General Linear Systems Research Articles

Performance analysis and control of fractional‐order positive systems

Unlike a general integer-order linear system, the L 1 and L 2 norms of its state and output variables may not be convergent in an infinite time interval even if the concerned fractional-order linear system is asymptotically stable or even Mittag-Leffler stable. Therefore, this study mainly investigates the analysis and synthesis of L 1 and L 2 performances in a finite time interval for a class of fractional-order positive systems. The L ∞ performance problem in the infinite time interval is also concerned since the boundness of the L ∞ norm of the state and output variables can be guaranteed in the infinite time interval if the concerned fractional-order positive system is asymptotical stable. The L 1 and L ∞ performance characterisations are expressed by linear programming and L 2 performance characterisation is expressed in terms of linear matrix inequality with a diagonal positive definite solution. Based on these performance conditions, the problems of stabilisation by a state feedback controller with L p (p ∈ {1, 2, ∞})-gain guaranteed are solved for fractional-order positive systems. An example is presented to show the effectiveness of the proposed methods in this study.

Stabilization of discrete-time linear systems by delay independent truncated predictor feedback

For a discrete-time linear system with input delay, the predictor feedback law is the product of a feedback gain matrix with the predicted state at a future time instant ahead of the current time instant by the amount of the delay, which is the sum of the zero input solution and the zero state solution of the system. The zero state solution is a finite summation that involves past input, requiring considerable memory in the digital implementation of the predictor feedback law. The truncated predictor feedback, which results from discarding the finite summation part of the predictor feedback law, reduces implementation complexity. The delay independent truncated predictor feedback law further discards the delay dependent transition matrix in the truncated predictor feedback law and is thus robust to unknown delays. It is known that such a delay independent truncated predictor feedback law stabilizes a discrete-time linear system with all its poles at z = 1 or inside the unit circle no matter how large the delay is. In this paper, we first construct an example to show that the delay independent truncated predictor feedback law cannot compensate too large a delay if the open loop system has poles on the unit circle at z ≠ 1. Then, a delay bound is provided for the stabilizability of a general linear system by the delay independent truncated predictor feedback.

Adaptive fault-tolerant time-varying formation tracking for multi-agent systems under actuator failure and input saturation

This paper studies the time-varying formation tracking problem for general linear multi-agent systems with multiple leaders in the presence of both actuator failure and input saturation. The followers are required to uniquely determine and track the convex combination of the states of leaders, while maintaining a predefined time-varying formation. A hyperbolic tangent function is firstly introduced to modify the actuator model with input saturation constraint. Then, an augmented plant for dynamics of each follower is constructed to derive the control protocol by exploiting the dynamic surface control technique. The proposed control protocol deals with faults of bias and unknown bounded loss of effectiveness by means of adaptive fault-tolerant strategies, while a formation feasible condition should be satisfied. With the control signal generated by the augmented plant, the time-varying formation error is proved to be semi-globally uniformly bounded under the faults and input saturation, based on standard Lyapunov theory. Finally, a numerical simulation is implemented to demonstrate the effectiveness of the proposed algorithm.

Time-Varying Formation Control for Time-Delayed Multi-Agent Systems with General Linear Dynamics and Switching Topologies

Time-varying formation analysis and design problems for general linear multi-agent systems with switching interaction topologies and time-varying delays are studied. Firstly, a consensus-based formation control protocol is constructed using local information of the neighboring agents. An algorithm with three steps is presented to design the proposed formation control protocol. Then, based on linear matrix inequality technique and common Lyapunove–Krasovskii stability theory, sufficient conditions for general linear multi-agent systems with switching topologies and time-varying delays to achieve time-varying formation are given together with a time-varying formation feasibility condition. Finally, a numerical simulation is given to demonstrate the effectiveness of the obtained theoretical results.

Guaranteed Cost Consensus for Multiple General Dynamic Systems With Switching Topologies

Guaranteed cost consensus problems for general linear multi-agent systems with switching topologies are investigated, where both the jointly-connected and connected switching topologies are considered. Each interaction topology in the interaction topology set may not be connected and the union topology in each finite time interval is connected for the jointly-connected switching topology, while each interaction topology is connected for the connected switching topology. In the guaranteed cost consensus problem, a guaranteed cost function is constructed to represent the consensus regulation performance and the control energy consumption. By the state space decomposition approach and some mathematics tools, several sufficient conditions for guaranteed cost consensus with switching topologies are given and design methods for the consensus control gain matrix are presented. When all the general linear dynamic systems achieve guaranteed cost consensus, an upper bound is given for the introduced guaranteed cost function and the consensus function represented the whole motion of all dynamic systems is determined. The effectiveness of the proposed results is demonstrated by numerical simulations.

Robust global consensus tracking of linear multi‐agent systems with input saturation via scheduled low‐and‐high gain feedback

This study deals with the protocol design for achieving robust global consensus tracking of multi-agent systems. Agent dynamics are described as general linear systems with actuator saturation and input additive uncertainties and disturbances. Via developing a scheduled low-and-high gain design technique, the state feedback protocol is proposed, under which global consensus tracking and disturbance rejection for such systems can be achieved under the mild assumptions on agent dynamics and network topology. This result is further extended to the case of the reduced-order observer-based protocol design, with the help of the special coordinate basis approach. Finally, the advantages of the proposed protocols are illustrated by a numerical simulation.

