Prescribed Decay Rate Research Articles

ASYMMETRIC FUNCTIONAL-BASED APPROACHTO EXPONENTIAL STABILITY OF LINEAR DISTRIBUTEDTIME-DELAY SYSTEMS

In this paper, the problem of exponential stability of linear time-delay systems with mixed discrete and distributed delays is studied. Based on an asymmetric Lyapunov-Krasovskii functional approach, sufficient conditions are derived in terms of linear matrix inequalities to guarantee the exponential convergence of the system state trajectories with a prescribed decay rate. The efficacy of the obtained results is demonstrated by a given numerical example and simulations.

Event-triggered fixed-time tracking of state-constrained surface ships under actuation saturation with prescribed control performance

This paper investigates the event-trigger-based fixed-time tracking control of a fully-actuated marine surface vehicle subject to full-state constraints, lumped disturbances, model uncertainties, and input saturation. The definition of the constrainedly practically fixed-time stability (CPFTS) is given first, and a sufficient condition for CPFTS is subsequently provided. To handle the input saturation, an auxiliary system is designed elaborately. Then applying the barrier Lyapunov function technology and blending an error transformation into backstepping scheme, the model-based adaptive controller and neural-network-based (NN-based) adaptive controller are proposed, respectively. Under these two controllers, the resulting closed-loop system signals are CPFTS and the settling time is not only independent of the system initial conditions but also assigned by users previously. Besides, it is proved that the tracking errors can converge to an arbitrarily small preset compact set within the preassigned time at a prescribed decay rate whilst all system states preserve the time-varying constraints.

Gain-Scheduled Control of LPV Systems with Structural Constraints

In large-scale dynamic systems, consisting of the interconnection of several subsystems, the control law must obey a certain distributed structure defined by the information exchange pattern on a communication network. In case the system exhibits parameters which vary over time, the Linear Parameter Varying (LPV) paradigm provides a control-oriented framework for the design of robust or gain-scheduled control laws. Since each subsystem has only access to a partial knowledge of the overall set of parameters, due to the constraint imposed by the communication network, the controller gain must satisfy some structural constraints which increase the complexity of the design problem. This paper proposes a novel approach based on Linear Matrix Inequalities (LMIs) to deal with the distributed control of large-scale systems described by a LPV dynamics. LMI conditions are established for the design of a gain-scheduled controller ensuring exponential stability of the overall system with a prescribed decay rate, while satisfying the structural constraints imposed by the communication network. A simulation case study is presented to show the effectiveness of the proposed approach.

Boundary controllability and stabilizability of a coupled first-order hyperbolic-elliptic system

In this paper, we study the controllability of a coupled first-order hyperbolic-elliptic system in the interval $ (0, 1) $ by a Dirichlet boundary control acting at the left endpoint of the hyperbolic component only. Using the multiplier approach and compactness-uniqueness argument, we establish the exact controllability for the hyperbolic component of the model, at any time $ T>1 $. We explore the method of moments to conclude the exact controllability at the critical time $ T = 1 $. For the case of small time, that is for $ T<1 $, we show that the system is not null controllable. Further, using a Gramian-based approach introduced by Urquiza, we prove the exponential stabilization of the corresponding closed-loop system with arbitrary prescribed decay rate by means of boundary feedback control law.

Dynamic output feedback robust [formula omitted] control of networked control systems with time-Varying delays via T-S fuzzy models

This paper addresses the problem of robust stabilization for networked control systems (NCS) where the stochastic behavior of the network does not allow correct and complete transmission of data to both the actuators and the controller. The objective is to model the NCS using Takagi-Sugeno (T-S) fuzzy systems with uncertainties and external disturbances and to design a time-varying delay fuzzy controller such that the closed-loop system is robustly exponentially stable with a prescribed decay rate. Sufficient conditions are derived using the delay-dependent Lyapunov-Krasovskii functional with a robust H∞ controller in terms of linear matrix inequalities (LMIs). Accordingly, desired fuzzy controllers are designed by using feasible solutions to the obtained LMIs. The proposed fuzzy controller can keep the T-S fuzzy system stable in the presence of time-varying delay, uncertainties, and external disturbance. Two numerical examples with different conditions on parameters are provided to illustrate the performance and robustness of the proposed approach.

