Papers placed here are preprints or reprints. Generally speaking, the published versions of preprints should be the same, except for possible minor copyediting. However, for citations to specific results, it is advisable to check out the actual published paper, since journals sometimes change the numbering style for theorems and such.
When a date appears on a preprint, it represents the compilation date of the version placed on the web. It doesn't necessarily bear any relationship to the date of submission or of publication.

Mathematical Control Theory: Deterministic Finite Dimensional Systems (Second Edition, Springer, New York, 1998)
(pdf file, 544 pages)

(Important note: This book is copyrighted by Springer-Verlag. Springer has kindly allowed me to place a copy on the web, as a reference and for ease of web searches. Please consider buying your own hardcopy.)

Input to state stability: Basic concepts and results
Publication data: in Nonlinear and Optimal Control Theory (P. Nistri and G. Stefani, eds.), Springer-Verlag, Berlin, 2006, pp. 163-220.
Keywords: input to state stability -- nonlinear systems -- detectability -- nonlinear regulation
Description: This is an up to date expository paper on ISS and related concepts, prepared for a CIME course in June 2004.

Structure and timescale analysis in genetic regulatory networks
Coauthor(s): M. Chaves and R. Albert
Publication data: in Proceedings of IEEE Conf on Decision and Control, San Diego, 2006, IEEE Publications, to appear.
Keywords: genetic regulatory networks - Boolean systems - hybrid systems
Description: This work is concerned with the study of the robustness and fragility of gene regulation networks to variability in the timescales of the distinct biological processes involved. It explores and compares two methods: introducing asynchronous updates in a Boolean model, or integrating the Boolean rules in a continuous, piecewise linear model. As an example, the segment polarity network of the fruit fly is analyzed. A theoretical characterization is given of the model's ability to predict the correct development of the segmented embryo, in terms of the specific timescales of the various regulation interactions.

On the structural monotonicity of chemical reaction networks
Coauthor(s): David Angeli and Patrick DeLeenheer
Publication data: in Proceedings of IEEE Conf on Decision and Control, San Diego, 2006, IEEE Publications, to appear.
Keywords: chemical networks - monotone systems - stability of dynamical systems
Description: This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence without making assumptions on the structure (e.g., mass-action kinetics) of the dynamical equations involved, and relying only on stoichiometric constraints. The key idea is to find a suitable set of coordinates under which the resulting system is cooperative. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property.

A remark on singular perturbations of strongly monotone systems
Publication data: in Proceedings of IEEE Conf on Decision and Control, San Diego, 2006, IEEE Publications, to appear.
Almost global convergence in singular perturbations of strongly monotone systems
Publication data: in Positive Systems (C. Commault and N. Marchand, eds.), Lecture Notes in Control and Information Sciences Volume 341 (Proceedings of the second Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 06) Grenoble, France, Aug. 30), Springer-Verlag, 2006, pp. 415-422.
Coauthor(s): Liming Wang
Keywords: singular perturbations - Hirsch generic convergence theorem - monotone systems
Description: These papers deal with global convergence to equilibria, and in particular Hirsch's generic convergence theorem for strongly monotone systems, for singular perturbations of monotone systems.

Signal detection and approximate adaptation implies an approximate internal model
Coauthor(s): Burton Andrews and Pablo Iglesias
Publication data: in Proceedings of IEEE Conf on Decision and Control, San Diego, 2006, IEEE Publications, to appear.
Keywords: internal model principle - biological adaptation
Description: The proper function of many biological systems requires that external perturbations be detected, allowing the system to adapt to these environmental changes. It is now well established that this dual detection and adaptation requires that the system have an internal model in the feedback loop. In this paper we relax the requirement that the response of the system adapt perfectly, but instead allow regulation to within a neighborhood of zero. We show that linear systems with the ability to detect input signals and approximately adapt require an approximate model of the input. We illustrate our results by analyzing two well-studied biological

Computational aspects of feedback in neural circuits
Coauthor(s): Wolfgang Maass and Prashant Joshi
Publication data: PLoS Computational Biology, to appear .
Keywords: neural networks - feedback linearization - computation by cortical microcircuits
Description: It had previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate in this article the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on time-varying inputs. But even with noise the resulting computational model can perform a large class of biologically relevant real-time computations that require a non-fading memory.

Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles
Publication data: submitted.
A note on monotone systems with positive translation invariance
Publication data: in Proceedings of 14th IEEE Mediterranean Conference on Control and Automation, June 28-30, 2006, Università Politecnica delle Marche, Ancona, Italy, paper FLA3-1, 6 pages.
Coauthor(s): D. Angeli
Description: Strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are shown to have the property that all projected solutions converge to a unique equilibrium. This result may be seen as a dual of a well-known theorem of Mierczynski for systems that satisfy a conservation law. As an application, it is shown that enzymatic futile cycles have a global convergence property.

A Petri net approach to the study of persistence in chemical reaction networks
Coauthor(s): Patrick de Leenheer, David Angeli
Publication data: submitted (and arXiv q-bio.MN/068019v2, 10 Aug 2006).
Keywords: Petri nets - chemical networks - persistence - nonlinear dynamics
Description: Persistency is the property, for differential equations in $\R^n$, that solutions starting in the positive orthant do not approach the boundary. For chemical reactions and population models, this translates into the non-extinction property: provided that every species is present at the start of the reaction, no species will tend to be eliminated in the course of the reaction. This paper provides checkable conditions for persistence of chemical species in reaction networks, using concepts and tools from Petri net theory, and verifies these conditions on various systems which arise in the modeling of cell signaling pathways.

Interconnections of monotone systems with steady-state characteristics
Coauthor(s): David Angeli
Publication data: in "Optimal Control, Stabilization, and Nonsmooth Analysis" (de Queiroz, M., M. Malisoff, and P. Wolenski, eds.), Springer-Verlag, Heidelberg, 2004, pp. 135-154.
Keywords: monotone systems - small-gain theorem - multi-stability - signaling cascades
Description: 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.

Monotone Control Systems (please click here for typos)
Coauthor(s): David Angeli
Publication data: IEEE Trans. Autom. Control 48(2003): 1684-1698.
Keywords: monotone systems - small-gain theorem - MAPK cascades
Description: 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 chemical reaction networks
Coauthor(s): Patrick de Leenheer, David Angeli
Publication data: J. of Mathematical Chemistry, to appear.
Keywords: monotone systems - chemical networks
Description: We analyze certain chemical reaction networks and show that every solution converges to some steady state. The reaction kinetics are assumed to be monotone but otherwise arbitrary. When diffusion effects are taken into account, the conclusions remain unchanged. The main tools used in our analysis come from the theory of monotone dynamical systems. We review some of the features of this theory and provide a self-contained proof of a particular attractivity result which is used in proving our main result.

