Robust Stability Conditions Research Articles

On stability for sampled-data nonlinear ADRC-based control system with application to the ball-beam problem

This paper mainly concerns with the stability analysis of the sampled-data nonlinear active disturbance rejection control (ADRC)-based control system. Firstly, a class of single-input-single-output (SISO) continuous plant is discretized using zero-order-hold (ZOH), and several kinds of digital implementation methods for the nonlinear extended state observer (NLESO) are newly proposed. Then the sampled-data nonlinear ADRC (NLADRC) based closed-loop system is transformed into a discrete-time Lurie-like system, to which linear matrix inequality (LMI)-based sufficient conditions for absolute stability and robust absolute stability are obtained. The sufficient conditions provide convenient and effective methods for determining the stability and its relationship with the parameters of the controller, the plant and the sampling period. Using the ball-beam system as an example, the proposed results are verified in both simulations and experiments.

Robust Stability Region of a Microgrid Under Parametric Uncertainty Using Bialternate Sum Matrix Approach

This paper analyzes the robust stability of a mixed source microgrid comprised of a synchronous generator-based distributed energy resource (DER) and an inverter-based DER. Few studies conducted recently indicated that power system stabilizers (PSSs) are advantageous for damping local oscillations of synchronous generator-based DERs. However, stability issues may arise due to interaction of the parameters of PSS and system. This paper proposes a method to determine the stability region for the system when any two parameters are considered uncertain based on robust control theory. A sufficient condition for robust stability is presented using bialternate sum matrix of the system with uncertain parameters. The bialternate sum matrix approach is compared with a branch and bound approach presented in earlier studies. These two methods are shown to yield identical results for a single machine infinite bus system. However, the proposed bialternate sum matrix approach requires far less computational effort. Therefore, it is applied to determine the stability region for the operation of mixed source microgrid in both grid-connected and islanded modes.

Dynamic output-feedback control for nonlinear continuous-time systems based on parametric uncertain subsystem and interval type-2 fuzzy model

This paper addresses the interval type-2 fuzzy robust dynamic output-feedback control problem for a class of nonlinear continuous-time systems with parametric uncertainties and immeasurable premise variables. First, the parametric uncertainties are assumed to be a subsystem based on the control input matrix and output matrix, and described as a linear fractional. Secondly, the nonlinear continuous-time systems are described by the interval type-2 fuzzy model. Thirdly, the new dynamic output feedback controller is designed based on the interval type-2 fuzzy model and the linear fractional (parametric uncertainties), the sufficient conditions for robust stabilization are given in the form of linear matrix inequalities (LMIs). Compared with previous work, the developed methods not only have abilities to handle the fuzzy system with immeasurable premise variables but also can deal with the parametric uncertainties effectively. The results are further extended to a mobile robot case and a chemical process case. Finally, two simulation examples are performed to show the effectiveness of the propose methods.

Non-fragile robust finite-time stabilization and $$H_{\infty }$$H∞ performance analysis for fractional-order delayed neural networks with discontinuous activations under the asynchronous switching

In this paper, the global non-fragile robust finite-time stabilization and $$H_{\infty }$$ performance analysis are investigated for uncertain switched fractional-order neural networks with discontinuous activations, time delay and external disturbance under the asynchronous switching. Firstly, a new inequality, which is concerned with the fractional derivative of the variable upper limit integral for the nonsmooth integrable functional, is developed. Secondly, the non-fragile switched controller with two types of gain perturbations is designed. Under the Filippov fractional differential inclusion framework, the global non-fragile robust finite-time stabilization conditions are addressed in terms of linear matrix inequalities (LMIs) by applying nonsmooth analysis theory, inequality analysis technique, average dwell-time method and Lyapunov functional approach. In addition, the global non-fragile robust finite-time $$H_{\infty }$$ performance analysis is performed, and the global non-fragile robust finite-time stabilization conditions with $$H_{\infty }$$ disturbance attenuation level are also derived in the form of LMIs. Finally, two numerical examples are given to illustrate the feasibility of the proposed non-fragile switched controller and the validity of the theoretical results.

