Infinite-dimensional State Space Research Articles

Parametric generation of robust chaos with time-delayed feedback and modulated pump source

We consider a chaos generator composed of two parametrically coupled oscillators whose natural frequencies differ by factor of two. The system is driven by modulated pump source on the third harmonic of the basic frequency, and on each next period of pumping the excitation of the oscillator of doubled frequency is stimulated by the signal from the oscillator of the basic frequency undergoing quadratic nonlinear transformation and time delay. Using qualitative analysis and numerical results, we argue that chaotic dynamics in the system corresponds to hyperbolic strange attractor. It is a kind of Smale–Williams solenoid embedded in the infinite-dimensional state space of the stroboscopic map of the time-delayed system.

Unambiguous state discrimination in high dimensions

We propose methods for the unambiguous discrimination of quantum states encoded in the spatial profile of light modes, in particular in the orbital angular momentum of light. Perfect discrimination between quantum states is possible only between orthogonal states of a known basis system. Generalised measurement strategies have been developed for the distinction between non-orthogonal states in a two-dimensional state space, e.g. the polarisation of light. Here we consider information encoded in superpositions of orbital angular momentum states of light, defined within an infinite-dimensional state space. Generalised measurements will be an important addition to quantum communication in high-dimensional spaces.

Pursuit of food versus pursuit of information in a Markovian perception–action loop model of foraging

Efficient coding, redundancy reduction, and other information theoretic optimization principles have successfully explained the organization of many biological phenomena, from the physiology of sensory receptive fields to the variability of certain DNA sequence ensembles. Here we examine the hypothesis that behavioral strategies that are optimal for survival must necessarily involve efficient information processing, and ask whether there can be circumstances in which deliberately sacrificing some information can lead to higher utility? To this end, we present an analytically tractable model for a particular instance of a perception–action loop: a creature searching for a randomly moving food source confined to a 1D ring world. The model incorporates the statistical structure of the creature's world, the effects of the creature's actions on that structure, and the creature's strategic decision process. The underlying model takes the form of a Markov process on an infinite dimensional state space. To analyze it we construct an exact coarse graining that reduces the model to a Markov process on a finite number of “information states”. This mathematical technique allows us to make quantitative comparisons between the performance of an information-theoretically optimal strategy with other candidate search strategies on a food gathering task. We find that1.Information optimal search does not necessarily optimize utility (expected food gain).2.The rank ordering of search strategies by information performance does not predict their ordering by expected food obtained.3.The relative advantage of different strategies depends on the statistical structure of the environment, in particular the variability of motion of the source.We conclude that there is no simple relationship between information and utility. Even in the absence of information processing costs or bandwidth constraints, behavioral optimality does not imply information efficiency, nor is there a simple tradeoff between the two objectives of gaining information about a food source versus obtaining the food itself. For a wide range of values of the food source's movement parameter, the strategy of collecting the most information possible about the unknown source location carries an ineliminable structural cost, leading to a situation in which a foraging creature could actually choose to be less well-informed while simultaneously being, on average, better fed.

LQ-boundary control of a diffusion-convection-reaction system

In this work, the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations (PDEs) with spatially varying coefficients is studied. An infinite-dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model. The evolutionary model which incorporates the spatially varying coefficients of the underlying set of the PDEs is used for subsequent linear quadratic regulator synthesis. The formulated linear quadratic-state feedback controller is applied to a nonlinear model of the reactor and its performance is studied.

Optimal LQ-Control of a PDAE Model of a Catalytic Distillation Process

In this contribution a linear quadratic (LQ) control design for a partial differential and algebraic equation (PDAE) system which represents a catalytic distillation process, is presented. The model involves a set of coupled partial differential equations (PDEs), ordinary differential equations (ODEs), and algebraic equations (AEs). The design is based on an infinite-dimensional state-space representation of the system in a Hilbert space and the well-known operator Riccati equation (ORE) method. The underlying PDE-ODE-AE system is converted to one containing coupled PDEs-ODEs, in which the PDE part involves a hyperbolic operator with a space-varying and non-symmetric velocity matrix whose eigenvalues are not necessarily negative through of the domain. Moreover, in the resulting PDE-ODE system, the control variable acts through the ODE part on the boundaries of the PDE section. Using the boundary control transformation method, the PDE-ODE system is represented in an infinite-dimensional state-space with a homogeneous boundary condition. The stabilizability and the detectability properties of the resulting system are explored, which provide guarantees of the existence and uniqueness of the solution to the resulting ORE. The ORE is solved by converting it to a set of equivalent matrix Riccati equations. The designed LQ controller is implemented on the process and its performance is evaluated.

