Delay-free System Research Articles

Semi-empirical modelling of the mass flow in a pilot-scale tubular reactor and tracking control

In this paper a model for a pilot-scale tubular reactor is developed using aggregated-parameters. The resulting model is a linear-varying system with time-varying delay. This model is simulated and compared to the results obtained from the experiments performed on the pilot-scale tubular reactor. In order to design a tracking control and considering the characteristics of the resulting model, predictive control techniques are used. The predictor is applied and a delay-free system is obtained. A tracking controller using the block control technique is proposed and is applied during simulation, the results are shown.

Switched systems approach to state bounding for time delay systems

This paper investigates the problems of reachable set estimation for dynamic systems with time-varying delay and random delay. Different from the traditional Lyapunov-Krasovskii functional method, a switched systems approach is adopted to study the reachable set estimation problem for time-varying delay systems. Through augmentation, time-varying delay systems can be transformed into switched delay-free systems. The reachable set of a time-varying delay system is estimated based on its corresponding augmented switched system. The number of decision variables in the obtained reachable set estimation condition is small compared with the Lyapunov-Krasovskii functional method. In addition, the problem of mean square estimation of the system state for random delay systems is also studied. The effectiveness of the theoretical findings is verified through several simulation examples.

A unified Razumikhin-type criterion on input-to-state stability of time-varying impulsive delayed systems

This paper discusses the problem of input-to-state stability (ISS) of time-varying impulsive delayed systems. By introducing a switching parameter and using the notion of average impulsive interval, a unified Razumikhin-type criterion on ISS which is simultaneously effective for stabilizing impulses and destabilizing impulses is derived. The condition which requires the coefficient of the estimated upper bound of the derivative of a Lyapunov function to be constant in the existing results on ISS of impulsive systems is weakened. The results in this paper allow the coefficient of the derivative of a Lyapunov function to be time-varying function which can be both positive and negative and may even be unbounded. Furthermore, the impulsive intervals of an impulsive sequence are allowed to have arbitrarily small lower bound and large enough upper bound simultaneously. As a by-product, a unified criterion on ISS for time-varying impulsive delay-free systems is also presented. Two examples are presented to illustrate the effectiveness of our results.

Delayed output feedback of discrete‐time time‐delay systems with applications to spacecraft rendezvous

This work studies the delayed output feedback (DOF) of discrete-time systems with input and output delays that are arbitrarily large, bounded, and exactly known. The significance of the DOF is that only the present and delayed inputs and outputs are used for feedback. For systems with a single input delay and multiple output delays, the idea of state prediction is utilised to construct the DOF controller. While for systems with multiple input and output delays, the DOF controller is designed by firstly reducing the original system to a delay-free one. It is shown in both cases that the closed-loop systems controlled by DOF are asymptotically stable and act like delay-free systems with the same dimensions as open-loop systems. Compared with the continuous-time case, the DOF controllers for discrete-time systems do not contain distributed delay terms and the implementation problems can be avoided. The DOF is used to stabilise the spacecraft rendezvous system with input and output delays and simulations show the effectiveness of the proposed approach.

New formulation of predictors for finite-dimensional linear control systems with input delay

This paper focuses on a prediction-based control for linear time invariant systems subject to a constant input delay, also known as the Artstein reduction approach. Standardly, this method consists in considering a predicted delay-free system, on which one can design straightforwardly a stabilizing controller. The resulting controller is then defined through an implicit integral equation, involving both the original system state and past values of the input. We propose here an alternative formulation which allows to write explicitly the Artstein transformation, and thus the corresponding controller, in terms of past values of the state only. This formal explicit formulation is the main contribution of the paper.

