Cross-coupled Algebraic Riccati Equations Research Articles

Pareto Optimality in Infinite Horizon Mean-Field Stochastic Cooperative Linear-Quadratic Difference Games

This paper is concerned with the mean-field stochastic cooperative linear-quadratic (LQ) dynamic difference game in an infinite time horizon. First, the necessary and sufficient conditions for the stability in the mean-square sense, and the stochastic Popov-Belevith-Hautus (PBH) eigenvector tests for exact observability and exact detectability of mean-field stochastic linear difference systems are derived by the <inline-formula><tex-math notation="LaTeX">$\mathscr {H}$</tex-math></inline-formula>-representation technique. Second, the relation between the solvability of the cross-coupled generalized Lyapunov equations&#x00A0;(CC-GLEs) and exact observability, exact detectability, and stability of the mean-field dynamic system is well characterized. It is then shown that the cross-coupled algebraic Riccati equations&#x00A0;(CC-AREs) admit a unique positive definite (positive semi-definite, respectively) solution under exact observability (exact detectability, respectively), which is also a feedback stabilizing solution. Furthermore, all Pareto optimal strategies and solutions can be respectively derived via the solutions to the weighted CC-AREs (WCC-AREs) and the weighted cross-coupled algebraic Lyapunov equations&#x00A0;(WCC-ALEs). Finally, a practical application on the computation offloading in the multi-access edge computing network (MECN) is presented to illustrate the proposed theoretical results.

A Game Approach to Distributed Robust Optimal Filtering With Consensus Through Multiple Sensors: Theory and Application

This article addresses the problem of distributed state estimation on the infinite-time horizon for discrete-time systems with multiple sensors. The desired filter, referred to as distributed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> Gaussian filter, reaching a tradeoff between optimality and robustness, constructs a local estimate based on its own observation and on those collected from its neighbors. A Nash game approach is used to deal with such a multiobjective distributed filtering problem, and causal bounded Nash equilibrium strategies, consisting of the optimal filter gains and the corresponding worst-case disturbance signals, are analytically conducted based on solutions to a set of cross-coupled algebraic Riccati equations. Moreover, to further improve the cohesiveness among local estimates, additional consensus objective is taken into consideration by exchanging prior estimates among neighboring nodes. A choice of the consensus term, with which the error dynamics are shown to be internally stable, is derived via a numerically tractable and efficient method. Finally, an application to speed monitoring through wireless sensor network is included to show the validity of the presented results.

Robust stabilization subject to structured uncertainties and mean power constraint

This paper deals with a robust stabilization problem for discrete-time systems subject to multiple disturbances occurring in controller and actuating channel, where both linear structured uncertainties and white Gaussian noises are included. The desired control law is aimed to robustly stabilize the system and to satisfy some pre-specified mean power constraint, simultaneously. By the philosophy of the mixed H2∕H∞ control, a solvability condition is first derived for single-input systems that reveals the intrinsic relation between the unstable poles of the plant and the disturbance parameters, together with two cross-coupled algebraic Riccati equations. The result is further generalized to multiple-input systems with a sufficient condition given again by the unstable poles of the plant. An example is included to illustrate the current results.

Composite Stackelberg Strategy for Singularly Perturbed Bilinear Quadratic Systems

AbstractBased on singularly perturbed bilinear quadratic problems, this paper proposes to decompose the full-order system into two subsystems of a slow-time and fast-time scale. Utilizing the fixed point iterative algorithm to solve cross-coupled algebraic Riccati equations, equilibrium strategies of the two subsystems can be obtained, and further the composite strategy of the original full-order system. It was proved that such a composite strategy formed ano(ε) (near) Stackelberg equilibrium, and a numerical result of the algorithm was presented in the end.

A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations

In this letter, a computational approach for solving cross-coupled algebraic Riccati equations (CAREs) is investigated. The main purpose of this letter is to propose a new algorithm that combines Newton's method with a gradient-based iterative (GI) algorithm for solving CAREs. In particular, it is noteworthy that both a quadratic convergence under an appropriate initial condition and reduction in dimensions for matrix computation are both achieved. A numerical example is provided to demonstrate the efficiency of this proposed algorithm.

Numerical computation of cross-coupled algebraic Riccati equations related to H∞-constrained LQG control problem

In this paper, the numerical algorithm for solving the state and output feedback H ∞-constrained LQG control problem is investigated. Although the equations that have to be solved to design the controller consist of the nonlinear cross-coupled algebraic Riccati equations (CAREs), it is newly proven that both the uniqueness and the positive semidefiniteness of the iterative solutions can be guaranteed when disturbance attenuation level γ is sufficiently large. The computational examples are given to demonstrate the efficiency of the proposed algorithm.

A Numerical Analysis of the Nash Strategy for Weakly Coupled Large-Scale Systems

This note discusses the feedback Nash equilibrium of linear quadratic N-player Nash games for infinite-horizon large-scale interconnected systems. The asymptotic structure along with the uniqueness and positive semidefiniteness of the solutions of the cross-coupled algebraic Riccati equations (CAREs) is newly established via the Newton-Kantorovich theorem. The main contribution of this study is the proposal of a new algorithm for solving the CAREs. In order to improve the convergence rate of the algorithm, Newton's method is combined with a new decoupling algorithm; it is shown that the proposed algorithm attains quadratic convergence. Moreover, it is shown for the first time that solutions to the CAREs can be obtained by solving the independent algebraic Lyapunov equation (ALE) by using the reduced-order calculation

Numerical computation of cross‐coupled algebraic Riccati equations related to H2/H∞ control problem for singularly perturbed systems

AbstractIn this paper, we present a numerical algorithm to the cross‐coupled algebraic Riccati equations(CARE) related to H2/H∞ control problems for singularly perturbed systems (SPS) by means of Newton's method. The resulting algorithm can be widely used to solve Nash game problems and robust control problems because the CARE is solvable even if the quadratic term has an indefinite sign. We prove that the resulting iterative algorithm has the property of the quadratic convergence. Using the solution of the CARE, we construct the high‐order approximate H2/H∞ controller. Copyright © 2004 John Wiley &amp; Sons, Ltd.