Early lumping observer design and sensor placement for transport-reaction systems

This paper deals with the observer design problem for a class of one-dimensional multi-species transport-reaction systems satisfying sector bounded nonlinearities and considering measurements distributed over the spatial domain. A design method is proposed based on a reduced-order model and a Lyapunov function, which provides sufficient conditions in terms of standard linear matrix inequalities (LMIs) to ensure the exponential convergence of the estimation error with a prescribed decay rate. The observer performance is further improved through an offline optimal sensor placement algorithm considering a parameterized reduced-order output matrix. A nonisothermal tubular reactor is presented to demonstrate the observer performance as well as the advantages of the proposed sensor placement optimization scheme.

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

Comparison of Two Formulas for Approximating Functions with a Prescribed Decay Rate

Comparison of Two Formulas for Approximating Functions with a Prescribed Decay Rate

Mixed Incremental H ∞ and Incremental Passivity Analysis for Markov Switched Stochastic Nonlinear Systems

The current study addresses the mixed incremental H∞ and incremental passivity analysis for Markov switched stochastic (MSS) nonlinear systems. The multiple Lyapunov functions approach and the structure of Markov framework are utilized to establish some sufficient conditions for the MSS nonlinear systems, which will be used for the incrementally globally asymptotically stable in the mean(IGASiM) and performance index analysis. Then, the mixed incremental H∞ and incremental passivity performance issues are solved for two instances: in the first case, all subsystems are not IGASiM, while in the second one, both of IGASiM and unstable subsystems exist. Hence, it is shown that when none of the subsystems is IGASiM, the MSS nonlinear systems are IGASiM and possess the mixed incremental H∞ and incremental passivity performance metric in the presence of specified conditions. The mathematical induction is selected to guarantee the robust incremental stability of MSS systems with IGASiM and non-IGASiM subsystems and the performance index can be exhibited a prescribed decay rate. The effectiveness of the proposed results is demonstrated by two simulation examples.

Delay-dependent and decay-rate-dependent conditions for exponential mean stability and non-fragile controller design of positive Markov jump linear systems with time-delay

This paper proposes some new sufficient conditions and necessary conditions for exponential mean stability with prescribed decay rate, and sufficient conditions for non-fragile controller design of both continuous-time and discrete-time positive Markov jump linear systems with time-delay. First, sufficient stability conditions are derived by constructing novel linear stochastic co-positive Lyapunov-Krasovskii functionals. Second, the corresponding necessary conditions are established by applying a model transformation technique and analyzing the relationship between stochastic stability of the transformed systems and exponential mean stability with given decay rate of the original systems. Compared with the existing conditions, the proposed stability conditions are not only delay-dependent but also decay-rate-dependent, and the sufficient conditions plus the necessary conditions can be used to judge the system stability more precisely. Third, based on the new sufficient conditions, non-fragile controllers are designed by solving linear programming problems such that the closed-loop systems are positive and exponentially mean stable with an expected decay rate. Finally, by numerical examples, the effects of both time-delay and decay rate on exponential mean stability are exploited and the validity of the non-fragile controller design conditions is demonstrated.

A novel observer-predictor control for uncertain systems with unknown time-varying input and output delays

In this paper, a novel observer-predictor control is proposed to stabilise uncertain systems subject to unknown but bounded arbitrarily-fast time-varying input and output delays. Due to the presence of time-varying model uncertainties and delay mismatches, the separation principle cannot be used to design the observer and control gains separately, which becomes a challenging problem. By applying small gain theory and Lyapunov-Krasovskii functionals (LKF), in combination with Bessel-Legendre inequalities and the Projection Lemma, we develop sufficient conditions based on Linear Matrix Inequalities (LMIs) for control design. As a result, the closed-loop robust performance can be improved whilst satisfying a prescribed decay rate in the closed-loop dynamics at the expense of introducing more LMI decision variables without increasing the complexity of the control scheme. Also, the Extended State Observer (ESO) is integrated in the proposed control scheme to improve the rejection performance of mismatched disturbances. The effectiveness of the proposed method is illustrated by simulation results.

