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This article considers the problem of output feedback stabilization for a class of stochastic high‐order nonlinear time‐delay systems with unknown output function. For stochastic high‐order nonlinear time‐delay systems, based on the Lyapunov stability theorem, by combining the addition of one power integrator and homogeneous domination method, the maximal open sector Δ$\Delta$ of output function is given. As long as output function belongs to any closed sector included in Δ$\Delta$, an output feedback controller can be developed to guarantee the closed‐loop system globally asymptotically stable in probability. [ABSTRACT FROM AUTHOR]

Abstract This article considers the problem of output feedback stabilization for a class of stochastic high‐order nonlinear time‐delay systems with unknown output function. For stochastic high‐order nonlinear time‐delay systems, based on the Lyapunov stability theorem, by combining the addition of one power integrator and homogeneous domination method, the maximal open sector Δ of output function is given. As long as output function belongs to any closed sector included in Δ, an output feedback controller can be developed to guarantee the closed‐loop system globally asymptotically stable in probability.

Abstract Aiming to improve the estimation and prediction accuracy of a target's position, this paper proposes a state estimation method for photoelectric tracking systems, based on the evaluation of the tracked target's motion intention. Traditional photoelectric tracking systems utilize external physical quantities such as the position, velocity, and acceleration of the target as the estimated states. While this method can output good results for pre‐modelled target positions, it struggles to maintain the accuracy when facing manoeuvering targets or complex motion patterns targets. Here, the relevant parameters of the tracked target's motion intention are directly estimated innovatively, like estimating the circling point position rather than the circular flying target's position and velocity. This approach enables recognizing the target's motion intention and leads to precise estimation, which specifically consists of an interacting multiple model approach, multiple unscented Kalman estimators, and a robust estimator. The effectiveness and stability of this estimator are validated through software simulations and experiments on a dual‐reflection mirror platform.

Abstract This paper extends the characteristic ratio approach using novel inequalities to ensure zero/low overshoot for linear‐time‐invariant systems with zeros. The extension provided by this paper is based on the maximally‐flatness property of a transfer function, where the square‐magnitude of the transfer function is ensured to be a low‐pass filter. In order to be able to design low‐order/fixed structure controllers, a partial pole‐assignment approach is used instead of the full pole‐assignment used in the Characteristic Ratio Assignment (CRA) method. The developed inequalities and additional stability conditions are combined into an optimization problem using time domain restrictions when necessary. Although the method given in the paper is general, particular inequalities are developed for PI and PI‐PD controller cases, due to their frequent use in industrial applications. Similarly, First‐Order‐Plus‐Delay‐Time (FOPDT) and Second‐Order‐Plus‐Delay‐Time (SOPDT) systems are considered specifically, since most of the practical systems can be approximated by one of these types. The study is extended to plants with uncertainties where a theorem is developed to decrease computation time dramatically. The benefits of the proposed methods are demonstrated by several examples.

This article discusses invariant subspaces of a matrix with a given partition structure. The existence of a nontrivial structured invariant subspace is equivalent to the possibility of decomposing the associated system with multiple feedback blocks such that the feedback operators are subject to a given constraint. The formulation is especially useful in the stability analysis of time‐delay systems using the Lyapunov–Krasovskii functional approach where computational efficiency is essential in order to achieve accuracy for large scale systems. The set of all structured invariant subspaces are obtained (thus all possible decompositions are obtained as a result) for the coupled differential‐difference equations (DDE) associated with the DDE of retarded and neutral types, as well as systems with a time‐varying delay. It was shown that the known ad hoc methods of reducing the dimensions of delay channels can be considered as special cases of decomposition where one subsystem has trivial dynamics. The reduction of computational cost is demonstrated by a numerical example. For the general case, a recursive procedure is developed to obtain the set of all structured invariant subspaces. Based on this procedure, a method is presented to obtain a nontrivial structured invariant subspace that considers computational efficiency and increased possibility of terminating in a finite number of steps. [ABSTRACT FROM AUTHOR]

