Multiple Input Delays Research Articles

Minimum-energy control for time-varying systems with multiple state and input delays

This paper considers the minimum-energy control problem for a class of time-varying systems with multiple state and input delays. First, a state transform matrix is presented. By using the transform matrix, a system with multiple state and input delays is converted to a formal equivalent one without delay. Then, the optimal problem of the novel system is solved by using the maximum principle. Analytical expressions of the optimal control law and optimal performance are given by two formulas with respect to the state transform matrix. At last, an algorithm is given to solve the analytical expression of the state transform matrix. Simulation results show that the design algorithm is efficient and easy to implement.

Stabilisation and consensus of linear systems with multiple input delays by truncated pseudo-predictor feedback

This paper provides a new approach referred to as pseudo-predictor feedback (PPF) for stabilisation of linear systems with multiple input delays. Differently from the traditional predictor feedback which is from the model reduction appoint of view, the proposed PPF utilises the idea of prediction by generalising the corresponding results for linear systems with a single input delay to the case of multiple input delays. Since the PPF will generally lead to distributed controllers, a truncated pseudo-predictor feedback (TPPF) approach is established instead, which gives finite dimensional controllers. It is shown that the TPPF can compensate arbitrarily large yet bounded delays as long as the open-loop system is only polynomially unstable. The proposed TPPF approach is then used to solve the consensus problems for multi-agent systems characterised by linear systems with multiple input delays. Numerical examples show the effectiveness of the proposed approach.

Optimal tracking control for discrete-time systems with multiple input delays under sinusoidal disturbances

This study researches the tracking control problem for discrete-time systems with multiple input delays affected by sinusoidal disturbances. This study is organized around the expression of sinusoidal and disturbances and the delay-free transformation. First, based on the periodic characteristic of the sinusoidal disturbance, the sinusoidal disturbances are considered as the output of an exosystem. By proposing a discrete variable transformation, the discrete-time system with multiple input delays and the quadratic performance index are transformed into equivalent delay-free ones. Then, by constructing an augmented system that comprises the states of the exosystems of sinusoidal disturbances, the reference input, and the delay-free transformation systems, the original tracking problem is transformed into the optimal tracking problem for a delay-free system with respect to the simplified performance index. The optimal tracking control (OTC) law is obtained from Riccati and Stein equations. The existent and uniqueness of the optimal control law is proved. A reduced-order observer is constructed to solve the problem of physically realizable for the items of the reference input and sinusoidal disturbances. Finally, the feasibility and effectiveness of the proposed approaches are validated by numerical examples.

Truncated Predictor Feedback Stabilization of Polynomially Unstable Linear Systems With Multiple Time-Varying Input Delays

This technical note is concerned with stabilization of polynomially unstable linear systems with multiple time-varying input delays. A truncated predictor feedback (TPF), which is obtained by truncating the distributed terms in the traditional predictor feedback, is proposed. The proposed TPF is a linear static partial state feedback with a time-varying feedback gain that is constructed from the steady solution to a class of parametric Lyapunov differential equations. Under the assumption that the open-loop system is only polynomially unstable and is uniformly controllable, it is shown that, for sufficiently small value of the design parameter, the resulting closed-loop system is exponentially stable no matter how large the time-varying delays are as long as they are bounded. Simulation demonstrates these theoretical results.

Lyapunov–Krasovskii functionals for predictor feedback control of linear systems with multiple input delays

This paper is concerned with the Lyapunov–Krasovskii functional construction of linear control systems with multiple input delays. By transforming the predictor feedback control systems into a delay-free linear system with external inputs, a Lyapunov–Krasovskii functional is constructed in terms of a set of linear matrix inequalities (LMIs). It is shown that the solvability of this set of LMIs is equivalent to the asymptotic stability of the delay-free linear system induced from the predictor feedback control system. The proposed Lyapunov–Krasovskii functional is also found to be an ISS Lyapunov–Krasovskii functional for the predictor feedback control systems. An example is worked out to validate the effectiveness of the proposed method.

Observer-based synchronization of memristive systems with multiple networked input and output delays

This paper investigates the synchronization problem of memristive systems with multiple networked input and output delays via observer-based control. A memristive system is set up, and the fuzzy method has been employed to linearize the dynamical system of the memristive system; the networked input and output delays are considered in the synchronization problem of this system. A truncated predictor feedback approach is employed to design the observers. Under certain restrictions, a class of finite-dimensional observer-based output feedback controllers is designed. A numerical example is carried out to demonstrate the effectiveness of the proposed methods.

