Interconnected Linear Systems Research Articles

Local stabilization of networked linear systems with decentralized switching controllers

In this paper, we consider the decentralized control design problem for a particular case of networked switched affine systems. The considered network consists of interconnected (identical) linear systems with the controlled input switching among a finite set of values. We address a challenging case where the network can be only locally stabilized. Scalable Lyapunov based conditions are proposed for designing such decentralized controllers. The derived conditions (theoretical and linear matrix inequalities) do not depend on the number of subsystems. Ellipsoidal estimates of the domain of attraction are given.

On the Admissibility and Stability of Multiagent Nonlinear Interconnected Positive Systems With Heterogeneous Delays

Many multi-agent interconnected systems include typical nonlinearities which are highly sensitive to inevitable communication delays. This makes their analysis challenging and the generalization of results from linear interconnected systems theory to those nonlinear interconnected systems very limited. This paper deals with the analysis of Multi-Agent Nonlinear Interconnected Positive Systems (MANIPS). The main contributions of this work are two fold. Based on Perron-Frobenius theorem, we first study the “admissibility” property for MANIPS, and show that it is a fundamental requirement for this category of systems. Then, using admissibility/positivity properties and sequences of functions theory, we propose a suitable Lyapunov function candidate to conduct the analysis of the dynamical behavior of such systems. We show that the stability of MANIPS is reduced to the positiveness property (i.e. negative or positive definiteness) of a new specific matrix-valued function ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {Z}$</tex-math></inline-formula> ) that we derive in this paper. Furthermore, the obtained results generalize the existing theory. The quality of the results achieved are demonstrated through the applications of the developed theory on cells with multi-stage maturation process dynamical models.

The Robust Model Predictive Control Design for Linear Interconnected Systems

In this paper, we discuss the design of robust model predictive control for linear interconnected systems by several constraints. Linear interconnected system is a complex systems which consist of several subsystems and the dynamic of subsystem is influenced by the states or outputs of the other subsystems. The systems are controlled by formulating and applying the model predictive control on each subsystems. We formulate the optimization problem that minimize the cost function for each subsystems. The optimization problem is solved by gradient method to get the minimum cost function. Numerical simulation was conducted to show the effectiveness and performance of robust model predictive control. Based on the results of the numerical simulation and test the robustness, it can be shown that the robust model predictive control is effective to control the linear interconnected systems.

Decentralized static output feedback controller design for linear interconnected systems

Decentralized static output feedback controller design for linear interconnected systems

Distributed Model Predictive Control of Time-Delay Systems

In this paper, a new approach to distributed MPC of linear interconnected time-delay systems is proposed. The general case of time-delay in both the states and the control inputs is considered. The approach includes representation of the interconnected time-delay dynamics in the augmented state space and reformulation of the centralized MPC problem into a Quadratic Programming framework. The latter this is solved distributedly by using the dual gradient method for optimization. The suggested approach would be appropriate for embedded distributed MPC and is illustrated on a simulation example of two interconnected time-delay systems.

Flatness of interconnected linear systems and applications to electrical systems

In this paper, we completely describe flatness of the simplest class of interconnected systems: we consider an interconnection of (controllable) single-input linear subsystems in dimension two with a star topology and a linear interconnection dynamics (with no inputs acting directly on the interconnection variable). First, we observe that even if each subsystem is flat, flatness of the global interconnected system is not necessarily preserved. Then, we give necessary and sufficient verifiable conditions for flatness of the interconnected system. When the interconnected system is flat, we analyze how its flat output depends on the interconnection variable and how it can be expressed in function of the flat outputs of each subsystem. Finally, we show how our results can be applied to electrical power systems.

Design of Decentralized Adaptive Integral Terminal Sliding Mode Controller for Linear Interconnected Mechanical Systems in the Presence of External Disturbance

Design of Decentralized Adaptive Integral Terminal Sliding Mode Controller for Linear Interconnected Mechanical Systems in the Presence of External Disturbance

Relaxed Stabilization Conditions for Interconnected Nonlinear Systems

This paper deals with the conservatism reduction of stability and stabilization conditions for nonlinear continuous-time interconnected systems. Based on Takagi–Sugeno modeling, the interconnected system is described by a convex combination of interconnected linear systems. Line integral fuzzy Lyapunov functions are considered to develop stability and stabilizability criteria for the considered class of interconnected nonlinear systems. All analysis and design conditions are formulated in LMI terms. The effect of the coupling due to interconnection terms is also investigated and taken into account to obtain less conservative conditions. Examples are given to illustrate the merit of the obtained results.

