## Backstepping-based output regulation for distributed-parameter systems

Besides the pure stabilization an additional basic property of a controller is to achieve the asymptotic tracking of online prescribed reference inputs in the presence of disturbances. If these exogenous signals are describable by a signal model, then the output regulation theory provides a systematic framework for the controller design. In this research project the backstepping approach for boundary controlled distributed-parameter systems is combined with results from output regulation theory. This yields new systematic procedures to determine output feedback regulators for distributed-parameter systems.

### Selected publications

*Deutscher, J.:***Backstepping design of robust state feedback regulators for linear 2x2 hyperbolic systems.**IEEE Trans. Autom. Control 62 (2017), pp. 5240-5247.*Deutscher, J.:***Output regulation for general linear heterodirectional hyperbolic systems with spatially-varying coefficients.**Automatica 85 (2017), pp. 34-42.*Deutscher, J.:***Finite-time output regulation for linear 2x2 hyperbolic systems using backstepping.**Automatica 75 (2017), pp. 54-62.*Deutscher, J. and Kerschbaum, S.:***Backstepping design of robust state feedback regulators for second order hyperbolic PIDEs.**Proc. CPDE 2016 in Bertinoro, Italy, pp. 81-86*.**Deutscher, J.:***Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs.**IEEE Trans. Autom. Control 61 (2016), pp. 2288-2294.*Deutscher, J.:***A backstepping approach to the output regulation of boundary controlled parabolic PDEs.**Automatica 57 (2015), pp. 56-64*.*

## Fault diagnosis and fault tolerant control for distributed-parameter systems

Due to the increasing complexity of technical systems the detection of faults and their consideration in the controller design becomes more and more important in order to assure the safety of various technical processes. Though this problem has been thoroughly investigated for finite-dimensional systems only a few results exist for distributed-parameter systems. In this research project new methods for the fault detection in infinite-dimensions are derived. In order to take faults in the closed-loop system into account new approaches are developed for fault tolerant control.

### Selected publications

*Fischer, F. and Deutscher, J.:***Fault detection for parabolic systems with distributed inputs and outputs using the modulation function approach.**IFAC World Congress 2017 in Toulouse, France, pp. 6968-6973.*Fischer, F. and Deutscher, J.:***Algebraic fault detection and isolation for parabolic distributed-parameter systems using modulation functions.**Proc. CPDE 2016 in Bertinoro, Italy, pp. 164-169.*Deutscher, J.:***Fault detection for distributed-parameter systems using finite-dimensional functional observers.**Int. J. Control 89 (2016), pp. 550-563*.*

**Structure preserving approximation of distributed-parameter systems**

Classical order reduction methods use an FE-model of the underlying distributed-parameter system. This has the disadvantage that a very high order FE-model is needed for an accurate system description, which leads to a high numerical effort in the order reduction. In this research project new methods are derived for the direct order reduction of PDE-models. This allows for an adaptive grid selection in the order reduction and thus increases the numerical efficiency of the corresponding approximation method. Furthermore, the degrees of freedom in the considered approximation techniques are determined such that system properties of interest are preserved.

### Selected publications

*Deutscher, J. and Harkort, Ch.:***Structure preserving approximation of distributed-parameter second order systems using Krylov subspaces.***Harkort, Ch. and**Deutscher, J.:***Stability and****passivity preserving Petrov-Galerkin approximation of linear****infinite-dimensional systems.**Automatica 48 (2012), pp. 1347-1352.*Harkort, Ch. and Deutscher, J.:***Krylov subspace methods for linear infinite-dimensional systems.**IEEE Trans. Autom. Control 56 (2011), pp. 441-447.

## Spillover prevention for early-lumping based compensator designs

A classical approach for the design of finite-dimensional compensators for infinite-dimensional systems is the early-lumping approach. Thereby, the compensator design is based on a finite-dimensional approximation of the plant. This can lead to the spillover problem, i. e. the resulting closed-loop system is unstable though the compensator stabilizes the corresponding approximation. In this research project new methods for the analysis of spillover are developed. Furthermore, easy to implement extensions are investigated for early-lumping-based compensators to prevent spillover.

### Selected publications

*Harkort, Ch. and**Deutscher, J.:***Discrete-time modal state reconstruction for infinite-dimensional systems using generalized sampling****.**Proc. IFAC World Congress 2011 in Milano, Italy.*Harkort, Ch. and Deutscher, J.:***Finite-dimensional observer-based control of linear distributed-parameter systems using cascaded output observers.**Int. J. Control 84 (2011), pp. 107-122*.*

## Direct design of finite-dimensional compensators

Typically the design of compensators for distributed-parameter systems is based on the late-lumping or the early-lumping approach. Thereby, the resulting finite-dimensional compensator is obtained by using an approximation of an infinite-dimensional controller or of the infinite-dimensional plant. Consequently, the properties of the resulting closed-loop system have to be investigated after the design. In this research project finite-dimensional compensators are directly determined for infinite-dimensional systems. Since their design is not based on any approximation they directly achieve the desired specifications for the closed-loop system.

### Selected publications

*Deutscher, J.*:**Finite-dimensional dual state feedback control of linear boundary control systems.**Int. J. Control 86 (2013), pp. 41-53.*Deutscher, J.*:**Output regulation for linear distributed-parameter systems using finite-dimensional dual observers.**Automatica 47 (2011), pp. 2468-2473.*Deutscher, J. and Harkort, Ch.*:**A parametric approach to finite-dimensional control of linear distributed-parameter systems.**Int. J. Control 83 (2010), pp. 1674-1685.

**Feedforward control for the ****shallow water equations**

In many practical applications the dynamics of a fluid can be modelled by the shallow water equations. In this research project new methods based on semi-discretizations and flatness-based techniques are developed for the feedforward control of these fluid models. The theoretical results are investigated by using a laboratory experiment that is shown below.

**Laboratory experiment:** round plexiglas cylinder with colored water. The simulation result for the water level is the yellow dashed line.