Solution Of Algebraic Riccati Equation Research Articles

Continuous Time Algebraic Riccati Equation Solution for Second Order Systems

Abstract The continuous time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second order structure, the ARE required to be solved to get the control values in standard control problems results in complex sub-equations in terms of the second order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second order system, due to the eigendecomposition of the 2n x 2n system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the 2n x 2n system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, positive semi-definite solution matrix).

Optimal Tracking Control of Heterogeneous MASs Using Event-Driven Adaptive Observer and Reinforcement Learning.

This article considers the output tracking control problem of nonidentical linear multiagent systems (MASs) using a model-free reinforcement learning (RL) algorithm, where partial followers have no prior knowledge of the leader's information. To lower the communication and computing burden among agents, an event-driven adaptive distributed observer is proposed to predict the leader's system matrix and state, which consists of the estimated value of relative states governed by an edge-based predictor. Meanwhile, the integral input-based triggering condition is exploited to decide whether to transmit its private control input to its neighbors. Then, an RL-based state feedback controller for each agent is developed to solve the output tracking control problem, which is further converted into the optimal control problem by introducing a discounted performance function. Inhomogeneous algebraic Riccati equations (AREs) are derived to obtain the optimal solution of AREs. An off-policy RL algorithm is used to learn the solution of inhomogeneous AREs online without requiring any knowledge of the system dynamics. Rigorous analysis shows that under the proposed event-driven adaptive observer mechanism and RL algorithm, all followers are able to synchronize the leader's output asymptotically. Finally, a numerical simulation is demonstrated to verify the proposed approach in theory.

Imaginary axis eigenvalues of Hamiltonian matrix: controllability, defectiveness and the ϵ-characteristic

The eigenstructure of imaginary axis eigenvalues of a Hamiltonian matrix is of importance in many fields of control systems, for example, in stability analysis of linear Hamiltonian systems, computation of the solutions of an algebraic Riccati equation (ARE), existence of a Lyapunov function for LTI systems, etc. The dynamical system consisting of all stationary trajectories for an optimal control problem – often called the Hamiltonian system – is known to admit, in its state space dynamical equation, a Hamiltonian matrix for the system matrix. In each of these cases, defectiveness of imaginary axis eigenvalues of a Hamiltonian matrix turns out to be of crucial importance. For example, defectiveness causes unboundedness of the oscillatory stationary trajectories in the Hamiltonian system. A characterisation of the ARE solutions in terms of Lagrangian invariant subspaces of the Hamiltonian matrix when the imaginary axis eigenvalues are defective has been addressed in the literature for the controllable case. This paper focuses on the general case of uncontrollable systems; we formulate conditions under which the imaginary axis eigenvalues of the Hamiltonian matrix are non-defective: this is central for solutions corresponding to imaginary axis eigenvalues to not become unbounded. We provide conditions on the so-called -characteristic of the non-defective imaginary axis eigenvalue: this helps in the characterisation of Lagrangian invariant subspaces. We formulate conditions under which a passivity-based Hamiltonian matrix is normal (i.e. it commutes with its transpose), and we link normality of such Hamiltonian matrices with all-pass behaviour. In summary, this paper formulates results that link defectiveness of imaginary axis eigenvalues of Hamiltonian matrix to solvabilities of Lyapunov and Algebraic Riccati equations, controllability/observability, -characteristic and sign-controllability. We consider examples in the area of bounded-real transfer functions and RLC circuits, both controllable and uncontrollable, to study applicability of the results of this paper.

Non-spillovercontroldesignof tall buildingsin modalspace

In this paper, a new algorithm for active control design of structures is proposed and investigated. The algorithm preserves the decoupling property of the modal vibration equation and eliminates the spillover problem, which is the main shortcoming in the independent modal space control (IMSC) algorithm. With linear quadratic regulator(LQR) control law, the analytical solution of algebraic Riccati equation and the optimal actuator control force are obtained, and the control design procedure is significantly simplified . A numerical example for the control design of a tall building subjected to wind loads demonstrates the effectiveness of the proposed algorithm in reducing the acceleration and displacement responses of tall buildings under wind actions.

KRYLOV SUBSPACE METHODS OF HESSENBERG BASED FOR ALGEBRAIC RICCATI EQUATION

In this paper, we propose a class of special Krylov subspace methods to solve continuous algebraic Riccati equation (CARE), i.e., the Hessenberg-based methods. The presented approaches can obtain efficiently the solution of algebraic Riccati equation to some extent. The main idea is to apply Kleinman-Newton's method to transform the process of solving algebraic Riccati equation into Lyapunov equation at every inner iteration. Further, the Hessenberg process of pivoting strategy combined with Petrov-Galerkin condition and minimal norm condition is discussed for solving the Lyapunov equation in detail, then we get two methods, namely global generalized Hessenberg (GHESS) and changing minimal residual methods based on the Hessenberg process (CMRH) for solving CARE, respectively. Numerical experiments illustrate the efficiency of the provided methods.

