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The continuous-time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second-order structure of size n, the ARE required to be solved to get the control values in standard control problems results in complex subequations in terms of the second-order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second-order system, due to the eigendecomposition of the 2n × 2n system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the 2n × 2n system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, and positive semidefinite solution matrix). [ABSTRACT FROM AUTHOR]

Nonsymmetric algebraic Riccati equations are a class of nonlinear matrix equations with wide applications. In this paper, numerical solution of nonsymmetric algebraic Riccati equations is studied. A generalized ALI iteration method is proposed to solve the equation, which is an extension of the ALI iteration method and the MALI iteration method. Theoretical analysis and numerical experiments show that the proposed method is feasible and is more effective than the ALI iteration method and the MALI iteration method. [ABSTRACT FROM AUTHOR]

A nonlinear optimal (H-infinity) control approach is proposed for the dynamic model of multi-DOF redundant robotic manipulators. Because of the complicated kinematics and dynamics and the high dimensionality of the state-space model of such robots, the related control problem is of elevated difficulty. In the present article, a three-link planar robotic manipulator it considered. The article's approach relies first on approximate linearization of the state-space model of the redundant robotic manipulator, according to first-order Taylor series expansion and the computation of the related Jacobian matrices. For the approximately linearized model of the manipulator, a stabilizing H-infinity feedback controller is designed. To compute the controller's gains an algebraic Riccati equation is solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. The proposed control method retains the advantages of typical optimal control, that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs. [ABSTRACT FROM AUTHOR]

In the first part of this article, we study feedback stabilization of a parabolic coupled system by using localized interior controls. The system is feedback stabilizable with exponential decay $ -\omega<0 $ for any $ \omega>0 $. A stabilizing control is found in feedback form by solving a suitable algebraic Riccati equation. In the second part, a conforming finite element method is employed to approximate the continuous system by a finite dimensional discrete system. The approximated system is also feedback stabilizable (uniformly) with exponential decay $ -\omega+\epsilon $, for any $ \epsilon>0 $ and the feedback control is obtained by solving a discrete algebraic Riccati equation. The error estimate of stabilized solutions as well as stabilizing feedback controls are obtained. We validate the theoretical results by numerical implementations. [ABSTRACT FROM AUTHOR]

In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions X j to the solution X of large-scale continuous-time algebraic Riccati equations (CAREs) 0 = R (X) : = A H X + X A + C H C − X B B H X generate the same sequence X j when used with the same parameters. The Hermitian low-rank approximations X j are of the form X j = Z j Y j Z j H , where Z j is a matrix with only few columns and Y j is a small square Hermitian matrix. Each X j generates a low-rank Riccati residual R (X j) such that the norm of the residual can be evaluated easily allowing for an efficient termination criterion. Here a new family of methods to generate such low-rank approximate solutions X j of CAREs is proposed. Each member of this family of algorithms proposed here generates the same sequence of X j as the four previously known algorithms. The approach is based on a block rational Arnoldi decomposition and an associated block rational Krylov subspace spanned by A H and C H. Two specific versions of the general algorithm will be considered; one will turn out to be a rediscovery of the RADI algorithm, the other one allows for a slightly more efficient implementation compared to the RADI algorithm (in case the Sherman-Morrision-Woodbury formula and a direct solver is used to solve the linear systems that occur). Moreover, our approach allows for adding more than one shift at a time. [ABSTRACT FROM AUTHOR]

Suppose Q (x) is a real n × n regular symmetric positive semidefinite matrix polynomial. Then it can be factored as Q (x) = G (x) T G (x) , where G (x) is a real n × n matrix polynomial with degree half that of Q (x) if and only if det ⁡ (Q (x)) is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equation X S X − X R + R T X + P = 0 , where P , R , S are real n × n matrices with P and S real symmetric. In addition, we provide a detailed algorithm for computing the factorization. [ABSTRACT FROM AUTHOR]

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called 'ZQ-ARE' that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor's flight control system faster than the traditional differential-algebraic Riccati equation solution. [ABSTRACT FROM AUTHOR]

