Rate of convergence for the Legendre pseudospectral optimal control of feedback linearizable systems

Pseudospectral (PS) computational methods for nonlinear constrained optimal control have been applied to many industrial-strength problems, notably, the recent zero-propellant-maneuvering of the international space station performed by NASA. In this paper, we prove a theorem on the rate of convergence for the optimal cost computed using a Legendre PS method. In addition to the high-order convergence rate, two theorems are proved for the existence and convergence of the approximate solutions. Relative to existing work on PS optimal control as well as some other direct computational methods, the proofs do not use necessary conditions of optimal control. Furthermore, we do not make coercivity type of assumptions. As a result, the theory does not require the local uniqueness of optimal solutions. In addition, a restrictive assumption on the cluster points of discrete solutions made in existing convergence theorems is removed.     

An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type

An eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented, which extends the recently developed continuous pole placement method for delay equations of retarded type. The approach consists of two steps. First the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots, which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation. No restrictions are made on the dimension of the system and the number of delays. Also the effect of small delay perturbations is explicitly taken into account. As a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots. As a second step, stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi-continuous way, by applying small changes to the controller parameters, and meanwhile monitoring other characteristic roots with a large real part. A numerical example is presented.     

Stability of PID-controlled second-order time-delay feedback systems

In the previous papers, the stability of PID-controlled first-order time-delay systems has been investigated by means of several methods, of which the Nyquist criterion, a generalization of the Hermite–Biehler Theorem, and the root location method are well known. Explicit expressions of the boundaries of the stability region, which is the set of controller parameters that give stable closed-loop systems, have been determined. From these studies, one can verify that not all plants can be made stable and then obtain the set of process parameters that allow stable closed-loop systems. With this set, one can implement the stability region of the process parameters. In a recent paper the stability conditions based on Pontryagin’s studies and valid for arbitrary-order plants have been presented. The procedure deduced for the controller parameters is exhaustive, but that deduced for the process parameters requires further mathematical evaluations, whose complexity is proportional to the number of process time constants. In the aforementioned recent paper these evaluations have been performed for a second-order time-delay plant whose transfer function has no zero. The aim of this paper is to execute these calculations for a second-order plant whose transfer function has one zero and to provide the related stability region.     

Further control synthesis for time-delay systems with actuator saturation

This paper is concerned with the control synthesis problem for systems with time-varying delay and actuator saturation. A new controller design method is proposed in which auxiliary feedback matrix method is adopted to handle the saturation term in the system. The improvement of the proposed method lies in the application of delay partitioning idea to further enlarge the estimated domain of attraction. All the results are given in terms of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness and superiority of our obtained results.     

A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.     

Design of fixed-point smoother algorithm for linear discrete time-delay systems

This paper investigates the fixed-point smoothing problems for linear discrete-time systems with multiple time-delays in the observations.The linear discrete-time systems considered have l + 1 output channels.One is instantaneous observation and the others are delayed.The fixed-point smoothers involving recursive algorithm and non-recursive algorithm are designed by using innovation analysis theory without relying on the system augmentation approach.Also, it is further shown that the design of fixed-point smoother comes down to solving l + 1 Riccati equations with the same dimensions as the original systems.     

Lyapunov stability tests for linear time-delay systems

An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proofs are presented in details for the case of systems with multiple delays. The state of the art, ongoing research and potential extensions to other classes of delay systems are discussed.     

Robust stability of interval time-delay systems with delay-dependence

This paper proposes a sufficient robust stability condition for interval time-delay systems with delay-dependence. The properties of the comparison theorem and matrix measure are employed to investigate the problem. The stability criteria are delay-dependent and less conservative than delay-independent stability criteria [5] , [6] , [12] and [16] and delay-dependent stability criteria 1, [14] , [15] and [17] when delay is small. However, the results of this paper indeed give us one more choice for the stability examination of the interval time-delay systems. Simulation examples are given to demonstrate the application of our result.     

A modified Chebyshev pseudospectral DD algorithm for the GBH equation

In this paper, a Chebyshev spectral collocation domain decomposition (DD) semi-discretization by using a grid mapping, derived by Kosloff and Tal-Ezer in space is applied to the numerical solution of the generalized Burger’s-Huxley (GBH) equation. To reduce roundoff error in computing derivatives we use the above mentioned grid mapping. In this work, we compose the Chebyshev spectral collocation domain decomposition and Kosloff and Tal-Ezer grid mapping, elaborately. Firstly, the theory of application of the Chebyshev spectral collocation method with grid mapping and DD on the GBH equation is presented. This method yields a system of ordinary differential algebraic equations (DAEs). Secondly, we use a fourth order Runge-Kutta formula for the numerical integration of the system of DAEs. Application of this modified method to the GBH equation show that this method (M-DD) is faster and more accurate than the standard Chebyshev spectral collocation DD (S-DD) method.     

线性时滞系统稳定性分析综述

时滞在工程领域广泛存在,对此综述了线性时滞系统的稳定性研究方法.从频域和时域两个角度详细介绍了各种方法的特点,着重讨论基于线性矩阵不等式(LMI)的分析方法,指出保守性是分析的重点.对现有结果的保守性进行比较和评述,并提出了改进的思路.     

