Robust consensus tracking of linear multiagent systems with input saturation and input‐additive uncertainties

The problem of robust global consensus tracking of linear multiagent systems with input saturation and input‐additive uncertainties is investigated in this paper. By using the parametric Lyapunov equation approach and an existing dynamic gain scheduling technique, a new distributed nonlinear‐gain scheduling consensus‐trackining algorithm is developed to solve this problem. Under the assumption that each agent is asymptotically null controllable with bounded control, it is shown that the robust global consensus tracking can be achieved under the undirected graph provided that its generated graph contains a directed spanning tree. Compared with the existing algebraic Riccati equation approach, which requires the online solution of a parameterized algebraic Riccati equation, all the parameters in the proposed nonlinear algorithm are offline determined a priori. Finally, numerical examples are provided to validate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic behaviour of the solution of the projection Riccatidifferential equation

The solution of the Riccati differential equation (RDE) is shown to be asymptotically close to the solution of the projection Riccati differential equation (PRDE). The asymptotic behaviour of the latter is analyzed in an explicit formula. The almost-periodic asymptote of the solution of the PRDE is computed by an algorithm based upon the concepts of an aperiodic/almost-periodic generator (APG) decomposition of a linear map and unit row-staircase form of a polynomial matrix. The analysis ultimately provides a convergence criterion. In particular, it is shown that the solution of the PRDE always converges in the aperiodic case     

Identification and filtering

A theory of the discrete Riccati equation asymptotic behavior for a degenerate filtering problem corresponding to prediction. As an application, ARMA identification is shown to be determined from the asymptotic behavior of the reflection coefficients. Explicitly the time constants of the reflection coefficient sequence determine the moving average portion of the process. This work completes earlier work of the author on “invariant directions” of the Riccati equation.     

A Riccati-type algorithm for solving generalized Hermitian eigenvalue problems

The paper describes a heuristic algorithm for solving a generalized Hermitian eigenvalue problem fast. The algorithm searches a subspace for an approximate solution of the problem. If the approximate solution is unacceptable, the subspace is expanded to a larger one, and then, in the expanded subspace a possibly better approximated solution is computed. The algorithm iterates these two steps alternately. Thus, the speed of the convergence of the algorithm depends on how to generate a subspace. In this paper, we derive a Riccati equation whose solution can correct the approximate solution of a generalized Hermitian eigenvalue problem to the exact one. In other words, the solution of the eigenvalue problem can be found if a subspace is expanded by the solution of the Riccati equation. This is a feature the existing algorithms such as the Krylov subspace algorithm implemented in the MATLAB and the Jacobi–Davidson algorithm do not have. However, similar to solving the eigenvalue problem, solving the Riccati equation is time-consuming. We consider solving the Riccati equation with low accuracy and use its approximate solution to expand a subspace. The implementation of this heuristic algorithm is discussed so that the computational cost of the algorithm can be saved. Some experimental results show that the heuristic algorithm converges within fewer iterations and thus requires lesser computational time comparing with the existing algorithms.

     

The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

In this paper based on a system of Riccati equations with variable coefficients, we present a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccati equation with constant coefficient expansion method to construct more new exact solutions of nonlinear differential equations in mathematical physics. A pair of generalized Hamiltonian equations is chosen to illustrate our algorithm such that more families of new exact solutions are obtained which contain soliton-like solution and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.     

Numerical integration of the differential matrix Riccati equation

Two new Bernoulli substitution methods for solving the Riccati differential equation are tested numerically against direct integration of the Riccati equation, the Chandrasekhar algorithm, and the Davison-Maki method on a large set of problems taken from the literature. The first of these new methods was developed for the time-invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric Riecati equation whose solution can be given explicitly in terms of a matrix exponential of ordern. This method is fast and accurate when the extremal solutions of the associated algebraic Riccati equation are well separated. The second new method was developed as a means of eliminating the instabilities associated with the Davison-Maki algorithm. By using reinitialization at each time step the Davison-Maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time-invariant and time-dependent problems. From the results of our study we conclude that the modified Davison-Maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the Chandrasekhar algorithm is better.     