Scalable distributed H2 controller synthesis for interconnected linear discrete-time systems

The current limitation in the synthesis of distributed H 2 controllers for linear interconnected systems is scalability due to non-convex or unstructured synthesis conditions. In this paper we develop convex and structured conditions for the existence of a distributed H 2 controller for discrete-time interconnected systems with an interconnection structure that corresponds to an arbitrary graph. Neutral interconnections and a storage function with a block-diagonal structure are utilized to attain coupling conditions that are of a considerably lower computational complexity compared to the corresponding centralized H 2 controller synthesis problem. The effectiveness and scalability of the developed distributed H 2 controller synthesis method is demonstrated for small- to large-scale oscillator networks on a cycle graph.

Handling unmeasured disturbances in data-driven distributed control with virtual reference feedback tuning

The data-driven synthesis of a distributed controller in the presence of noise is considered, via the distributed virtual reference feedback tuning (DVRFT) framework. The analysis is performed for a linear interconnected system on an arbitrary graph that is subject to unmeasured exogenous inputs. By solving a dynamic network identification problem with prediction-error filtering and a tailor-made noise model, we show that the distributed model-reference control problem can be solved directly from data. Sufficient conditions are provided for which the local controller estimates are consistent. Moreover, it is shown how the method can be applied in the single-input-single-output case, leading to consistent estimates with standard virtual reference feedback tuning as well. The effectiveness of the method is demonstrated via a small network example with two interconnected systems.

Distributed Computational Framework for Large-Scale Stochastic Convex Optimization

This paper presents a distributed computational framework for stochastic convex optimization problems using the so-called scenario approach. Such a problem arises, for example, in a large-scale network of interconnected linear systems with local and common uncertainties. Due to the large number of required scenarios to approximate the stochasticity of these problems, the stochastic optimization involves formulating a large-scale scenario program, which is in general computationally demanding. We present two novel ideas in this paper to address this issue. We first develop a technique to decompose the large-scale scenario program into distributed scenario programs that exchange a certain number of scenarios with each other to compute local decisions using the alternating direction method of multipliers (ADMM). We show the exactness of the decomposition with a-priori probabilistic guarantees for the desired level of constraint fulfillment for both local and common uncertainty sources. As our second contribution, we develop a so-called soft communication scheme based on a set parametrization technique together with the notion of probabilistically reliable sets to reduce the required communication between the subproblems. We show how to incorporate the probabilistic reliability notion into existing results and provide new guarantees for the desired level of constraint violations. Two different simulation studies of two types of interconnected network, namely dynamically coupled and coupling constraints, are presented to illustrate advantages of the proposed distributed framework.

Trajectory tracking in networks of linear systems

This paper proposes a control method for trajectory tracking in interconnected linear systems. The physical couplings among all subsystems are considered by a modification of the reference signals of the individual subsystems. Feedforward controllers are designed for the modified isolated systems. This way of solution leads to a set of differential–algebraic equations whose solution depends on the invariant zeros of the subsystems. The design of the controllers needs only a single communication step for the information exchange between neighboring subsystems. The main results of the paper are a new representation of the internal dynamics as differential–algebraic equations that lead to a relation between the tracked flat output and the original subsystem output and the proof that the proposed networked controllers ensure trajectory tracking.

An Emerging Dynamics Approach for Synchronization of Linear Heterogeneous Agents Interconnected Over Switching Topologies

In this letter, we propose an emergent dynamics based tool to approximate the synchronized behavior in large networks of interconnected linear systems with switching interconnection topologies and linear coupling. We assume that the networks are heterogeneous, but the states of all systems are of the same dimension. The goal of this letter is to show that, in the case of sufficiently large coupling gains, the network is practically synchronized, and its' synchronized behavior can be approximated by a reduced order switching system independent of the control gains. The results are ensured for strongly connected networks under fairly mild assumptions by introducing a minimum dwell-time between two consecutive switches. Numerical simulations illustrate the theoretical results.

Decentralized Suboptimal State Feedback Integral Tracking Control Design for Coupled Linear Time-Varying Systems

In this paper, a suboptimal state feedback integral decentralized tracking control synthesis for interconnected linear time-variant systems is proposed by using orthogonal polynomials. Particularly, the use of operational matrices allows, by expanding the subsystem input states and outputs over a shifted Legendre polynomial basis, the conversion of time-varying parameter differential state equations to a set of time-independent algebraic ones. Hence, optimal open-loop state and control input coefficients are forwardly determined. These data are used to formulate a least-square problem, allowing the synthesis of decentralized state feedback integral control gains. Closed-loop asymptotic stability LMI conditions are given. The proposed approach effectiveness is proved by solving a nonconstant reference tracking problem for coupled inverted pendulums.

