Control Of Linear Time-delay Systems Research Articles

Learning-based adaptive optimal control of linear time-delay systems: A value iteration approach

This paper proposes a novel learning-based adaptive optimal controller design method for a class of continuous-time linear time-delay systems. A key strategy is to exploit the state-of-the-art reinforcement learning (RL) techniques and adaptive dynamic programming (ADP), and propose a data-driven method to learn the near-optimal controller without the precise knowledge of system dynamics. Specifically, a value iteration (VI) algorithm is proposed to solve the infinite-dimensional Riccati equation for the linear quadratic optimal control problem of time-delay systems using finite samples of input-state trajectory data. It is rigorously proved that the proposed VI algorithm converges to the near-optimal solution. Compared with the previous literature, the nice features of the proposed VI algorithm are that it is directly developed for continuous-time systems without discretization and an initial admissible controller is not required for implementing the algorithm. The efficacy of the proposed methodology is demonstrated by two practical examples of metal cutting and autonomous driving.

Dissipative control for linear time-delay systems based a modified equivalent-input-disturbance approach

Dissipative control for linear time-delay systems based a modified equivalent-input-disturbance approach

A Value Iteration Approach to Adaptive Optimal Control of Linear Time-Delay Systems

This paper studies the adaptive optimal control for linear time-delay systems described by delay differential equations (DDEs). A key strategy is to exploit the value iteration (VI) approach to solve the linear quadratic optimal control problem for time-delay systems. However, previous learning-based control methods are all exclusively devoted to discrete-time time-delay systems. In this article, we aim to fill in the gap by developing a learning-based VI approach to solve the infinite-dimensional algebraic Riccati equation (ARE) for continuous-time time-delay systems. One nice feature of the proposed VI approach is that an initial admissible controller is not required to start the algorithm. The efficacy of the proposed methodology is demonstrated by the example of autonomous driving.

Application of periodic Bernoulli polynomials in approximating the solution of linear time-delay optimal control problems

In this paper, a numerical method based on application of periodic Bernoulli polynomials for solving optimal control of linear time-delay systems with quadratic performance index is presented. Some important properties of Bernoulli polynomials are reviewed and a new operational matrix of delay based on these functions is introduced. Also an error bound for approximating a function with periodic Bernoulli polynomials is presented. Using operational matrices of differentiation, delay and the integration of the cross product of two Bernoulli polynomials, the linear time delay optimal control problem with quadratic objective function is converted to a nonlinear optimization problem. Finally, numerous illustrative examples are studied to show the efficiency and accuracy of the proposed method compared with some other methods.

New results on robust sliding mode control for linear time-delay systems

AbstractThis work focuses on the sliding mode control (SMC) for a family of linear systems with uncertainties and time-varying delays. First, an integral switching surface is constructed to verify the robust asymptotic stability of the considered system and the results are extended to analyse the mixed $\mathscr{H}_{\infty }\big /$Passivity performance index. Thereafter, a suitable SMC law is developed to force the system state onto the predefined switching surface in short time. By using Lyapunov stability theory, some novel results are obtained, and the required stability conditions are established in terms of linear matrix inequalities which can be solved by standard Matlab toolbox. Finally, the results are validated over a Chua’s circuit model, which describes the practical application of the developed results.

Explicit Reference Governor for the Constrained Control of Linear Time-Delay Systems

This paper introduces an explicit reference governor to supervise closed-loop linear time-delay systems. The proposed scheme enforces state and input constraints by modifying the reference of the supervised system so that the state vector always belongs to admissible sublevel sets of a suitably defined Lyapunov–Krasovskii functional. To accomplish this, this paper extends the existing definition of “dynamic safety margin” to a time-delay setting and illustrates how to employ classic Lyapunov–Krasovskii functionals even though the reference is time varying. Constraint enforcement for arbitrary reference signals and asymptotic convergence to any strictly steady-state admissible set point is rigorously proven. Experimental results are reported to demonstrate the simplicity, practicality, and robustness of the proposed method.

A finite frequency domain-based approach for active fault tolerant control of linear time-delay systems with residual feedback

This paper presents a finite frequency-based active fault tolerant control approach for the compensation of unknown failures in linear time-delay plants. An integration of fault detection filter based on observer technology and [Formula: see text] controller in residual feedback form is considered in the active fault tolerant control system. Different from the traditional schemes, exact fault estimation is not necessary in the proposed active fault tolerant control. To achieve the desired system performance when a fault occurs, the residual is directly embedded in the control loop as a feedback term to compensate the influence of fault. By employing the Generalized Kalman–Yakubovich–Popov lemma, we derive the sufficient conditions of the existence of such an active fault tolerant control plant, and iterative algorithms are applied to obtain the solutions to the fault detection filter and controller parameter matrices. Finally, simulation results are proposed to demonstrate the effectiveness of the developed scheme.

