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A robust adaptive finite-time controller for satellite fast attitude maneuver is proposed in this paper. The finite-time sliding mode is proposed based on standard sliding mode, hence the terminal convergence rate could be largely improved, and the inherent robustness to some typical perturbations is maintained. A new method is proposed to deal with the singularity issue based on the property of Euler rotation. In order to deal with inertia matrix uncertainty, finite-time adaptive law for inertia matrix estimation variables is proposed. Considering that the variable estimation system has no direct feedback, an auxiliary state converging slower than system is designed to achieve finite-time stability. The overall global finite-time stability is proved by Lyapunov method and the performance of the controller is demonstrated by numerical simulation results.

A smooth version of Sylvester's law of inertia is presented for symmetric matrix functions of constant rank. The techniques used in the proof are constructive but the resulting numerical approaches are unstable, and therefore require stabilization. Two different stabilization techniques are suggested, one based on a descent method and one based on Newton's method. Some numerical tests are included to demonstrate the applicability of the obtained numerical methods.

This paper studies the problem of trajectory tracking control for a class of vehicles (underwater vehicles, some horizontally moving vehicles, indoor airships). The control development is based on some velocity transformation arising from the inertia matrix decomposition, Lyapunov’s direct method and a non-adaptive nonlinear tracking controller in terms of the Generalized Velocity Components (GVC). In the nonlinear controller the control gains are strictly related to the vehicle dynamics (especially dynamical couplings). The general algorithm is presented for a 6 DOF vehicle. In the simulation two trajectories were tested. Moreover, one robustness test was done (corresponding to robustness issue considered in this work). The simulation results obtained for a full airship model show that the proposed control scheme guarantees satisfactory performance.

To analyze the rigid–flexible coupling effects on the dynamic performance of a robot system, a dynamic model of a parallel robot with flexible spatial links is derived in detail using a floating frame of reference (FFR) formulation. Compared to the previous rigid–flexible coupling model where the kinematic chains are all flexible links or where the joints are all flexible components, the inertia matrix and the stiffness matrix are not constant matrix which leading to the differences in respect of dynamic performance in model. To verify the correctness of the derived dynamics equations, the dynamics solutions of the spatial parallel robot from an ideal rigid–body model and the FFR model containing rigid and flexible coordinates were established by an FFR formulation. Furthermore, a finite element analysis (FEA) model, which included rigid links and flexible spatial links, was constructed for comparison. The comparison of the three models showed that the trajectory trends were the same, but the motion trajectories of the end-effector obtained by the FFR and FEA models varied within a certain range, and the maximum variations occurred at the peaks of the trajectories. However, since the FFR model considered the coupling effects of rigid and flexible links and the micro-displacement of the end-effector, the amount of deformation was the largest.

Interconnection and damping assignment passivity-based control scheme has been used to stabilize many physical systems such as underactuated mechanical systems through total energy shaping. In this method, some partial differential equations (PDEs) related to kinetic and potential energy shaping shall be solved analytically. Finding a suitable desired inertia matrix as the solution of nonlinear PDEs relevant to kinetic energy shaping is a challenging problem. In this paper, a systematic approach to solving this matching equation for systems with one degree of underactuation is proposed. A special structure for desired inertia matrix is proposed to simplify the solution of the corresponding PDE. It is shown that the proposed method is more general than that of some reported methods in the literature. In order to derive a suitable desired inertia matrix, a necessary condition is also derived. The proposed method is applied to three examples, including pendubot, VTOL aircraft, and 2D SpiderCrane.