Multistability in monotone input/output systems
Coauthor(s): David Angeli
Publication data: Systems and Control Letters 51(2004): 185-202.
Keywords: monotone systems -- bistability -- hysteresis -- multiple steady states
Description: This paper studies the emergence of multi-stability and hysteresis in those systems that arise, under positive feedback, from monotone systems with well-defined steady-state responses. Such feedback configurations appear routinely in several fields of application, and especially in biology. The results are stated in terms of directly checkable conditions which do not involve explicit knowledge of basins of attractions of each equilibria.

Detection of multi-stability, bifurcations, and hysteresis in a large class of biological positive-feedback systems (please click here for a typo and here for revised Suppl. Fig. 7(b))
Coauthor(s): David Angeli and Jim Ferrell, Jr.
Publication data: Proc. Natl. Acad. Sci. USA 101(2004): 1822-1827.
Keywords: monotone systems -- bistability -- hysteresis -- multiple steady states
Description: Multistability is an important recurring theme in cell signaling, of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or "remember" transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex than this. Here we show that for a class of feedback systems of arbitrary order, the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this "open loop," feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems, and illustrate the method with two examples taken from recent experimental studies of bistable systems - a two-variable Cdc2/Wee1 system and a more complicated five-variable MAPK cascade.

Robustness and fragility of Boolean models for genetic regulatory networks
Coauthor(s): M. Chaves and R. Albert
Publication data: Journal of Theoretical Biology 235(2005): 431-449.
Keywords: Boolean models -- gene and protein networks -- asynchronous computation
Description: Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. Here we propose a framework for gene regulatory networks that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild type pattern. On the other hand, separation of timescales between pre- and post-translational processes and a minimal prepattern ensure convergence to the wild type expression pattern regardless of fluctuations.

Methods of robustness analysis for Boolean models of gene control networks
Coauthor(s): M. Chaves and R. Albert
Publication data: IEE Proceedings Systems Biology 153 (2006): 154-167.
Keywords: Boolean models -- gene and protein networks -- hybrid systems -- asynchronous computation
Description: As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuous-time character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model's gene pattern prediction as a function of the timescales of the various processes.

Inferring dynamic architecture of cellular networks using time series of gene expression, protein and metabolite data (supplementary materials here)
Coauthor(s): Boris Kholodenko and Anatoly Kiyatkin
Publication data: Bioinformatics 20(2004): 1877-1886.
Keywords: reverse engineering -- gene and protein networks
Description: High-throughput technologies have facilitated the acquisition of large genomics and proteomics data sets. However, these data provide snapshots of cellular behavior, rather than help us reveal causal relations. Here, we propose how these technologies can be utilized to infer the topology and strengths of connections among genes, proteins, and metabolites by monitoring time-dependent responses of cellular networks to experimental interventions. We show that all connections leading to a given network node, e.g., to a particular gene, can be deduced from responses to perturbations none of which directly influences that node, e.g., using strains with knock-outs to other genes. To infer all interactions from stationary data, each node should be perturbed separately or in combination with other nodes. Monitoring time series provides richer information and does not require perturbations to all nodes. (See also the original "unravelling algorithm" paper for motivation, as well as these papers for: computational complexity and noise issues.)

Non-monotone systems decomposable into monotone systems with negative feedback
Coauthor(s): G.A. Enciso and H.L. Smith
Publication data: J. of Differential Equations 224(2006): 205-227.
Keywords: monotone systems -- small-gain theorem -- reaction-diffusion partial differential equations
Description: Motivated by the theory of monotone i/o systems, this paper shows that certain finite and infinite dimensional semi-dynamical systems with negative feedback can be decomposed into a monotone open loop system with inputs and a decreasing output function. The original system is reconstituted by plugging the output into the input. By embedding the system into a larger symmetric monotone system, this paper obtains finer information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence. An important new result is the extension of the "small gain theorem" of monotone i/o theory to reaction-diffusion partial differential equations: adding diffusion preserves the global attraction of the ODE equilibrium.

Passivity gains and the `secant condition' for stability
Publication data: Systems & Control Letters 55(2006): 177-183.
Description: A generalization of the classical secant condition for the stability of cascades of scalar linear systems is provided for passive systems. The key is the introduction of a quantity that combines gain and phase information for each system in the cascade. For linear one-dimensional systems, the known result is recovered exactly.

Diagonal stability for a class of cyclic systems and applications
Coauthor(s): M. Arcak
Publication data: Automatica 42(2006): 1531-1537.
Description: This paper considers a class of systems with a cyclic structure that arises, among other examples, in dynamic models for certain biochemical reactions. We first show that a criterion for local stability, derived earlier in the literature, is in fact a necessary and sufficient condition for diagonal stability of the corresponding class of matrices. We then revisit a recent generalization of this criterion to output strictly passive systems, and recover the same stability condition using our diagonal stability result as a tool for constructing a Lyapunov function. Using this procedure for Lyapunov construction we exhibit classes of cyclic systems with sector nonlinearities and characterize their global stability properties.

Parameter estimation in models combining signal transduction and metabolic pathways: The dependent input approach
Coauthor(s): N. van Riel
Publication data: IEE Proc. Systems Biology 153 (2006): 263-274.
Description: Biological complexity and limited quantitative measurements impose severe challenges to standard engineering methodologies for systems identification. This paper presents an approach, justified by the theory of universal inputs for distinguishability, based on replacing unmodeled dynamics by fictitious `dependent inputs'. The approach is particularly useful in validation experiments, because it allows one to fit model parameters to experimental data generated by a reference (wild-type) organism and then testing this model on data generated by a variation (mutant), so long as the mutations only affect the unmodeled dynamics that produce the dependent inputs. As a case study, this paper addresses the pathways that control the nitrogen uptake fluxes in baker's yeast Saccharomyces cerevisiae enabling it to optimally respond to changes in nitrogen availability. Well-defined perturbation experiments were performed on cells growing in steady-state. Time-series data of extracellular and intracellular metabolites were obtained, as well as mRNA levels. A nonlinear model was proposed, and shown to be structurally identifiable given input/output data. The identified model correctly predicted the responses of different yeast strains and different perturbations.

A cooperative system which does not satisfy the limit set dichotomy
Coauthor(s): Y. Wang
Publication data: J. of Differential Equations 224(2006): 373-384.
Description: The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x < y, then either Omega(x) < Omega (y), or Omega(x) = Omega(y) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems.

