This study investigates the adaptive fuzzy fault-tolerant control (FTC) problem for a class of non-linear multi-input multi-output systems with interconnections among subsystems and external unknown disturbance. The introduced multiple coupling terms have unknown non-linear forms relying on the system outputs and the corresponding subsystems, which cause that the system is no longer pure-feedback. Moreover, the considered actuator failures have a lock-in-place type and a type of loss of effectiveness. To solve the problem of actuator faults, compensative adaptive laws are constructed by taking advantages of the global fuzzy approximation property and a novel backstepping design procedure. According to the Lyapunov stability theorem, it is shown that all the signals in the closed-loop system are bounded and the outputs can converge to a giving compact set. In addition, simulation examples are presented to validate the effectiveness of the proposed adaptive fuzzy FTC protocol. Compared with previous researches on non-linear systems with unknown coupling terms, the proposed adaptive fuzzy FTC approach is more general, which may provide promising prospects of implementing in practical industrial systems.

In this study, a class of models governed by a second-order (in time) evolution differential equation with damping and positive definite, self-adjoint system operator is studied. Damping term has a special, but practically important, form of a finite sum of fractional powers of the system operator. This type of damping operator can be used to model a variety of physical phenomena related to dissipation of energy empirically observed in physical systems. It may be used to model dissipation mechanisms resulting from air damping, internal structural damping, internal viscous damping, etc. Using spectral theory of linear unbounded operators and semi-group theory necessary and sufficient conditions of approximate controllability for second order infinite dimensional system with damping are formulated and proved. Some important, from the practical point of view, remarks and comments are also provided. In particular, necessary condition for approximate controllability is formulated and discussed. Finally, an illustrative example related to approximate controllability of distributed parameter system described by the partial differential equation with higher order spatial differential operators is presented. The example refers to a system describing dynamical behaviour of damped Kirchhoff–Love plate. The study extends earlier results on approximate controllability of damped second-order abstract evolution dynamical systems.

Adaptive controllers with closed loop reference models have guaranteed transient performance due to the additional design freedom available in them. However, robustness in the presence of both matched perturbations and time delay is not guaranteed, which is a necessary requirement for most of the safety-critical systems, such as, aerospace applications. This study attempts to achieve guaranteed transient and steady state performance with improved robustness in the presence of both matched perturbations and input delay by incorporating a proportional–integral (PI) compensator in the feedback loop of the closed loop reference model reference adaptive control (MRAC) architecture. The proposed system also reduces high frequency oscillations in control channel without disturbing the relative order of plant. The parameters of PI compensator have been tuned using genetic algorithm. Effectiveness of the proposed work has been validated on standard numerical examples using simulation results and the performance has been compared with similar and recent works.

This study presents a new configuration and a design method for the control of a switched neutral-delay system. The equivalent-input-disturbance (EID) approach is used to construct an EID estimator to estimate the influence of exogenous disturbances on the system. A new control law is constructed that incorporates an EID estimate to ensure satisfactory disturbance rejection performance. A sufficient condition for uniformly ultimate boundedness is derived in term of linear matrix inequalities (LMIs) to guarantee the stability of the system. Moreover, state feedback control laws and state observers are designed based on the LMIs for arbitrary switching rules. A numerical example demonstrates the validity of the method.

This study deals with the problem of aperiodic-disturbance rejection for a class of plants with non-linear state-dependent uncertainty in a modified repetitive-control system (MRCS). Since both the non-linear uncertainty and the disturbances are often unknown, an equivalent-input-disturbance (EID) estimator is constructed using a full-order state observer with variant system matrix to estimate them. The EID estimate, which exhibits the overall effect on the output of the non-linear uncertainty and all types of disturbances, is incorporated into a linear repetitive control law to compensate for the non-linear uncertainty and the disturbances. A continuous-discrete two-dimensional model of the EID-based MRCS is built that enables the preferential adjustment of the control and learning actions. A sufficient condition for the robust stability for the EID-based MRCS is given in terms of a linear matrix inequality. It yields the parameters of the repetitive controller and the EID estimator. Finally, a numerical example demonstrates the design procedures and illustrates the effectiveness of the method.