In this paper we construct a general observer theory for differential delay systems based on different types of symmetries (exact symmetry or asymptotic symmetry), ending up with a certain number of semi‐global and global observers, with bounded or unbounded system's solutions. We introduce the notions of symmetry for a differential delay system, being inspired by well‐known definitions of symmetry for an ordinary or partial differential system, and variational symmetry for the associated variational differential delay system. We illustrate observer design procedures in details, by proving that the existence of a (variational asymptotic) symmetry with system's detectability in the first approximation have a central role in the design of a state observer. The symmetry is a one‐parameter group of transformations which maps the system into itself (exact symmetry) or into a different system (asymptotic symmetry), approximating the original one with better and better accuracy as the parameter of the symmetry is larger. The types of symmetries we consider here show an important contractive action on the state and input spaces for which the system's solutions are mapped into arbitrarily small neighbourhoods of the origin in which the transformed system can be well‐approximated by its linearization. The parameter of the symmetry may be constant (semiglobal observers) or updated on‐line by a state norm estimator (global observers). [ABSTRACT FROM AUTHOR]

We consider the problem of enforcing safety in multi‐input systems with distinct input delays via the use of Control Barrier Functions (CBFs). For systems with input delay(s), a popular approach for enforcing safety is the combination of a CBF designed for the delay‐free system with state‐predictors that compensate the input delays. Typically, this comes with the assumption that the system does not violate safety constraints before all input delays have been compensated and the system is fully under control. In this paper, we introduce two control approaches that enforce safety before all input delays have been compensated, whenever it is possible to do so. We do this by utilizing a robust CBF formulation that treats longer‐delayed inputs as known disturbances when determining control effort for shorter delayed inputs. This formulation ensures that, whenever possible, a subset of input channels with shorter delays will be utilized for keeping the system in the admissible safe set before longer input delays have been compensated. The effectiveness of our approaches is demonstrated with two numerical examples. [ABSTRACT FROM AUTHOR]

Aiming to improve the estimation and prediction accuracy of a target's position, this paper proposes a state estimation method for photoelectric tracking systems, based on the evaluation of the tracked target's motion intention. Traditional photoelectric tracking systems utilize external physical quantities such as the position, velocity, and acceleration of the target as the estimated states. While this method can output good results for pre‐modelled target positions, it struggles to maintain the accuracy when facing manoeuvering targets or complex motion patterns targets. Here, the relevant parameters of the tracked target's motion intention are directly estimated innovatively, like estimating the circling point position rather than the circular flying target's position and velocity. This approach enables recognizing the target's motion intention and leads to precise estimation, which specifically consists of an interacting multiple model approach, multiple unscented Kalman estimators, and a robust estimator. The effectiveness and stability of this estimator are validated through software simulations and experiments on a dual‐reflection mirror platform. [ABSTRACT FROM AUTHOR]

This paper extends the characteristic ratio approach using novel inequalities to ensure zero/low overshoot for linear‐time‐invariant systems with zeros. The extension provided by this paper is based on the maximally‐flatness property of a transfer function, where the square‐magnitude of the transfer function is ensured to be a low‐pass filter. In order to be able to design low‐order/fixed structure controllers, a partial pole‐assignment approach is used instead of the full pole‐assignment used in the Characteristic Ratio Assignment (CRA) method. The developed inequalities and additional stability conditions are combined into an optimization problem using time domain restrictions when necessary. Although the method given in the paper is general, particular inequalities are developed for PI and PI‐PD controller cases, due to their frequent use in industrial applications. Similarly, First‐Order‐Plus‐Delay‐Time (FOPDT) and Second‐Order‐Plus‐Delay‐Time (SOPDT) systems are considered specifically, since most of the practical systems can be approximated by one of these types. The study is extended to plants with uncertainties where a theorem is developed to decrease computation time dramatically. The benefits of the proposed methods are demonstrated by several examples. [ABSTRACT FROM AUTHOR]

In this article, we explored the controllability of fractional dynamical systems with a single delay in the control function with the Caputo fractional derivative. It is the first work in which the author studies the controllability of a time-varying fractional dynamical system with a delay in the control function. We develop the necessary and sufficient criteria for the solution representation of controllability of time-varying fractional linear dynamical systems by utilizing the Grammian matrix. We use Schauder's fixed point theorem to establish sufficient conditions for the controllability of time-varying nonlinear fractional dynamical systems. With the help of successive approximation techniques, numerical examples validate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the stability analysis of discrete‐time systems with a time‐varying delay. The conservatism and computation burden are two important factors to evaluate a stability condition. By taking the relationship of two reciprocally convex parts into consideration, a new combined matrix‐separation‐based inequality is proposed that involves only a few free matrices. Moreover, an improved matrix‐injection‐based transformation lemma with the parameter varying within a closed interval is proposed by introducing only one free matrix. By constructing an appropriate Lyapunov–Krasovskii functional and applying the improved methods, a relaxed stability condition is consequently obtained with a small number of decision variables. Two numerical examples are given to show the merits of the proposed methods.