Prescribed performance control of strict‐feedback systems under actuation saturation and output constraint via event‐triggered approach

SummaryIn this work, we study the performance‐guaranteed event‐triggered control for a class of uncertain nonlinear systems in strict‐feedback form subject to input saturation and output constraint. The prescribed performance (ie, convergence rate, tracking error accuracy) and output constraint are firstly taken into account for nonlinear systems with event‐triggered input. By blending a speed transformation into the barrier Lyapunov function and introducing an intermediate variable to the system, two different event‐triggered control schemes are proposed for systems with and without saturation, respectively. Each scheme has two rules to determine triggering time sequences, one for control signal updating and the other for control signal transmission with the latter being a subsequence of the first. Meanwhile, it is proved that the tracking error converges to a preset compact set around zero at the prescribed decay rate and the output is maintained within a given bound at all times. Simulation verification also confirms the effectiveness of the proposed approach.

Optimization of Communication Network Topology in Distributed Control Systems Subject to Prescribed Decay Rate.

In this paper, we propose a simple cohesive framework to find an optimal directed control network topology with minimum number of links while a prescribed decay rate is satisfied in the transient response of a distributed control system. In order to guarantee the system's decay rate to be faster than a prespecified value, a constraint on the dominant eigenvalue of the system is required to be considered. This results in a nonconvex optimization problem as eigenvalue of a parametric nonsymmetric matrix is a nonconvex, nonsmooth, and even non-Lipschitz function. Here, we present a convex equivalent optimization problem whose minimizer also solves this eigenvalue optimization problem. This optimization problem proposes a state-feedback matrix which results in a decay rate faster than a given value while input signal costs are considered. The equivalent optimization problem in combination with sparsity-promoting optimal control constitutes a combinatorial optimization problem. Using alternating direction method of multipliers, the problem is decomposed into a chain of analytically solvable subproblems which are differentiable and separable. The proposed optimization framework includes relative preference between the topology of the control network and the decay rate of the system. The simulation results show the effectiveness of the proposed framework.

Nonlocal equations with regular varying decay solutions

We present a new approach for studying the large time asymptotic behavior to the nonlocal equation ∂tu=Ju−χ0u in the whole Rn, where Ju=∫RnJ(⋅,y)u(y,t)dy. We obtain a condition so that the solutions decay in time at the rate of a regular varying function R, even for non-symmetric J, for a wide class of R. This condition involves a sharp bound for Jku0 (u0=u|t=0) or the kernels of Jk, and the decay rate is proved by analyzing an exponential series with regular varying coefficients. Classical cases, including the fractional Laplacian, the compactly support regular kernels, and algebraic decaying kernels are shown to satisfy our condition by employing the sharp Young's inequality and the Fourier transform technique. In particular, we have generalized the pioneer work by Chasseigne et al. (2006) [11]. Finally, we exhibit nonlocal equation that has solutions with a prescribed decay rate, for a wide class of regular varying function.

Exponential State Estimation for Memristor-Based Discrete-Time BAM Neural Networks With Additive Delay Components.

This paper focuses on the dynamical behavior for a class of memristor-based bidirectional associative memory neural networks (BAMNNs) with additive time-varying delays in discrete-time case. The necessity of the proposed problem is to design a proper state estimator such that the dynamics of the corresponding estimation error is exponentially stable with a prescribed decay rate. By constructing an appropriate Lyapunov-Krasovskii functional (LKF) and utilizing Cauchy-Schwartz-based summation inequality, the delay-dependent sufficient conditions for the existence of the desired estimator are derived in the absence of uncertainties which are further extended to available uncertain parameters of the prescribed memristor-based BAMNNs in terms of linear matrix inequalities (LMIs). By solving the proposed LMI conditions the estimation gain matrices are obtained. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed results.

