Immersion‐ and invariance‐based adaptive stabilization of switched nonlinear systems

Abstract

SummaryThis paper presents a novel framework to asymptotically adaptively stabilize a class of switched nonlinear systems with constant linearly parameterized uncertainty. By exploiting the generalized multiple Lyapunov functions method and the recently developed immersion and invariance (I&I) technique, which does not invoke certainty equivalence, we design the error estimator, continuous state feedback controllers for subsystems, and a switching law to ensure boundedness of all closed-loop signals and global asymptotical regulation of the states, where the solvability of the I&I adaptive stabilization problem for individual subsystems is not required. Then, along with the backstepping method, the proposed design technique is further applied to a class of switched nonlinear systems in strict-feedback form with an unknown constant parameter so that the I&I adaptive stabilization controllers for the system is developed. Finally, simulation results are also provided to demonstrate the effectiveness of the proposed design method.

A Simultaneous Adaptation Law for a Class of Nonlinearly Parametrized Switched Systems

“…Therefore, using (30), an ultimate bound b on the tracking error ξ can be found as (14). Remark 6: The existence ofᾱ iσ > 0 andβ iσ > 0 to establish (25) justifies the reason for the selection of α i > max σ∈Ω ( σ /2) and β i > max σ∈Ω ( σ /2).…”

“…Remark 1: Property 1 holds for many practical NLIP systems such as Monod kinetic [22], EL systems [20] etc. For such systems, the existing LIP-based adaptive control solutions [14]- [16] are inapplicable. Determination of ν in (2) does not require structural knowledge of the dynamics, as ν can be determined from the first law of physics.…”

“…Switched systems represent an important class of hybrid systems consisting of subsystems with continuous dynamics together with a logic that orchestrates the switching action between them [1]- [9]. While some adaptive control approaches have been proposed to deal with the relevant problem of having parametric uncertainties in the subsystem dynamics ( [10]- [13] for linear and [14]- [18] for nonlinear subsystems), only few approaches, namely [17], [18], address some classes of uncertain switched systems whose subsystem dynamics have nonlinear in parameters (NLIP) form.…”

Singular L2‐gain control for switched nonlinear control systems under arbitrary switching

“…The study of switched systems has received a great attention for their ability in describing a wide range of physical and engineering systems and for the existence of control systems that cannot be asymptotically stabilized by a single smooth feedback control law 1,2 . Recently, more and more results for stability analysis and control synthesis of switched nonlinear systems have been proposed 3‐22 . For switched nonlinear systems in some particular forms (canonical form, strict‐feedback form, etc), the design of asymptotically stabilizing controllers has been studied in some literature 4,5,7‐9,14‐18 .…”

“…Recently, more and more results for stability analysis and control synthesis of switched nonlinear systems have been proposed 3‐22 . For switched nonlinear systems in some particular forms (canonical form, strict‐feedback form, etc), the design of asymptotically stabilizing controllers has been studied in some literature 4,5,7‐9,14‐18 . Moreover, the L 2 ‐gain control problem of switched nonlinear systems in particular forms has also been considered in several studies 10,12,19 .…”