Closed-loop Equilibrium Research Articles

Some Remarks on Linear-Quadratic Closed-Loop Games with Many Players

AbstractWe identify structural assumptions which provide solvability of the Nash system arising from a linear-quadratic closed-loop game, with stable properties with respect to the number of players. In a setting of interactions governed by a sparse graph, both short-time and long-time existence of a classical solution for the Nash system set in infinitely many dimensions are addressed, as well as convergence to the solution to the respective ergodic problem as the time horizon goes to infinity; in addition, equilibria for the infinite-dimensional game are shown to provide $$\epsilon $$ ϵ -Nash closed-loop equilibria for the N-player game. In a setting of generalized mean-field type (where the number of interactions is large but not necessarily symmetric), directly from the N-player Nash system estimates on the value functions are deduced on an arbitrary large time horizon, which should pave the way for a convergence result as N goes to infinity.

Passivity-based fractionated payload 6-DOF maneuver control using electromagnetic actuation

Employing fractionated spacecraft architecture with electromagnetic actuation into space missions can enhance resilience and flexibility since all subsystems of conventional monolithic spacecraft are separated as several modules to be easily replaced on-orbit when failure happens. Previous electromagnetic configuration used in fractionated spacecraft would render unexpected forces and torques which may undermine the stability of entire system. This paper considers the maneuver of a separated payload in a fractionated spacecraft architecture (FSA) using electromagnetic actuation with adaptive passivity-based control. The FSA consists of a fractionated payload module (FPM) and a supporting spacecraft module (SSM). Relative motion dynamics of FPM and SSM are formulated on SE(3). An adaptive passivity-based 6-DOF motion control scheme is established for FPM's translation and rotation tracking control simultaneously. The closed-loop equilibrium can be attained by the proposed passivity-based 6-DOF control scheme. The controller has no requirement for the precise inertia knowledge of FPM, which processes the robustness of the modelling errors. Numerical simulations are presented to show the effectiveness of the proposed control scheme.

Global Stabilization of Antipodal Points on $n$-Sphere With Application to Attitude Tracking

Existing approaches to robust global asymptotic stabilization of a pair of antipodal points on unit $n$-sphere $\mathbb{S}^n$ typically involve the non-centrally synergistic hybrid controllers for attitude tracking on unit quaternion space. However, when switching faults occur due to parameter errors, the non-centrally synergistic property can lead to the unwinding problem or in some cases, destabilize the desired set. In this work, a hybrid controller is first proposed based on a novel centrally synergistic family of potential functions on $\mathbb{S}^n$, which is generated from a basic potential function through angular warping. The synergistic parameter can be explicitly expressed if the warping angle has a positive lower bound at the undesired critical points of the family. Next, the proposed approach induces a new quaternion-based controller for global attitude tracking. It has three advantageous features over existing synergistic designs: 1) it is consistent, i.e., free from the ambiguity of unit quaternion representation; 2) it is switching-fault-tolerant, i.e., the desired closed-loop equilibria remain asymptotically stable even when the switching mechanism does not work; 3) it relaxes the assumption on the parameter of the basic potential function in literature. Comprehensive simulation confirms the high robustness of the proposed centrally synergistic approach compared with existing non-centrally synergistic approaches.

Receding Horizon Games for Modeling Competitive Supply Chains⋆

The vast majority of products we use daily are supplied to us through complex global supply chains that transform raw materials into finished goods and distribute them to end consumers. This paper proposes a modeling methodology for dynamic competitive supply chains based on game theory and model predictive control. We model each manufacturer in the supply chain as a rational utility maximizing agent that selects their actions by finding an open-loop generalized Nash equilibrium of a multi-stage game. To react to competitors and the state of the market, every agent re-plans their actions in a receding horizon manner based on estimates of market and supplier parameters thereby creating an approximate closed-loop equilibrium policy. We demonstrate through numerical simulations that this modeling approach is computationally tractable and generates economically interpretable behaviors in a variety of settings such as demand spikes, supply shocks, and information asymmetry.

Time-inconsistent linear quadratic optimal control problem for forward–backward stochastic differential equations

We study the time-inconsistent linear quadratic optimal control problem for forward–backward stochastic differential equations with potentially indefinite cost weighting matrices for both the state and the control variables. Our research makes two contributions. Firstly, we introduce a novel type of Riccati equation system with parameters and constraint conditions, known as the generalized equilibrium Riccati equation. This equation system offers a comprehensive solution for the closedloop equilibrium strategy of the problem at hand. Secondly, we establish the well-posedness of the generalized equilibrium Riccati equation for the one-dimensional case, provided certain conditions are met.