Performance analysis of robust stable PID controllers using dominant pole placement for SOPTD process models

This paper derives new formulations for designing dominant pole placement based proportional-integral-derivative (PID) controllers to handle second order processes with time delays (SOPTD). Previously, similar attempts have been made for pole placement in delay-free systems. The presence of the time delay term manifests itself as a higher order system with variable number of interlaced poles and zeros upon Pade approximation, which makes it difficult to achieve precise pole placement control. We here report the analytical expressions to constrain the closed loop dominant and non-dominant poles at the desired locations in the complex s-plane, using a third order Pade approximation for the delay term. However, invariance of the closed loop performance with different time delay approximation has also been verified using increasing order of Pade, representing a closed to reality higher order delay dynamics. The choice of the nature of non-dominant poles e.g. all being complex, real or a combination of them modifies the characteristic equation and influences the achievable stability regions. The effect of different types of non-dominant poles and the corresponding stability regions are obtained for nine test-bench processes indicating different levels of open-loop damping and lag to delay ratio. Next, we investigate which expression yields a wider stability region in the design parameter space by using Monte Carlo simulations while uniformly sampling a chosen design parameter space. The accepted data-points from the stabilizing region in the design parameter space can then be mapped on to the PID controller parameter space, relating these two sets of parameters. The widest stability region is then used to find out the most robust solution which are investigated using an unsupervised data clustering algorithm yielding the optimal centroid location of the arbitrary shaped stability regions. Various time and frequency domain control performance parameters are investigated next, as well as their deviations with uncertain process parameters, using thousands of Monte Carlo simulations, around the robust stable solution for each of the nine test-bench processes. We also report, PID controller tuning rules for the robust stable solutions using the test-bench processes while also providing computational complexity analysis of the algorithm and carry out hypothesis testing for the distribution of sampled data-points for different classes of process dynamics and non-dominant pole types.

Robust PID Controller Design for Both Delay-Free and Time-Delay Systems

Robust PID controller design methods are proposed for linear single-input single-output systems. Non-parametric models represented by a finite number of frequency responses are used for them. Sufficient conditions for closed-loop stability are derived based on the Nyquist stability criterion and the sufficient conditions are reduced to convex constraints. Together with the convex constraints and closed-loop model matching problems the robust PID controller design problems are formulated as convex optimization problems. A important feature of the proposed design methods is that they can be applied to delay-free and time-delay systems in the same manner. Moreover, the proposed methods are extended to two-degree-of-freedom PID controller design methods.

Approximation-based Transient Behavior Analysis of Multi-Agent Systems with Delay

This paper proposes an approximation-based analysis method for the asymptotic behavior of a class of multi-agent systems with communication delay among the agents. It is considered that the agents implement the consensus protocol. The communication delay affects the asymptotic behavior of the agents that cannot be analyzed by using the methods developed for ordinary differential equations. The method proposed in this paper explores that, if the delay satisfies a “smallness” condition, then the delayed system has a number of dominant eigenvalues, which can be computed using numerical methods. The result of the proposed approximation method is a delay-free multi-agent system with weighted consensus protocol which has the same order as the original system, and it approximates well the dynamic behavior of the original multi-agent system with communication delay.

Optimal filter for MJL system with delayed modes and observations

This study investigates the optimal filter for Markov jump linear (MJL) system with multiple delayed modes and observations, in which the data losses are introduced naturally and the Markov chains are assumed to be known up to the present time. To deal with the mode and observation delays, a reorganised observation sequence is defined. Moreover, the optimal filter problem is transformed into one of the delay-free system which is just with jumping parameters and multiplicative noises. An optimal MJL filter is presented based on the mean squared method, where the filter gains are determined by solving a set of generalised coupled Riccati difference equations based on a set of coupled Lyapunov equations. Alternatively, an optimal stationary MJL filter is developed, where the filter gains are derived by solving a set of generalised coupled algebraic Riccati equations based on coupled algebraic Lyapunov equations. It can be shown that the difference Riccati equations developed in the former filter converge to the stationary ones under appropriate conditions.

ISS Stabilization of Discrete-time LPV Systems with Interval Time-varying State Delay and Saturating Actuators

Abstract We address the problem of input-to-state stabilization of linear parameter-varying (LPV) discrete-time systems with delayed state and subject to saturated actuators and to l2-limited disturbance. It is assumed that the time-varying delay belongs to a known interval and its maximum variation between two consecutive instants has limited rate. We propose convex delay-dependent conditions for the synthesis of LPV state feedback controllers that do not require the real-time knowledge of the delay and ensure the local input-to-state stability of the closed-loop system for a set of initial conditions and subject to energy bounded disturbance input signals. The approach is based on rewriting the delayed system as a switched uncertain augmented delay-free system with a dead-zone non-linearity and on the application of the generalized sector condition to deal with the saturating actuators. To illustrate the efficiency of our approach, we present a numerical example.

