New stability and stabilization conditions for nonlinear systems with time-varying delay based on delay-partitioning approach

This paper focuses on stability analysis and stabilization of nonlinear systems with interval time-varying delay, modeled by Takagi-Sugeno (T-S) fuzzy approach. To achieve more relaxation in the feasibility region, delay-partitioning approach is used for all integral terms in the Lyapunov-Krasovskii functional (LKF). A fuzzy Lyapunov function is proposed instead of non-integral term in LKF, and moreover, some slack matrices variables are offered to enlarge the design space. By doing this, new delay-dependent stability criteria are obtained. During the derivation of stability conditions, Jensen’s integral inequality is applied to deal with integral terms. Furthermore, in this paper the problem of controller design via the parallel distributed compensation (PDC) scheme is studied. Stability and stabilization conditions with less conservative are achieved in terms of linear matrix inequality (LMI). Finally, two numerical examples are presented to show the effectiveness of the proposed results.     

New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations

This paper studies the problem of the stability analysis of interval time-varying delay systems with nonlinear perturbations. Based on the Lyapunov–Krasovskii functional (LKF), a sufficient delay-range-dependent criterion for asymptotic stability is derived in terms of linear matrix inequality (LMI) and integral inequality approach (IIA) and delayed decomposition approach (DDA). Further, the delay range is divided into two equal segments for stability analysis. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. Two well-known examples are given to show less conservatism of our obtained results and the effectiveness of the proposed method.     

Computation of stabilizing PI and PID controllers for processes with time delay

In this paper, a new method for the computation of all stabilizing PI controllers for processes with time delay is given. The proposed method is based on plotting the stability boundary locus in the (kp, ki) plane and then computing the stabilizing values of the parameters of a PI controller for a given time delay system. The technique presented does not need to use Pade approximation and does not require sweeping over the parameters and also does not use linear programming to solve a set of inequalities. Thus it offers several important advantages over existing results obtained in this direction. Beyond stabilization, the method is used to compute stabilizing PI controllers which achieve user specified gain and phase margins. The proposed method is also used to design PID controllers for control systems with time delay. The limiting values of a PID controller which stabilize a given system with time delay are obtained in the (kp, ki) plane, (kp, kd) plane, and (ki, kd) plane. Examples are given to show the benefits of the method presented.     

Robust absolute stability criteria for uncertain Lurie interval time-varying delay systems of neutral type

This study investigates the delay-dependent robust absolute stability analysis for uncertain Lurie systems with interval time-varying delays of neutral type. First, we divide the whole delay interval into two segmentations with an unequal width and checking the variation of the Lyapunov–Krasovskii functional (LKF) for each subinterval of delay, much less conservative delay-dependent absolute and robust stability criteria are derived. Second, a new delay-dependent robust stability condition for uncertain Lurie neutral systems with interval time-varying delays, which expressed in terms of quadratic forms of linear matrix inequalities (LMIs), and has been derived by constructing the LKF from the delayed decomposition approach (DDA) and integral inequality approach (IIA). Finally, three numerical examples are given to show the effectiveness of the proposed stability criteria.     

An analytical method for PID controller tuning with specified gain and phase margins for integral plus time delay processes

In this paper, an analytical method is proposed for proportional-integral/proportional-derivative/proportional-integral-derivative (PI/PD/PID) controller tuning with specified gain and phase margins (GPMs) for integral plus time delay (IPTD) processes. Explicit formulas are also obtained for estimating the GPMs resulting from given PI/PD/PID controllers. The proposed method indicates a general form of the PID parameters and unifies a large number of existing rules as PI/PD/PID controller tuning with various GPM specifications. The GPMs realized by existing PID tuning rules are computed and documented as a reference for control engineers to tune the PID controllers.     

Further results on stabilization for interval time-delay systems via new integral inequality approach

This paper investigates the stability and stabilization problems for interval time-delay systems. By introducing a new delay partitioning approach, various Lyapunov-Krasovskii functionals with triple-integral terms are established to make full use of system information. In order to reduce the conservatism, improved integral inequalities are developed for estimation of double integrals, which show remarkable outperformance over the Jensen and Wirtinger ones. Particularly, the relationship between the time-delay and each subinterval is taken into consideration. The resulting stability criteria are less conservative than some recent methods. Based on the derived condition, the state-feedback controller design approach is also given. Finally, the numerical examples and the application to inverted pendulum system are provided to illustrate the effectiveness of the proposed approaches.     

Further improved stability criteria for uncertain T–S fuzzy systems with interval time-varying delay by delay-partitioning approach

This paper focuses on further improved stability criteria for uncertain T–S fuzzy systems with interval time-varying delay by a delay-partitioning approach. A modified augmented Lyapunov–Krasovskii functional (LKF) is established by partitioning the delay in all integral terms. Then some tighter bounding inequalities, i.e., Peng–Park׳s integral inequality (reciprocally convex approach) and the Free-Matrix-Based integral inequality (which yields less conservative stability criteria than the use of Wirtinger-based inequality does) are introduced to reduce the enlargement in bounding the derivative of LKF as much as possible, therefore, less conservative results can be expected in terms of e_s and LMIs. Finally, a numerical example is included to show that the proposed methods are less conservative than existing ones.     

