On higher‐order truncated predictor feedback for linear systems with input delay

This paper is concerned with the problem of stabilizing a linear system with input delay. Motivated by the first‐order truncated predictor feedback (TPF) approach recently developed by the authors, a general higher‐order TPF controller that contains higher‐order terms of the nominal feedback gains is proposed. It is shown that this higher‐order TPF can also globally and semi‐globally stabilize the concerned time‐delay systems in the absence and in the presence of input saturation, respectively. Safe implementation via numerical approximation of this higher‐order TPF is also established. However, in spite of the fact that the higher‐order TPF utilizes more information of the state, numerical examples have demonstrated that the first‐order TPF outperforms the higher‐order TPF, indicating that the intuition of higher‐order approximation leading to better results is incorrect in this case.Copyright © 2013 John Wiley & Sons, Ltd.     

Observer‐predictor feedback for consensus of discrete‐time multiagent systems with both state and input delays

This article is concerned with the consensus problem for discrete‐time multiagent systems with both state and input delays. Single observer‐predictor‐based protocols and multiple observer‐predictors feedback protocols are simultaneously established to predict the future state such that the input delay that can be arbitrarily large yet bounded is completely compensated. It is shown that the consensus of the multiagent system can be achieved by the single/multiple observer‐predictors feedback protocol. Moreover, sufficient conditions guaranteeing the consensus of the multiagent system are provided in terms of the stability of some simple observer‐error systems, and the separation principle is discovered. Finally, a numerical example is worked out to illustrate the effectiveness of the proposed approaches.     

Compensation of time‐varying input delay for discrete‐time nonlinear systems

We consider general discrete‐time nonlinear systems (of arbitrary nonlinear growth) with time‐varying input delays and design an explicit predictor feedback controller to compensate the input delay. Such results have been achieved in continuous time, but only under the restriction that the delay rate is bounded by unity, which ensures that the input signal flow does not get reversed, namely, that old inputs are not felt multiple times by the plant (because on such subsequent occasions, the control input acts as a disturbance). For discrete‐time systems, an analogous restriction would be that the input delay is non‐increasing. In this work, we do not impose such a restriction. We provide a design and a global stability analysis that allow the input delay to be arbitrary (containing intervals of increase, decrease, or stagnation) over an arbitrarily long finite period of time. Unlike in the continuous‐time case, the predictor feedback law in the discrete‐time case is explicit. We specialize the result to linear time‐invariant systems and provide an explicit estimate of the exponential decay rate. Carefully constructed examples are provided to illustrate the design and analytical challenges. Copyright © 2015 John Wiley & Sons, Ltd.     

Global stabilization of discrete‐time feedforward time‐delay systems by bounded controls

This paper studies the problem of global stabilization of a family of discrete‐time feedforward time‐delay systems with bounded controls. Two classes of nonlinear control laws are established based on a special canonical form of the considered system. The proposed control laws use not only the current states but also the delayed states for feedback and, moreover, contain some free parameters. These advantages can help to improve the transient performance of the closed‐loop system significantly. A practical example is given for illustration.     

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

It 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.     

Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay

The stabilization of exponentially unstable linear systems with time-varying input delay is considered in this paper. We extend the truncated predictor feedback (TPF) design method, which was recently developed for systems with all poles on the closed left-half plane, to be applicable to exponentially unstable linear systems. Assuming that the time-varying delay is known and bounded, the design approach of a time-varying state feedback controller is developed based on the solution of a parametric Lyapunov equation. An explicit condition is derived for which the stability of the closed-loop system with the proposed controller is guaranteed. It is shown that, for the stability of the closed-loop system, the maximum allowable time-delay in the input is inversely proportional to the sum of the unstable poles in the plant. The effectiveness of the proposed method is demonstrated through numerical examples.     

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 $z\neq 1$. Then, a delay bound is provided for the stabilizability of a general linear system by the delay independent truncated predictor feedback.     

Reachable set estimation for discrete‐time linear systems with time delays

This paper is concerned with the reachable set estimation problem for discrete‐time linear systems with multiple constant delays and bounded peak inputs. The objective is to check whether there exists a bounded set that contains all the system states under zero initial conditions. First, delay‐dependent conditions for the solvability of the addressed problem are derived by employing a novel Lyapunov–Krasovskii functional. The obtained conditions are expressed in terms of matrix inequalities, which are linear when only one scalar variable is fixed. On the basis of these conditions, an ellipsoid containing the reachable set of the considered system is obtained. An approach for determining the smallest ellipsoid is also provided. Second, the approach and results developed in the first stage are generalized to the case of systems with polytopic parameter uncertainties, and delay‐dependent conditions are given in the form of relaxed matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust state‐feedback control of stochastic state‐multiplicative discrete‐time linear switched systems with dwell time

Linear discrete‐time switched stochastic systems are considered, where the problems of mean square stability, stochastic l₂‐gain and state‐feedback control design are treated and solved. Solutions are obtained for both nominal and polytopic‐type uncertain systems. In all these problems, the switching obeys a dwell time constraint. In our solution, to each subsystem of the switched system, a Lyapunov function is assigned that is nonincreasing at the switching instants. The latter function is allowed to vary piecewise linearly, starting at the end of the previous switch instant, and it becomes time invariant after the dwell. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain Linear Matrix Inequalities conditions. Based on the solution of the stochastic l₂‐gain problem, we derive a solution to the state‐feedback control design, where we treat a variety of special cases. Being affine in the system matrices, all the aforementioned solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a practical example taken from the field of flight control. Copyright © 2015 John Wiley & Sons, Ltd.     