Discrete Near-Optimal Wave Disturbances Rejection Control for Offshore Platforms With Delays in Measurement and Actuator

The wave disturbances rejection control problem for linear discrete jacket-type offshore platforms with delays in measurement and actuator is studied. Taken the delays in measurement and actuator into account, the linear model of the jacket-type offshore platform is established in discrete domain, in which the external wave forces are generated from a designed exosystem based on the JONSWAP wave spectrum and linearized wave theory. A delay-free model from the model of the jacket-type offshore platform with delays in measurement and actuator is formulated based on a designed discrete transformation vector, and the system output with delays in measurement is reformed into a corresponding delay-free form. Then, the discrete near-optimal wave disturbances rejection controller (NOWDRC) is derived from the reformed delay-free system and the corresponding system output in respect of an average quadratic performance index, which is made up of a system state feedback item, a wave disturbances feedforward item and a controller memory item. Finally, simulation results demonstrate that the system output under proposed NOWDRC, including the displacement and velocity of the jacket-type offshore platform, can be offset to smaller values than those under a classic feedback optimal vibration controller, and the delays in measurement and actuator can be compensated effectively.

Predictor‐based output feedback control design for sampled systems with input delay subject to disturbance

In this study, a predictor-based output feedback control design is proposed for sampled systems with input delay subject to disturbance. An extended state observer (ESO) is first constructed to simultaneously estimate the system state and disturbance based on only the output measurement. Then a filtered predictor is constructed by using the estimated state and disturbance to compensate the input delay so as to improve the disturbance rejection performance. For the presence of a constant or asymptotically stable disturbance, a robust H infinity feedback control design based on state prediction and disturbance estimation is proposed to realise no steady-state output error, despite mismatched disturbance. When there exists a time-varying disturbance with deterministic dynamics, such as a sinusoidal type, the proposed predictor can achieve a small prediction error by properly tuning the filter, such that the existing observer-based feedback control methods for delay-free systems can be directly adopted to stabilise such a system with input delay. Moreover, the output error bounds in the presence of time-varying disturbance are quantitatively analysed for using the ESO-based predictor feedback and anti-disturbance predictor feedback, respectively. Two illustrative examples are given to demonstrate the effectiveness and merit of the proposed method.

A generalized smith predictor for unstable time-delay SISO systems

In this work, a generalization of the Smith Predictor (SP) is proposed to control linear time-invariant (LTI) time-delay single-input single-output (SISO) systems. Similarly to the SP, the combination of any stabilizing output-feedback controller for the delay-free system with the proposed predictor leads to a stabilizing controller for the delayed system. Furthermore, the tracking performance and the steady-state disturbance rejection capabilities of the equivalent delay-free loop are preserved. In order to place this contribution in context, some modifications of the SP are revisited and recast under the same structure. The features of the proposed scheme are illustrated through simulations, showing a comparison with respect to the corresponding delay-free loop, which is here considered to be the ideal scenario. In order to emphasize the feasibility of this approach, a successful experimental implementation in a laboratory platform is also reported.

Optimal LQ tracking control for continuous-time systems with pointwise time-varying input delay

This paper investigates the linear quadratic (LQ) tracking problem for linear continuous-time systems with pointwise time-varying input delays. It is assumed that the time-varying input delay takes only one value from a finite set at a given time instant. By defining an indicator, the investigated LQ tracking problem is firstly transformed into a special optimal control problem for continuous-time systems with multiple delays in a single input channel. Then the duality between the LQ tracking for the system with multiple input delay and the smoothing for an associated backward delay free system is established. Finally, an explicit analytical solution to the proposed LQ tracking problem is presented by solving the smoothing problem.

