Lyapunov Equation Research Articles

Gramians, Energy Functionals, and Balanced Truncation for Linear Dynamical Systems With Quadratic Outputs

In this article, we investigate a balancing-based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. With this aim, we propose a new algebraic observability Gramian for the system based on the Hilbert space adjoint theory. We then show that the proposed Gramian satisfies a particular type of generalized Lyapunov equation and we investigate its connection to energy functionals. It allows one to find states that are hard to control and hard to observe via an appropriate balancing transformation. Truncation of such states yields reduced-order models. Finally, based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_2$</tex-math></inline-formula> energy considerations, we derive error bounds, depending on the neglected singular values. We demonstrate the efficiency of the proposed methodology using a numerical example.

Fixed-time nonlinear homogeneous sliding mode approach for robust tracking control of multirotor aircraft: Experimental validation

This paper presents a robust scheme for fixed-time tracking control of a multirotor system. The aircraft is subjected to matched lumped disturbances, i.e., unmodeled dynamics, parameters uncertainties, and external perturbations besides measurement noise. Firstly, a novel Nonlinear Homogeneous Continuous Terminal Sliding Manifold (NHCTSM) based on the weighted homogeneity theory is presented. The sliding manifold is designed with prescribed dynamics featuring Global Asymptotic Stability (GAS) and fixed-time convergence. Then, a novel Fixed-time Non-switching Homogeneous Nonsingular Terminal Sliding Mode Control (FNHNTSMC) is proposed for the position and attitude loops by employing the developed NHCTSM and an appropriate reaching law. Moreover, the control framework incorporates a disturbance observer to feedforward and compensate for the disturbances. The designed control scheme can drive the states of the system to the desired references in fixed-time irrespective of the values of the Initial Conditions (ICs). Since the existing works on homogeneous controllers rely on the bi-limit homogeneity concept in the convergence proofs, the estimate of the settling-time or its upper-bound cannot be given explicitly. In contrast, this study employs Lyapunov Quadratic Function (LQF) and Algebraic Lyapunov Equation (ALE) in the stability analysis of both controller and observer. Following this method, an expression of the upper-bound of the settling-time is explicitly derived. Furthermore, to assure the Uniform Ultimate Boundedness (UUB) of all signals in the feedback system, the dynamics of the observer and controller are jointly analyzed. Simulations and experiments are conducted to quantify the control performance. The proposed approach achieves superior performance compared with recent literature on fixed-time/finite-time control and a commercially available PID controller. The comparative results witness that the developed control scheme improves the convergence-time, accuracy, and robustness while overcoming the singularity issue and mitigating the chattering effect of conventional SMC.

Adaptive frequency‐limited ‐model order reduction

AbstractIn this paper, we present an adaptive framework for constructing a pseudo‐optimal reduced model for the frequency‐limited ‐optimal model order reduction problem. We show that the frequency‐limited pseudo‐optimal reduced‐order model has an inherent property of monotonic decay in error if the interpolation points and tangential directions are selected appropriately. We also show that this property can be used to make an automatic selection of the order of the reduced model for an allowable tolerance in error. The proposed algorithm adaptively increases the order of the reduced model such that the frequency‐limited ‐norm error decays monotonically irrespective of the choice of interpolation points and tangential directions. The stability of the reduced‐order model is also guaranteed. Additionally, it also generates the approximations of the frequency‐limited system Gramians that monotonically approach the original solution. Further, we show that the low‐rank alternating direction implicit iteration method for solving large‐scale frequency‐limited Lyapunov equations implicitly performs frequency‐limited pseudo‐optimal model order reduction. We consider two numerical examples to validate the theory presented in the paper.

On The Neutrosophic Formula of Some Matrix Equations Derived from Data Mining Theory and Control Systems

This paper is dedicated to studying the neutrosophic formula of some famous matrix equations used in theoretical data mining algorithms and control systems by using neutrosophic matrices and refined neutrosophic matrices over neutrosophic real fields. On the other hand, we concentrate on the neutrosophic formula of the Sylvester equation, and Lyapunov equation, where we study their formulas and properties in terms of theorems in the neutrosophic real number field and refined real number field. Also, we illustrate many different examples to clarify the validity of our work.

