Delay-free Case Research Articles

Robust String Stability and Safety of CTH Predictor-Feedback CACC

We establish robustness of string stability to delay uncertainty as well as positivity of spacing and speed states, for homogeneous vehicular platoons under predictor-feedback Cooperative Adaptive Cruise Control (CACC). Each individual vehicle’s dynamics are described by a second-order linear system with delayed desired acceleration, under acceleration information transmitted to the ego vehicle from a single, preceding vehicle. The nominal design (in the delay-free case) is a constant time-headway (CTH) policy and no restriction on the delay size, in relation with the desired time headway, is imposed. The proofs rely on combination of an input-output approach (on the frequency domain) and on deriving estimates on explicit, closed-loop solutions; under specific, sufficient conditions that are derived on initial conditions and parameters of the baseline, CTH controller. We illustrate in simulation and numerical examples, the guarantees of robust stability and string stability as well as of collisions avoidance, of CTH predictor-feedback CACC design. We also present extensions of our design and analysis approach to heterogeneous, third-order dynamic models of vehicles.

Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay

Abstract In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval $[0,2\pi ]$ and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of $L^{2}(0,2\pi )$-norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.

On Semi-Global Exponential Stability Under Sampling for Locally Lipschitz Time-Delay Systems

In this article, the stability preservation problem for locally Lipschitz nonlinear time-delay systems, by emulation of continuous-time dynamic output feedback controllers, is studied. Sufficient Lyapunov-like conditions are provided such that, by suitably fast sampling, the Euler emulation of continuous-time dynamic output feedback controllers ensures the semiglobal exponential stability of the related sampled-data closed-loop system. If other emulation schemes than the Euler one are used, practical semiglobal exponential stability is guaranteed, with arbitrarily small final target ball, by suitably fast sampling. The intersampling system behavior, as well as time-varying sampling intervals, is taken into account. In the provided results, the delay-free case is included as a special case. The practical applicability of the provided results is shown through the study of a nonlinear chemical reactor system with recycle and a glucose–insulin system.

Razumikhin and Krasovskii approaches for safe stabilization

This paper studies the stabilization and safety problems of nonlinear time-delay systems. Following both Razumikhin and Krasovskii approaches, we propose novel control Lyapunov functions/functionals for the stabilization problem and novel control barrier functions/functionals for the safety problem. The proposed control Lyapunov and barrier functions/functionals extend the existing ones from the delay-free case to the time-delay case, and allow for designing the stabilizing and safety controllers in closed-form. Since analytical solutions to time-delay optimal control problems are hard to be achieved, a sliding mode control based approach is developed to merge the proposed control Lyapunov and barrier functions/functionals. Based on the sliding surface functional, a feedback control law is established to investigate the stabilization and safety objectives simultaneously. In particular, the properties of the sliding surface functional are analyzed, and further how to construct the sliding surface functional is discussed. Finally, the proposed approaches are illustrated via two numerical examples from the connected cruise control problem of automotive systems and the synchronization problem of multi-agent systems.

Quantized Stabilization of Continuous-Time Switched Systems With Delay and Disturbance

This article addresses the problem of stabilizing a continuous-time switched system affected by a completely unknown disturbance, data quantization, and time-varying delay. In the sense of combined dwell-time and average dwell-time, it is assumed that the switching is slow enough. Suppose that the bound of delay is known but the one of disturbance is unknown. An estimation for the bound of disturbance is used to counteract the unknown disturbance. By extending the approach of the delay-free case, a communication and control strategy is developed by introducing a virtual system. On this basis, the exponential decay and practical stability of the closed-loop system are guaranteed by using a Lyapunov function. Two examples are illustrated to show the usefulness of the proposed framework for stability analysis of some classes of switched systems.

