Truncated Predictor Feedback Research Articles

Truncated predictive control for delayed cyber–physical systems under deception attacks

In this work, the state tracking, input delay prediction, uncertainty and disturbance rejection problems are studied for the delayed cyber–physical systems under uncertainties, nonlinearities and external disturbances in which the addressed model is described by the fuzzy technique. More specifically, such an input-delayed system is linearized through an interval type-2 fuzzy technique accompanied by a parallel distribution compensation method. Further, the presence of constant input delays in the examined system is proficiently compensated with the assistance of a truncated predictor feedback approach. In parallel, the impacts of external disturbances and uncertainties that are occurred in the system are estimated by means of uncertainty and disturbance estimator method in an efficacious way. With the help of this estimated information, an uncertainty and disturbance estimator-based truncated predictive tracking control protocol is formulated to acquire the state tracking results of the underlying system in the presence of external disturbances and uncertainties. Remarkably, in this study, the interval type-2 fuzzy is blended with the designed reference system. Furthermore, the secure tracking control law is designed for the considered system under the random deception attacks. To address the deception attacks, the Bernoulli stochastic distribution is considered in this study. Moreover, the constraints are expressed for computing the truncated predictive feedback control gain matrices. Conclusively, the numerical simulation analysis is carried out to demonstrate the usefulness of the devised approach.

Scaled Consensus of Exponentially Unstable Networked Systems With Time-Varying Input Delay

This work studies the scaled consensus problems of exponentially unstable networked systems with time-varying input delays. First, a distributed consensus algorithm is designed under the truncated predictor feedback technique. Then, a delay bound is evaluated by utilizing the Lyapunov function-based method. Moreover, the consensus can be guaranteed when the upper bound of the time-varying input delay is no more than the delay bound. Furthermore, the relation between the delay bound and the network synchronizability is established. In other words, better network synchronizability implies a bigger delay bound. Finally, in the simulation examples, the feasibility of the theoretical results is verified, and the actual delay bounds are evaluated.

Event-triggered stabilization of linear input-delay systems with state quantization and unknown time-varying delay: A Razumikhin stability theorem-based approach

Benefiting from the input-to-state stability (ISS) version of the Razumikhin stability theorem, this paper deals with the stabilization of continuous linear input-delay systems with state quantization and without any knowledge about the delay. An event-triggered delay independent truncated predictor feedback (ET-DITPF) controller with a switched feedback control law is proposed to guarantee the globally uniformly asymptotic stability (GUAS)/globally uniformly exponential stability (GUES) of the closed-loop system. As a prominent feature, this controller uses no information about the time delay (neither its value nor its upper bound). In the proposed event-triggered framework, a lower bound for each two consecutive events is derived to show the non-occurrence of the Zeno behavior. Numerical simulations confirm the effectiveness of the obtained theoretical results.

Event-triggered delay-independent truncated predictor feedback stabilization of Lipschitz nonlinear input delay systems with state quantization

This article addresses the event-triggered control design for stabilization of Lipschitz nonlinear systems with time-varying input delay and state quantization constraints. The stabilization is done by an event-triggered delay-independent truncated predictor feedback controller using the information of the upper bound of time-varying input delay (and not its exact value). A sufficient condition on the value of this upper bound in order to guarantee the stability of the closed-loop system under the state quantization constraints is derived. Also, the avoidance of Zeno behavior is proved by finding a lower bound for each two successive events. Various numerical examples illustrate the effectiveness of the obtained results.

Stabilization of discrete-time linear systems with infinite distributed input delays

This paper investigates the stabilization problem of discrete-time linear systems with infinite distributed input delays. A novel framework is adopted to analyze the stability of the concerned systems. Under this framework, two truncated predictor feedback controllers are developed for two classes of discrete-time linear systems with infinite distributed input delays via the low gain method respectively. It is shown that under the designed controllers, those two classes systems are globally exponentially stable. To the best of our knowledge, this is the first time that the stabilization problem of discrete-time linear systems with infinite distributed input delays is considered. Two simulation examples are provided to illustrate the effectiveness of the proposed controllers.

Synchronization of chaotic multi-agent systems with communication delay

In this paper, we analyze the problem of consensus control in chaotic multi-agent systems with communication delays. Each agent is considered to access not only its own information but also the information of neighbors with time delays. This is different to the literature where there is only delay in the information received from the neighbors. Using the truncated predictor feedback approach (TPF) and several lemmas, a linear matrix inequality (LMI), and a Lyapunov function, a consensus control protocol has been derived to ensure synchronization of agents. Our results have also been verified by several simulations demonstrating the capabilities of our control protocol in ensuring the consensus of agents.

