Marco Kletting

Interval methods for simulation of dynamical systems with state-dependent switching characteristics

In this paper, an interval arithmetic simulation algorithm is introduced for simulation of continuous-time systems with state-dependent switching between different dynamical models. For that purpose, the conditions for all possible transitions between these models have to be evaluated during simulation to determine the switching times and hence to obtain guaranteed enclosures for all state variables. In contrast to other simulation techniques, all system parameters are defined as interval variables to analyze the effect of uncertainties on the switching times and the dynamical behavior of the complete system

Interval Observer Design Based on Taylor Models for Nonlinear Uncertain Continuous-Time Systems

In most applications in control engineering not all state variables can be measured. Consequently, state estimation is performed to reconstruct the non-measurable states taking into account both system dynamics and the measurement model. If the system is subject to interval bounded uncertainties, an interval observer provides a guaranteed estimation of all states. The estimation consists of a recursive application of prediction and correction steps. The prediction step corresponds to a verified integration of the system model describing the system dynamics between two points of time at which measured data is available. In this paper, a Taylor model based integrator is used. Considering the state enclosures obtained in the prediction step, the correction step reconstructs states and parameters from the uncertain measurements with the help of a measurement model. The enclosures of states and parameters determined by the interval observer are consistent with both system and measurement models as well as all uncertainties.

CONSISTENCY TECHNIQUES FOR SIMULATION OF WASTEWATER TREATMENT PROCESSES WITH UNCERTAINTIES

Abstract In this paper, interval simulation methods are presented to determine guaranteed enclosures of state variables of an activated sludge process in biological wastewater treatment. This process is characterized by nonlinearities and uncertain but bounded parameters. In uncertain systems an axis-parallel interval box is mapped to a complexly shaped region in the state space that represents sets of possible combinations of state variables. The approximation of this complex region by a single interval box causes accumulation of overestimation over simulation time. The algorithm presented in this paper minimizes this so called wrapping effect by applying consistency techniques, to avoid safety-critical states.

Robust controller design for bounded state and control variables and uncertain parameters using interval methods

In this paper, a new interval arithmetic approach for analysis and design of robust controllers of systems with bounded state and control variables is presented. Based on guaranteed simulations of nonlinear dynamical systems with uncertain parameters, bounds of the admissible range of the control variables are calculated. During the computation of these bounds, time-dependant restrictions of the state variables arc considered as well as limitations of the control variables. To point out the advantages of this approach, it is applied to a subsystem model of the activated sludge process in biological wastewater treatment describing the reduction of biodegradable organic matter.

Robust analysis of flatness based control using interval methods

In this paper a new approach to the robustness analysis of flatness based tracking controllers using interval methods is proposed. This methodology allows us to explicitely determine admissible intervals for the uncertain parameters such that specified error bounds for the state space trajectories are not violated. In contrast to earlier approaches no additional controllers have to be considered to take robustness properties into account. The presented approach also poses no a priori restrictions on the velocity of the reference trajectory. The application of the robustness analysis is demonstrated for a feedforward as well as a feedback tracking controller for the Van de Vusse type continuous stirred tank reactor (CSTR).

Guaranteed Robust Tracking with Flatness Based Controllers Applying Interval Methods

Flatness based tracking controller design (see e.g. [4, 5, 16]) is one of the most important tools for the control of nonlinear systems. A drawback of this approach is the lack of methods for the robustness analysis of such controllers with respect to uncertain parameters in the plant. In [1] the application of interval methods has been proposed for the guaranteed robustness analysis of flatness based tracking controllers. This approach allows to explicitly calculate the deviations from the reference trajectory which are caused by uncertain parameters in the plant in a guaranteed way. In this contribution the analysis using interval methods is extended to the case when a nonlinear tracking observer is necessary to estimate unmeasured states. Furthermore it is shown that unknown sensor offsets can be included into this robustness framework. The approach is illustrated for a magnetic levitation system.

INTERVAL ANALYSIS AND NONLINEAR CONTROL OF WASTEWATER PLANTS WITH PARAMETER UNCERTAINTY

Abstract This paper presents the successful application of interval arithmetics to a simplified activated sludge model that describes the reduction of biodegradable substrate in biological wastewater treatment. Reliable analysis of the steady-state behaviour as well as plant control have to account for the dominant uncertain system parameter given by the maximum specific growth rate of biomass. The proposed control strategy consists of nonlinear control of oxygen concentration using desired trajectories derived from interval evaluations of the uncertain steady-state substrate concentration. By this, plant operating costs can be significantly reduced resulting in superior plant efficiency.

Modelling the glucose metabolism with backpropagation through time trained Elman nets

Type-I diabetes mellitus patients can not produce the hormone insulin endogenously. As this hormone is necessary to control the blood sugar level, which is raised by eating, insulin must be delivered exogeneously. Delivering insulin exogeneously demands correct dosage to avoid an extremely high or low blood glucose level. Most patients are not able to administer the adequate insulin dose because they are not able to predict the evolution of their own glucose level after a meal. Therefore, a model of the glucose metabolism is of crucial interest to help patients to determine correct insulin doses. These models shall be capable of predicting the course of the blood glucose level for a couple of hours with reasonable precision. In this paper a computer aided assistance system for diabetes patients running on a mobile handheld device is presented. This assistance system mainly consists of a model of the glucose metabolism, implemented by a modified Elman net. The training is performed through the BPTT algorithm where the training data were generated with an analytical non-linear glucose metabolism model that is quite realistic but cannot be adapted to every single patient. The glucose metabolism process is defined by two inputs, injected insulin and ingested glucose, and one output, namely the blood glucose. Due to the fact that metabolic processes in general have large time constants this process is characterized by the fact that the current net output, that is the blood glucose level, heavily depends on data that are not present in the current input layer any more. The Elman nets context-layer is capable of storing this information. Simulation results demonstrate that the output of this type of neural network closely follows the reference.

ROBUST FLATNESS BASED CONTROLLER DESIGN USING INTERVAL METHODS

Abstract Flatness based tracking controller design is one of the most important tools for the control of nonlinear systems. However, the tracking performance may become very poor if the actual plant parameters deviate too much from the nominal ones. In this contribution a robustness analysis using interval methods is applied in order to determine an optimal assignment for the desired tracking error dynamics such that the trajectories of the controlled system remain within specified tolerances around the reference trajectories for a maximum uncertainty of the unknown parameters. The robustness analysis is demonstrated for the Van de Vusse type continuous stirred tank reactor (CSTR).

Flatness-based control of a simplified wastewater treatment plant

This paper presents a nonlinear control approach for a simplified activated sludge model that covers the reduction of biodegradable substrate in biological wastewater treatment. In the proposed control scheme, the volume flow of oxygen into the aeration tank and the volume flow of excess sludge out of the settler tank are utilized as manipulated variables. As the oxygen concentration as well as the substrate concentration represent flat outputs, the proposed control strategy takes advantage of differential flatness. The inverse dynamics is evaluated in a feedforward manner using only desired values from a trajectory planning module in combination with stabilizing proportional-integral feedback of the substrate concentration as well as the oxygen concentration, respectively. Consequently, accurate tracking of desired trajectories for the flat outputs becomes possible without cost-intensive and often unreliable measurements of the bacteria concentration. As shown by simulation results, steady-state accuracy concerning the substrate concentration is guaranteed due to the Integral control part despite dominant parameter uncertainties concerning the maximum growth rate of bacteria and variations of substrate concentration in the influent volume flow as disturbance. Hence, regulations on the admissible substrate concentration at the plant output can be easily met and plant operating costs can be significantly reduced resulting in superior plant efficiency