Stability of delay integro-differential equations using a spectral element method

This paper describes a spectral element approach for studying the stability of delay integro-differential equations (DIDEs). In contrast to delay differential equations (DDEs) with discrete delays that act point-wise, the delays in DIDEs are distributed over a period of time through an integral term. Although both types of delays lead to an infinite dimensional state-space, the analysis of DDEs with distributed delays is far more involved. Nevertheless, the approach that we describe here is applicable to both autonomous and non-autonomous DIDEs with smooth bounded kernel functions. We also describe the stability analysis of DIDEs with special kernels (gamma-type kernel functions) via converting the DIDE into a higher order DDE with only discrete delays. This case of DIDEs is of practical importance, e.g., in modeling wheel shimmy phenomenon. A set of case studies are then provided to show the effectiveness of the proposed approach.

Linear quadratic optimal boundary control of a diffusion-convection-reaction system

AbstractIn this work, the boundary control of a distributed parameter system (DPS) modeled by parabolic partial differential equations with spatially varying coefficients is studied. An infinite dimensional state space setting is formulated and an exact transformation of the boundary actuation is realized to obtain an evolutionary model. The evolutionary model is used for subsequent linear quadratic regulator synthesis which incorporates the spatially varying coefficients of the underlying set of the PDEs. The formulated LQR controller is applied to the nonlinear model of the system and its performance is studied.

Linear quadratic regulator for time-varying hyperbolic distributed parameter systems

This paper addresses the linear quadratic (LQ) problem for a class of time-varying hyperbolic partial differential equation (PDEs) systems. The control method is based on two main ingredients: infinite-dimensional state space description and the well-known Riccati equation approach. First, the dynamical properties are studied, where the existence and uniqueness of the solution and exponential stability are proved. Next, an LQ-control feedback is computed by using the corresponding operator Riccati differential equation, whose solution can be obtained via a matrix Riccati PDE. The proposed method is applied to a catalytic fixed-bed reactor control problem. An optimal controller is designed for the linearized model and numerical simulations are performed to show the performance of the controller.

An Incomplete Information Inventory Model with Presence of Inventories or Backorders as Only Observations

In many real-life contexts, inventory levels are only incompletely observed due to nonobservation of demand, discrepancies in transmitting sales data, transaction errors, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system where the demand is not observed and the unmet demand is backordered. As a result, the inventory manager cannot tell the exact quantities of inventories or backorders. However, by looking at the shelf, he knows whether the inventory is positive or non-positive. Only with this information, the inventory manager must determine the order quantity in each period that would minimize the expected total discounted cost over an infinite-horizon. The dynamic programming formulation of this problem has an infinitedimensional state space. We use the concept of unnormalized probability to establish the existence of an optimal feedback policy and the uniqueness of the solution of the dynamic programming equation when the periodic cost has linear growth.

On asymptotic stability of semi-linear distributed parameter dissipative systems

The concept of asymptotic stability is studied for a class of semi-linear distributed parameter dissipative systems with nonlinearity defined on a convex subset of the state space. This is achieved by using infinite-dimensional Banach state space description. Stability criteria are established, which are based on a weaker technical condition of the m -dissipativity. These theoretical results are applied to a class of transport–reaction processes. Different types of nonlinearities are studied by adapting the criteria given in the early portions of the paper.

