Open-loop Information Structure Research Articles

Real-time coordinated control strategy for the hybrid electric propulsion system of a flying car with model adaptation and game theory

Coordination among the multiple power components is essential for the hybrid electric propulsion systems since the decision made by one impacts the state and decisions of others due to the coupling mechanical and electrical dynamics. Traditional model-based approaches with weight-sum objectives may result in misleading optimization or even unstable situations because of the inherent poor scalability and synergy. To overcome this issue, this paper proposes a novel game theory-based control strategy with model adaptation mechanism for the hybrid electric flying car to enhance the control performance, system stability, and robustness. Firstly, a control-oriented model of the system is derived with the utilization of the recursive least square parameter estimation method. The non-cooperative game framework is then established with the turboshaft engine subsystem and electric supply subsystem treated as two independent players. The performance of Nash equilibrium solutions with closed-loop and open-loop information structures are investigated where the problem is iteratively solved by exploiting Pontryagin’s Minimum Principle and dynamic programming, respectively. The simulation results demonstrate that the game theory-based controllers can outperform MPC with the weight-sum objectives in terms of control efficiency and robustness improvement and the game theory-based controller with open-loop information structure shows excellent computation efficiency. Moreover, the result of the hardware-in-the-loop experiment demonstrates the real-time performance of the proposed controller.

A game approach to multi-dimensional opinion dynamics in social networks with stubborn strategist agents

In a social network, individuals express their opinions on several interdependent topics, and therefore the evolution of their opinions on these topics is also mutually dependent. In this work, we propose a differential game model for the multi-dimensional opinion formation of a social network whose population of agents interacts according to a communication graph. Each individual’s opinion evolves according to an aggregation of disagreements between the agent’s opinions and its graph neighbors on multiple interdependent topics exposed to an unknown extraneous disturbance. For a social network with strategist agents, the opinions evolve over time with respect to the minimization of a quadratic cost function that solely represents each individual’s motives against the disturbance. We find the unique Nash/worst-case equilibrium solution for the proposed differential game model of coupled multi-dimensional opinions under an open-loop information structure. Moreover, we propose a distributed implementation of the Nash/worst-case equilibrium solution. We examine the non-distributed and proposed distributed open-loop Nash/worst-case strategies on a small social network with strategist agents in a two-dimensional opinion space. Then we compare the evolved opinions based on the Nash/worst-case strategy with the opinions corresponding to social optimality actions for non-strategist agents.

Linear quadratic open-loop Stackelberg game for stochastic systems with Poisson jumps

This paper is concerned with the finite horizon linear quadratic (LQ) Stackelberg game for stochastic systems with Poisson jumps under the open-loop information structure. First, the follower solves a LQ stochastic optimal control problem with Poisson jumps. With the aid of an introduced generalized differential Riccati equation with Poisson jumps (GDREP), the sufficient conditions for the optimization of the follower are put forward. Then, the leader faces an optimal control problem for a forward-backward stochastic differential equation with Poisson jumps (FBSDEP). By introducing new state and costate variables, a sufficient condition for the existence and uniqueness of the open-loop Stackelberg strategies is presented in terms of the solvability of two differential Riccati equations and a convexity condition. In addition, the state feedback representation of the open-loop Stackelberg strategies is obtained via the related differential Riccati equation. Finally, two examples shed light on the effectiveness of the obtained results.

Stackelberg solution for a two-agent rational expectations model

In this paper, we derive the Stackelberg solution associated with a leader–follower rational expectations model. By solving the dynamic rational expectations optimization problem subject to forward and backward stochastic difference equations (FBSDEs), we obtain the Stackelberg strategy of the leader with an adapted open-loop information structure, in terms of the FBSDEs. Furthermore, we solve the associated FBSDEs by establishing a nonhomogeneous relationship between backward and forward stochastic processes. This leads to an explicit form of the Stackelberg strategy of the leader.

