Early Lumping Approach Research Articles

Reduced order in domain control of distributed parameter port-Hamiltonian systems via energy shaping

An in-domain finite dimensional controller for a class of distributed parameter systems on a one-dimensional spatial domain formulated under the port-Hamiltonian framework is presented. Based on (Trenchant et al. 2017) where positive feedback and a late lumping approach is used, we extend the Control by Interconnection method and propose a new energy shaping methodology with an early lumping approach on the distributed spatial domain of the system. Our two main control objectives are to stabilize the closed-loop system, as well as to improve the closed-loop dynamic performances. With the early lumping approach, we investigate two cases of the controller design, the ideal case where each distributed controller acts independently on the spatial domain (fully-actuated), and the more realistic case where the control action is piecewise constant over certain intervals (under-actuated). We then analyze the asymptotic stability of the closed-loop system when the infinite dimensional plant system is connected with the finite dimensional controller. Furthermore we provide simulation results comparing the performance of the fully-actuated case and the under-actuated case with an example of an elastic vibrating string.

Finite dimensional shape control design of linear port-Hamiltonian systems with in-domain pointwise inputs

This paper is concerned with shape control of a class of infinite-dimensional port-Hamiltonian system, using an early lumping approach, i.e. the dynamic controller synthesized from a low-order discretized version of the system. The approach provides an optimal criterion for choosing a free parameter of the controller so that the closed-loop system converges to the best approximation of the desired imposed shape. The methodology is based on the so called control by interconnection method and structural invariant from which an analytical expression is obtained for the shapes that the system can actually achieve. An Euler-Bernoulli beam model with pointwise inputs is used as example to illustrate the proposed methodology.

Observer-based boundary control of distributed port-Hamiltonian systems

An observer-based boundary controller for infinite-dimensional port-Hamiltonian systems defined on 1D spatial domains is proposed. The design is based on an early-lumping approach in which a finite-dimensional approximation of the infinite-dimensional system derived by spatial discretization is used to design the observer and the controller. As long as the finite-dimensional approximation approaches the infinite-dimensional model, the performances also do. The main contribution is a constructive method which guarantees that the interconnection between the controller and the infinite-dimensional system is asymptotically stable. A Timoshenko beam model has been used to illustrate the approach.

Observer-Based State Feedback Controller for a class of Distributed Parameter Systems

Abstract This paper aims to propose a finite-dimensional observer-based state feedback controller to stabilize a class of boundary controlled system. To this end, we propose to use an early-lumping approach, where the infinite-dimensional port-Hamiltonian system is first discretized using a structure-preserving method. Then, we build a passive observed-based controller using a Linear Matrix Inequality (LMI) and finally, the controller is interconnected with the infinite-dimensional system in a passive way. Due to its passivity and Hamiltonian structure, this observer-based controller can stabilize not only the discretized lumped parameter system but also the original distributed parameter system. This approach avoids the intrinsic drawback of early lumping approach and spillover effects. Finally, the boundary controlled undamped wave equation is used to illustrate the effectiveness of the proposed controller.

A control-oriented thermal model for a moving dual-layer substrate

This paper describes an early lumping approach for generating a mathematical model of the heating process of a moving dual-layer substrate. The heat is supplied by convection and nonlinearly distributed over the whole considered spatial extend of the substrate. Using CFD simulations as a reference, two different modelling approaches have been investigated in order to achieve the most suitable model type. It is shown that due to the possibility of using the transition matrix for time discretization, an equivalent circuit model achieves superior results when compared to the Crank-Nicolson method. In order to maintain a constant sampling time for the in-visioned-control strategies, the effect of variable speed is transformed into a system description, where the state vector has constant length but a variable number of non-zero entries. The handling of the variable transport speed during the heating process is considered as the main contribution of this work. The result is a model, suitable for being used in future control strategies.