Elementwise decoupling and convergence of the Riccati equation in the SG algorithm

It is shown that the difference Riccati equation of the Stenlund-Gustafsson (SG) algorithm for estimation of linear regression models can be solved elementwise. Convergence estimates for the elements of the solution to the Riccati equation are provided, directly relating convergence rate to the signal-to-noise ratio in the regression model. It is demonstrated that the elements of the solution lying in the direction of excitation exponentially converge to a stationary point while the other elements experience bounded excursions around their current values.     

Comments on "On some bounds in the algebraic Riccati and Lyapunov equations"

Results presented by Moil in the paper, which give some bounds for the trace and the determinant of the positive definite solution to the algebraic Riccati matrix equation, are shown to be erroneous, and suggestions to remedy these errors are made.     

Bounds on the solution of the algebraic Riccati equation under perturbations in the coefficients

This paper is concerned with bounds on symmetric solutions of the continuous algebraic matrix Riccati equation under perturbations on its coefficients. Special emphasis is given to the question of estimating the ‘size’ of the perturbations on the stabilizing solution of the Riccati equation when one or all its coefficients are subject to small perturbations. Upper bounds on the norm of the perturbations on the stabilizing solution are presented. Moreover, it is shown that these perturbations can be determined as a single-valued continuous function of the perturbations on the coefficient of the Riccati equation.     

Solution of the singular discrete regulator problem using eigenvector methods

The discrete regulator problem with a singular state matrix cannot be solved using either the eigenvectors or the sign function of the associated compound matrix, since then, the implicit equation defining the feedback matrix cannot be transformed into the regular discrete Riccati equation. In this paper, it is shown that the solution of a continuous Riccati equation, whose matrix parameters depend on the matrices of both discrete system and cost functional, is the solution of the implicit equation. Since the continuous Riccati equation has symmetric spectrum with respect to the imaginary axis, it has no singularity problem and therefore the matrix sign function method can be used for its solution.     

Optimal boundary control of a tracking problem for a parabolic distributed parameter system

A comparative study is made of five methods for calculating the optimal control function for a linear parabolic tracking problem with boundary control. Questions of computational instability, numerical accuracy and economic computer usage are investigated. Open-loop methods based upon the variational equations are shown to have the advantages of efficiency, accuracy and ease of programming. Methods based on the method of lines and the Riccati equation are shown to be less straightforward in use. The fourth-order explicit Runge-Kutta algorithm has little scope for application because of its restricted stability range. The Kalman-Englar algorithm is much more robust but a Crank-Nicholson algorithm for the associated auxiliary equation is not very satisfactory in some cases. The auxiliary variable method may then confer some benefits.     

Systolic算法和结构求解代数Riccati方程

本文基于矩阵的符号函数法，提出了一种Ｕ－Ｄ分解算法和脉动（Ｓｙｓｔｏｌｉｃ）结构有效地求解代数Ｒｉｃｃａｔｉ方程以及用固定大小的方形阵列解决大型问题的方法。     

Systolic算洁和结构求解代数Riccati方程

本文基于矩阵的符号函数法，提出了一种Ｕ－Ｄ分解算法和脉动（Ｓｙｓｔｏｌｉｃ）结构有效地求解代数Ｒｉｃｃａｔｉ方程以及用固定大小的方形阵列解决大型问题的方法．     

A stabilization algorithm for a class of uncertain linear systems

This paper presents an algorithm for the stabilization of a class of uncertain linear systems. The uncertain systems under consideration are described by state equations which depend on time-varying unknown-but-bounded uncertain parameters. The construction of the stabilizing controller involves solving a certain algebraic Riccati equation. Furthermore, the solution to this Riccati equation defines a quadratic Lyapunov function which is used to establish the stability of the closed-loop system. This leads to a notion of ‘quadratic stabilizability’. It is shown that the stabilization procedure will succeed if and only if the given uncertain linear system is quadratically stabilizable.The paper also deals with a notion of ‘overbounding’ for uncertain linear systems. This procedure enables the stabilization algorithm to be applied to a larger class of uncertain linear systems. Also included in the paper are results which indicate the degree of conservativeness introduced by this overbounding process.     

