Feedback mechanisms for global oscillations in Lure systems

The paper presents two mechanisms for global oscillations in feedback systems, based on bifurcations in absolutely stable systems. The external characterization of the oscillators provides the basis for a (energy-based) dissipativity theory for oscillators, thereby opening new possibilities for rigorous stability analysis of high-dimensional systems and interconnected oscillators.     

一类组合类摆系统的双态性

The property of dichotomy of interconnected second-order pendulum-like systems with multiple equilibria is investigated. This interconnection can be viewed as harmonic control of independent subsystems. Linear interconnections and a class of input and output interconnections are considered in this paper. The effects of input and output interconnections are shown through a permutation matrix. Frequency domain and linear matrix inequality (LMI) conditions of dichotomy of interconnected pendulum-like systems are derived. It is shown that global properties of two coupled systems can be changed significantly through interconnections. Examples are given to illustrate the results.     

Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback

Motivated by the recent development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of mean-field feedback and analyze the robustness of its phase-locking with respect to general inputs. Under standard assumptions, this system can be reduced to a modified version of the Kuramoto model of coupled nonlinear oscillators. In the first part of the paper we present an analytical study on the existence of phase-locked solutions under generic interconnection and feedback configurations. In particular we show that, in general, no oscillating phase-locked solutions can co-exist with any non-zero proportional mean-field feedback. In the second part we prove some robustness properties of phase-locked solutions (namely total stability). This general result allows in particular to justify the persistence of practically phase-locked states if sufficiently small feedback gains are applied, and to give explicit necessary conditions on the intensity of a desynchronizing mean-field feedback. Furthermore, the Lyapunov function used in the analysis provides a new characterization of the robust phase-locked configurations in the Kuramoto system with symmetric interconnections.     

Multistability in Systems With Counter-Clockwise Input–Output Dynamics

The notion of counterclockwise (CCW) input-output dynamics is extended and used for the global analysis of multistability in positive feedback interconnections. A library of examples is provided to illustrate the usefulness of this concept and help recognizing CCW dynamics in specific applications, especially those arising in systems biology     

Decentralized global robust stabilization of a class of interconnected minimum-phase nonlinear systems

This paper focuses on a class of large-scale interconnected minimum-phase nonlinear systems with parameter uncertainty and nonlinear interconnections. The uncertain parameters are allowed to be time-varying and enter the systems nonlinearly. The interconnections are bounded by nonlinear functions of states. The problem we address is to design a decentralized robust controller such that the closed-loop large-scale interconnected nonlinear system is globally asymptotically stable for all admissible uncertain parameters and interconnections. It is shown that decentralized global robust stabilization of the system can be achieved using a control law obtained by a recursive design method together with an appropriate Lyapunov function.     

The global convergence analysis of a class of nonlinear network systems

In this paper, the global convergence for a class of nonlinear network systems with multiple equilibriums was studied, which can be viewed as interconnected systems composed of nonlinear systems through linear input and output interconnections. Frequency-domain conditions were established for global convergence and convergence of bounded solutions. The effects of the input and output interconnections can be studied through a nonsingular inner coupling matrix and nonzero scales, representing the interconnection, which takes values according to the eigenvalues of the nonsingular outer coupling matrix. Then, the design method based on linear matrix inequality was presented by using the KYP Lemma and Schur complement.     

A multilevel optimization of large-scale dynamic systems

A multilevel feedback control scheme is proposed for optimization of large-scale systems composed of a number of (not necessarily weakly coupled) subsystems. Local controllers are used to optimize each subsystem, ignoring the interconnections. Then, a global controller may be applied to minimize the effect of interconnections and improve the performance of the overall system. At the cost of suboptimal performance, this optimization strategy ensures invariance of suboptimality and stability of the systems under structural perturbations whereby subsystems are disconnected and again connected during operation.     