Internally Positive Representations and Stability Analysis of Linear Difference Systems with Multiple Delays

This work introduces the Internally Positive Representation of linear continuous-time difference systems with multiple delays. The technique, originally developed for delay-free linear systems, and recently extended to differential systems with multiple delays, proved to be a useful tool to exploit results available for positive systems also for systems that are not positive. As a result, a sufficient delay-independent stability condition for difference systems with constant delays is given, which is shown to be less conservative than similar existing results.

Distributed Optimal Consensus Control for Multiagent Systems With Input Delay.

This paper addresses the problem of distributed optimal consensus control for a continuous-time heterogeneous linear multiagent system subject to time varying input delays. First, by discretization and model transformation, the continuous-time input-delayed system is converted into a discrete-time delay-free system. Two delicate performance index functions are defined for these two systems. It is shown that the performance index functions are equivalent and the optimal consensus control problem of the input-delayed system can be cast into that of the delay-free system. Second, by virtue of the Hamilton-Jacobi-Bellman (HJB) equations, an optimal control policy for each agent is designed based on the delay-free system and a novel value iteration algorithm is proposed to learn the solutions to the HJB equations online. The proposed adaptive dynamic programming algorithm is implemented on the basis of a critic-action neural network (NN) structure. Third, it is proved that local consensus errors of the two systems and weight estimation errors of the critic-action NNs are uniformly ultimately bounded while the approximated control policies converge to their target values. Finally, two simulation examples are presented to illustrate the effectiveness of the developed method.

Robust stability bounds for multi-delay networked control systems

ABSTRACTIn this paper, the robust stability of a perturbed linear continuous-time system is examined when controlled using a sampled-data networked control system (NCS) framework. Three new robust stability bounds on the time-invariant perturbations to the original continuous-time plant matrix are presented guaranteeing stability for the corresponding discrete closed-loop augmented delay-free system (ADFS) with multiple time-varying sensor and actuator delays. The bounds are differentiated from previous work by accounting for the sampled-data nature of the NCS and for separate communication delays for each sensor and actuator, not a single delay. Therefore, this paper expands the knowledge base in multiple inputs multiple outputs (MIMO) sampled-data time delay systems. Bounds are presented for unstructured, semi-structured, and structured perturbations.

Sliding Mode Control Based On SVM for Fractional Order Time-delay System

An approach of fractional order sliding mode control based on support vector machine (SVM) was proposed to deal with fractional order system affected by both state delays and control delays. The original system is transformed into linear delay-free system by linear transformation to eliminate the impact of time-delay. Furthermore, a fractional-order sliding mode surface is designed and the SVM is applied to adjust the parameters of reaching law in order to overcome the defects in the conventional sliding mode control reaching law parameters should be set in advance. Compared with integer-order sliding mode control, the proposed method not only enables system to reach switching surface within shorter time, but also guarantees faster response, stronger robustness, lower chattering, and the control performance is improved. The simulation results show that the validity of the proposed scheme.

Robust output regulation problem for discrete-time linear systems with both input and communication delays

Recently, the robust output regulation problem for continuous-time linear systems with both input and communication time-delays was studied. This paper will further present the results on the robust output regulation problem for discrete-time linear systems with input and communication delays. The motivation of this paper comes from two aspects. First, it is known that the solvability of the output regulation problem for linear systems is dictated by two matrix equations. While, for delay-free systems, these two matrix equations are same for both continuous-time systems and discrete-time systems, they are different for continuous-time time-delay systems and discrete-time time-delay systems. Second, the stabilization methods for continuous-time timedelay systems and discrete-time time-delay systems are also somehow different. Thus, an independent treatment of the robust output regulation problem for discrete-time time-delay systems will be useful and necessary.

Exponential Stability of Switched Positive Homogeneous Systems

This paper studies the exponential stability of switched positive nonlinear systems defined by cooperative and homogeneous vector fields. In order to capture the decay rate of such systems, we first consider the subsystems. A sufficient condition for exponential stability of subsystems with time-varying delays is derived. In particular, for the corresponding delay-free systems, we prove that this sufficient condition is also necessary. Then, we present a sufficient condition of exponential stability under minimum dwell time switching for the switched positive nonlinear systems. Some results in the previous literature are extended. Finally, a numerical example is given to demonstrate the effectiveness of the obtained results.