Global analysis of energy-based swing-up control for soft robot

In this paper, we explore the energy-based swing-up control for a soft robot equipped with an actuated constant curvature soft pendulum and an unactuated rotational base joint. The aim is to swing the robot up towards its upright equilibrium point (UEP) with the pendulum at an upright position. We establish a necessary and sufficient condition to protect the control law from singularities and demonstrate the robot’s potential motion towards a homoclinic orbit or a closed-loop equilibrium point. After examining the robot’s closed-loop equilibrium points, we generate formulae to compute all such points and introduce two conditions concerning control parameters, eliminating all but the UEP and downward equilibrium point (DEP) with the robot in a downward position. We prove the instability of the equilibrium points with negative gravitational potential energy. This obviates the need for one of the two conditions concerning such equilibrium points. We also prove the robot’s linear controllability at the UEP. Our obtained results reveal that, regardless of its initial state, the soft robot can be swung-up towards its UEP using the energy-based controller, provided that it meets the proposed control parameter conditions, and can subsequently be balanced around the UEP using a locally stabilizing controller. Our simulation investigations validate the presented theoretical results.

Distributed event-triggered fuzzy control for nonlinear interconnected systems

This paper deals with the asynchronous distributed event-triggered control of continuous-time nonlinear interconnected systems. The nonlinear dynamics of the subsystems are represented by Takagi–Sugeno fuzzy models via the sector nonlinearity approach. Moreover, the nonlinear interconnections among the subsystems are assumed to be known and sector-bounded. In the event-triggered control setup, where the state information is asynchronously available to the local controllers only in specific time instants, it is necessary to deal with the asynchronous premise variables because they introduce extra difficulties to derive suitable co-design conditions. To deal with this issue, we propose a new triggering strategy such that the local event-triggering mechanisms (ETMs) counteract the effects of asynchronous premise variables. With this new distributed ETM, a co-design condition is proposed, and the existence of a strictly positive minimum inter-event time is proved to exclude Zeno behaviour. Moreover, a multi-objective optimization procedure is introduced to enlarge the estimate of the domain of attraction of the closed-loop equilibrium and minimize the number of transmissions provided by the ETMs. Finally, the approach is validated through the synchronization of interconnected oscillators.

Risk Aversion and Hydro-energy Reservoirs Management under Monopolistic and Oligopolistic Industrial Structures

This article analyses management of hydropower dams within monopolistic and oligopolistic competition and when hydroelectricity producers are risk averse and face demand uncertainty. In each type of market structure we analytically determine the water release path at the closed-loop equilibrium. In the monopolistic case, we demonstrate that intertemporal allocation of hydro resources is done by additional pumping or storage depending on the relative abundance of water between different regions in order both to internalize the risk and to smooth the effect of uncertainty on electricity prices. In the oligopolistic case, we determine the conditions under which the relative scarcity (abundance) of water in the dam of a hydroelectric operator can favor additional strategic pumping (storage) in its competitor’s dams. Finally, we compare the effects of risk aversion on the output industry. We show, in monopolistic case, that the greater the generators’ risk aversion the larger the current water storage for future demands. In the oligopolistic case, however, the risk aversion increasing may be either in favor of reducing or increasing the total hydro output depending on demand uncertainty and market power parameters. We deduce that risk aversion coupled with market power may induce, under some condition, a misallocation of hydraulic resource across time.

Equilibria, Stability, and Sensitivity for the Aerial Suspended Beam Robotic System Subject to Parameter Uncertainty

This article studies how parametric uncertainties affect the cooperative manipulation of a cable-suspended beam-shaped load by means of two aerial robots not explicitly communicating with each other. In particular, this article sheds light on the impact of the uncertain knowledge of the model parameters available to an established communicationless force-based controller. First, we find the closed-loop equilibrium configurations in the presence of the aforementioned uncertainties, and then, we study their stability. Hence, we show the fundamental role played in the robustness of the load attitude control by the internal force induced in the manipulated object by nonvertical cables. Furthermore, we formally study the sensitivity of the attitude error to such parametric variations, and we provide a method to act on the load position error in the presence of uncertainties. Eventually, we validate the results through an extensive set of numerical tests in a realistic simulation environment, including underactuated aerial vehicles and sagging-prone cables, and through hardware experiments.