Decentralized adaptive output feedback design for large-scalenonlinear systems

In this paper, we present a global, decentralized adaptive design procedure for a class of large-scale nonlinear systems, which utilizes only local output feedback. The advocated scheme guarantees robustness to parametric and dynamic uncertainties in the interconnections and also rejects any bounded disturbances entering the system. The systems belonging to this class are those which can be transformed using a global diffeomorphism to the output feedback canonical form, where the interconnections are a function of subsystem outputs only. The uncertainties are assumed to be bounded by an unknown pth-order polynomial in the outputs. The resulting controller maintains global robustness and disturbance rejection properties. The output tracking error is shown to be bounded within a compact set, the size of which can be made arbitrarily small by appropriate choice of the control gains. For the case where the objective is regulation, global asymptotic regulation of all the states of the closed-loop system is achieved     

Local ISS of large-scale interconnections and estimates for stability regions

We consider interconnections of locally input-to-state stable (LISS) systems. The class of LISS systems is quite large, in particular it contains input-to-state stable (ISS) and integral input-to-state stable (iISS) systems.Local small-gain conditions both for LISS trajectory and Lyapunov formulations guaranteeing LISS of the composite system are provided in this paper. Notably, estimates for the resulting stability region of the composite system are also given. This in particular provides an advantage over the linearization approach, as will be discussed.     

Decentralized stabilization of a class of interconnected stochasticnonlinear systems

This paper focuses on a class of large-scale interconnected stochastic nonlinear systems. The interconnections are bounded by strong nonlinear functions that contain first order and higher order polynomials as special cases. The problem we address is to design a decentralized controller such that the closed-loop, large-scale, interconnected stochastic nonlinear system is globally asymptotically stable in probability for all admissible interconnections. It is shown that the decentralized global stabilization via both state feedback and output feedback can be solved by a Lyapunov-based recursive design method     

Indirect adaptive control for interconnected systems

An indirect adaptive controller for interconnected systems is introduced. Each subsystem is subject to bounded disturbances and to possibly unbounded interconnections with other subsystems. A variable dead zone is incorporated into a gradient estimation scheme to limit the effects of the interconnections and disturbances. An adaptive state feedback control law is described which stabilizes the interconnected system, forcing the input and output signals of each subsystem to remain bounded     

Global identification of complex static systems with hierarchical structure: Deterministic problem

The paper deals with a deterministic case of the identification problem of static complex systems with hierarchical structure, in which the globally optimal model of the whole system as a hierarchical configuration of models of individual subsystems is obtained

The considerations are confined to hierarchical complex systems, in which

(a) subsystems belonging to the same levels arc not interconnected

(b) the individual subsystems can be connected through their outputs only with the others from lower levels

Moreover only the hierarchical systems without, external factors (e.g. external controls) affecting particular subsystems (except for the subsystem from the highest level) are examined

It. is shown that for some global identification quality criteria, some structures of interconnections between individual subsystems in the system identified, and for some forms of models assumed for individual subsystems, the problem considered can be reduced to several identification problems of complex systems with cascade structure. The optimal solution of these problems based on the dynamic programming method is discussed.     

Global H infinty control and almost disturbance decoupling for a class of interconnected non-linear systems

This paper is concerned with a global H infinity control problem for a class of interconnected non-linear systems. We first consider a fairly general clas of large-scale non-linear systems with strong non-linear interconnections. It is shown that the decentralized H infinity control problem for the system can be converted into the centralized control problems associated with a set of auxiliary systems. The solutions to the latter problems in general rely on solutions of their associated Hamilton-Jacobi-Isaacs (HJI) inequalities. Realizing that finding a global solution of the HJI inequality is usually impossible, we then consider a global decentralized almost disturbance decoupling problem (DADDP) and a global decentralized inverse H infinity control problem (DIHCP) for a class of interconnected systems with lower triangular structure. The DADDP is concerned with the design of decentralized control laws that achieve an arbitrarily small disturbance input to the controlled output. The DIHCP involves seeking not only control laws but also state-dependent weights of the control inputs such that the associated global decentralized control problem is solvable. It is shown that the solutions to both the DADDP and DIHCP can be obtained via a recursive design technique. L2-gain from the     

Bounded control of feedforward time‐delay systems with linearized systems consisting of chain of oscillators

For the global stabilization of a family of feedforward nonlinear time‐delay systems whose linearized systems consist of a chain of identical oscillators, a saturated feedback control is established based on a special canonical form. The proposed control laws use not only the current states but also the delayed states for feedback, which helps maintain the decoupling property in the recursive design. Moreover, explicit conditions guaranteeing the stability of the closed‐loop system are provided. When the nonlinear terms vanish and even the oscillators are distinct, a modified saturated feedback control can still be established for the corresponding global stabilization problem. Two numerical examples are given to illustrate the effectiveness of the proposed approaches.     