Algorithmic and complexity results for decompositions of biological networks into monotone subsystems
Coauthor(s): B. DasGupta, G. Enciso, Y. Zhang
Publication data: BioSystems, to appear.
Description: A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graph-theoretic language, the problems can be recast in terms of maximal sign-consistent subgraphs. The theoretical results include polynomial-time approximation algorithms as well as constant-ratio inapproximability results. One of the algorithms, which has a worst-case guarantee of 87.9% from optimality, is based on the semidefinite programming relaxation approach of Goemans-Williamson. The algorithm was implemented and tested on a Drosophila segmentation network and an Epidermal Growth Factor Receptor pathway model.

Honey-pot constrained searching with local sensory information
Coauthor(s): Bhaskar DasGupta, Joao P. Hespanha, James Riehl
Publication data: Nonlinear Analysis 65(2006): 1773-1793.
Description: This paper investigates the problem of searching for a hidden target in a bounded region of the plane by an autonomous robot which is only able to use limited local sensory information. It proposes an aggregation-based approach to solve this problem, in which the continuous search space is partitioned into a finite collection of regions on which we define a discrete search problem and a solution to the original problem is obtained through a refinement procedure that lifts the discrete path into a continuous one. The resulting solution is in general not optimal but one can construct bounds to gauge the cost penalty incurred. The discrete version is formalized and an optimization problem is stated as a `reward-collecting' bounded-length path problem. NP-completeness and efficient approximation algorithms for various cases of this problem are discussed.

Adaptation and regulation with signal detection implies internal model
Publication data: Systems and Control Letters 50 (2003): 119-126.
Keywords: regulation - internal model - adaptation
Description: This note provides a simple result showing, under suitable technical assumptions, that if a system S adapts to a class of external signals U, then S must necessarily contain a subsystem which is capable of generating all the signals in U. It is not assumed that regulation is robust, nor is there a prior requirement for the system to be partitioned into separate plant and controller components. Instead, a "signal detection" capability is imposed. These weaker assumptions make the result better applicable to cellular phenomena such as the adaptation of E-coli chemotactic tumbling rate to constant concentrations.

Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2 (supplementary materials 2-4)
Coauthor(s): J.R. Pomerening and J.E. Ferrell, Jr
Publication data: Nature Cell Biology 5(2003): 346-351.
Keywords: cell cycle - oscillations - bistabilit y
Description: In the early embryonic cell cycle, Cdc2-cyclin B functions like an autonomous oscillator, at whose core is a negative feedback loop: cyclins accumulate and produce active mitotic Cdc2-cyclin B Cdc2 activates the anaphase-promoting complex (APC); the APC then promotes cyclin degradation and resets Cdc2 to its inactive, interphase state. Cdc2 regulation also involves positive feedback4, with active Cdc2-cyclin B stimulating its activator Cdc25 and inactivating its inhibitors Wee1 and Myt1. Under the correct circumstances, these positive feedback loops could function as a bistable trigger for mitosis, and oscillators with bistable triggers may be particularly relevant to biological applications such as cell cycle regulation. This paper examined whether Cdc2 activation is bistable, confirming that the response of Cdc2 to non-degradable cyclin B is temporally abrupt and switchlike, as would be expected if Cdc2 activation were bistable. It is also shown that Cdc2 activation exhibits hysteresis, a property of bistable systems with particular relevance to biochemical oscillators. These findings help establish the basic systems-level logic of the mitotic oscillator.

On the representation of switched systems with inputs by perturbed control systems
Coauthor(s): J.L. Mancilla-Aguilar, R. Garcia, Y. Wang
Publication data: Nonlinear Analysis: Theory, Methods & Applications 60(2005): 1111-1150.
Description: This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem.

Uniform stability properties of switched systems with switchings governed by digraphs
Coauthor(s): J.L. Mancilla-Aguilar, R. Garcia, Y. Wang
Publication data: Nonlinear Analysis: Theory, Methods Applications 63(2005): 472-490.
Description: This paper develops characterizations of various uniform stability properties of switched systems described by differential inclusions, and whose switchings are governed by a digraph. These characterizations are given in terms of stability properties of the system with restricted switchings and also in terms of Lyapunov functions.

Global attractivity, I/O monotone small-gain theorems, and biological delay systems
Coauthor(s): G. Enciso
Publication data: Discrete and Continuous Dynamical Systems 14(2006): 549-578.
Description: This paper further develops a method, originally introduced in a paper by Angeli and Sontag, for proving global attractivity of steady states in certain classes of dynamical systems. In this aproach, one views the given system as a negative feedback loop of a monotone controlled system. An auxiliary discrete system, whose global attractivity implies that of the original system, plays a key role in the theory, which is presented in a general Banach space setting. Applications are given to delay systems, as well as to systems with multiple inputs and outputs, and the question of expressing a given system in the required negative feedback form is addressed.

Global stabilization for systems evolving on manifolds
Coauthor(s): M. Malisoff and M. Krichman
Publication data: Journal of Dynamical and Control Systems 12(2006): 161-184.
Description: This paper shows that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since discontinuous feedbacks are allowed, solutions are understood in the ``sample and hold'' sense introduced by Clarke-Ledyaev-Sontag-Subbotin (CLSS). This work generalizes the CLSS Theorem, which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds relative to actuator errors, under small observation noise.

Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks
Coauthor(s): P. Berman and B. Dasgupta
Publication data: Discrete Applied Mathematics (Special Series on Computational Molecular Biology), to appear. (Preliminary conference version had appeared in Proc. 7th. Int. Workshop on Approximation Algorithms for Combinatorial Optimization Problems, Cambridge, MA, Aug. 2004, (K. Jansen, S. Khanna, J.D P. Rolim and D. Ron, eds.), LNCS 3122, Springer-Verlag, NY, 2004, pp. 39-50.)
Description: This paper investigates computational complexity aspects of a combinatorial problem that arises in the reverse engineering of protein and gene networks, showing relations to an appropriate set multicover problem with large "coverage" factor, and providing a non-trivial analysis of a simple randomized polynomial-time approximation algorithm for the problem. (See also the original "unravelling algorithm" paper for motivation, as well as these papers for: time-varying data, and noise issues.)

Monotone systems under positive feedback: Multistability and a reduction theorem
Coauthor(s): G. Enciso
Publication data: Systems and Control Letters 54 (2005): 159-168.
Description: For feedback loops involving single input, single output monotone systems with well-defined I/O characteristics, a previous paper 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. New results characterizing strong monotonicity of feedback loops involving cascades are also presented.

