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This article investigates the global stabilization problem of the circular orbit rendezvous system with actuator saturation and time-delay. By decomposing the linearized relative motion equations into a cascade of neutral stable linear systems, linear state feedback controllers are proposed in the presence of both actuator saturation and/or time-delay. The global stability of the closed-loop system is proved. Optimal feedback gain is also obtained in the delay-free case. Simulation results are given to show the effectiveness of the presented methods.

This paper addresses a version of the stochastic linear quadratic control problem on time scales S Δ LQ , which includes the discrete time and continuous time as special cases. Riccati equations on time scales are given, and the optimal control can be expressed as a linear state feedback. Furthermore, we present the uniqueness and existence of the solution to the Riccati equation on time scales. Furthermore, we give an example to illustrate the theoretical results.

In this paper, the global Mittag–Leffler stabilization of fractional-order BAM neural networks is investigated. First, a new lemma is proposed by using basic inequality to broaden the selection of Lyapunov function. Second, linear state feedback control strategies are designed to induce the stability of fractional-order BAM neural networks. Third, based on constructed Lyapunov function, generalized Gronwall-like inequality, and control strategies, several sufficient conditions for the global Mittag–Leffler stabilization of fractional-order BAM neural networks are established. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows us to obtain necessary and sufficient conditions assuring the existence of a sampled-data Nash equilibrium strategy, extending earlier results to a general context with more than two players. Furthermore, we provide a numerical algorithm for calculating the feedback matrices of the Nash equilibrium strategies. Finally, we illustrate the effectiveness of the proposed algorithm by two numerical examples. As both situations highlight a stabilization effect, this confirms the efficiency of our approach.

Nonautonomous discrete-time control systems with uncertain nonlinearities and bounded external disturbances are considered. Based on the method of matrix comparison systems and the technique of difference linear matrix inequalities, an approach to solve the problems of state estimation, finite time boundedness with respect to given sets, the suppression of initial deviations and uncertain disturbances using a linear state feedback controller is developed. A method to design a controller with variable coefficients that guarantees the transition from one given ellipsoid to another under any disturbances bounded by the L∞ norm is proposed.

This paper presents a Model Reference Adaptive Control (MRAC) based framework for vehicle adaptive cruise control systems. The linear longitudinal dynamic model is created with uncertainties in the state and input matrix. The controller structure aims to handle these uncertainties and also maintains the desired distance with the lead vehicle. To evaluate the performance of the proposed controller, several simulation studies are conducted by comparing the results with a linear state feedback controller via MATLAB/Simulink. Whereas the linear controller underperforms under uncertain conditions, it is seen that MRAC achieves the desired performance despite model uncertainties.

This paper deals with the global Mittag-Leffler synchronization of fractional-order memristive neural networks (FMNNs) with time delay. Since the FMNNs are essentially a class of switched systems with irregular switching laws, it is more difficult to achieve synchronization than with the traditional neural networks. First, under the framework of fractional-order differential inclusions and set-valued maps, the FMNNs are transformed into a continuous system with uncertainties. Then a linear state feedback combined with switching control law is designed in order to achieve the Mittag-Leffler synchronization. In addition, several synchronization criteria are obtained by constructing appropriate Lyapunov functionals, together with the help of some inequality techniques. Finally, an example is given to demonstrate the effectiveness of the obtained results.

In this article, we propose a new computationally efficient algorithm for computing an outer convex robust positively invariant (RPI) approximation to the minimal robust positively invariant (mRPI) set for polytypic linear difference inclusion (PLDI) systems with additive disturbances. The LDI modelling framework is useful to analyse parametrically uncertain, time-varying linear system or switching linear discrete-time systems. The disturbance which is considered in this paper, is bounded by a polytypic set and acts additively on the state of the system. It is also assumed that the nominal LDI system is absolutely asymptotically stable by a stabilizing linear state feedback. The accuracy of the approximation can be set in advance. The proposed algorithm has far less computational burden in comparison with existing algorithms.

In this paper, we aim to solve the stabilization problem for a large class of fractional-order nonautonomous systems via linear state feedback control and adaptive control. By constructing quadratic Lyapunov functions and utilizing a new property for Caputo fractional derivative we derive some sufficient conditions for the global asymptotical stabilization of a class of fractional-order nonautonomous systems. We give two illustrative examples to validate the effectiveness of the theoretical results.

