Publications

Complete list of publications (pdf)

OPTIMAL CONTROL BOOLEAN CONTROL NETWORKS SWITCHED SYSTEMS SWITCHED SYSTEMS and POWER CONVERTERS FUZZY MODELING and CONTROL

SYSTEMS BIOLOGY ARTIFICIAL NEURAL NETWORKS

OPTIMAL CONTROL

Fainshil, L. and Margaliot, M. "A maximum principle for the stability analysis of positive bilinear control systems with applications to positive linear switched systems" , SIAM J. Control and Optimization, 50:2193-2215, 2012.

Abstract:

We consider a continuous-time bilinear control system with Metzler matrices. Each entry in the transition matrix of such a system is non-negative, making the positive orthant an invariant set of the dynamics. Motivated by the stability analysis of positive linear switched systems (PLSSs), we define a control as optimal if, for a fixed final time, it maximizes the spectral radius of the transition matrix. Our main result is a first-order necessary condition for optimality in the form of a maximum principle (MP). The proof of this MP combines the standard needle variation with a basic result from the Perron-Frobenius theory of non-negative matrices. We describe several applications of this MP to the stability analysis of PLSSs under arbitrary switching.

Laschov, D. and Margaliot, M. "A Pontryagin maximum principle for multi-input Boolean control networks" , in Recent Advances in Dynamics and Control of Neural Networks, (E. Kaslik and S. Sivasundaram, eds.), to appear.

Abstract:

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 state-space representation for Boolean control networks using the semi--tensor product of matrices. This representation proved quite useful for studying Boolean control networks in a control-theoretic framework. Using this representation, we consider a Mayer-type 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.

Laschov, D. and Margaliot, M. "A maximum principle for single-input Boolean control networks" , IEEE Trans. Automatic Control, 56: 913-917, 2011.

Abstract:

Boolean networks are recently 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 Pontryagin's maximum principle for single-input Boolean networks.

Ratmansky, I. and Margaliot, M. "A simplification of the Agrachev-Gamkrelidze second-order variation for bang-bang controls", Systems and Control Letters, 59: 25-32, 2010.

Abstract:

We consider an expression for the second-order variation (SOV) of bang-bang controls developed by Agrachev and Gamkrelidze. The SOV plays an important role in both necessary and sufficient second-order optimality conditions for bang-bang controls. These conditions are stronger than the one provided by the first-order Pontryagin maximum principle (PMP). For a bang-bang control with k switching points, the SOV contains k(k+1)/2 Lie-algebraic 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, 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 second-order sufficient condition for optimality.

Margaliot, M. and Branicky, M. S. ``Nice reachability for planar bilinear control systems with applications to planar linear switched systems'', IEEE Trans. Automatic Control, 54: 1430-1435, 2009.

Abstract:

We derive a new reachability with nice controls type result for planar bilinear control systems. We show that any point in the reachable set can be reached by 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 main result. We demonstrate this using one example.

Sharon, Y. and Margaliot, M. "Third-order nilpotency, nice reachability and asymptotic stability", Journal of Differential Equations, 233: 136-150, 2007.

Abstract:

We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [1].

BOOLEAN CONTROL NETWORKS

Laschov, D. and Margaliot, M. "Controllability of Boolean control networks via the Perron-Frobenius theory" , Automatica , 48: 1218-1223, 2012.

Abstract:

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.

Laschov, D. and Margaliot, M. "A Pontryagin maximum principle for multi-input Boolean control networks" , in Recent Advances in Dynamics and Control of Neural Networks , (E. Kaslik and S. Sivasundaram, eds.), to appear.

Abstract:

SWITCHED SYSTEMS

Margaliot, M. and Langholz, G. "Necessary and sufficient conditions for absolute stability: the case of second-order systems", IEEE Trans. Circuits Systems-I, 50(2):227-234, 2003.

Abstract:

We consider the problem of absolute stability of linear feedback systems in which the control is a sector-bounded time-varying nonlinearity. Absolute stability entails not only the characterization of the "most destabilizing" nonlinearity, but also determining the parametric value of the nonlinearity that yields instability of the feedback system. The problem was first formulated in the 1940s, however, finding easily verifiable necessary and sufficient conditions for absolute stability remained an open problem all along. Recently, the problem gained renewed interest in the context of stability of hybrid dynamical systems, since solving the absolute stability problem implies stability analysis of switched linear systems. In this paper, we introduce the concept of generalized first integrals and use it to characterize the "most destabilizing" nonlinearity and to explicitly construct a Lyapunov function that yields an easily verifiable, necessary and sufficient condition for absolute stability of second-order systems.