Krein Orthogonal Entire Matrix Functions and Related Lyapunov Equations: A State Space Approach

For a class of entire matrix valued functions of exponential type new necessary and sufficient conditions are derived in order that these functions are Krein orthogonal functions. The conditions are stated in terms of certain operator Lyapunov equations. These equations arise by using infinite dimensional state space representations of the entire matrix functions involved. As a corollary, using a recent operator inertia theorem, we give a new proof of the Ellis-Gohberg-Lay theorem which relates the number of zeros of a Krein orthogonal function in the open upper half plane to the number of negative eigenvalues of the corresponding selfadjoint convolution operator.

Control under quantization, saturation and delay: An LMI approach

This paper studies quantized and delayed state-feedback control of linear systems with given constant bounds on the quantization error and on the time-varying delay. The quantizer is supposed to be saturated. We consider two types of quantizations: quantized control input and quantized state. The controller is designed with the following property: all the states of the closed-loop system starting from a neighborhood of the origin exponentially converge to some bounded region (both, in R n and in some infinite-dimensional state space). Under suitable conditions the attractive region is inside the initial one. We propose decomposition of the quantization into a sum of a saturation and of a uniformly bounded (by the quantization error bound) disturbance. A Linear Matrix Inequalities (LMIs) approach via Lyapunov–Krasovskii method originating in the earlier work [Fridman, E., Dambrine, M., & Yeganefar, N. (2008). On input-to-state stability of systems with time-delay: A matrix inequalities approach. Automatica, 44, 2364–2369] is extended to the case of saturated quantizer and of quantized state and is based on the simplified and improved Lyapunov–Krasovskii technique.

Spectral Bound and Reproduction Number for Infinite-Dimensional Population Structure and Time Heterogeneity

Spectral bounds of quasi-positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ordinary differential equations: the sign of the spectral bound of the variational matrix at 0 decides whether, at low density, the population becomes extinct or grows. Another important threshold parameter is the reproduction number $\mathcal{R}$, which is the spectral radius of a related positive matrix. As is well known, the spectral bound and $\mathcal{R}-1$ have the same sign provided that the matrices have a particular form. The relation between spectral bound and reproduction number extends to models with infinite-dimensional state space and then holds between the spectral bound of a resolvent-positive closed linear operator and the spectral radius of a positive bounded linear operator. We also extend an analogous relation between the spectral radii of two positive linear operators which is relevant for discrete-time models. We illustrate the general theory by applying it to an epidemic model with distributed susceptibility, population models with age structure, and, using evolution semigroups, to time-heterogeneous population models.

Multi-feed attainable region construction using the Shrink–Wrap algorithm

An implementation of the Infinite DimEnsionAl State-space (IDEAS) based Shrink–Wrap algorithm for attainable region (AR) construction is developed, which identifies the multi-feed AR in reduced dimension (ARrd). The proof of the validity of the method is based on the stoichiometric characteristics of the reaction system and the properties of well-connected reactor networks, which allow the establishment of necessary and sufficient conditions for a point in mass fraction space to be in the multi-feed AR. The two-feed AR for methane steam reforming at isothermal and isobaric conditions is identified in the three-dimensional mass fraction space, and it is shown that for feeds consisting of pure methane and pure water, the equilibrium curves define the boundaries of the projections of the two-feed AR.

Controllability and Observability of Infinite-Dimensional Descriptor Systems

In this paper, the classical systems theoretic concepts of controllability and observability are considered for descriptor systems with an infinite-dimensional state space. These are systems of the form Exdot(t) = Ax(t) + Bu(t), y(t) = Cx(t), where x(ldr), u(ldr), and y(ldr) are functions with values in separable Hilbert spaces X, U, and Y. For the operators, we assume that E : XrarrZ, B : UrarrZ, and C : XrarrY are bounded, where Z is another Hilbert space. The operator A is assumed to be closed and defined on some dense subspace D(A) sub X. Mappings are defined that induce the notions of controllability and observability. The controllable states and the unobservable states are characterized by invariant subspaces. Based on that, Kalman decompositions of infinite-dimensional descriptor systems are presented.

