt), aggressive (agg), and paradoxical (par) types. It is assumed that in the course of the game players can switch their behavior from one type to another. Using the possibility of such switches in a repeated bimatrix 2 × 2 game in [5, 6] allowed to obtain new solutions of this game. In the present paper, extension of this approach to NPDGs leads to a new formulation of the problem. In particular, of interest is the question of how players' outcomes at Nash solutions are transformed. An urgent problem is minimizing the time of "abnormal" behavior while achieving a good result. The paper proposes a formalization of an NPDG with behavior types (NPDGwBT). It is assumed that in an NPDGwBT each player, simultaneously with choosing a positional strategy, chooses also his own indicator function defined on the whole game horizon and taking values in the set {normal, altruistic, aggressive, paradoxical}. The indicator function of a player shows the dynamics of changes in the behavior type demonstrated by the player. Thus, in this NPDGwBT each player controls the choice of a pair {positional strategy, indicator function}. We define the notion of a BT-solution of such a game. It is expected that using behavior types in the NPDGwBT which differ from the normal one (so-called abnormal types) in some cases may lead to more favorable outcomes for the players than in the NPDG. We consider two examples of an NPDGwBT with simple dynamics in the plane in each of which one player keeps to altruistic behavior type over some time period. It is shown that in the first example payoffs of both players increase on a BT-solution as compared to the game with the normal behavior type, and in the second example, the sum of players' payoffs is increased.

Distributed cooperative H∞ optimal tracking control of MIMO nonlinear multi-agent systems in strict-feedback form via adaptive dynamic programming

ABSTRACTThe design of distributed cooperative H∞ optimal controllers for multi-agent systems is a major challenge when the agents’ models are uncertain multi-input and multi-output nonlinear systems in strict-feedback form in the presence of external disturbances. In this paper, first, the distributed cooperative H∞ optimal tracking problem is transformed into controlling the cooperative tracking error dynamics in affine form. Second, control schemes and online algorithms are proposed via adaptive dynamic programming (ADP) and the theory of zero-sum differential graphical games. The schemes use only one neural network (NN) for each agent instead of three from ADP to reduce computational complexity as well as avoid choosing initial NN weights for stabilising controllers. It is shown that despite not using knowledge of cooperative internal dynamics, the proposed algorithms not only approximate values to Nash equilibrium but also guarantee all signals, such as the NN weight approximation errors and the cooperative tracking errors in the closed-loop system, to be uniformly ultimately bounded. Finally, the effectiveness of the proposed method is shown by simulation results of an application to wheeled mobile multi-robot systems.

Model-Based Reinforcement Learning in Differential Graphical Games

This paper seeks to combine differential game theory with the actor-critic-identifier architecture to determine forward-in-time, approximate optimal controllers for formation tracking in multi-agent systems, where the agents have uncertain heterogeneous nonlinear dynamics. A continuous control strategy is proposed, using communication feedback from extended neighbors on a communication topology that has a spanning tree. A model-based reinforcement learning technique is developed to cooperatively control a group of agents to track a trajectory in a desired formation. Simulation results are presented to demonstrate the performance of the developed technique.

Game Control of maritime objects

ABSTRACTThe paper presents the application of the theory of differential games and multi-stage static and multi-matrix static games for the automation of the process of controlling maritime objects – merchant ships, ships or unmanned vessels. It has been shown that through the appropriate formulation of the plotter quality control, consisting of an integrated and final pay out, it is possible to implement collision and collision games as well as non-operational and cooperative games. On the example of safe control of one's own ship in situations of passing with many ships encountered at sea, algorithms of defining a safe trajectory of a ship supporting the manoeuvring decision of the navigator were presented. Considerations are illustrated by examples of computer simulation algorithms in Matlab/Simulink software for safe ship trajectories in real-time maritime situations.

Supply chain decisions with reference quality effect under the O2O environment

The reference quality effect is an important factor that influences consumer purchasing behavior. To investigate how firms should incorporate the reference quality effect under different business models, we focus on a supply chain consisting of a supplier and a retailer where the retailer can be an offline store, a pure online store or a combination of offline and online stores within an offline to online model. We formulate the reference quality effect with a modified Nerlove-Arrow model and use two different sales functions to reflect the fact that the reference quality effect will affect consumers in various ways when the business model varies. Utilizing differential game theory, the equilibrium decisions of the channel members are derived, and the analysis illustrates how the firms should adjust their decisions when the retailers use different retail patterns. The basic model does not allow product returns, but then this assumption is relaxed. Comparison between the two cases shows under what conditions consumers will benefit from the retailer’s allowing product returns.

Three Important Applications of Mathematics in Financial Mathematics

This paper analyzes the basic connotation of financial mathematics, financial mathematics through research development, control theory, differential game theory and capital asset pricing model from stochastic optimal, and discusses three important applications of mathematics in the financial field.

Game Approach for H∞ Robust Control Strategy to Follow the Production in the Singularly Perturbed Bilinear Dynamic Input-Output Systems

For simulating and analyzing the input and output problem of national economy more accurately, this paper considers the fast and slow production processes during the course of social production development, takes stochastic economic risks into consideration, and constructs a H∞ robust control model to follow the production in the singularly perturbed dynamic input-output systems. Further introducing ideas of noncooperative differential game theory, the H∞ robust control model is transformed into a saddle-point equilibrium game model, and a new method for solving dynamic input-output problem by using saddle-point equilibrium strategies is obtained. A numerical result is presented in the end to illustrate the effectiveness of the method.