Accessibility of Nonlinear Time-Delay Systems

A full characterization of accessibility is provided for nonlinear time-delay systems. It generalizes the rank condition which is known for weak controllability of linear time-delay systems, as well as the celebrated geometric approach for delay-free nonlinear systems and the characterization of their accessibility. Besides, fundamental results are derived on integrability and basis completion which are of major importance for a number of general control problems for nonlinear time-delay systems. They are shown to impact preconceived ideas about canonical forms for nonlinear time-delay systems.

A novel feedforward–feedback suboptimal control of linear time-delay systems

This paper presents a new approach for solving the optimal control problem of linear time-delay systems with a quadratic cost functional. In this approach, a method of successive substitution is employed to convert the original time-delay optimal control problem into a sequence of linear time-invariant ordinary differential equations (ODEs) without delay and advance terms. The obtained optimal control consists of a linear state feedback term and a forward term. The feedback term is determined by solving a matrix Riccati differential equation. The forward term is an infinite sum of adjoint vectors, which can be obtained by solving recursively the above-mentioned sequence of linear non-delay ODEs. A fast-converging iterative algorithm for this purpose is presented which provides a promising possible reduction of computational efforts. Numerical examples demonstrating the efficiency, simplicity and high accuracy of the suggested technique have been included. Simulation results reveal that just a few iterations of the proposed algorithm are required to find an accurate enough feedforward–feedback suboptimal control.

Assessment of He's homotopy perturbation method for optimal control of linear time-delay systems

This paper renders an optimal regulator for a system with state time-delay and quadratic performance index. First, by means of Maximum Principle and necessary optimal conditions, a continues-time two-point boundary-value problem (TPBVP) including both time-delay and time-advance is derived. Then, using Homotopy Perturbation Method (HPM), the obtained TPBVP is transformed into a sequence of linear time-invariant TPBVP without any time-delay or timeadvance. Solving the resulted linear TPBVP sequence in a recursive manner, concludes that the optimal control law in the form of rapid convergent series. At the end, an iterative algorithm with low complexity in computation and fast convergence rate is proposed to achieve an accurate enough suboptimal control law. It's worth noting that simulation based on application of a harmonic oscillator is conducted to demonstrate efficiency of this method.

Direct data‐driven control of linear time‐delay systems

AbstractThis paper presents an extension of the Virtual Reference Feedback Tuning (VRFT) methodology dedicated to linear time‐delay systems with known delay and unknown dynamics. The standard VRFT is not well suited for systems with dominant time‐delay as it yields high order controllers. The proposed direct approach, relying on a Smith Predictor structure, guarantees the same level of performance as the standard VRFT but with lower order controllers. The joint direct data‐driven design of the controller and the predictor is facilitated by the introduction of an ad‐hoc optimization initialization. Effectiveness and robustness to uncertainty in the time‐delay estimation are shown in a vehicle dynamics control problem.Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Robust adaptive control of linear time-delay systems with point time-varying delays via multiestimation

This paper presents an adaptive pole-placement based controller for continuous-time linear systems with unknown and eventually time-varying point delays under uncertainties consisting of unmodeled dynamics and eventual bounded disturbances. A multiestimation scheme is designed for improving the identification error performance and then to deal with possibly errors between the true basic delay compared to that used in regressor vector of measurements of the adaptive scheme and also to prevent the closed-loop system against potential instability. Each estimation scheme in the parallel disposal possesses a relative dead-zone which freezes the adaptation process for small sizes of the adaptation error compared to the estimated size of the absolute value of the contribution of the uncertainties to the filtered output versus time. All the estimation schemes run in parallel but only that which is currently in operation parameterizes the adaptive controller to generate the plant input at each time. A supervisor chooses the appropriate estimator in real time which respects a prescribed minimum residence time at each estimation algorithm in operation. That strategy is the main tool used to ensure the closed-loop stability under estimates switching. The relative dead-zone in the adaptation mechanism prevents the closed-loop system against potential instability caused by uncertainties.

Optimal on-line control of linear time-delay systems

In this paper a problem of synthesis of optimal feedbacks for a constrained linear time-delay system is considered. A novel concept of a state of the optimal control problem is introduced. The state of the problem allows to define a feedback as a function of a finite-dimensional position of the problem rather than a functional of the infinite-dimensional position of the time-delay system. Two types of optimal feedbacks are studied: one is constructed on the base of the deterministic model, the second one explicitly takes into account the additive disturbances in the model of the system. Algorithms of constructing in real time realizations of both types of feedbacks are justified. Copyright © 2007 IFAC.