State and output feedback control of time‐delay switched linear systems

In this paper, we propose contributions on the stabilization and control of switched linear systems subject to time‐delays through the assignment of the switching law. As a first step, based on previous results related to switched linear systems with no time‐delays and exploiting the concept of piecewise quadratic Lyapunov–Krasovskii functionals, we solve the problem of finding suitable state‐dependent switching laws ensuring the prescribed control objectives. Secondly, we extend such results and present a strategy to construct an output feedback switching law, based on the available measurements made on the system. In both cases, the design of the control strategy is done by computing a feasible solution to a set of matrix inequalities associated to the modes of the switched linear system. Copyright © 2011 John Wiley & Sons, Ltd.     

Non‐uniform in time robust global asymptotic output stability for discrete‐time systems

In this paper the notions of non‐uniform in time robust global asymptotic output stability (RGAOS) and input‐to‐output stability (IOS) for discrete‐time systems are studied. Characterizations as well as links between these notions are provided. Particularly, it is shown that a discrete‐time system with continuous dynamics satisfies the non‐uniform in time IOS property if and only if the corresponding unforced system is non‐uniformly in time RGAOS. Necessary and sufficient conditions for the solvability of the robust output feedback stabilization (ROFS) problem are also given. Copyright © 2005 John Wiley & Sons, Ltd.     

Stabilization of stochastic coupled systems with Markovian switching via feedback control based on discrete‐time state observations

This study investigates the stabilization issue of stochastic coupled systems with Markovian switching via feedback control. A state feedback controller based on the discrete‐time observations is applied for the stabilization purpose. By making use of the graph theory and the Lyapunov method, we establish both Lyapunov‐ and coefficient‐type sufficient criteria to guarantee the stabilization in the sense of stability, and then, we further develop the mean‐square asymptotical stability. In particular, the upper bound of the duration between 2 consecutive state observations is well formulated. Applications to a concrete stabilization problem of stochastic coupled oscillators with Markovian switching and some numerical analyses are presented to illustrate and to demonstrate the easy verifiability, effectivity, and efficiency of our theoretical findings.     

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

This 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.     

Stabilization strategy for unstable first order linear systems with large time‐delay

This work considers the problem of stabilization of a class of unstable first order linear systems subject to a large input‐output time delay. Necessary and sufficient conditions are stated to guarantee the stability of the closed loop delayed system by means of a compensation scheme based on two static gains and an induced delay term. The proof of the main result is derived by considering a discrete time approach that, under adequate assumptions, allows to conclude the stability condition for the continuous time case. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Cooperative robust output regulation problem for discrete‐time linear time‐delay multi‐agent systems

In this paper, we study the cooperative robust output regulation problem for discrete‐time linear multi‐agent systems with both communication and input delays by a distributed internal model approach. We first introduce the distributed internal model for discrete‐time multi‐agent systems with both communication and input delays. Then, we define the so‐called auxiliary system and auxiliary augmented system. Finally, we solve our problem by showing, under some standard assumptions, that if a distributed state feedback control or a distributed output feedback control solves the robust output regulation problem of the auxiliary system, then the same control law solves the cooperative robust output regulation problem of the original multi‐agent systems.     

Output feedback preview control for polytopic uncertain discrete‐time systems with time‐varying delay

This paper considers the preview tracking control problem of polytopic uncertain discrete‐time systems with a time‐varying delay subject to a previewable reference signal. First, a model transformation is employed and a discrete‐time system with a time‐invariant delay and an external disturbance is obtained. The difference operator method can be extended to derive an augmented error system that includes future information on the reference signal. Then, a previewable reference signal is fully utilized through reformulation of the output equation while considering the output feedback. Based on the small gain theorem, a static output feedback controller with preview actions is designed such that the output can asymptotically track the reference signal. Finally, numerical simulation examples also illustrate the superiority of the desired preview controller for the uncertain system in the paper.     

Stability analysis and output‐feedback stabilization of discrete‐time systems with a time‐varying state delay and nonlinear perturbation

This paper aims to derive stability conditions and an output‐feedback stabilization method for discrete‐time systems with a time‐varying state delay and nonlinear perturbation. With a new way of handling the Lyapunov stability criterion, linear matrix inequality conditions are obtained for estimating bounds on delay to ensure the asymptotic stability. Based on the conditions, a synthesis procedure is developed for finding stabilizing output‐feedback gains, which are formulated as direct design variables. Three numerical examples are employed to demonstrate the effectiveness and advantages of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Delay‐dependent robust control for singular discrete‐time Markovian jump systems with time‐varying delay

The problem of delay‐dependent robust stabilization for uncertain singular discrete‐time systems with Markovian jumping parameters and time‐varying delay is investigated. In terms of free‐weighting‐matrix approach and linear matrix inequalities, a delay‐dependent condition is presented to ensure a singular discrete‐time system to be regular, causal and stochastically stable based on which the stability analysis and robust stabilization problem are studied. An explicit expression for the desired state‐feedback controller is also given. Some numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Output feedback stabilization for a class of stochastic non‐linear systems with delays in input

In this paper, constructive techniques are developed for a class of stochastic non‐linear systems with delays in input. Non‐linear terms considered in this paper are more general than those satisfying linear growth conditions. The purpose is to design an output feedback controller such that the resulting closed‐loop system is globally asymptotically stable in probability. The desired output feedback controller is explicitly constructed using the Lyapunov method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Stability and stabilization of discrete‐time singular Markov jump systems with time‐varying delay

The stochastic stability and stochastic stabilization of time‐varying delay discrete‐time singular Markov jump systems are discussed. For full and partial knowledge of transition probabilities cases, delay‐dependent linear matrix inequalities (LMIs) conditions for the systems to be regular, causal and stochastically stable are given. Sufficient conditions are proposed for the existence of state feedback controller in terms of LMIs. Finally, two numerical examples to illustrate the effectiveness of the method are given. Copyright © 2009 John Wiley & Sons, Ltd.