This paper presents a unifying dynamics formulation for nonsmooth multibody systems (MBSs) subject to changing topology and multiple impacts based on a linear projection operator. An oblique projection matrix ubiquitously yields all characteristic variables of such systems as follows: (i) the constrained acceleration before jump discontinuity from the projection of unconstrained acceleration, (ii) post-impact velocity from the projection of pre-impact velocity, (iii) impulse during impact from the projection of pre-impact momentum, (iv) generalized constraint force from the projection of generalized input force, and (v) post-impact kinetic energy from pre-impact kinetic energy based on projected inertia matrix. All solutions are presented in closed-form with elegant geometrical interpretations. The formulation is general enough to be applicable to MBSs subject to simultaneous multiple impacts with nonidentical restitution coefficients, changing topology, i.e., unilateral constraints becoming inactive or vice versa, or even when the overall constraint Jacobian becomes singular. Not only do the solutions always exist regardless of the constraint condition, but also the condition number for a generalized constraint inertia matrix is minimized in order to reduce numerical sensitivity in computation of the projection matrix to roundoff errors. The model is proven to be energetically consistent if a global restitution coefficient is assumed. In the case of nonidentical restitution coefficients, the set of energetically consistent restitution matrices is characterized by using a linear matrix inequality (LMI).

This paper presents a redundantly actuated and over-constrained 2RPU-2SPR parallel manipulator with two rotational and one translational coupling degrees of freedom. The kinematics analysis is firstly carried out and the mapping relationship of the velocity, acceleration and the independent parameters between the actuator joint and the moving platform are deduced by using the vector dot product and cross product operation. By employing d′Alembert′s principle and the principle of virtual work, the dynamics equilibrium equation is derived, and the simplified dynamics mathematical model of the parallel manipulator is further derived. Simultaneously, the generalized inertia matrix which can characterize the acceleration performance between joint space and operation space is further separated, and the performance indices including the dynamics dexterity, inertia coupling characteristics, energy transmission efficiency and driving force/torque balance are introduced. The analysis results show that the proposed redundantly actuated and over-constrained 2RPU-2SPR parallel manipulator in comparison with the existing non-redundant one has better dynamic comprehensive performance, which can be demonstrated practically by the successful application of the parallel kinematic machine head module of the hybrid machine tool.

This paper discusses the range of the eigenvalues of a class of matrices. By using the eigenvalues range of a class of matrices, an extension of the inner product type Cauchy–Schwarz inequality is obtained, the convergence proof of the least squares based iterative algorithm for solving the coupled Sylvester matrix equations is given and the best convergence factor is determined. Moreover, by using the eigenvalues range of this class of matrices, an iterative algorithm for solving linear matrix equation is established. Three numerical examples are offered to illustrate the effectiveness of the results suggested in this paper.

Perpetual points in mathematics defined recently, their significance, in systems dynamics is ongoing research. In linear unforced mechanical systems, the perpetual points are associated with rigid body motion, and they are not just a few points, but they form the perpetual manifolds. The mechanical systems with perpetual manifolds the rigid body motion are called perpetual mechanical systems. The concept of perpetual manifolds has been extended to the augmented perpetual manifolds of multidegree of freedom system, that the accelerations are equal but not necessarily zero. This is the case of externally forced perpetual mechanical systems that are moving in rigid body motion. A theorem defines the conditions that an externally forced mechanical system moves as a rigid body with state–space the exact augmented perpetual manifolds. Herein following this theorem, a corollary is written and proved. The corollary is about the perpetual mechanical system with nonlinear forces, and the perpetual one with linear internal forces. More precisely for the same time and state-dependent inertia matrix, and the same external forcing they have the same solution or otherwise stated their exact augmented perpetual manifolds that define their state space are the same. Therefore for the same initial conditions, they have the same motion. The theory, analytically and numerically, with two examples, is verified with excellent agreement. The significance of this work is that there is no need for complicated modeling and model update of nonlinear internal forces of mechanical systems when rigid body motions are the target.

The problem of solving periodic Sylvester matrix equations is discussed in this paper. A new kind of iterative algorithm is proposed for constructing the least square solution for the equations. The basic idea is to develop the solution matrices in the least square sense. Two numerical examples are presented to illustrate the convergence and performance of the iterative method.