Exact computation of amplification for a class of nonlinear systems arising from cellular signaling pathways
Coauthor(s): Madalena Chaves
Publication data: Automatica, in press.
Description: A commonly employed measure of the signal amplification properties of an input/output system is its induced L2 norm, sometimes also known as H-infinity gain. In general, however, it is extremely difficult to compute the numerical value for this norm, or even to check that it is finite, unless the system being studied is linear. This paper describes a class of systems for which it is possible to reduce this computation to that of finding the norm of an associated linear system. In contrast to linearization approaches, a precise value, not an estimate, is obtained for the full nonlinear model. The class of systems that we study arose from the modeling of certain biological intracellular signaling cascades, but the results should be of wider applicability.

Steady-states of receptor-ligand dynamics: A theoretical framework
Coauthor(s): Madalena Chaves and Robert J. Dinerstein
Publication data: Journal of Theoretical Biology, 227(2004): 413-428.
Description: This paper studies aspects of the dynamics of a conventional mechanism of ligand-receptor interactions, with a focus on the stability and location of steady-states. A theoretical framework is developed, and, as an application, a minimal parametrization is provided for models for two- or multi-state receptor interaction with ligand. In addition, an "affinity quotient" is introduced, which allows an elegant classification of ligands into agonists, neutral agonists, and inverse agonists.
Keywords: multi-state receptor models --- agonist classes --- biochemical networks

Optimal length and signal amplification in weakly activated signal transduction cascades
Coauthor(s): Madalena Chaves and Robert J. Dinerstein
Publication data: J. Physical Chemistry 108(2004): 15311-15320.
Description: Weakly activated signaling cascades can be modeled as linear systems. The input-to-output transfer function and the internal gain of a linear system, provide natural measures for the propagation of the input signal down the cascade and for the characterization of the final outcome. The most efficient design of a cascade for generating sharp signals, is obtained by choosing all the off rates equal, and a "universal" finite optimal length.
Keywords: signal transduction pathways -- signal amplification and optimization -- H-infinity gain

On the stability of a model of testosterone dynamics
Coauthor(s): G. Enciso
Publication data: Journal of Mathematical Biology 49 (2004): 627-634.
Description: We prove the global asymptotic stability of a well-known delayed negative-feedback model of testosterone dynamics, which has been proposed as a model of oscillatory behavior. We establish stability (and hence the impossibility of oscillations) even in the presence of delays of arbitrary length.

Inference of signaling and gene regulatory networks by steady-state perturbation experiments: Structure and accuracy (supplementary materials here)
Coauthor(s): M. Andrec, B.N. Kholodenko, and R.M. Levy
Publication data: J. Theoretical Biology 232 (2005): 427-441.
Description: One of the fundamental problems of cell biology is the understanding of complex regulatory networks. Such networks are ubiquitous in cells, and knowledge of their properties is essential for the understanding of cellular behavior. This paper studies the effect of experimental uncertainty on the accuracy of the inferred structure of the networks determined using the "unravelling algorithm" (see also these papers on: time-varying data and computational complexity issues).

An analysis of a circadian model using the small-gain approach to monotone systems
Coauthor(s): D. Angeli
Publication data: Proc. IEEE Conf. Decision and Control, Bahamas, Dec. 2004, IEEE Publications, 2004, pp. 575-578.
Description: We show how certain properties of Goldbeter's original 1995 model for circadian oscillations can be proved mathematically. We establish global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, but, on the other hand, this stability persists even under arbitrary delays in the feedback loop. We are mainly interested in illustrating certain mathematical techniques, including the use of theorems concerning tridiagonal cooperative systems and the recently developed theory of monotone systems with inputs and outputs.

A tutorial on monotone systems - with an application to chemical reaction networks
Coauthor(s): P. De Leenheer and D. Angeli
Publication data: in Proc. 16th Int. Symp. Mathematical Theory of Networks and Systems (MTNS 2004)}, CD-ROM, WP9.1, Katholieke Universiteit Leuven.
Description: Monotone systems are dynamical systems for which the flow preserves a partial order. Some applications will be briefly reviewed in this paper. Much of the appeal of the class of monotone systems stems from the fact that roughly, most solutions converge to the set of equilibria. However, this usually requires a stronger monotonicity property which is not always satisfied or easy to check in applications. Following work of J.F. Jiang, we show that monotonicity is enough to conclude global attractivity if there is a unique equilibrium and if the state space satisfies a particular condition. The proof given here is self-contained and does not require the use of any of the results from the theory of monotone systems. We will illustrate it on a class of chemical reaction networks with monotone, but otherwise arbitrary, reaction kinetics.

Computational complexities of honey-pot searching with local sensory information
Coauthor(s): B. DasGupta and J.P. Hespanha
Publication data: Proceedings American Control Conf., Boston, June 2004, CD-ROM, ThA06.1, IEEE Publications, Piscataway.
Description: In this paper we investigate the problem of searching for a hidden target in a bounded region of the plane, by an autonomous robot which is only able to use limited local sensory information. We formalize a discrete version of the problem as a "reward-collecting" path problem and provide efficient approximation algorithms for various cases.

Aggregation-based approaches to honey-pot searching with local sensory information
Coauthor(s): B. DasGupta and J.P. Hespanha
Publication data: Proceedings American Control Conf., Boston, June 2004, CD-ROM, WeM17.4, IEEE Publications, Piscataway.
Description: We investigate the problem of searching for a hidden target in a bounded region by an autonomous agent that is only able to use limited local sensory information. We propose an aggregation-based approach to solve this problem, in which the continuous search space is partitioned into a finite collection of regions on which we define a discrete search problem. A solution to the original problem is then obtained through a refinement procedure that lifts the discrete path into a continuous one. The resulting solution is in general not optimal but one can construct bounds to gauge the cost penalty incurred.

Separation principles for input-output and integral-input to state stability
Coauthor(s): D. Angeli, B. Ingalls, and Y. Wang
Publication data: SIAM J. Control and Optimization 43(2004): 256-276.
Keywords: IOSS - iISS
Description: We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of "separation principles" for input-output-to-state stability -- characterizations of the property in terms of weaker stability notions. When applied to the closely related notion of integral ISS, these characterizations yield analogous results.

Asymptotic controllability implies input to state stabilization
Coauthor(s): M. Malisoff and L. Rifford
Publication data: Siam J. Control and Optimization, 42(2004): 2221-2238.
Keywords: ISS - feedback - control-Lyapunov functions
Description: The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed. Our controllers have the property that they tolerate small observation noise as well.

