The standard H/sub ∞/ problem in systems with a single input delay

Rotational Motion Control by Feedback with Minimum L1-Norm

Finite-horizon and infinite-horizon linear quadratic optimal control problems: A data-driven Euler scheme

ABSTRACTThis paper focuses on the optimal control design for a system of coupled parabolic–hypebolic partial differential equations by using the infinite-dimensional state-space description and the corresponding operator Riccati equation. Some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear–quadratic (LQ)-optimal control problem. A state LQ-feedback operator is computed by solving the operator Riccati equation, which is converted into a set of algebraic and differential Riccati equations, thanks to the eigenvalues and the eigenvectors of the parabolic operator. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ-optimal controller designed in the early portion of the paper is implemented for the original nonlinear model. Numerical simulations are performed to show the controller performances.

We present a unified theory of matrix pencil techniques for solving both continuous and discrete-time algebraic Riccati equations (AREs) under fairly general conditions on the coefficient matrices. The theory applies to a large class of AREs and Riccati-like equations arising from the singular H/sup /spl infin//- and H/sup 2/-control problems, singular linear quadratic control, the 4-block Nehari problem, or from singular J-spectral factorizations. The underlying concept is the so-called proper deflating subspace of a (possibly singular) matrix pencil in terms of which necessary and sufficient conditions for the solvability of Riccati equations are given. It is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm.

New results in the Lyapunov-type algorithm for algebraic MCV Riccati equations

In this paper, finite- and infinite-horizon H/sub /spl infin// optimal control problems for a class of linear time-invariant descriptor systems are studied. Using a dynamic game theoretic approach, we provide upper and lower bounds of H/sub /spl infin//-optimal performance, which are characterized in terms of a parameterized reduced-order Riccati-like algebraic equation and a parameterized reduced-order game Riccati differential or algebraic equation. Moreover, we show that, if a prespecified performance level is achievable, the central H/sub /spl infin//-controller to guarantee such a performance level is not unique. A numerical example is included to illustrate the results obtained in the paper.

: Numerical issues related to the computational solution of the algebraic matrix Riccati equation were studied. The approach used the generalized eigenproblem formulation for the solution of general forms of algebraic Riccati equations arising in both continuous- and discrete-time applications. These general forms result from control and filtering problems for systems in generalized state space form. A Newton-type iterative refinement procedure for the generalized Riccati solution was derived. Balancing to improve numerical condition was studied. A Fortran package called RICPACK was developed. Numerical experiments with RICPACK were performed to investigate a number of proposed condition numbers. Experience with RICPACK to date indicates that it is the most powerful software yet developed to solve general classes of Riccati equations reliably for a few hundred or less problems. The special structure of models of physical systems given in linear second-order form was also examined. Exploiting that structure in solving associated Riccati equations was studied. Tests for controllability and observability were derived in terms of the original second-order-model matrices. Originator-supplied keywords include: Riccati equations, Generalized matrix Riccati equations, Discrete Riccati equation, Kalman filtering, Numerical condition, Balancing, Generalized eigneproblem, Numerical software, Second-order models.

We prove a fundamental representation formula for all solutions of the matrix Riccati differential equation and of the corresponding algebraic Riccati equation. This formula contains the complete information on the phase portrait of the matrix equation and on the structure of the set F of all solutions of the corresponding algebraic equation. In particular we describe all constant, periodic and almost-periodic solutions of the matrix Riccati differential equation. Further we give an application of the fundamental representation formula to the investigation of non-autonomous Riccati equations.

Generalized Riccati equations arising in stochastic games

The paper presents a state-space equation for the two-link and three-link fingers and proposed a solution of finite-time, nonlinear and closed-loop optimal control tracking problem using the state-dependent Riccati equations. The technique involves the use of the solution of the algebraic Riccati equation for the infinite-time case (hence the technique is approximate) and the change of variables that converts a state-dependent, nonlinear, differential Riccati equation (SD-DRE) to a linear differential Lyapunov equation (DLE) which can be solved in analytical form. The effectiveness is validated by using software simulation and hardware experiment.

A Riccati-equation based state-space solution to the H/sub /spl infin//-optimal control problem for periodic multirate sampled-data systems is presented. The solution is characterized by two standard discrete algebraic Riccati equations with a set of associated matrix positive definiteness conditions and coupling criteria. The equations are derived by discretizing a pair of coupled periodic Riccati differential equations with jumps, which characterize the H/sub /spl infin//-optimal controller. An important feature of the proposed solution is the fact that the so-called causality constraint, which is imposed on the controller structure in lifting-based solutions, is automatically satisfied. Instead, the solutions to the algebraic Riccati equations are required to satisfy a set of matrix positive definiteness and coupling conditions, which are easy to check.

On the regularity of solutions of an operator Riccati equation arising in linear quadratic optimal control problems for hereditary differential systems

In the present paper, we consider large-scale continuous-time differential matrix Riccati equations having low rank right-hand sides. These equations are generally solved by Backward Differentiation Formula (BDF) or Rosenbrock methods leading to a large scale algebraic Riccati equation which has to be solved for each timestep. We propose a new approach, based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimentional differential algebraic Riccati equation. The latter matrix differential problem is then solved by Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments will be given.

A necessary and sufficient condition for the H/sub /spl infin// filtering problem to be solvable is that the H/sub /spl infin// algebraic Riccati equation (ARE) has a positive semi-definite stabilizing solution with an additional condition that a certain matrix is positive definite. It is shown that such a solution is a monotonically non-increasing convex function of the prescribed H/sub /spl infin// norm bound /spl gamma/. Moreover, in this paper, the degree of freedom contained in the H/sub /spl infin// filter is investigated based upon this property of the Riccati solution. It turns out that the degree of freedom reduces to zero as /spl gamma/ tends to the optimal value in a certain case. These results provide a guideline for the design of an H/sub /spl infin// filter.

The behaviour of most modern multi-agent networked systems, used for distributed learning and control tasks, is describable by a set of interacting algebraic Riccati equations. However, due to the complexity of their behaviour, these interacting Riccati equations have, to this point, not been subject to rigorous scrutiny. To this end, a general class of algebraic Riccati equations is considered, their behaviour is analysed, and conditions for convergence to a unique set of stabilising solutions is established. The class of algebraic Riccati equations considered in this work is selected so that obtained results would be generalisable for a wide range of statistical learning and control purposes. Finally, application of the obtained results in distributed Kalman filtering and decentralised linear control is demonstrated.