On optimal e ∞ to e ∞ filtering

The ultimate proof of our understanding of natural or technological systems is reflected in our ability to control them. Although control theory offers mathematical tools for steering engineered and natural systems towards a desired state, a framework to control complex self-organized systems is lacking. Here we develop analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system's entire dynamics. We apply these tools to several real networks, finding that the number of driver nodes is determined mainly by the network's degree distribution. We show that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control, but that dense and homogeneous networks can be controlled using a few driver nodes. Counterintuitively, we find that in both model and real systems the driver nodes tend to avoid the high-degree nodes.

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Controllability of complex networks

The complex structured singular value

Introduction Fundamentals of biochemical modeling Balance equations Rate laws Generalized mass-action kinetics Various enzyme kinetic rate laws Thermodynamic flow-force relationships Power-law approximation Steady states of biochemical networks General considerations Stable and unstable steady states Multiple steady states Metabolic oscillations Background Mathematical conditions for oscillations Glycolytic oscillations Models of intracellular calcium oscillations A simple three-variable model with only monomolecular and bimolecular reactions Possible physiological significance of oscillations Stoichiometric analysis Conservation relations Linear dependencies between the rows of the stoichiometry matrix Non-negative flux vectors Elementary flux modes Thermodynamic aspects A generalized Wegscheider condition Strictly detailed balanced subnetworks Onsager's reciprocity reactions for coupled enyme reactions Time hierarchy in metabolism Time constants The quasi-steady-state approximation The Rapid equilibrium approximation Modal analysis Metabolic control analysis Basic definitions A systematic approach Theorems of metabolic control analysis Summation theorems Connectivity theorems Calculation of control coefficients using the theorems Geometrical interpretation Control analysis of various systems General remarks Elasticity coefficients for specific rate laws Control coefficients for simple hypothetical pathways Unbranched chains A branched system Control of erythrocyte energy metabolism The reaction system Basic model Interplay of ATP production and ATP consumption Glycolytic energy metabolism and osmotic states A simple model of oxidative phosphorylation A three-step model of serine biosynthesis Time-dependent control coefficients Are control coefficients always parameter independent? Posing the problem A system without conserved moieties A system with a conserved moiety A system including dynamic channeling Normalized versus non-normalized coefficients Analysis in terms of variables other than steady-state concentrations and fluxes General analysis Concentration ratios and free-energy-differences as state variables Entropy production as response variable Control of transient times Control of oscillations A second-order approach A quantitative approach to metabolic regulations Co-response coefficients Fluctuations of internal variables versus parameter perturbations Internal response coefficients Rephrasing the basic equations of metabolic control analysis in terms of co-response coefficients and internal response coefficients Control within and between subsystems Modular approach Overall elasticities Overall control coefficients Flux control insusceptibility Control exerted by elementary steps in enzyme catalysis Control analysis of metabolic channeling Comparison of metabolic control analysis and power-law formalism Computational aspects Application of optimization methods and the interrelation with evolution Optimization of the catalytic properties of single enzymes Basic assumptions Optimal values of elementary rate constants Optimal Michaelis constants Optimization of multienzyme systems Maximization of steady-state flux Influence of osmotic constraints and minimization of intermediate concentrations Minimization of transient times Optimal stoichiometries.

The Regulation of Cellular Systems

Necessary and sufficient conditions are found for there to exist a robust controller for a linear, time-invariant, multivariable system (plant) so that asymptotic tracking/regulation occurs independent of input disturbances and arbitrary perturbations in the plant parameters of the system. In this problem, the class of plant parameter perturbations allowed is quite large; in particular, any perturbations in the plant data are allowed as long as the resultant closed-loop system remains stable. A complete characterization of all such robust controllers is made. It is shown that any robust controller must consist of two devices 1) a servocompensator and 2) a stabilizing compensator. The servocompensator is a feedback compensator with error input consisting of a number of unstable subsystems (equal to the number of outputs to be regulated) with identical dynamics which depend on the disturbances and reference inputs to the system. The sorvocompensator is a compensator in its own right, quite distinct from an observer and corresponds to a generalization of the integral controller of classical control theory. The sole purpose of the stabilizing compensator is to stabilize the resultant system obtained by applying the servocompensator to the plant. It is shown that there exists a robust controller for "almost all" systems provided that the number of independent plant inputs is not less than the number of independent plant outputs to be regulated, and that the outputs to be regulated are contained in the measurable outputs of the system; if either of these two conditions is not satisfied, there exists no robust controller for the system.

The robust control of a servomechanism problem for linear time-invariant multivariable systems

The authors present a framework for dealing with continuous-time periodic systems. The main tool is a lifting technique which provides a strong correspondence between continuous-time periodic systems and certain types of discrete-time time-invariant systems with infinite-dimensional input and output spaces. Despite the infinite dimensionality of the input and output spaces, a lifted system has a finite-dimensional state space if the original system does. This fact permits rather constructive methods for analyzing these systems. As a demonstration of the utility of this framework, the authors use it to describe the continuous-time (i.e., intersample) behavior of sampled-data systems, and to obtain a complete solution to the problem of parameterizing all controllers that constrain the L/sup 2/-induced norm of a sampled-data system to within a certain bound. >

A general framework for linear periodic systems with applications to H/sup infinity / sampled-data control

Linear multivariable control, a geometric approach

This paper extends the results of Lin on structural controllability of single-input linear systems to multiinput linear systems. An interesting byproduct of this extension is an application of generic analysis to the determination of the rank of a structured matrix. The algorithm for computing generic rank is applicable to various problems other than the controllability problem studied here.

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Structural controllability of multiinput linear systems

The problem of designing a compensator to obtain arbitrary pole placement in the system consisting of the plant and compensator in cascade is considered. The design uses only those state variables which can be measured. It is shown that for a controllable observable plant a compensator of order \beta = \min(\nu_{c} - 1, \nu_{o} - 1) is sufficient to achieve this result. Here \nu_{c}(\nu_{o}) is the controllability (observability) index of the plant. This result is obtained by first showing that any multi-input multi-output linear time-invariant system may be made controllable (observable) from a single input (output) using only output feedback. The main result is then proved in a constructive manner which explicitly relates the compensator parameters to the coefficients of the desired characteristic polynomial.

Pole placement using dynamic compensators

Given an interconnection of a nominal discrete-time plant and a stabilizing controller together with structured, norm-bounded, nonlinear/time-varying perturbations, necessary and sufficient conditions for robust stability and performance of the system are provided. It is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem. The resulting stability robustness problem is shown to be equivalent to a simple algebraic one, the solution of which provides the desired necessary and sufficient conditions for performance/stability robustness. These conditions provide an effective tool for robustness analysis and can be applied to a large class of problems. In particular, it is shown that some known results can be obtained immediately as special cases of these conditions. >

J.B. Pearson

Papers

A general framework for linear periodic systems with applications to H/sup infinity / sampled-data control

Linear multivariable control, a geometric approach

Structural controllability of multiinput linear systems

Pole placement using dynamic compensators

Performance robustness of discrete-time systems with structured uncertainty