Holcman, D. and Margaliot, M. ``Stability analysis of switched homogeneous systems in the plane'', SIAM J. Control and Optimization, 41(5):1609-1625, 2003.

Abstract:

We study the stability of second-order switched homogeneous systems. Using the concept of generalized first integrals we explicitly characterize the "most destabilizing" switching-law and construct a Lyapunov function that yields an easily verifiable, necessary and sufficient condition for asymptotic stability. Using the duality between stability analysis and control synthesis, this also leads to a novel algorithm for designing a stabilizing switching controller.

Margaliot, M. and Gitizadeh, R. ``The problem of absolute stability: a dynamic programming approach'', Automatica, 40(7):1247-1252, 2004.

Abstract:

We consider the problem of absolute stability of a feedback system composed of a linear plant and a single sector-bounded nonlinearity. Pyatnitskiy and Rapoport used a variational approach and the Maximum Principle to derive an implicit characterization of the "most destabilizing" nonlinearity. In this paper, we address the same problem using a dynamic programming approach. We show that the corresponding value function is composed of simple building blocks which are the generalized first integrals of appropriate linear systems. We demonstrate how the results can be used to design stabilizing switched controllers.

Margaliot, M. and Liberzon, D. ``Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions'', Systems and Control Letters, 55(1): 8-16, 2006.

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, Unsolved Problems in Mathematical Systems Theory and Control, 2004]. 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.

Margaliot, M. and Yfoulis, C. ``Absolute stability of third-order systems - a numerical algorithm'' , Automatica, 42(10): 1705-1711, 2006.

Abstract:

The problem of absolute stability is one of the oldest open problems in the theory of control. Even for the particular case of second-order systems a complete solution was presented only very recently. For third-order systems, the most general theoretical results were obtained by Barabanov. He derived an implicit characterization of the "most destabilizing" nonlinearity using a variational approach. In this paper, we show that his approach yields a simple and efficient numerical scheme for solving the problem in the case of third-order systems. This allows the determination of the critical value where stability is lost in a tractable and accurate fashion. This value is important in many practical applications and we believe that it can also be used to develop a deeper theoretical understanding of this interesting problem.

Margaliot, M. ``Stability analysis of switched systems using variational principles: an introduction'', Automatica, 42(12): 2059-2077, 2006.

Abstract:

Many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode. Switched systems provide a suitable mathematical model for such processes, and their stability analysis is important for both theoretical and practical reasons. We review a specific approach for stability analysis based on using variational principles to characterize the "most unstable" solution of the switched system. We also discuss a link between the variational approach and the stability analysis of switched systems using Lie-algebraic considerations. Both approaches require the use of sophisticated tools from many different fields of applied mathematics. The purpose of this paper is to provide an accessible and self-contained review of these topics, emphasizing the intuitive and geometric underlying ideas.

Sharon, Y. and Margaliot, M. ``Third-order nilpotency, nice reachability and asymptotic stability'', Journal of Differential Equations, 233: 136-150, 2007.

Abstract:

We consider products of matrix exponentials under the assumption that the matrices span a nilpotent Lie algebra. In 1995, Leonid Gurvits conjectured that nilpotency implies that these products are, in some sense, simple. More precisely, there exists a uniform bound l such that any product can be represented as a product of no more than l matrix exponentials. This conjecture has important applications in the analysis of linear switched systems, as it is closely related to the problem of reachability using a uniformly bounded number of switches. It is also closely related to the concept of nice reachability for bilinear control systems. The conjecture is trivially true for the case of first-order nilpotency. Gurvits proved the conjecture for the case of second-order nilpotency using the Baker-Campbell-Hausdorff formula. We show that the conjecture is false for the third-order nilpotent case using an explicit counterexample. Yet, the underlying philosophy behind Gurvits' conjecture is valid in the case of third-order nilpotency. Namely, such systems do satisfy the following nice reachability property: any point in the reachable set can be reached using a piecewise constant control with no more than four switches. We show that even this form of finite reachability is no longer true for the case of fifth-order nilpotency.

Margaliot, M. ``On the Analysis of Nonlinear Nilpotent Switched Systems using the Hall-Sussmann System'', Systems and Control Letters, 58: 766-772, 2009.

Abstract:

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 Hall-Sussmann 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 Lie-algebraic structure of the nilpotent Hall-Sussmann system.

Margaliot, M. and Yfoulis, C. ``A Lie-algebraic analysis of the problem of absolute stability'', submitted.