This paper studies the anti-windup (AW) problem for a certain class of non-linear systems, in which the plant is globally quadratically stable and also partially linearisable by a suitably chosen non-linear feedback control law. Three types of AW compensators are proposed for this type of non-linear system: the first one is a non-linear extension of the popular linear internal model control (IMC) scheme; the second one has a similar structure to the IMC AW compensator yet is of reduced order and has entirely linear dynamics; and the third one is again a linear AW compensator, but can endow the closed-loop system with some sub-optimal performance properties. All three AW compensators are able to provide global exponential stability guarantees for the aforementioned class of systems. This work was inspired by a wave energy application whose dynamics fall into the class of systems studied in this study. Simulation results show the efficacy of the three AW compensators when applied to the wave energy application.

This study proposes a sampled-data design method for robust control of open two-level quantum systems with operator errors. The required control performance is characterised using the concept of a sliding mode domain related to fidelity, coherence or purity. The authors have designed a control law offline and then utilise it online for a two-level system subject to decoherence with operator errors in the system model. They analyse three cases of approximate amplitude damping decoherence, approximate phase damping decoherence and approximate depolarising decoherence. They design specific sampling periods for these cases that can guarantee the required control performance.

In this study, the fault-tolerant control problem is investigated for a class of non-affine output-constrained multi-input multiple-output non-linear systems with actuator faults. The controlled systems contain unknown non-linear functions, unknown external disturbance and immeasurable states. The fuzzy logic systems and a robust compensation function are employed to deal with the unknown non-linear functions and non-affine non-linear actuator fault problems. Then, the state observer and the non-linear disturbance observer are developed for estimating the immeasurable states and unknown compounded disturbance, respectively. By using the barrier Lyapunov function method and in the unified framework of adaptive backstepping output feedback control design, a novel adaptive fuzzy fault-tolerant controller design scheme is developed. It is shown that all the variables of the closed-loop system are semi-globally uniformly bounded. Moreover, the system outputs cannot violate their predefined bounds in the presence of the non-affine non-linear faults. A simulation is given to validate the effectiveness of the proposed approach.

This study investigates the problem of global decentralised fault-tolerant control for a class of non-linear large-scale systems with unknown time-delayed interconnections. The authors propose a novel time-delay replacement technique. With this technique, a delay-assumption independent controller is then developed. In addition, the delayed input variables of the fuzzy approximator are replaced by the bounded reference signals so that the global stability is achieved for the approximation-based fault tolerant control field. In doing so, it will eliminate the limitation which is posted in estimating unknown functions in the traditional approach. The proposed controller ensures that all the signals in the closed-loop system are globally uniformly ultimately bounded, and the tracking errors can converge to a small neighbourhood of the original point by appropriate choice of the design parameters. Simulation results are provided to show the effectiveness of the control scheme.

In this study, a new method for designing fractional-order robust lead–lag controllers in order to ensure robust stability in uncertain systems with unstructured uncertainty, has been presented. In this method, at first by the use of pole placement and solving Diophantine equation, a fractional-order controller, which has a new form, is designed in order to stabilise the nominal system. Then a constrained optimisation problem is solved, in which, the cost function consists of the numerator and denominator coefficients of the proposed fractional-order controller and the constraints are the robust stability conditions. Finally, the parameters of the robust fractional-order controller are acquired. This method is implemented on an unstable system and simulation results are presented.

This study is concerned with the fault estimation problem of discrete-time Takagi–Sugeno fuzzy systems with low-frequency faults. A fault estimation filter is designed to solve this problem. To make the estimation error as small as possible in the presence of high-frequency disturbances, the filter is designed to satisfy two finite-frequency H _∞ performance indices simultaneously. An algorithm is proposed to calculate the parameters of the desired filter. The obtained filter can estimate the faults in low-frequency domain effectively, e.g. stuck faults. For comparison, a full-frequency fault estimation method is also given. An example is given to illustrate advantages of the proposed finite-frequency method.