Optimal Time for the Controllability of Linear Hyperbolic Systems in One-Dimensional Space

We are concerned about the controllability of a general linear hyperbolic system of the form $\partial_t w (t, x) = \Sigma(x) \partial_x w (t, x) + \gamma C(x) w(t, x) $ ($\gamma \in \mathbb{R}$) i...

Delay-dependent distributed event-triggered tracking control for multi-agent systems with input time delay

This paper addresses delay-dependent distributed event-triggered tracking control for general linear multi-agent systems. First, both fixed and time-varying input delays are taken into consideration of tracking consensus protocol design due to the network transmission delay among agents, respectively. Correspondingly, a new delay-dependent triggering mechanism is proposed, by which the protocols of agent are only updated at the violation of the pre-defined triggering mechanism. One of its advantages is that the triggering instant can not be affected by its neighbours’ state of last triggering instant, which can reduce effectively the frequency of control updates. Then, the co-design method of the control protocol gains and the triggering parameters are developed and delay-dependent event-triggered tracking consensus criteria are obtained by Lyapunov stability theory, respectively. Moreover, the Zeno-behavior of event-triggered mechanisms is excluded effectively by showing the existence of the strictly positive consecutive triggering interval. Last, two comparison examples are provided to demonstrate the effectiveness of the developed method.

Robust Control of Uncertain Linear Systems Based on Reinforcement Learning Principles

In this paper, a reinforcement learning (RL) approach is developed to solve the robust control for uncertain continuous-time linear systems. The objective is to find a feedback control law for the uncertain linear system using an online policy iteration algorithm. The robust control problem is solved by constructing an extended algebraic Riccati equation with properly defined weighting matrices for a general uncertain linear system. An online policy iteration algorithm is developed to solve the robust control problem based on RL principles without knowing the nominal system matrix. The convergence of the algorithm to the robust control solution for uncertain linear systems is proved. The simulation examples are given to demonstrate the effectiveness of the proposed algorithm. The results extend the design method of robust control to uncertain linear systems.

Sampled-Data Control for Fuzzy Markovian Jump Systems With Actuator Saturation

In this article, the problem of sampled-data control is considered for fuzzy Markovian jump systems (FMJSs) with actuator saturation. Firstly, the mode-dependent lyapunov functional is constructed, which consists of a two-sided closed-loop function. On the basis of this function, stochastically stable criteria are presented in the form of linear matrix inequalities (LMIs). Then, based on the stochastically stable criteria, mode-dependent sampled-data controllers are designed for FMJSs. As a corollary, stable criteria and controller design are also proposed for the general aperiodic sampled-data linear system with actuator saturation. The improved LMI-based optimization conditions are obtained to estimate the region of attraction. Finally, numerical examples are provided to show the significant improvement of the presented method.

H∞ Containment Control of Multiagent Systems Under Event-Triggered Communication Scheduling: The Finite-Horizon Case.

This paper investigates the finite-horizon H∞ containment control issue for a general discrete time-varying linear multiagent systems with multileaders. All followers in such a system are driven into a convex hull spanned by multiple leaders, which can be transformed into a problem of tracking a virtual trajectory generated by these leaders. For this purpose, a local state observer is put forward to estimate the state of each agent itself. Then, the estimated state is transmitted to corresponding neighbors governing by an innovation-based event-triggered scheduling protocol. The purpose of the addressed problem is to design both an event-based distributed controller and a state observer such that a prescribed H∞ containment index can be achieved over a given finite horizon. First, with the help of the completing the square method, a sufficient condition is established to ensure the desired H∞ containment performance. Then, by resort to a novel nominal energy cost index combined with Moore-Penrose pseudoinverse method, the desired controller and observer parameters are obtained by solving two coupled backward recursive Riccati difference equations. Two positive scalars in proposed nominal energy cost index provide a tradeoff among the controlled tracking errors, the energy of transformed control inputs, and the precision of estimated states. Finally, a simulation example is given to illustrate the usefulness of the proposed theoretical results.

Stabilisability of discrete-time multi-agent systems under fixed and switching topologies

ABSTRACTThis paper studies the problem of stabilisability for general discrete-time linear multi-agent systems. For fixed topology, a necessary and sufficient graph-theoretic condition is proposed. Based on this, a general linear form of the external control input is given, and two methods to design the feedback gain matrix by Riccati equality/inequality are given. For switching topology, a sufficient graph-theoretic condition is presented, under which the stabilisability of multi-agent systems is achieved by introducing an averaging analysis approach. Additionally, both results under fixed and switching topologies are applied for solving the tracking and formation control of discrete-time multi-agent systems, and some sufficient and/or necessary conditions are presented. Finally, numerical examples are given to illustrate the theoretical results.