The problem of global stabilization of the circular orbit spacecraft rendezvous system subject to actuator saturation is studied in this brief. Based on the linearized Clohessy–Wiltshire equations, a class of saturated linear state feedback controllers is designed by utilizing Lyapunov functions that are frequently used in the absolute stability theory. The global stability of the closed-loop system with the proposed saturated linear feedback is proved by providing explicit conditions on the parameters in the feedback gains. Optimal linear feedback gains are also discussed. The most significant advantages of the proposed controllers are that the controllers are linear and the global stability of the resulting closed-loop system can be guaranteed. Numerical simulations show the effectiveness of the proposed approaches.

This paper addresses a version of the linear quadratic control problem for mean-field stochastic differential equations with deterministic coefficients on time scales, which includes the discrete time and continuous time as special cases. Two coupled Riccati equations on time scales are given and the optimal control can be expressed as a linear state feedback. Furthermore, we give a numerical example.

For discrete linear time-varying systems with bounded system matrices we discuss the pole assignment problem utilizing linear state feedback. It is shown that uniform complete controllability is sufficient for the Lyapunov exponents being arbitrarily assignable by choosing a suitable feedback.

This paper is a note on the stability and robustness of the Nash Game (Vamvoudakis et al. in Adaptive optimal control algorithm for zero-sum nash games with integral reinforcement learning, 2012; Ning et al. in Optim Control Appl Methods. doi: 10.1002/oca.2042 , 2012; Bouyer et al. in Concurrent games with ordered objectives, 2012) based mixed $${\mathcal {H}}_{2}/{\mathcal {H}}_{\infty }$$ Model Predictive Controllers (Aadaleesan and Saha in Mixed $$\mathcal {H}_{2}/\mathcal {H}_{\infty }$$ Model Predictive Control for Unstable and Non-Minimum Constrained Processes, 2008; Aadaleesan in Nash Game based Mixed $$\mathcal {H}_{2}/\mathcal {H}_{\infty }$$ Model Predictive Control applied with Laguerre-Wavelet Network Model, 2011) for linear state feedback systems addressed in Part 1 (Aadaleesan and Saha in Int J Dyn Control, 2016) of this series. The mixed $${\mathcal {H}}_{2}/{\mathcal {H}}_{\infty }$$ MPC proposed in Part 1 (Aadaleesan and Saha in Int J Dyn Control, 2016) and that developed by Orukpe et al. (Model predictive control based on mixed $$\mathcal {H}_2/\mathcal {H}_{\infty }$$ control approach, 2007) are compared in this Part 2. The issues of stability and robustness of the multi-criterion optimal control are dealt in this paper using set theoretic concepts.

This paper takes advantage of wide-area measurements to propose a centralized nonlinear controller that acts on power system stabilizers (PSSs), to cooperatively increase the damping of problematic small-signal oscillations all over the system. Details are given to utilize existing stabilizers while causing minimum changes to the equipment and warranting improvement, or at least no detriment, of current system behavior. A set of gains is designed by linear state feedback control to respond to representative system conditions, and these are coordinated with an equal number of decision trees in charge of scheduling by switching. The resulting nonlinear controller is implemented on a model that resembles the characteristics of the Central American System, exemplifying its feasibility to be applied in large systems. The effect of time delay in the control loop is assessed, and the limits for tolerance to latency in this application are determined.

A new 4D chaotic system with infinitely many equilibria is proposed using a linear state feedback controller in the Sprott C system. Although the new 4D chaotic system has only two nonlinear terms, it has rich dynamic characteristics, such as hidden attractors and coexisting attractors. Besides, the freedom of offset boosting of a variable is achieved by adjusting a controlled constant. The dynamic characteristics of the new chaotic system are fully analyzed from the aspects of phase portraits, bifurcation diagrams, Lyapunov exponents and Poincaré maps. The corresponding analogue electronic circuit is designed and implemented to verify the new 4D chaotic system. By taking advantage of the complex dynamic properties of the new chaotic system, a random number generator algorithm is proposed.

A number of methods for estimating the DoA and properties of a null controllable region have been presented for continuous-time and discrete-time systems with actuator saturation [47, 52, 108] and the references therein. Some properties of the null controllable region have been presented for DOSs with actuator saturation in the previous chapters. It has been shown in [49] that a boundary of the DoA under a saturated stabilizing linear state feedback is a unique stable limit cycle of its time-reversed system .