Abstract:

The absolute stability problem (ASP) entails determining a critical parameter value for which the stability of a feedback system, composed of an n'th-order linear system and a sector-bounded nonlinear function, loses its stability. The ASP is one of the oldest open problems in the theory of stability and control. Recently, it is attracting considerable interest, as solving it is equivalent to providing a necessary and sufficient condition for the stability of linear switched systems under arbitrary switching. Pyatnitsky pioneered the most promising approach for addressing the ASP. His approach is based on using optimal control techniques for characterizing the "most destabilizing" sector-bounded nonlinear function. This is equivalent to characterizing the "most destabilizing" switching law for a linear switched system under arbitrary switching. In this paper, we develop a new approach to the ASP which is based on a Lie-algebraic analysis of the switching function that determines the optimal control. We show that the finiteness of the associated Lie algebra implies that the switching function itself is the solution of a switched linear system of order at most n^2. Furthermore, the switching function has a special and symmetric structure. This makes it possible to obtain an explicit analytic expression for the switching function for low orders of n. We demonstrate this using two examples.

Margaliot, M. and Hespanha, J. P. ``Root-mean-square gains of switched linear systems: a variational approach'', Automatica, 44: 2398-2402, 2008.

Abstract:

We consider the problem of computing the root-mean-square (RMS) gain of switched linear systems. We develop a new approach which is based on an attempt to characterize the ``worst-case'' switching law (WCSL), that is, the switching law that yields the maximal possible gain. Our main result provides a sufficient condition guaranteeing that the WCSL can be characterized explicitly using the differential Riccati equations corresponding to the linear subsystems. This condition automatically holds for first-order systems, so we obtain a complete solution to the RMS gain problem in this case. In particular, we show that in the first-order case there always exists a WCSL with no more than two switches.

Fainshil, L., Margaliot, M. and Chigansky, P. ``On the stability of positive linear switched systems under arbitrary switching laws'', IEEE Trans. Automatic Control, 54: 897-899, 2009.

Abstract:

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 n_p such that the conjecture is true for all n< n_p, but is not true for n= n_p. We show that n_p=3.

Monovich, T. and Margaliot, M. "Analysis of discrete-time linear switched systems: a variational approach", SIAM J. Control and Optimization, 49: 808-829, 2011.

Abstract:

A powerful approach for analyzing the stability of continuous-time switched systems is based on using tools from optimal control theory to characterize the "most unstable" switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific "most unstable" switching law. More generally, this so-called variational approach was successfully applied to derive nice-reachability-type results for both linear and nonlinear continuous-time switched systems. Motivated by this, we develop in this paper an analogous approach for discrete-time linear switched systems. We derive and prove a necessary condition for optimality of the "most unstable" switching law. This yields a type of discrete-time maximum principle (MP). We demonstrate using an example that this MP is in fact weaker than its continuous-time counterpart. To overcome this, we introduce the auxiliary system of a discrete-time linear switched system, and show that regularity properties of time-optimal controls for the auxiliary system imply nice-reachability results for the original discrete-time linear switched system. Using this approach, we derive several new Lie-algebraic conditions guaranteeing nice-reachability results. These results, and their proofs, turn out to be quite different from their continuous-time counterparts.

Monovich, T. and Margaliot, M. "A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems", Automatica, 47: 1489-1495, 2011.

Abstract:

A powerful approach for analyzing the stability of continuous-time switched systems is based on using tools from optimal control theory to characterize the "most unstable" switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific "most unstable" switching law. For discrete-time (DT) switched systems, the variational approach received considerably less attention. This approach is based on using a first-order necessary optimality condition in the form of a maximum principle (MP), and typically this is not enough to completely characterize the "most unstable" switching law. In this paper, we provide a simple and self-contained derivation of a second-order necessary optimality condition for DT bilinear control systems. This provides new information on the optimal controls that cannot be derived using the first-order MP. We demonstrate several applications of this second-order MP to the stability analysis of DT linear switched systems.

Teichner, R. and Margaliot, M. "Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems", Automatica , 48: 95-101, 2012.

Abstract:

We consider the problem of stability under arbitrary switching of a discrete-time linear switched system. A powerful approach for addressing this problem is based on studying the "most unstable" switching law (MUSL). If the solution of the switched system corresponding to the MUSL converges to the origin, then the switched system is stable for any switching law. The MUSL can be characterized using optimal control techniques. This variational approach leads to a Hamilton-Jacobi-Bellman equation characterizing the behavior of the switched system under the MUSL. The solution of this equation is sometimes referred to as a Barabanov norm of the switched system. Although the Barabanov norm was studied extensively, it seems that there are few examples where it was actually computed in closed-form. In this paper, we consider a special class of positive planar discrete-time linear switched systems and provide a closed-form expression for a corresponding Barabanov norm. The unit circle in this norm is a parallelogram.