Convergence of the doubling algorithm for the discrete-time algebraic Riccati equation

If the doubling algorithm (DA) for the discrete-time algebraic Riccati equation converges, the speed of convergence is high. However, its convergence has not yet been examined exactly. Firstly, a certain matrix appearing in the DA is shown to be non-singular and therefore the algorithm is well defined. Secondly, it is found that the loss of significant digits hardly occurs in the DA. Finally, it is proved that if the time-invariant discrete-time linear system, whose state is estimated by the steady-state Kalman filter, is reachable and detectable, or stabilizable and observable, then all three matrix sequences in the DA converge.     

H∞滤波问题数值求解的精细积分算法

有限时间H∞滤波的Riccati方程和滤波方程分别为非线性矩阵微分方程和线性变系 数微分方程,而且Riccati微分方程解的存在性还依赖于参数 γ-2,因此求这些方程的数值解一 般比较困难.按照结构力学与最优控制的模拟关系,Riccati方程解存在的临界参数 γ-2cr对应于 广义Rayleigh商的一阶本征值.因此可以用精细积分法结合扩展的Wittrick-Williams(W-W) 算法计算 γ-2cr .并求解Ricclati方程,滤波微分方程的解也可以由精细积分法计算.     

Existence of discrete-time LQG-controllers

Existence results for the LQG-controller are investigated. An infimal Riccati equation based controller may potentially give closed loop eigenvalues on the unit circle. Assuming left and right invertibility it is shown that there exists an optimal controller if and only if the Riccati equation based controller stabilizes the closed loop system after removal of all its unobservable and uncontrollable modes. Furthermore this reduced controller is the optimal controller, and its transfer function is unique. This existence condition is a considerable simplification of the more general geometric condition recently derived by Trentelman and Stoorvogel.     

An imbedded initialization of Newton's algorithm for matrix Riccati equation

An effective numerical computation of the steady-state Riccati matrix is based on the successive solutions of a Lyapunov equation using Newton's method. The requirements of this algorithm are an initial stabilizing matrix and the numerical solution of the associated Lyapunov equation. Computationally, the first requirement is the more influencing factor in solving the Riccati equation with reasonable accuracy and speed. In this paper an initial matrix, based on the parameter imbedded solution of the Riccati equation, is introduced for the Newton's algorithm. The imbedding Newton algorithm has been applied to a variety of system, both stable and unstable as well as high-dimensional, A matrices, one of which is reported here. The proposed modification has improved the required CPU time of previous initialization schemes by as much as a factor of 6 times for the same order of accuracy.     

Multilayer dynamic neural networks for non-linear system on-line identification

To identify on-line a quite general class of non-linear systems, this paper proposes a new stable learning law of the multilayer dynamic neural networks. A Lyapunov-like analysis is used to derive this stable learning procedure for the hidden layer as well as for the output layer. An algebraic Riccati equation is considered to construct a bound for the identification error. The suggested learning algorithm is similar to the well-known backpropagation rule of the multilayer perceptrons but with an additional term which assure the stability property of the identification error.     

Stability of a Riccati equation arising in recursive parameter estimation under lack of excitation

Stability properties of the Riccati equation in a recently suggested antiwindup algorithm for recursive parameter estimation are analyzed. Convergence of the resulting dynamic system is implied by that of a linear time-varying difference matrix equation. By means of converging matrix products theory, the linear mapping associated with the system is shown to be a paracontraction with respect to a certain norm. Therefore, measured in that norm, the solution to the matrix equation will not diverge notwithstanding excitation properties of the data. Thus the purpose of anti-windup is achieved.