Permanent magnet synchronous motor-powered electric vehicles control is the subject of various research work. However, their global dynamic models are nonlinear and coupled. Therefore, to achieve efficient operation, an effective control system is crucial. In this study, we propose and compare linear H-Infinity and Galerkin approximation approach for Nonlinear H-Infinity control strategies to improve the durability and performance of post-driven speed in electric vehicles. In the case of linear systems, the linear H-Infinity controller is found by solving the algebraic equation known as the Riccati equation. On the other hand, the control problem based nonlinear H-Infinity poses challenges because it involves solving a nonlinear partial differential equation known as name of the Hamilton-Jacobi-Isaacs equation, which is difficult or even impossible to solve by using analytical methods. In these situations, the Galerkin approximation approach provides an approximation to the Hamilton-Jacobi equation solution. In order to evaluate the performance of Galerkin approximation approach and linear H-Infinity controllers, electric vehicle feedback simulations will be conducted, taking into account different constraints. The goal is to ensure efficient operation in different situations. The results demonstrate that the Galerkin approximation Approach for nonlinear H-Infinity controller reveals a similar performance and durability as the H-Infinity controller, and stands out for its ability to optimize the control system performance of the EV, providing a faster response, reducing undesirable ripples, and enhancing overall stability and precision. Generally, this comparative study brings to light the effectiveness of linear and Galerkin approximations for H-Infinity control in permanent magnet synchronous motor-powered electric vehicles. The results contribute to the advancement of control strategies and provide valuable information for the conception and employment of efficient electric vehicle control systems. [ABSTRACT FROM AUTHOR]

This paper describes a Q-learning based algorithm to design the linear quadratic tracker (LQT) for linear time invariant (LTI) continuous-time systems with partially unknown dynamics. The proposed approach uses a fixed-point equation in terms of Q-function in order to estimate the unknown optimal gain parameters. The fixedpoint equation, which is derived by applying the Pontryagin's minimum principle in Q-learning, is based on the modified algebraic Riccati equation (ARE) for LQT problem. The online adaptation of the optimal parameters are achieved by using the gradient descent based parameter update laws by minimizing the Bellman's error term which is derived from fixed-point equation mentioned earlier. A persistence of excitation condition has been used to establish the desired optimal convergence of the estimated control parameters. Simulation results have been shown to validate the efficiency of the proposed Q-learning approach. [ABSTRACT FROM AUTHOR]

The focus of this paper is on solving the state estimation problem for general continuous-time linear systems through the use of distributed networked observers. To better reflect the communication environment, stochastic noises are considered when observers exchange information. In the networked observers, each local observer measures only part of the system output, and the state estimation can not be accomplished within a single observer. Then, all observers communicate through a pre-specified graph to make up information in the remaining system output. By solving a parametric algebraic Riccati equation (ARE), a simple method to calculate parameters in the observers is proposed. Furthermore, using the stability theory of stochastic differential equations, state omniscience is discussed in almost sure sense and in the mean square sense for the cases of state-dependent noises and non-state-dependent noises, respectively. It is shown that, for observable linear systems, the resulting observers work in a coordinated mode to reach state omniscience under the connected graph. Illustrative examples are provided to show the effectiveness of the distributed observers. [ABSTRACT FROM AUTHOR]

Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients, there is no approach available to compute solutions in the large-scale sparse setting. In this paper, we develop an iterative method to compute low-rank approximations of stabilizing solutions of large-scale sparse continuous-time algebraic Riccati equations with indefinite quadratic terms. We test the developed approach for dense examples in comparison to other established matrix equation solvers, and investigate the applicability and performance in large-scale sparse examples. [ABSTRACT FROM AUTHOR]

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called ’ZQ-ARE’ that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor’s flight control system faster than the traditional differential-algebraic Riccati equation solution.