SWITCHED SYSTEMS and POWER CONVERTERS

Reznikov, B., Ruderman, A. and Margaliot, M. Analysis of a Flying Capacitor Converter: A Switched Systems Approach, Int. J. Circuit Theory Applications , to appear.

Abstract:

Flying capacitor converters (FCCs) attract considerable interest because of their inherent natural voltage balancing property. Several researchers analyzed the voltage balance dynamics in FCCs using frequency domain methods. Recently, considerable research attention has been devoted to switched systems, i.e. systems composed of several subsystems, and a switching law that determines which subsystem is active at every time instant. In this paper, we propose a new approach to the analysis of an FCC. The analysis is performed in the time-domain, treating the FCC as a switched system. The subsystems are the various configurations obtained for each state of the circuit switches, and the switching law is determined by the modulation. We demonstrate this new approach by using it to analyze a single-phase single-leg three-level FCC. The switched systems approach provides simple closed-form expressions for the system behavior. We show that the natural balancing property is equivalent to the asymptotic stability of a certain matrix. We also show that it is possible to rigorously analyze properties such as the capacitor time constant and relate them to the parameter values of the load, carrier frequency, and duty ratio.

Margaliot, M. and Weiss, G. " The low frequency distortion in D-class amplifiers", IEEE Trans. Circuits and Systems II, 57: 772-776, 2010.

Abstract:

D-class amplifiers are non linear amplifiers that transform an input signal u into an output signal y. The transformation is based on using pulse-width modulation to create a rectangular signal p whose duty cycle depends linearly on the input signal u. After low-pass filtering p to eliminate the switching frequency in its harmonics, we obtain the filtered output signal y that is a good approximation of u. This kind of amplifier is gaining popularity (as compared to the classical A-class and B-class amplifiers) due to its high efficiency. A buck power supply may be regarded as a particular instance of a D-class amplifier. While the principle of the D-class amplifier is intuitively clear, a rigorous mathematical analysis of this system is not trivial, as the amplifier is time-varying and non-linear. In this paper we provide a rigorous proof of the fact that the signal p is a good approximation of u in the low frequency range, and provide precise error bounds for this approximation.

FUZZY MODELING and CONTROL

Margaliot, M. and Langholz, G. New Approaches to Fuzzy Modeling and Control - Design and Analysis, World Scientific, 2000.
TOC Chapter 1 purchase this book here

Margaliot, M. and Langholz, G. ``Fuzzy Lyapunov based approach to the design of fuzzy controllers'', Fuzzy Sets Systems, 106(1): 49-59, 1999.

Abstract:

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

Margaliot, M. and Langholz, G. ``Hyperbolic optimal control and fuzzy control'', IEEE Trans. Systems, Man, Cybernetics: Part A, 29(1): 1-10, 1999.

Abstract:

In this paper, we consider a new approach to fuzzy control which entails the formulation of a novel state-space representation and a new form of optimal control problem. Basically, in this new formulation, linear functions in the conventional state-space representation and cost functional are replaced by hyperbolic functions. We give a solution for this new, infinite-time, optimal control problem, which we call hyperbolic optimal control. Furthermore, we show that the resulting optimal controller is in fact a Mamdani-type fuzzy controller with Gaussian membership functions and center of gravity defuzzification. These results enable us to investigate analytically important issues, such as stability and robustness, pertaining to fuzzy controllers as well as add a powerful theoretical framework to the field of fuzzy control.

Margaliot, M. and Langholz, G. ``Design and analysis of fuzzy schedulers using fuzzy Lyapunov synthesis'', Engineering Applications of Artificial Intelligence, 14(2):183-188, 2001.

Abstract:

Recently, a new approach to the design of fuzzy control rules was suggested. The method, referred to as fuzzy Lyapunov synthesis, extends classical Lyapunov synthesis to the domain of "computing with words", and allows the systematic, instead of heuristic, design and analysis of fuzzy controllers given linguistic information about the plant. In this paper, we use fuzzy Lyapunov synthesis to design and analyze the rule-base of a fuzzy scheduler. Here, too, rather than use heuristics, we can derive the fuzzy rule-base systematically. This suggests that the process of deriving the rules can be automated. Our approach may lead to a novel computing with words algorithm: the input is linguistic information concerning the "plant" and the "control" objective, and the output is a suitable fuzzy rule-base.

Margaliot, M. and Langholz, G. ``A new approach to fuzzy modeling and control of discrete-time systems'' , IEEE Trans. Fuzzy Systems, 11(4):486-494, 2003.

