A small-gain theorem for almost global convergence of monotone systems (original) (raw)
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Fixed points and convergence in monotone systems under positive or negative feedback
International Journal of Control, 2014
In this paper a theorem is discussed that unifies two lines of work in I/O monotone control systems. Under a generalized small gain hypothesis, it is shown that almost all solutions of closed loops of MIMO monotone systems are convergent, regardless of whether the feedback is positive or negative. This result is based on a topological argument showing that any monotonically decreasing n-dimensional map that has convergent iterations must have a unique fixed point. The paper also generalizes the standard small gain theorem by replacing the small gain condition with a weaker hypothesis. An example and simulations are given involving a simple cyclic system under arbitrary feedback.
A small-gain result for orthant-monotone systems in feedback: The non sign-definite case
IEEE Conference on Decision and Control and European Control Conference, 2011
This note introduces a small-gain result for interconnected MIMO orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces of the considered subsystems. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations corresponded to positive-feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders.
IEEE Transactions on Automatic Control, 2003
Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to systems with inputs and outputs, a necessary first step in trying to understand interconnections, especially including feedback loops, built up out of monotone components. Basic definitions and theorems are provided, as well as an application to the study of a model of one of the cell's most important subsystems.
Monotone systems under positive feedback: multistability and a reduction theorem
Systems & Control Letters, 2005
For feedback loops involving single input, single output monotone systems with well-defined I/O characteristics, a recent paper by Angeli and Sontag provided an approach to determining the location and stability of steady states. A result on global convergence for multistable systems followed as a consequence of the technique. The present paper extends the approach to multiple inputs and outputs. A key idea is the introduction of a reduced system which preserves local stability properties.
Interconnections of Monotone Systems with Steady-State Characteristics
Lecture Notes in Control and Information Sciences, 2004
One of the key ideas in control theory is that of viewing a complex dynamical system as an interconnection of simpler subsystems, thus deriving conclusions regarding the complete system from properties of its building blocks. Following this paradigm, and motivated by questions in molecular biology modeling, the authors have recently developed an approach based on components which are monotone systems with respect to partial orders in state and signal spaces. This paper presents a brief exposition of recent results, with an emphasis on small gain theorems for negative feedback, and the emergence of multi-stability and associated hysteresis effects under positive feedback.
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.