Abstract In this paper, a novel nonlinear unknown input observer is proposed in order to fault estimation for nonlinear uncertain systems with time delays. By the estimation of the faults, the features are detected such as shape, size occurrence time etc. The time delay is considered a constant and known parameter in the states. The disturbances are investigated in the states and outputs and also, and sensor and actuator faults are considered. The stability of the closed‐loop system is guaranteed by Lyapunov–Krasovskii theory and some feasible Linear matrix inequalities (LMI). The proposed method is simulated on a continuous‐stirred tank reactor (CSTR) with uncertainties and time delay. Simulation results show the appropriate efficiency of the proposed method.

This paper is concerned with the stability analysis of discrete‐time systems with a time‐varying delay. The conservatism and computation burden are two important factors to evaluate a stability condition. By taking the relationship of two reciprocally convex parts into consideration, a new combined matrix‐separation‐based inequality is proposed that involves only a few free matrices. Moreover, an improved matrix‐injection‐based transformation lemma with the parameter varying within a closed interval is proposed by introducing only one free matrix. By constructing an appropriate Lyapunov–Krasovskii functional and applying the improved methods, a relaxed stability condition is consequently obtained with a small number of decision variables. Two numerical examples are given to show the merits of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, a novel nonlinear unknown input observer is proposed in order to fault estimation for nonlinear uncertain systems with time delays. By the estimation of the faults, the features are detected such as shape, size occurrence time etc. The time delay is considered a constant and known parameter in the states. The disturbances are investigated in the states and outputs and also, and sensor and actuator faults are considered. The stability of the closed‐loop system is guaranteed by Lyapunov–Krasovskii theory and some feasible Linear matrix inequalities (LMI). The proposed method is simulated on a continuous‐stirred tank reactor (CSTR) with uncertainties and time delay. Simulation results show the appropriate efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Summary: We are considering the problem of designing observers for heat partial differential equations (PDEs) that are subject to sensor delay and parameter uncertainty. In order to get finite‐dimensional observers, described by ordinary differential equations (ODE), we develop a design method based on the modal decomposition approach. The approach is extended so that both parameter uncertainty and sensor delay effects are compensated for. To cope more effectively with sensor delay, an output predictor is designed and the online provided output predictions are substituted to the future output values in the observer. To compensate for parameter uncertainty, we design a parameter estimator providing online parameter estimates, which are substituted to the unknown parameters in the observer. The parameter estimator design is made decoupled from the observer gain design by using an appropriate decoupling transformation. Using an analysis of the small‐gain‐theorem type, the whole (state and parameter) estimation error system is shown to be exponentially stable, under well‐defined conditions on the observer dimension, the sensor delay, and signal persistent excitation (PE). [ABSTRACT FROM AUTHOR]

This article investigates the finite‐time tracking control problem with guaranteed performance for a class of nonlinear pure‐feedback systems subject to both time‐varying state delays and control input delays. Both the state delays and control input delays are assumed to be unknown yet bounded by some known constants. A new performance function is introduced that ensures the finite‐time convergence of tracking errors and guaranteed performance. The key to handling both the states and input delays lies in a novel construction of the Lyapunov–Krasovskii (LK) function, which also avoids the reconstruction of the LK function in stability analysis. In addition, a continuous package function is introduced, which allows the unknown nonlinear terms arising from the uncertain perturbation and the derivation of the LK function to be eliminated. Under the proposed method, the semi‐global uniform ultimate boundedness of all signals in the closed‐loop system can also be ensured. Simulation examples are provided to demonstrate the effectiveness of the proposed approach in this article. [ABSTRACT FROM AUTHOR]

This paper concentrates on presenting a reliable procedure to compute the stabilizing region of fractional‐order proportional integral derivative (FOPID) controllers for interval fractional‐order plants having an interval time delay. An interval fractional‐order plant is defined as a fractional‐order transfer function whose denominator and numerator coefficients are all uncertain and lie in specified intervals. Also, an interval time delay points to a delay term whose value varies in a specific interval. The D‐decomposition technique and the value set concept are employed to determine the stabilizing region of FOPID controllers. In this study, first, a theorem is presented to compute the boundary of the value sets of systems having interval time day. Then, a lemma is provided for robust stability analysis of the given closed‐loop control system. For a convenient use of the paper results, an algorithm is proposed to solve the problem of robustly stabilizing interval fractional‐order plants with an interval time delay using FOPID controllers. Finally, four examples are provided to illustrate the proposed procedure. [ABSTRACT FROM AUTHOR]

