On non-fragility of controllers for time delay systems: A numerical approach

This paper proposes a new stability condition for a class of time delay systems. Firstly, a generalized double integral inequality is obtained. Then, a less conservative stability criterion is proposed by using the double integral inequality and choosing some new Lyapunov-Krasovskii functionals. Finally, two numerical examples are proposed to show the effectiveness of our method.

The problem of anti-saturation control for a class of time-delay systems with actuator saturation is considered in this paper. By introducing appropriate variable substitution, a new delay time-delay systems model with actuator saturation systems is established. Based on the Lyapunov stability theory, the stability condition and the anti-saturation controller design method are obtained by using the linear matrix inequality approach. By introducing the matrix into the Lyapunov function, the proposed conditions are less conservative than the previous results. Finally, a simulation example shows the validity and rationality of the method.

This paper reveals novel properties of spectrum of linear delay differential systems, and concerns the stability of a general class of time delay systems. The stability issue can be well characterized by the distribution of spectrum of the associated characteristic equation. Benefit properties on finite and bounded are formulated. And the finiteness condition and distribution boundary are obtained in a practical way. Based on the argument principle, an analytical formula for computing the number of unstable zeros (with positive real part) is deduced, which is determined by definite integral. Moreover, a procedure put no restriction on system except for the delay engendering purely imaginary roots is proposed to investigate the stability of linear time delay systems. At last, typical examples are given to illustrate that the method carried out is reliable and efficient.

A complex system describing interaction of subsystems of the second order with delay in connections between them is studied. Necessary and sufficient conditions of the existence of a diagonal Lyapunov–Krasovskii functional for the considered system are derived. The obtained results are applied for the stability a nalysis of a mechanical system and a model of population dynamics. In addition, it is shown that they can be used in a problem of formation control.

In this work, we consider the design problem of a full-order and also reduced-order unknown input observers for a particular class of time-delay systems. Asymptotic stability and existence conditions for the designed observers are established. The quadruple-tank process is used as a benchmark to prove the efficiency of the proposed algorithms.

Stability is usually in the sense of Lyapunov's asymptotical stability, thus the solutions starting from points close to a stable equilibrium may have a very long transient. In the applications of time-delayed feedback controls, it is important not only to determine the stable regions in the gain plane or gain space, but also to find out the abscissa that can be used as an index of stability. Based on the D-subdivision method, this paper proposes a simple algorithm for finding and labeling the stable regions in feedback gain plane with abscissa.The labeled sub-regions with smaller abscissa are better in applications. The main results are presented for the controlled pendulum or inverted pendulum under a delayed feedback, and are illustrated with two case studies.

Nonlinear discrete-time systems with delayed control: A reduction

ABSTRACTIn this contribution, we study time-delay systems with distributed delay whose kernel is piecewise constant. It is shown that the problem of finding a Lyapunov matrix for this class of time-delay systems can be reduced to the computation of solutions to an auxiliary system of linear differential equations with boundary conditions. The problem of solving the auxiliary system is then reduced to the solution of a system of linear algebraic equations. It is established that the auxiliary system with boundary conditions admits a unique solution if and only if there exists a unique Lyapunov matrix.

In this note, the analysis of time delay systems (TDSs) using Lambert W function approach is reassessed. A common canonical (CC) form of time delay systems is defined. We extended the recent results of Cepeda–Gomez and Michiels (2015, “Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using the Matrix Lambert W Function,” Automatica, 53, pp. 339–345) for second-order into nth order system. The eigenvalues of a time delay system are either real or complex conjugate pairs and therefore, the whole eigenspectrum can be associated with only two real branches of the Lambert W function. A new class of time delay systems is characterized to extend the applicability of the above-said method. Moreover, this approach has been exploited to design a controller which places a subset of eigenvalues at desired locations. Stability is guaranteed by using a new algorithm developed in this paper, which is based on the Nyquist plot. The approach is validated through numerical examples.

