Linear System Subject Research Articles

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.

Linear state estimation for Markov jump linear system with multi-channel observation delays and packet dropouts

ABSTRACTThis paper is to investigate the linear minimum mean square error estimation for Markovian jump linear system subject to unknown Markov chains, multi-channel mode and observation delays, and packet losses. The reorganisation method is employed to convert the delayed measurement system into an equivalent delay-free one and a new state variable is introduced, by which the original state estimation with transmission delays and data losses is transformed into the new state estimation for the reorganised delay-free system with jumping parameters and multiplicative noises. The new state estimation is derived via the innovation analysis method, and an analytical solution to the estimator is given in terms of a set of generalised Riccati difference equations based on a set of coupled Lyapunov equations. Then the original state estimation will be obtained via the jumping property. Finally, we show that the difference Riccati equations converge to a set of generalised algebraic Riccati equations under appropriate assumptions, which result in an optimal stationary filter.

Self-Validated Time-Domain Analysis of Linear Systems With Bounded Uncertain Parameters

This brief presents a novel approach to predict the bounds of the time-domain response of a linear system subject to multiple bounded uncertain input parameters. The method leverages the framework of Taylor models in conjunction with the numerical inversion of Laplace transform (NILT). Different formulations of the NILT are reviewed, and their advantages and limitations are discussed. An implementation relying on an inverse fast Fourier transform turns out to be the most efficient and accurate alternative. The feasibility of the technique is validated based on several diverse application examples, namely a control loop, a lossy transmission-line network, and an active low-pass filter.

Auxiliary Fault Tolerant Control With Actuator Amplitude Saturation and Limited Rate

In this paper, the problem of fault tolerant tracking control for a linear time-invariant system subject to actuator faults and saturations is addressed. An auxiliary system is developed to ensure actuators behave within amplitude and rate limits under the influence of partial loss of control effectiveness. Based on the auxiliary system, a fault tolerant compensation controller is constructed to guarantee tracking errors to converge to a small region. Some relationships among tracking errors, command signals, actuator faults, amplitude and rate limits as well as controller parameters are comprehensively studied and explicitly illustrated with formulas. An example of rudder-roll damping control for a cruise keeping ship is included to illustrate the proposed procedures and their effectiveness.

Observability and Controllability Analysis of Linear Systems Subject to Data Losses

We provide algorithmically verifiable necessary and sufficient conditions for fundamental system theoretic properties of discrete-time linear, systems subject to data losses. More precisely, the systems in our modeling framework are subject to disruptions (data losses) in the feedback loop, where the set of possible data loss sequences is captured by an automaton. As such, the results are applicable in the context of shared (wireless) communication networks and/or embedded architectures where some information on the data loss behavior is available a priori . We propose an algorithm for deciding observability (or the absence of it) for such systems, and show how this algorithm can be used also to decide other properties including constructibility, controllability, reachability, null-controllability, detectability, and stabilizability by means of relations that we establish among these properties. The main apparatus for our analysis is the celebrated Skolem's Theorem from Linear Algebra. Moreover, we study the relation between the model adopted in this paper and a previously introduced model where, instead of allowing dropouts in the feedback loop, one allows for time-varying delays.

Uniting two local output controllers for linear system subject to input saturation: LMI Approach

In this paper, we solve the problem of hybrid output unification of two local output feedback controllerU0 and U1 for linear systems subject to input saturation. The closed-loop system with the controller U0 has a smaller region of attraction R0 and higher decay rate σ0, while the closed-loop system with U1 has a larger region of attraction R1 and a slower decay rate σ1, R0⊂R1 and σ0 > σ1. To the best of our knowledge, the problem of unification of two local output feedback controller, has been solved only in the recent papers [17] and [2]. In [17], the problem of unification of two output controller is solved by using two norm estimators and a trigger time τ* to switch from the “global” controller to the “local” one, where it is defined in [17] and chosen sufficiently large. The work [2] proves that it can be selected arbitrarily small. Such choice of τ* minimizes the use of the slow controller U1 and, thus, ameliorates the performance significantly. In this paper, the solution of the unification problem is formulated by means of linear matrix inequalities (LMIs), which can be easily verified numerically. The numerical example given in this paper compares the performance of the proposed hybrid controller with hybrid controllers in [17] and [2].

Regional stability and stabilization of a class of linear hyperbolic systems with nonlinear quadratic dynamic boundary conditions

This paper addresses the boundary control problem of fluid transport in a Poiseuille flow taking the actuator dynamics into account. More precisely, sufficient stability conditions are derived to guarantee the exponential stability of a linear hyperbolic differential equation system subject to nonlinear quadratic dynamic boundary conditions by means of Lyapunov based techniques. Then, convex optimization problems in terms of linear matrix inequality constraints are derived to either estimate the closed-loop stability region or synthesize a robust control law ensuring the local closed-loop stability while estimating an admissible set of initial states. The proposed results are then applied to application-oriented examples to illustrate local stability and stabilization tools.