Enhancement of disturbance‐rejection performance of uncertain input‐delay systems: a disturbance predictor approach

To suppress disturbances in uncertain tracking control systems with an input-delay, a disturbance predictor based on a high-order generalised extended-state observer is devised in this study. A smooth enough approximation of the disturbance is first made, then the approximation ahead of delay-time is predicted. The construction of the disturbance predictor is based on a truncated Taylor polynomial. To facilitate the analysis and design of the closed-loop control system, control gains for the nominal plant are designed in advance. Then the stability analysis for the closed-loop system is conducted, which yields a robust stability condition. As the disturbance predictor takes into consideration the influence of the input-delay, the presented method enables the enhancement of the disturbance-rejection performance. Finally, comparisons of the developed method with major methods in this field are conducted to validate the developed method and to demonstrate its advantages.

Fuzzy sliding mode control design of Markovian jump systems with time-varying delay

This paper studies the robust stochastic stabilization problem for a class of fuzzy Markovian jump systems with time-varying delay and external disturbances via sliding mode control scheme. Based on the equivalent-input-disturbance (EID) approach, an online disturbance estimator is implemented to reject the unknown disturbance effect on the considered system. Specifically, to obtain exact EID estimation Luenberger fuzzy state observer and a low-pass filter incorporated to the closed-loop system. Moreover, novel fuzzy EID-based sliding mode control law is constructed to ensure the stability of the closed-loop system with satisfactory disturbance rejection performance. By employing Lyapunov stability theory and some integral inequalities, a new set of delay-dependent robust stability conditions is derived in terms of linear matrix inequalities (LMIs). The resulting LMI is used to find the gains of the state-feedback controller and the state observer a for the resulting closed-loop system. At last, numerical simulations based on the single-link arm robot model are provided to illustrate the proposed design technique.

An Iterative Method of the Structural-Parametric Design of Robust Systems with State Controllers

An iterative method for designing a robust control system with state controllers is proposed that includes a transformation of the plant structure to improve its controllability by correcting the singular values of the controllability Gramian and the subsequent synthesis of the basic controller for the corrected plant using modal control. This approach allows us to design systems satisfying the robust stability condition for the given intervals of variation of the plant parameters.

Guaranteed-cost active disturbance rejection control for uncertain systems with an integral controller

Guaranteed-cost active disturbance rejection control (ADRC) for uncertain systems is investigated in this study. Firstly, an integral action is introduced in the framework of ADRC to measure and reduce the tracking error. Then, a robust stability condition is presented, and a quadratic cost function where the tracking error is appearing explicitly is used for ADRC performance assessment. The cost bound is formulated by linear matrix inequality and optimised to obtain controller parameters. Full-dimension extended state observer is used, and thus, the proposed strategy is applicable to an uncertain system that allows relative-degree varying or right-half-plane zero. Finally, the validity of the proposed method and its advantages is demonstrated through the simulations of comparative examples and experiments on a motor speed control system.

Robust Tracking and Disturbance Rejection for Linear Uncertain System With Unknown State Delay and Disturbance

A robust tracking control method is presented in this paper for an uncertain plant with an unknown state delay and an exogenous disturbance. The effects of the uncertainties, the delay, and the exogenous disturbance are treated as a total disturbance; thus, the construction of the observer does not need the delay information. The system design is divided into the design of the gains of the state-feedback controller as well as the design of the gains of the observer and the improved equivalent-input-disturbance (EID) estimator. The pole-assignment method is used to design the gains of the state-feedback controller of a simplified system. A robust-stability condition in the form of a linear matrix inequality is derived to determine the gains of the observer and the improved EID estimator. Since the devised Lyapunov functional is of a more general form than those in existing EID-based methods and the restrictive commutative condition is avoided in this design, the developed design method is less conservative. Finally, comparisons of the developed method with a sliding-mode control method for a matched disturbance and a conventional EID-based method for an unmatched disturbance illustrate the validity and superiority of the developed method.

On the robust state-feedback stabilisation of nD hybrid Roesser models with implicit LFR uncertainty

ABSTRACTThis paper addresses the problem of robust stabilisation of nD hybrid systems described by Roesser model. The model is in a descriptor form where all the system matrices are corrupted by uncertainty in a general form as implicit linear fractional representations . Based on the S-procedure technique, robust stability and robust stabilisation conditions are established for hybrid systems. Simulation example is provided in order to illustrate the relevance of our approach.

