Eigenvector approach for order-reduction of singularly perturbed linear-quadratic optimal control problems

doi:10.1016/S0005-1098(98)00141-1

Automatica

Volume 35, Issue 1, January 1999, Pages 151-158

https://doi.org/10.1016/S0005-1098(98)00141-1 Get rights and content

Abstract

In this paper we show how to decompose the singularly perturbed algebraic Riccati equation and the corresponding linear-quadratic optimal control problem at steady state in terms of reduced-order pure-slow and pure-fast problems by using the eigenvector approach. The eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small. In such cases the decomposition methods based on series expansions, fixed point iterations, subspace iterations, and Newton iterations, either fail to produce solutions of the corresponding algebraic equations or display very slow convergence.

References (0)

Cited by (34)

Slow and fast dynamics of a natural gas hydrogen reformer
2013, International Journal of Hydrogen Energy
It has been known for some time that fuel cells operate in multiple time scales due to phenomena of different nature that govern their dynamics (electrical, chemical, mechanical, electrochemical, thermodynamic). It is expected that multi-time scales will be presented in the hydrogen gas reformer (produces hydrogen from hydrogen rich fuels) due to its complex chemical and physical nature. In this paper, we show how to present a linearized hydrogen gas reformer dynamic model in the coordinates in which slow and fast variables are explicitly defined and exactly separated. This result is important in its own right to get a better understanding of slow and fast dynamics of particular variables, study their interactions, and as well for designing reduced-order controllers and estimators for the hydrogen gas reformer that will independently operate in different time scales.
Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation
2010, Applied Mathematics and Computation
In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension $n_{1} + n_{2}$ into the pure-slow regular matrix differential Riccati equation of dimension $n_{1}$ and the pure-fast stiff matrix differential Riccati equation of dimension $n_{2}$ . A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.
A method for optimal control and filtering of multitime-scale linear singularly-perturbed stochastic systems
2008, Automatica
In this paper we introduce a transformation for the exact closed-loop decomposition of the optimal Kalman filter and the linear quadratic optimal controller of multi time scale continuous-time, linear, singularly-perturbed stochastic systems. The solution of the corresponding algebraic regulator and filter Riccati equations are obtained in terms of solutions of reduced-order subsystem, algebraic, Riccati equations corresponding to the system time scales. We have also obtained $N$ completely independent reduced-order subsystem Kalman filters working in parallel in different time scales. This allows parallel processing of information with lower-order, different rates Kalman filters consistent with the system time scales.
Discrete system order-reduction via balancing transformation using singular perturbations
2005, IFAC Proceedings Volumes (IFAC-PapersOnline)
Order-reduction techniques based on system balancing are developed for linear discrete systems using the method of singular perturbations. The generalized residualization technique is introduced with three variants as improved tools for order-reduction at low and medium frequencies. The newly introduced techniques have the same theoretical upper error bound with respect to the H_∞ norm of the reduced-order system as the existing techniques, but simulation results show the superiority of the new techniques at low and medium frequencies. Next, new techniques are proposed to perform order-reduction of lightly damped linear discrete-time systems by employing the method of singular perturbations and balancing transformation.
Improvement of system order reduction via balancing using the method of singular perturbations
2001, Automatica
In this paper we first present several alternative techniques for system order reduction based on system balancing by employing the method of singular perturbations. The techniques obtained have the same robustness accuracy evaluated with respect to the H_∞ norm of the reduced-order system as the two techniques existing in the control literature known as the direct truncation and balancing residualization methods. The newly proposed techniques preserve the exact DC gain as the original system, and produce very good to excellent accuracy at low and medium frequencies. We also present a simple formula that fixes the DC gain for the direct truncation method. In the second part of the paper we propose how to perform order-reduction of a balanced system using theory of singular perturbations that can produce very good accuracy at high frequencies, particularly for systems that have lightly damped highly oscillatory modes. The result obtained for such kind of systems requires that the modes with higher Hankel singular values should be neglected and those with smaller Hankel singular values should be retained. This is a counter-intuitive result in the spirit of H_∞ theory, but an intuitive result in the spirit of theory of singular perturbations. The second part of the paper raises an important question about validity of the system robustness estimate based on the H_∞ norm for linear systems with lightly damped highly oscillatory modes.
Singularly perturbed modeling and LQR controller design for a fuel cell system
2020, Energies

View all citing articles on Scopus

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Tamer Başar.

View full text

Brief PaperEigenvector approach for order-reduction of singularly perturbed linear-quadratic optimal control problems☆

Abstract

Brief Paper
Eigenvector approach for order-reduction of singularly perturbed linear-quadratic optimal control problems☆