Model Predictive Control and Moving Horizon Estimation of a Large-Scale Chemical Reactor Model

The presented contribution concerns the model predictive control (MPC) and moving horizon estimation (MHE) of a catalytic fixed-bed reactor model. Rigorous modeling for these systems leads to systems of (transport) partial differential equations. Following a so-called early lumping approach for the model predictive control and estimation yields high-dimensional systems of ordinary differential equations and therefore the need to solve large-scale dynamic optimization problems online. It is shown how a tailored gradient method and efficient numerical integration can be combined to solve the concerned optimization methods in a time-efficient way.

Feedback Tracking Control of Continuous Reheating Furnaces

AbstractA Lyapunov-based MIMO state feedback controller is developed for slab temperatures in a continuous, fuel-fired reheating furnace. Following an early lumping approach, the computationally simple tracking controller is designed for a nonlinear, switched dynamic model that captures both conductive and radiative heat transfer. The controller modifies reference trajectories of furnace temperatures and is part of a cascade control scheme. Given some nonrestrictive conditions, exponential stability is ensured, even under non-steady state operating conditions. The capabilities of the controller are demonstrated by means of an example problem.

Discrete-Time Modal State Reconstruction for Infinite-Dimensional Systems Using Generalized Sampling

AbstractIn this contribution a novel approach for discrete-time state reconstruction for infinite-dimensional systems is presented. It is shown that an exact reconstruction of a finite number of modal states of the infinite-dimensional system can be obtained at the sampling time instances. This is achieved by aid of generalized sampling which generates the output samples by weighted averaging over one sampling interval. The reconstruction is done in finite time and is thus non-asymptotic. Since the method makes it possible to implement state feedback controls without involving a Luenberger observer, the spillover problem arising from the early-lumping approach can be avoided.

Finite-dimensional observer-based control of linear distributed parameter systems using cascaded output observers

The synthesis of compensators for linear distributed-parameter systems on the basis of a finite-dimensional approximation is a classical technique. This so-called early-lumping approach suffers from the occurrence of spillover which means that the closed-loop dynamics is deteriorated by the neglected dynamics. In this contribution, an early-lumping compensator design is presented that overcomes the spillover problem by using certain fictitious outputs which can be reconstructed without spillover and that suppress the contributions of the unmodelled dynamics. When the new outputs are used for the compensator instead of the available measurements, the perturbation of the closed-loop spectrum, caused by spillover, can be reduced to an arbitrary extent. It is shown that the worst-case eigenvalue perturbation decreases exponentially with respect to the compensator order so that the spillover can be reduced systematically. In addition, an a priori estimate for the compensator order, that guarantees a prescribed maximal eigenvalue perturbation, is presented. The proposed design procedure is demonstrated for the control of an Euler–Bernoulli beam with Kelvin–Voigt damping.

Parametric approach to the decoupling of linear distributed-parameter systems

In this study the input–output decoupling of linear distributed-parameter systems with distributed control and distributed measurement is considered. The decoupling problem is formulated as an eigenstructure assignment problem for distributed-parameter systems, which is solved using the parametric approach. The resulting static state feedback controller exponentially stabilises the closed-loop system and decouples the reference transfer behaviour with respect to the first p dominant closed-loop modal states. The residual dynamics are taken into account by compensating further dominant eigenvalues with transmission zeros, so that they do not influence the reference transfer behaviour. A heat conductor is used to demonstrate the design procedure which is also compared with an early lumping approach.

State estimation of a stratified storage tank

Two approaches for the state estimation of the spatial temperature profile in a stratified storage tank, i.e. the state of the art domestic hot water storage system, are presented. There are a distributed parameter observer as a late lumping and an Unscented Kalman filter (UKF) as an early lumping approach which are investigated with respect to applicability and reconstruction convergence. Both, the distributed parameter observer and the UKF are designed based on a hybrid distributed parameter model of the stratified storage tank. The hybrid model is composed of a finite state automaton interacting with an underlying thermal heat conduction–convection system described by a quasi-linear partial differential equation. The performance of the estimation algorithms is illustrated by simulation studies and measurement data, showing excellent convergence results.