Robust H∞ control of an observer-based repetitive-control system

This paper concerns the problem of designing a robust observer-based modified repetitive-control system with a prescribed H∞ disturbance rejection level for a class of strictly proper linear plants with unknown aperiodic disturbances and time-varying structural uncertainties. A correction to the amount of the delay in the repetitive controller is introduced that leads to a significant improvement in tracking performance. An integrated performance index is defined to quantify the overall effect of rejecting the aperiodic disturbances and tracking the periodic reference input. A Lyapunov functional with two tuning parameters is used to derive a linear-matrix-inequality based robust stability condition for the system with a prescribed disturbance-rejection bound. Combining the performance indices, an optimization algorithm that searches for the best combination of state-observer gain and the feedback control gains is developed. A numerical example illustrates the design procedure and demonstrates the effectiveness of the method.

Robust disturbance rejection for a fractional-order system based on equivalent-input-disturbance approach

This paper presents a disturbance rejection method that is based on the equivalent-input-disturbance approach for uncertain fractional-order (FO) systems. An FO observer is used to reconstruct the plant states and estimate the disturbances. A disturbance estimator is incorporated in the construction of the FO control system to actively compensate for the entire disturbance. A robust stability condition for the control system and the parameters of the controller are derived using an indirect Lyapunov method. The presented method effectively rejects disturbances and handles modeling uncertainties without requiring prior knowledge of the disturbance. Comparison simulations on both numerical and practical examples demonstrate the effectiveness of the proposed method.

Robust Control of Positive Fractional‐Order Interconnected Systems with Heterogeneous Delays

AbstractThe problem of robust decentralized control of positive fractional‐order interconnected systems with heterogeneous time‐varying delays is studied in this paper. Necessary and sufficient conditions are first derived for internal positiveness of the system. By exploiting the monotonicity induced from positivity of the system, robust stability conditions subject to uncertain system parameters are derived. The derived stability conditions are then utilized to address the controller synthesis problem. The design conditions for obtaining controller gains of stabilizing decentralized controllers are formulated using linear programming, which can be effectively solved by various convex optimization algorithms. Finally, the effectiveness of the obtained results is validated by two numerical examples.

Design and trajectory tracking control of a piezoelectric nano-manipulator with actuator saturations

Abstract This paper presents the design of an XYZ nano-manipulator as well as the model based high precision tracking control of the nano-manipulating system. Aiming at large range and high mechanical bandwidth, the proposed mechanical design employs compound bridge-type amplifiers to increase the workspace without significant drop of stiffness. To further improve the system tracking performance and avoid possible actuator saturations, a robust anti-windup tracking control architecture combining a parallel internal-model based controller and an anti-windup compensator is adopted for the trajectory tracking of the designed nano-manipulating system. As a theoretical extension on a recent result [17] , we further investigate the robust stability condition of the closed-loop system and formulate the optimization design of the anti-windup compensators as a two block H ∞ optimization problem solvable with the Nehari approach. Real time control experiments demonstrate excellent tracking performance and saturation compensation capability with tracking precision error less than 0.28 % , which significantly outperforms relevant algorithms in the literature.

Novel global robust exponential stability criterion for uncertain inertial-type BAM neural networks with discrete and distributed time-varying delays via Lagrange sense

In this paper, the global robust exponential stability problem for a class of uncertain inertial-type BAM neural networks with both time-varying delays is focused through Lagrange sense. The existence of time-varying delays in discrete and distributed terms is explored with the availability of lower and upper bounds of time-varying delays. Firstly, we transform the proposed inertial BAM neural networks to usual one. Secondly, by the aid of LKF, stability theory, integral inequality, some novel sufficient conditions for the global robust exponential stability of the addressed neural networks are obtained in terms of linear matrix inequalities, which can be easily tested in practice by utilizing LMI control toolbox in MATLAB software. Furthermore, many comparisons of proposed work are listed with some existing literatures to get less conservatism. Finally, two numerical examples are provided to demonstrate the advantages and superiority of our theoretical outcomes.

