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In this paper, the problem of observer design for a class of 1D nonlinear heat equations with pointwise in-domain temperature measurements is addressed. A pointwise measurement injection observer is designed and the robust convergence of its estimation error in presence of bounded distributed perturbations is established by verifying input-to-state stability. The obtained convergence conditions express the underlying interplay between heat conduction and radiation and include specific dependencies on the sensor locations which are the main degrees of freedom in the design approach. The theoretical results are experimentally validated on a semiconductor wafer processing unit.

Due to climate change, heavy rainfall events are occurring more frequently and at a higher rate. At the same time, the sewer system is often not designed to handle such a large amount of precipitation, which often leads to flooding of certain critical areas. However, modern sensor technology for monitoring the filling and flow rates in the sewer system, as well as for monitoring and forecasting rainfall probabilities, including quantity information, form a good basis for developing modern multifunctional early warning systems, which can serve as an aid for decisions on action. Urban water drainage systems represent complex networks with nonlinear dynamics and different types of interactions. This yields an involved modelling problem for which different off-line simulation approaches are available. Nevertheless, these approaches cannot be used for real-time simulations, i.e., running in parallel to weather now- and forecasts and enabling the monitoring and automatic control of urban water drainage systems. Alternative approaches, used commonly for automation purposes, involve parameterized linear delay systems, which can be used in real-time but lack the necessary level of detail, which is required for adequate flood risk prognostics. Given this setup, an approach for the effective modelling of detailed water drainage systems for real-time applications implemented with the open-source Storm Water Management Model (SWMM) software is addressed and exemplified for a part of the water drainage system of the city of Flensburg in northern Germany. Additionally, a freely available early-warning system prototype is introduced and used to combine weather forecast information on a 2-h prediction horizon with the developed model and available measurements. This prototype is subsequently used for data assimilation using the ensemble Kalman filter (EnKF) for the considered area in Flensburg. The project presented here is part of the NEPTUN project. NEPTUN is financed by Interreg Deutschland-Danmark with means from the European Regional Development Fund.

We prove the existence of a multidimensional nontrivial invariant toroidal manifold for the Kuramoto network with adaptive coupling. The constructed invariant manifold corresponds to the multicluster behavior of the oscillators phases. Contrary to the static coupling, the adaptive coupling strengths exhibit quasiperiodic oscillations preserving zero phase-difference within clusters. The derived sufficient conditions for the existence of the invariant manifold provide a tradeoff between the natural frequencies of the oscillators, coupling plasticity parameters, and the interconnection structure of the network. Furthermore, we study the robustness of the invariant manifold with respect to the perturbations of the interconnection topology, and establish structural, and quantitative constraints on the perturbation adjacency matrix preserving the invariant manifold. Additionally, we demonstrate the application of the new results to the problem of interconnection topology design, which consists in endowing the desired multicluster behavior to the network by controlling its interconnection structure.

The nonlinear stochastic dynamics of a class of two-state bioreactors with isotonic or nonisotonic kinetics is analytically characterized with Fokker–Planck (FP) theory, with emphasis on: (i) the spatiotemporal geometry of the two-state probability density function (PDF) motion, (ii) conditions for metastability-based bio-extinction/revival (inexistent in deterministic systems), and (iii) state PDF behavior of the optimal (maximum yield) operation. It is found that, depending on the kind of kinetics: (i) the stationary state PDF is mono or bimodal, (ii) the state PDF motions can be either non-metastable along deterministic-diffusion time scale, or metastable towards probabilistic extinction/revival along deterministic-diffusion-escape time scale, and (iii) the optimal operation can have robust (or fragile) stationary PDF, depending on the particular kinetics and operation condition. The developments and results are illustrated with representative examples with Monod and Haldane kinetics, and put in perspective with the ones drawn before with Monte Carlo (MC) and FP methods.

The observer design problem for the cell population estimation in a yeast batch process is addressed. The cell population balance model is described by a partial integro-differential equation coupled with a set of ordinary differential equations. Based on the observability property of the first moment of the cell distribution and the structural observability of the discretized cell population balance model, an extended Kalman Filter operating on online biomass measurements is designed for the model equations. The observer is validated using experimental data.

The stochastic dynamics of a two-state bioreactor model with random feed flow fluctuations and non-monotonic specific growth rate is analyzed. Using the Fokker-Planck equation approach for describing the probability density function (PDF) evolution the lack of stochastic robustness due to deterministic bifurcation phenomena for the open-loop reactor operating under optimal (maximum production) operation condition is established, and the associated stochastic stabilization problem is addressed. Inherent differences between the presence of multiplicative noise, due to the feed flow fluctuations, and additive background noise are analytically established. Numerical simulation results illustrate these inherent differences, the stochastic fragility of the open-loop operation yielding a stochastic extinction phenomenon, as well as the stochastic PDF stabilization with a proportional feedback control.

The parameter identification problem of cell population balance models in a yeast batch process is addressed. The model is described by a partial integro-differential equation coupled with a set of ordinary differential equations. Based on the solution equivalence of the mass balance model and the first moment of the cell population balance model, the identification process is carried out in two steps. The growth related parameters are identified by only taking mass balance based measurements into account. The growth parameters are then utilized to identify the parameters related to cell division in the cell population balance model by means of optimization. The results are verified with an R2 analysis.