Stabilization of Discrete-Time Linear Systems Subject to Input Saturation and Multiple Unknown Constant Delays

In this technical note, we study the semi-global stabilization of general linear discrete-time critically unstable systems subject to input saturation and multiple unknown input delays. Based on a simple frequency-domain stability criterion, we find upper bounds for delays that are inversely proportional to the argument of open-loop eigenvalues on the unit circle. For delays satisfying these upper bounds, linear low-gain state and finite dimensional dynamic measurement feedbacks are constructed to solve the semi-global stabilization problems.

Observer-based output feedback control of discrete-time linear systems with input and output delays

In this paper, we study observer-based output feedback control of discrete-time linear systems with both multiple input and output delays. By generalising our recently developed truncated predictor feedback approach for state feedback stabilisation of discrete-time time-delay systems to the design of observer-based output feedback, two types of observer-based output feedback controllers, one being memory and the other memoryless, are constructed. Both full-order and reduced-order observer-based controllers are established in both the memory and memoryless schemes. It is shown that the separation principle holds for the memory observer-based output feedback controllers, but does not hold for the memoryless ones. We further show that the proposed observer-based output feedback controllers solve both the l2 and l∞ semi-global stabilisation problems. A numerical example is given to illustrate the effectiveness of the proposed approaches.

Input delay compensation of linear systems with both state and input delays by nested prediction

In this paper we consider the input delay compensation of linear systems with both state and input delays. Under the condition that the open-loop time-delay systems in the absence of input delay can be stabilized by (delayed) state feedback, a nested predictor based approach is established to predict the future states so that the delays in the input are compensated. It is shown that the compensated closed-loop system in the presence of input delay possesses the same characteristic equation as the closed-loop system in the absence of input delay. The proposed approach can be generalized to general linear systems having multiple state delays and/or multiple input delays. To implement the nested predictor based controllers by applying numerical integration approaches, an augmented nested predictor based approach by adding input filters is proposed. Two numerical examples are worked out to show the effectiveness of the proposed approach.

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be "smaller" than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.

Stabilization of linear system with input saturation and unknown constant delays

In this paper, we study the stabilization of linear critically unstable systems subject to input saturation and multiple unknown input delays. We find tight upper bounds for delays which are inversely proportional to the maximal magnitude of open-loop eigenvalues on the imaginary axis. For delays satisfying these upper bounds, linear low-gain state and finite dimensional dynamic measurement feedbacks are constructed to solve the semi-global stabilization problem. The effectiveness of the proposed design is illustrated by an example.

Observer based output feedback control of linear systems with input and output delays

This paper is concerned with observer based output feedback control of linear systems with both (multiple) input and output delays. Our recently developed truncated predictor feedback (TPF) approach for state feedback stabilization of time-delay systems is generalized to the design of observers. By imposing some restrictions on the open-loop system, two classes of observer based output feedback controllers, one being finite dimensional and the other infinite dimensional, are constructed. It is further shown that, the infinite dimensional observer based output feedback controllers can be generalized to linear systems with both time-varying input and output delays. It is also shown that the separation principle holds for the infinite dimensional observer based output feedback controllers, but does not hold for the finite dimensional ones. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches.

Integral sliding mode controller design for congestion problem in ATM networks

A new integral sliding mode control strategy for high-speed networks operating in asynchronous transfer mode (ATM) is derived to cope with the congestion problem in network flow control. The novelty of the proposed design solution stem from allowing high-speed networks to have simultaneously acting multiple time-varying input delays, uncertainties and time-varying perturbations. A delay-dependent condition is derived via the linear matrix inequality approach that guarantees both the asymptotic stability and a prescribed H ∞-performance level of the closed-loop system in sliding mode dynamics when perturbations of available bit-rate bandwidth occur. These new results are obtained with no restriction on the derivative of the time-varying delays hence are less conservative than the existing ones. This control scheme does achieve both of the main goals in ATM network traffic, namely convergence of the queue length to the desired steady-state status and the weighted fairness condition. The application to benchmark bottleneck example and the simulation results demonstrate the applicability, efficiency and robustness of the proposed synthesis method.

Multiple Input Delays Estimation Using an Artificial Bee Colony Algorithm

This paper focuses on the time delay estimation of the system described in the form of discrete-time state equation with multiple input delays. To estimate the input delays, a new evolutionary computation called the artificial bee colony (ABC) algorithm is utilized. This algorithm is originally motivated from the social behaviors of honeybee organization, and it has been proven to be a powerful means for solving the optimized problem. Based on the proposed algorithm, the unknown system input delays can be further solved by minimizing a quadratic cost function of the system. Two illustrative examples are provided to verify the potential of the presented method in the time delay estimation. Some simulations containing different initial condition examinations and appearance of noises are further given. Numerical results show that the proposed method can do well in the multiple inputs delay estimation of discrete-time state equations.