Adaptive Cooperative Tracking and Parameter Estimation of an Uncertain Leader over General Directed Graphs

This paper studies the cooperative tracking problem of heterogeneous Euler-Lagrange systems with an uncertain leader. Different from most existing works, system dynamic knowledge of the leader node is unaccessible to any follower node in our paper. Distributed adaptive observers are designed for all follower nodes, simultaneously estimate the state and parameters of the leader node. The observer design does not rely on the frequency knowledge of the leader node, and the estimation errors are shown to converge to zero exponentially. Moreover, the results are applied to general directed graphs, where the symmetry of Laplacian matrix does not hold. This is due to two newly developed Lyapunov equations, which solely depend on communication network topologies. Interestingly, using these Lyapunov equations, many results of multi-agent systems over undirected graphs can be extended to general directed graphs. Finally, this paper also advances the knowledge base of adaptive control systems by providing a main tool in the analysis of parameter convergence for adaptive observers.

Pareto Optimality in Infinite Horizon Mean-Field Stochastic Cooperative Linear-Quadratic Difference Games

This paper is concerned with the mean-field stochastic cooperative linear-quadratic (LQ) dynamic difference game in an infinite time horizon. First, the necessary and sufficient conditions for the stability in the mean-square sense, and the stochastic Popov-Belevith-Hautus (PBH) eigenvector tests for exact observability and exact detectability of mean-field stochastic linear difference systems are derived by the <inline-formula><tex-math notation="LaTeX">$\mathscr {H}$</tex-math></inline-formula>-representation technique. Second, the relation between the solvability of the cross-coupled generalized Lyapunov equations&#x00A0;(CC-GLEs) and exact observability, exact detectability, and stability of the mean-field dynamic system is well characterized. It is then shown that the cross-coupled algebraic Riccati equations&#x00A0;(CC-AREs) admit a unique positive definite (positive semi-definite, respectively) solution under exact observability (exact detectability, respectively), which is also a feedback stabilizing solution. Furthermore, all Pareto optimal strategies and solutions can be respectively derived via the solutions to the weighted CC-AREs (WCC-AREs) and the weighted cross-coupled algebraic Lyapunov equations&#x00A0;(WCC-ALEs). Finally, a practical application on the computation offloading in the multi-access edge computing network (MECN) is presented to illustrate the proposed theoretical results.

SOLVING SYSTEMS OF SECOND-ORDER MATRIX EQUATIONS

Matrix equations and systems of matrix equations are widely used in control system optimization problems. However, there are developed only methods for solving the most popular matrix equations, like Riccati and Lyapunov equations .There is no universal approach to solving all problems of this class. The previously considered method of solving systems of algebraic equations over a field of real numbers [1] is summarized in this paper and a scheme for systems of polynomial matrix equations of the second degree with many unknowns is proposed. Also, the recurrent formula for developing a solution into a chain matrix fraction is given. The convergence of the proposed method is investigated. The results of numerical experiments that confirm the validity of theoretical calculations and the effectiveness of the proposed scheme are presented.

H-Infinity Control With Nearly Symmetric State Matrix

In this paper, we give an upper bound on the deviation from H-infinity optimality of a class of controllers as a function of the deviation from symmetry in the state matrix. We further suggest a scalar measure of symmetry which is shown to be directly relevant for estimating nearness to optimality. In connection to this, we give a simple analytical solution to a class of Lyapunov equations for two dimensional state matrices. Finally, we demonstrate how a well-chosen symmetric part for nearly symmetric state matrices may lead not only to near-optimality, but also to controller sparsity, a desirable property for large-scale systems. In the special case that the state matrix is symmetric and Hurwitz, our main result simplifies to give an H-infinity optimal controller with several benefits, a result which has recently appeared in the literature. In this sense, the above is a significant generalization which considers a much wider class of systems, yet allows one to retain the benefits of symmetric state matrices, while offering means of quantifying the effect of this on the H-infinity norm.

Practical prescribed-time sampled-data control of triangular nonlinear systems satisfying a linear growth condition by linear time-varying feedback

In this paper, the practical prescribed-time sampled-data control problem of a class of triangular nonlinear systems satisfying a linear growth condition is studied. With the help of some properties of a class of parametric Lyapunov equations (PLEs) and by constructing a time-varying Lyapunov-like function, a linear time-varying feedback controller is designed to make the states of the closed-loop system converge to an arbitrarily small neighbourhood of the origin in any prescribed time, while the control siginals are guaranteed to be bounded. A numerical example validates the effectiveness of the proposed method.

Selective LQ wide area damping of power networks based on the spectral decomposition of Gramians

Inter-area oscillations in large power network are poorly controlled via local decentralized control. One promising strategy for wide-area damping of these oscillations is to minimize the quadratic state and control cost function for a linearized system model. In this paper, we improve this linear-quadratic method by introducing novel selective approach. Namely, instead of using the full state we minimize in the cost function only the selected spectral part of the state. The proposed selective LQ optimal problem is reformulated via the selected spectral part of the closed-loop observability Gramian. This spectral part is a solution to the modal Lyapunov equation specially designed for a chosen spectral range. Novel approach allows to take into account in global optimization only the selected set of oscillations and therefore to improve flexibility and accuracy of controllers setting. Numerical experiments with the IEEE New England test system confirm that the selective approach improves the performance of the conventional LQ method for suppressing low frequency inter-area oscillations.