This paper introduces another version of biconjugate residual method (BCR) for solving the generalized coupled Sylvester matrix equations over centrosymmetric or anti-centrosymmetric matrix. We prove this version of BCR algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round-off errors. Furthermore, a method is provided for choosing the initial matrices to obtain the least norm solution of the problem. At last, some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed.

In this paper, the problem of adaptive fixed-time tracking control for non-cooperative target spacecraft fly-around mission in the presence of the parameter uncertainties and external disturbances is investigated. Firstly, a novel and effective 6 DOF relative motion dynamic model constructed by relative position motion model expressed in the LOS frame and attitude dynamic model described by Modified Rodriguez parameters (MRPs) is firstly established. Subsequently, a new modified nonsingular fast terminal sliding mode (MNFTSM) with bounded convergence time in regardless of the initial states is designed, which not only can circumvent the singularity problem effectively, but also has the advantages of FTSM. By employing MNFTSM and adaptive techniques, two continuous adaptive fixed-time nonsingular fast terminal sliding mode (AFNFTSM) control laws are proposed, which can eliminate the chattering phenomenon and do not require precise knowledge of the mass, inertia matrix. Rigorous proofs show that both of two AFNFTSM control laws can guarantee that the relative position and attitude errors can converge to the small regions containing the origin in fixed time. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed control schemes.

This paper introduces an alternating direction method of multipliers (ADMM) for finding solutions to a class of Sylvester matrix equation AXB = E subject to a linear matrix inequality constraint CX...

Solving the well-known Lyapunov and Sylvester matrix equations appears in a wide range of applications such as in control theory and signal processing. This article establishes the matrix form of the biconjugate residual (BCR) algorithm for computing the generalized reflexive solution X and the generalized anti-reflexive solution Y of the generalized Sylvester matrix equation It is proven that the proposed BCR algorithm converges within a finite number of iterations in the absence of round-off errors. At the end, various numerical implementations illustrating the effectiveness and accuracy of the proposed BCR algorithm are presented and discussed.

The purpose of this paper is deriving the minimal residual (MINIRES) algorithm for finding the symmetric least squares solution on a class of Sylvester matrix equations. We prove that if the system is inconsistent, the symmetric least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrix to obtain the minimum norm least squares symmetric solution of the problem. Finally, we give some numerical examples to illustrate the performance of MINIRES algorithm.

By keeping in mind the great number of applications of generalized Sylvester matrix equations in systems and control theory, in this paper we establish some necessary and sufficient conditions for the solvability to a system of eight generalized Sylvester matrix equations over the quaternion algebra. The general solution to this system is also constructed when it is solvable. Moreover, an algorithm and a numerical example are also given to make the results of this paper more practical in various fields of engineering. The findings of this paper generalize previous results in the literature.

In this paper, some theorems are stated, which allow to design robust control laws without chattering, of type PD or PI, for uncertain nonlinear MIMO systems having a quite general structure, to track sufficiently regular trajectories with preassigned maximum error. The proposed control laws are easy to design and implement, above all for the robotic systems, because these laws can also be decentralized and they are based on two design parameters: the first related to the maximum eigenvalue of the inertia matrix from which the practical stability depends on, and the second related to the practical region of asymptotic stability (RAS), to the precision of the tracking error and to the convergence velocity of the tracking error to the desired neighborhood. If the trajectories to track are not sufficiently regular, suitable filtering laws are proposed for these trajectories, so as to facilitate the implementation of the controller and reduce the control action especially during the transient phase. Three significant examples of application in the terrestrial, sea transportation and robotic areas, well showing the simplicity of design and implementation of the controllers and their effectiveness, are reported.

In this paper, A fuzzy Sylvester matrix equation with crisp coefficient matrices is considered. We use the arithmetic operation rule of fuzzy number to transfer the equation into two crisp Sylvester matrix equations, which avoids using Kronecker operation and which makes it possible to apply some existing methods to solve Sylvester matrix equation. Since the two transferred equations keep the number of unknowns unchanged, numerical operations needed in our method are much less than the operations in the method using Kronecker product. At last, we use several small-scale examples to illustrate the correctness of our method and several large-scale examples to illustrate the efficiency of our method.