International audience; We propose a hybrid solution for the stabilization of the origin of a linear time-invariant stabilizable system with the property that a suitable neighborhood of a pre-defined unsafe point in the state space is avoided by the closed-loop solutions. Hybrid tools are motivated by the fact that the task at hand cannot be solved with continuous feedback, whereas the proposed hybrid solution induces nominal and robust asymptotic stability of the origin. More specifically, we formulate a semiglobal version of the problem at hand and describe a fully constructive approach under the assumption that the unsafe point to be avoided does not belong to the equilibrium subspace induced by the control input on the linear dynamics. The approach is illustrated on a numerical example

In this paper, stabilization control of fractional-order bidirectional associative memory neural networks is formulated and studied. By estimating Mittag-Leffler function and some novel analysis techniques of fractional calculation, a generalized Gronwall-like inequality of Caputo fractional derivative is established. Then by applying Lyapunov approach, linear state feedback control law and partial state feedback control law are presented to stabilize the fractional-order bidirectional associative memory neural networks. This analysis framework can be applied to closed-loop control of fractional-order systems. A numerical example is given to show the effectiveness of the derived results via computer simulations.

In this note, the exponential synchronization of a class of chaotic uncertain systems is addressed. A linear state feedback control is used which guarantees the synchronization when uncertainties are present both in linear and nonlinear parts of the slave system dynamic. It is shown that, with some a-priori rough bounds on the slave uncertainties, the closed loop stability can be guaranteed. Numerical simulations are presented to show the effectiveness of the proposed method.

A new approach is proposed to the rejection of bounded exogenous disturbances in linear control systems via use of the so-called combined feedback of the form u = Kx + K1w. Along with the static linear state feedback, this control law contains a linear feedback from the vector of disturbances (or some of its components) whose instantaneous values are assumed to be available. The control design procedure is based on the linear matrix inequality technique; it is characterized by simplicity and ease of implementation and reduces to solving convex optimization problems. The combined feedback design is also performed in the sparse formulation, which can be thought of as a desire to reduce the control resource required to handle the system.

In this paper, we study quantized and delayed state feedback control of the linear time invariant (LTI) systems by considering the logarithmic quantizer and the quantized control input. The concepts of quadratic stabilization and control Lyapunov functionals (CLFs) are extended to the LTI systems with delays and quantized signals. After generalizing the sector bounded approach into LTI systems, we propose that a LTI time-delay system via the linear state feedback and quantization is quadratic stabilization if and only if the corresponding uncertain auxiliary system is quadratic stabilization under the case of the single control input or the multiple control inputs. Two examples are provided to illustrate the effectiveness of the proposed methods.

This paper studies the H ∞ control problem for a class of neural networks with discontinuous activations and nonlinear external disturbance. Firstly, under the framework of Filippov solution, we investigate the existence of the solution for the discontinuous perturbation neural networks. Secondly, we design linear state feedback controllers and adaptive controllers to ensure the stabilization of the discontinuous neural networks with a generalized H ∞ disturbance attenuation level ρ. Finally, we give two examples to show the effectiveness of the designed controllers and the correction of the theoretical results.

International audience; In this paper, a new controller is proposed, based on a well known homogeneous controller. The suggested controller gives rise to an efficient trade-off between the standard linear state feedback and twisting algorithm. As a consequence, the obtained controller has the advantages of both controllers with their drawbacks reduced. To achieve this objective, a parameter on the exponent terms of the homogeneous controller is adapted. The convergence of the closed loop system to a vicinity of the origin is given. Finally, some simulations validate the effectiveness of the proposed controller.

In this paper time-variant systems with sector nonlinearities tending to be affected by bounded external disturbances are considered. We present an approach to solving problems of state estimation, stability, boundedness relatively to given sets, rejection of bounded external disturbances and initial deviations by using static linear state feedback, based on matrix comparison systems method and differential linear matrix inequalities.