Application of Differential Games in Mechatronic Control System

Differential games are a combination of game theory and optimum control methods. Their solutions are based on Bellman's principle of optimality. In this paper, the zero-sum differential game theory has been used for the purposes of controlling a mechatronic object: a single-link manipulator. In this case, analytical solutions are unavailable, thus approximate solutions were used. Two approximation methods were compared with the use of numerical simulations and selected quality indicators. The results confirm previous assumptions and the connection between the differential game theory and <i>H</i><sub>∞</sub> control problems.

A Global Optimal Path Planning and Controller Design Algorithm for Intelligent Vehicles

Autonomous vehicle guidance and trajectory planning is one of the key technologies in the autonomous control system for intelligent vehicles. Firstly, the target pursuit model for intelligent vehicles was established and described in this text. Then, the research work for global motion planning was carried out based on Stackelberg Differential Game Theory, and the global optimal solution was obtained by using the survival type differential game. Finally, to overcome errors, we use a polynomial method to achieve the smooth motion planning. So, based on Terminal Sliding Mode method, the Active Front Steering controller design was used to calculate the desired active wheel angle for intelligent vehicle path tracking. The simulation and experiment results demonstrate the feasibility and effectiveness of this method for intelligent vehicles' path planning and tracking.

Foraging motion of swarms with leaders as Nash equilibria

The consequences of having a leader in a swarm are investigated using differential game theory. We model foraging swarms with leader and followers as a non-cooperative, multi-agent differential game. The agents in the game start from a set of initial positions and migrate towards a target. The agents are assumed to have no desire, partial desire or full desire to reach the target. We consider two types of leadership structures, namely hierarchical leadership and a single leader. In both games, the type of leadership is assumed to be passive. We identify the realistic assumptions under which a unique Nash equilibrium exists in each game and derive the properties of the Nash solutions in detail. It is shown that having a passive leader economizes in the total information exchange at the expense of aggregation stability in a swarm. It turns out that, the leader is able to organize the non-identical followers into harmony under missing information.

Programmed iteration method in packages of spaces

An abstract version of the programmed iteration method (used in the theory of differential games) is considered that is intended for the problem of observing phase constraints on a nonempty subset of the real line. It is assumed that trajectories and uncontrolled disturbances are realized in packages of mutually inconsistent spaces. This consistency is postulated to investigate solvability conditions for the problem in the class of quasi-strategies. As a result, the set of successful solvability is constructed and the structure of resolving strategies is determined.

Supply chain coordination in quality improvement with reference effects

With increasing public attention on product quality, supply chain quality management has been the focus of considerable research attention in recent years. However, only a few published articles have considered reference effects in supply chain quality management. In practice, reference effects influence consumer purchasing behaviours. Therefore, we have developed a dynamic model that incorporates the effects of the reference quality and reference price on the demand function of a ‘single-supplier single-manufacturer’ supply chain system. In our model, the demand function is assumed to be influenced by the reference price and reference quality. The reference price formation is assumed to be affected by the gap between quality and the reference quality. By utilizing differential game theory, we calculate and compare the open-loop equilibrium quality levels in two different scenarios: (i) Each channel member makes decisions within a subsidy programme and (ii) the supplier vertically integrates with the manufacturer. In addition, a total cost-sharing contract based on quality improvement efforts is proposed and proven to be effective in the supply chain system that is under consideration.

Stochastic cooperative advertising in a manufacturer\u2013retailer decentralized supply channel

This work considers cooperative advertising in a manufacturer–retailer supply chain. While the manufacturer is the Stackelberg leader, the retailer is the follower. Using Sethi model it models the dynamic effect of the manufacturer and retailer’s advertising efforts on sale. It uses optimal control technique and stochastic differential game theory to obtain the players’ advertising strategies and the long-run value of the awareness share. Further, it models the relationship between the payoffs of both players and the awareness share. The work shows that with the provision of subsidy the retail advertising effort increases while the manufacturer’s advertising effort reduces. It further shows that the total channel payoff is higher for subsidised retail advertising. However, the subsidy can only be possible if the rate of growth of the manufacturer’s payoff is twice higher than that of the retailer.

Crowd-Averse Cyber-Physical Systems: The Paradigm of Robust Mean-Field Games

For a networked controlled system, we illustrate the paradigm of robust mean-field games. This is a modeling framework at the interface of differential game theory, mathematical physics, and $H_{\infty}$ - optimal control that tries to capture the mutual influence between a crowd and its individuals. First, we establish a mean-field system for such games including the effects of adversarial disturbances. Second, we identify the optimal response of the individuals for a given population behavior. Third, we provide an analysis of equilibria and their stability.

Nonlinear elliptic systems and mean-field games

We consider a class of quasilinear elliptic systems of PDEs consisting of N Hamilton–Jacobi–Bellman equations coupled with N divergence form equations, generalising to N > 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. We show the connection of these systems with the classical strongly coupled systems of Hamilton–Jacobi–Bellman equations of the theory of N-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some N-person games.