This study investigates the controllability problems for linear and nonlinear fractional control systems with multiple control delays in the context of the ψ$$ \psi $$‐Caputo fractional derivative. For the linear case, using the positive definiteness of the Gramian matrix, the necessary and sufficient conditions are obtained. For the nonlinear case, utilizing the iterative technique together with the completeness of the space, sufficient conditions for the existence of a solution are obtained. Just a few examples are included to make theoretical findings easier to grasp, and their respective graphs are plotted. [ABSTRACT FROM AUTHOR]

We complete the first step toward the resolution of several decades-old challenges in disturbance-robust adaptive control. For a scalar system with an unknown parameter for which no a priori bound is given, with a disturbance that is of unlimited magnitude and possibly persistent (not square integrable), and without a persistence of excitation necessarily verified by the state, we consider the problems of (practical) gain assignment relative to the disturbance. We provide a solution to these heretofore unsolved feedback design problems with the aid of infinite-dimensional nonlinear feedback employing distributed delay of the state and input itself. Specifically, in addition to (0) the global boundedness of the infinite-dimensional state of the closed-loop system when the disturbance is present, we establish (1) practical input-to-output stability with assignable asymptotic gain from the disturbance to the plant state; (2) assignable exponential convergence rate; and (3) assignable radius of the residual set. The accompanying identifier in our adaptive controller guarantees (4) boundedness of the parameter estimate even when disturbances are present; (5) an ultimate estimation error which is proportional to the magnitude of the disturbance with assignable gain when there exists sufficient excitation of the state; and (6) exact parameter estimation in finite time when the disturbance is absent and there is sufficient excitation. Among our results, one reveals a trade-off between "learning capacity"" and "disturbance robustness"": the less sensitive the identifier is to the disturbance, the less likely it is to learn the parameter. [ABSTRACT FROM AUTHOR]

This research investigates the controllability of linear and non‐linear fractional dynamical systems with distributed delays in control using the ψ$$ \psi $$‐Caputo fractional derivative. For controllability of linear systems, the positive definiteness of Grammian matrix, which is characterized by Mittag–Leffler functions, is used to provide necessary and sufficient conditions. For the controllability of non‐linear systems, the iterative technique with the completeness of X$$ X $$ is used to obtain sufficient conditions. Using the ψ$$ \psi $$‐Caputo fractional derivative, this study is new since it investigates the ideas of controllability. A couple of numerical results are offered to explain the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the operator spectrum theory is applied to study the general stabilisation issues for mean‐field stochastic Markov jump systems (MF‐SMJSs), where multiple delays, multiplicative noises and homogeneous Markov chain exist simultaneously. The innovative contributions are described as follows. On the one hand, a feasible model augmented strategy is adopted to transform the dynamics into an auxiliary delay‐free form. By introducing a delay‐dependent linear Lyapunov operator (DDLLO), the Lyapunov/spectrum stabilising conditions are constructed, which are both necessary and sufficient. On the other hand, in terms of spectral analysis technique, the notions of interval stabilisation and essential destabilisation are generalised to MF‐SMJSs for the first time. The necessary and sufficient stabilisation conditions are derived, respectively, which can be verified availably by LMI feasibility tests. To confirm the effectiveness of the theoretic results, two illustrative examples are included.

The current study investigates the stability analysis based on gain and phase margin (GPM) using fractional-order proportional-integral (FOPI) controller in a time-delayed single-area load frequency control (LFC) system with demand response (DR). The DR control loop is introduced into the classical LFC system to improve the frequency deviation. Although the DR enhances the system’s reliability, the excessive use of open communication networks in the control of the LFC results in time delays that make the system unstable. A frequency-domain approach is proposed to compute the time delay that destabilizes the system using GPM values and different parameter values of the FOPI controller. This method converts the equation into an ordinary polynomial with no exponential terms by eliminating the exponential terms from the system’s characteristic equation. The maximum time-delay values at which the system is marginally stable are calculated analytically using the new polynomial. Finally, the verification of the time delays calculated is demonstrated by simulation studies in the Matlab/Simulink environment and the root finder (quasi-polynomial mapping-based root finder, QPmR) algorithm to define the roots of polynomials with exponential terms providing information about their locations.