Predictor-Feedback Prescribed-Time Stabilization of LTI Systems With Input Delay

This article first deals with the problem of prescribed-time stability of linear systems without delay. The analysis and design involve the <i>Bell polynomials</i>, <i>the generalized Laguerre polynomials</i>, <i>the Lah numbers</i>, and a suitable <i>polynomial-based Vandermonde matrix</i>. The results can be used to design a new controller&#x2014;with time-varying gains&#x2014;ensuring prescribed-time stabilization of controllable linear time-invariant (LTI) systems. The approach leads to similar results compared to Holloway <i>et al.</i> 2019, but offers an alternative and compact control design (especially for the choice of the time-varying gains). Based on the preliminary results for the delay-free case, we then study controllable LTI systems with single input and subject to a constant input delay. We design a predictor feedback with time-varying gains. To achieve this, we model the input delay as a transport partial differential equation (PDE) and build on the cascade PDE&#x2013;ordinary differential equation setting (inspired by Krstic 2009) so as the design of the prescribed-time predictor feedback is carried out based on the backstepping approach, which makes use of <i>time-varying kernels</i>. We guarantee the bounded invertibility of the backstepping transformation, and we prove that the closed-loop solution converges to the equilibrium in a prescribed time. A simulation example illustrates the results.

On the External Positivity of SISO Linear Dynamic Systems under a Class of Nonzero and Possibly Negative Initial Conditions Eventually Subject to Incommensurate Point Internal and External Delays

The property of external positivity of dynamic systems is commonly defined as the non-negativity of the output for all time under zero initial conditions and any given non-negative input for all time. This paper investigates the extension of that property for a structured class of initial conditions of a single-input single-output (SISO) linear dynamic system which can include, in general, certain negative initial conditions. The above class of initial conditions is characterized analytically based on the structure of the transfer function. The basic study is performed in the delay-free case, but extensions are then given for systems subject to a finite number of internal and external, in general incommensurate, point delays and for the closed-loop dynamic systems which incorporate a feedback compensator. The formulation relies on calculating the output based on the impulse responses by considering the relation of the mentioned sets of structured initial conditions with the zero-state response which allows to keep the non-negativity of the zero-input response and that of the total response provided the non-negativity for all time of the zero-state response.

Stability Switches and Hopf Bifurcation Induced by Nutrient Recycling Delay in a Reaction-Diffusion Nutrient-Phytoplankton Model

In this paper, we study a reaction-diffusion nutrient-phytoplankton model with nutrient recycling delay and toxin-producing phytoplankton. For the delay-free case, we prove that the release rate of toxic chemicals by the toxin-producing phytoplankton can destabilize the system and cause periodic oscillation. For the time-delayed case, we observe that nutrient recycling delay may bring about stability switches and Hopf bifurcation. Moreover, we derive the formula for determining the direction and stability of the bifurcating periodic solutions. Finally, we give some numerical simulations to support the theoretical analyses.

Integral reinforcement learning-based optimal output feedback control for linear continuous-time systems with input delay

In this paper, an integral reinforcement learning (IRL)-based model-free optimal output-feedback (OPFB) control scheme is developed for linear continuous-time systems with input delay, where the input and past output data are employed rather than the system dynamic model. First, the equivalence between the delayed optimal control and delay-free case is analyzed. Subsequently, the system state is constructed with output signal and the Bellman equation is written in the form of past outputs. Therefore, the IRL algorithm is developed to learn the OPFB control policy, where the iterative policy is evaluated and improved simultaneously. It is proved that the obtained optimal OPFB controller gives the same solution as the optimal state-feedback. Finally, the presented simulation results illustrate the effectiveness of the developed control method.

Decentralized Cooperative Online Estimation With Random Observation Matrices, Communication Graphs and Time Delays

We analyze convergence of decentralized cooperative online estimation algorithms by a network of multiple nodes via information exchanging in an uncertain environment. Each node has a linear observation of an unknown parameter with randomly time-varying observation matrices. The underlying communication network is modeled by a sequence of random digraphs and is subjected to nonuniform random time-varying delays in channels. Each node runs an online estimation algorithm consisting of a consensus term taking a weighted sum of its own estimate and neighbours' delayed estimates, and an innovation term processing its own new measurement at each time step. By stochastic time-varying system, martingale convergence theories and the binomial expansion of random matrix products, we transform the convergence analysis of the algorithm into that of the mathematical expectation of random matrix products. Firstly, for the delay-free case, we show that the algorithm gains can be designed properly such that all nodes' estimates converge to the true parameter in mean square and almost surely if the observation matrices and communication graphs satisfy the stochastic spatio-temporal persistence of excitation condition. Secondly, for the case with time delays, we introduce delay matrices to model the random time-varying communication delays between nodes. It is shown that under the stochastic spatio-temporal persistence of excitation condition, for any given bounded delays, proper algorithm gains can be designed to guarantee mean square convergence for the case with conditionally balanced digraphs.