Families of solutions of algebraic Riccati equations

We consider Homogeneous Algebraic Riccati Equations in the general situation when the matrix of the dynamics can be mixed. We show that in this case the equation may have infinitely many families of solutions. An analysis of these families is carried over and an explicit formula is derived. When the matrix of the dynamics is unmixed, this formula provides an explicit parametrization of the whole set of the solutions. We finally show that there are situations in which the equation has spurious solutions that do not belong to any of the families and derive sufficient conditions under which, on the contrary, the union of the families covers the whole set of solutions.

Negative imaginary synthesis via dynamic output feedback and static state feedback: A Riccati approach

This paper is concerned with the study of strictly negative imaginary (SNI) systems and the problem of synthesizing a controller rendering a linear fractional system SNI. First, a set of necessary and sufficient conditions are developed for an SNI lemma without imposing minimality assumptions. The proposed lemma supports both the analysis and synthesis of SNI systems. Then, a set of sufficient conditions, based on the solution of algebraic Riccati equations, is provided for the solution of the SNI synthesis problem. Finally, an illustrative example demonstrates the usefulness of the results given herein.

An iterative Q‐learning scheme for the global stabilization of discrete‐time linear systems subject to actuator saturation

SummaryIn this paper, we propose a model‐free algorithm for global stabilization of linear systems subject to actuator saturation. The idea of gain‐scheduled low gain feedback is applied to develop control laws that avoid saturation and achieve global stabilization. To design these control laws, we employ the framework of parameterized algebraic Riccati equations (AREs). Reinforcement learning techniques are developed to find the solution of the parameterized ARE without requiring any knowledge of the system dynamics. In particular, we present an iterative Q‐learning scheme that searches for a low gain parameter and iteratively solves the parameterized ARE using the Bellman equation. Both state feedback and output feedback algorithms are developed. It is shown that the proposed scheme achieves model‐free global stabilization under bounded controls and convergence to the optimal solution of the ARE is achieved. Simulation results are presented that confirm the effectiveness of the proposed method.

Output Consensus of Heterogeneous Multiagent Systems with Physical and Communication Graphs

This paper deals with the output consensus problem of heterogeneous networks in a leader-follower manner that are interconnected by a physical coupling graph. The network under consideration consists of linear agents with different dynamics/dimensions. Both the state-feedback and output-feedback control protocols based upon information flow prescribed by a separate communication graph are developed, using the internal model principle and relative outputs of neighboring agents. With the small-gain theorem, we convert the consensus problem into a H∞ control problem of decoupled linear systems having the same dimensions as a single agent, where the disturbance attenuation constraints depend on the largest singular value related to the global information of physical and communication graphs. Then, we provide local synthesis procedures for control gains in terms of feasible solutions of algebraic Riccati equations. Finally, simulation examples are presented to verify the performance of the theoretical results.

Numerical solution of algebraic Riccati equations by Newton's method

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.

A GPU‐aware mixed‐precision solver for low‐rank algebraic Riccati equations

SummaryWe investigate different alternatives for the solution of algebraic Riccati equations on hybrid hardware platforms (ie, CPUs+GPUs). We evaluate a mixed‐precision approach that uses single‐precision arithmetic to obtain an approximation to the solution and later improve it to the desired precision, applying some steps of an economic iterative refinement. This method exploits the higher performance of the hardware to accelerate the solver when single‐precision arithmetic is employed and simultaneously obtains a high‐accuracy solution with the iterative refinement. We extend this approach to exploit the low‐rank property of the equation, when possible, to further improve its efficiency. The experimental evaluation shows that the mixed‐precision approach reports time and energy savings and provides similar or even more accurate solutions than well‐known methods like the sign function iteration or the structure‐preserving doubling algorithm.

Fast verified computation for solutions of algebraic Riccati equations arising in transport theory

SummaryA fast algorithm for enclosing the solution of the nonsymmetric algebraic Riccati equation arising in transport theory is proposed. The equation has a special structure, which is taken into account to reduce the complexity. By exploiting the structure, the enclosing process involves only quadratic complexity under a reasonable assumption. The algorithm moreover verifies the uniqueness and minimal positiveness of the enclosed solution. Numerical results show the efficiency of the algorithm.

A POD projection method for large-scale algebraic Riccati equations

The solution of large-scale matrix algebraic Riccati equations is important for instance in control design and model reduction and remains an active area of research. We consider a class of matrix algebraic Riccati equations (AREs) arising from a linear system along with a weighted inner product. This problem class often arises from a spatial discretization of a partial differential equation system. We propose a projection method to obtain low rank solutions of AREs based on simulations of linear systems coupled with proper orthogonal decomposition. The method can take advantage of existing (black box) simulation code to generate the projection matrices. We also develop new weighted norm residual computations and error bounds. We present numerical results demonstrating that the proposed approach can produce highly accurate approximate solutions. We also briefly discuss making the proposed approach completely data-based so that one can use existing simulation codes without accessing system matrices.