Truncated Predictor Feedback Control for Switched Linear Systems Subject to Input Delay and Saturation

Truncated Predictor Feedback Control for Switched Linear Systems Subject to Input Delay and Saturation

Robust consensus of heterogeneous multiagent systems with switching topologies: A time‐delay case

AbstractThe robust leader‐following consensus of heterogeneous multiagent systems under switching communication topologies is investigated in this paper. Especially, the input delay is considered in each follower. In order to access the information of leader through time‐varying communication, a type of distributed dynamic compensator is proposed for every follower firstly, which get rid of the dependence on the global spectrum information and is proved that it can be viewed as a asymptotic observer of the leader. Then combined with the internal model principle, two types of distributed control law are proposed based on the compensator. By applying the truncated predictor feedback scheme to synthesize the closed‐loop systems, it is proved that the consensus can be achieved by both of the control laws despite the existence of the input delay and the plant parameters perturbations. Finally, numerical simulations are illustrated to demonstrate the results.

Stability analysis and predictor feedback control for systems with unbounded delays

This paper proposes a novel general framework with new Lyapunov stability theorems for control problems of systems with unbounded delays. Under this framework, the delayed input can be captured in the stability estimate by augmenting it as a state with initial conditions. Then by applying the novel framework, we consider the stabilization problems of linear systems with distributed unbounded input delays via the so-called predictor feedback controllers. Both time-invariant and time-varying cases are considered. It is shown that under the new framework, the previous works on low-gain based truncated predictor feedback controllers can be improved by removing an implicit assumption on initial conditions. It is also shown that the predictor feedback controllers can exponentially stabilize both time-invariant and time-varying linear systems with distributed unbounded input delays. Several examples are provided to illustrate the effectiveness of the predictor feedback controllers.

Asymptotic stabilization of nonlinear systems with long input delay via memoryless feedback: A linearization method

This paper considers the problem of how to control nonlinear systems with a large delay in the input. A linearization method is presented to achieve local asymptotic stabilization (LAS) via memoryless state and/or output feedback, for multiple-input/multiple-output (MIMO) nonlinear systems with a large input delay. The development of the linearization technique is based on the Razumikhin theorem, a forward completeness lemma of the FDE (functional differential equation) on a closed and bounded interval, and the previous work on truncated predictor feedback control of linear systems with input delay.

Stabilization of linear systems with time‐varying input delay by event‐triggered delay independent truncated predictor feedback

SummaryThis article deals with the problem of stabilization of linear systems with time‐varying input delay by an event‐triggered delay independent truncated predictor feedback law, either of the state feedback type or the output feedback type. Only the information of a delay bound rather than the delay itself is required in the design of both control laws and event‐triggering strategies. For both the state feedback case and the output feedback case, an admissible delay bound that guarantees the stabilizability of a general linear system is established, and the Zeno behavior is shown to be excluded. For linear systems with all open‐loop poles at the origin or in the open left‐half plane, stabilization can be achieved for a delay under an arbitrarily large bound.

Consensus of Multiagent Systems With Time-Varying Input Delay via Truncated Predictor Feedback

This article investigates the consensus tracking of exponentially unstable multiagent systems with time-varying input delay. The truncated predictor feedback approach is utilized for designing delay-dependent state and output feedback protocols. And the explicit conditions that can realize consensus tracking are established in terms of parametric Lyapunov equations and scalar inequalities. Besides, the protocol design algorithms under the maximum allowable delay and the maximum convergence rate are, respectively, provided. Compared with the existing results, the salient characteristic of the current results is to reveal the quantitative relationship among the time delay, unstable plant, network topologies, and the convergence rate. Specifically, for achieving the consensus tracking, the synchronization force from network connectivity needs to dominate the anti-synchronization force from unstable open-loop poles and input delays; the maximum allowable input delay is inversely proportional to the sum of the unstable open-loop poles; the convergence rate becomes smaller as the input delay bounds or/and the sum of the unstable poles in plant increase. Numerical simulations confirm the effectiveness of the proposed theoretical design.

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.

Adaptation in truncated predictor feedback to overcome uncertainty in the delay

SummaryIt is well known in the literature that delay‐adaptive control schemes for general linear systems with input delay, induced from predictor feedback, typically incorporate distributed terms in both the control law and the update law for the delay estimator, leading to the infinite‐dimension characteristic of the schemes. Furthermore, when the full actuator state is not employed, only local regulation has been achieved in the sense that the initial state has to be sufficiently small and, at the same time, the initial guess of the delay has to be sufficiently close to its actual value. In this paper, by restricting our attention to systems without exponentially unstable poles, we avoid implementational issues arising from typical infinite‐dimensional control schemes by developing a finite‐dimensional scheme inspired by the truncated predictor feedback (TPF) design technique. Our scheme achieves global regulation as long as a sufficiently small neighborhood of the actual delay is identified. An adaptation‐free version of the robust TPF, which is inferred from the delay‐adaptive control scheme, is also included to reveal a second level of robustness to the uncertainty in the value of the delay that appears in the state transition matrix of the TPF.