Adaptive reliable H ∞ control for linear time-delay systems via memory state feedback

The reliable controller design problem for continuous-time linear systems with time delay and actuator faults is considered, based on a linear matrix inequality (LMI) technique and an adaptive method. A new delay-dependent memory state-feedback reliable controller is established in a parameter-dependent form, in which fault parameters are adjusted online based on an adaptive method to compensate automatically the fault effect on system. In the framework of the LMI technique, the stability and H∞ performance of closed-loop systems are guaranteed in normal and faulty cases. A numerical example and its simulations are given to illustrate the effectiveness of the proposed method.

RETRACTED: Observer-based passive control of linear time-delay systems with parametric uncertainty

This article has been retracted at the request of the Authors. Please see Elsevier Policy on Article Withdrawal ( http://www.elsevier.com/locate/withdrawalpolicy ). Reason: The authors have plagiarized part of a paper that had already appeared in Syst. Control Lett. , 42 (2001) 303–312, doi: 10.1016/S0167-6911(00)00100-6 . One of the conditions of submission of a paper for publication is that authors declare explicitly that their work is original and has not appeared in a publication elsewhere. Re-use of any data should be appropriately cited. As such this article represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and we apologize to readers of the journal that this was not detected during the submission process.

On [formula omitted] control for linear systems with interval time-varying delay

This paper deals with the problem of delay-dependent robust H ∞ control for linear time-delay systems with norm-bounded, and possibly time-varying, uncertainty. The time-delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, and no restriction on the derivative of the time-varying delay is needed, which allows the time-delay to be a fast time-varying function. Based on an integral inequality, which is introduced in this paper, and Lyapunov–Krasovskii functional approach, a delay-dependent bounded real lemma (BRL) is first established without using model transformation and bounding techniques on the related cross product terms. Then employing the obtained BRL, a delay-dependent condition for the existence of a state feedback controller, which ensures asymptotic stability and a prescribed H ∞ performance level of the closed-loop systems for all admissible uncertainties, is proposed in terms of a linear matrix inequality (LMI). A numerical example is also given to illustrate the effectiveness of the proposed method.

General receding horizon control for linear time-delay systems

A general receding horizon control (RHC), or model predictive control (MPC), for time-delay systems is proposed. The proposed RHC is obtained by minimizing a new cost function that includes two terminal weighting terms, which are closely related to the closed-loop stability. The general solution of the proposed RHC is derived using the generalized Riccati method. Furthermore, an explicit solution is obtained for the case where the horizon length is less than or equal to the delay size. A linear matrix inequality (LMI) condition on the terminal weighting matrices is proposed, under which the optimal cost is guaranteed to be monotonically non-increasing. It is shown that the monotonic condition of the optimal cost guarantees closed-loop stability of the RHC. Simulations demonstrate that the proposed RHC effectively stabilizes time-delay systems.

Memory observer-based output feedback control for linear time-delay systems

This paper considers synthesis problems of stabilizing dynamic output feedback controllers for linear time-delay systems via infinite-dimensional Linear Matrix Inequality (LMI) approach. We derive an existence condition and an explicit formula of dynamic output feedback controllers for linear time-delay systems, which guarantee the internal stability of the closed loop systems. The derived dynamic output feedback controllers can be interpreted as controllers which consist of memory state feedback controllers and memory observers. Next, we introduce a technique to reduce conditions for synthesis in the form of infinite-dimensional LMIs to a finite number of LMIs, and present a feasible algorithm for synthesis of controllers based on the finite-dimensional LMIs. Finally we demonstrate the efficacy of the proposed dynamic output feedback controllers by a numerical case study.

Comments and further results on "A descriptor system approach to H/sub /spl infin// control of linear time-delay systems"

This paper presents some comments and further results concerning the descriptor system approach to H/sub /spl infin// control of linear time-delay systems. Upon the system model of the paper by Fridman and Shaked (2001), we propose a new iterative state-feedback controller design procedure, which is based on a new bounded real lemma derived upon an inequality recently proposed by Moon (2001). The proposed design solves both the instantaneous and delayed feedback problems in a unified framework, and is illustrated by a numerical example to be much less conservative than the above paper and other relevant references.

Delay-dependent dissipative control for linear time-delay systems

This paper deals with the problem of delay-dependent dissipative control for a class of linear time-delay systems. We develop the design methods of dissipative static state feedback and dynamic output feedback controllers such that the closed-loop system is quadratically stable and strictly ( Q, S, R)-dissipative. Sufficient conditions for the existence of the quadratic dissipative controllers are obtained by using linear matrix inequality (LMI) approach. Furthermore, a procedure of constructing such controllers from the solutions of LMIs is given. It is shown that the solvability of a dissipative controller design problem is implied by the feasibility of LMIs. The main results of this paper unify the existing results on H ∞ control and passive control.