Abstract:

We present a new approach to fuzzy modeling and control of discrete-time systems which is based on the formulation of a novel state-space representation using the hyperbolic tangent function. The new representation, designated the hyperbolic model, combines the advantages of fuzzy system theory and classical control theory. On the one hand, the hyperbolic model is easily derived from a set of Mamdani-type fuzzy rules. On the other hand, classical control theory can be applied to design controllers for the hyperbolic model that not only guarantee stability and robustness but are themselves equivalent to a set of Mamdani-type fuzzy rules. Thus, this new approach combines the best of two worlds. It enables linguistic interpretability of both the model and the controller, and guarantees closed-loop stability and robustness.

Margaliot, M. and Langholz, G. ``Fuzzy control of a benchmark problem: a computing with words approach'', IEEE Trans. Fuzzy Systems, 12(2):230-235, 2004.

Abstract:

The rotational/translational proof-mass actuator (RTAC) system is a well-known benchmark for nonlinear control for which a number of controllers were designed. In this paper, we apply fuzzy Lyapunov synthesis, which is a computing with words version of classical Lyapunov synthesis, to design fuzzy controllers for the RTAC system. This allows us to systematically design state-feedback and output-feedback controllers using only a linguistic description of the RTAC system. The designed fuzzy controllers yield a globally asymptotically stable closed-loop system. We demonstrate their advantages in comparison with a previously designed linear controller for the RTAC system.

SYSTEMS BIOLOGY

Margaliot, M. and Tuller, T. "Analysis of the steady state translation rate in the infinite-dimensional homogeneous ribosome flow model", submitted, 2013.

Abstract:

Gene translation is a central stage in the intra-cellular process of protein synthesis. Gene translation proceeds in three major stages: initiation, elongation, and termination. During the elongation step ribosomes (intra cellular macro-molecules) link amino acids together in the order specified by messenger RNA (mRNA) molecules. The homogeneous ribosome flow model (HRFM) is a mathematical model of translation elongation under the assumption of constant elongation rate along the mRNA sequence. The HRFM includes n first-order nonlinear ODEs, where n represents the length of the mRNA sequence, and two positive parameters: ribosomal initiation rate and the (constant) elongation rate. Here we analyze the HRFM when n goes to infinity and derive a simple expression for the steady-state protein synthesis rate. Simulations suggest that the behavior of the HRFM for n larger than 15 is already in good agreement with the behavior in the case of infinite n studied here. Thus, our result may be used in practice for estimating the translation rates of different genes or for engineering genes to produce a desired translation rate.

Margaliot, M. and Tuller, T. "Ribosome flow model with positive feedback", submitted, 2013.

Abstract:

Eukaryotic mRNAs usually form a circular structure; thus, ribosomes that terminate translation at the 3’ end can diffuse with increased probability to the 5’end of the transcript, initiating another cycle of translation. This phenomenon describes ribosomal flow with positive feedback–an increase in the flow of ribosomes terminating translating the ORF increases the ribosomal initiation rate. The aim of the current study is to mathematically model this phenomenon and rigorously analyze it. To this end, we suggest a modified version of the Ribosome Flow Model (RFM) [1], called the Ribosome Flow Model with Input and Output (RFMIO). In this model the input is the initiation rate and the output is the translation rate. We analyze this model after closing the loop with a positive linear feedback. We show that the closed-loop system admits a unique globally asymptotically stable equilibrium point. From a biophysical point of view, this means that there exists a unique steady-state of ribosome distributions along the mRNA, and thus a unique steady-state translation rate. The solution from any initial distribution will converge to this steady-state. The steady-state distribution demonstrates a decrease in ribosome density along the coding sequence. For the case of constant elongation rates, we obtain closed-form expressions relating the model parameters with the location of the equilibrium point. These results may perhaps be used to modify the biological system in order to obtain a desired translation rate.

Margaliot, M. and Tuller, T. "Stability analysis of the ribosome flow model", IEEE/ACM Trans. Computational Biology and Bioinformatics, 9(5):1545-1552, 2012.

Abstract:

Gene translation is a central process in all living organisms. Developing a better understanding of this complex process may have ramifications to almost every biomedical discipline. Reuveni et al. recently proposed a new computational model of gene translation called the Ribosome Flow Model (RFM). In this paper, we consider a particular case of this model, called the Homogeneous Ribosome Flow Model (HRFM). From a biological viewpoint, this corresponds to the case where the elongation rates of all the coding sequence codons are identical. This regime has been observed in experiments when the concentration of the tRNAs is not rate limiting as their abundances are very high relatively to the concentrations of elongation factors. We consider the steady-state distribution of the HRFM. We derive formulas that relate the different parameters when the system is in this steady-state. We prove the following properties: 1) the ribosomal density profile is monotonically decreasing along the coding sequence; 2) the ribosomal density at each codon monotonically increases with the initiation rate; and 3) for a constant initiation rate, the translation rate monotonically decreases with the length of the coding sequence. In addition, we analyze the translation rate of the HRFM at the limit of very high and very low initiation rate, and provide explicit formulas for the translation rate in these two cases. We discuss the relationship between these theoretical results and biological findings on the translation process.

