Infinite-dimensional Discrete-time Research Articles

Moving horizon estimation for pipeline leak detection, localization, and constrained size estimation

Advanced pipeline leak detection and localization techniques are needed to reduce greenhouse gas emissions from hydrocarbon transportation pipelines. Developing effective leak detection and localization methods is challenging due to the spatiotemporal dynamics of process variables, the presence of process/measurement disturbances and constraints, and the limited measurement data. To address this issue, this manuscript proposes a novel moving horizon estimation design for pipeline leak detection, constrained estimation of leak size and location by using an infinite-dimensional pipeline hydraulic model. Based on the mass and momentum balance laws and the Cayley–Tustin time-discretization method, an infinite-dimensional discrete-time pipeline hydraulic model is proposed considering (unknown but bounded) disturbance and leak. By introducing a coordinate transformation, we decouple the leak size and location estimation problems. The implementable discrete-time moving horizon estimator and observer are designed for constrained leak size and location estimation. The effectiveness of the proposed designs is validated via simulation examples.

Stability Radii of Infinite-Dimensional Discrete-Time Systems Discomfited by Stochastic Perturbations

Stability Radii of Infinite-Dimensional Discrete-Time Systems Discomfited by Stochastic Perturbations

Approximate moving horizon estimation for switching conservative linear infinite-dimensional systems

This work addresses moving horizon estimation for switching conservative linear infinite-dimensional systems described by partial differential equations (PDE), where the plant and measurement equations are corrupted with bounded disturbances, and the system mode is regarded as an unknown and unpredictable discrete state to be estimated. To address the issues associated with unbounded operators (induced by boundary or point observation and disturbance) and facilitate discrete-time moving horizon estimator design, the Cayley–Tustin transformation is deployed for model time-discretization without any spatial discretization or model reduction while preserving model essential properties. A series of observability concepts along with corresponding properties are proposed and analyzed for the switching linear infinite-dimensional discrete-time systems. A moving horizon estimation algorithm that accounts for state/output and mode estimation and constraint handling is proposed. Based on the proposed observability properties, we prove the stability of the proposed moving horizon estimator. The derived results are demonstrated by an undamped Schrödinger equation with switching group-velocity dispersion.

Stability radii of infinite-dimensional discrete-time systems discomfited by stochastic perturbations

Stability radii of infinite-dimensional discrete-time systems discomfited by stochastic perturbations

Nonlinear discrete time systems with inputs in Banach spaces*

In this paper we study ISS-like properties of infinite dimensional discrete time systems and compare them with their continuous time counterparts available in the literature. New characterizations of such properties are derived in this work. We discuss differences between discrete and continuous time systems in view of their robust stability and demonstrate the corresponding properties by means of examples.

ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems

<p style='text-indent:20px;'>Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.</p><p style='text-indent:20px;'>We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.</p>

Sampled-data output regulation of unstable well-posed infinite-dimensional systems with constant reference and disturbance signals

We study the sample-data control problem of output tracking and disturbance rejection for unstable well-posed linear infinite-dimensional systems with constant reference and disturbance signals. We obtain a sufficient condition for the existence of finite-dimensional sampled-data controllers that are solutions of this control problem. To this end, we study the problem of output tracking and disturbance rejection for infinite-dimensional discrete-time systems and propose a design method of finite-dimensional controllers by using a solution of the Nevanlinna-Pick interpolation problem with both interior and boundary conditions. We apply our results to systems with state and output delays.

On the Optimal Speed Problem for the Class of Linear Autonomous Infinite-Dimensional Discrete-Time Systems with Bounded Control and Degenerate Operator

The optimal speed problem for the class of linear discrete-time systems with the infinite-dimensional state vector and degenerate operator is solved. Some properties of convex sets are formulated and proved. Necessary and sufficient conditions under which this problem has a solution in the case of the zero point located on the boundary of the reachability set are established. The optimality conditions are written as a discrete-time maximum principle. For the inner points, the degenerate character of the maximum principle is demonstrated. For an inner point, the optimal speed problem is solved by developing an algorithm with reduction to the admissible case of the boundary point. Some examples are given.

Estimation of the Distribution of Random Parameters in Discrete Time Abstract Parabolic Systems with Unbounded Input and Output: Approximation and Convergence.

A finite dimensional abstract approximation and convergence theory is developed for estimation of the distribution of random parameters in infinite dimensional discrete time linear systems with dynamics described by regularly dissipative operators and involving, in general, unbounded input and output operators. By taking expectations, the system is re-cast as an equivalent abstract parabolic system in a Gelfand triple of Bochner spaces wherein the random parameters become new space-like variables. Estimating their distribution is now analogous to estimating a spatially varying coefficient in a standard deterministic parabolic system. The estimation problems are approximated by a sequence of finite dimensional problems. Convergence is established using a state space-varying version of the Trotter-Kato semigroup approximation theorem. Numerical results for a number of examples involving the estimation of exponential families of densities for random parameters in a diffusion equation with boundary input and output are presented and discussed.