Exact recursive updating of state uncertainty sets for linear SISO systems

This paper addresses the classical problem of determining the set of possible states of a linear discrete-time SISO system subject to bounded disturbances, from measurements corrupted by bounded noise. These so-called uncertainty sets evolve with time as new measurements become available. We present two theorems which give a complete description of the relationship between uncertainty sets at two successive time instants, and this yields an efficient algorithm for recursively updating uncertainty sets. Numerical simulations demonstrate performance improvements over existing exact methods.

Fault Detection for Linear Discrete Time-Varying Systems Subject to Random Sensor Delay: A Riccati Equation Approach

This paper mainly studies the fault detection (FD) problem for linear discrete time-varying systems subject to random sensor delay. By assuming that the measurement channel is with Transmission Control Protocol (TCP), an observer-based FD filter (FDF) is provided as a residual generator via embedding the packet indicator into the filter. To construct this FDF, its design issue is formulated into two sub-problems. One is to maximize the $\mathcal {H}_{-}/\mathcal {H}_{\infty }$ or $\mathcal {H}_{\infty }/\mathcal {H}_{\infty }$ FD performance index, which aims to enhance the ratio of fault sensitivity/disturbance attenuation. The other one is to find the filter parameter matrices such that the error between the residual and the fault is minimized in the $\mathcal {H}_{\infty }$ sense. By employing stochastic analysis and introducing some adjoint operator based optimization approaches, analytical solutions to the aforementioned FDF design problem are derived via solving recursive Riccati equations. An illustrative example is given to show the effectiveness of the proposed methodologies.

Robust compensation of a chattering time-varying input delay with jumps

We investigate the design of a prediction-based controller for a linear system subject to a time-varying input delay, not necessarily First-In/First-Out (FIFO). This means that the input signals can be reordered. The feedback law uses the current delay value in the prediction. It does not exactly compensate for the delay in the closed-loop dynamics but does not require to predict future delay values, contrary to the standard prediction technique. Modeling the input delay as a transport Partial Differential Equation, we prove asymptotic stabilization of the system state, that is, robust delay compensation, providing that the average ℒ2-norm of the delay time-derivative over some time-window is sufficiently small and that the average time between two discontinuities (average dwell time) is sufficiently large.

A geometric approach to structural model matching by output feedback in linear impulsive systems

Abstract This paper provides a complete characterization of solvability of the problem of structural model matching by output feedback in linear impulsive systems with nonuniformly spaced state jumps. Namely, given a linear impulsive plant and a linear impulsive model, both subject to sequences of state jumps which are assumed to be simultaneous and measurable, the problem consists in finding a linear impulsive compensator that achieves exact matching between the respective forced responses of the linear impulsive plant and of the linear impulsive model, by means of a dynamic feedback of the plant output, for all the admissible input functions and for all the admissible sequences of jump times. The solution of the stated problem is achieved by reducing it to an equivalent problem of structural disturbance decoupling by dynamic feedforward. Indeed, this latter problem is formulated for the so-called extended linear impulsive system, which consists of a suitable connection between the given plant and a modified model. A necessary and sufficient condition for the solution of the structural disturbance decoupling problem is first shown. The proof of sufficiency is constructive, since it is based on the synthesis of the compensator that solves the problem. The proof of necessity is based on the definition and the geometric properties of the unobservable subspace of a linear impulsive system subject to unequally spaced state jumps. Finally, the equivalence between the two structural problems is formally established and proven.

L∞ Observer for Uncertain Linear Systems

AbstractIn this paper, an L∞ observer design method is proposed for linear system subject to parameter uncertainty and bounded disturbance. The proposed L∞ observer, which satisfies a peak‐to‐peak disturbances attenuation performance, is designed to overbound the estimation error. Moreover, sufficient conditions for the design of L∞ observer are derived and expressed in terms of linear matrix inequalities (LMIs). The novelty of the proposed method is that we develop an L∞ observer that not only can attenuate bounded disturbance but also provides an upper bound of estimation error norm. Simulation results are presented to illustrate the effectiveness of the proposed method.

Hands-off Control for Discrete-time Linear Systems subject to Polytopic Uncertainties

Abstract This paper develops approaches to the hands-off control problem subject to performance constraints for discrete-time linear systems. The approaches minimize the l1-norm of the control input to acquire the hands-off property, while satisfying the performance constraints that are given in terms of the quadratic cost of states and inputs with respect to the optimal solution to the finite-horizon linear quadratic regulator problem. We consider three kinds of the input and state matrices for the system; 1) known, 2) uncertain but contained in a known discrete set, and 3) uncertain but contained in a known polytopic uncertainty set. For the first two cases, we show that each problem is formulated as an l1 optimization that is expressed as a second-order cone programming. We also show that the last case leads to a second-order cone programming after relaxations. A numerical example is included to illustrate the validity of the proposed approach.