We present a novel approach to address integro-differential systems incorporating state, input, and output delays. Our approach leverages product spaces and employs a boundary perturbation technique. Initially, we focus on state-delay equations, wherein we introduce a variation of constants formula for the mild solution. Additionally, we establish spectral properties using a characteristic equation. Subsequently, we extend our analysis to integro-differential systems affected by state, input, and output delays. Notably, we demonstrate the equivalence between a such delay system and a regular free-delay system within the Salamon–Weiss framework. This equivalence sheds valuable insights on the nature of the integro-differential system under consideration. [ABSTRACT FROM AUTHOR]

Active disturbance rejection control (ADRC) has been gaining attention in recent years and has shown its performance in multiple applications including non‐linear ones, without the need of accurate models. Despite the good results of this technique, time delay can deteriorate the performance of ADRC, limiting its application. Here, the effect of time delay on the stability of a linear ADRC is analysed, using an alternative mathematical description, and a new effective design technique, based on a modified ADRC scheme, is proposed to overcome the delay effect while maintaining the disturbance rejection properties of the ADRC. An experimental example is discussed considering a system with low damped mechanical resonances, showing good results using the proposed technique. [ABSTRACT FROM AUTHOR]

Abstarct: This work presents a theoretical framework for the safety‐critical control of time delay systems. The theory of control barrier functions, that provides formal safety guarantees for delay‐free systems, is extended to systems with state delay. The notion of control barrier functionals is introduced, to attain formal safety guarantees by enforcing the forward invariance of safe sets defined in the infinite dimensional state space. The proposed framework is able to handle multiple delays and distributed delays both in the dynamics and in the safety condition, and provides an affine constraint on the control input that yields provable safety. This constraint can be incorporated into optimization problems to synthesize pointwise optimal and provable safe controllers. The applicability of the proposed method is demonstrated by numerical simulation examples. [ABSTRACT FROM AUTHOR]

Abstract Cloud energy storage is considered a promising application in future power systems. It focuses on optimally leveraging the capacity of centralized large‐scale energy storage compared with the requirements of small‐scale localized users. In this paper, to satisfy the small‐ and medium‐scale timely energy storage requirement from localized users, the concept of the cloud‐based location sharing energy storage is proposed. The modular mobile energy storage system is flexibly configured and deployed at different sites to fulfil the long‐term seasonally dynamic transformer capacity increment and short‐term daily energy arbitrage based on economic values. To optimize the overall incomes of the energy storage investment, a two‐step user potential identification algorithm is proposed to discover the most valuable users at different time scales from the regional power usage profile. Then, the number/capacity optimal planning algorithm is proposed to optimally share the mobile energy storage system among users seasonally and total profits are further increased through daily energy arbitrage at less valuable seasons. Finally, the structure and dispatching strategy of the cloud‐based location sharing energy storage management system is illustrated. Case study results prove that the proposed identification algorithm can excavate the most valuable users at different time scales and the optimal planning and operation strategy is able to guarantee the overall income benefits.

Investigation of the mutual synchronizatin system with delays, composed of n (n= 2p +1, p ∈ N) oscillators joined into a onedirectional ring, is carried out. The investigation is based on the use of eigenvectors of the matrix, which describes the structure of the internal links of the system.

Investigation of the threedirectional forced synhronization system, composed of four oscillators, is carried out. The exact expressions of the step responses functions for the synchronization system are derived and the transition process in the system is analysed.

In this paper, the operator spectrum theory is applied to study the general stabilisation issues for mean‐field stochastic Markov jump systems (MF‐SMJSs), where multiple delays, multiplicative noises and homogeneous Markov chain exist simultaneously. The innovative contributions are described as follows. On the one hand, a feasible model augmented strategy is adopted to transform the dynamics into an auxiliary delay‐free form. By introducing a delay‐dependent linear Lyapunov operator (DDLLO), the Lyapunov/spectrum stabilising conditions are constructed, which are both necessary and sufficient. On the other hand, in terms of spectral analysis technique, the notions of interval stabilisation and essential destabilisation are generalised to MF‐SMJSs for the first time. The necessary and sufficient stabilisation conditions are derived, respectively, which can be verified availably by LMI feasibility tests. To confirm the effectiveness of the theoretic results, two illustrative examples are included. [ABSTRACT FROM AUTHOR]

This paper considers an optimal sliding mode control based on the cost control guaranteed approach using the linear quadratic regulator method to stabilize delay fractional under involved disturbance. We propose an approach to an open research problem in the design of an LMI-based sliding mode controller in which there are some constraints such as optimizing system performance. The sliding mode technique is well-known as an effective tool for calculating the transient response of the system and achieving robust system performance. LQR classic techniques are less effective for studying an optimal fractional system in the presence of disturbance due to nonlinearity, so we use the optimal sliding mode approach control law designed for the nominal system and, then, combined it with a fractional sliding mode controller. By using the Razumikhin theorem for the stability of fractional order systems with delay and linear matrix inequality, conditions on asymptotically stabilization were obtained . The presented controller stabilizes the nominal system and guarantees an adequate level of system performance. The sliding mode controller presented in the article, in addition to eliminating the effect of disturbance in the system, is independent of the delay A numerical example was provided to illustrate the effectiveness of the main results.