Balancing at the border of instability
Coauthor(s): L. Moreau
Publication data: Physical Review E 68(2003): 020901(1-4).
Keywords: bifurcations - adaptive control - neural integration
Description: Some biological systems operate at the critical point between stability and instability and this requires a fine-tuning of parameters. We bring together two examples from the literature that illustrate this: neural integration in the nervous system and hair cell oscillations in the auditory system. In both examples the question arises as to how the required fine-tuning may be achieved and maintained in a robust and reliable way. We study this question using tools from nonlinear and adaptive control theory. We illustrate our approach on a simple model which captures some of the essential features of neural integration. As a result, we propose a large class of feedback adaptation rules that may be responsible for the experimentally observed robustness of neural integration. We mention extensions of our approach to the case of hair cell oscillations in the ear.

Feedback tuning of bifurcations
Coauthor(s): L. Moreau and M. Arcak
Publication data: Systems and Control Letters 50 (2003) 229-239.
Keywords: bifurcations - adaptive control
Description: This paper studies a feedback regulation problem that arises in at least two different biological applications. The feedback regulation problem under consideration may be interpreted as an adaptive control problem for tuning bifurcation parameters, and it has not been studied in the control literature. The goal of the paper is to formulate this problem and to present some preliminary results.

On predator-prey systems and small-gain theorems
Coauthor(s): P. De Leenheer and D. Angeli
Publication data: Mathematical Biosciences and Engineering 2(2005): 25-42.
Keywords: small-gain theorem - Lotka-Volterra models
Description: This paper deals with an almost global attractivity result for Lotka-Volterra systems with predator-prey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular input-output properties. We use a small-gain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global attractivity result. It provides sufficient conditions to rule out oscillatory or more complicated behavior which is often observed in predator-prey systems.

Untangling the wires: a novel strategy to trace functional interactions in signaling and gene networks
Coauthor(s): B.N. Kholodenko, A. Kiyatkin, F. Bruggeman, H. Westerhoff, and J. Hoek
Publication data: Proc. Natl. Acad. Sci. USA 99(2002): 12841-12846.
Keywords: protein networks - gene networks - structure identification
Description: Emerging technologies have enabled the acquisition of large genomics and proteomics data sets. This paper proposes a novel quantitative method for determining functional interactions in cellular signaling and gene networks. It can be used to explore cell systems at a mechanistic level, or applied within a modular framework, which dramatically decreases the number of variables to be assayed. The topology and strength of network connections are retrieved from experimentally measured network responses to successive perturbations of all modules. In addition, the method can reveal functional interactions even when the components of the system are not all known, in which casesome connections retrieved by the analysis will not be direct but correspond to the interaction routes through unidentified elements. The method is tested and illustrated using computer-generated responses of a modeled MAPK cascade and gene network. (See also these papers on: time-varying data, noise, and computational complexity issues).

Asymptotic amplitudes and cauchy gains: A small-gain principle and an application to inhibitory biological feedback
Publication data: Systems and Control Letters 47(2002): 167-179.
Keywords: small-gain theorem - inhibitory feedback - stability - oscillations - MAPK cascades
Description: The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops. An application in biology (MAPK signaling) is worked out in detail.

For differential equations with r parameters, 2r+1 experiments are enough for identification
Publication data: J. Nonlinear Science 12 (2002): 553-583.
Keywords: observability - identification - dynamical systems with parameters - chemical reaction constants
Description: Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical kinetics, and supposing that one may at any time measure the values of some of the variables and possibly apply external inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially available about the parameters? This paper shows that the best possible answer (assuming exact measurements) is 2r+1 experiments, where r is the number of parameters.

Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction
Publication data: IEEE Trans. Autom. Control 46(2001): 1028-1047.
Erratum (pdf file; appeared in IEEE Trans. Autom. Control 47(2002):705)
Keywords: kinetic proofreading - chemical reactors - stability - deficiency-zero networks
Description: This paper deals with the theory of structure, stability, robustness, and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks. The results given here extend, but are also heavily based upon, certain previous work by Feinberg, Horn, and Jackson, of which a self-contained and streamlined exposition is included. The theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by McKeithan for T-cell receptor signal transduction.

Stability and stabilization: Discontinuities and the effect of disturbances
Publication data: Nonlinear Analysis, Differential Equations, and Control (Proc. NATO Advanced Study Institute, Montreal, Jul/Aug 1998; F.H. Clarke and R.J. Stern, eds.), Kluwer, 1999, pp. 551-598.
Keywords: nonlinear stabilization - control-Lyapunov functions - feedback - input-to-state stability - disturbances - measurement errors
Description: In this expository paper, we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows:
Review of Stability and Asymptotic Controllability
The Problem of Stabilization
Obstructions to Continuous Stabilization
Control-Lyapunov Functions and Artstein's Theorem
Discontinuous Feedback
Nonsmooth CLF's
Insensitivity to Small Measurement and Actuator Errors
Effect of Large Disturbances: Input-to-State Stability
Comments on Notions Related to ISS
Note: if you are looking for the following paper: Nonlinear Feedback Stabilization Revisited in Dynamical Systems, Control, Coding, Computer Vision, Proc. Math. Theory of Networks and Systems (MTNS98), Padova, July 1998" (G. Picci and D.S. Gilliam, eds.), Birkhauser Verlag, Basel, 1999, pp. 223-262, please look at this paper instead, as it has a more complete exposition of the same results.

Asymptotic controllability and input-to-state stabilization: The effect of actuator errors
Publication data: in "Optimal Control, Stabilization, and Nonsmooth Analysis" (de Queiroz, M., M. Malisoff, and P. Wolenski, eds.), Springer-Verlag, Heidelberg, 2004, pp. 155-171.
Coauthor(s): M. Malisoff
Keywords: ISS - feedback - control-Lyapunov functions
Description: We discuss several issues related to the stabilizability of nonlinear systems. First, for continuously stabilizable systems, we review constructions of feedbacks that render the system input-to-state stable with respect to actuator errors. Then, we discuss a recent paper which provides a new feedback design that makes globally asymptotically controllable systems input-to-state stable to actuator errors and small observation noise. We illustrate our constructions using the nonholonomic integrator, and discuss a related feedback design for systems with disturbances.

Well-defined steady-state response does not imply CICS
Coauthor(s): E.P. Ryan
Publication data: Systems and Control Letters 55(2006): 707-710.
Keywords: - nonlinear stability - steady-state response
Description: Systems for which each constant input gives rise to a unique globally attracting equilibrium are considered. A counterexample is provided to show that inputs which are only asymptotically constant may not result in states converging to equilibria (failure of the converging-input converging state, or ``CICS'' property).