Hybrid Stabilization of Linear Systems With Reverse Polytopic Input Constraints

This paper addresses the problem of globally uniformly exponentially stabilizing a linear system to the origin by output feedback while avoiding a prescribed set of input values. We consider that the set of input values to avoid is given by the union of a finite number of closed polytopes that do not contain the origin and we refer to this restriction as a reverse polytopic input constraint. We show that the synthesis of the hybrid controller can be performed by solving a set of linear matrix inequalities for the full-state feedback case, and by solving a set of bilinear matrix inequalities for the output feedback case. The resulting closed-loop hybrid system is shown to satisfy key conditions for well-posedness and robustness to small measurement noise. Furthermore, we apply the proposed hybrid controller to the stabilization of a single-input linear system subject to reverse polytopic constraints on the norm of the input. The behavior of the corresponding closed-loop hybrid system is validated by simulations.

Recursive Robust Regulator for Discrete-Time Markovian Jump Linear Systems

In this paper, we propose a robust regulator for discrete-time Markovian jump linear systems (DMJLS) subject to structured parameter uncertainties. States of Markov chain are considered known at each instant and the transition probabilities matrix is time-varying. The framework proposed to deal with this problem is developed through a regularization approach based on robust least-squares and penalty functions. Recursive solutions in terms of coupled Riccati equations, which do not depend on any auxiliary parameter to be tuned, are provided. These equations resemble the recursive solutions of DMJLS standard regulators when the system is not subject to uncertainties. Conditions for convergence and stability of this robust regulator are established. Numerical examples present comparative studies considering the proposed robust regulator, the standard linear quadratic regulator, and an $H_{\infty }$ -controller for DMJLS.

An asymmetric Lyapunov function for linear systems with asymmetric actuator saturation

SummaryThis paper proposes an asymmetric Lyapunov function approach to the estimation of the domain of attraction and the domain with a guaranteed regional gain for a linear system subject to asymmetric actuator saturation. Depending on the sign of each of the m inputs, the input space is divided into 2m regions. In each region, the linear system with asymmetrically saturated inputs can be expressed as a linear system with symmetric dead zone. A quadratic function of the augmented state vector containing the system state and the symmetric dead‐zone function is constructed for each region. From these quadratic functions, an asymmetric Lyapunov function is composed. Furthermore, based on the special properties of the intersections between regions, 2 generalized asymmetric Lyapunov functions are proposed that lead to reduced conservativeness. A set of conditions are established under which the level sets of these asymmetric Lyapunov functions are contractively invariant and are thus estimates of the domain of attraction. Another set of conditions are derived under which the level sets are subsets of the domain with a guaranteed regional gain. Based on these conditions, LMI‐based optimization problems are formulated and solved to obtain the largest level sets as the estimates of the domain of attraction and of the domain with a guaranteed regional gain. Simulation results demonstrate the effectiveness of the proposed approach.

Low-Gain Integral Control for Multi-Input Multioutput Linear Systems With Input Nonlinearities

We consider the inclusion of a static antiwindup component in a continuous-time low-gain integral controller in feedback with a multi-input multi-output stable linear system subject to an input nonlinearity (from a class of functions that includes componentwise diagonal saturation). We demonstrate that the output of the closed-loop system asymptotically tracks every constant reference vector, which is “feasible” in a natural sense, provided that the integrator gain is sufficiently small. Robustness properties of the proposed control scheme are investigated and three examples are discussed in detail.

Self-triggered Control for Uniform Ultimate Boundedness using Skewed Structured Singular Values

This paper proposes a self-triggered control approach that assures that the system state is uniformly ultimately bounded. In particular, a linear discrete-time system subject to both parametric uncertainties and disturbances is considered. The approach uses the skewed structured singular value for predictions to determine the next execution time. A numerical example is provided to illustrate the approach.

Event-triggered Consensus Seeking under Non-uniform Time-Varying Delays

In this paper, we study consensus seeking for a class of linear multi-agent systems (MAS) subject to the inevitable imperfections of packet-based networked communication. These imperfections include unknown non-uniform time-varying transmission delays and limited communication resources. To reduce the utilization of communication resources, we propose a dynamic event-triggered control scheme resulting in aperiodic transmission of information between agents. The proposed framework leads to event-triggered controllers that guarantee the MAS to asymptotically reach consensus, strictly positive lower bounds on the inter-event times and robustness for unknown non-uniform time-varying delays in terms of maximum allowable delays.

H2 and H∞ dynamic output feedback control of continuous-time Markov systems with uncertain rates using LMIs

The aim of this paper is to investigate the problems of H∞ and H2 dynamic output feedback control for continuous-time Markov jump linear systems (MJLS) subject to uncertain transition rates. Synthesis conditions are proposed in terms of parameter-dependent matrix inequalities with a scalar parameter and polynomial variables. The conditions become linear for fixed values of the scalar parameter, which provides an extra degree of freedom to search for feasible solutions and less conservative H∞ and H2 bounds. Differently from the existing approaches, the proposed method provides full order mode-dependent controllers constructed in terms of the state space matrices of the operation modes and partitions of the slack variables introduced in the design conditions. Therefore, the resulting controller does not depend explicitly on the Lyapunov matrix, being robust with respect to the uncertain transition rates. Moreover, the closed-loop MJLS stability and the bounds to the H∞ or H2 norm are certified by means of polynomially parameter-dependent Lyapunov matrices. Numerical examples illustrate the applicability and the advantages of the proposed technique when compared with other approaches from the literature.