Time Delay Estimation Control of Permanent Magnet Spherical Actuator Based on Gradient Compensation

A multi-degree-of-freedom Permanent Magnet Spherical Actuator (PMSpA) has a special mechanical structure and electromagnetic fields, and is easily affected by nonlinear disturbances such as modeling errors and friction. Therefore, the quality of a PMSpA control system may be deteriorated. In order to keep the PMSpA with good trajectory tracking performance, this paper designs a time delay estimation controller based on gradient compensation. Firstly, the dynamic model of the PMSpA with nonlinear terms is derived. The nonlinear terms in the complex dynamic model can be simplified and estimated by the time delay estimation method. Secondly, for the estimation errors caused by time delay control, a gradient compensator is introduced to further correct and compensate for it. Furthermore, the stability of the designed controller is proved by the Lyapunov equation. Finally, the correctness and effectiveness of the controller are validated by comparison with other controllers through simulation. In addition, experimental results have also shown that the control accuracy of the spherical motor can be effectively improved using the proposed controller.

On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that the Gramians of controllability and observability of a bilinear system can be divided into parts (sub-Gramians) that characterize the measure of these generalized modes and their interactions. Furthermore, the properties of sub-Gramians were investigated in relation to modal controllability and observability. We also propose an algorithm for computing the Gramians and sub-Gramians based on the element-wise computation of the solution matrix. Based on the proposed algorithm, a novel criterion for the existence of solutions to the generalized Lyapunov equation is proposed, which allows, in some cases, to expand the domain of guaranteed existence of a solution of bilinear equations. Examples are provided that illustrate the application and practical use of the considered spectral decompositions.

Analysis and Prediction of Electric Power System’s Stability Based on Virtual State Estimators

The stability of bilinear systems is investigated using spectral techniques such as selective modal analysis. Predictive models of bilinear systems based on inductive knowledge extracted by big data mining techniques are applied with associative search of statistical patterns. A method and an algorithm for the elementwise solution of the generalized matrix Lyapunov equation are developed for discrete bilinear systems. The method is based on calculating the sequence of values of a fixed element of the solution matrix, which depends on the product of the eigenvalues of the dynamics matrix of the linear part and the elements of the nonlinearity matrixes. A sufficient condition for the convergence of all sequences is obtained, which is also a BIBO (bounded input bounded output) systems stability condition for the bilinear system.

On some estimation problems for nonlinear dynamic systems

Two problems of nonlinear guaranteed estimation for states of dynamical systems are considered. It is supposed that unknown measurable in $t$ disturbances are linearly included in the equation of motion and are additive in the measurement equations. These disturbances are constrained by nonlinear integral functionals, one of which is analog of functional of the generalized work. The studied problem consists in creation of the information sets according to measurement data containing the true position of the trajectory. The dynamic programming approach is used. If the first functional requires solving a nonlinear equation in partial derivatives of the first order which is not always possible, then for functional of the generalized work it is enough to find a solution of the linear Lyapunov equation of the first order that significantly simplifies the problem. Nevertheless, even in this case it is necessary to impose additional conditions on the system parameters in order for the system trajectory of the observed signal to exist. If the motion equation is linear in state variable, then many assumptions are carried out automatically. For this case the issue of mutual approximation of information sets via inclusion for different functionals is discussed. In conclusion, the most transparent linear quadratic case is considered. The statement is illustrated by examples.

Synchronization Control of Dynamic Positioning Ships Using Model Predictive Control

Underway replenishment is essential for ships performing long-term missions at sea, which can be formulated into the problem of leader-tracking configuration. Not only the position and orientation but also the velocities are required to be controlled for ensuring the synchronization; additionally, the control inputs are constrained. On this basis, in this paper, a novel synchronization controller on account of model predictive control (MPC) for dynamic positioning (DP) ships is devised to achieve underway replenishment. Firstly, a novel synchronization controller based on MPC is devised to ensure the synchronization of not only the position and orientation but the velocities; furthermore, it is a beneficial solution for its advantages in handling the control input constraints ignored in most studies of underway replenishment. Secondly, a neurodynamic optimization system is applied to the implementation of MPC, which can improve the computational efficiency and shorten the simulation time. Thirdly, stability, frequently neglected by traditional MPC, is ensured by the means of adding a terminal cost function exported from the Lyapunov equation into the objective function. Finally, the effectiveness and advantages of the proposed control design are illustrated by extensive simulations.

General Lyapunov-Based Iterative Algorithm for Linear Quadratic Regulator Problem of Stochastic Systems with Markovian Jump

This paper investigates the linear quadratic regulator(LQR) problem of linear stochastic systems with Markovian jump. Firstly, two iterative algorithms are proposed for solving the corresponding coupled algebraic Riccati equa- tions (CAREs) based on the general-type Lyapunov equation derived from linear stochastic systems. It is verified that the second algorithm adding an adjustable factor converges faster than the first one without it. Secondly, a monotonic convergence theorem is established for the proposed iterative algorithms under certain initial conditions. In the end, a numerical example is given to verify the efficiency of the proposed algorithms.

General Lyapunov-Based Iterative Algorithm for Linear Quadratic Regulator Problem of Stochastic Systems with Markovian Jump

This paper investigates the linear quadratic regulator(LQR) problem of linear stochastic systems with Markovian jump. Firstly, two iterative algorithms are proposed for solving the corresponding coupled algebraic Riccati equa- tions (CAREs) based on the general-type Lyapunov equation derived from linear stochastic systems. It is verified that the second algorithm adding an adjustable factor converges faster than the first one without it. Secondly, a monotonic convergence theorem is established for the proposed iterative algorithms under certain initial conditions. In the end, a numerical example is given to verify the efficiency of the proposed algorithms.

Robust Stabilization and Observer-Based Stabilization for a Class of Singularly Perturbed Bilinear Systems

An infinite-bound stabilization of a system modeled as singularly perturbed bilinear systems is examined. First, we present a Lyapunov equation approach for the stabilization of singularly perturbed bilinear systems for all ε∈(0, ∞). The method is based on the Lyapunov stability theorem. The state feedback constant gain can be determined from the admissible region of the convex polygon. Secondly, we extend this technique to study the observer and observer-based controller of singularly perturbed bilinear systems for all ε∈(0, ∞). Concerning this problem, there are two different methods to design the observer and observer-based controller: one is that the estimator gain can be calculated with known bounded input, the other is that the input gain can be calculated with known observer gain. The main advantage of this approach is that we can preserve the characteristic of the composite controller, i.e., the whole dimensional process can be separated into two subsystems. Moreover, the presented stabilization design ensures the stability for all ε∈(0, ∞). A numeral example is given to compare the new ε-bound with that of previous literature.