A neural net-based time-delay compensation scheme and disturbance rejection for pneumatic systems

In practical electro-pneumatic or hydraulic servo mechanism, there is usually an unavoidable operating delay time in the solenoid valves. This may sometimes cause closed-loop instability if the time delay is not treated carefully in the control system designs. Furthermore, pneumatic actuators can offer high performance at low cost, but are often suffered from external disturbances depending on the operating conditions. This paper proposes a non-model-based design approach for pneumatic actuating systems. In the proposed control system, a gain scheduled fuzzy-PID controller ensures tracking performance an adaptable wavelet neuro compensator compensates for time-delay of the control valve, and a disturbance rejecter diminishes influence from load changes. A condition ensuring the closed-loop stability is derived. The proposed design is experimentally verified to show its effectiveness.     

时滞系统稳定性分析:一种积分等式方法

提出一种积分等式方法研究时滞系统稳定性问题.该方法是已有积分不等式方法的改进.结合自由权矩阵构建积分等式,且自由权矩阵的选取由Lyapunov-Krasovskii泛函的导数确定,从而在稳定条件的推导过程中,不引入任何模型转换及限界方法.基于该方法,可得到具有更低保守性的稳定条件.仿真例子说明了所提方法的有效性.     

Improved delay-dependent bounded real lemma for uncertain time-delay systems

This paper concerns reducing the conservatism of delay-dependent bounded real lemma (BRL) of time-delay systems with uncertainty described by a linear fractional form. By introducing an augmented Lyapunov functional, an improved BRL is established in terms of linear matrix inequalities (LMIs). Theoretical comparisons show that the obtained BRL is less conservative than some previous ones and the recent results in [S.Y. Xu, L. James, Y. Zou, New results on delay-dependent robust H_∞ control for systems with time-varying delays, Automatica 42 (2006) 343-348; T. Li, L. Guo, C. Lin, Y.M. Zhang, Delay-range-dependent bounded real lemma for time-delay systems, Asian Journal of Control 10 (2008) 708-717] are actually equivalent to the special cases of the obtained BRL. Numerical examples are also given to demonstrate the effectiveness of the proposed result.     

Synthesized sliding mode and time-delay control for a class of uncertain systems

In this note, the sliding mode control (SMC) is incorporated with time-delay control (TDC) during the sliding phase to reduce the switching gain. TDC identifies the unknown system dynamics and disturbance directly for every time-delay. For a system with a lumped perturbation which is relatively slow varying with respect to the sampling interval, a much low switching gain can be used if a reasonably good estimate of the derivative of system state can be obtained using the past information, hence chattering can be reduced or eliminated while retaining the tracking accuracy.     

A numerical algorithm for stability testing of fractional delay systems

This paper presents an effective numerical algorithm for testing the BIBO stability of fractional delay systems described by fractional-order delay-differential equations. It is based on using Cauchy's integral theorem and solving an initial-value problem. The algorithm has a reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of feedback control for both integer- and fractional-order systems having time delays.     

On spectral controllability of multi-input time-delay systems

It is proved that under certain verifiable conditions, spectral controllability of a linear system with commensurate state delays and with m inputs, m > 1, implies the existence of an n × 1 vector b contained in the image of the control matrix B such that the system with B replaced by b is spectrally controllable. Both the necessity and the sufficiency of these conditions are shown, a procedure to construct b is given, and an example is discussed.     

Hierarchy of LMI conditions for the stability analysis of time-delay systems

Assessing stability of time-delay systems based on the Lyapunov–Krasovskii functionals has been the subject of many contributions. Most of the results are based, first, on an a priori design of functionals and, finally, on the use of the famous Jensen’s inequality. In contrast with this design process, the present paper aims at providing a generic set of integral inequalities which are asymptotically non conservative and then to design functionals driven by these inequalities. The resulting stability conditions form a hierarchy of LMI which is competitive with the most efficient existing methods (delay-partitioning, discretization and sum of squares), in terms of conservatism and of complexity. Finally, some examples show the efficiency of the method.     

Complete stability robustness of third-order LTI multiple time-delay systems

A unique procedure is presented in this paper, for a complete stability robustness of the third-order LTI multiple time-delay systems (LTI-MTDS). The uniqueness of the treatment is simply due to the fact that there is no comparable methodology, presently, in the literature. The end result of this procedure is an exhaustive and precise determination of the stable regions in the domain of time delays. The backbone of the method is a novel framework called “the cluster treatment of characteristic roots, (CTCR)”. CTCR is constructed over two fundamental propositions. The first proposition claims the existence of a bounded number of so-called “kernel curves”, where the only imaginary characteristic roots occur. The second proposition is on an interesting directional invariance property of the crossing tendencies of these imaginary roots. For simplicity of conveyance and without loss of generality, the number of time delays is taken as two in this document. The new methodology is expandable to higher-order dynamics with more time delays than two, as the authors intend to demonstrate in future publications.     

Stability of Runge–Kutta methods for neutral delay-integro-differential-algebraic system

Stability properties of Runge–Kutta methods for the linear neutral delay-integro-differential-algebraic system are considered. It is proved that every A-stable natural Runge–Kutta method preserves the delay-independent stability of the exact solution. Some numerical experiments are given.