Cloud energy storage is considered a promising application in future power systems. It focuses on optimally leveraging the capacity of centralized large‐scale energy storage compared with the requirements of small‐scale localized users. In this paper, to satisfy the small‐ and medium‐scale timely energy storage requirement from localized users, the concept of the cloud‐based location sharing energy storage is proposed. The modular mobile energy storage system is flexibly configured and deployed at different sites to fulfil the long‐term seasonally dynamic transformer capacity increment and short‐term daily energy arbitrage based on economic values. To optimize the overall incomes of the energy storage investment, a two‐step user potential identification algorithm is proposed to discover the most valuable users at different time scales from the regional power usage profile. Then, the number/capacity optimal planning algorithm is proposed to optimally share the mobile energy storage system among users seasonally and total profits are further increased through daily energy arbitrage at less valuable seasons. Finally, the structure and dispatching strategy of the cloud‐based location sharing energy storage management system is illustrated. Case study results prove that the proposed identification algorithm can excavate the most valuable users at different time scales and the optimal planning and operation strategy is able to guarantee the overall income benefits. [ABSTRACT FROM AUTHOR]

This article deals with observer design for a class of nonlinear systems subject to delayed output measurements. Using an observer structure borrowed from Targui et al., we propose novel linear matrix inequality conditions ensuring the asymptotic convergence of the estimation error towards zero. We demonstrate analytically that the established linear matrix inequalities are less conservative than that of Targui et al., from a feasibility viewpoint, in the sense that they tolerate larger values of the upper bounds of the delay while guaranteeing the asymptotic convergence of the observer. Such linear matrix inequality conditions are obtained due to the use of a specific Lyapunov–Krasovskii functional, the Young inequality in a judicious way and a reformulation of the Lipschitz condition in a convenient way. We provide two illustrative examples to support the efficiency and superiority of the proposed linear matrix inequality–based techniques. [ABSTRACT FROM AUTHOR]

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and Ω . We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam–Hyers stability of a second-order semi-linear time-delayed system. [ABSTRACT FROM AUTHOR]

In this article, we address the problem of adaptive state observation of linear time‐varying systems with delayed measurements and unknown parameters. Our new developments extend the results reported in our recently works. The case with known parameters has been studied by many researchers. However in this article we show that the generalized parameter estimation‐based observer design provides a very simple solution for the unknown parameter case. Moreover, when this observer design technique is combined with the dynamic regressor extension and mixing estimation procedure the estimated state and parameters converge in fixed‐time imposing extremely weak excitation assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we study controllability properties of time-invariant linear delay-differential (d-d) systems with a single delay in the pseudo-state. We adopt a topological and geometric point of view and derive a formula for the distance of an approximately controllable system from uncontrollability. We demonstrate by examples that approximately controllable d-d systems may have zero distance from uncontrollability. Thus, the set of approximately controllable d-d systems is not open in the parameter space. To remedy this anomaly, we propose a slight strengthening of the controllability concept and introduce the well-posed property of strict controllability. We show that strictly controllable d-d systems form an open dense subset in the parameter space. Moreover, we show that a d-d system is strictly controllable if and only if it has a positive distance from uncontrollability. Finally, we prove that the d-d systems that are not strictly controllable form a closed set of Lebesgue measure zero and can be represented as the union of a proper algebraic variety and a proper local analytic variety in the parameter space. [ABSTRACT FROM AUTHOR]

For time-invariant finite-dimensional systems, it is known that global asymptotic stability (GAS) is equivalent to uniform GAS (UGAS), in which the decay rate and transient overshoot of solutions are requested to be uniform on bounded sets of initial states. This paper investigates this relationship for time-invariant delay systems. We show that UGAS and GAS are equivalent for this class of systems under the assumption of robust forward completeness, i.e., under the assumption that the reachable set from any bounded set of initial states on any finite time horizon is bounded. We also show that, if the state space is a space in a particular family of Sobolev or H\"older spaces, then GAS is equivalent to UGAS and that robust forward completeness holds. Based on these equivalences, we provide a novel Lyapunov characterization of GAS (and UGAS) in the aforementioned spaces. [ABSTRACT FROM AUTHOR]

