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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.

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.

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.

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.

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 .

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.

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.

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.

Selection of control parameters is an important issue in the field of control design. This selection depends on the initial condition of the systems. On the other hand, controller parameters should be adjusted under all initial conditions to reach the optimal performance. To overcome this problem, in this paper, an online optimal linear state feedback control is introduced. Firstly, a straightforward particle swarm optimization is used to optimize the state feedback control parameters under some certain initial conditions. Then, in order to adapt the optimal controller to different initial conditions, the moving least squares approximation is used. The proposed technique is applied to an inverted pendulum system. The feasibility and efficiency of the proposed controller are assessed in simulation results.

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.

This paper considers the stabilization and robust stabilization of fractional-order linear systems subject to input saturation with the fractional order α : 0 < α < 1 case. Using a linear state feedback control, a sufficient condition based on linear matrix inequality (LMI) for the asymptotical stabilization of fractional-order linear systems subject to input saturation is presented. Further, based on the proposed stabilization condition, a condition based on LMI for robust stabilization of fractional-order linear systems subject to input saturation is derived. Finally, a numerical example is given to prove the applicability of the proposed method.

The main contribution of this paper is to completely characterize the dynamic behavior of the discrete-time double integrator with a saturated locally stabilizing linear state feedback law. In continuous-time setting, any linear state feedback control law that locally stabilizes the double integrator also globally stabilizes the system in the presence of actuator saturation. In discrete-time setting, the equivalent of the double integrator does not have the same property. In this paper, we completely characterize the global behavior of saturated locally stabilizing linear state feedback laws for the discrete-time double integrator.

We consider the use of linear multivariable feedback control to achieve a nonovershooting and nonundershooting step response. Recently, Schmid and Ntogramatzidis (2010) [13] introduced a linear state feedback controller design method to avoid overshoot. In this paper, we describe conditions under which the design method may be modified to avoid undershoot. The method is applicable to square and nonsquare systems, minimum and nonminimum phase systems, and also strictly proper and bi-proper systems.

This paper is concerned with local stabilization for unstable bilinear systems with input saturation. Given a prespecified polytope \(\mathcal{P}\) of the state space containing the zero equilibrium point, a linear state feedback is designed to guarantee that the closed-loop system is asymptotically stable in \(\mathcal{P}\) and \(\mathcal{P}\) is enclosed in the domain of attraction of the zero equilibrium point. Such sufficient conditions are derived via linear matrix inequalities (LMIs). Finally, an example illustrates the effectiveness of the proposed method.

Decentralized robust control problem is investigated for a class of large scale neutral delay systems. The considered systems have mismatches in time delay functions. A state coordinate transformation is first employed to change the original system into a cascade system. Then the virtual linear state feedback controller is developed to stabilize the first subsystem. Based on the virtual controller, a memoryless state feedback controller is constructed for the second subsystem.

The stabilization of switched linear systems with constrained inputs (SLSCI) is considered. The authors design admissible linear state feedbacks and the switching rule which has a minimal dwell time (MDT) to stabilized the system. First, for each subsystem with constrained inputs, a stabilizing linear state feedback and an invariant set of the closed-loop system are simultaneously constructed, such that the input constraints are satisfied if and only if the closed-loop system’s states lie inside this set. Then, by constructing a quadratic Lyapunov function for each closed-loop subsystem, an MDT is deduced and an MDT-based switching strategy is presented to ensure the stability of the switched system.

This paper discusses the control problems of singular linear time-invariant delay systems with uncertainties. A new adaptive variable structure control strategy is given by Lyapunov stability theory. The invariance of sliding mode is proved in the circumstances of the external disturbance and parameter perturbation with matching conditions. Through memory and memoryless of the linear state feedback, the control strategy ensure that system reaches sliding mode in finite time and the closed-loop system is global asymptotic stable. Simulation results further show that the strategy is feasible and effective.