An algorithm for computing the antitriangular factorization of symmetric matrices, relying only on orthogonal transformations, was recently proposed. The computed antitriangular form straightforwardly reveals the inertia of the matrix. A block version of the latter algorithm was described in a different paper, where it was noticed that the algorithm sometimes fails to compute the correct inertia of the matrix. In this paper we analyze a possible cause of the failure of detecting the inertia and propose a procedure to recover it. Furthermore, we propose a different algorithm to compute the antitriangular factorization of a symmetric matrix that handles most of the singularities of the matrix at the very end of the algorithm. Numerical results are also given showing the reliability of the proposed algorithm.

The development of a computational adaptive optimal controller for regulation and tracking of spacecraft attitude with unknown moment of inertia is the subject of this paper. The control technique accomplishes this by sequentially prioritizing higher goals of stabilization, optimization and identification. The proposed controller assumes measurement of angular velocities, and control torques. Conditions are derived that ensure asymptotic stability of the states. The proposed methodology uses an integral reinforcement learning based approach to drive the system to a linear quadratic optimal control solution. A novel identification method is derived that follows from the optimal controller formulation. Simulations of a spinning satellite with unknown moment of inertia are considered which shows successful optimal regulation and tracking of angular velocities along with accurate identification of unknown inertia.

In recent years, several linear matrix equations such as Lyapunov and Sylvester matrix equations have received considerable attention due to their important applications in engineering and applied mathematics. In this work, an iterative technique based on the Hestenes–Stiefel (HS) version of biconjugate residual (BCR) algorithm is introduced to solve the generalized Sylvester matrix equation ∑ i = 1 f ( A i XB i ) + ∑ j = 1 g ( C j YD j ) = E , over the generalized reflexive matrices X and Y. We show that the proposed iterative technique converges to the generalized reflexive solutions within a finite number of iterations in the absence of round-off errors. Numerical examples confirm the efficiency and accuracy of the proposed iterative technique and also comparisons with other algorithms are made.

The problem considered in this paper is to solve periodic Sylvester matrix equations. A new algorithm is presented to derive the least squares solution of the equations. The proposed iteration can converge to the unique solution of the considered matrix equations at finite steps with arbitrary initial condition. A numerical test result is provided to illustrate the convergence and efficiency of the iterative algorithm.

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations AkXk + YkBk = Mk, CkXk+1 + YkDk = Nk (k = 1, 2) over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

Robust control of attitude tracking system between two rigid spacecraft is addressed using nonlinear disturbance observer-based control technique. A relative attitude dynamics model is derived for spacecraft tracking maneuver, where parameter uncertainty and environmental disturbances are considered as disturbance torques. A composite control technique is proposed for robust attitude tracking of a rigid spacecraft about a spacecraft under multiple disturbances by combining a nonlinear disturbance observer with an asymptotic tracking control. The proposed nonlinear disturbance observer is used to enhance the disturbance attenuation ability and robustness performance against uncertain inertia parameter and disturbances by estimating and compensating for the disturbances through feedforward. Stability and tracking performance of the nonlinear disturbance observer are analyzed. Furthermore, the stability of the composed control approach consisting of the asymptotic tracking control and nonlinear disturbance observer is established through Lyapunov method. Simulation results show that the composite control technique can significantly enhance disturbance attenuation ability, robust dynamics performance and the desired relative attitude tracking accuracy of a rigid spacecraft under external disturbances and uncertain inertia matrix.

An approach to forming analytical solutions of the discrete and continuous Sylvester and Lyapunov linear algebraic matrix equations is described. The approach is based on reducing the square matrix to the Jordan normal form. Examples, algorithms, and implementations in Matlab are presented.

The conditions of regularization and the structure of generalized Green’s operator for a re-gularized linear matrix differential-algebraic boundary-value problem are found. To solve the problem of regularization of a generalized matrix differential-algebraic boundary-value problem, the original conditions of solvability and the structure of the general solution of a matrix equation of the Sylvester type are used.