The ISS philosophy as a unifying framework for stability-like behavior
Publication data: in Nonlinear Control in the Year 2000 (Volume 2) (Lecture Notes in Control and Information Sciences, A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, eds.), Springer-Verlag, Berlin, 2000, pp. 443-468.
Keywords: ISS - input/output stability - detectability - nonlinear control - Lyapunov functions - dissipation
Description: (This is an expository paper prepared for a plenary talk given at the Second Nonlinear Control Network Workshop, Paris, June 9, 2000.) The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography is also included, listing extensions, applications, and other current work.

A small-gain theorem for almost global convergence of monotone systems
Coauthor(s): David Angeli and Patrick de Leenheer
Publication data: Systems and Control Letters 52(2004): 407-414.
Description: 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 flow and potentially destabilizes the resulting closed-loop system.
Keywords: almost global stability -- monotone control systems

Uniform global asymptotic stability of differential inclusions
Coauthor(s): David Angeli, Brian Ingalls and Yuan Wang
Publication data: Journal of Dynamical and Control Systems 10(2004): 391-412.
Description: The stability of differential inclusions defined by locally Lipschitz compact valued maps is addressed. It is shown that if such a differential inclusion is globally asymptotically stable, then in fact it is uniformly globally asymptotically stable (with respect to initial states in compacts). This statement is trivial for differential equations, but here we provide the extension to compact (not necessarily convex) valued differential inclusions. The main result is presented in a context which is useful for control-theoretic applications: a differential inclusion with two outputs is considered, and the result applies to the property of global error detectability.
Keywords: stability -- differential inclusions -- asymptotic stability

Global stability in a chemostat with multiple nutrients
Coauthor(s): Patrick De Leenheer, Simon A. Levin, Christopher A. Klausmeier
Publication data: J. Mathematical Biology 52(2006): 419-438.
Description: We study a single species in a chemostat, limited by two nutrients, and separate nutrient uptake from growth. For a broad class of uptake and growth functions it is proved that a nontrivial equilibrium may exist. Moreover, if it exists it is unique and globally stable, generalizing a previous result by Legovic and Cruzado.

Some new directions in control theory inspired by systems biology
Publication data: Systems Biology 1(2004): 9-18.
Keywords: systems biology - control theory - cellular signaling
Description: This paper, addressed primarily to engineers and mathematicians with an interest in control theory, argues that entirely new theoretical problems arise naturally when addressing questions in the field of systems biology. Examples from the author's recent work are used to illustrate this point.

Crowding effects promote coexistence in the chemostat
Coauthor(s): David Angeli and Patrick de Leenheer
Publication data: Journal of Mathematical Analysis and Applications 319 (2006): 48-60.
(Summary in: First Multidisciplinary International Symposium on Positive Systems: Theory And Applications (Posta 2003), Rome, August 2003 (L. Benvenuti, A. De Santis, and L. Farina, eds.), Springer-Verlag, Heidelberg, 2003 pp. 167-174.)
Description: We provide an almost-global stability result for a particular chemostat model, in which crowding effects are taken into consideration. The model can be rewritten as a negative feedback interconnection of two monotone i/o systems with well-defined characteristics, which allows the use of a small-gain theorem for feedback interconnections of monotone systems. This leads to a sufficient condition for almost-global stability, and we show that coexistence occurs in this model if the crowding effects are large enough.
Keywords: chemostat -- small-gain theorems -- stability

Measurement to error stability: a notion of partial detectability for nonlinear systems
Coauthor(s): Brian Ingalls and Yuan Wang
Publication data: in Proc. IEEE Conf. Decision and Control, Las Vegas, Dec. 2002, IEEE Publications, 2002, pp. 3946-3951.
Description: For systems whose output is to be kept small (thought of as an error output), the notion of input to output stability (IOS) arises. Alternatively, when considering a system whose output is meant to provide information about the state (i.e. a measurement output), one arrives at the detectability notion of output to state stability (OSS). Combining these concepts, one may consider a system with two types of outputs, an error and a measurement. This leads naturally to a notion of partial detectability which we call measurement to error stability (MES). This property characterizes systems in which the error signal is detectable through the measurement signal. This paper provides a partial Lyapunov characterization of the MES property. A closely related property of stability in three measures (SIT) is introduced, which characterizes systems for which the error decays whenever it dominates the measurement. The SIT property is shown to imply MES, and the two are shown to be equivalent under an additional boundedness assumption. A nonsmooth Lyapunov characterization of the SIT property is provided, which yields the partial characterization of MES. The analysis is carried out on systems described by differential inclusions -- implicitly incorporating a disturbance input with compact value-set.
Keywords: regulation ISS input to state stability

An infinite-time relaxation theorem for differential inclusions Erratum
Coauthor(s): Brian Ingalls and Yuan Wang
Publication data: Proceedings of the AMS 131(2003): 487-499.
Keywords: Lipschitz differential inclusions - relaxation - Filippov's Lemma
Description: The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition.

A relaxation theorem for differential inclusions with applications to stability properties
Coauthor(s): Brian Ingalls and Yuan Wang
Publication data: in Mathematical Theory of Networks and Systems (D. Gilliam and J. Rosenthal, eds.), Electronic Proceedings of MTNS-2002 Symposium held at the University of Notre Dame, August 2002.
Description: The fundamental Filippov--Wazwski Relaxation Theorem states that the solution set of an initial value problem for a locally Lipschitz inclusion is dense in the solution set of the same initial value problem for the corresponding relaxation inclusion on compact intervals. In a recent paper of ours, a complementary result was provided for inclusions with finite dimensional state spaces which says that the approximation can be carried out over non-compact or infinite intervals provided one does not insist on the same initial values. This note extends the infinite-time relaxation theorem to the inclusions whose state spaces are Banach spaces. To illustrate the motivations for studying such approximation results, we briefly discuss a quick application of the result to output stability and uniform output stability properties.
Keywords: differential inclusions -- relaxation theorems

Nonlinear observability notions and stability of switched systems
Coauthor(s): Joao P. Hespanha, Daniel Liberzon, and David Angeli
Publication data: IEEE Trans. Autom. Control 50 (2005): 154- 168.
(Preliminary revsion: Nonlinear observability and an invariance principle for switched systems
with Joao P. Hespanha and Daniel Liberzon, in Proc. IEEE Conf. Decision and Control, Las Vegas, Dec. 2002, IEEE Publications, 2002, pp. 4300-4305.)
Description: This paper proposes several definitions of observability for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunov-like sufficient condition for observability is also obtained. As an application, we prove several variants of LaSalle's stability theorem for switched nonlinear systems. These results are demonstrated to be useful for control design in the presence of switching as well as for developing stability results of Popov type for switched feedback systems.
Keywords: observability -- switched nonlinear systems -- LaSalle invariance

Output-input stability and minimum-phase nonlinear systems
Coauthor(s): Daniel Liberzon and A. Stephen Morse
Publication data: IEEE Trans. Autom. Control 47(2002): 422 -436.
Keywords: nonlinear control - minimum phase - adaptive control - ISS - detectability
Description: This paper introduces and studies a new definition of the minimum-phase property for general smooth nonlinear control systems.
The definition does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the ``input-to-state stability'' philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions.
The class of minimum-phase systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts.
As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control.