Semi-global containment control of discrete-time linear systems with actuator position and rate saturation

In this paper, we study semi-global containment control problem for a multi-agent system in the presence of multiple leader agents. We describe the dynamics of each follower agent in multi-agent system by a general discrete-time linear system subject to both actuator position and rate saturation. By using low gain approach, we construct both a linear state feedback control law and an observer-based output feedback control law for each follower agent. Under some standard assumptions, we provide a sufficient condition to guarantee that the states of follower agents converge to the convex hull formed by the leader agents asymptotically. Finally, we illustrate the theoretical results by numerical simulation results.

Observer-Based Consensus for Multiagent Systems Under Stochastic Sampling Mechanism

This paper is concerned with the consensus problem of general linear dynamic multiagent systems with stochastic sampling. In this paper, the sampling intervals randomly switch between two different values. The communication topology between agents is fixed and directed. Full- and reduced-order observers are designed based on neighbor agents’ relative output information. The algorithms to construct such observers are also provided. By using the estimated states of the agents, the observer-based consensus protocol with stochastic sampling are presented. Sufficient conditions to ensure consensus in mean square are derived by using Lyapunov stability theory. Finally, simulations are given to examine the effectiveness of the proposed methods.

Exponential Consensus of Multiagent Systems With Lipschitz Nonlinearities Using Sampled-Data Information

In this paper, exponential consensus of general linear multiagent systems with Lipschitz nonlinear dynamics using sampled-data information is investigated. Both leaderless and leader-following consensuses are considered. Using input-delay approach, the resulting sampled-data closed-loop systems are first reformulated as continuous systems with time-varying delay in the control input. Then, decoupled conditions in terms of linear matrix equality (LMI) on the Lipschitz constant, the decay rate, the communication graph parameters, and the control gain matrix to guarantee exponential consensus are derived using novel Lyapunov functionals. Based on the sufficient conditions, controller design methods are also provided in the form of decoupled LMIs. Finally, simulation examples including the consensus of Chua’s circuit systems are given to illustrate the effectiveness of the obtained results.

Distributed algebraic tearing and interconnecting techniques

A class of novel parallel preconditioning schemes in conjunction with a Krylov subspace iterative method for solving general sparse linear systems is presented. The proposed preconditioning schemes are domain decomposition methods that enforce the continuity of the solution on the subdomain interfaces using Lagrange multipliers, without requiring geometric information, namely algebraic tearing and interconnecting methods. Hence, they are applicable to a wide variety of problems as they are based only on the adjacency graph corresponding to the coefficient matrix. A modification of the proposed schemes, which improves performance while reducing the required operations is also presented. The algebraic tearing and interconnecting methods are designed for distributed systems with multicore nodes. Numerical results concerning the convergence behavior and the parallel performance of the proposed schemes are given along with discussions.

Consensus of Linear Multi-Agent Systems with Persistent Disturbances via Distributed Output Feedback

This paper addresses the consensus problem of general linear multi-agent systems with persistent disturbances by distributed output feedback. Suppose that states of agents can not be obtained directly. Several estimators are designed to observe states of agents and the unknown disturbances. A protocol is proposed to drive all agents achieve consensus. Based on the method of model transformation and the property of permutation matrix, sufficient conditions for consensus are obtained in terms of linear matrix inequalities. Finally, simulations are given to show the effectiveness of presented results.

Self-triggered leader-following consensus of multi-agent systems with input time delay

In this paper, a self-triggered algorithm for leader-following consensus of general linear multi-agent systems with input time delay is proposed. It is shown that the leader-following consensus can be reached asymptotically and the Zeno-behaviour of triggering time sequences can be excluded. Under this algorithm, the next exact triggering time instant of each agent can be computed by the local information at the previous triggering time instant. Continuous communication, monitoring of measurement error and updating of controller input are thus avoided.

Event-triggered consensus control for uncertain multi-agent systems with external disturbance

This paper investigates the event-triggered consensus control for a general linear multi-agent system with model uncertainties and external disturbances. A consensus protocol is proposed using the local state information at asynchronous event-triggering time instants, where each agent determines when to perform sampling and control updating using the designed event-triggered condition. Then the robust consensus condition and the feedback matrix design method are derived to make the uncertain multi-agent system reach consensus with a desired disturbance attenuation ability. Furthermore, it is demonstrated that the Zeno behaviour will never occur by giving the lower bound of inter-event time intervals. Finally, a simulation example is given to verify the obtained theoretical results, with the comparison simulation conducted by using the periodic triggering strategy.

Semi‐global containment control for linear systems in the presence of actuator position and rate saturation

SummaryThis paper studies the semi‐global containment control problem for a group of general linear systems in the presence of actuator position and rate saturation. Both a state feedback containment control algorithm and an output feedback containment algorithm are constructed for each follower agent in the system by using low gain approach. We show that the states of all follower agents will converge to the convex hull formed by the leader agents asymptotically under these control algorithms when the communication topology among follower agents is a connected undirected graph and each leader agent is a neighbor of at least one follower agent. Simulation results illustrate the theoretical results.