The solution of the generalized Sylvester real matrix equation XB − AX = CY is important in stability analysis and controller design in linear systems. This paper presents an explicit solution to the generalized Sylvester real matrix equation XB − AX = CY . Based on the derived explicit solution to the considered generalized Sylvester real matrix equation, a new approach is provided for obtaining the solutions to the generalized Sylvester quaternion j-conjugate matrix equation XB − A X ^ = CY using the real representation of a quaternion matrix. The closed form solution is established in an explicit form for this generalized Sylvester quaternion j-conjugate matrix equation. The existing complex representation method requires the coefficient matrix A to be a block diagonal matrix over complex field, while it is any suitable dimension quaternion matrix in the present paper. Therefore, we generalize the existing results.

In this note, we dedicate to solving the forward-time discrete periodic Sylvester matrix equations (FDPSE) and the reverse-time discrete periodic Sylvester matrix equations (RDPSE), which are frequently encountered in the analysis and design of linear discrete-time periodic systems. Explicit and complete parametric solutions to these equations are derived without any constraints on the coefficient matrices. As one of the most important applications of the periodic Sylvester matrix equations, periodic Luenberger-type observers design problem for linear discrete-time periodic systems is solved in the paper. Necessary and sufficient conditions for the existence of Luenberger observers are established, and a parametric design algorithm for Luenberger observers is presented. A numerical example is worked out to illustrate the effectiveness of our results.

We present an iterative method based on the Hermitian and skew-Hermitian splitting HSS for solving the continuous Sylvester equation. By using the HSS of the coefficient matrices A and B, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations CGNR-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matrices and which are the skew-Hermitian part of A and B, respectively is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods.

Recently, Xue etc. (37) discussed the Smith method for solving Sylvester equation AX + XB = C, where one of the matrices A and B is at least a nonsingular M-matrix and the other is an (singular or nonsingular) M-matrix. Furthermore, in order to nd the minimal non-negative solution of a certain class of non-symmetric algebraic Riccati equations, Gao and Bai (17) considered a doubling iteration scheme to inexactly solve the Sylvester equations. This paper discusses the iterative error of the standard Smith method used in (17) and presents the prior estimations of the accurate solution X for the Sylvester equation. Furthermore, we give a new version of the Smith method for solving discrete-time Sylvester equation or Stein equation AXB + X = C, while the new version of the Smith method can also be used to solve Sylvester equation AX + XB = C, where both A and B are positive denite. We also study the convergence rate of the new Smith method. At last, numerical examples are given to illustrate the eectiveness of our methods.

In this paper, we present an accelerated gradient based algorithm by minimizing certain criterion quadratic function for solving the generalized Sylvester-transpose matrix equation A X B + C X T D = F . The idea is from (Ding and Chen, 2005; Niu et?al., 2011; Wang et?al., 2012) in which some efficient algorithms were developed for solving the Sylvester matrix equation and the Lyapunov matrix equation. On the basis of the information generated in the previous half-step, we further introduce a relaxation factor to obtain the solution of the generalized Sylvester-transpose matrix equation. We show that the iterative solution converges to the exact solution for any initial value provided that some appropriate assumptions. Finally, some numerical examples are given to illustrate that the introduced iterative algorithm is efficient.