Time delay estimation in discrete-time state-space models

This paper proposes a method for estimating multiple input delays in systems described by discrete-time state-space models. The proposed method is based on the minimization of a quadratic cost function related to the prediction error of the state variables over a given time frame. It is assumed that the system inputs are known and that noisy measurements of the states are available. The statistical properties of the resulting estimate are used to determine the delay values that are consistent with the observations at a given significance level. Simulation experiments are presented for illustration. As expected, the actual error rate is found to be consistent with the significance level employed in the estimation procedure. The relation between the precision of the estimation and the noise level is also investigated.

Exact Stability Analysis of a Second-Order Leaderless Consensus Protocol with Multiple Communication and Input Delays

An earlier investigation on a leaderless consensus protocol for double integrators with multiple time delays (Meng et al., 2011) is revisited in this paper from two completely different and novel perspectives. First, the crucial stability analysis of time delayed system is replaced with a recent technique called the Cluster Treatment of Characteristic Roots (CTCR). CTCR paradigm is pursued after a block-diagonalization (mode decoupling) transformation on the system. This treatment produces the unique stability tables for the dynamics in the space of the delays. Furthermore they are exact and exhaustive. Secondly, the novel concept of Spectral Delay Space is presented, as an overture to the CTCR for the determination of the needed potential stability crossing (switching) hypersurfaces in the delay space. Examples cases are provided to display the strengths, efficiency and explicitness of this new stability analysis mechanism.

시간 지연 연속 시간 시스템의 디지털 모델링

Control Theory for continuous-time system has been well developed. Due to the development of computer technology, digital control scheme are employed in many areas. When delays are in control systems, it is hard to control the system efficiently. Delays by controller-to-actuator and sensor-to-controller deteriorate control performance and could possibly destabilize the overall system. In this paper, a new approximated discretization method and digital design for control systems with multiple state, input and output delays and a generalized bilinear transformation method with a tunable parameter are also provided, which can re-transform the integer time-delayed discrete-time model to its continuous-time model. Illustrative example is given to demonstrate the effectiveness of the developed method.

Digital modeling and control of multiple time-delayed distributed power grid

Distributed power grid (DPG) control systems are so highly interconnected that the effects of local disturbances as well as transmission time delays can be amplified as they propagate through a complex network of transmission lines. These effects deteriorate control performance and could possibly destabilize the overall system. In this paper, a new approximated discretization method and digital design for DPG control systems with multiple state, input and output delays as well as a generalized bilinear transformation method are presented. Based on a procedure for the generation of impulse response data, the multiple fractional/integer time-delayed continuous-time system is transformed to a discrete-time model with multiple integer time delays. To implement the digital modeling, the singular value decomposition (SVD) of a Hankel matrix together with an energy loss level is employed to obtain an extended discrete-time state space model. Then, the extended discrete-time state space model of the DPG control system is reformulated as an integer time-delayed discrete-time system by computing its observable canonical form. The proposed method can closely approximate the step response of the original continuous time-delayed DPG control system by choosing various energy loss levels. For completeness, an optimal digital controller design for the DPG control system and a generalized bilinear transformation method with a tunable parameter are also provided, which can re-transform the integer time-delayed discrete-time model to its continuous-time model. Illustrative examples are given to demonstrate the effectiveness of the developed method.

Delay-dependent [formula omitted] control of linear systems with multiple time-varying state and input delays

This paper is concerned with the L 2 – L ∞ controller design problem for delayed linear systems. We first propose a novel class of continuous-time multiple delayed linear systems. The time-delays are assumed to be time-varying continuously differentiable functions, which appear in the state and control input simultaneously. By using the Lyapunov–Krasovskii functional approach and Jensen integral inequality technique, a sufficient condition for the existence of an L 2 – L ∞ controller is formulated in the form of linear matrix inequalities (LMIs). When these LMIs are feasible, the L 2 – L ∞ controller is explicitly presented. Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the developed controller design procedures.

Indefinite linear quadratic optimal control problem for singular discrete-time system with multiple input delays

An indefinite linear quadratic (ILQ) optimal control problem is discussed for singular discrete-time-varying linear systems with multiple input delays. The problem is transformed to the one for standard systems by normalizability decomposition. An explicit controller is obtained by computing the gain of the smoothing estimation of dual systems. Necessary and sufficient conditions guaranteeing the existence of unique solution are given simultaneously. A numerical example illustrates the presented method.