Guaranteed cost consensus protocol design for linear multi-agent systems with sampled-data information: An input delay approach

To investigate the energy consumption involved in a sampled-data consensus process, the problem of guaranteed cost consensus for sampled-data linear multi-agent systems is considered. By using an input delay approach, an equivalent system is constructed to convert the guaranteed cost consensus problem to a guaranteed cost stabilization problem. A sufficient condition for guaranteed cost consensus is given in terms of linear matrix inequalities (LMIs), based on a refined time-dependent Lyapunov functional analysis. Reduced-order protocol design methodologies are proposed, with further discussions on determining sub-optimal protocol gain and enlarging allowable sampling interval bound made as a complement. Simulation results illustrate the effectiveness of the theoretical results.     

Further results on delay-range-dependent stability with additive time-varying delay systems

In this paper, new conditions for the delay-range-dependent stability analysis of time-varying delay systems are proposed in a Lyapunov–Krasovskii framework. Time delay is considered to be time-varying and has lower and upper bounds. A new method is first presented for a system with two time delays, integral inequality approach (IIA) used to express relationships among terms of Leibniz–Newton formula. Constructing a novel Lyapunov–Krasovskii functional includes information belonging to a given range; new delay-range-dependent criterion is established in term of linear matrix inequality (LMI). The advantage of that criterion lies in its simplicity and less conservative. This paper also presents a new result of stability analysis for continuous systems with two additive time-variant components representing a general class of delay with strong application background in network-based control systems. Resulting criteria are then expressed in terms of convex optimization with LMI constraints, allowing for use of efficient solvers. Finally, three numerical examples show these methods reducing conservatism and improving maximal allowable delay.     

Robust stabilizing first-order controllers for a class of time delay systems

In this paper, stabilizing regions of a first-order controller for an all poles system with time delay are computed via parametric methods. First, the admissible ranges of one of the controller’s parameters are obtained. Then, for a fixed value of this parameter, stabilizing regions in the remaining two parameters are determined using the D-decomposition method. Phase and gain margin specifications are then included in the design. Finally, robust stabilizing first-order controllers are determined for uncertain plants with an interval type uncertainty in the coefficients. Examples are given to illustrate the effectiveness of the proposed method.     

A novel approach to delay-fractional-dependent stability criterion for linear systems with interval delay

This paper considers the problem of delay-fractional-dependent stability analysis of linear systems with interval time-varying state delay. By developing a delay variable decomposition approach, both the information of the variable dividing subinterval delay, and the information of the lower and upper bound of delay can be taken into full consideration. Then a new delay-fractional-dependent stability criterion is derived without involving any direct approximation in the time-derivative of the Lyapunov–Krasovskii (LK) functional via some suitable Jensen integral inequalities and convex combination technique. The merits of the proposed result lie in less conservatism, which are realized by choosing different Lyapunov matrices in the variable delay subintervals and estimating the upper bound of some cross term in LK functional more exactly. At last, two well-known numerical examples are employed to show the effectiveness and less conservatism of the proposed method.     

Delay-dependent global exponential robust stability for delayed cellular neural networks with time-varying delay

This paper investigates a class of delayed cellular neural networks (DCNN) with time-varying delay. Based on the Lyapunov–Krasovski functional and integral inequality approach (IIA), a uniformly asymptotic stability criterion in terms of only one simple linear matrix inequality (LMI) is addressed, which guarantees stability for such time-varying delay systems. This LMI can be easily solved by convex optimization techniques. Unlike previous methods, the upper bound of the delay derivative is taken into consideration, even if larger than or equal to 1. It is proven that results obtained are less conservative than existing ones. Four numerical examples illustrate efficacy of the proposed methods.     

New delay dependent stability criteria for recurrent neural networks with interval time-varying delay

This paper is concerned with the delay dependent stability criteria for a class of static recurrent neural networks with interval time-varying delay. By choosing an appropriate Lyapunov–Krasovskii functional and employing a delay partitioning method, the less conservative condition is obtained. Furthermore, the LMIs-based condition depend on the lower and upper bounds of time delay. Finally, a numerical example is also designated to verify the reduced conservatism of developed criteria.     

Discrete-time domain two-degree-of-freedom control design for integrating and unstable processes with time delay

A discrete-time domain two-degree-of-freedom (2DOF) design method is proposed for integrating and unstable processes with time delay. Based on a 2DOF control structure recently developed, a controller is analytically designed in terms of the H₂ optimal control performance specification for the set-point tracking, and another controller is derived by proposing the desired closed-loop transfer function for load disturbance rejection. Both controllers can be tuned relatively independent to realize control optimization. Analytical expression of the set-point response is given for quantitatively tuning the single adjustable parameter in the set-point tracking controller. At the meantime, sufficient and necessary conditions for holding robust stability of the closed-loop control system are established for tuning another adjustable parameter in the disturbance rejection controller, along with numerical tuning guidelines. Illustrative examples from the literature are used to demonstrate the effectiveness of the proposed method.     

