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13
On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws
"... We consider ndimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this conditi ..."
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Cited by 25 (5 self)
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We consider ndimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true for the case n = 2, but is not true in general. Their results imply that there exists some minimal integer np such that the conjecture is true for all n < np, but is not true for n = np. We show that np = 3.
Controllability of Boolean control networks via the PerronFrobenius theory
 AUTOMATICA
, 2012
"... Boolean control networks (BCNs) are recently attracting considerable interest as computational models for genetic and cellular networks. Addressing controltheoretic problems in BCNs may lead to a better understanding of the intrinsic control in biological systems, as well as to developing suitable ..."
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Cited by 13 (3 self)
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Boolean control networks (BCNs) are recently attracting considerable interest as computational models for genetic and cellular networks. Addressing controltheoretic problems in BCNs may lead to a better understanding of the intrinsic control in biological systems, as well as to developing suitable protocols for manipulating biological systems using exogenous inputs. We introduce two definitions for controllability of a BCN, and show that a necessary and sufficient condition for each form of controllability is that a certain nonnegative matrix is irreducible or primitive, respectively. Our analysis is based on a result that may be of independent interest, namely, a simple algebraic formula for the number of different control sequences that steer a BCN between given initial and final states in a given number of time steps, while avoiding a set of forbidden states.
A Pontryagin Maximum Principle for Multi–Input Boolean Control Networks
"... A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. Boolean networks have been studied extensively as models for simple artificial neural networks. Recently, Boolean networks gained considerable interest as models for biological syst ..."
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Cited by 9 (3 self)
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A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. Boolean networks have been studied extensively as models for simple artificial neural networks. Recently, Boolean networks gained considerable interest as models for biological systems composed of elements that can be in one of two possible states. Examples include genetic regulation networks, where the ON (OFF) state corresponds to the transcribed (quiescent) state of a gene, and cellular networks where the two possible logic states may represent the open/closed state of an ion channel, basal/high activity of an enzyme, two possible conformational states of a protein, etc. Daizhan Cheng developed an algebraic statespace representation for Boolean control networks using the semi–tensor product of matrices. This representation proved quite useful for studying Boolean control networks in a controltheoretic framework. Using this representation, we consider a Mayertype optimal control problem for Boolean control networks. Our main result is a necessary condition for optimality. This provides a parallel of Pontryagin’s maximum principle to Boolean control networks.
MinimumTime Control of Boolean Networks
, 2012
"... Boolean networks (BNs) are discretetime dynamical systems with Boolean statevariables. BNs are recently attracting considerable interest as computational models for biological systems and, in particular, as models of gene regulating networks. Boolean control networks (BCNs) are Boolean networks wi ..."
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Cited by 5 (3 self)
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Boolean networks (BNs) are discretetime dynamical systems with Boolean statevariables. BNs are recently attracting considerable interest as computational models for biological systems and, in particular, as models of gene regulating networks. Boolean control networks (BCNs) are Boolean networks with Boolean inputs. We consider the problem of steering a BCN from a given state to a desired state in minimal time. Using the algebraic statespace representation (ASSR) of BCNs we derive several necessary conditions, stated in the form of maximum principles (MPs), for a control to be timeoptimal. In the ASSR every state and input vector is a canonical vector. Using this special structure yields an explicit statefeedback formula for all timeoptimal controls. To demonstrate the theoretical results, we develop a BCN model for the genetic switch controlling the lambda phage development upon infection of a bacteria. Our results suggest that this biological switch is designed in a way that guarantees minimal time response to important environmental signals.
On the analysis of nonlinear nilpotent switched systems using the HallSussmann system
 Systems Control Lett
, 2009
"... We consider an open problem on the stability of nonlinear nilpotent switched systems posed by Daniel Liberzon. Partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems. The global structure is crucial in establishing stability. ..."
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Cited by 2 (2 self)
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We consider an open problem on the stability of nonlinear nilpotent switched systems posed by Daniel Liberzon. Partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems. The global structure is crucial in establishing stability. We show that nice reachability analysis may be reduced to the reachability analysis of a specific canonical system, the nilpotent HallSussmann system. Furthermore, local nice reachability properties for this specific system imply global nice reachability for general nilpotent systems. We derive several new results revealing the elegant Liealgebraic structure of the nilpotent HallSussmann system.
Continuousdiscrete time observer design for lipschitz systems with sampled measurements
 IEEE Transactions on Automatic Control
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A Simplification of the AgrachevGamkrelidze SecondOrder Variation for BangBang Controls
, 2009
"... We consider an expression for the second–order variation (SOV) of bangbang controls derived by Agrachev and Gamkrelidze. The SOV plays an important role in both necessary and sufficient second–order optimality conditions for bangbang controls. These conditions are stronger than the one provided by ..."
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We consider an expression for the second–order variation (SOV) of bangbang controls derived by Agrachev and Gamkrelidze. The SOV plays an important role in both necessary and sufficient second–order optimality conditions for bangbang controls. These conditions are stronger than the one provided by the first–order Pontryagin maximum principle (PMP). For a bangbang control with k switching points, the SOV contains k(k + 1)/2 Liealgebraic terms. We derive a simplification of the SOV by relating k of these terms to the derivative of the switching function, defined in the PMP, evaluated at the switching points. We prove that this simplification can be used to reduce the computational burden associated with applying the SOV to analyze optimal controls. We demonstrate this by using the simplified expression for the SOV to show that the chattering control in Fuller’s problem satisfies a secondorder sufficient condition for optimality.
A remark about linear switched systems in the plane
"... Abstract: In this note we prove that if a switched system F formed by a pair of linear vector fields of R 2 is asymptotically controllable, then the discrete time operator associated to F admits at least one real eigenvalue λ, with λ < 1. For the particular case at hand, this is an improvement ..."
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Abstract: In this note we prove that if a switched system F formed by a pair of linear vector fields of R 2 is asymptotically controllable, then the discrete time operator associated to F admits at least one real eigenvalue λ, with λ < 1. For the particular case at hand, this is an improvement of previous existing results.
Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications
"... AbstractIn this paper, a unified framework is proposed to study the exponential stability of discretetime switched linear systems, and more generally, the exponential growth rates of their trajectories, under three types of switching rules: arbitrary switching, optimal switching, and random switc ..."
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AbstractIn this paper, a unified framework is proposed to study the exponential stability of discretetime switched linear systems, and more generally, the exponential growth rates of their trajectories, under three types of switching rules: arbitrary switching, optimal switching, and random switching. It is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong, the weak, and the mean generating functions, respectively. In particular, necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence. Various properties of the generating functions are established and their relations are discussed. Algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples.
Positive Linear Switched Systems Are Not Uniformly Asymptotically Stable, Even For n = 3
, 2008
"... We consider ndimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this conditi ..."
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We consider ndimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true for the case n = 2, but is not true in general. Their results imply that there exists some minimal integer np such that the conjecture is true for all n < np, but is not true for n = np. We show that np = 3.