A New Global Robust Exponential Stability Criterion for H∞ Control of Uncertain Stochastic Neutral-type Neural Networks with Both Timevarying Delays

This paper mainly focuses on a novel H∞ control design to handle the global robust exponential stability problem for uncertain stochastic neutral-type neural networks (USNNNs) with mixed time-varying delays. Here the delays are assumed to be both discrete and distributed, which means that the lower and upper bounds can be derived. Firstly, we draw a control law for stabilized and stability of the neutral-type neural networks (NNNs). Secondly, by employing the Lyapunov-Krasovskii functional(LKF) theory, Jensen’s integral inequality, new required sufficient conditions for the global robust exponential stability of the given neural networks (NNs) are established in terms of delay-dependent linear matrix inequalities (LMIs), which can be easily checked in practice. The conditions obtained are expressed in terms of LMIs whose feasibility can be verified easily by MATLAB LMI control toolbox. Moreover, we have compared our work with previous one in the existing literature and showed that it reduces conservatism. Finally, one numerical example with their simulations is given to validate the effectiveness of our proposed theoretical results.

Receptance-based stability criterion for second-order linear systems with time-varying delay

Abstract This paper presents a simple receptance-based criterion to analyze the robust stability of second-order systems with time-varying delay. The proposed approach is based on the closed-loop receptance which is directly related to the open-loop one by using the Sherman-Morrison equation. In this kind of problem, stability analysis cannot be performed from the closed-loop eigenvalues due to the time-variant nature of the system. In this context, a new robust stability condition is proposed by using the Small-Gain Theorem for second-order systems with either single or multiple inputs. The main contribution can be interpreted as a receptance-based generalization of the Single-Input Single-Output (SISO) first order Small-Gain Theorem condition. Moreover, the proposed stability criterion is combined with a detuning strategy to deal with the trade-off between performance and robustness with respect to delay variation. No limitation is imposed to the time-varying delay derivative which is a general result. Moreover, the proposed approach can also be used to analyze delay uncertainty due to the implementation simplicity since closed-loop poles are not computed in this criterion. Numerical examples are given to illustrate the effectiveness of the proposed approach.

New criteria for finite-time stability of nonlinear fractional-order delay systems: A Gronwall inequality approach

This paper investigates robust finite-time stability problem of fractional-order systems (FOSs) with time-varying delay and nonlinear perturbation. Based on a generalized Gronwall inequality, a delay-dependent sufficient condition for robust finite-time stability of such systems is established in terms of the Mittag-Leffler function. Compared with the finite-time stability conditions in the literature, our condition is simple and convenient for testing and application. The obtained results are applied to finite-time stability of linear uncertain FOSs and linear non-autonomous FOSs with time-varying delays. A numerical example is given to show the effectiveness of the obtained result.

Stochastic stability of positive Markov jump linear systems with completely/partially known transition rates

This paper addresses the stochastic stability problem of positive Markov jump linear systems (PMJLSs). A necessary and sufficient condition of stochastic stability for PMJLSs is given which can be checked by solving linear programming feasibility problems due to the positivity property. A common co-positive Lyapunov function is constructed to solve the problem, and the equivalence among stochastic stability, 1-moment stability and exponential mean stability is proved afterwards. Considering the uncertain transition rates situation, PMJLSs with partially known transition rate matrix are investigated, and a necessary and sufficient condition for robust stochastic stability is proposed in linear programming form. Numerical examples are presented to show the effectiveness of the proposed results.

Quantized feedback sliding-mode control: An event-triggered approach

This paper is concerned with the robust stabilization of quantized feedback sliding-mode control (SMC) for a class of uncertain linear systems via an event-triggered approach. By introducing an event-triggered mechanism, the event-triggered state instead of the state itself is quantized. And the generated finite event-triggered quantized state signal is sent through a digital network to the decoder and used for constructing a sliding mode controller to stabilize the uncertain linear systems. The relation among the lower bound of the quantized measurement saturating parameter, the upper bound of the event-triggered threshold parameter and the robust stabilization condition of linear uncertain systems, is first presented. Then, under the relation and by the combination of the proposed event-triggered mechanism for the system state and the discrete on-line adjustment policy for the quantizer sensitivity, the established quantized and event-triggered SMC laws are shown to ensure the reachability of the desired switching surface and global robust stabilization of uncertain linear systems is achieved. The theoretical result is verified via simulation illustration finally.