Nicole Gehring

Controllability and Prediction-Free Control of Coupled Transport Processes Viewed As Linear Systems with Distributed Delays

Abstract The solution of transport equations results in functional differential equations with time-delays. This papers deals with the control of linear systems with lumped and distributed delays that represent a coupled system of transport processes and ordinary differential equations. These time-delay systems can be viewed as modules over a ring of entire functions. It is shown that spectral controllability and freeness of the module over an associated ring are necessary and sufficient for the module to be free. Using a module basis, a flatness-based tracking controller is derived that is infinite-dimensional, in general, due to the distributed delays. However, no (explicit) predictions are required to assign a finite spectrum to the delay system. Two examples illustrate the results, one of which being a neutral type system.

Control of Linear Delay Systems: An Approach without Explicit Predictions

The control of linear time-invariant systems with incommensurate lumped and distributed delays is addressed. Using a module-theoretic point of view where these systems are modules over the ring of entire functions in ℝ(s)[e− τ s ] necessary and sufficient conditions for the freeness of these modules are presented. If these conditions hold a module basis can be used to design a tracking controller that assigns an arbitrary finite spectrum to the closed loop. Though the controller is infinite dimensional, in general, it does not involve any explicit predictions. This generalizes the so-called reduction approach, by which for certain state representations predictions can be calculated exactly and thus finite spectrum assignment can be achieved. Examples illustrate the main results.

Algebraic parameter identification for infinite dimensional fluid transmission line models

This article addresses the problem of parameter identification in fluid transmission lines. The recently developed algebraic approach to parameter identification in linear distributed parameter systems is applied to three different transmission line models: the inviscid model, the linear friction model, and the frequency-dependent friction model. The main idea is to derive exact equations for parameter identification based on the infinite dimensional system models using only spatially concentrated measurements. To this end, the transmission line models are subjected to several transformation steps including the Laplace transform. Using further algebraic manipulations, equations are derived, which allow for the calculation of system parameters. The identification method presented is illustrated by means of numerical simulations covering two different scenarios of pressure and flow rate measurements taken at the boundaries of a pipeline.

Prediction-free tracking control for systems with incommensurate lumped and distributed delays: Two examples

Abstract Linear systems with lumped and distributed delays can be represented by modules over the ring of entire functions in Ĉ( s )[e – τs ]. While in the case of commensurate delays spectral controllability is sufficient for the existence of a basis of this module, in the incommensurate case addressed here additional conditions are required. Exploiting the relations between the (known) delay amplitudes a new module with favorable freeness properties can be defined. Based on that, necessary and sufficient conditions for the freeness of this module are presented. If these conditions are satisfied a basis can be used to derive a flatness-based tracking control without any explicit predictions. The approach is illustrated on a neutral system and on a system with distributed delays.

Algebraic identification of heavy rope parameters1

Abstract An algebraic approach to the identification of parameters for a heavy rope model is proposed. It is based on operational calculus. The parameters are calculated solely from the measurements of the lower and the upper deflection of the rope. Two different sets of boundary conditions are discussed.

Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

ABSTRACT This paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially varying coefficients. Thereby, the ODE is coupled to the PDE in-domain and at the uncontrolled boundary, whereas the ODE is coupled with the latter boundary. For the state feedback design, a two-step backstepping approach is developed, which yields the conventional kernel equations and additional decoupling equations of simple form. In order to implement the state feedback controller, the design of observers for the PDE-ODE systems in question is considered, whereby anti-collocated measurements are assumed. Exponential stability with a prescribed convergence rate is verified for the closed-system pointwise in space. The resulting compensator design is illustrated for a 4 × 4 heterodirectional hyperbolic system coupled with a third-order ODE modelling a dynamic boundary condition.

Tracking control for a long pneumatic transmission line

This paper presents a tracking controller for a long pneumatic transmission line modeled by a bidirectionally coupled system of partial differential equations (PDEs) and ordinary differential equations (ODEs). The feedforward part of the controller is designed by applying a flatness-based approach for hyperbolic PDEs to a second-order quasilinear model of the pneumatic system. The stabilizing feedback is derived by the application of a recently developed backstepping approach for coupled PDE-ODE systems to a different, more simple, linear model of the line. In simulations, the tracking controller is used to asymptotically stabilize a complex quasilinear third-order distributed parameter model of the pneumatic transmission line along a desired trajectory. This complex model has previously been shown to accurately reproduce the behavior of the pneumatic test bench considered.

Algebraische Methodenzur Parameteridentifikation für das schwere Seil

Zusammenfassung Vorgestellt wird eine Methode zur Identifikation von Parameter eines linearen, verteiltparametrischen Modells für das schwere Seil unter ausschließlicher Verwendung gemessener Randgrößen. Die Methode basiert auf der Operatordarstellung der Lösung des zugehörigen Randwertproblems. Aus dieser erhält man durch sukzessives Falten der gemessenen Trajektorien ein System algebraischer Gleichungen in den gesuchten Parametern. Simulationsergebnisse illustrieren den Ansatz. Abstract A method is presented that allows for the identification of parameters in a linear infinite-dimensional model for the heavy rope using only boundary measurements. The approach relies on an operational representation of the solution of the corresponding boundary value problem. A system of algebraic equations in the parameters is generated by repeated convolution of the measured trajectories. Simulations illustrate the results.

Output feedback control of general linear heterodirectional hyperbolic ODE–PDE–ODE systems

This paper considers the backstepping design of observer-based compensators for general linear heterodirectional hyperbolic ODE–PDE–ODE systems, where the ODEs are coupled to the PDEs at both boundaries and the input appears in an ODE. A state feedback controller is designed by mapping the closed-loop system into a stable ODE–PDE–ODE cascade. This is achieved by representing the ODE at the actuated boundary in Byrnes–Isidori normal form. The resulting state feedback is implemented by an observer for a collocated measurement of the PDE state, for which a systematic backstepping approach is presented. The exponential stability of the closed-loop system is verified in the ∞-norm. It is shown that all design equations can be traced back to kernel equations known from the literature, to simple Volterra integral equations of the second kind and to explicitly solvable boundary value problems. This leads to a systematic approach for the boundary stabilization of the considered class of ODE–PDE–ODE systems by output feedback control. The results of the paper are illustrated by a numerical example.

Parameter identification, fault detection and localization for an electrical transmission line

The current and voltage on an electrical transmission line, e.g. a coaxial cable, can be modeled by means of the telegraphers equations. While most methods for parameter identification and fault detection for such an infinite dimensional model rely on some finite dimensional approximation, this is not necessary for the algebraic approach presented here. It derives simple polynomial equations relating the concentrated measurements and the unknown parameters by using the Laplace transform and computing characteristic sets for differential ideals. In the end, the identification of parameters and the detection of faults - e.g. a broken isolation between two conductors - requires only the calculation of convolutions of measured signals. The results are illustrated for a signal transmission problem using both experimental and simulation data.