A Differential Game Model of Opinion Dynamics: Accord and Discord as Nash Equilibria

This paper presents a noncooperative differential (dynamic) game model of opinion dynamics with open-loop information structure. In this game, the agents’ motives are shaped by their expectations of the nature of others’ opinions as well as how susceptible they are to get influenced by others, how stubborn they are, and how quick they are willing to change their opinions on a set of issues in a prescribed time interval. These motives are independently formed by all agents. The existence of a Nash equilibrium in the network means that a collective behavior emerges out of local interaction rules and these individual motives. We prove that a unique Nash equilibrium may exist in the game under quite different circumstances. It may exist not only if there is a harmony of perceptions among the agents of the network, but also when agents have different views about the correlation among issues. The first leads to an accord in the network usually expressed as a partial consensus, and the second to a discord in the form of oscillating opinions. In the case of an accord, the harmony in the network may be in the form of similarity in pairwise conceptions about the issues but may also be an agreement on the status of a “leader” in the network. A Nash equilibrium may fail to exist only if the network is in a state of discord.

Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

In this article, we consider the linear-quadratic time-inconsistent mean-field type leader-follower Stackelberg differential game with an adapted open-loop information structure. The objective functionals of the leader and the follower include conditional expectations of state and control (mean field) variables, and the cost parameters could be general nonexponential discounting depending on the initial time. As stated in the existing literature, these two general settings of the objective functionals induce time inconsistency in the optimal solutions. Given an arbitrary control of the leader, we first obtain the follower's (time consistent) equilibrium control and its state feedback representation in terms of the nonsymmetric coupled Riccati differential equations (RDEs) and the backward stochastic differential equation (SDE). This provides the rational behavior of the follower, characterized by the forward-backward SDE (FBSDE). We then obtain the leader's explicit (time consistent) equilibrium control and its state feedback representation in terms of the nonsymmetric coupled RDEs under the FBSDE constraint induced by the follower. With the solvability of the nonsymmetric coupled RDEs, the equilibrium controls of the leader and the follower constitute the time-consistent Stackelberg equilibrium. Finally, the numerical examples are provided to check the solvability of the nonsymmetric coupled RDEs.

Linear quadratic mean field Stackelberg differential games

We consider linear–quadratic mean field Stackelberg differential games with the adapted open-loop information structure of the leader. There are one leader and N followers, where N is arbitrarily large. The leader holds a dominating position in the game in the sense that the leader first chooses and then announces the optimal strategy to which the N followers respond by playing a Nash game, i.e., choosing their optimal strategies noncooperatively and simultaneously based on the leader’s observed strategy. In our setting, the followers are coupled with each other through the mean field term included in their cost functions, and are strongly influenced by the leader’s open-loop strategy included in their cost functions and dynamics. From the leader’s perspective, he is coupled with the N followers through the mean field term included in his cost function. To circumvent the complexity brought about by the coupling nature among the leader and the followers with large N, which makes the use of the direct approach almost impossible, our approach in this paper is to characterize an approximated stochastic mean field process by solving a local optimal control problem of the followers with leader’s control taken as an exogenous stochastic process. We show that for each fixed strategy of the leader, the followers’ local optimal decentralized strategies lead to an ϵ-Nash equilibrium. The paper then solves the leader’s local optimal control problem, as a nonstandard constrained optimization problem, with constraints being induced by the approximated mean field process determined by Nash followers (which also depend on the leader’s control). We show that the local optimal decentralized controllers for the leader and the followers constitute an (ϵ1,ϵ2)-Stackelberg–Nash equilibrium for the original game, where ϵ1 and ϵ2 both converge to zero as N→∞. Numerical examples are provided to illustrate the theoretical results.

Titik Setimbang Nash pada Permainan Linear Kuadratik Non-kooperatif dengan Asumsi Keseluruhan Pemain dapat Menstabilkan Sistem

We discuss about Nash equilibria for the linear quadratic differential game for an infinite planning horizon. We consider an open-loop information structure. In the standard literature this problem is solved under the assumption and provide both necessary and sufficient conditions for existence of Nash equilibria for this game under the assumption that the system as a whole is stabilizable. Keywords: linear quadratic differential games, open-loop information structure

Investment decisions in imperfect power markets with hydrogen storage and large share of intermittent electricity