a = positive parameter atx aty atz T = linear acceleration due to the tether tension, m · s−2 ax ay az T = linear acceleration of the gripper’s thruster force, m · s−2 C k = matrix in coordinated desaturation controller d = position vector of the capture position, m Fl = tether tension, N I = inertia matrix of the combination, kg · m I0 = nominal value of combination’s inertia matrix, kg · m Kξ = positive-definite design matrix k2 = positive-definite design matrix m = mass of tethered space robot–target combination, kg Oxlylzl = space tether frame Oxpypzp = space platform orbital frame Oxtytzt = combination orbital frame Ox 0 t y 0 t z 0 t = combination body frame P = positive-definite design matrix R = transformationmatrix from platform orbital frame to combination body frame S k = constant positive weighting matrix Tl = tether control torque, Nm x y z T = centroid position of the combination in the platform orbital frame, m ΔI = inertia matrix uncertainty, kg · m e = positive parameter λ k = Lagrange multiplier λL = upper bound of disturbance λL = estimation values of disturbance μ = positive design parameter ξ = state of the auxiliary design system σ = modified Rodrigues parameters σd = desired modified Rodrigues parameters τ k = vector of optimal thruster force and tether tension τc = control torque of the combination, N · m τd = disturbing torques, N · m τt = control torque of the thruster, N · m τl = control torque of the tether, N · m ω = absolute angular velocity of the combination, rad · s−1 ωd = desired angular velocity of the combination, rad · s−1

By applying the generalized singular-value decompositions (GSVDs) of matrix pairs, a necessary and sufficient solvability condition for mixed Sylvester equations is established, the explicit representation of the general solution is given. Also, the minimum-norm solution of the matrix equations is discussed.

This paper presents the finite-time adaptive fuzzy control to regulate thruster faults in rigid spacecraft system. The considered faults are that the thrusters losing power partially or even totally. Fuzzy logic systems are used to approximate the unknown nonlinear function which is produced by the external disturbances, unknown inertia matrix and thrusters faults. Uniting fast nonsingular terminal sliding mode surface (FNTSMS) and the fault-tolerant control (FTC), two novel adaptive fuzzy finite-time FTC schemes are developed. It has been testified that this control method can make sure that the rigid spacecraft has excellent property of fault-tolerant and all the signals are finite-time convergence. Simulation examples on a rigid body spacecraft with three thrusters and six thrusters are done to indicate that the finite-time FTC schemes are available in achieving high attitude tracking with faults and uncertainty of the system.

In this paper, a decentralized adaptive fuzzy control has been developed for two cooperating robotic manipulators moving an object with impedance interaction. The contact forces are described using gradients of nonlinear potentials; then, the deformations of the contact surface can be obtained by an impedance approach. The cooperating manipulators are considered as a combination of subsystems, and the decentralized local dynamics coupled with physical interactions among the subsystems are developed. To compensate for the effect of dynamics uncertainties and external disturbances, decentralized fuzzy control combining parameter adaptations and disturbance observers is constructed. It guarantees the motion trajectories and impedance forces of the constrained object converging to the desired manifolds. It is theoretically established that the disturbance observers compensate for unparameterizable uncertainties, while the adaptive fuzzy mechanism compensates for the fast-changing components of the uncertainties that go beyond the disturbance observers. Moreover, unknown nonlinear dynamics such as the inertia matrix, Coriolis/centripetal matrix, and frictions, as well as interconnections with nonlinear bounds, can be accommodated through online learning. The experiments on two real robots have been carried out to verify the effectiveness of the proposed theoretical results.

This paper proposes a nonlinear adaptive position and attitude-tracking controller for satellite proximity operations between a target and a chaser satellite. The controller requires no information about the mass and inertia matrix of the chaser satellite and takes into account the gravitational acceleration, the gravity-gradient torque, the perturbing acceleration due to Earth’s oblateness, and constant (but otherwise unknown) disturbance forces and torques. Sufficient conditions to identify the mass and inertia matrix of the chaser satellite are also given. The controller is shown to ensure almost global asymptotical stability of the translational and rotational position and velocity tracking errors. Unit dual quaternions are used to simultaneously represent the absolute and relative attitude and position of the target and chaser satellites. The analogies between quaternions and dual quaternions are explored in the development of the controller.