A New Approach to the Design of Sampled-Data Dynamic Output Feedback Stabilizers

In this article, a new methodology for the design of sampled-data dynamic output feedback stabilizers, by means of Lyapunov–Krasovskii functionals, for nonlinear systems with state-delays, is presented. The notion of dynamic output steepest descent feedback (DOSDF), induced by a general class of Lyapunov–Krasovskii functionals, is introduced. Then, it is shown that DOSDFs, no matter whether continuous or not, are stabilizers in the sample-and-hold sense. The main advantage of the proposed methodology is that all kinds of discontinuities, in the function describing the DOSDF, are here accommodated. This fact greatly enlarges the possibilities of designing sampled-data dynamic output feedback stabilizers for nonlinear systems with state-delays. The intersampling system behavior as well as time-varying sampling intervals are taken into account. The stabilization in the sample-and-hold sense theory is used to prove the results. The proposed methodology is novel as well for the delay-free case, which is here dealt with as a special case.

A Robust Control Approach to Event-Triggered Networked Control Systems With Time-Varying Delays

A robust control approach is proposed to solve the asymptotic stabilization problem of networked control systems, where the data transmission is mediated by a triggering function, and is subject to time-varying transmission delays. We show that the triggering mechanism and the delay effects could be regarded as uncertain components interconnected with a stable linear time-invariant system. We then use two operators to describe the uncertainties, and find the corresponding integral quadratic constraints of the operators. The stability of the networked control system is analyzed from a robust control point of view. Beyond the stability analysis, a co-design algorithm is devised to design the control gain and detection parameters jointly to maximize the inter-event interval as well as the maximum allowable time-delay. Theoretically, the proposed scheme guarantees that out-of-order packet delivery does not happen. The robust control framework allows the algorithm developed for time-varying delays to be applied to the constant delay and the delay free cases. Finally, the theoretical results are demonstrated and verified by numerical simulations.

Finite-Time Stabilization and Energy Consumption Estimation for Delayed Nonlinear Systems

This article concentrates on finite-time stabilization and energy consumption estimation for nonlinear systems with and without delay. By constructing an appropriate controller and utilizing inequality techniques, sufficient conditions are proposed to guarantee the finite-time stability of the delayed nonlinear system. Furthermore, the energy consumption produced in system controlling is estimated by inequality techniques. Then, we formulate similar results for the delay-free case. Finally, numerical examples are presented to demonstrate the effectiveness of our theoretical results.

Standard L1 Stability for Singular Switched Positive Systems with Time-Varying Distributed Delay Under MDRDT Constraint

This paper is concerned with the standard [Formula: see text] stability with respect to singular switched positive systems (SSPSs) with time-varying distributed delays (TVDDs) under mode-dependent ranged dwell time (MDRDT) constraint. Firstly, for SSPSs with TVDDs under MDRDT constraint, a sufficient condition of asymptotical stability is proposed. Next, to analyze disturbance attenuation performance, a sufficient condition of standard [Formula: see text] stability is presented for SSPSs with TVDDs under MDRDT constraint. Via convex optimization approach, the corresponding standard [Formula: see text]-gain performance optimum can be received. In addition, the standard [Formula: see text] stability issue is also extended to the SSPSs in nonsingular case, in delay-free case and in RDT constraint case, respectively. Finally, the validity is proved by giving three examples.