Tron, E. and Margaliot, M. ``Mathematical modeling of observed natural behavior: a fuzzy logic approach'', Fuzzy Sets Systems, 146(3):437-450, 2004.

Abstract:

Fuzzy modeling is routinely used to transform the knowledge of an expert, be it a physician or a process operator, into a mathematical model. The emphasis is on constructing a fuzzy expert system that replaces the human expert. In this paper, we advocate a different application of fuzzy modeling, namely, as a tool that can assist human observers in the difficult task of transforming their observations into a mathematical model. In many fields of science, including biology, psychology and economy, human observers have provided linguistic descriptions and explanations of various systems. However, to study these phenomena in a systematic manner, there is a need to construct a suitable mathematical model, a process that usually requires subtle mathematical understanding. Fuzzy modeling is a simple, direct, and natural approach for transforming the linguistic description into a mathematical model. Furthermore, fuzzy modeling offers a unique advantage-the close relationship between the linguistic description and the mathematical model can be used to verify the validity of the verbal explanation suggested by the observer. We demonstrate this using an example of territorial behavior of fish.

Tron, E. and Margaliot, M. ``How does the Dendrocoleum lacteum orient to light? a fuzzy modeling approach'', Fuzzy Sets Systems, 155(2): 236-251, 2005.

Abstract:

We apply fuzzy modeling to derive a mathematical model for a biological phenomena: the orientation to light of the planarian Dendrocoleum lacteum. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a mathematical model. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. This seems to indicate that the verbal explanations suggested by the ethologists are indeed correct.

Rozin, V. and Margaliot, M. ``The fuzzy ant'' , IEEE Computational Intelligence Magazine, 2(4): 18-28, 2007.

Abstract:

The design of artificial systems inspired by biological behavior is recently attracting considerable interest. Many biological agents such as plants or animals were forced to develop sophisticated mechanisms in order to tackle various problems they encounter in their habitat. For example, animals must develop efficient mechanisms for orienting themselves in space. Similar problems arise in the design of artificial systems. For example, planning and realizing oriented movements is a crucial problem in the design of autonomous robots. Thus, lessons from biological behavior may inspire suitable artificial designs. In some cases, ethologists provided verbal descriptions of the relevant animal behavior. Fuzzy modeling is the most suitable tool for transforming these verbal descriptions into mathematical models or computer algorithms that can be used in artificial systems. We demonstrate this by using fuzzy modeling to develop a mathematical model for the foraging behavior of ants. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature.

Rashkovsky, Y. and Margaliot, M. ``Nicholson's blowflies revisited - a fuzzy modeling approach'', Fuzzy Sets Systems, 158(10): 1083-1096, 2007.

Abstract:

We apply fuzzy modeling to derive a mathematical model for a biological phenomenon: the regulation of population size in the Australian sheep-blowfly Lucilia cuprina. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a well-defined mathematical model. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. We believe that the fuzzy modeling approach demonstrated here may supply a suitable framework for biomimicry, that is, the design of artificial systems based on mimicking natural behavior.

Margaliot, M. ``Mathematical modeling of natural phenomena: a fuzzy logic approach'', in Fuzzy Logic - A Spectrum of Theoretical and Practical Issues (P. P. Wang, Da Ruan, and E. E. Kerre, eds.) Springer, pp. 113-134, 2007.

Abstract:

In many fields of science human observers have provided verbal descriptions and explanations of various systems. A formal mathematical model is indispensable when we wish to rigorously analyze these systems. In this chapter, we survey some recent results on transforming verbal descriptions into mathematical models using fuzzy modeling. This is a simple and direct approach that offers a unique advantage: the close relationship between the verbal description and the mathematical model can be used to verify the validity of the verbal explanation suggested by the observer. We review two applications of this approach from the field of ethology: fuzzy modeling of the territorial behavior of fish, and of the orientation to light of a flat worm. The fuzzy modeling approach demonstrated here may supply a suitable framework for biomimicry, that is, the design of artificial systems based on mimicking a natural behavior observed in nature.

