SDRE Control Stability Criteria and Convergence Issues: Where Are We Today Addressing Practitioners' Concerns?

Abstract

This paper has three purposes: 1) to provide a survey on the State-Dependent Riccati Equation (SDRE) stability analysis methodologies developed to date; these stability analysis techniques produce either a guarantee or a high degree of confidence that the closed-loop system is asymptotically stable over a domain of interest, 2) to present an argument that practical rules of thumb can be just as important as theoretical stability proofs with regard to real world implementation, and 3) to justify support of the above argument using some forms of actual implementation. The paper neither favors any particular stability analysis technique nor introduces a new stability analysis framework. Rather it presents a view of stability reasoned judgment and practical justification for actual implementation. The reasoned judgment and justification are mainly based on the space access vehicle control and the satellite attitude control examples whose performance becomes unstable in the presence of high gain magnitude conditions. The fundamental argument of this new view (i.e., reasoned judgment and justification) is that for even linear and static gain controllers such as the Linear Quadratic Regulator (LQR), stability of the system depends on the operational domain under practical implementation with nonlinear limiters in the loop. Therefore, for a variable gain or nonlinear controller like the SDRE method, the task of determining a region of stability either practically or theoretically is much more difficult as compared to a linear LQRbased approach. The SDRE designs of a space access vehicle control, a helicopter flight control, and a satellite attitude control are presented with a set of practical rules defined and applied to it as a practical design benchmark problem on how to pass the SDRE stability concern gate. The paper concludes with the recommendation of some practical design rules of thumb for practitioners

Bulletin of the Polish Academy of Sciences: Technical Sciences

“…Asymptotic stability of the closed-loop system (18) implies that it is possible to control the states from the initial values to the final ones. However, the global stability property is difficult to prove [1,4,7,[17][18][19][20]. The controlled system with the SDRE compensator-based feedback is locally asymptotically stable.…”

Finite-time SDRE control of F16 aircraft dynamics

“…The differential SDRE equations using the SDC matrices is then solved on-line (in real time) to give the suboptimum control law. The technique for the finite-time nonlinear optimal control problem in the multivariable case is locally asymptotically stable and locally asymptotically optimal as described in following theoretical contributions [2,4,6,16,20].…”

SDRE-based high performance feedback control for nonlinear mechatronic systems

“…An algebraic Riccati equation (ARE) using the SDC matrices is then solved on-line to give the suboptimum control law. The SDRE feedback scheme for the infinite-time nonlinear optimal control problem in the multivariable case is locally asymptotically stable and locally asymptotically optimal, as described in first solid theoretical contributions (Cloutier et al, 1996;Mracek and Cloutier, 1998) and more recent works (Bogdanov and Wan, 2007;Bracci et al, 2006;Çimen, 2010;Erdem and Alleyne, 2004;Hammett et al, 1998;Lam et al, 2012;Liang and Lin, 2011;Shamma and Cloutier, 2003;Sznaier et al, 2000).…”