We identify a relationship between the solutions of a nonsymmetric algebraic T -Riccati equation (T -NARE) and the deflating subspaces of a palindromic matrix pencil, obtained by arranging the coefficients of the T -NARE. The interplay between T -NAREs and palindromic pencils allows one to derive both theoretical properties of the solutions of the equation, and new methods for its numerical solution. In particular, we propose methods based on the (palindromic) QZ algorithm and the doubling algorithm, whose effectiveness is demonstrated by several numerical tests. [ABSTRACT FROM AUTHOR]

This article introduces a new design of a decentralized robust optimal feedback controller for a class of large‐scale systems using the modified algebraic Riccati equations. The studied class of large‐scale systems is characterized by structured uncertainties that are bounded. Our main contribution in this work is the use of the Lyapunov direct method for the stability analysis, associated with a quadratic function, in order to determine a new sufficient condition to guarantee the global asymptotic stability of the class of the studied uncertain interconnected systems. The proposed control approach is then implemented on a large‐scale power systems composed of three interconnected generators. Simulation results show that the proposed robust optimal controller is quite effective and has an outstanding performance. [ABSTRACT FROM AUTHOR]

In this article, we study the distributed control problem of vehicular platooning by taking into account network imperfections, namely the input delay and the packet dropout. The delay occurs within the inter network of each vehicle from its controller to its actuators and the packet dropout occurs within the intra network of vehicles when information is exchanged between vehicles. Relative state-feedback and relative input-output feedback control strategies are adopted for the vehicular platooning under network imperfection. To this end, and to avoid the requirement of relative state information, the relative state measurements are constructed for platoon systems based on a distributed observer via only relative input-output measurements and in the presence of delay, packet dropout and noise. Conditions for synchronization are provided in terms of the unique positive-definite solutions to some algebraic Riccati equations. Simulations validate the theoretical results. [ABSTRACT FROM AUTHOR]

The paper proposes an approach to constructing the general solution of a -dimensional system of differential equations with constant coefficients whose fundamental system consists of bounded and unbounded solutions as . The algorithm is based on preliminary solution of two matrix algebraic Riccati equations and stable linear Cauchy problems. The method is implemented for a finite-dimensional equation of radiative transfer and for a linear Hamiltonian system. [ABSTRACT FROM AUTHOR]

This paper investigates the static-feedback guaranteed cost control problem for linear systems with actuator faults including outage and loss of effectiveness. Under the actuator redundancy condition, theoretical analysis shows that a static-feedback guaranteed cost controller can always be well designed to ensure that the resulting closed-loop system is stable with desirable quadratic performance. In particular, the feedback gain can be determined through the solution of a modified algebraic Riccati equation. Furthermore, extension to the system with uncertainties is further studied. Compared with the dynamic feedback controller, the static-feedback controller consists only of logical gates/modules and it does not require any memory element, and hence it is simplest in a design perspective. Different from the existing results, the severe and time-varying actuator outage faults can be handled very well by the proposed control strategy. Finally, simulation on a linearised reduced-order aircraft system is provided for verifying the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, a novel method is proposed to solve the adaptive optimal tracking algorithm for a class of Markov jump systems. First, the augmented system with the tracking signal is built under the decoupling Markov jump systems and it is proved that the selected performance index satisfies the algebraic Riccati equation which can be solved by policy iteration schemes. Then, a reinforcement learning (RL) algorithm is used to solve the coupled algebraic Riccati equations by using partial knowledge of system dynamics. The convergence of the partial model‐free integral RL iteration algorithm is also proved. Finally, a simulation example is given to show the better tracking effectiveness and accuracy of the online iteration algorithm comparing with the offline one. [ABSTRACT FROM AUTHOR]