Multi-loop PI controller design based on the direct synthesis for interacting multi-time delay processes

In this article, a new analytical method based on the direct synthesis approach is proposed for the design of a multi-loop proportional-integral (PI) controller. The proposed design method is aimed at achieving the desired closed-loop response for multiple-input, multiple-output (MIMO) processes with multiple time delays. The ideal multi-loop controller is firstly designed in terms of the relative gain and desired closed-loop transfer function. Then, the standard multi-loop PI controller is obtained by approximating the ideal multi-loop controller using the Maclaurin series expansion. The simulation study demonstrates the effectiveness of the proposed method for the design of multi-loop PI controllers. The multi-loop PI controller designed by the proposed method shows a fast, well-balanced, and robust response with the minimum integral absolute error (IAE).     

Stability analysis and dynamic regulation of multi-dimensional Taylor network controller for SISO nonlinear systems with time-varying delay

Though many studies are focused on the stabilization of nonlinear systems with time-varying delay, they fail to involve the dynamic regulation without on-line optimization commonly. For this sake, feedback linearization, Lyapunov-Razumikhin theorem and polynomial approximation theorem are employed here to verify that the multi-dimensional Taylor network (MTN) controller can stabilize the single input single output (SISO) nonlinear time-varying delay systems through dynamic regulation of the system output with no need for on-line optimization. Here, the design of the controller is transformed into a convex optimization problem, which is tackled by means of the appropriate optimization method. Like its PD-like controller peers, the MTN controller functions well in eliminating the dependence on the system model. The effectiveness of the proposed approach is demonstrated and confirmed via two examples.     

A delay decomposition approach to robust stability analysis of uncertain systems with time-varying delay

This paper is concerned with delay-dependent robust stability for uncertain systems with time-varying delays. The proposed method employs a suitable Lyapunov-Krasovskii's functional for new augmented system. Then, based on the Lyapunov method, a delay-dependent robust criterion is devised by taking the relationship between the terms in the Leibniz-Newton formula into account. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Numerical examples are included to show that the proposed method is effective and can provide less conservative results.     

Stabilization of multi-input hybrid fractional-order systems with state delay

In this paper, the stabilization of a particular class of multi-input linear systems of fractional order differential inclusions with state delay using variable structure control is considered. First, the sliding surface with a fractional order integral formula is defined, and then the sufficient conditions for stability of the sliding surface are derived. Also, the concepts related to sliding control stabilization of differential inclusion systems with integer order are extended to differential inclusion systems with fractional order 0<q<1. Furthermore, for stabilization of fractional order systems with delay in the state space, a feedback and the concept of the norm of a state space is exploited. Finally, simulation results are provided to show the effectiveness of the proposed method.     

Novel delay-partitioning stabilization approach for networked control system via Wirtinger-based inequalities

This paper studies the problems of stability analysis and state feedback stabilization for networked control system. By developing a novel delay-partitioning approach, the information on both the range of network-induced delay and the maximum number of consecutive data packet dropouts can be taken into full consideration. Various augmented Lyapunov–Krasovskii functionals (LKFs) with triple-integral terms are constructed for the two delay subintervals. Moreover, the Wirtinger-based inequalities in combination with an improved reciprocal convexity are utilized to estimate the derivatives of LKFs more accurately. The proposed approaches have improved the stability conditions without increasing much computational complexity. Based on the obtained stability criterion, a stabilization controller design approach is also given. Finally, four numerical examples are presented to illustrate the effectiveness and outperformance of the proposed approaches.     

Synchronization of complex dynamical networks with random coupling delay and actuator faults

This paper addresses the issue of passivity-based synchronization problem for a family of Markovian jump neutral complex dynamical networks (NCDNs) with coupling delay and actuator faults. Also, by considering the effect of random fluctuation in complex dynamical network systems, the occurrence of coupling delay are taken in terms of a stochastic distribution, which obeys the Bernoulli distribution. To handle the fault effects in actuators of proposed complex network systems, an actuator fault model is considered. The main objective of this paper is to develop a robust state feedback controller such that for all possible actuator failures and random coupling delays, all nodes of the proposed Markovian jump NCDNs is globally asymptotically synchronized to the reference node in mean square sense and guarantee the output strict passivity performance. By developing a suitable Lyapunov–Krasovskii functional and utilizing the Wirtinger-based integral inequality, the required a set of sufficient conditions for the synchronization of proposed system is established in form of linear matrix inequalities. Finally, three numerical examples including a 3-dimensional Lorenz chaotic model are provided to demonstrate the correctness and superiority of the proposed control scheme.