TY - JOUR

T1 - Analysis of the error in an iterative algorithm for asymptotic regulation of linear distributed parameter control systems

AU - Aulisa, Eugenio

AU - Gilliam, David S.

AU - Pathiranage, Thanuka W.

N1 - Funding Information:
★ The research of the first and third authors was partially supported by the National Science Foundation grant DMS-1412796.
Publisher Copyright:
© EDP Sciences, SMAI 2019.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - Applications of regulator theory are ubiquitous in control theory, encompassing almost all areas of systems and control engineering. Examples include active noise suppression [Banks et al., Decision and Control, Active Noise Control: Piezoceramic Actuators in Fluid/structure Interaction Models, IEEE, Los Alamitos, CA (1991) 2328-2333], design and control of energy efficient buildings [Borggaard et al., Control, Estimation and Optimization of Energy Efficient Buildings. Riverfront, St. Louis, MO (2009) 837-841.] and control of heat exchangers [Aulisa et al., IFAC-PapersOnLine 49 (2016) 104-109.]. Numerous other examples can be found in [Aulisa and Gilliam, A Practical Guide to Geometric Regulation for Distributed Parameter Systems. Chapman and Hall/CRC, Boca Raton (2015).]. In the geometric approach to asymptotic regulation the main object of interest is a pair of operator equations called the regulator equations, whose solution provides a control solving the tracking/disturbance rejection regulation problem. In this paper we present an iterative algorithm, called the β-iteration method, which is based on the geometric methodology, and delivers accurate control laws for approximate asymptotic regulation. This iterative scheme has been successfully applied to a wide range of linear and nonlinear multi-physics examples and in practice only one or two iterations are usually required to deliver sufficiently accurate results. One drawback to these research efforts is that no proof was given of the convergence of the method. This work contains a detailed analysis of the error in the iterative scheme for a large class of linear distributed parameter systems. In particular we show that the iterative errors converge at a geometric rate. We demonstrate our estimates on three control problems in multi-physics applications.

AB - Applications of regulator theory are ubiquitous in control theory, encompassing almost all areas of systems and control engineering. Examples include active noise suppression [Banks et al., Decision and Control, Active Noise Control: Piezoceramic Actuators in Fluid/structure Interaction Models, IEEE, Los Alamitos, CA (1991) 2328-2333], design and control of energy efficient buildings [Borggaard et al., Control, Estimation and Optimization of Energy Efficient Buildings. Riverfront, St. Louis, MO (2009) 837-841.] and control of heat exchangers [Aulisa et al., IFAC-PapersOnLine 49 (2016) 104-109.]. Numerous other examples can be found in [Aulisa and Gilliam, A Practical Guide to Geometric Regulation for Distributed Parameter Systems. Chapman and Hall/CRC, Boca Raton (2015).]. In the geometric approach to asymptotic regulation the main object of interest is a pair of operator equations called the regulator equations, whose solution provides a control solving the tracking/disturbance rejection regulation problem. In this paper we present an iterative algorithm, called the β-iteration method, which is based on the geometric methodology, and delivers accurate control laws for approximate asymptotic regulation. This iterative scheme has been successfully applied to a wide range of linear and nonlinear multi-physics examples and in practice only one or two iterations are usually required to deliver sufficiently accurate results. One drawback to these research efforts is that no proof was given of the convergence of the method. This work contains a detailed analysis of the error in the iterative scheme for a large class of linear distributed parameter systems. In particular we show that the iterative errors converge at a geometric rate. We demonstrate our estimates on three control problems in multi-physics applications.

KW - Distributed parameter systems

KW - Dynamic regulator equations

KW - Geometric control

KW - Iterative numerical algorithm

KW - Regulator problem

UR - http://www.scopus.com/inward/record.url?scp=85101840432&partnerID=8YFLogxK

U2 - 10.1051/m2an/2019021

DO - 10.1051/m2an/2019021

M3 - Article

AN - SCOPUS:85101840432

VL - 53

SP - 1577

EP - 1606

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 5

ER -