Featured Application: This proposed method is suitable for the cooperative consensus control of various homogeneous Multiple Autonomous Unmanned linear systems, such as an underwater robot swarm and aerial UAV swarm. This paper addresses the distributed optimal decoupling synchronous control of multiple autonomous unmanned linear systems (MAUS) subject to complex network dynamic coupling. The leader–follower mechanism based on neighborhood error dynamics is established and the network coupling term is regarded as the external disturbance to realize the decoupling cooperative control of each agent. The Bounded L2-Gain problem for the network coupling term is formulated into a multi-player zero-sum differential game. It is shown that the solution to the multi-player zero-sum differential game requires the solution to coupled Hamilton–Jacobi (HJ) equations. The coupled HJ equations are transformed into an algebraic Riccati equation (ARE), which can be solved to obtain the Nash equilibrium of a multi-player zero-sum game. It is shown that the bounded L2-Gain for coupling attenuation can be realized by applying the zero-sum game solution as the control protocol and the ultimately uniform boundedness (UUB) of a local neighborhood error vector under conservative conditions is proved. A simulation example is provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a novel suboptimal attitude tracking controller based on the algebraic Riccati equation for a near-space hypersonic vehicle (NSHV). Since the NSHV's attitude dynamics is complexly nonlinear, it is hard to directly construct an appropriate algebraic Riccati equation. We design the construction based on the Chebyshev series and the Koopman operator theory, which includes three steps. First, the Chebyshev series are considered to transform the error dynamics of the NSHV's attitude into a polynomial system. Second, the Koopman operator is used to obtain a series of high-dimensional linear dynamics to approximate each of the polynomial system's vector fields. In this step, our contribution is to determine a well-posed linear dynamics with the minimal dimension to approximate the original nonlinear vector field, which helps to design the control law and analyze the control performance. Third, based on the high-dimensional dynamics, the NSHV's attitude error dynamics is separated into the linear part and the nonlinear part, such that the algebraic Riccati equation can be constructed according to the linear part. Then, the suboptimal error feedback control law is derived from the algebraic Riccati equation. The closed-loop control system is proved to be locally exponentially stable. Finally, the numerical simulation demonstrates the effectiveness of the suboptimal control law. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the semilocal convergence issue of two-step Newton method for solving nonlinear operator equation in Banach spaces. Under the assumption that the first derivative of the operator satisfies a generalized Lipschitz condition, a new semilocal convergence analysis for the two-step Newton method is presented. The Q-cubic convergence is obtained by an additional condition. This analysis also allows us to obtain three important spacial cases about the convergence results based on the premises of Kantorovich, Smale and Nesterov-Nemirovskii types. As applications of our convergence results, a nonsymmetric algebraic Riccati equation arising from transport theory and a two-dimensional nonlinear convection-diffusion equation are provided. [ABSTRACT FROM AUTHOR]

The article proposes a nonlinear optimal (H-infinity) control approach for a type of underactuated power-line inspection robots. To implement this control scheme, the state-space model of the power-line inspection robots undergoes first approximate linearization around a temporary operating point, through first-order Taylor series expansion and through the computation of the associated Jacobian matrices. To select the feedback gains of the controller an algebraic Riccati equation is solved at each time step of the control method. The global stability properties of the control loop are proven through Lyapunov analysis. The significance of the article's results is outlined in the following: (i) the proposed control method is suitable for treating underactuated robotic systems and in general nonlinear dynamical systems with control inputs gain matrices which are in a nonquadratic form, (ii) by achieving stabilization of the power-line inspection robots in underactuation conditions the proposed control method ensures the reliable functioning of these robotic systems in the case of actuators' failures or enables the complete removal of certain actuators and the reduction of the weight of these robotic systems, (iii) the proposed control method offers a solution to the nonlinear optimal control problem which is of proven global stability while also remaining computationally tractable, (iv) the proposed nonlinear optimal control method retains the advantages of linear optimal control that is fast and accurate tracking of reference setpoints under moderate variations of the control inputs, and (v) by minimizing the amount of energy that is dispersed by the actuators of the power-line inspection robots the proposed control method improves the autonomy and operational capacity of such robotic systems. [ABSTRACT FROM AUTHOR]

The focus of this paper is on solving the state estimation problem for general continuous-time linear systems through the use of distributed networked observers. To better reflect the communication environment, stochastic noises are considered when observers exchange information. In the networked observers, each local observer measures only part of the system output, and the state estimation can not be accomplished within a single observer. Then, all observers communicate through a pre-specified graph to make up information in the remaining system output. By solving a parametric algebraic Riccati equation (ARE), a simple method to calculate parameters in the observers is proposed. Furthermore, using the stability theory of stochastic differential equations, state omniscience is discussed in almost sure sense and in the mean square sense for the cases of state-dependent noises and non-state-dependent noises, respectively. It is shown that, for observable linear systems, the resulting observers work in a coordinated mode to reach state omniscience under the connected graph. Illustrative examples are provided to show the effectiveness of the distributed observers.