We consider the use of linear multivariable state feedback control to achieve a nonundershooting step response. The recently introduced nonovershooting linear state feedback controller design method of Schmid and Ntogramatzidis (2010) is here modified to yield a step response that is nonundershooting. The design method is applicable to square and non-square systems, minimum and nonminimum phase systems, and also strictly proper and bi-proper systems.

This letter introduces a hyperchaotic system from the Lü system [Lü et al., 2004] with a linear state feedback controller. This hyperchaotic system has more complex dynamical behaviors, and can generate 4-scroll hyperchaotic attractor and 2-scroll chaotic attractor under different control parameters. In particular, the system can also exhibit novel coexisting intermittent chaotic orbits. Theoretical analyses and simulation experiments are conducted to investigate the dynamical behaviors of the proposed hyperchaotic system.

In this paper, based on the stability properties of a passive system, a simple linear state feedback controller is proposed to realize the stability control of a unified chaotic system. Using this method, we can render the non-passive unified chaotic system to be equivalent to a passive one. Simulation results are shown to verify the effectiveness of the proposed controller.

The main contribution of this paper is to point out the vast difference between a discrete-time double integrator subject to actuator saturation and a continuous-time double integrator subject to actuator saturation. In continuous-time, any linear state feedback control law which locally stabilizes the double integrator, also globally stabilizes the double integrator in the presence of actuator saturation. In discrete-time, the equivalent of the double integrator has intrinsically different behavior. Some linear feedbacks which locally stabilize the double integrator, also globally stabilize the double integrator in the presence of actuator saturation. However, other linear feedbacks which locally stabilize the double integrator, do not globally stabilize the double integrator in the presence of actuator saturation. We classify a class of linear feedback laws which achieve local stability but also yield nonzero periodic solutions in the presence of actuator saturation and therefore cannot achieve global asymptotic stability.

This paper shows new convergence properties of constrained linear discrete time system with bounded disturbances under Model Predictive Control (MPC) law. The MPC control law is obtained using an affine disturbance feedback parametrization with an additional linear state feedback term. This parametrization has the same representative ability as some recent disturbance feedback parametrization, but its choice together with an appropriate cost function results in a different closed-loop convergence property. More exactly, the state of the closed-loop system converges to a minimal invariant set with probability one. Deterministic convergence to the same minimal invariant set is also possible if a less intuitive cost function is used. Numerical experiments are provided that validate the results.

This paper investigates the synchronization of coupled chaotic systems with time delay in the presence of parameter mismatches by using intermittent linear state feedback control. Quasi-synchronization criteria are obtained by means of a Lyapunov function and the differential inequality method. Numerical simulations on the chaotic systems are presented to demonstrate the effectiveness of the theoretical results.

A strategy to control the complex dynamical networks for global synchronization as well as to tune their synchronous states is presented based on linear state feedback controllers. The synchronous state can be changed from one system to other similar system just by choosing a suitable tuning matrix. Especially, some of the criteria are expressed in normal inequalities instead of matrix inequalities. Simulations are also done to verify the theoretical results.

International audience; In this chapter we consider the stability analysis of bilinear systems controlled via a sampled-data state feedback controller. Sampling periods may be time-varying and subject to uncertainties. The goal of this study is to find a constructive manner to estimate the maximum allowable sampling period (MASP) that guarantees the local stability of the system. Stability criteria are proposed in terms of linear matrix inequalities (LMI).

The implementation, testing and performance evaluation of a partial feedback linearising force tracking controller on an electrohydraulic actuator is described. The underlying assumptions necessary for the development of the controller are highlighted, and the control law is derived in detail. Performance comparisons are conducted against a linear state feedback with integral controller and a standard PID controller, the latter being the most common industrial solution. Results show that the nonlinear partial feedback linearising controller has improved tracking properties, as might be expected from its use of more modelling and feedback information. Multiple experiments are also conducted to investigate the robustness of the system to certain model parameters; it is shown that the controller tolerates a measurable shift in these parameters.