Margaliot, M. ``Biomimicry and fuzzy modeling: a match made in heaven'' , IEEE Computational Intelligence Magazine, 3(3): 38-48, 2008.

Abstract:

Biomimicry, the design of artificial systems that mimic natural behavior, is recently attracting considerable interest. An important component in biomimicry is the ability to perform reverse engineering of the functioning of a biological agent, in order to implement this behavior in an artificial system. In some cases, biologists already studied the relevant behavior and provided a detailed verbal description of it. Indeed, biological theories are generally descriptive in nature. For example, Darwin's original presentation of his evolutionary theory did not include a single equation. In this case, mimicking the natural behavior is reduced to the following problem: how can we convert the given verbal description into a well-defined mathematical formula or algorithm that can be implemented by an artificial system? Fuzzy modeling, with its ability to handle and manipulate verbal information, provides the most adequate approach for solving this problem. Thus, fuzzy modeling may be very suitable for addressing biomimicry in a systematic manner: start with a verbal description of an animal's behavior (e.g., the foraging behavior of ants) and, using fuzzy logic theory, obtain a mathematical model of this behavior which can be implemented by artificial systems (e.g., autonomous robots). The purpose of this position paper is to highlight these issues and to alert the attention of the computational intelligence community to this new and potentially very promising application of fuzzy modeling.

Laschov, D. and Margaliot, M. ``Mathematical modeling of the lambda switch: a fuzzy logic approach'' , J. Theoretical Biology, 260: 475-489, 2009.

Abstract:

Gene regulation plays a central role in the development and functioning of living organisms. Developing a deeper qualitative and quantitative understanding of gene regulation is an important scientific challenge. The lambda switch is commonly used as a paradigm of gene regulation. Verbal descriptions of the structure and functioning of the lambda switch have appeared in biological textbooks. We apply fuzzy modeling to transform one such verbal description into a well-defined mathematical model. The resulting model is a piecewise quadratic second-order differential equation. It demonstrates functional fidelity with known results while being simple enough to allow a rather detailed analysis. Properties such as the number, location, and domain of attraction of equilibrium points can be analyzed analytically. Furthermore, the model provides a rigorous explanation for the so-called stability puzzle of the lambda switch.

ARTIFICIAL NEURAL NETWORKS

Kazas, G. and Margaliot, M. "Visualizing the topology of mental disorders using self organizing feature maps", submitted.

Abstract:

Mental disorders have a large impact on individuals, families, and communities, and are one of the main causes worldwide of disability and distress. Correct diagnosis of mental disorders is essential in clinical practice, pharmacological research, and successful treatment. Unfortunately, the aetiology and pathogenesis of many mental disorders are still unknown. Psychiatrists must thus resort to classifying disorders according to their symptoms. This provides little information on the topology of the disorders, that is, how one disorder relates to another and why certain disorders bunch together. Here we propose an algorithmic approach for visualizing the topology of mental disorders using a self organizing feature map (SOFM). A SOFM is a specific type of artificial neural network that can be trained to produce a low-dimensional representation (called a map) of a high-dimensional input space. This map preserves the topology of the original input space. We trained a SOFM to produce a two-dimensional map of 27 relatively wellknown mental disorders. Each disorder is represented by an 82-dimensional input vector describing the symptoms associated with the disorder, as described in the Diagnostic and Statistical Manual of Mental Disorders (DSM-IV-TR). The map shows distinct clusters of mental disorders. Each cluster contains disorders that are similar to one another, and separate from those in other clusters. This provides a way to clearly visualize the topology of these mental disorders. We demonstrate three more applications of the resulting map: automatic classification of disorders based on apparent symptoms; rigorous estimation of the relative importance of various symptoms in the diagnostic process; and adding a dimensional component to psychiatric diagnoses.

Kolman, E. and Margaliot, M. Knowledge-Based Neurocomputing: A Fuzzy Logic Approach, Springer, 2009.
TOC Chapters 2 and 3 purchase this book here

Kolman, E. and Margaliot, M. ``Neural networks=fuzzy rule bases'', in Applied Computational Intelligence: Proceedings of the 6th International FLINS Conference (D. Ruan et al, eds.), World Scientific, pp. 111-117, 2004.

Kolman, E. and Margaliot, M. ``Are artificial neural networks white boxes?'', IEEE Trans. Neural Networks, 16(4): 844-852, 2005.

Abstract:

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.

Kolman, E. and Margaliot, M. ``Knowledge extraction from neural networks using the all-permutations fuzzy rule base: the LED display recognition problem'', IEEE Trans. Neural Networks, 18(3): 925-931, 2007.

