Modelling of distributed-parameter systems for control purposes

Thirty years ago, control design was mainly based on linear finite-dimensional models sometimes closely but more often less closely connected to the real character of the plant. In many applications, these linear controllers are sufficient in terms of accuracy and dynamic performance and hence are still common in industry. If the plant under consideration exhibits significant nonlinearities and/or has a distributed-parameter structure, an increase in the performance of the closed-loop system can only be achieved by taking into account the real nature of the system to be controlled. Beginning in the 80s, modern control methods were introduced to systematically account for the essential nonlinearities in the control design process for lumped-parameter nonlinear systems. Since that time a variety of modelbased nonlinear control design techniques have been developed, among which the sliding mode control, the passivity-based design approach, backstepping, differential geometric methods, the flatness-based control and many others can be found. Nowadays, the application of nonlinear control design methods is even standard in many industrial applications. Clearly, apart from the advances in control theory, the availability of modern computer programs for numeric and symbolic computation, together with the steadily increasing power of automation hardware being used in real-time applications, make the practical use of these modern control concepts possible. Usually, the so-called early lumping approach is used in control engineering practice if one deals with distributed-parameter systems (DPSs), i.e. their mathematical models are given in the form of partial differential equations. Thereby, the DPS is reduced to a system of (nonlinear) ordinary differential equations by means of Galerkin or Rayleigh-Ritz approximation, finite-difference or finite-element schemes or by other model approximation techniques. Thus, the distributed-parameter nature of the system is not fully taken into account in a systematic way. In this case, it is well known that the performance of the resulting closed-loop system may be degraded, or in the worst case, the system can even be destabilized. Therefore, in the last years the control community has made much effort to extend certain concepts known from finite-dimensional control theory to the distributedparameter case. Within the so-called late lumping approach, the controller is directly designed on the basis of the distributed-parameter model and the control law is then (numerically) approximated for the purpose of implementation on the real system. In general, it turns out that an appropriate formulation of the mathematical model of distributed-parameter systems drastically simplifies the subsequent control design task. This special issue is concerned with this latter aspect where we aim at presenting theoretical and practical results in the modelling of distributed-parameter systems in view of a subsequent controller design. The papers are based on selected contributions that were presented in a special session of the 5th MATHMOD conference in February 2006, Vienna, Austria. Mathematical and Computer Modelling of Dynamical Systems Vol. 14, No. 3, June 2008, 177–178