R. Kern

Physical modelling of a long pneumatic transmission line: models of successively decreasing complexity and their experimental validation

ABSTRACT There exist a significant number of models, which describe the dynamics of pneumatic transmission lines. The models are based on different assumptions and, thereby, vary in the physical phenomena they incorporate. These assumptions made are not always stated clearly and the models are rarely validated with measurement data. The aim of this article is to present multiple distributed parameter models that, starting from a physical system description, successively decrease in complexity and finally result in a rather simple system representation. Data, both from simulation studies as well as from a pneumatic test bench, serve as a quantitative validation of these assumptions. Based on a detailed discussion of the different models, this article aims at facilitating the choice of an appropriate model for a given task where the effect of long pneumatic transmission lines cannot be neglected and a trade-off between accuracy and complexity is required.

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.

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.