In this note, we revisit the problem of global practical stabilization for planar linear systems subject to actuator saturation and input additive disturbances. A parameterized linear state feedback law is designed such that, by tuning the value of the parameter, all trajectories of the closed-loop system converge to an arbitrarily small neighborhood of the origin in a finite time and remain in there.

In this paper we consider infinite dimensional systems which are subjected to stochastic structured multiperturbations. We first characterize the stability radii of these systems in terms of a Lyapunov equation and the corresponding Lyapunov inequalities. Then we investigate the problem of maximizing the stability radius by linear state feedback. We show that the supremal achievable stability radius can be determined via the resolution of a parametrized Riccati equation. Illustrative examples are included. Copyright © 2006 John Wiley & Sons, Ltd.

This paper addresses the robust input-output energy decoupling problem for uncertain singular systems in which all parameter matrices except E exist as time-varying uncertainties. By means of linear matrix inequalities (LMIs), sufficient conditions are derived for the existence of linear state feedback and input transformation control laws, such that the resulting closed-loop uncertain singular system is generalized quadratically stable and the energy of every input controls mainly the energy of a corresponding output, and influences the energy of other outputs as weakly as possible.

The paper extends the concept of robust controllability via linear state feedback to stochastic uncertain systems. We show that the controllability of a stochastic uncertain system can be characterized using solutions to a game-type differential Riccati equation.

The problem of decentralized control is considered for a class of time-varying large scale systems with uncertainties and external disturbances in the interconnections. In this paper, the upper bounds of the uncertainties and external disturbances are assumed to be unknown. The adaptation laws are proposed to estimate such unknown bounds, and by making use of the updated values of these unknown bounds, a class of decentralized linear and non-linear state feedback controllers are constructed. It is shown that by employing the proposed decentralized non-linear state feedback controllers, the solutions of the resulting adaptive closed-loop large scale system can be guaranteed to be uniformly bounded, and the states are uniformly asymptotically stable. By using the decentralized linear state feedback controllers, one can guarantee the uniform ultimate boundedness of the resulting adaptive closed-loop large scale system. Finally, a numerical example is given to demonstrate the validity of the results.

Although E 1 -optimal linear state feedback controllers are known to be dynamic, it has been shown that suboptimal performance arbitrarily closed to optimal can be achieved by using a static non-linear feedback law. In this paper, this fact is established by using a novel approach which shows that the result is a natural consequence of elementary state-space theory. The approach is motivated by recent works in active vision systems, which have considered a state-feedback problem tightly connected with E 1 optimization. This problem, which has independent interest, is discussed in some detail. The new formulation of the problem provides additional insight in E 1 state-feedback. In particular, it leads naturally to some extensions which do not follow in a straightforward manner from previous works on the subject.

This paper investigates the stabilization problem for a class of Lipschitz nonlinear time-delay systems. With the useful information extracted from the Lipschitz nonlinearity of the system and employing a Lyapunov-Krasovskii functional, Linear State Feedback Controller design conditions have been derived and expressed in terms of linear matrix inequalities. The designed state feedback controller ensures closed loop stability of the nonlinear time-delay system. To test the effectiveness of the proposed control methodology, control law is employed for a liquid monopropellant rocket motor with a pressure feeding system.

The model-based control system design approach based on pole assignment introduced in Chap. 4 is extended beyond systems of second order first by means of linear state feedback control and subsequently by means of polynomial control.

A nonlinear dynamic model for a crane lifter system is derived using the Lagrangian method. The developed dynamic model has two inputs and four outputs. Several properties of the model such as passivity are presented. An energy based nonlinear control scheme is then proposed for the system. The developed scheme guarantees the asymptotic convergence of the trolley position and the hoist position to their desired values. However, the hoist and the load sway angles will either converge to zero or towards a stable trajectory. To ensure that the hoist and the load sway angles always converge to zero, a linear state feedback controller is used in conjunction with the nonlinear control scheme. The derived model and the control scheme are simulated using a digital computer; the simulation results illustrate the theoretical developments very well.