The study deals with an observer-based event-triggered control strategy for linear systems using only local (i.e. available) variables. Sufficient conditions based on linear matrix inequalities associated to convex optimisation problems are proposed to ensure the asymptotic stability of the closed loop and the output convergence to a constant reference in both emulation and co-design contexts. Indeed, the proposed approach allows either to design the event-triggering rules or co-design the event-triggering rule along with the controller gain.

This study deals with the properties of magnitude–frequency responses in fractional order systems. Using Phragmén–Lindelöf theorem in complex analysis, it is shown that the supremum of the magnitude–frequency response of a fractional system with a commensurate order less than one cannot be greater than that of its integer order bounded-input, bounded-output stable counterpart. Further results are also obtained on magnitude–frequency response of stable/unstable fractional order systems. Moreover, it is found that the supremum (infimum) of the magnitude-scaling frequency of the family of fractional order systems having a fixed structure and different orders in the range (0, 2) is a piecewise logarithmically convex (concave) function of the scaling frequency. On the basis of the properties of magnitude–frequency responses in fractional order systems, some sample consequent results which are useful in analysis of fractional order circuits and systems are derived. These results are validated by various numerical examples.

The bounded-input bounded-output stability of fractional delay systems of neutral type is investigated in this study. The proposed method calculates the number of unstable poles of the system for each delay value. All time-delay intervals in time-delay space where system is stable are precisely determined. The stability of systems with not only indefinitely large values of time-delay, but also with multiple poles on the imaginary axis has been studied. The proposed method is employed for investigating the stability of a fractional delay system when its fractional orders approach to the integer numbers or zero. It is proved that simple case of neutral types and time-delay systems have the same conditions for the independent stability. Two numerical examples are provided to illustrate the proficiencies of the proposed method.

Distributed cooperative droop control consisting of the primary decentralised droop control and the secondary distributed control is studied, which aims to achieve an exact current sharing between generators, worked in the voltage control mode, of DC microgrids. For the DC microgrids with the distributed cooperative droop control, the dynamic stability has not been well investigated although its steady performance has been widely reported. This study focuses on the stability problem of DC microgrids with fixed topology and shows it is equivalent to the semistability problem of a class of second-order matrix systems. Some further sufficient conditions as well followed. The steady state is analysed deeply for some special cases. A DC microgrid of four nodes is simulated on the Matlab/Simulink platform to illustrate the efficacy of analytic results.

This study deals with the fixed-time stabilisation of a class of non-linear systems. A non-linear controller, based on sliding mode, is proposed to ensure the fixed-time stabilisation of a non-holonomic system in chained form. An upper bound of the settling time, independent of the initial system state, is estimated. Then, a distributed switched strategy, based on local information, is introduced to solve the fixed-time consensus problem for multiple non-holonomic agents. Using Lyapunov functions, the switching times are explicitly given and only depend on the controller gains and the Laplacian matrix. Simulation results highlight the efficiency of the proposed method.

The least mean square methods include two typical parameter estimation algorithms, which are the projection algorithm and the stochastic gradient algorithm, the former is sensitive to noise and the latter is not capable of tracking the time-varying parameters. On the basis of these two typical algorithms, this study presents a generalised projection identification algorithm (or a finite data window stochastic gradient identification algorithm) for time-varying systems and studies its convergence by using the stochastic process theory. The analysis indicates that the generalised projection algorithm can track the time-varying parameters and requires less computational effort compared with the forgetting factor recursive least squares algorithm. The way of choosing the data window length is stated so that the minimum parameter estimation error upper bound can be obtained. The numerical examples are provided.

In this study, Mittag–Leffler stability and the fractional Lyapunov direct method are used to study fractional-order leaderless and leader-following consensus algorithms with non-linear dynamics under a directed network topology. The sufficient conditions are given to guarantee that the consensus can be achieved in the non-linear fractional-order systems with and without a virtual leader, and the adaptive feedback control protocols are also given in this study. Finally, some numerical simulations are presented to demonstrate the effectiveness of the theoretical results.