Lurie控制系统的关联绝对稳定性-双线性矩阵不等式方法

研究任意两个相互独立的Lurie控制系统能否通过关联或协调控制组成绝对稳定大系统的问题,给出两个Lurie控制系统可关联绝对稳定的充分条件,并给出了计算关联矩阵的双线性矩阵不等式方法.研究结果表明:两个非绝对稳定的系统可以通过关联或协调控制来实现关联大系统的绝对稳定性,取消了大系统稳定性分析中对关联的不合理限制.文末给出了本文结果的数值例子.     

The influence of limit cycle topology on the phase resetting curve

Understanding the phenomenology of phase resetting is an essential step toward developing a formalism for the analysis of circuits composed of bursting neurons that receive multiple, and sometimes overlapping, inputs. If we are to use phase-resetting methods to analyze these circuits, we can either generate phase-resetting curves (PRCs) for all possible inputs and combinations of inputs, or we can develop an understanding of how to construct PRCs for arbitrary perturbations of a given neuron. The latter strategy is the goal of this study. We present a geometrical derivation of phase resetting of neural limit cycle oscillators in response to short current pulses. A geometrical phase is defined as the distance traveled along the limit cycle in the appropriate phase space. The perturbations in current are treated as displacements in the direction corresponding to membrane voltage. We show that for type I oscillators, the direction of a perturbation in current is nearly tangent to the limit cycle; hence, the projection of the displacement in voltage onto the limit cycle is sufficient to give the geometrical phase resetting. In order to obtain the phase resetting in terms of elapsed time or temporal phase, a mapping between geometrical and temporal phase is obtained empirically and used to make the conversion. This mapping is shown to be an invariant of the dynamics. Perturbations in current applied to type II oscillators produce significant normal displacements from the limit cycle, so the difference in angular velocity at displaced points compared to the angular velocity on the limit cycle must be taken into account. Empirical attempts to correct for differences in angular velocity (amplitude versus phase effects in terms of a circular coordinate system) during relaxation back to the limit cycle achieved some success in the construction of phase-resetting curves for type II model oscillators. The ultimate goal of this work is the extension of these techniques to biological circuits comprising type II neural oscillators, which appear frequently in identified central pattern-generating circuits.     

Constant and switching gains in semi-active damping of vibrating structures

A limit cycle is the stability boundary for linear and non-linear control systems. Hamiltonian mechanics and power flow control are employed to demonstrate this property of limit cycles. The presentation begins with the concept of linear limit cycles which is extended to non-linear limit cycles. Many examples are used to demonstrate these concepts including linear and non-linear oscillators, power engineering, and an extension to a class of plane differential systems. Power flow control based on Hamiltonian mechanics is shown to be applicable to a large class of non-linear systems. Finally, eigenanalysis and flight stability for linear systems are extended to non-linear systems and is referred to as ‘the power flow principle of stability for non-linear systems’.     

A theory of global concurrency control in multidatabase systems

This article presents a theoretical basis for global concurrency control to maintain global serializability in multidatabase systems. Three correctness criteria are formulated that utilize the intrinsic characteristics of global transactions to determine the serialization order of global subtransactions at each local site. In particular, two new types of serializability, chain-conflicting serializability and sharing serializability, are proposed and hybrid serializability, which combines these two basic criteria, is discussed. These criteria offer the advantage of imposing no restrictions on local sites other than local serializability while retaining global serializability. The graph testing techniques of the three criteria are provided as guidance for global transaction scheduling. In addition, an optimal property of global transactions for determinating the serialization order of global subtransactions at local sites is formulated. This property defines the upper limit on global serializability in multidatabase systems.     

Criteria for global pinning-controllability of complex networks

In this paper, we study pinning-controllability of networks of coupled dynamical systems. In particular, we study the problem of asymptotically driving a network of coupled identical oscillators onto some desired common reference trajectory by actively controlling only a limited subset of the whole network. The reference trajectory is generated by an exogenous independent oscillator, and pinned nodes are coupled to it through a linear state feedback. We describe the time evolution of the complex dynamical system in terms of the error dynamics. Thereby, we reformulate the pinning-controllability problem as a global asymptotic stability problem. By using Lyapunov-stability theory and algebraic graph theory, we establish tractable sufficient conditions for global pinning-controllability in terms of the network topology, the oscillator dynamics, and the linear state feedback.