Method Of Invariant Ellipsoids Research Articles

Decentralized leader-following consensus control design for discrete-time multi-agent systems with switching topology

The problem of consensus control of linear discrete-time multi-agent systems (MASs) with switching topology is considered in the presence of a leader. The goal of consensus control is to bring the states of all agents to the leader state while providing stability for local agents, as well as the MAS as a whole. In contrast to the traditional approach, which uses the concept of an extended dynamic multi-agent system model and communication topology graph Laplacian, this paper proposes a decomposition approach, which provides a separate design of local controllers. The control law is chosen in the form of distributed feedback with discrete PID controllers. The problem of local controllers’ design is reduced to a set of semidefinite programming problems using the method of invariant ellipsoids. Sufficient conditions for agents’ stabilization and global consensus condition fulfillment are obtained using the linear matrix inequality technique. The availability of information about a finite set of possible configurations between agents allows us to design local controllers offline at the design stage. A numerical example demonstrates the effectiveness of the proposed approach.

Sparse Filtering Under Norm-Bounded Exogenous Disturbances Using Observers

The paper considers the sparse filtering problem under arbitrary norm-bounded exogenous disturbances. We propose a simple and universal observer-based approach to its solution, based on the LMI technique and the method of invariant ellipsoids; it allows the use of a reduced number of system outputs. From a technical point of view of application, we reduce the original problem to semi-definite programming, which is easily solved numerically. The proposed simple approach is easy to implement and can be equally extended to systems in continuous and discrete time.

Optimization of Invariant Ellipsoid Technique for Sparse Controllers Design

The paper deals with the method for the design of linear controllers with sparse state feedback matrices for control the plants under conditions of unknown and bounded disturbances. The importance of the sparsity property in feedback can be explained by two factors. First, by minimizing the columnar norm of the feedback matrix in the control it becomes pos- sible to use a minimum number of measuring devices. Secondly, by minimizing the row norm of the feedback matrix, the required number of executive (control) devices is minimized. Both properties, if they are achievable in the synthesis of the controller, reduce the cost of the system and improve the fault tolerance and quality of regulation by reducing the structural complexity. The search algorithm for sparse matrices is based on the method of invariant ellipsoids and is formulated as a solution to a system of linear matrix inequalities with additional constraints. A special set of optimization conditions is proposed which for a disturbed system minimizes overshoot and overshoots in transient processes of the disturbed closed- loop system simultaneously with minimizing errors in the steady state. The proposed method also assumes the possibility of minimizing both the row norm of the feedback matrix and the column one, while preserving the robustness properties, which makes it possible to solve the sparse control problem (a sparse control is understood as a linear controller with a sparse feedback matrix). The efficiency of the proposed control scheme is confirmed by the results of computer modeling and comparison with some existing ones.

Robust decentralized plug and play voltage tracker of islanded microgrids under loads and lines uncertainties by the invariant ellipsoids

Microgrids (MGs) comprising of distributed generation units (DGs) are subjected to plug-and-play operation (PnP), lines (connecting the DGs) parameters uncertainty, and load changes. Robust stability and an authentic operation for islanded microgrids can be guaranteed through a robust decentralized voltage tracker developed in this paper. The proposed controller design has the following merits: 1) fully decentralized, 2) scalable, and 3) maintains robust stability against PnP of DGs, load changes, and lines parameters’ uncertainties. A sufficient condition is developed by linear matrix inequality convex optimization is exploited to solve the problem. The MGs’ changes in load and the line's parameters are modelled as norm-bounded uncertainties. The suggested controller uses local measurements from DGs, i.e., decentralized by the decomposition of the global system into subsystems. For each subsystem, the rest of the system's impact is considered disturbances, whose influence must be minimized. The proposed disturbance rejection control algorithm is based on the method of invariant-ellipsoids. Several time-domain scenarios such as connecting and disconnecting a number of DGs, local load changes, and variation of transmission line parameters are executed to assess the suggested controller's effectiveness, using MATLAB /Sim PowerSystems Toolbox.

Disturbance‐rejection voltage control of an isolated microgrid by invariant sets

New disturbance-rejection control for an islanded microgrid (MG) is presented in this study. The proposed method can be used with MGs consisting of several distributed generation (DG) units and local loads with both grid-connected and islanded modes capability. The proposed controller utilises state feedback with integral control to track the desired voltage. The uncertainty of the load parameters is tackled as a disturbance. The proposed controller is designed using the method of invariant ellipsoids to guarantee robust stability, fast transient response and zero steady-state error. The controller synthesis is formulated as a convex optimisation problem that is effectively solved using linear matrix inequality methods. The performance of the proposed MG voltage controller is assessed by several simulations in the presence of random load variations, load unbalances and step changes in the voltage reference. Moreover, the power mismatch during accidental islanding event is also considered.