One step on the way to approach human performance in robotics is to provide joint torque sensing and control for better interaction capabilities with the environment, and a large number of actuated degrees of freedom (DOFs) for improved versatility. However, the increasing complexity also raises the question of how to resolve the kinematic redundancy which is a direct consequence of the large number of DOFs. Here we give an overview of the most practical and frequently used torque control solutions based on null space projections. Two fundamental structures of task hierarchies are reviewed and compared, namely the successive and the augmented method. Then the projector itself is investigated in terms of its consistency. We analyze static, dynamic, and the new concept of stiffness consistency. In the latter case, stiffness information is used in the pseudoinversion instead of the inertia matrix. In terms of dynamic consistency, we generalize the weighting matrix from the classical operational space approach and show that an infinite number of weighting matrices exist to obtain dynamic consistency. In this context we also analyze another dynamically consistent null space projector with slightly different structure and properties. The redundancy resolutions are finally compared in several simulations and experiments. A thorough discussion of the theoretical and empirical results completes this survey.

© 2017 Informa UK Limited, trading as Taylor & Francis Group In many areas of applied linear algebra, it is necessary to work with matrix approximations. A usual situation occurs when a matrix obtained from experimental or simulated data is needed to be approximated by a matrix that lies in a corresponding statistical model and satisfies some specific properties. In this short note, we focus on symmetric and positive-semidefinite approximations and we show that the positive and negative indices of inertia of the symmetric approximation and the rank of the positive-semidefinite approximation are always bounded from above by the rank of the original matrix.

In this article, we develop a real representation method for computing the solution pair (X, Y) to the non-homogeneous generalized Sylvester quaternion j-conjugate matrix equation XB − AX = CY + R. Compared to the existing complex representation method [C. Song, G. Chen, Acta Mathematica Scientia 2012, 32(B)(5):1967–1982], the advantage of this new approach is that there is no special requirement on the any coefficient matrix. In this sense, we generalize the existing results. Finally, a numerical example is provided to support the theoretical findings and to testify the effectiveness and usefulness of the developed algorithm.

In order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

This paper is concerned with some generalizations of the Hermitian and skew-Hermitian splitting (HSS) iteration for solving continuous Sylvester equations. The main contents we will introduce are the normal and skew-Hermitian splitting (NSS) iteration methods for the continuous Sylvester equations. It is shown that the new schemes can outperform the standard HSS method in some situations. Theoretical analysis shows that the NSS methods converge unconditionally to the exact solution of the continuous Sylvester equations. Moreover, we derive the upper bound of the contraction factor of the NSS iterations. Numerical experiments further show the effectiveness of our new methods.

This paper is concerned with the problem of attitude control and disturbance rejection of rigid spacecraft in the presence of parameter uncertainty. It is assumed that the external disturbance is generated by some time varying exosystems. The unit quaternion is used as the kinematic variables since it is free of singularity. An internal model and an adaptive control law are proposed. The parameter uncertainty caused by the unknown inertia matrix is handled by combining the semi-tensor product and adaptive control method. The asymptotical stability of the closed-loop system is given via backstepping and Lyapunov analysis. Finally, an illustrative example is provided to show the effectiveness of the proposed approach.

Algorithms of the Bartels-Stewart type for numerically solving Sylvester matrix equations of modest size are modified for the case where the linear operators associated with these equations are self-adjoint. The superiority of the modified algorithms over the original ones is illustrated by numerical results.

By analyzing the eigenvalues of the related matrices, the convergence analysis of the least squares based iteration is given for solving the coupled Sylvester equationsAX+YB=CandDX+YE=Fin this paper. The analysis shows that the optimal convergence factor of this iterative algorithm is 1. In addition, the proposed iterative algorithm can solve the generalized Sylvester equationAXB+CXD=F. The analysis demonstrates that if the matrix equation has a unique solution then the least squares based iterative solution converges to the exact solution for any initial values. A numerical example illustrates the effectiveness of the proposed algorithm.

In this paper, we discuss the properties of the eigenvalues related to the symmetric positive definite matrices. Several new results are established to express the structures and bounds of the eigenvalues. Using these results, a family of iterative algorithms are presented for the matrix equation AX=F and the coupled Sylvester matrix equations. The analysis shows that the iterative solutions given by the least squares based iterative algorithms converge to their true values for any initial conditions. The effectiveness of the proposed iterative algorithm is illustrated by a numerical example.