An example of a GAS system which can be destabilized by an integrable perturbation
Coauthor(s): M. Krichman
Publication data: IEEE Transactions on Automatic Control 48(2003): 1046-1049.
Keywords: integral stability
Description: A construction is given of a globally asymptotically stable time-invariant system which can be destabilized by some integrable perturbation. Besides its intrinsic interest, this serves to provide counterexamples to an open question regarding Lyapunov functions.

A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections
Coauthor(s): Brian Ingalls
Publication data: J. Franklin Institute 339(2002): 211-229.
Keywords: small-gain theorem - ISS - interconnections
Description: A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.

A remark on the converging-input converging-state property
Publication data: IEEE Trans. Autom. Control 48(2003): 313-314.
Description: Suppose that an equilibrium is asymptotically stable when external inputs vanish. Then, every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system, must also converge to the equilibrium. This "well-known" but hard-to-cite fact is proved and slightly generalized here.

Input-output-to-state stability
Coauthor(s): Mikhail Krichman and Yuan Wang
Publication data: SIAM J Control 39(2001): 1874-1928.
Keywords: detectability - norm observers - Lyapunov functions - ISS
Description: This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the input-output-to-state stability property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of "norm-estimators", and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.

Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation
Coauthor(s): Daniel Liberzon and Yuan Wang
Publication data: Systems and Control Letters 46(2002): 111-127. (Preliminary version was in a 99ACC paper.)
Erratum
Keywords: integral input to state stability - control-Lyapunov functions
Description: We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed.

Learning complexity dimensions for a continuous-time control system
Coauthor(s): Pirkko Kuusela and Daniel Ocone
Publication data: SIAM J. Control and Optimization 43(2004): 872-898.
Keywords: linear systems identification, learning theory, VC dimension
Description: This paper takes a computational learning theory approach to a problem of linear systems identification. It is assumed that input signals have only a finite number k of frequency components, and systems to be identified have dimension no greater than n. The main result establishes that the sample complexity needed for identification scales polynomially with n and logarithmically with k.

A unifying integral ISS framework for stability of nonlinear cascades
Coauthor(s): Murat Arcak and David Angeli
Publication data: SIAM Journal Control and Optimization 40(2002): 1888-1904.
Description: We analyze nonlinear cascades in which the driven subsystem is integral ISS, and characterize the admissible integral ISS gains for stability. This characterization makes use of the convergence speed of the driving subsystem, and allows a larger class of gain functions when the convergence is faster. We show that our integral ISS gain characterization unifies different approaches in the literature which restrict the nonlinear growth of the driven subsystem and the convergence speed of the driving subsystem.

Characterizations of detectability notions in terms of discontinuous dissipation functions
Coauthor(s): Mikhail Krichman
Publication data: Int. J. Control, 75(2002): 882 - 900.
Keywords: detectability - IOSS - nonsmooth Lyapunov
Description: We consider a new Lyapunov-type characterization of detectability for nonlinear systems without controls, in terms of lower-semicontinuous (not necessarily smooth, or even continuous) dissipation functions, and prove its equivalence to the GASMO (global asymptotic stability modulo outputs) and UOSS (uniform output-to-state stability) properties studied in previous work.
The result is then extended to provide a construction of a discontinuous dissipation function characterization of the IOSS (input-to-state stability) property for systems with controls. This paper complements a recent result on smooth Lyapunov characterizations of IOSS.
The utility of non-smooth Lyapunov characterizations is illustrated by application to a well-known transistor network example.

Lyapunov characterizations of input to output stability
Coauthor(s): Yuan Wang
Publication data: SIAM J. Control and Optimization 39 (2001) 226-249.
Keywords: input/output stability -- ISS -- nonlinear control -- robust control
Description: This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated.

Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems
Coauthor(s): D. Nesic and A. Teel
Publication data: Systems and Control Letters 38(1999): 49-60.
Keywords: ISS - sampling - KL functions
Description: We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time system and a K function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a closed-loop system, implies uniform ISS of the sampled-data nonlinear system; our results serve to quantify these facts by means of comparison functions. Our results can be used as an alternative to prove and extend results of Aeyels et al and extend some results by Chen et al to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate.

Clocks and insensitivity to small measurement errors
Publication data: Control, Optimisation and Calculus of Variations 4(1999): 537-557. (Preliminary version had appeared as "Feedback insensitive to small measurement errors" in Proc. IEEE Conf. Decision and Control, Phoenix, Dec. 1999, IEEE Publications, 1999, pp. 2661-2666.)
Keywords: hybrid systems - discontinuous feedback - measurement noise
Description: This paper provides a precise result which shows that insensitivity to small measurement errors in closed-loop stabilization can be attained provided that the feedback controller ignores observations during small time intervals.

A polynomial-time algorithm for checking equivalence under certain semiring congruences motivated by the state-space isomorphism problem for hybrid systems
Coauthor(s): Bhaskar DasGupta
Publication data: Theoretical Computer Science 262(2001): 161-189. (Summarized version: "A polynomial-time algorithm for an equivalence problem which arises in hybrid systems theory", in Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1998)
Keywords: hybrid systems - complexity - state-space equivalence
Description: The area of hybrid systems concerns issues of modeling, computation, and control for systems which combine discrete and continuous components. The subclass of piecewise linear (PL) systems provides one systematic approach to discrete-time hybrid systems, naturally blending switching mechanisms with classical linear components. PL systems model arbitrary interconnections of finite automata and linear systems.
Tools from automata theory, logic, and related areas of computer science and finite mathematics are used in the study of PL systems, in conjunction with linear algebra techniques, all in the context of a "PL algebra" formalism. PL systems are of interest as controllers as well as identification models.
Basic questions for any class of systems are those of equivalence, and, in particular, if state spaces are equivalent under a change of variables. This paper studies this state-space equivalence problem for PL systems. The problem was known to be decidable, but its computational complexity was potentially exponential; here it is shown to be solvable in polynomial-time.

Input-to-state stability for discrete-time nonlinear systems
Coauthor(s): Zhong-Ping and Yuan Wang
Publication data: Proc. 14th IFAC World Congress (Beijing), Vol E, pp. 277-282, 1999.
Keywords: input to state stability - discrete-time
Description: This paper studies the input-to-state stability (ISS) property for discrete-time nonlinear systems. We show that many standard ISS results may be extended to the discrete-time case. More precisely, we provide a Lyapunov-like sufficient condition for ISS, and we show the equivalence between the ISS property and various other properties, as well as provide a small gain theorem.