This paper aims to propose a constructive methodology for determining the matrix sign function for a stable variant of the Kung-Traub method. It analytically shows that the new scheme is asymptotically stable. Different numerical experiments compare the new scheme's behavior with the existing matrix iteration of the same type. Finally, the given approach applies to solve the algebraic Riccati equation and the Sylvester equation. [ABSTRACT FROM AUTHOR]

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Many control problems require solving the algebraic Riccati equation (ARE). Previous studies have focused more on solving the time-invariant ARE than on solving the time-varying ARE (TVARE). This paper proposes a typical recurrent neural network called zeroing neural network (ZNN) to determine the solution of TVARE. Specifically, the ZNN model, which is formulated as an implicit dynamic equation, is developed by defining an indefinite error function and using an exponential decay formula. Then, such a model is theoretically analyzed and proven to be effective in solving the TVARE. Computer simulation results with two examples also validate the efficacy of the proposed ZNN model.

In this paper, a reinforcement learning (RL) approach is developed to solve the robust control for uncertain continuous-time linear systems. The objective is to find a feedback control law for the uncertain linear system using an online policy iteration algorithm. The robust control problem is solved by constructing an extended algebraic Riccati equation with properly defined weighting matrices for a general uncertain linear system. An online policy iteration algorithm is developed to solve the robust control problem based on RL principles without knowing the nominal system matrix. The convergence of the algorithm to the robust control solution for uncertain linear systems is proved. The simulation examples are given to demonstrate the effectiveness of the proposed algorithm. The results extend the design method of robust control to uncertain linear systems.

The present study uses linear quadratic regulator (LQR) theory to control a vibratory system modeled by a fractional-order differential equation. First, as an example of such a vibratory system, a viscoelastically damped structure is selected. Second, a fractional-order LQR is designed for a system in which fractional-order differential terms are contained in the equation of motion. An iteration-based method for solving the algebraic Riccati equation is proposed in order to obtain the feedback gains for the fractional-order LQR. Third, a fractional-order state observer is constructed in order to estimate the states originating from the fractional-order derivative term. Fourth, numerical simulations are presented using a numerical calculation method corresponding to a fractional-order state equation. Finally, the numerical simulation results demonstrate that the fractional-order LQR control can suppress vibrations occurring in the vibratory system with viscoelastic damping. [ABSTRACT FROM AUTHOR]

Synchronization of distributed hydropower units is necessary for ensuring the quality of the electric power produced by renewable sources. In this article, a nonlinear optimal control approach is proposed for stabilization and synchronization of distributed hydropower generators. The dynamic model of the interacting hydropower generation units undergoes approximate linearization with the use of first-order Taylor series expansion and the computation of the associated Jacobian matrices. The linearization point is updated at each time-step of the control method. For the approximately linearized model of the distributed hydropower system an H-infinity feedback controller is designed. This controller achieves solution of the related optimal control problem under model uncertainty and external perturbations. For the computation of the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to achieve state estimation-based control for the system of the distributed hydropower generators the H-infinity Kalman Filter is used as a robust state estimator. [ABSTRACT FROM AUTHOR]

An optimal ergodic control (EC) problem is investigated for a linear stochastic differential equation with quadratic cost functional. Constant nonhomogeneous terms, not all zero, appear in the state equation, which lead to the asymptotic limit of the state nonzero. Under the stabilizability condition, for any (admissible) closed-loop strategy, an invariant measure is proved to exist, which makes the ergodic cost functional well-defined and the EC problem well-formulated. Sufficient conditions, including those allowing the weighting matrices of cost functional to be indefinite, are introduced for finiteness and solvability for the EC problem. Some comparisons are made between the solvability of EC problem and the closed-loop solvability of stochastic linear-quadratic optimal control problem in the infinite horizon. The regularized EC problem is introduced to be used to obtain the optimal value of the EC problem. [ABSTRACT FROM AUTHOR]

The article proposes a non-linear optimal control approach for four-wheel omnidirectional mobile robots. The method has been successfully tested so-far on the control problem of several types of autonomous ground vehicles and the present article shows that it can also provide the only optimal solution to the control problem of four-wheel omnidirectional robotic vehicles. To implement this control scheme, the state-space model of the robotic vehicle undergoes first approximate linearisation around a temporary operating point, through first-order Taylor series expansion and through the computation of the associated Jacobian matrices. To select the feedback gains of the H-infinity controller an algebraic Riccati equation is repetitively solved at each time-step of the control method. The global stability properties of the control loop are proven through Lyapunov analysis. Finally, to implement state estimation-based feedback control, the H-infinity Kalman Filter is used as a robust state estimator. [ABSTRACT FROM AUTHOR]

