SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

On the existence of maximal solution for generalized algebraic Riccati equations arising in stochastic control

Existence of maximal solution is proved for a generalized version of the well-known standard algebraic Riccati equations which arise in certain stochastic optimal control problems.     

Closed-form solutions to discrete-time LQ optimal control and disturbance attenuation

This paper presents straightforward derivations of closed-form solutions to two discrete-time problems: linear-quadratic optimal control and disturbance attenuation. Then, as the main results, it shows how to reduce directly the closed-form solutions to their recursive forms. These recursive solutions are in the form of Riccati equation and, hence, establish a close relation between the closed-form solutions and the Riccati equations used in both problems.     

Iterative algorithm for minimal norm least squares solution to general linear matrix equations

This paper is concerned with minimal norm least squares solution to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Two iterative algorithms are proposed to solve this problem. The first method is based on the gradient search principle for solving optimization problem and the second one can be regarded as its dual form. For both algorithms, necessary and sufficient conditions guaranteeing the convergence of the algorithms are presented. The optimal step sizes such that the convergence rates of the algorithms are maximized are established in terms of the singular values of some coefficient matrix. It is believed that the proposed methods can perform important functions in many analysis and design problems in systems theory.     

A non-recursive method for solving the general discrete-time riccati equations related to the H∞ control problem

In this paper we propose a nonrecursive method for solving the general discrete-time algebraic Riccati equation related to the H_∞ control problem (H_∞-DARE). We have achieved this by casting the problem of solving a given H_∞-DARE to the problem of solving an auxiliary continuous-time algebraic Riccati equation associated with the H_∞ control problem (H_∞-CARE) for which the well known nonrecursive methods of solving are available. The advantages of our approach are: it reduces the computation involved in the recursive algorithms while giving much more accurate solutions, and it readily provides the properties of the general H_∞-DARE.     

The dual algebraic Riccati equations and the set of all solutions of the discrete-time Riccati equation

In this paper, two new pairs of dual continuous-time algebraic Riccati equations (CAREs) and dual discrete-time algebraic Riccati equations (DAREs) are proposed. The dual DAREs are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix. Then, in the case of singular matrices, a generalised inverse is introduced to deal with the dual DARE problem. These dual AREs can easily lead us to an iterative procedure for finding the anti-stabilising solutions, especially to DARE, by means of that for the stabilising solutions. Furthermore, we provide the counterpart results on the set of all solutions to DARE inspired by the results for CARE. Two examples are presented to illustrate the theoretical results.     

Composite control of a class of nonlinear singularly perturbed discrete-time systems via D-SDRE

In this paper, the regulation problem of a class of nonlinear singularly perturbed discrete-time systems is investigated. Using the theory of singular perturbations and time scales, the nonlinear system is decoupled into reduced-order slow and fast (boundary layer) subsystems. Then, a composite controller consisting of two sub-controllers for the slow and fast subsystems is developed using the discrete-time state-dependent Riccati equation (D-SDRE). It is proved that the equilibrium point of the original closed-loop system with a composite controller is locally asymptotically stable. Moreover, the region of attraction of the closed-loop system is estimated by using linear matrix inequality. One example is given to illustrate the effectiveness of the results obtained.     

求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

Closed-form solution for a class of continuous-time algebraic Riccati equations

In the present paper we obtain a closed-form solution for a class of continuous-time algebraic Riccati equations (AREs) with vanishing state weight. The ARE in such a class solves a minimum energy control problem. The obtained closed-form solution is used to prove a link between two independent fundamental limitation results in control over networks.     

An Optimized Iterative Method for Numerical Solution of Large Systems of Equations Based on the Extremal Property of Zeroes of Chebyshev Polynomials

An iterative method suitable for numerical solution of large systems of equations is presented. An extremal property of the Chebyshev polynomials is established, providing a logical foundation for the proposed procedure. A modification of the method is applicable for evaluation of the maximal eigenvalue of a matrix with real eigenvalues and of the associated eigenvector.     

