Reduced-order Algebraic Riccati Equations Research Articles

Linear quadratic tracking control of unknown systems: A two-phase reinforcement learning method

This paper considers the problem of linear quadratic tracking control (LQTC) with a discounted cost function for unknown systems. The existing design methods often require the discount factor to be small enough to guarantee the closed-loop stability. However, solving the discounted algebraic Riccati equation (ARE) may lead to ill-conditioned numerical issues if the discount factor is too small. By singular perturbation theory, we decompose the full-order discounted ARE into a reduced-order ARE and a Sylvester equation, which facilitate designing the feedback and feedforward control gains. The obtained controller is proved to be a stabilizing and near-optimal solution to the original LQTC problem. In the framework of reinforcement learning, both on-policy and off-policy two-phase learning algorithms are derived to design the near-optimal tracking control policy without knowing the discount factor. The advantages of the developed results are illustrated by comparative simulation results.

Optimal Control of Wind Turbine Systems via Time-Scale Decomposition

In this paper, we design an optimal controller for a wind turbine (WT) with doubly-fed induction generator (DFIG) by decomposing the algebraic Riccati equation (ARE) of the singularly perturbed wind turbine system into two reduced-order AREs that correspond to the slow and fast time scales. In addition, we derive a mathematical expression to obtain the optimal regulator gains with respect to the optimal pure-slow and pure-fast, reduced-order Kalman filters and linear quadratic Gaussian (LQG) controllers. Using this method allows the design of the linear controllers for slow and fast subsystems independently, thus, achieving complete separation and parallelism in the design process. This solves the corresponding ill-conditioned problem and reduces the complexity that arises when the number of wind turbines integrated to the power system increases. The reduced-order systems are compared to the original full-order system to validate the performance of the proposed method when a wind turbulence and a large-signal disturbance are applied to the system. In addition, we show that the similarity transformation does not preserve the performance index value in case of Kalman filter and the corresponding LQG controller.

The Linear Quadratic Regulator Problem for a Class of Controlled Systems Modeled by Singularly Perturbed Itô Differential Equations

This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of $O(\sqrt{\varepsilon})$ approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an $O(\varepsilon)$ approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter $\varepsilon$ is not precisely known.

コスト保証制御理論によるマルチモデルシステムのロバスト安定化

This paper considers the robust stabilization of uncertain multimodeling systems. The guaranteed cost control technique is used to choose the design parameter which is included in the control gain. The e-independent quadratically stabilizing controller is newly proposed. It is shown that if the reduced-order algebraic Riccati equations (AREs) have positive definite stabilizing solutions, then closed-loop uncertain multimodeling systems with the proposed controller are quadratically stable and have a cost bound. To show the effectiveness of the proposed algorithm, a numerical example is included. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn, 160(4): 49– 59, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20271

H 2 guaranteed cost control problem of singularly perturbed systems with uncertainties

This paper studies the H 2 guaranteed cost control problem for singularly perturbed linear uncertain systems. This is an extension of the work of Garcia et al . (1998) in the sense that the state aud input matrices are allowed to be uncertain and the new method of calculation for the H 2 norm is presented. Thus, the new technique is applicable to more realistic singularly perturbed linear uncertain systems. Moreover, we propose the k -independent controller by solving the reduced-order algebraic Riccati equation only. Therefore, since we need not solve the full-order algebraic Riccati equation or separate the slow aud fast subsystems based on the existing method, it is easy to construct the stabilizing controller. As another significant improvement, some assumptions made previously are not necessary here. A numerical example is given to show the potential of the proposed technique.

The Guaranteed Cost Control Problem of Uncertain Singularly Perturbed Systems

In this paper we study the algebraic Riccati equation corresponding to the guaranteed cost control theory for an uncertain singularly perturbed system. The construction of the controller involves solving the full-order algebraic Riccati equation with small parameter ε. Under control-oriented assumptions, we first provide the sufficient conditions such that the full-order algebraic Riccati equation has a positive semi-definite stabilizing solution. Next we propose an iterative algorithm based on the Kleinman algorithm to solve the algebraic Riccati equation which depends on the parameter ε. Our new idea is to use the solutions of the reduced-order algebraic Riccati equations for the initial condition. By using the iterative algorithm, we can easily obtain a required solution of the algebraic Riccati equation. Moreover, using the initial conditions without ε, we show that there exists an ε̃ such that the proposed algorithm has quadratic convergence. Finally, in order to show the effectiveness of the proposed algorithm, numerical examples are included.

Decomposition of Optimal Control and Filtering Tasks for Linear Continuous Stochastic Systems

The algebraic regulator and filter Riccati equations are decomposed into reduced-order algebraic Riccati equations. That is, the exact solution of the global algebraic Riccati equation is found in terms of the reduced-order nonsymmetric algebraic Riccati equations. Using the separation principle the corresponding LQG optimal problem is obtained in terms of the local subsystems.

The exact slow-fast decomposition of the algebraic Ricatti equation of singularly perturbed systems

The algebraic Riccati equation for singularly perturbed control systems is completely and exactly decomposed into two reduced-order algebraic Riccati equations corresponding to the slow and fast time scales. The pure-slow and pure-fast algebraic Riccati equations are asymmetric ones, but their O( epsilon ) perturbations are symmetric. It is shown that the Newton method is very efficient for solving the obtained asymmetric algebraic Riccati equations. The method presented is very suitable for parallel computations. Due to the complete and exact decomposition of the Riccati equation, this procedure might produce new insight into the two-time-scale optimal filtering and control problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>