A delay‐independent output feedback for linear systems with time‐varying input delay

SummaryIt is well known that a delay‐dependent or delay‐independent truncated predictor feedback law stabilizes a general linear system in the presence of a certain amount of input delay. Results also exist on estimating the maximum delay bound that guarantees stability. In the face of a time‐varying or unknown delay, delay‐independent feedback laws are preferable over delay‐dependent feedback laws as the former provide robustness to the uncertainties in the delay. In the light of few results on the construction of delay‐independent output feedback laws for general linear systems with input delay, we present in this paper a delay‐independent observer–based output feedback law that stabilizes the system. Our design is based on the truncated predictor feedback design. We establish an estimate of the maximum allowable delay bound through the Razumikhin‐type stability analysis. An implication of the delay bound result reveals the capability of the proposed output feedback law in handling an arbitrarily large input delay in linear systems with all open‐loop poles at the origin or in the open left‐half plane. Compared with that of the delay‐dependent output feedback laws in the literature, this same level of stabilization result is not sacrificed by the absence of the prior knowledge of the delay.

Stability criteria of linear systems with multiple input delays under truncated predictor feedback

The model reduction technique as a feedback design approach for linear systems with input delays achieves finite spectrum assignment of the closed-loop system in a delay free form. The resulting predictor-type feedback law contains a distributed term that involves the convolution between the past input and the state transition matrix and hence causes difficulty in its implementation. For the purpose of easy implementation, the truncated predictor feedback (TPF) simplifies the predictor feedback by discarding the distributed term, and delay independent truncated predictor feedback further eliminates the delay-dependent exponential factor. In this paper, we consider the stabilization of a general linear system with multiple time-varying input delays by TPF or delay independent TPF. Stability criteria expressed in terms of input delays and the feedback parameter are derived through Lyapunov–Krasovskii stability analysis, and then numerically studied under both feedback laws.

Maximum delay bounds of linear systems under delay independent truncated predictor feedback

In a predictor feedback law for a linear system with input delay, the future state is predicted as the state solution of the linear system. The zero input solution contains the transition matrix. The zero state solution gives rise to the distributed nature of the feedback law. In a 2007 IEEE TAC paper, it is established that, when the system is not exponentially unstable, low gain feedback can be designed such that the predictor feedback law, with the distributed term truncated, still achieves stabilization for an arbitrarily large delay. Furthermore, in the absence of purely imaginary poles, the transition matrix in the truncated predictor feedback (TPF) can be safely dropped, resulting in a delay independent TPF law, which is simply a delay independent linear state feedback. In this paper, we first construct an example to show that, in the presence of purely imaginary poles, the linear delay independent TPF in general cannot stabilize the system for an arbitrarily large delay. By using the Lyapunov–Krasovskii Stability Theorem, we derive a bound on the delay under which the delay independent truncated predictor feedback law achieves stabilization for a general system that may be exponentially unstable.

Truncated Predictor Feedback Control for Active Magnetic Bearing Systems With Input Delay

In this brief, we investigate the control of linear systems with input delay with applications to active magnetic bearings (AMBs). This study is motivated by emerging energy explorations that require compressors and other rotating machines to operate in remote locations. The separation between the machine and the control site in these applications, which may extend for several kilometers, introduces a time delay in the transmission of the control signals. Here, we study the effect of input delay in the control of AMBs in remote rotating machines and we present a truncated predictor feedback (TPF) control design to maximize the input delay that the stable closed-loop system can tolerate. The controller corresponding to the maximum input delay is obtained iteratively from the solutions to local linear matrix inequality problems. The capabilities of the designed TPF controller are first verified through simulation and then experimentally validated on an AMB test rig.

Stabilization of exponentially unstable discrete-time linear systems by truncated predictor feedback

Predictor state feedback solves the problem of stabilizing a discrete-time linear system with input delay by predicting the future state with the solution of the state equation and thus rendering the closed-loop system free of delay. The solution of the state equation contains a term that is the convolution of the past control input with the state transition matrix. Thus, the implementation of the resulting predictor state feedback law involves iterative calculation of the control signal. A truncated predictor feedback law results when the convolution term in the state prediction is discarded. When the feedback gain is constructed from the solution of a certain parameterized Lyapunov equation, the truncated predictor feedback law has been shown to achieve asymptotic stabilization of a system that is not exponentially unstable in the presence of an arbitrarily large delay by tuning the value of the parameter small enough. In this paper, we extend this result to exponentially unstable systems. Stability analysis leads to a bound on the delay and a range of the values of the parameter for which the closed-loop system is asymptotically stable as long as the delay is within the bound. The corresponding output feedback result is also derived.

Truncated Prediction Output Feedback Control of a Class of Lipschitz Nonlinear Systems With Input Delay

This brief addresses an output feedback control design for Lipschitz nonlinear systems in the presence of input delay. A nonlinear observer is introduced to estimate the system states. A truncated predictor state feedback, which is the conventional predictor state feedback law with the distributed term dropped, is implemented with the estimated states. Lyapunov-Krasovskii functionals are constructed to establish conditions under which this observer-based truncated predictor feedback law asymptotically stabilize the system at the origin. The effectiveness of the proposed method is demonstrated on the Chua's circuit.