The problem of solving matrix equations has many applications in control and system theory. This paper is concerned with the iterative solutions of the Sylvester-transpose matrix equation ∑ i = 1 k ( A i XB i + C i X T D i ) = E , and the periodic Sylvester matrix equation A ^ j X ^ j B ^ j + C ^ j X ^ j + 1 D ^ j = E ^ j for j = 1 , 2 , … , λ . The basic idea is to develop the conjugate gradients squared (CGS) and bi-conjugate gradient stabilized (Bi-CGSTAB) methods for obtaining matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. Numerical test results are given to compare matrix iterative methods with other well-known methods.

This paper investigates the design and experimental development of planar programmable inertia generators. An inertia generator is a hand-held haptic device that has a programmable inertia. By moving internal masses in reaction to accelerations induced by the user, the effective inertia of the device is modified in order to render a prescribed inertia. In this paper, a one-degree-of-freedom device with one internal moving mass is first proposed. The corresponding dynamic model is developed and the rendering capabilities of the device are investigated. Then, a controller is designed to produce the appropriate motion of the internal mass in reaction to the acceleration induced by the user. A prototype is presented and experimental results are discussed. A mechanical architecture is then proposed for the design of a planar three-degree-of-freedom inertia generator. The corresponding dynamic model is derived, and it is shown that the generalized inertia matrix of the proposed mechanism is always of full rank. The rendering capabilities of the device are also investigated. Finally, simulation results obtained with the three-degree-of-freedom inertia generator are reported and discussed.

The bi-conjugate gradient stabilised (Bi-CGSTAB) method is one of the efficient computational tools to solve the non-Hermitian linear systems Ax = b. By employing Kronecker product and vectorisation operator, this study investigates the matrix form of the Bi-CGSTAB method for solving the coupled Sylvester matrix equations ∑ i=1 k (A i XB i + C i YD i ) = M, ∑ i=1 k (E i XF i + G i YH i ) = N [including (second-order) Sylvester and Lyapunov matrix equations as special cases] encountered in many systems and control applications. Several numerical examples are given to compare the efficiency and performance of the investigated method with some existing algorithms.

This paper is focused on the dynamic formulation of mechanical joints using different approaches that lead to different models with different numbers of degrees of freedom. Some of these formulations allow for capturing the joint deformations using a discrete elastic model while the others are continuum-based and capture joint deformation modes that cannot be captured using the discrete elastic joint models. Specifically, three types of joint formulations are considered in this investigation; the ideal, compliant discrete element, and compliant continuum-based joint models. The ideal joint formulation, which does not allow for deformation degrees of freedom in the case of rigid body or small deformation analysis, requires introducing a set of algebraic constraint equations that can be handled in computational multibody system (MBS) algorithms using two fundamentally different approaches: constrained dynamics approach and penalty method. When the constrained dynamics approach is used, the constraint equations must be satisfied at the position, velocity, and acceleration levels. The penalty method, on the other hand, ensures that the algebraic equations are satisfied at the position level only. In the compliant discrete element joint formulation, no constraint conditions are used; instead the connectivity conditions between bodies are enforced using forces that can be defined in their most general form in MBS algorithms using bushing elements that allow for the definition of general nonlinear forces and moments. The new compliant continuum-based joint formulation, which is based on the finite element (FE) absolute nodal coordinate formulation (ANCF), has several advantages: (1) It captures modes of joint deformations that cannot be captured using the compliant discrete joint models; (2) It leads to linear connectivity conditions, thereby allowing for the elimination of the dependent variables at a preprocessing stage; (3) It leads to a constant inertia matrix in the case of chain like structure; and (4) It automatically captures the deformation of the bodies using distributed inertia and elasticity. The formulations of these three different joint models are compared in order to shed light on the fundamental differences between them. Numerical results of a detailed tracked vehicle model are presented in order to demonstrate the implementation of some of the formulations discussed in this investigation.