Direct Adaptive Control for Nonlinear Matrix Second‐order Systems With Time‐varying and Sign‐indefinite Damping and Stiffness Operators

Abstract

A direct adaptive control framework for a class of nonlinear matrix second‐order systems with time‐varying and sign‐indefinite damping and stiffness operators is developed. The proposed framework guarantees global asymptotic stability of the closed‐loop system states associated with the plant dynamics without requiring any knowledge of the system nonlinearities other than the assumption that they are continuous and bounded. The proposed adaptive control approach is used to design adaptive controllers for suppressing thermoacoustic oscillations in combustion chambers.

Quadratic stability and stabilization of matrix second‐order time‐varying systems

“…However, they cannot be used for the global stabilization results of the time-varying cases. The stabilization problem of matrix second-order time-varying systems was addressed in [22] and [23], in which the coefficient matrix M.t/ is assumed to be a constant matrix. There are few results about the stabilization problem of matrix second-order time-varying systems, where all the system matrices including the second-order differential coefficient matrix M.t/ are time-varying.…”

Lyapunov-stable adaptive stabilization of non-linear time-varying systems with matched uncertainty

“…In addition, Ryan (1991) considers only single-input systems whereas this paper considers multi-input systems. Notably, the present paper differs from Ryan (1991), Corless and Ryan (1993), Bernstein (2001, 2004), Ilchmann and Ryan (2003), Chellaboina et al (2003) and Haddad et al (inpress 2007), in both method of proof and resulting parameter-monotonic adaptive law. More specifically, the current paper's proofs utilize new tools presented in appendix A, and the current paper's parametermonotonic adaptive law incorporates an exponentially decaying factor, which has no counterpart in Ryan (1991), Corless and Ryan (1993), Roup and Bernstein (2001, 2004), Chellaboina et al (2003, Ilchmann and Ryan (2003), and Haddad et al (submitted 2005).…”

“…More specifically, the current paper's proofs utilize new tools presented in appendix A, and the current paper's parametermonotonic adaptive law incorporates an exponentially decaying factor, which has no counterpart in Ryan (1991), Corless and Ryan (1993), Roup and Bernstein (2001, 2004), Chellaboina et al (2003, Ilchmann and Ryan (2003), and Haddad et al (submitted 2005). Nevertheless, the present paper and the previous work (Ryan 1991, Corless and Ryan 1993, Roup and Bernstein 2001, Chellaboina et al 2003, Ilchmann and Ryan 2003, Roup and Bemstein 2004, Haddad et al (inpress 2007, both require the assumption of matched uncertainty. The problem of adaptive stabilization of non-linear time-varying systems with unmatched uncertainty is open.…”