Abstract:

A major drawback of artificial neural networks (ANNs) is their black-box character. Even when the trained network performs adequately, it is very difficult to understand its operation. In this paper, we use the mathematical equivalence between artificial neural networks and a specific fuzzy rule base to extract the knowledge embedded in the network. We demonstrate this using a benchmark problem: the recognition of digits produced by a LED device. The method provides a symbolic and comprehensible description of the knowledge learned by the network during its training.

Kolman, E. and Margaliot, M. ``Extracting symbolic knowledge from recurrent neural networks-a fuzzy logic approach'', Fuzzy Sets and Systems, 160: 145-161, 2009.

Abstract:

Considerable research has been devoted to the integration of fuzzy logic (FL) tools with classic artificial intelligence (AI) paradigms. One reason for this is that FL provides powerful mechanisms for handling and processing symbolic information stated using natural language. In this respect, fuzzy rule-based systems are white-boxes, as they process information in a form that is easy to understand, verify and, if necessary, refine. The synergy between artificial neural networks (ANNs), which are notorious for their black-box character, and FL proved to be particularly successful. Such a synergy allows combining the powerful learning-from-examples capability of ANNs with the high-level symbolic information processing of FL systems. In this paper, we present a new approach for extracting symbolic information from recurrent neural networks (RNNs). The approach is based on the mathematical equivalence between a specific fuzzy rule-base and functions composed of sums of sigmoids. We show that this equivalence can be used to provide a comprehensible explanation of the RNN's functioning. We demonstrate the applicability of our approach by using it to extract the knowledge embedded within an RNN trained to recognize a formal language.

Kolman, E. and Margaliot, M. ``A new approach to knowledge-based design of recurrent neural networks'', IEEE Trans. Neural Networks, 19: 1389-1401, 2008.

Abstract:

We develop a new approach for designing a recurrent neural network (RNN) that is suitable for solving a given problem. Initial information on the problem domain is stated in terms of symbolic If-Then rules. These rules have a special structure and inferring them yields a mapping that is equivalent to that of a net of sigmoid activated neurons with feedback connections. Thus, inferring the rules automatically yields a suitable RNN. We demonstrate the efficiency of our approach by using it to design an RNN that recognizes a formal language.

Roth, I. and Margaliot, M. ``Analysis of learning near temporary minima using the all permutations fuzzy rule-base'', submitted.

Abstract:

An important problem in learning using gradient descent algorithms (such as backprop) is the slowdown incurred by temporary minima~(TM). We consider this problem for a network trained to solve the XOR problem. The network is transformed into the equivalent all permutations fuzzy rule-base which provides a symbolic representation of the knowledge embedded in the network. We develop a mathematical model for the evolution of the fuzzy rule-base parameters during learning in the vicinity of TM. We show that the rule-base becomes singular and tends to remain singular in the vicinity of TM. The analysis of the fuzzy rule-base suggests a simple remedy for overcoming the slowdown in the learning process incurred by TM. This is based on slightly perturbing the values of the training examples, so that they are no longer symmetric. Simulations demonstrate the effectiveness of this approach in reducing the time spent in the vicinity of TM.

Duenyas, S. and Margaliot, M. "Knowledge Extraction From a Class of Support Vector Machines: A Fuzzy Logic Approach", submitted.

Abstract:

Support vector machines (SVMs) proved to be highly efficient computational tools in various classification tasks. However, the knowledge learned by an SVM is encoded in a long list of parameter values, and it is not easy to comprehend what the SVM is actually computing. We show that certain types of SVMs are mathematically equivalent to a specific fuzzy-rule base, the fuzzy all-permutations rule base (FARB). The equivalent FARB provides a symbolic representation of the SVM functioning. This leads to a new approach for knowledge extraction from SVMs. Several examples demonstrate the effectiveness of this approach.

MISCELLANEOUS

Margaliot, M. and Gotsman, C. ``Smooth surface approximation and adaptive sampling by piecewise linear surfaces'', in Computer Graphics - Developments in Virtual Environments (R. A. Earnshaw and J. A. Vince, eds.), Academic Press, pp. 17-27, 1995.

Margaliot, M. and Langholz, G. ``The Routh-Hurwitz array and realization of characteristic polynomials'', IEEE Trans. Automatic Control, 45(12): 2424-2427, 2000.

Margaliot, M. and Langholz, G. ``Some nonlinear optimal control problems with closed-form solutions'', Int. J. Robust Nonlinear Control, 11: 1365-1374, 2001.

Margaliot, M. Book review of Pattern Recognition (3rd Edition) by S. Theodoridis and K. Koutroumbas, IEEE Trans. Neural Networks, 19: p. 376, 2008.