A switching approach to event-triggered control systems under denial-of-service attacks

This paper addresses the resilient event-triggered control (ETC) of networked control systems under denial-of-service (DoS) attacks. The scenario in which the presence of DoS attacks limits the data rate is studied considering a deterministic DoS attack model that constrains the frequency and duration of the DoS attack sequence in an average sense. Based on a switching approach, constructive sufficient conditions are proposed to design the resilient ETC scheme in both emulation and co-design approaches. The conditions establish a limitation to the average DoS frequency and duration such that the exponential stability of the closed-loop equilibrium and the Zeno-freeness is guaranteed even in the presence of DoS attacks. Moreover, optimization problems subject to LMI constraints are formulated to solve the emulation and co-design problems by enlarging the resilience to the considered class of DoS attacks while the number of transmissions is reduced. This establishes a clear trade-off between resilience and resource awareness in networked control systems under DoS attacks. The effectiveness of the proposed conditions is illustrated in two numerical examples.

Mean–variance portfolio selection with dynamic attention behavior in a hidden Markov model

In this paper, we study closed-loop equilibrium strategies for mean–variance portfolio selection problems in a hidden Markov model with dynamic attention behavior. In addition to the investment strategy, the investor’s attention to news is introduced as a control of the accuracy of the news signal process. The objective is to find equilibrium strategies by numerically solving an extended HJB equation by using Markov chain approximation method. An iterative algorithm is constructed and its convergence is established. Numerical examples are provided to illustrate the results.

Optimal Distributed Frequency and Voltage Control for Zonal Electricity Markets

Zonal pricing is a well-suited mechanism to incentivize grid-supporting behavior of profit-maximizing producers and consumers operating on a large-scale power system. In zonal electricity markets, local system operators create individual price zones, which provide appropriate price signals depending on local grid conditions such as an excess or shortage of electrical energy in certain regions. In this paper, a real-time zonal pricing controller for AC power networks is presented that ensures frequency and voltage stability as well as Pareto efficiency of the resulting closed-loop equilibria. Based on a dynamic network model that takes line losses and power exchange with adjacent price zones into account, distributed continuous-time control laws are derived which require only neighbor-to-neighbor communication. Application to a real-time congestion management strategy illustrates how spatially and temporally differentiated prices enable grid-supportive operation of the participants without interventions by a superordinate control authority. Effectiveness of different zonal pricing concepts compared to an isolated grid operation is demonstrated through simulations on the IEEE-57 bus system.

Global asymptotic stability of input-saturated one degree-of-freedom Euler–Lagrange systems with Rayleigh dissipation under nonlinear control

ABSTRACT In this manuscript, the regulation of one degree-of-freedom Euler–Lagrange systems subject to input saturation is addressed. In particular, the design and analysis of a nonlinear static state feedback controller is presented. As a result, it is proven via Lyapunov's direct method that, in the presence of Rayleigh dissipation, the closed-loop equilibrium point is globally asymptotically stable with a strict Lyapunov function. Since saturation occurs in the system which contains the actuator model, the proposed control law is unconstrained and can be simplified to a proportional-derivative with desired gravity compensation algorithm. As a by-product global asymptotic stability is also proven for the case where Rayleigh dissipation is null. Numerical simulations on a crank-slider mechanism are presented. Moreover, experimental results on a DC-DC buck power converter are also shown and confirm the viability of our approach.

Hierarchical Robust Adaptive Control for Wind Turbines With Actuator Fault

Abstract This article solves the problem of regulating the rotor speed tracking error for wind turbines in the full-load region by an effective robust adaptive control strategy. The developed controller compensates for the uncertainty in the control input effectiveness caused by a pitch actuator fault, unmeasurable wind disturbance, and nonlinearity in the model. Wind turbines have multilayer structures such that the high-level structure is nonlinearly coupled through an aggregation of the low-level control authorities. Hence, the control design is divided into two stages. First, an L2 controller is designed to attenuate the influence of wind disturbance fluctuations on the rotor speed. Then, in the low-level layer, a controller is designed using a proposed adaptation mechanism to compensate for actuator faults. The theoretical results show that the closed-loop equilibrium point of the regulated rotor speed tracking error dynamics in the high level is finite-gain L2 stable, and the closed-loop error dynamics in the low level is globally asymptotically stable. Simulation results show that the developed controller significantly reduces the root mean square of the rotor speed error compared to some well-known works, despite the largely fluctuating wind disturbance, and the time-varying uncertainty in the control input effectiveness.