We review a family of algorithms for Lyapunov‐ and Riccati‐type equations which are all related to each other by the idea of doubling: they construct the iterate Qk=X2k of another naturally‐arising fixed‐point iteration (Xh) via a sort of repeated squaring. The equations we consider are Stein equations X − A∗X A = Q, Lyapunov equations A∗X + X A + Q = 0, discrete‐time algebraic Riccati equations X = Q + A∗X(I + G X)−1A, continuous‐time algebraic Riccati equations Q + A∗X + X A − X G X = 0, palindromic quadratic matrix equations A + Q Y + A∗Y2 = 0, and nonlinear matrix equations X + A∗X−1A = Q. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory. [ABSTRACT FROM AUTHOR]

A non-linear optimal (H-infinity) control approach is proposed for the dynamic model of multi-degree-of-freedom (DOF) electro-hydraulic robotic manipulators. Control of electro-hydraulic manipulators is a non-trivial problem because of their non-linear and multi-variable dynamics. In this study, the considered robotic system consists of a multi-link robotic manipulator that receives actuation from rotary electro-hydraulic drives. The article's approach relies first on approximate linearisation of the state-space model of the electro-hydraulic manipulator, according to first-order Taylor series expansion and the computation of the related Jacobian matrices. For the approximately linearised model of the manipulator, a stabilising H-infinity feedback controller is designed. To compute the controller's gains, an algebraic Riccati equation is solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. The proposed control method retains the advantages of typical optimal control, i.e. fast and accurate tracking of the reference setpoints under moderate variations of the control inputs.

The use of robotic limb exoskeletons is growing fast either for rehabilitation purposes or in an aim to enhance human ability for lifting heavy objects or for walking for long distances without fatigue. The paper proposes a nonlinear optimal control approach for a lower-limb robotic exoskeleton. The method has been successfully tested so far on the control problem of several types of robotic manipulators and this paper shows that it can also provide an optimal solution to the control problem of limb robotic exoskeletons. To implement this control scheme, the state-space model of the lower-limb robotic exoskeleton undergoes first approximate linearization around a temporary operating point, through first-order Taylor series expansion and through the computation of the associated Jacobian matrices. To select the feedback gains of the H-infinity controller an algebraic Riccati equation is solved at each time-step of the control method. The global stability properties of the control loop are proven through Lyapunov analysis. Finally, to implement state estimation-based feedback control, the H-infinity Kalman Filter is used as a robust state estimator. [ABSTRACT FROM AUTHOR]

Based on a modified Newton method in the Banach algebra of square matrices, we construct here a simple iterative algorithm that converges to the unique positive solution of the algebraic Riccati equation X C X = D . Numerical examples illustrating the theoretical study and showing the speed of convergence of our approach are discussed as well. At the end, we put some open questions as purposes for future research. [ABSTRACT FROM AUTHOR]

Summary: Robust time‐varying formation design problems for second‐order multiagent systems subjected to external disturbances are investigated. First, by constructing an extended state observer, the disturbance compensation is estimated, which is a critical term in the proposed robust time‐varying formation control protocol. Then, an explicit expression of the formation center function is determined and impacts of disturbance compensations on the formation center function are presented. With the formation feasibility conditions, robust time‐varying formation design criteria are derived to determine the gain matrix of the formation control protocol by utilizing the algebraic Riccati equation technique. Furthermore, the tracking performance and the robustness property of multiagent systems are analyzed. Finally, the numerical simulation is provided to illustrate the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript. The first is to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. The second goal is to analyze and compare the main computational ingredients of these algorithms and to detect their strong points and their potential bottlenecks. Finally, we want to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of large-scale algebraic Riccati equations with high-rank constant terms. The solutions are not numerically low-rank, so the previously successful methods based on low-rank representations are not directly applicable. We modify the doubling algorithm, making use of the low-rank in the input matrix B. We also solve the challenging problems in the estimation of residuals and relative errors, convergence control and the output of the modified algorithm. Illustrative numerical examples are presented. [ABSTRACT FROM AUTHOR]