Corrective controllers can modify the stable-state behaviour of asynchronous sequential machines in a desirable way. This study addresses a static corrective control scheme formulated as a static feedback law. As no internal states are needed, the use of static corrective controllers greatly reduces the implementation cost. The existence condition for static corrective controllers is found to be more restrictive than that of dynamic controllers. Based on the existence condition, the output of the static corrective controller is assigned so as to match the closed-loop system to a reference model. The proposed static controller is applied to the model matching problem of on-board computer error counters implemented on field-programmable gate array systems.

The study deals with parametrisation of input shapers and smoothers with distributed delays, which are a common tool used for pre-compensating oscillatory modes of flexible systems. These filtering structures can be easily parametrised, if the oscillatory mode is undamped, leading to fully analytical formulas. For the damped case, however, the parametrisation needs to be done numerically as a rule. Utilising a straightforward complex domain transformation, as the main results, the structure of the filters is turned to the closed-form that can be parametrised analytically for the damped case too. The adjusted smoothers and shapers accommodate the reference and the system output signals without vibration at the same time lengths like the pre-forms for the undamped case. This methodology is applied to Trapezoidal, S-curve and Trigonometric smoothers, Jerk Limited shaper and recently proposed Zero Vibration shaper with a distributed delay. The proposed new types of smoothers and shapers, which are essentially based on exponential distribution of delays, are investigated in time and frequency domains. Subsequently, the basic properties, i.e. response performances, spectrum distribution and robustness analysis, are demonstrated and cross-compared in a case study example.

In this study, an introduced auxiliary error signal-based robust control method and an adaptive control strategy are synthesised through backstepping method for high precise motion tracking control of double-rod hydraulic servo systems. The controller takes the non-linear behaviour in flow, unmatched disturbances in inertia load dynamics, and matched uncertainties in pressure dynamics into consideration. In addition, the developed controller does not require a priori knowledge on the bounds of the lumped disturbances and the gain of the devised robust control law can be tuned itself. As a result, the obtained control strategy can achieve an exact tracking performance for the whole closed-loop hydraulic system in the presence of matched and unmatched uncertainties simultaneously. Extensive experiments are carried out for the trajectory tracking control of a double-rod hydraulic servo system and the results reveal the high-performance of the presented controller.

This study deals with the H _∞ filtering problem for switched linear parameter-varying (LPV) systems using the multiple parameter-dependent Lyapunov function technique. The parameter and mode-dependent switching law and the filter state-relied controllers are designed to ensure the H _∞ performance without requiring the stability of each individual subsystem. An H _∞ filtering synthesis condition is obtained by a matrix optimisation iterating algorithm. Simulation result on a switched LPV aero-engine model shows the feasibility and the validity of the H _∞ filtering scheme.

This study is concerned with the stability problem for a class of fractional-order two-dimensional (2D) non-linear continuous-time systems. A Caputo fractional-order 2D non-linear continuous-time system represented by the Roesser model is proposed for the first time. By using the Lyapunov function method, an asymptotic stability criterion is derived. Two numerical examples are provided to show the effectiveness of the proposed results.

In this study, the authors aim to study explicit iterative algorithms for solving coupled discrete-time Lyapunov matrix equations. First, an explicit iterative algorithm based on fixed point theory of dynamic equations is presented via adding a tuning parameter. Second, a necessary and sufficient condition is provided for the convergence of the proposed algorithm. Moreover, the optimal value of the tuning parameter is derived for the fastest convergence of the algorithm. Third, by using the latest updated information, a modified version of the presented explicit iterative algorithm is also established with a necessary and sufficient condition being provided to guarantee the convergence of the modified algorithm. Finally, a numerical example is given to demonstrate the effectiveness of the proposed algorithms.

The authors propose a consensus control framework for multiple heterogeneous general single-input single-output linear systems with no zeros at s = 0 on a directed information graph with constant, yet, non-uniform and unknown delays. The proposed consensus control is easy to design via loop-shaping like graphical approach, robust against plant uncertainty and constant delay, and completely decentralisable (i.e. locally synthesisable without consulting other agents). Consensus proof using multi-input multi-output Nyquist theorem and algebraic graph theory is given, with a numerical example to illustrate the theory.