측정 민감도 및 외란 하에서 쿼드로터의 강인 자세 제어

A robust attitude control scheme for quadrotor is proposed under measurement sensitivity and external disturbance. In order to treat measurement sensitivity, a new determination process of a compact with allowed measurement sensitivity is presented by utilizing Lyapunov equation. With our control scheme, all controlled system states are shown to be bounded. Moreover, the ultimate bounds of all states can be made arbitrarily small by adjusting the gain-scaling factor in the presence of external disturbance. The validity of our control method is first checked via simulation, and then is also experimentally shown in the attitude control of quadrotor system.

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.

Convergence and robustness of bounded recurrent neural networks for solving dynamic Lyapunov equations

Recurrent neural networks have been reported as an effective approach to solve dynamic Lyapunov equations, which widely exist in various application fields. Considering that a bounded activation function should be imposed on recurrent neural networks to solve the dynamic Lyapunov equation in certain situations, a novel bounded recurrent neural network is defined in this paper. Following the definition, several bounded activation functions are proposed, and two of them are used to construct the bounded recurrent neural network for demonstration, where one activation function has a finite-time convergence property and the other achieves robustness against noise. Moreover, theoretical analyses provide rigorous and detailed proof of these superior properties. Finally, extensive simulation results, including comparative numerical simulations and two application examples, are demonstrated to verify the effectiveness and feasibility of the proposed bounded recurrent neural network.

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.

Autonomous Nonlinear Boundary-Value Problems for the Lyapunov Equation in the Hilbert Space

Autonomous Nonlinear Boundary-Value Problems for the Lyapunov Equation in the Hilbert Space

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.

On hyper-exponential stability and stabilization of linear systems by bounded linear time-varying controllers

Hyper-exponential stability analysis and hyper-exponential stabilization of linear systems by bounded linear time-varying feedback are investigated in this paper. On the one hand, we propose some Lyapunov-like hyper-exponential stability theorems (both global and local) based on the comparison principle and the concepts of hyper-exponentially stable functions and hyper-exponentially increasing functions. On the other hand, we establish methods to design bounded linear time-varying controllers such that hyper-exponential stability of linear time-invariant systems can be guaranteed. The key design tool is the utilization of a time-varying parameter contained in the controller and the properties of solution to a parametric Lyapunov equation. Both state feedback and observer-based output feedback are accommodated. As a further result, hyper-exponential semi-global stabilization for linear systems by bounded controls is discussed. Finally, the validity of the proposed schemes is illustrated through numerical simulations on spacecraft rendezvous control system.

Stability analysis of 2‐D discrete and continuous state‐space systems

This paper presents new stability conditions for two-dimensional (2-D) systems in state-space description. Both discrete and continuous systems are studied. These results are based on the criteria first presented by Huang, De Carlo, Strintzis, Murray, Delsarte, et al. and on the discrete Lyapunov equation with complex elements for 2-D systems. The stability properties of the Mansour matrix are also used for stability testing in state-space. Criteria for the VSHP property of 2-D polynomials are further presented using the continuous Lyapunov equation with complex elements and the stability properties of the Schwarz matrix form. The stability properties of the Schwarz matrix are also used for testing the VSHP property of 2-D polynomials in state-space. The proposed new criteria are non-conservative for the stability analysis of 2-D discrete and continuous systems and achieve the aim of reducing the original 2-D problem as much as possible to a set of 1-D stability tests. Numerical examples are given to illustrate the utility of the proposed conditions.

Numerical method based on fiber bundle for solving Lyapunov matrix equation

In this paper, we firstly introduce the origin of Lyapunov matrix equation, and then the geometric methods for solving Lyapunov equation are given by using the Log-Euclidean metric and the fiber bundle-based Riemannian metric based on the manifold of positive definite Hermitian matrices. Then, the solution of Lyapunov matrix equation is presented by providing a natural gradient descent algorithm (NGDA), a Log-Euclidean descent algorithm (LGDA) and a Riemannian gradient algorithm based on fiber bundle (RGA). At last, the convergence speeds of the RGA, the NGDA and the LGDA are compared via two simulation examples. Simulation results show that the convergence speed of the RGA is faster than both of the LGDA and the NGDA.