Output regulation for coupled linear parabolic PIDEs

This paper presents a backstepping-based solution of the output regulation problem for coupled parabolic partial integro-differential equations (PIDEs) with spatially-varying coefficients and distinct diffusion coefficients. The considered setup assumes in-domain as well as boundary disturbances, while the output to be controlled can be defined distributed in-domain, pointwise in-domain or at the boundaries and need not be measured. By assuming a finite-dimensional signal model, which may also be non-diagonalizable, a systematic solution of the output regulation problem is presented by making use of observer-based feedforward control. Existence conditions for the corresponding regulator are formulated in terms of the plant transfer behaviour. For the resulting closed-loop system, exponential stability with a prescribed decay rate is verified. The regulator design for two unstable coupled parabolic PIDEs demonstrates the results of the paper.

A novel approach to exponential stability of continuous-time Roesser systems with directional time-varying delays

This paper addresses the problem of exponential stability analysis of two-dimensional (2D) linear continuous-time systems with directional time-varying delays. An abstract Lyapunov-like theorem which ensures that a 2D linear system with delays is exponentially stable for a prescribed decay rate is exploited for the first time. In light of the abstract theorem, and by utilizing new 2D weighted integral inequalities proposed in this paper, new delay-dependent exponential stability conditions are derived in terms of tractable matrix inequalities which can be solved by various computational tools to obtain maximum allowable bound of delays and exponential decay rate. Two numerical examples are given to illustrate the effectiveness of the obtained results.

Reset stabilisation of positive linear systems

In this paper, the problems of reset stabilisation for positive linear systems (PLSs) are investigated. Some properties relating to reset control of PLSs are first revealed. It is shown that these properties are different from the corresponding ones of general linear systems. Second, a class of periodic reset scheme is designed to exponentially stabilise an unstable PLS with a prescribed decay rate. Then, for a given PLS with reset control, some discussions on the upper bound of its decay rate are presented. Meanwhile, the reset stabilisation for PLSs in a special case is probed as well. Finally, two numerical examples are used to demonstrate the correctness and effectiveness of the obtained theoretical results.

Markov Chain Approach to Probabilistic Guidance for Swarms of Autonomous Agents

AbstractMotivated by biological swarms occurring in nature, there is recent interest in developing swarms comprised completely of engineered agents. The main challenge for developing swarm guidance laws compared to earlier formation flying and multi‐vehicle coordination approaches is the sheer number of agents involved. While formation flying applications might involve up to 10 to 20 agents, swarms are desired to contain hundreds to many thousands of agents. In order to deal with the sheer size, the present paper makes a break with past deterministic methods, and considers the swarm as a statistical ensemble for which guidance can be performed from a probabilistic point of view. The probability‐based approach takes advantage of the law of large numbers, and leads to computationally tractable and implementable swarm guidance laws. Agents following a probabilistic guidance algorithm make statistically independent probabilistic decisions based solely on their own state, which ultimately guides the swarm to the desired density distribution in the configuration space. Two different synthesis methods are introduced for designing probabilistic guidance laws. The first is based on the Metropolis‐Hastings (M‐H) algorithm, and the second is based on using linear matrix inequalities (LMIs). The M‐H approach ensures convergent swarm behavior subject to enforced desired motion constraints, while the LMI approach additionally ensures exponential convergence with a prescribed decay rate, and allows minimization of a cost function that reflects fuel expenditure. In addition, both algorithms endow the swarm with the property of self‐repair, and the capability to strictly enforce zero‐probability keep‐out regions. This last property requires a slight generalization of the Perron‐Frobenius theory, and can be very useful in swarm applications that contain regions where no agents are allowed to go. Simulation examples are given to illustrate the methods and demonstrate desired properties of the guided swarm.

Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment

We consider the problem of controlling a linear system when the state is available with a known time-varying delay (delayed-state feedback control) or the actuator is affected by a delay. The solution proposed in this paper consists in partially assigning the spectrum of the closed-loop system to guarantee the exponential zero-state stability with a prescribed decay rate by means of a finite-dimensional control law. A non conservative bound on the maximum allowed delay for the prescribed decay rate is presented, which holds for both cases of constant and time-varying delays. An advantage over recent and similar approaches is that differentiability or continuity of the delay function is not required. We compare the performance of our approach, in terms of delay bound and input signal, with another recent approach.