This paper considers an optimal sliding mode control based on the cost control guaranteed approach using the linear quadratic regulator method to stabilize delay fractional under involved disturbance. We propose an approach to an open research problem in the design of an LMI-based sliding mode controller in which there are some constraints such as optimizing system performance. The sliding mode technique is well-known as an effective tool for calculating the transient response of the system and achieving robust system performance. LQR classic techniques are less effective for studying an optimal fractional system in the presence of disturbance due to nonlinearity, so we use the optimal sliding mode approach control law designed for the nominal system and, then, combined it with a fractional sliding mode controller. By using the Razumikhin theorem for the stability of fractional order systems with delay and linear matrix inequality, conditions on asymptotically stabilization were obtained. The presented controller stabilizes the nominal system and guarantees an adequate level of system performance. The sliding mode controller presented in the article, in addition to eliminating the effect of disturbance in the system, is independent of the delay A numerical example was provided to illustrate the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

Within the current paper, we design a sliding-based control law to stabilize a set of systems that are nonlinear, fractional order involve delay, perturbation, and uncertainty. A control law-based sliding mode is considered in such a way that the variables of the closed loop system reach the sliding surface in a limited time and stay on it for later times. Then, using the Razomokhin stability theorem, the stability of the systems is proved and in the end, a calculation is found to search for useful methods. [ABSTRACT FROM AUTHOR]

Aiming at efficient solution of optimal control problems for continuous nonlinear time-delay systems, a multiple shooting algorithm with a lifted continuous Runge-Kutta integrator is proposed. This integrator is in implicit form to remove the restriction of smaller integration step sizes compared with delays. A tangential predictor is applied in the integrator such that Newton iterations required can be reduced considerably. If one Newton iteration is applied, the algorithm has the same structure as direct collocation algorithms whereas derives a condensed nonlinear programming problem. Then, the solution of variational sensitivity equation is decoupled from forward simulation by utilizing the implicit function theorem. Under certain conditions, this function evaluation and derivative computation procedure is proved to be convergent with a global order. Complexity analysis shows that the computational cost can be largely reduced by this lifted multiple shooting algorithm. Then, parallelizable optimal control solver can be constructed by embedding this algorithm in a general-purpose nonlinear programming solver. Simulations on a numerical example demonstrate that the computational speed of multi-threading implementation of this algorithm is increased by 36 % compared with non-lifted one, and increased by a factor of 6.64 compared with traditional sequential algorithm; meanwhile, the accuracy loss is negligible. [ABSTRACT FROM AUTHOR]

This article studies the weighted memory-event-triggered $H_{\infty}$ static output control issue of Takagi–Sugeno fuzzy wind turbine systems with uncertainty. To decrease the frequency of data communication, a novel adaptive memory-event-triggered mechanism is presented to choose the “necessary” control signals, which has the following two benefits. First, a weighted average signal over a historic period is utilized as the input of event-triggered scheme, instead of the current system information in the conventional one. This could reduce the control signal updating rate and avoid the false triggering events incurred by stochastic environment noises and disturbances. Second, a dynamic triggering threshold is adopted to adaptively regulate the control signal updating frequency along with the average signal. By applying the distributed delay system method to describe the weighted historic signal, a new uncertain T–S fuzzy wind turbine system with distributed delay is established. With the aid of the measured outputs and the integral inequality based on the weighting function of the average signal, the memory-event-triggered static output controller design conditions are obtained to ensure the system exponential stability and the $H_{\infty}$ performance. Lastly, an experiment platform integrating Zigbee modules as the wireless network is set up to illustrate the advantages of the proposed strategy. [ABSTRACT FROM AUTHOR]