This paper analyzes the impact of hydrogen as energy storage on production and investment decisions in an electricity market when individual participants behave strategically. We develop a game-theoretic model on investment and generation game à la Cournot under the open-loop information structure. This framework is implemented as a mixed complementarity problem and applied to the German case assuming the phase-out of the German nuclear power plants, rising renewable energy supply and increasing energy demand for electric vehicles. The numerical results of our analysis indicate that utilization of energy storage has a positive effect on energy systems with large amount of intermittent electricity and inelastic demand. We find that additional hydrogen storage capacities improve system reliability, increase overall welfare and decrease GHG emissions. Adding demand for hydrogen as a fuel for FCEVs allows for a synergetic use of the technology and changes the investment incentives for energy storage. Although the power-to-gas technology has a price-smoothing effect the overall generation capacity is higher with energy storage providing additional supply security in markets with a large amount of intermittent energy production.

Dynamical Methods Applied in Natural Resource Economics

Abstract. This paper presents, in brief, the fundamentals of optimal control theory together with some notes for differential games, which is the game theoretic analogue of the optimal control. As it is recommended by literature references the main tool of analysis in open loop information structure for environmental models is the Pontryagin’s Maximum Principle, while the Hamilton–Jacobi–Bellman equation is the tool of analysis for any closed loop informational structure. As applications of the above theoretic considerations we present some environmental economic models which are solved both as optimal control problems and as differential games as well. Keywords. Optimal control; Differential games; Renewable resources; Environnemental and Resource Economies. JEL. C61; C62; D43; Q0; Q2; Q20; Q50; Q52; Q53.

Open-Loop Nash Equilibria in a Class of Linear-Quadratic Difference Games With Constraints

We study a class of $N$ -player finite-horizon linear-quadratic difference games with linear constraints. We introduce constrained open-loop information structure and derive necessary conditions for the existence of constrained open-loop Nash equilibria. We show that these conditions lead to a weakly coupled system of parametric two-point boundary value problem and a set of linear complementarity problems. By restricting the costate variables to be affine in the state variable, we show that these necessary conditions can be reformulated as a single large-scale linear complementarity problem. Then we provide sufficient conditions under which a solution of the linear complementarity problem constitutes a constrained open-loop Nash equilibrium.

Optimal firm growth under the threat of entry

The paper studies the incumbent-entrant problem in a fully dynamic setting. We find that under an open-loop information structure the incumbent anticipates entry by overinvesting, whereas in the Markov perfect equilibrium the incumbent slightly underinvests in the period before the entry. The entry cost level where entry accommodation passes into entry deterrence is lower in the Markov perfect equilibrium. Further we find that the incumbent’s capital stock level needed to deter entry is hump shaped as a function of the entry time, whereas the corresponding entry cost, where the entrant is indifferent between entry and non-entry, is U-shaped.

Economic Growth and Choice of Energy: A Simplistic Strategic Approach

In this paper, we present a simplistic dynamic model to analyze the tradeoff between the use of environment-friendly energy and the use of environment-polluting energy in a dynamic world characterized by players that have different interests. Our approach builds upon a neoclassical growth model, integrating different economies, and including pollution as a separate growth factor. The model is cast in a differential game framework with the open-loop information structure. In this setting, we analyze the impact of pollution over time on the fossil fuel/“green” energy ratio. Furthermore, based on a simulation study for the linearized model, we consider the question whether certain cooperative arrangements between countries having different interests can be expected in case pollution shocks occur.

Necessary and Sufficient Conditions for Pareto Optimality in Infinite Horizon Cooperative Differential Games

In this technical note, we derive necessary and sufficient conditions for the existence of Pareto solutions in infinite horizon cooperative differential games with open loop information structure. First, we reformulate the notion of Pareto optimality which entails to solving several constrained optimal control problems with a specific constraint structure. Then, we provide weak sufficient conditions under which all Pareto candidates can be obtained by solving a weighted sum optimal control problem, with weightvectors belonging to the unit simplex. We supplement our results with relevant examples.