On computing the stabilizing solution of a class of discrete‐time periodic Riccati equations

This paper addresses the problem of solving a class of periodic discrete‐time Riccati equation with an indefinite sign of its quadratic term. Such an equation is closely related to the so‐called full‐information H _∞ control of discrete‐time periodic systems. A globally convergent iterative algorithm with a local quadratic convergence rate is proposed for this purpose. An application to the problem of H _∞ filtering of discrete‐time periodic systems is also developed and illustrated via a numerical example. Copyright © 2013 John Wiley & Sons, Ltd.     

Sliding mode control for a class of nonlinear discrete-time networked systems with multiple stochastic communication delays

In this article, a sliding mode control problem is studied for a class of uncertain nonlinear networked systems with multiple communication delays. A sequence of stochastic variables obeying Bernoulli distribution is applied in the system model to describe the randomly occurring communication delays. The discrete-time system considered is also subject to parameter uncertainties and state-dependent stochastic disturbances. A novel discrete switching function is proposed to facilitate the sliding mode controller design. The sufficient conditions are derived by means of the linear matrix inequality (LMI) approach. It is shown that the system dynamics in the specified sliding surface is robustly exponentially stable in the mean square if two LMIs with an equality constraint are feasible. A discrete-time SMC controller is designed that is capable of guaranteeing the discrete-time sliding-mode reaching condition of the specified sliding surface. Finally, a simulation example is given to show the effectiveness of the proposed method.     

Stochastic stability of a class of unbounded delay neutral stochastic differential equations with general decay rate

Without the linear growth condition on the drift coefficient, this article examines the existence and uniqueness of global solutions of a class of neutral stochastic differential equations with unbounded delay and their asymptotic stabilities with general decay rate. To illustrate the application of our results, this article gives a two-dimensional system as an example.     

Continuity properties of solutions to H2 and H∞ Riccati equations

In H₂ and H_∞ optimal control (semi-)stabilizing solutions of algebraic Riccati equations play an essential role. It is well-known that these solutions might have discontinuities as a function of the system parameters. The paper shows that these discontinuities are directly linked to non-left-invertibility and, in contrast to what one might think, unrelated to zeros on the imaginary axis.     

Existence and multiplicity of positive periodic solutions for a class of higher-dimension functional differential equations with impulses

This paper deals with the existence of multiple periodic solutions for n-dimensional functional differential equations with impulses. By employing the Krasnoselskii fixed point theorem, we obtain some easily verifiable sufficient criteria which extend previous results.     

The optimal control related to Riemannian manifolds and the viscosity solutions to Hamilton–Jacobi–Bellman equations

In this paper we study the optimal stochastic control problem for stochastic differential equations on Riemannian manifolds. The cost functional is specified by controlled backward stochastic differential equations in Euclidean space. Under some suitable assumptions, we conclude that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation which is a fully nonlinear parabolic partial differential equation on Riemannian manifolds.     

A novel optimal tracking control scheme for a class of discrete-time nonlinear systems using generalised policy iteration adaptive dynamic programming algorithm

In this paper, a novel iterative adaptive dynamic programming (ADP) algorithm, called generalised policy iteration ADP algorithm, is developed to solve optimal tracking control problems for discrete-time nonlinear systems. The idea is to use two iteration procedures, including an i-iteration and a j-iteration, to obtain the iterative tracking control laws and the iterative value functions. By system transformation, we first convert the optimal tracking control problem into an optimal regulation problem. Then the generalised policy iteration ADP algorithm, which is a general idea of interacting policy and value iteration algorithms, is introduced to deal with the optimal regulation problem. The convergence and optimality properties of the generalised policy iteration algorithm are analysed. Three neural networks are used to implement the developed algorithm. Finally, simulation examples are given to illustrate the performance of the present algorithm.     

H 2 optimal control for a wide class of discrete-time linear stochastic systems

In this article, the problem of H ₂-control of a discrete-time linear system subject to Markovian jumping and independent random perturbations is considered. Different H ₂ performance criteria (often called H ₂-norms) are introduced and characterised via solutions of some suitable linear equations on certain spaces of symmetric matrices. Some aspects specific to the discrete-time framework are revealed. The problem of optimisation of H ₂-norms is solved under the assumption that full state vector is available for measurements. One shows that among all stabilising controllers of higher dimension, the best performance is achieved by a zero-order controller. The corresponding feedback gain of the optimal controller is constructed based on the stabilising solution of a system of discrete-time generalised Riccati equations.