Resilient decentralized sampled-data [formula omitted] filter design for linear interconnected systems subject to denial-of-service attacks

This paper investigates the design problem of the resilient decentralized sampled-data H∞ filter for linear interconnected systems subject to the denial-of-service attacks. A piecewise Lyapunov function-based method is first developed to determine the filter parameters and the sampled-data communication schemes. Then the obtained results are utilized to facilitate the design of the resilient decentralized sampled-data H∞ filter against the denial-of-service attacks. Compared with the existing results, the communication channels from the subsystems of interconnected systems to the filters are allowed to be independent and be compromised by different adversaries. Finally, numerical simulations on the double-inverted pendulums system and the three-machine power system are provided to illustrate the efficacy of the proposed method.

Adaptive sliding mode observer–based decentralized control design for linear systems with unknown interconnections

In this study, a class of linear interconnected systems with unknown interconnections between subsystems is considered. Primarily, an observer is designed to reconstruct an equivalent of unknown interconnections that are considered uncertain. A decentralized controller will be thereafter designed to keep the stability of the original system when the estimated system is at risk of instability due to the lack of information on interconnections. To design a decentralized observer and to estimate states of each subsystem, without knowing the relations between subsystems, a combination of Luenberger observer along with the adaptive sliding mode technique is used. Because the interconnected system might generally be unstable, a state feedback controller is used to stabilize each subsystem using estimated states together with the output of other subsystems. The stability of the system and the convergence of the discrepancy between real states and that of estimated are guaranteed, gaining the Lyapunov theory. Simulation results signify that the proposed decentralized controller based on a new adaptive sliding mode observer is highly efficient for linear interconnected systems with unknown interconnections.

Decentralized finite-time control for linear interconnected fractional-order systems with input saturation

This paper investigates decentralized finite-time boundedness and control problems for a linear interconnected fractional-order system with input saturation. By Lyapunov function, generalized Gronwall inequality, and fractional-order Caputo derivative inequality, sufficient conditions for finite-time boundedness of the linear interconnected fractional-order system are established. Both the decentralized state feedback controller and the decentralized output feedback controller are designed to guarantee the finite-time boundedness of the closed-loop system, and the seeking methods of controller gains are given by solving linear matrix inequalities. Finally, two simulation examples are employed to illustrate the effectiveness of the obtained results.

Output feedback stabilization of an underactuated cascade network of interconnected linear PDE systems using a backstepping approach

We detail in this article the development of a robust stabilizing output feedback control law for an underactuated cascade network of n systems of two heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the subsystems is actuated. The proposed approach combines successive backstepping transformations that present a specific cascade structure. With these transformations it is possible to rewrite the original network system as a simple system for which all the in-domain coupling terms have been removed. One can then design a strictly proper stabilizing control law. The proposed control law is proved to be robust to small delays in the actuation. Finally, a boundary observer is designed, enabling stabilization by output feedback.

Consensusability of linear interconnected multi-agent systems

Consensusability is an important property for many multi-agent systems (MASs) as it implies the existence of networked controllers driving the states of MAS subsystems to the same value. Consensusability is of interest even when the MAS subsystems are physically coupled, which is the case for real-world systems such as power networks. In this paper, we study necessary and sufficient conditions for the consensusability of linear interconnected MASs. These conditions are given in terms of the parameters of the subsystem matrices, as well as the eigenvalues of the physical and communication graph Laplacians. Our results show that weak coupling between subsystems and fast information diffusion rates in the physical and communication graphs favor consensusability. Technical results are verified through computer simulations.

Fault-tolerant control for a class of linear interconnected hyperbolic systems by boundary feedback

This paper considers a fault-tolerant control problem for a class of interconnected linear hyperbolic partial differential equation systems. Both subsystem faults and coupling faults are considered. Firstly, the well-posedness of the faulty system is analyzed by using semigroup theory. Secondly, for the fault-free case, a stabilizing boundary feedback control based on small-gain theorem is proposed. Consequently, in the presence of faults, fault recoverability conditions are established that maintain the stability of the faulty systems. The fault-tolerant control strategies are also provided. A heat exchanger example is taken to illustrate the effectiveness and practicality of the proposed methods.