Disturbance Attenuation with Minimization of Ellipsoids Restricting Phase Trajectories in Transition and Steady State

Abstract A new method for attenuation of external unknown bounded disturbances in linear dynamical systems with known parameters is proposed. In contrast to the well known results, the developed static control law ensures that the phase trajectories of the system are located in an ellipsoid, which is close enough to the ball in which the initial conditions are located, as well as provides the best control accuracy in the steady state. To solve the problem, the method of Lyapunov functions and the technique of linear matrix inequalities are used. The linear matrix inequalities allow one to find optimal controller. In addition to the solvability of linear matrix inequalities, a matrix search scheme is proposed that provides the smallest ellipsoid in transition mode and steady state with a small error. The proposed control scheme extends to control linear systems under conditions of large disturbances, for the attenuation of which the integral control law is used. Comparative examples of the proposed method and the method of invariant ellipsoids are given. It is shown that under certain conditions the phase trajectories of a closed-loop system obtained on the basis of the invariant ellipsoid method are close to the boundaries of the smallest ellipsoid for the transition mode, while the obtained control law guarantees the convergence of phase trajectories to the smallest ellipsoid in the steady state.

Decentralized Robust Saturated Control of Power Systems Using Reachable Sets

Electric power grids are highly nonlinear complex systems. This manuscript presents a novel approach to the stabilization of large power systems. The proposed control satisfied three constraints: decentralization, input saturation imposed in practice, and robustness against load changes. The large power system is decomposed into subsystems, for each a decentralized controller is designed. The effect of the rest of the system on each subsystem is considered as an external disturbance and represented in norm-bounded form. A new approach to solve this problem is proposed in the present paper. The approach is based on the method of invariant ellipsoids, and the tool of linear matrix inequalities (LMI) is utilized to solve the resulting optimization problem. Control of multimachine power system is studied using the proposed control. Comparison with other techniques is also given.

Optimization of nonlinear cascade systems in Lurie form with bounded external disturbances

Optimization of nonlinear cascade systems in the form of Lurie with bounded disturbances is considered. For the solution of the problem the method of invariant ellipsoids is used. The obtained result is compared with the previously obtained classic result based on Lyapunov function.

Settling time in a linear dynamic system with bounded external disturbances

The article deals with the problem of estimating the settling time for a linear dynamic system subjected to the action of nonrandom bounded external disturbances. The suggested approach is based on the method of invariant ellipsoids and the technique of linear matrix inequalities. Both the continuous and the discrete version of the problem are considered. As an example the problem of control of a two-mass system is investigated.

Designing suboptimal discrete output control laws for damping bounded perturbations

The suboptimal linear discrete output control laws were designed which are capable of minimizing the upper estimate of the maximal value of the norm of controlled output for any bounded perturbations and all initial system conditions from some domain. The laws were designed using the method of invariant ellipsoids and the apparatus of linear matrix inequalities.

Optimization of linear systems subject to bounded exogenous disturbances: The invariant ellipsoid technique

This survey covers a variety of results associated with control of systems subjected to arbitrary bounded exogenous disturbances. The method of invariant ellipsoids reduces the design of optimal controllers to finding the smallest invariant ellipsoid of the closed-loop dynamical system. The main tool of this approach is the linear matrix inequality technique. This simple yet versatile approach has high potential in extensions and generalizations; it is equally applicable to both the continuous and discrete time versions of the problems.

Suppression of bounded exogenous disturbances: A linear dynamic output controller

In the paper, we study a problem of constructing a linear dynamic output controller for suppressing bounded exogenous perturbations in a linear control system. We propose an approach based on a method of invariant ellipsoids and technique of linear matrix inequalities. A control of gyroplatform and two-mass system is given as an example.

Synthesis of a suboptimal controller by output for dampening limited disturbances

In the framework of the method of invariant ellipsoids, for a linear dynamic object a suboptimal linear dynamic controller by output is synthesized, at which the upper estimate of a maximum value of the norm of the controllable output reaches a minimum at all initial states of a system from a certain domain and at any disturbances bounded by the norm.

Using the method of invariant ellipsoids for linear robust output stabilization of spacecraft

Consideration was given to the problem of robust output stabilization of the nonlinear system using the method of invariant ellipsoids on assumption that the system satisfies a Lipschitz-like condition and the system output is subjected to bounded external disturbances. It was reduced to a special optimization problem with solution based on the theory of linear matrix inequalities. The proposed approach was illustrated by an example of stabilization of a spacecraft with elastic dynamic elements. The designed control represents a combination of the "accelerate-decelerate" algorithm and a linear feedback obtained using the method of invariant ellipsoids.

Robust filtering under nonrandom disturbances: The invariant ellipsoid approach

We present a simple and universal observer-based approach to solving the problem of robust filtering of unknown-but-bounded exogenous disturbances. The heart of this approach is the method of invariant ellipsoids. Application of this technique allows for a reformulation of the original problem in terms of linear matrix inequalities with reduction to semidefinite programming and one-dimensional optimization, which are easy to solve numerically. Continuous-time and discrete-time cases are studied in equal detail. The efficacy of the approach is demonstrated via the double pendulum example.

Rejection of bounded exogenous disturbances by the method of invariant ellipsoids

Rejection of the bounded exogenous disturbances was first studied by the l 1-optimization theory. A new approach to this problem was proposed in the present paper on the basis of the method of invariant ellipsoids where the technique of linear matrix inequalities was the main tool. Consideration was given to the continuous and discrete variants of the problem. Control of the "double pendulum" was studied by way of example.