Open-loop Nash equilibria in the non-cooperative infinite-planning horizon LQ game

In this paper we reconsider Nash equilibria for the affine linear quadratic differential game for an infinite planning horizon. We consider an open-loop information structure. In the standard literature this problem is solved under the assumption that every player can stabilize the system on his own. In this note we relax this assumption and provide both necessary and sufficient conditions for the existence of Nash equilibria for this game under the assumption that the system as a whole is stabilizable. Basically, it is shown that to see whether equilibria exist for all initial states one has to restrict the analysis to the controllable modes of the different players.

Differential Game-Theoretic Analysis on Information Availability in Decentralized Demand-Side Energy Management Systems

SUMMARY Differential games are considered an extension of optimal control problems, which are used to formulate centralized control problems in smart grids. Optimal control theory is used to study systems consisting of one agent with one objective, whereas differential games are used to formulate systems consisting of multiple agents with multiple objectives. Therefore, a differential-game-theoretic approach is appropriate for formulating decentralized demand-side energy management systems where there are multiple decision-making entities interacting with each other. Moreover, in many smart grid applications, we need to obtain information for control via communication systems. To formulate the influence of communication availability, differential game theory is also promising because the availability of communication is considered as part of an information structure (i.e., feedback or open-loop) in differential games. The feedback information structure is adopted when information for control can be obtained, whereas the open-loop information structure is applied when the information cannot be obtained because of communication failure. This paper proposes a comprehensive framework for evaluating the performance of demand-side actors in a demand-side management system using each control scheme according to both communication availability and sampling frequency. Numerical analysis shows that the proposed comprehensive framework allows for an analysis of trade-off for decentralized and centralized control schemes.

Debt stabilization games in the presence of risk premia

As a result of the recent financial crisis and the ensuing economic recession, fiscal deficits have soared in many OECD countries. As a consequence, government debt has been on the rise again after a period of stable or declining government debt. In this paper we analyze debt stabilization in a country that features endogenous risk premia, imposed by financial markets that evaluate the probability of debt default by governments. Endogenous risk premia arise by assuming, e.g., simple linear relations between risk premia and the level of debt. As a result the real interest rate on government debt can be written as a constant (measuring the risk-free real interest rate corrected for real output growth) plus an endogenous risk premium that depends on the debt level. We bring such an endogenous risk premium into Tabellini (1986) model and analyze the impact of it. This gives rise to a non-linear differential game. We solve this game for both a cooperative setting and a non-cooperative setting. The non-cooperative game is solved under an open-loop information structure. We present a bifurcation analysis w.r.t. the risk premium parameter.

A gametheoretic approach for non-uniform pole shifting and pole homothety

This paper investigates the problems of non-uniform pole shifting or pole homothety in multivariable linear systems by using a gametheoretic approach and more specifically a particular Nash strategy with an open-loop information structure for a game including a time preference rate in the quadratic criteria. Such a result is possible due to properties of non-symmetric Algebraic Riccati equations associated with an open-loop Nash strategy. Numerical examples allow to illustrate the efficiency of the proposed approach.

Open-Loop Nash Equilibria in the Non-cooperative Infinite-planning Horizon LQ Game

In this note we reconsider Nash equilibria for the linear quadratic differential game for an infinite planning horizon. We consider an open-loop information structure. In the standard literature this problem is solved under the assumption that every player can stabilize the system on his own. In this note we relax this assumption and provide both necessary and sufficient conditions for existence of Nash equilibria for this game under the assumption that the system as a whole is stabilizable.

Continuous-Time Non-Symmetric Algebraic Riccati Theory: A Matrix Pencil Approach

A continuous-time non-symmetric algebraic Riccati system which incorporates as a particular case the non-symmetric algebraic Riccati equation is studied under assumptions on the matrix coefficients relaxed as far as possible. Necessary and sufficient existence conditions together with computable formulas for the stabilizing solution are given in terms of proper deflating subspaces of an associated matrix pencil. A numerically-sound algorithm able to decide existence and to compute the stabilizing solutions, if any, to the algebraic Riccati system is recalled. The whole development may be applied to fluid queues, transport theory, and game theory with an open-loop information structure without assuming the classical invertibility hypothesis on the quadratic matrix coefficient.