Michael Margaliot

Brief paper: Controllability of Boolean control networks via the Perron-Frobenius theory

Boolean control networks (BCNs) are recently attracting considerable interest as computational models for genetic and cellular networks. Addressing control-theoretic 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 Maximum Principle for Single-Input Boolean Control Networks

Boolean networks have recently been attracting considerable interest as computational models for genetic and cellular networks. We consider a Mayer-type optimal control problem for a single-input Boolean network, and derive a necessary condition for a control to be optimal. This provides an analog of Pontryagins maximum principle for single-input Boolean networks.

On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws

We consider n-dimensional 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.

Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions

Abstract We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton University Press, NJ, 2004, pp. 203–207.]. To prove the result, we consider an optimal control problem which consists in finding the “most unstable” trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. This property is obtained as a special case of a reachability result by bang-bang controls which is of independent interest. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

Are artificial neural networks white boxes

In this paper, we introduce a novel Mamdani-type fuzzy model, referred to as the all-permutations fuzzy rule base (APFRB), and show that it is mathematically equivalent to a standard feedforward neural network. We describe several applications of this equivalence between a neural network and our fuzzy rule base (FRB), including knowledge extraction from and knowledge insertion into neural networks.

Fuzzy Lyapunov-based approach to the design of fuzzy controllers

In this paper we extend the classical Lyapunov synthesis method to the domain of computing with words. This new approach is used to design fuzzy controllers. Assuming minimal knowledge about the plant to be controlled, the proposed method enables us to systematically derive the fuzzy rules that constitute the rule base of the controller. We demonstrate the approach by designing Mamdani-type and Takagi-Sugeno-Kang-type fuzzy controllers for two well-known plants.

Observability of Boolean networks: A graph-theoretic approach

Boolean networks (BNs) are discrete-time dynamical systems with Boolean state-variables and outputs. BNs are recently attracting considerable interest as computational models for genetic and cellular networks. We consider the observability of BNs, that is, the possibility of uniquely determining the initial state given a time sequence of outputs. Our main result is that determining whether a BN is observable is NP-hard. This holds for both synchronous and asynchronous BNs. Thus, unless P=NP, there does not exist an algorithm with polynomial time complexity that solves the observability problem. We also give two simple algorithms, with exponential complexity, that solve this problem. Our results are based on combining the algebraic representation of BNs derived by D. Cheng with a graph-theoretic approach. Some of the theoretical results are applied to study the observability of a BN model of the mammalian cell cycle.

New Approaches to Fuzzy Modeling and Control: Design and Analysis

Fuzzy Lyapunov synthesis fuzzy Lyapunov synthesis and stability analysis adaptive fuzzy controller design inverse optimality for fuzzy controllers hyperbolic approach to fuzzy modelling fuzzy controllers for the hyperbolic state-space model.

Minimum-Time Control of Boolean Networks

Boolean networks (BNs) are discrete-time dynamical systems with Boolean state variables. BNs have recently been attracting considerable interest as computational models for biological systems and, in particular, 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 state-space representation (ASSR) of BCNs we derive several necessary conditions, stated in the form of maximum principles (MPs), for a control to be time-optimal. In the ASSR every state and input vector is a canonical vector. Using this special structure yields an explicit state-feedback formula for all time-optimal 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 respon...

Nice reachability for planar bilinear control systems with applications to planar linear switched systems [corrected]

We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a ldquonicerdquo control, specifically, a control that is a concatenation of a bang arc with either 1) a bang-bang control that is periodic after the third switch; or 2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or invariant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either 1) a bang-bang control with no more than two discontinuities; or 2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result. We demonstrate this with an example.