Designing a stable cycle in weakly coupled identical systems

Consideration was given to a dynamic model containing weakly coupled identical subsystems. The subsystem was assumed to admit a family of periodic solutions where the period is a monotonic function of one parameter. Requirements on the coupling under which the model has an asymptotic orbital stable cycle were established. The problem of stabilization of the model oscillations by a small smooth autonomous coupling control was solved using the results obtained. The system of two coupled conservative systems with one degree of freedom was considered individually.     

Optimal control of weakly coupled bilinear systems

The optimization of the time-invariant bilinear weakly coupled system with a quadratic performance criterion is considered. A sequence of linear state and costate equations is constructed such that the open-loop solution of the optimization problem is obtained in terms of the reduced-order subsystems. This leads to a reduction in the size of the required computations and allows parallel processing of information. The near-optimal closed-loop control is obtained in the form of a linear feedback law, with the feedback gains calculated from two reduced-order independent time-varying linear-quadratic optimal control problems.     

Decoupling transformation for weakly coupled linear systems

The non-singular transformation that completely decouples a weakly coupled linear system under non-restrictive conditions is defined. The transformation matrices are obtained from two algebraic matrix equations. Algorithms that efficiently generate solutions to these equations are derived. The proposed transformation also completely decouples the corresponding Lyapunov matrix differential equation.     

Soft-constrained stochastic Nash games for weakly coupled large-scale systems

In this paper, we discuss infinite-horizon soft-constrained stochastic Nash games involving state-dependent noise in weakly coupled large-scale systems. First, we formulate linear quadratic differential games in which robustness is attained against model uncertainty. It is noteworthy that this is the first time conditions for the existence of robust equilibria have been derived based on the solutions of sets of cross-coupled stochastic algebraic Riccati equations (CSAREs). After establishing an asymptotic structure with positive definiteness for CSAREs solutions, we derive a recursive algorithm by means of Newton’s method so that it can be used to obtain solutions for CSAREs. As another important feature, we propose a high-order approximate Nash strategy based on iterative solutions. Finally, we provide a numerical example to verify the efficiency of the proposed algorithms.     

Distributed model predictive control of constrained weakly coupled nonlinear systems

This paper proposes a distributed model predictive control (MPC) strategy for a large-scale system that consists of several dynamically coupled nonlinear systems with decoupled control constraints and disturbances. In the proposed strategy, all subsystems compute their control signals by solving local optimizations constrained by their nominal decoupled dynamics. The dynamic couplings and the disturbances are accommodated through new robustness constraints in the local optimizations. The paper derives relationships among, and designs procedures for, the parameters involved in the proposed distributed MPC strategy based on the analysis of the recursive feasibility and the robust stability of the overall system. The paper shows that, for a given bound on the disturbances, the recursive feasibility is guaranteed if the sampling interval is properly chosen. Moreover, it establishes sufficient conditions for the overall system state to converge to a robust positively invariant set. The paper illustrates the effectiveness of the proposed distributed MPC strategy by applying it to three coupled cart-(nonlinear) spring–damper subsystems.     

Reduced-order solution to the finite-time optimal-control problemsof linear weakly coupled systems

The optimal solution to the finite-time optimal-control problem of weakly coupled linear systems is found in terms of completely decoupled reduced-order differential equations for both closed-loop and open-loop control. This is achieved through the use of the decoupling transformation that block diagonalizes the Hamiltonian matrix of the weakly coupled linear-quadratic control problem. The convergence to the optimal solution is pretty rapid. The proposed technique is very well suited for parallel computations     

Sliding mode control of continuous-time weakly coupled linear systems with external disturbances

We investigate the problem of sliding mode control for continuous-time weakly coupled linear systems with external disturbances. This is the first study of sliding mode control for continuous-time weakly coupled systems. Using a decoupling transformation, a weakly coupled system is first internally decoupled into two subsystems that are weakly coupled through control inputs only. Then, two sliding mode surfaces are designed for each subsystem using two sliding mode control techniques. The presented procedure greatly simplifies design due to reduced dimensions of subsystems, while providing good approximate results with the accuracy of $O\left(?\right)$, where $?$ is a small weak coupling parameter.     

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 sign-indefinite linear quadratic differential games for weakly coupled large-scale systems

In this paper, N-player linear quadratic differential games that are sign-indefinite for infinite horizon weakly coupled large-scale systems are discussed. After establishing the asymptotic structure and local uniqueness of the solution for cross-coupled sign-indefinite algebraic Riccati equations (CSARE), a new algorithm for solving CSARE is provided. It is shown that the proposed algorithm attains linear convergence. Moreover, in order to reduce the computational workspace, the recursive algorithm is combined. Finally, a high-order approximation strategy based on the proposed iterative solutions is described. As a result, it was recently proved that the numerical strategy achieves a high-order approximation of the equilibrium value. As another important feature, when the small parameters are unknown, a parameter-independent strategy is developed.     