We consider $\dot {\textrm {X}}(\textrm {t})=\textrm {X(t)}^{2}+\textrm {U}(\textrm {t}-\textrm {D}(\textrm {t}))$ , where D(t) is a long time-varying delay. If D(t) = 0, $\textrm {U(t)}=-\textrm {X(t)}^{2}-\textrm {cX(t)},\,\,\textrm {c}>0$ is a simply control, but it just delays finite time escape for this system. We design a predictor control and prove that the attraction region is $\textrm {X}(0) + \sup _{\theta \in [\varphi (0),\,0]} \int _{\varphi (0)}^{\theta } {\frac {\textrm {U}(\theta)}{\varphi ^{\prime }(\varphi ^{-1}(\theta))}\textrm {d}\theta } < \frac {1}{\sigma (0)}$ , with $\varphi (\theta)=\theta -\textrm {D}(\theta)$ , and $\sigma (\theta)=\varphi ^{-1}(\theta)$. Further, the predictor control locally exponentially stabilizes this system. [ABSTRACT FROM AUTHOR]

This brief work focuses on the stability of a novel hybrid dynamical system formulated by hybrid phenomena, finite memory and initial data. To solve this issue, the main idea is to explore the pre-asymptotic stability and uniform pre-asymptotic stability of a hybrid delay model with initial data by developing the unique properties of hybrid memory domain, hybrid memory arc, generalized hybrid memory solution and continuation. Then, by introducing the Razumikhin-type function method, new pre-asymptotic stability and uniform pre-asymptotic stability theorems for hybrid delay systems with initial data are proposed in the framework of generalized hybrid solutions containing finite memory and initial data. Finally, an example analysis is conducted to reveal the validity of the proposed criteria. [ABSTRACT FROM AUTHOR]

The paper presents a stability analysis of a closed-loop system with an active disturbance rejection control (ADRC) algorithm and a reduced-order extended state observer (RESO). The controller is designed for first-order plus delay time processes, and one of the input signals to the RESO is delayed to compensate for the effect of the system delay. If the process delay is known, perfect synchronization between the observer input signals can be achieved. In a more realistic case, the process delay is not precisely known, and the resulting imperfect synchronization may lead to instability of the closed-loop system. The perfect synchronization case shows that the closed-loop system can be stabilized for any delay. For the imperfect synchronization case, delay-dependent stability conditions are derived for a simple tuning method. • Stability of delayed systems with active disturbance rejection control is studied. • Stability conditions for the closed-loop system with two delays are derived. • The effect of imperfect delay time compensation on the system's stability is analyzed. • Sufficient delay-dependent stability conditions for a simple tuning method are given. [ABSTRACT FROM AUTHOR]

In this study, the Takagi–Sugeno (T–S) fuzzy impulsive control problem is investigated for a class of nonlinear positive systems with time delay. The time delay under consideration is both in the continuous-time dynamics and at the impulsive instants, which can model practical systems more accurately. An impulse-time-dependent copositive Lyapunov function (IDCLF) is constructed, and the Razumikhin technique is adopted to develop conditions that ensure the globally exponential stability of T–S fuzzy positive systems with delayed impulses. The size constraint between the impulse delay and the bound of impulsive intervals is removed. A T–S fuzzy impulsive controller is designed in terms of the solutions to certain vector inequalities that are readily solvable. Numerical examples and a practical example of lipoprotein metabolism and potassium ion transfer model are given to demonstrate the effectiveness, advantages, and practicality of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, a sliding mode control for systems that are nonlinear, having fractional order, and involve delay is used. The aim of this paper is to design a sliding mode controller, such that the closedlooped nonlinear system becomes asymptotically stable and its trajectory can be driven onto the sliding surface in finite time. By using the fractional Razumikhin theorem for the stability of fractional-order systems including delay and a linear matrix inequality, necessary conditions on asymptotic stabilization are obtained. Some numerical examples are given to illustrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and Ω. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam–Hyers stability of a second-order semi-linear time-delayed system.

The main advantage of the incremental dynamics formulation is its property of replacing state-dependent aspects of the system dynamics by measurements, solely requiring modeling the input-dependent dynamics. However, incremental controllers assume that state derivative measurements are available. The lack of accurate sensors and communication delay may appear as limitations for the use of incremental techniques. This article applies a second-order differentiator (SOD) for estimating state derivatives and attenuate measurement noise, and an input scaling gain (ISG) for attenuating the effects of high-frequency noise and delay in the feedback loop. A complete analysis of the combined solution is carried out using an incremental nonlinear dynamic inversion (INDI) strategy. First, a simple linear single-input single-output model is used to evaluate the influence of design parameters in the closed-loop response and stability. Then, the combined SOD and ISG approach is illustrated in a more complex case application, where INDI is used to control the lateral motion of an autonomous airship, results of which corroborate the ISG as an asset to increase the maximum delay allowed in the feedback loop. [ABSTRACT FROM AUTHOR]