This paper reports on an experimental study on use of static full state linear feedback for link tracking control of a compliant humanoid leg, cCub. Passive elasticity is used in pitch joints of the cCub legs, offering many benefits in terms of safe interaction as opposed to rigid robots. However, the elasticity makes the link tracking problem more challenging in particular tasks when the robots joints are under large gravitational forces or external disturbances, such as going up the stairs. Therefore, this study explores the experimental results of applying LQR control, formulated using Linear Matrix Inequality in discrete time and implemented on a humanoid robot. A summary of the theory used and a numerical simulation are presented. The proposed LQR controller provides a systematic way of designing the feedback gains with desired closed loop dynamics and it is shown to have good tracking performance in the experiments.

Laser cladding is a technique that is frequently used for the coating and repair of metallic components. More recently, the technology is used in the additive manufacturing domain for building freeform three-dimensional parts. A lot of attention is dedicated to the optimization of process parameters and to real-time feedback control strategies. This paper presents a feedback control scheme in which a hyperspectral camera is used to provide absolute temperature measurements of the melt pool surface with a high spatial resolution of 12 ${\mu }$ m/pixel. A combined linear state feedback and proportional-integral controller actuates the laser based on the measured temperature profile in order to maintain a constant melt pool width. The controller includes a model-based state observer for suppressing the noise introduced by powder particles that are present in the laser cladding process with coaxial powder feeding. The performance of the controller is evaluated by creating beads with varying thicknesses on a base plate of AISI 316L stainless steel. The experimental results demonstrate that the controller is able to successfully regulate the melt pool size with a standard deviation that is smaller than 10 ${\mu }$ m during laser melting (without powder) and 50 ${\mu }$ m during laser cladding (with powder).

This paper presents a novel integral sliding mode control (ISMC) for singularly perturbed systems (SPSs) with matched disturbances. Firstly, a linear state feedback control law for the fast subsystem of SPS is designed, and the slow subsystem can be obtained. Secondly, an ISMC incorporating the disturbance estimate provided by a disturbance observer is designed. In order to attenuate matched disturbances, the Lyapunov function is employed to determine the gain of ISMC by solving a linear matrix inequality (LMI). As a result, the closed-loop slow subsystem under the ISMC is exponentially stable, and the full-order SPSs are exponentially stable under the composite control law. Finally, the effectiveness and advantages of the proposed method are demonstrated by simulations.

The dynamic linear state feedback control problem is addressed for a class of nonlinear systems subject to time-delay. First, using the dynamic change of coordinates, the problem of global state feedback stabilization is solved for a class of time-delay systems under a type of nonhomogeneous growth conditions. With the aid of an appropriate Lyapunov-Krasovskii functional and the adaptive strategy used in coordinates, the closed-loop system can be globally asymptotically stabilized by the dynamic linear state feedback controller. The growth condition in perturbations are more general than that in the existing results. The correctness of the theoretical results are illustrated with an academic simulation example.

In this paper we consider the problem of stabilizing a bilinear system with time- varying delay via linear state feedback control. Based on the Lyapunov method, a delay-dependent criterion for determining the stabilization of system is obtained in terms of linear matrix inequalities (LMIs) and used to express the relationships between the terms in the Leibniz-Newton formula, which can be easily solved by efficient convex optimization algorithms. From the numerical examples, the obtained results have some significant improvements over the recent literatures.

This paper investigates the maximal perturbation bound problem for robust stabilizability of the fractional-order system with two-norm bounded perturbations or infinity-norm bounded perturbations. Firstly, a necessary condition and several sufficient conditions for robust stabilization are derived. Secondly, linear matrix inequality approaches for computing the maximal robust stabilizability perturbation bound of such perturbed fractional-order system with a linear state feedback controller, simultaneously obtaining the corresponding linear state feedback stabilizing controller are presented. With the help of the linear matrix inequality solvers, we can easily obtain the maximal robust stabilizability perturbation bound and the corresponding linear state feedback stabilizing controller. Finally, simulation examples are given to demonstrate the effectiveness of the proposed approaches.

This paper reports a new fractional-order four-dimensional (4D) smooth autonomous system which is special since it can generate a four-wing hyperchaotic attractor but has only one zero equilibrium. The issue of robust control for the fractional-order hyperchaotic system with uncertainties and interference is considered. On the basis of the fractional Lyapunov stability theorem, a linear state feedback controller is derived to guarantee robust Mittag–Leffler stability of the fractional-order hyperchaotic system. The control strength matrix is obtained by solving the linear matrix inequality (LMI). The effectiveness and robustness of the proposed scheme is verified via numerical simulation.