Genetic Algorithm based Decentralized PI Type Controller: Load Frequency Control

This work presents a design of decentralized PI type Linear Quadratic (LQ) controller based on genetic algorithm (GA). The proposed design technique allows considerable flexibility in defining the control objectives and it does not consider any knowledge of the system matrices and moreover it avoids the solution of algebraic Riccati equation. To illustrate the results of this work, a load–frequency control problem is considered. Simulation results reveal that the proposed scheme based on GA is an alternative and attractive approach to solve load–frequency control problem from both performance and design point of views.

Reduced basis method for H2 optimal feedback control problems

In this paper we examine the application of recently introduced parametric model reduction techniques to the H2 optimal feedback control problem. The H2 control problem provides a realistic framework for control applications, since it considers disturbances in the system and in the measurement outputs. Furthermore it employs state-estimation to reconstruct the unknown state from the noisy measurements. It turns out, that the controller is a dynamical system and two solutions of algebraic Riccati equations (AREs) are required to form it. We apply parametric model order reduction techniques to the AREs and to the state equation of the observer and show by numerical examples, that this approach can yield a significant speed-up in multi-query scenarios for large scale parametric problems for the control of partial differential equations (PDE).

Yet another characterization of solutions of the Algebraic Riccati Equation

This paper deals with a characterization of the solution set of algebraic Riccati equation (ARE) (over reals) for both controllable and uncontrollable systems. We characterize all solutions using simple linear algebraic arguments. It turns out that solutions of ARE of maximal rank have lower rank solutions encoded within it. We demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution. We characterize situations where there are no full rank solutions to the ARE. We also characterize situations when the number of solutions to the ARE is finite, when they are infinite and when they are bounded. We also explore the poset structure on the solution set of ARE, which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix. We provide several examples that bring out the essence of these results.

Taylor Series Method for Continuous Linear-Quadratic Regulators

A simple and eficient algorithm for finite-horizon continuous time LQR feedback control computation is presented. The method uses only basic linear algebra and does not require the solution of algebraic Riccati equation. The developed algorithm is applied to the benchmark problem of double inverted pendulum stabilization.

Application of policy iteration technique based adaptive optimal control design for automatic voltage regulator of power system

This paper presents an adaptive optimal control design approach for automatic voltage regulator (AVR) system using policy iteration technique based adaptive critic scheme. The infinite horizon optimal control design using linear quadratic regulator (LQR) by solution of algebraic Riccati equation (ARE) and Hamilton–Jacobi–Bellman (HJB) equation require the complete knowledge of the system dynamics, and giving solution offline. Policy iteration technique which is based on actor–critic structure consists of two-step iteration: policy evaluation and policy improvement. The online policy iteration technique based control scheme gives online the continuous-time adaptive optimal control solution without using the complete knowledge of the system’s internal dynamics. The knowledge of systems internal dynamics (i.e. matrix A) is not needed for evaluation of cost or the update of control policy; only the knowledge of input matrix B is required for updating the control policy. Thus this control scheme becomes partially model-free. The proposed control scheme has been implemented for AVR system for its both models neglecting and including sensor dynamics. The simulation results and performance analysis are presented to justify the robustness & effectiveness of control scheme. The comparative performance analysis of adaptive critic control scheme and LQR is also presented.

Solution of large generalized [formula omitted] algebraic Riccati equations

Although synthesis of H∞ controllers via solution of algebraic Riccati equations (AREs) has been known for several decades, there is not a generally accepted algorithm that is suitable for large-scale systems. However, systems of large model order arise in a number of situations, for instance in approximation of partial differential equation models and power systems. Often the system descriptions are sparse when in generalized state-space form but not when converted to standard first-order form. An extension of a game-theoretic iterative algorithm to solve the H∞-algebraic Riccati equations for large generalized systems is presented. The highlight of this work is to preserve the sparse structure of the system description by using the generalized form instead of converting it into standard form. The proposed method is compared to other methods using several examples. It is shown that this method is accurate, even near optimal attenuation where the standard method cannot be used. The proposed method also performs well for a problem of structural control where other methods cannot be used. A method for calculating optimal attenuation that is significantly faster than the usual bisection procedure is also described.

Convergence of the standard RLS method and UDUT factorisation of covariance matrix for solving the algebraic Riccati equation of the DLQR via heuristic approximate dynamic programming

The main focus of this article is to present a proposal to solve, via UDUT factorisation, the convergence and numerical stability problems that are related to the covariance matrix ill-conditioning of the recursive least squares (RLS) approach for online approximations of the algebraic Riccati equation (ARE) solution associated with the discrete linear quadratic regulator (DLQR) problem formulated in the actor–critic reinforcement learning and approximate dynamic programming context. The parameterisations of the Bellman equation, utility function and dynamic system as well as the algebra of Kronecker product assemble a framework for the solution of the DLQR problem. The condition number and the positivity parameter of the covariance matrix are associated with statistical metrics for evaluating the approximation performance of the ARE solution via RLS-based estimators. The performance of RLS approximators is also evaluated in terms of consistence and polarisation when associated with reinforcement learning methods. The used methodology contemplates realisations of online designs for DLQR controllers that is evaluated in a multivariable dynamic system model.