Oscillation analysis of linearly coupled piecewise affine systems: Application to spatio-temporal neuron dynamics

This paper discusses oscillation analysis of (a large number of) linearly coupled piecewise affine (PWA) systems, motivated by various kinds of reaction–diffusion systems including cell-signaling dynamics and neural dynamics. We derive a sufficient condition under which the system shows an oscillatory behavior called Y-oscillation. It is known that the analysis of PWA systems is difficult due to their switching nature. An important feature of the result obtained is that, under the assumption that every subsystem has a specific property in common, the criteria can be rewritten in terms of coupling topology in an easily checkable way, so it is applicable to large scale systems. The results obtained are applied to theoretical investigation of the cardiac action potential generation/propagation represented by spatio-temporal FitzHugh–Nagumo equations.     

Method of some inverse geophysical problem solution by using weakly coupled connected distributed computing systems

In this work a possible model for organizing a grid-based application that performs the solution of several inverse geophysical problems is described. As an example, we consider the problem of determining the parameters of seismic anisotropy in the Earth’s mantle by the inversion of seismic waveforms. It is shown that this class of problems is reduced to the tabulation of a complicated multidimensional function. In this approach, the calculation at each point in a definition interval is calculated independently, so this is ideally appropriate for calculations that use a loosely connected distributed computing infrastructure.     

Fault containment in weakly stabilizing systems

Research on fine tuning stabilization properties has received attention for more than a decade. This paper presents probabilistic algorithms for fault containment. We demonstrate two exercises in fault containment in a weakly stabilizing system, which expedite recovery from single failures, and confine the effect of any single fault to the constant-distance neighborhood of the faulty process. The most significant aspect of the algorithms is that the fault gap, defined as the smallest interval after which the system is ready to handle the next single fault with the same efficiency, is independent of the network size. We argue that a small fault gap increases the availability of the fault-free system.     

Synchronization in networks of identical linear systems

The paper investigates the synchronization of a network of identical linear state-space models under a possibly time-varying and directed interconnection structure. The main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms. Stronger conditions are shown to be sufficient–but to some extent, also necessary–to ensure synchronization with the diffusive static output coupling often considered in the literature.     

Priority systems with many identical processes

In this paper, we consider the termination problem for systems with an arbitrary number of identical priority finite-state processes. In our model, the number of finite-state processes involved in the computation is arbitrary, and a priority is assigned to each transition of a process to indicate the degree of importance or urgency. We show that the termination problem is undecidable for such systems, even when the underlying interprocess communication structure is a star. The undecidability result holds for systems with acyclic processes as well. However, if we require that no two processes reside in the same state (except the starting state) during the course of the computation, the termination problem is PSPACE-complete. Finally, we show that if the priority relation is empty, the problem becomes PTIME-complete.     

Optimal reduced-order solution of the weakly coupled discreteRiccati equation

The optimal solution of the weakly coupled algebraic discrete Riccati equation is obtained in terms of a reduced-order continuous-type algebraic Riccati equation via the use of a bilinear transformation. The proposed method has a rate of convergence of O(ε² ) where ε represents a small coupling parameter. A real-world physical example (a chemical plant model) demonstrates the efficiency of the proposed method. Simulation results obtained using a package for a computer-aided control system are presented. For this specific real-world example, the algorithm perfectly matches the presented theory, since convergence, with an accuracy of 10^-4, is achieved after nine iterations (i.e., 0.68¹⁸=10^-4)     

Optimization of weakly coupled subsystems by a two-level method

A two-level method to find the optimal control of a linear system is presented. We consider that the system is composed of coupled subsystems. And we solve two decoupled first-level problems and one coordinating second-level problem iteratively. When the system can be decoupled into weakly coupled subsystems, the solution fast converges to the optimal.     

Asymptotic solutions to weakly coupled stochastic teams withnonclassical information

The authors develop a new iterative approach toward the solution of a class of two-agent dynamic stochastic teams with nonclassical information when the coupling between the agents is weak, either through the state dynamics or through the information channel. In each case, the weak coupling is characterized in terms of a small (perturbation) parameter. When this parameter value (say, ∈) is set equal to zero, the original fairly complex dynamic team, with a nonclassical information pattern, is decomposed into or converted to relatively simple stochastic control or team problems, the solution of which makes up the zeroth-order approximation (in a function space) to the team-optical solution of the original problem. The fact that the zeroth-order solution approximates the optimal cost up to at least O(∈) is shown. It is also shown that approximations of all orders can be obtained by solving a sequence of stochastic control and/or simpler team problems     

On synchrony of weakly coupled neurons at low firing rate

The dynamics of a pair of weakly interacting conductance-based neurons, firing at low frequency, nu, is investigated in the framework of the phase-reduction method. The stability of the antiphase and the in-phase locked state is studied. It is found that for a large class of conductance-based models, the antiphase state is stable (resp., unstable) for excitatory (resp., inhibitory) interactions if the synaptic time constant is above a critical value tau(c)(s), which scales as the absolute value of log nu when nu goes to zero.     

Quantum coupled oscillators in two-dimensional systems

Quantum coupled one- and two-well oscillators describing a single-electron dynamics in two-dimensional systems have been investigated. Time realizations for mean position and velocity of wave packet, frequency spectra and signal transmission are discussed in detail.