On integral-input-to-state stabilization
Coauthor(s): Daniel Liberzon and Yuan Wang
Publication data: Proc. American Control Conf, San Diego, June 1999, pp. 1598-1602.
Keywords: integral input to state stability - control-Lyapunov functions
Description: This paper continues the investigation of the recently introduced integral version of input-to-state stability (iISS). We study the problem of designing control laws that achieve iISS disturbance attenuation. The main contribution is an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The results are compared and contrasted with the ones available for the ISS case.

Systems and Control Letters 40(2000): 247-60. (Preliminary version in: Universal formulas for CLF's with respect to Minkowski balls, Proc. American Control Conf, San Diego, June 1999, pp. 3033-3037.)

Finite gain stabilization of discrete-time linear systems subject to actuator saturation
Coauthor(s): Xiangyu Bao and Zongli Lin
Publication data: Automatica 36(2000): 269-277.
Keywords: discrete-time systems - saturation - input-to-state stability - ISS
Description: It is shown that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain lp stabilization can be achieved by linear output feedback, for all p>1. An explicit construction of the corresponding feedback laws is given. The feedback laws constructed also result in a closed-loop system that is globally asymptotically stable, and in an input-to-state estimate.

New characterizations of input to state stability
Coauthor(s): Y. Wang
Publication data: IEEE Trans. Autom. Control 41(1996): 1283-1294.
Keywords: - nonlinear control - input-to-state stability - continuous-time systems
Description: We present new characterizations of the Input to State Stability property. As a consequence of these results, we show the equivalence between the ISS property and several (apparent) variations proposed in the literature.

Comments on integral variants of ISS
Publication data: Systems and Control Letters 34 (1998): 93-100.
Keywords: - nonlinear control - input-to-state stability - continuous-time systems
Description: This note discusses two integral variants of the input-to-state stability (ISS) property, which represent nonlinear generalizations of L2 stability, in much the same way that ISS generalizes L-infinity stability. Both variants are equivalent to ISS for linear systems. For general nonlinear systems, it is shown that one of the new properties is strictly weaker than ISS, while the other one is equivalent to it. For bilinear systems, a complete characterization is provided of the weaker property. An interesting fact about functions of type KL is proved as well.

Input-to-state stabilization of linear systems with positive outputs
Coauthor(s): D. Nesic
Publication data: Systems and Control Letters 35 (1998), pp. 245-255.
Keywords: linear systems - stabilization - input-to-state stability
Description: This paper considers the problem of stabilization of linear systems for which only the magnitudes of outputs are measured. It is shown that, if a system is controllable and observable, then one can find a stabilizing controller, which is robust with respect to observation noise (in the ISS sense).

Meagre functions and asymptotic behaviour of dynamical systems
Coauthor(s): W. Desch, H. Logemann, and E.P. Ryan
Publication data: J. Nonlinear Analysis 44(2001): 1087-1109.
Keywords: - nonlinear control - invariance principle - continuous-time systems
Description: A measurable function x from a subset J of R into a metric space X is said to be C-meagre if C is non-empty subset of X and, for every closed subset K of X disjoint from C, the preimage of K under x has finite Lebesgue measure. This concept of meagreness, applied to trajectories, is shown to provide a unifying framework which facilitates a variety of characterizations, extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems.

Asymptotic controllability implies feedback stabilization
Coauthor(s): F.H. Clarke, Yu.S. Ledyaev, A.I. Subbotin
Publication data: IEEE Trans. Autom. Control 42 (1997): 1394-1407.
(Also available: preliminary version appeared in Proc. Conf. on Information Sciences and Systems (CISS 96), Princeton, NJ, 1996, pp. 1232-1237.)
Keywords: - nonlinear control - control Lyapunov functions - nonsmooth analysis - continuous-time systems
Description: It is shown that every asymptotically controllable system can be stabilized by means of some (discontinuous) feedback law. One of the contributions of the paper is in defining precisely the meaning of stabilization when the feedback rule is not continuous. The main ingredients in our construction are: (a) the notion of control-Lyapunov function, (b) methods of nonsmooth analysis, and (c) techniques from positional differential games.

A Lyapunov characterization of robust stabilization
Coauthor(s): Yu.S. Ledyaev
Publication data: J. Nonlinear Analysis 37(1999): 813-840.
Keywords: - nonlinear control - control Lyapunov functions - nonsmooth analysis - robust control - continuous-time systems
Description: One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of (in general, discontinuous) feedback stabilizers which are insensitive to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (possibly) discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances.

Analog neural nets with Gaussian or other common noise distributions cannot recognize arbitrary regular languages
Coauthor(s): W. Maass
Publication data: Neural Computation 11(1999): 771-782.
Keywords: - recurrent neural networks - probabilistic languages
Description: We consider recurrent analog neural nets where the output of each gate is subject to Gaussian noise, or any other common noise distribution that is nonzero on a large set. We show that many regular languages cannot be recognized by networks of this type, and we give a precise characterization of those languages which can be recognized. This result implies severe constraints on possibilities for constructing recurrent analog neural nets that are robust against realistic types of analog noise. On the other hand we present a method for constructing feedforward analog neural nets that are robust with regard to analog noise of this type.

Recurrent neural networks: Some systems-theoretic aspects
Publication data: in Dealing with Complexity: a Neural Network Approach (M. Karny, K. Warwick, and V. Kurkova, eds.), Springer-Verlag, London, 1997, pp. 1-12.
Keywords: - nonlinear systems theory (realization, observability, etc) - neural networks - recurrent (neural nets)
Description: This paper provides an exposition of some recent results regarding system-theoretic aspects of continuous-time recurrent (dynamic) neural networks with sigmoidal activation functions. The class of systems is introduced and discussed, and a result is cited regarding their universal approximation properties. Known characterizations of controllability, observability, and parameter identifiability are reviewed, as well as a result on minimality. Facts regarding the computational power of recurrent nets are also mentioned.

A learning result for continuous-time recurrent neural networks
Publication data: Systems and Control Letters 34 (1998): 151-158.
Keywords: - PAC learning - VC dimension - identification - neural networks - recurrent (neural nets)
Description: The following learning problem is considered, for continuous-time recurrent neural networks having sigmoidal activation functions. Given a ``black box'' representing an unknown system, measurements of output derivatives are collected, for a set of randomly generated inputs, and a network is used to approximate the observed behavior. It is shown that the