Iterative Refinement Procedure for Solutions to Algebraic Riccati Equations

Rotational Motion Control by Feedback with Minimum L1-Norm

Algebraic Riccati Equations (AREs) are fundamental to optimal control design, particularly Linear Quadratic Regulator (LQR), Linear Quadratic Gaussian (LQG), and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> control. Traditional numerical methods for solving AREs, such as Schur decomposition, eigenvector methods, and Newton's method, while reliable, require computationally intensive iterative algorithms. For large systems or applications requiring real-time control decisions, these computational demands can become impractical. At times, finding a stabilizing solution can be challenging. This paper explores the use of neural networks as a fast, data-driven alternative for approximating ARE solutions. In this work, we trained a neural network to predict the solution matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> directly from system parameters, bypassing traditional iterative solvers. The network architecture consists of four hidden layers with ReLU activation, dropout regularization, and batch normalization. As input, a 30-dimensional vector was used which consisted of the flattened system matrix and five additional stability-related features. Using a motivating aircraft control dynamical system from the literature as our benchmark, we generated a training dataset of 50,000 system-solution pairs through controlled perturbations of the motivating system, maintaining the original sparsity structure while varying numerical values. The trained network achieved a relative error of 6.730% when predicting the ARE solution for the original aircraft system. Closedloop simulations demonstrate that this approximation error does not significantly degrade control performance, with system trajectories closely matching those obtained using traditional ARE solvers. Once trained, the neural network provides near-instantaneous predictions, offering substantial computational advantages over iterative methods. This preliminary study explores the feasibility of neural network-based ARE solvers and provides a foundation for future work extending the approach to higher-dimensional systems and infinite-dimensional systems governed by partial differential equations.

Chapter 13 - Numerical Solutions and Conditioning of Algebraic Riccati Equations

This paper is concerned with the discrete time $H_\infty $ control problem with measurement feedback. It follows that, as in the continuous time case, the existence of an internally stabilizing controller that makes the $H_\infty $ norm strictly less than 1 is related to the existence of stabilizing solutions to two algebraic Riccati equations. However, in the discrete time case, the solutions of these algebraic Riccati equations must satisfy extra conditions.

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

In this paper, we propose a method for estimating the algebraic Riccati equation (ARE) with respect to an unknown discrete-time system from the system state and input observation. The inverse optimal control (IOC) problem asks, “What objective function is optimized by a given control system?” The inverse linear quadratic regulator (ILQR) problem is an IOC problem that assumes a linear system and quadratic objective function. The ILQR problem can be solved by solving a linear matrix inequality that contains the ARE. However, the system model is required to obtain the ARE, and it is often unknown in fields in which the IOC problem occurs, for example, biological system analysis. Our method directly estimates the ARE from the observation data without identifying the system. This feature enables us to economize the observation data using prior information about the objective function. We provide a data condition that is sufficient for our method to estimate the ARE. We conducted a numerical experiment to demonstrate that our method can estimate the ARE with less data than system identification if the prior information is sufficient.

Algebraic Riccati Equation (ARE) solutions play an important role in many optimal/suboptimal control problems. However, a key assumption in formulation and solution of the ARE is a certain ‘regularity condition’ on the feedthrough term D of the system. For example, formulation of the ARE requires nonsingularity of \(D+D^T\) in positive real dissipative systems and, in the case of bounded real dissipative systems, one requires nonsingularity of \(I-D^T D\). Note that for lossless systems \(D+D^T=0\), while for all-pass systems \(I-D^TD=0\); this rules out the formulation of the ARE. Noting that the ARE solutions are also extremal “storage functions” for dissipative systems, one can speak of storage function for the lossless case too. This contributed chapter formulates new properties of the ARE solution; we then show that this property is satisfied by the storage function for the lossless case too. The formulation of this property is via the set of trajectories of minimal dissipation. We show that the states in a first-order representation of this set satisfy static relations that are closely linked to ARE solutions; this property holds for the lossless case too. Using this property, we propose an algorithm to compute the storage function for the lossless case.

In the problem of infimizing a not necessarily positive semidefinite quadratic cost subject to a linear dynamical constraint, it is usually expected that the existence of a lower bound to the cost is equivalent to the existence of a negative semidefinite, antistabilizing solution to the algebraic Riccati equation. By a counterexample, it is shown that this equivalence breaks down in the discrete-time case. This phenomenon, as well as the whole question of the existence of the appropriate solution to the algebraic Riccati equation, are investigated in detail.

In this study, we define new solution bounds for the infinite-dimensional algebraic Riccati equation. We first suggest an upper bound solution in the continuous case. After that, we develop upper and lower bounds for the algebraic Riccati equation solution in discrete time. We provide a robust control application.

OPTIMAL SENSOR/ACTUATOR PLACEMENT FOR ACTIVE VIBRATION CONTROL USING EXPLICIT SOLUTION OF ALGEBRAIC RICCATI EQUATION

Undoubtedly one of the most important concepts in linear systems and control, both from a theoretical as well as from a practical point of view, is the algebraic Riccati equation. Since its introduction in control theory by Kaiman [16] the beginning of the sixties, the algebraic Riccati equation has known an impressive range of applications, such as linear quadratic optimal control, stability theory, stochastic filtering and stochastic control, stochastic realization theory, synthesis of linear passive networks, differential games and, most recently, H ∞ optimal control and robust stabilization. The purpose of the present paper is to give an expository survey of the main concepts, results and applications related to the algebraic Riccati equation.

: 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.

Improved Newton solvers, with or without line search, for both continuous- and discrete-time algebraic Riccati equations (AREs) are discussed. The basic theory and conceptual algorithm are briefly presented. Algorithmic details, computational steps, and convergence tests are described. The main results of an extensive performance investigation of the Newton solvers are summarized and compared with those obtained with the widelyused MATLAB solvers, care and dare. Randomly generated systems with orders till 2000, as well as the systems from the large COMPle ib collection of examples, are considered. Significantly improved accuracy, in terms of normalized and relative residuals, and sometimes greater efficiency than for care/dare have been obtained. The results strongly recommend the use of Newton solvers, especially for improving the solutions computed by other solvers.

This book provides a careful treatment of the theory of algebraic Riccati equations. It consists of four parts: the first part is a comprehensive account of necessary background material in matrix theory including careful accounts of recent developments involving indefinite scalar products and rational matrix functions. The second and third parts form the core of the book and concern the solutions of algebraic Riccati equations arising from continuous and discrete systems. The geometric theory and iterative analysis are both developed in detail. The last part of the book is an exciting collection of eight problem areas in which algebraic Riccati equations play a crucial role. These applications range from introductions to the classical linear quadratic regulator problems and the discrete Kalman filter to modern developments in HD*W*w control and total least squares methods.

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.