On the problem of operation speed for the class of linear infinite-dimensional discrete-time systems with bounded control

Consideration was given to the problem of operation speed for the class of linear autonomous systems with infinite-dimensional state vector. The statements about the properties of the Minkowsky sum for the convex sets were formulated and proved. For the boundary points of the 0-controllability set, the necessary and sufficient conditions for solvability of the problem of operation speed were established. The optimality conditions were set down for the boundary points in terms of the discreet principle of maximum. Nonuniqueness of the optimal control and degenerate nature of the principle of maximum were proved for the internal points. An algorithm to solve the problem of operation speed for the internal points was developed by reducing it to the allowed case of the boundary points. Examples were given.

A Finite-time Condition for Exponential Trichotomy in Infinite Dynamical Systems

AbstractIn this article we study exponential trichotomy for infinite dimensional discrete time dynamical systems. The goal of this article is to prove that finite time exponential trichotomy conditions allow us to derive exponential trichotomy for arbitrary times. We present an application to the case of pseudo orbits in some neighborhood of a normally hyperbolic set.

Model predictive control of axial dispersion chemical reactor

This paper discusses the development of model predictive control algorithm which accounts for the input and state constraints applied to the parabolic partial differential equations (PDEs) system describing the axial dispersion chemical reactor. Spatially varying terms arising from the nonlinear PDEs model are accounted for in model development. Finite-dimensional modal representation capturing the dominant dynamics of the PDEs system is derived for controller design through Galerkin's method and modal decomposition technique. Tustin's discretization and Cayley transform are used to obtain infinite-dimensional discrete-time dynamic modal representations which are used in subsequent constrained controller design. The proposed discrete-time constrained model predictive control synthesis is constructed in a way that the objective function is only based on the low-order modal representation of the PDEs system, while higher-order modes are utilized only in the constraints of the PDEs state. Finally, the MPC formulations are successfully applied, via simulation results, to the PDEs system with input and state constraints.

Asymptotic distribution for periodograms of infinite dimensional discrete time periodically correlated processes

In this article we shall consider a class of strongly T -periodically correlated processes with values in a separable complex Hilbert space H . The periodograms of these processes and their statistical properties were previously studied by the authors. In this paper we derive the asymptotic distribution of the periodogram, that appears to be a certain Wishart distribution on H T .

Absolute stability and integral control for infinite-dimensional discrete-time systems

We derive absolute stability results of Popov and circle-criterion types for infinite-dimensional discrete-time systems in an input–output setting. Our results apply to feedback systems in which the linear part is the series interconnection of an l 2 -stable linear system and an integrator and the nonlinearity satisfies a sector condition which allows for saturation and deadzone effects. The absolute stability theory is then used to prove tracking and disturbance rejection results for integral control schemes in the presence of input and output nonlinearities. Applications of the input–output theory to state-space systems are also provided.

Inference on periodograms of infinite dimensional discrete time periodically correlated processes

In this work we shall consider two classes of weakly second-order periodically correlated and strongly second-order periodically correlated processes with values in separable Hilbert spaces. The periodogram for these processes is introduced and its statistical properties are studied. In particular, it is proved that the periodogram is asymptotically unbiased for the spectral density of the processes, where the type of the convergence is fully specified.

On Infinite Dimensional Discrete Time Periodically Correlated Processes

On Infinite Dimensional Discrete Time Periodically Correlated Processes

Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis

The notion of non-uniform Robust Global Asymptotic Stability (RGAS) presented in this paper generalizes the notion of non-uniform in time RGAS for finiteor infinite-dimensional discrete-time systems. Lyapunov characterizations for this stability notion are provided. The results are applied to finitedimensional discrete-time systems obtained by time discretization of continuous-time systems by the explicit Euler method.

Residual mode filters and H ∞ control for a class of infinite-dimensional discrete-time systems

An H ∞, control problem with measurement feedback.for infinite-dimensional discrete-time (IDDT) systems whose homogeneous parts are described by Riesz-spectra operators is considered. The aim is to construct a finite-dimensional stabilizing controller for the IDDT system that makes the H ∞ norm of the closed-loop transfer function less than a given positive number δ. For that purpose, we first formulate the IDDT system as an IDDT system in l2 and derive a finite-dimensional reduced-order system for the IDDT system in l2. A stabilizing controller that makes the H ∞ norm of the closed-loop transfer function less than another positive number is then constructed for the reduced-order model. The finite-dimensional controller together with a residual mode Jilter plays a role of a finite-dimensional stabilizing controller that makes the H ∞ norm of the closed-loop transfer function less than δ for the original IDDT system, if the order of the residual mode filter is chosen suficiently large.

Residual mode filters and H∞ control for a class of infinite-dimensional discrete-time systems

Residual mode filters and H∞ control for a class of infinite-dimensional discrete-time systems

Null-Controllability of Discrete-Time Systems with Restrained Controls in Asplund Spaces

This paper is concerned with the null-controllability of the infinite-dimensional discrete-time linear system described by $$(A, B, \Omega ) x_{n + 1} = Ax_n + Bu_n$$ where xn ∈ X, un ∈ Ω ⊂ U, X and U are Asplund spaces, A ∈ L(X, X), B ∈ L (U, X), Ω is a convex set, int Ω ≠ 0 and 0 ∈ Ω.