The spot and balancing markets for electricity: open- and closed-loop equilibrium models

The spot and balancing markets for electricity: open- and closed-loop equilibrium models

Robust state-dependent mean–variance portfolio selection: a closed-loop approach

This paper studies a class of robust mean–variance portfolio selection problems with state-dependent risk aversion. Model uncertainty, in the sense of considering alternative dominated models, is introduced to the problem to reflect the investor’s uncertainty-averse preference. To characterise the robust portfolios, we consider closed-loop equilibrium control and spike variation approaches. Moreover, we show that a closed-loop equilibrium strategy exists and is unique under some technical conditions. This partially addresses open problems left in Björk et al. (Finance Stoch. 21:331–360, 2017) and Pun (Automatica 94:249–257, 2018). By using a necessary and sufficient condition for the equilibrium, we manage to derive the analytical form of the equilibrium strategy via the unique solution to a nonlinear ordinary differential equation system. To validate the proposed closed-loop control framework, we show that when there is no uncertainty, our equilibrium strategy is reduced to the strategy in Björk et al. (Math. Finance 24:1–24, 2014), which cannot be deduced under the open-loop control framework.

Optimal control of port-Hamiltonian systems: A continuous-time learning approach

In this paper, we propose a continuous-time adaptive feedback controller for the optimal control of input-state-output port-Hamiltonian systems with respect to general Lagrangian performance indices. The proposed control law implements an online learning procedure which uses the Hamiltonian of the system as an initial value function candidate. The continuous-time learning of the value function is achieved by means of a certain Lagrange multiplier that allows to evaluate the optimality of the current solution. In particular, constructive conditions for stabilizing initial value function candidates are stated and asymptotic stability of the closed-loop equilibrium is proven. Simulations of an exemplary nonlinear optimal control problem demonstrate the performance of the controller resulting from the proposed online learning procedure.

Optimal Emergency Frequency Control Based on Coordinated Droop in Multi-Infeed Hybrid AC-DC System

In multi-infeed hybrid AC-DC (MIDC) systems, the asynchronous interconnection between regional grids, complicated system dynamics and possible emergency faults have enormous effects on the frequency stability. To address the frequency instability problems in emergency situations, this paper proposes a decentralized emergency frequency control (EFC) strategy based on coordinated droop for the MIDC system. First, a P-f droop control for LCC-HVDC systems is introduced, and the coordinated droop mechanism among LCC-HVDC systems and generators is designed. Then, to appropriately allocate the power imbalance among the LCC-HVDC systems and generators, an optimal emergency frequency control (OEFC) problem is formulated, and the optimal droop coefficients are selected in a decentralized way, which can deal with various control objectives. A Lyapunov stability analysis shows that the closed-loop equilibrium is locally asymptotically stable considering the LCC-HVDC dynamics. The effectiveness of the proposed emergency control strategy is verified through simulations.

Closed-Loop Equilibrium Reinsurance-Investment Strategy with Insider Information and Default Risk

This paper studies the closed-loop equilibrium reinsurance-investment problem with insider information and default risk. The financial market consists of one risky asset, one defaultable bond, and one risk-free asset. The surplus process is governed by a jump-diffusion process. Two kinds of dependencies between the insurance market and the financial market are considered. In addition, the insurer has some extra claims information available from the beginning of the trading interval. The objective of the insurer is to choose a time-consistent reinsurance-investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. Since this problem is time-inconsistent, using closed-loop control approach from the perspective of game theory, we establish the extended Hamilton–Jacobi–Bellman (HJB) equations for the postdefault case and the predefault case, respectively. Closed-form solutions for the closed-loop equilibrium reinsurance-investment strategy and the corresponding value function are obtained. Finally, we provide a series of numerical examples to illustrate the effects of insider information and other some important model parameters on the closed-loop equilibrium reinsurance and investment strategies. The result analyses reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.

Controlled Markov chains with non-exponential discounting and distribution-dependent costs

This paper deals with a controlled Markov chain in continuous time with a non-exponential discounting and distribution-dependent cost functional. A definition of closed-loop equilibrium is given and its existence and uniqueness are established. Due to the time-inconsistency brought by the non-exponential discounting and distribution dependence, it is proved that the equilibrium is locally optimal in some appropriate sense. Moreover, it is shown that our problem is equivalent to a mean-field game for infinite-many symmetric players with a non-exponential discounting cost.