A small-gain theorem for almost global convergence of monotone systems (original) (raw)
A note on some sufficient conditions for mixed monotone systems
2017
Mixed monotone systems form an important class of nonlinear systems that have recently received attention in abstraction-based control design area. Slightly different definitions exist in the literature, and it remains a challenge to verify mixed monotonicity of a system in general. In this paper, we first clarify the relation between different existing definitions of mixed monotone system, and then provide a new and more general sufficient condition for mixed monotonicity. Discussions are provided regarding to these two main contributions.
A Remark on Multistability for Monotone Systems II
Proceedings of the 44th IEEE Conference on Decision and Control, 2005
A recent paper by Angeli and Sontag presented a stability criterium for feedback loops involving single input, single output models which admit a well-defined I/O characteristic and satisfy a monotonicity condition. This paper generalizes the argument to arbitrary strongly monotone systems. The main idea is to study the asymptotic behavior of a strongly monotone system by relating it to a graph on the plane which contains information about the equilibria of the system and their stability properties.
A small-gain theorem is presented for almost global stability of monotone control systems which are open-loop almost globally stable, when constant inputs are applied. The theorem assumes "negative feedback" interconnections. This typically destroys the monotonicity of the original ow and potentially destabilizes the resulting closed-loop system.
Multi-Stability in Monotone Input/Output Systems
2003
This paper studies the emergence of multi-stability and hysteresis in those systems that arise, under positive feedback, starting from monotone systems with well-defined steady-state responses. Such feedback configurations appear routinely in several fields of application, and especially in biology. Characterizations of global stability behavior are stated in terms of easily checkable graphical conditions. An example of a signaling cascade under positive feedback is presented.
Novel Stability Conditions for Nonlinear Monotone Systems and Consensus in Multi-Agent Networks
IEEE Transactions on Automatic Control
We introduce a novel definition of monotonicity, termed "type-K" in honor of Kamke, and study nonlinear type-K monotone dynamical systems possessing the plussubhomogeneity property, which we call "K-subtopical" systems after Gunawardena and Keane. We show that type-K monotonicity, which is weaker than strong monotonicity, is also equivalent to monotonicity for smooth systems evolving in continuous-time, but not in discrete-time. K-subtopical systems are proved to converge toward equilibrium points, if any exists, generalizing the result of Angeli and Sontag about convergence of topical systems' trajectories toward the unique equilibrium point when strong monotonicity is considered. The theory provides an new methodology to study the consensus problem in nonlinear multi-agent systems (MASs). Necessary and sufficient conditions on the local interaction rule of the agents ensuring the K-subtopicality of MASs are provided, and consensus is proven to be achieved asymptotically by the agents under given connectivity assumptions on directed graphs. Examples in continuous-time and discrete-time corroborate the relevance of our results in different applications.
Existence of linear feedback control for certain types of global stability
[1991 Proceedings] The Twenty-Third Southeastern Symposium on System Theory, 2000
We are concerned with a class of nonlinear control systems of the form x' = Ax + f(x) + Bu where the nonlinear term f(x) is quadratic and has the orthogonality property xTf(x) = 0 for all x. In the context of Lyapunov's second method, the existence of a linear feedback control U = Kx is examined. Sufficient conditions are discussed for the system to be controlled to a system with the origin as a global asymptotic stable point or to a system which is point dissipative. A system is point dissipative if there exists a bounded region into which every trajectory eventually enters and remains.
A global convergence result for strongly monotone systems with positive translation invariance
2006
We show that strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are so that all solutions converge to a unique equilibrium. The result may be seen as a dual of a well-known theorem of Mierczynski for systems that satisfy a conservation law. An application to a reaction of interest in biochemistry is provided as an illustration.
Stability of Linear Positive Systems
Ukrainian Mathematical Journal
We establish criteria of asymptotic stability for positive differential systems in the form of conditions of monotone invertibility of linear operators. The structure of monotone and monotonically invertible operators in the space of matrices is investigated.
Proceedings of the 45th IEEE Conference on Decision and Control, 2006
We present a converse Lyapunov result for nonlinear time-varying systems that are uniformly semiglobally asymptotically stable. This stability property pertains to the case when the size of initial conditions may be arbitrarily enlarged and the solutions of the system converge, in a stable way, to a closed ball that may be arbitrarily diminished by tuning a design parameter of the system (typically but not exclusively, a control gain). This result is notably useful in cascaded-based control when uniform practical asymptotic stability is established without a Lyapunov function, , e.g. via averaging. We provide a concrete example by solving the stabilization problem of a hovercraft.