Hierarchy of stability criterion for time-delay systems based on multiple integral approach

Sliding Mode Control of Time-Delay Systems with Delayed Nonlinear Uncertainties

Results on finite-gain mathcal{L_{infty}} stability from a disturbance to the output of a time-variant delay system are presented via a delay decomposition approach. By constructing an appropriate Lyapunov-Krasovskii functional and a novel integral inequality, which gives a tighter upper bound than Jensen’s inequality and Bessel-Legendre inequality, some sufficient conditions are established and desired feedback controllers are designed in terms of the solution to certain LMIs. Compared with the existing results, the obtained criteria are more effective due to the tuning scalars and free-weighting matrices. Numerical examples and their simulations are given to demonstrate the effectiveness of the proposed method.

Robust H∞ consensus control problem is investigated for multiagent systems. Each agent is tackled in a more generalized form, which includes parameter uncertainties, external disturbances, nonidentical time-varying state, and input delays. Firstly, a distributed control protocol based on state feedback of neighbors is designed. By a decoupling method, H∞ consensus control problem for multiagent systems is transformed into H∞ control problem for the decoupling subsystems. Then employing Lyapunov-Krasovskii functional and free-weighting matrices, a lower conservative bounded real lemma (BRL) is derived in terms of linear matrix inequalities (LMIs) such that a class of time-delay system is guaranteed to be globally asymptotically stable with the desired H∞ performance index. Extending BRL, a sufficient delay-dependent condition of lower complexity in terms of the matrix inequalities is obtained to make all agents asymptotically reach consensus with the desired H∞ performance index. Furthermore, an algorithm is elaborately designed to get feasible solution to this condition. Extending this algorithm, an optimization algorithm for control protocol parameter is proposed to improve the disturbance attenuation capacity or allowable delay bounds. Finally, simulation results are provided to illustrate the correctness of the theoretical results and the effectiveness of the algorithms.

Nonlinear stabilization for a class of time delay systems via inverse optimality approach

Guaranteed cost reliable controller for T-S uncertain time-delay systems with actuator failure is studied in this paper. For a class of time-delay systems with norm bounded uncertainties, the sufficient existence condition of guaranteed cost reliable controller is deduced with the theory of Lyapunov and linear matrix inequality. Meanwhile, singular value decomposition is used to construct the design method of the controller. Finally, a numerical example is given to show the steps and method of the theories proposed in the paper, also show the correctness of the theories.

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Abstract In this paper, hybrid technique proposed for tuning Time Delays System with proportional–integral–derivative (PID) controller. So, the performance and the robustness for a class of Time Delay System are improved. The proposed hybrid technique is the combination of the iterative algorithm and curve fitting technique. The proposed iterative algorithm is improved, performance of the feedback tuning iterative technique; so the computational complexity of the iterative algorithm reduced. By using the iterative technique, the best polynomial coefficients of curve fitting technique is determined. Using the curve fitting technique, the Time Delay System is tuned and the stability parameters of the system is maintained. The curve fitting technique is one of the non linear programming techniques which can be constructed that approximately fits the data from the extract data. The proposed hybrid technique is implemented in MATLAB working platform and the tuning performance is evaluated. Then, the system performance of the proposed hybrid technique is compared with classical PID controller, Ziegler–Nichols tuning method.

This paper proposes an H∞ output tracking control for a class of time-delay systems with uncertainties based on Takagi–Sugeno simple-structure cerebellar model articulation controller and virtual desired variables. The advantages of the proposed methodology are threefold: i) allow adaptive tuning for linear matrix inequality-based control; ii) guarantee H∞ control performance of the overall output tracking system; and iii) relax the condition of equilibrium point knowledge. We implement the controller on a stirred tank reactor system as an example. The validity of the proposed controller is verified by satisfactory numerical results.

A non-linear adaptive observer is proposed to estimate the unmeasured states of a class of time-delay non-linear systems. Irrespective of the size of the constant delay, it is shown that the observer is globally convergent if the growth rate of the system non-linearities is output dependent. The presented results generalize previous results on observation of Lipschitz time-delay systems. Subsequently, the observerbased control problem of time-delay systems, having triangular structure, is discussed for arbitrary large constant delay.