Identification of the Attainable Region for Batch Reactor Networks

In this work, we describe a method for automatically identifying the set of all points in concentration space that represent outlet compositions of some network of discretely fed batch reactors for a given reaction set with known kinetics. This so-called batch attainable region (BAR) is dependent on the batch network's feed and total operating time, and it is shown to be quantifiable using the Infinite DimEnsionAl State-space (IDEAS) framework. We first establish that a simple batch reactor model possesses the properties that allow application of the IDEAS framework. We then formulate the resulting IDEAS Infinite Linear Program (ILP) whose solution is guaranteed to identify the globally optimal network of batch reactors. We subsequently use a simple transformation of this IDEAS ILP that leads us to propose two algorithms that are related to the construction of the true BAR. The first is a “Shrink-Wrap”-like algorithm that is similar to that previously reported [Manousiouthakis et al. The Shrink-Wrap Algorithm for the Construction of the Attainable Region: Application of the IDEAS Framework. Comput. Chem. Eng. 2004, 28, 1563] and creates increasingly accurate approximations of a set guaranteed to contain the true BAR for all network operating times. The second is a breadth-first algorithm that creates increasingly accurate inner approximations to the BAR for a given network operating time. These two algorithms are applied to an example from the literature and are shown analytically to converge in the limit to the true BAR.

Global Capital/Total Annualized Cost Minimization of Homogeneous and Isothermal Reactor Networks

The minimum capital cost/total annualized cost (MCC/MTAC) problem is investigated, within the infinite-dimensional state-space (IDEAS) framework, for homogeneous and isothermal reactor networks. The resulting mathematical formulation is an infinite-dimensional program with linear constraints and a separable concave objective function to be minimized. The global optimum of this infinite program is approximated through global solution of a series of finite-dimensional programs with concave separable objective functions and linear feasible regions. A branch-and-bound algorithm is used to solve globally each of these finite programs. The proposed methodology is illustrated with a reactor network synthesis case study aiming at capital cost minimization.

LQ-feedback control of a reverse flow reactor

In this paper, a linear-quadratic controller is formulated for a catalytic flow reversal reactor (CFRR) using an infinite-dimensional Hilbert space representation of the system. The controller is developed on the basis of the catalytic reactor model with unidirectional flow and is formulated to keep the distribution of the temperature along the axis of the reactor at stationary state by using the fluid flow velocity. To formulate the controller, a linear infinite-dimensional state space description is used. Using the infinite-dimensional state space representation, a state LQ-feedback operator is computed via the solution of a Riccati differential equation. The controller is formulated and tested numerically for the catalytic combustion of lean methane emissions.

Partially Observed Inventory Systems: The Case of Rain Checks

In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to non-observation of demand, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system where the unmet demand is backordered. When inventory level is nonnegative, the inventory manager does not know the exact inventory level. Otherwise, inventory shortages occur and the inventory manager issues rain checks to customers. The shortages are fully observable via the rain checks. The inventory manager determines the order quantity based on the partial information on the inventory level. The objective is to minimize the expected total discounted cost over an infinite horizon. The dynamic programming formulation of this problem has an infinite dimensional state space. We use the methodology of the unnormalized probability to establish the existence of an optimal feedback policy when the periodic cost has linear growth. Moreover, uniqueness and continuity of the solution to dynamic programming equations are proved when the discount factor is sufficiently small.

Partially Observed Inventory Systems: The Case of Rain Checks

In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to nonobservation of demand, spoilage, misplacement, or theft of inventory. We study a discrete-time periodic-review inventory system where the unmet demand is backordered. When the inventory level is nonnegative, the inventory manager does not know the exact inventory level. Otherwise, inventory shortages occur, and the inventory manager issues rain checks to customers. The shortages are fully observed via the rain checks. The inventory manager determines the order quantity based on the partial information on the inventory level. The objective is to minimize the expected total discounted cost over an infinite horizon. The dynamic programming formulation of this problem has an infinite dimensional state space. We use the methodology of the unnormalized probability to establish the existence of an optimal feedback policy when the periodic cost has linear growth. Moreover, uniqueness and continuity of the solution to dynamic programming equations are proved when the discount factor is sufficiently small.