Exponential stability analysis for singular switched positive systems under dwell-time constraints

The article addresses the exponential stability analysis for singular switched positive systems(SSPSs) under dwell-time constraints which include mode-dependent minimum dwell time(MDMDT), mode-dependent constant dwell time(MDCDT) and mode-dependent ranged dwell time(MDRDT) constraints. For SSPSs in both delay-free case and time-varying delay case, a sufficient exponential stability condition is proposed with MDMDT, MDCDT and MDRDT constraints, respectively, and the exponential decay rate can be set as a free parameter based on diverse circumstances. To analyze the dwell-time stability, a novel discretized linear copositive Lyapunov function(DLCLF) approach is introduced in the article, and compared with the general copositive and homogeneous Lyapunov function approach, the main advantage of the DLCLF technique is itself affine dependence on conditions of system matrix, which will be extended and utilized to systems with uncertainties or/and time-varying parameters quite easily. Meanwhile, the proposed condition of exponential stability under MDMDT(or MDCDT or MDRDT) constraint will be degenerated into the one under minimum dwell time(or constant dwell time or ranged dwell time) constraint for some specific situations, which implies that the considered MDMDT(or MDCDT or MDRDT) case is more general and practical. Finally, the validity and significance of the results are illustrated by seven examples.

Accessibility and System Reduction of Nonlinear Time-Delay Control Systems

This article addresses the problem of checking the accessibility property for nonlinear single-input-single-output retarded time-delay systems with constant commensurable delays, described by higher order input-output functional differential equations. Necessary and sufficient conditions are found to check the accessibility property, and sufficient conditions are found to decompose the system equations into accessible and non-accessible subsystems. The conditions are presented in terms of autonomous elements, i.e., functions that cannot be influenced by system input. The results are generalizations from the delay-free case. However, the proofs do not transfer to the time-delay case easily, since algebraic structures with different properties are used here.

Global Stabilization of the Spacecraft Rendezvous System by Delayed and Bounded Linear Feedback

This article investigates the global stabilization problem of the circular orbit rendezvous system with actuator saturation and time-delay. By decomposing the linearized relative motion equations into a cascade of neutral stable linear systems, linear state feedback controllers are proposed in the presence of both actuator saturation and/or time-delay. The global stability of the closed-loop system is proved. Optimal feedback gain is also obtained in the delay-free case. Simulation results are given to show the effectiveness of the presented methods.

Projective Group Consensus of Multi-Agent Systems with Arbitrary Parameter

In this paper, the projective group consensus issue for second order multi-agent systems (MASs) in directed graphs with a dynamic leader is investigated. The proposed projective group consensus with arbitrary parameter includes traditional consensus, reverse group consensus and cluster consensus as its special cases. Novel distributed control protocols are designed to obtain projective group consensus without analyzing signed directed graph as in most current literatures on bipartite consensus problem. On the basis of Lyapunov stability property, algebraic graph and some necessary matrix theory, sufficient conditions for delay and delay-free cases are derived. Finally, simulations of nonlinear chaotic MASs are adopted to testify the theoretical results.

Dwell-time conditions for exponential stability and standard [formula omitted]-gain performance of discrete-time singular switched positive systems with time-varying delays

This paper addresses the dwell-time issues on exponential stability and standard L1-gain performance of discrete-time singular switched positive systems with time-varying delays. The dwell-time constraints refer to mode-dependent minimum dwell time(MDMDT), mode-dependent constant dwell time(MDCDT) and mode-dependent ranged dwell time(MDRDT). At first, a sufficient delay-dependent condition of exponential stability for delayed singular switched positive systems under three different kinds of dwell time constraints: MDMDT, MDCDT and MDRDT, is provided, respectively. Then, considering the disturbance attenuation performance, a sufficient condition of standard L1-gain performance for delayed SSPSs with dwell-time constraints: MDMDT, MDCDT and MDRDT, is proposed, respectively. Meanwhile, the corresponding dwell-time results of exponential stability and standard L1-gain performance for the system in delay-free case is also presented. Four examples are finally presented to show the feasibility and effectiveness of the obtained results.

Delay‐dependent stability of a class of stochastic delay systems driven by G‐Brownian motion

This study discusses delay-dependent stability of a class of stochastic delay systems driven by G-Brownian motion in the sublinear expectation space. With the help of the degenerate Lyapunov functional, the mean square exponential stability and quasi-sure exponential stability criteria for stochastic delay systems driven by G-Brownian motion are established and an explicit upper bound of time delay is derived. In particular, for the delay-free case, the corresponding sufficient conditions are also obtained. Here, the stability conditions are directly related to the coefficients of the stochastic delay systems and are different from the existing stability conditions which are presented in terms of the G-Lyapunov function. Some examples are introduced to illustrate the delay-dependent stability criteria.