A dissipativity-based observer design for a class of cell population balance models is presented and evaluated using experimental data. The population balance model is described by a partial integro-differential equation coupled with an ordinary differential equation for a batch bioreactor with biomass measurement via an optical density sensor. The dissipative observer is proven to exponentially converge in the first moment of the cell size distribution and the substrate concentration in the absence of modeling and measurement errors. The theoretical results are evaluated using experimental data from a 2 liter lab-scale reactor for anaerobic yeast fermentation on glucose using nitrogen gas supply, showing a good performance of the dissipativity-based estimation scheme.

Synchronization has been associated with fundamental brain functions, especially learning behavior. Electrical circuits are candidates to mimic the tremendous information processing ability of simple brain structures due to their inherent massive parallelism. This paper gives an electrical interpretation of synchronization behavior in linear, identical subsystems with diffusive couplings. We consider a general linear state-space model for which we synthesize a minimal, generic electrical circuit. A conductance models the couplings between subsystems to form the overall electrical system. To investigate the synchronization behavior, we show how a beneficial placement of transformers decouples the overall circuit and consequently obtain many smaller circuits that are easier to examine. It is shown that the asymptotic stability in these decoupled circuits leads to a vanishing synchronization error over time. Based on this observation, we are able to formulate a synchronization condition that is entirely dependent on electrical quantities. One benefit, among others, is that by the notion of passivity, the asymptotic stability in some electrical circuits becomes evident without any further calculation. Lastly, we apply these insights to a network of interconnected Chua circuits mimicking neuron populations and their synaptic coupling structure. We investigate different topologies, such as a two ring-topology with a bridge-synapse connection.

The multiple–input multiple–output (MIMO) output feedback (OF) control problem of an exothermic multi-jacket tubular open-loop unstable reactor is addressed. Over its axial length, the reactor has several equally sized cooling jackets. The controller must adjust the jacket temperatures on the basis of per jacket temperature measurements so that the closed-loop system is robustly stable. The problem is solved within a constructive framework, by combining notions and tools from chemical reactor engineering and partial differential equations (PDEs) control systems theory. The result is a MIMO nonlinear OF dynamic control design with (i) a decentralized MIMO passive state feedback (SF) controller implemented with a pointwise observer (PWO), (ii) closed-loop stability conditions in terms of sensor set and control gains, and (iii) efficient late lumping-based on-line implementation. The design is put in perspective with industrial PI and inventory control, and applied to a representative example through numerical simulation with favorable comparison against adaptive controllers.

Zusammenfassung Für gekoppelte semilineare Diffusions-Reaktionssysteme wird die Kombination der Backstepping-Methodik mit Dissipativitätskonzepten im Rahmen des Beobachterentwurfs vorgestellt. Dazu werden hinreichende Bedingungen für die Konvergenz des Beobachters unter Zuhilfenahme des intrinsischen Zusammenhangs zwischen Dissipativität und Lyapunovs direkter Methode hergeleitet. Numerische Simulationen für ein Beispiel nichtlinear gekoppelter, instabiler Diffusions-Reakti-onsgleichungen bestätigen die exponentielle Konvergenz der vorgestellten Verallgemeinerung des Backstepping-Konzepts für verteilt-parametrische Systeme.

Batteries are complex systems involving spatially distributed microscopic mechanisms on different time scales whose adequate interplay is essential to ensure a desired functioning. Describing these phenomena yields nonlinearly coupled partial differential equations whose numerical solution requires considerable effort and computation time, making it an infeasible solution for real-time applications. Anyway, having information about the internal electrochemical states of the battery can pave the way for many different advanced monitoring and control strategies with a big potential for improving efficiency and longevity. For such purposes, in the present paper, a combination of a low-order representation of the essential dynamics associated to the internal electrochemical mechanisms based on Dynamic Mode Decomposition for control (DMDc) is proposed to obtain an improved equivalent circuit model (ECM) representation with continuously updated parameters and combined with an extended Kalman Filter (EKF). The model-order reduction step extensively exploits the model structure, yielding a well structured low-order representation without artificial numerical correlations. The performance of the proposed method is illustrated with numerical simulations based on a well-established reference model, showing its potential usefulness in real-time applications requiring knowledge of the internal electrochemical states besides the state-of-charge.

The problem of dissipative proportional-integral (PI) output-feedback control for a class of semilinear heat equations with actuator disturbance is addressed. Sufficient conditions for strict dissipativity in the state and thus exponential stability of the closed-loop system are derived which are stated in terms of the dissipavity properties of the nonlinearity, the controller gains and the actuator and sensor shape functions (i.e., their form and localization). Based on these conditions a dissipation maximization procedure is proposed to appropriately choose the degrees of freedom using optimization algorithms. Numerical simulation results illustrate the performance of the proposed controller.

This chapter studies the input-to-state stability (ISS) property for nonlinear control systems with impulsive jumps at fixed moments. Sufficient conditions for the ISS are formulated in terms of a candidate ISS-Lyapunov function equipped with nonlinear rate functions which characterize the evolution of this function along the discontinuous trajectories of the system. For the case of unstable continuous dynamics, we derive new sufficient conditions for ISS under average-type dwell-time that provide a lower bound for the frequency of stabilizing jumps sufficient for the ISS.

The observability and detectability of a class of population balance models described by a partial integro-differential equation coupled with an ordinary differential equation is characterized for a simple two-species bioreactor model. By a direct calculation of the indistinguishable trajectories and employing the direct method of Lyapunov sufficient conditions for detectability are derived in terms of the cell division characteristic distributions. The cases of monotonic and non-monotonic growth rate functions are considered and the corresponding differences highlighted. An observer is designed with ensured convergence rate in the first moment of the distribution. Numerical simulation results illustrate the theoretical assessments.

The problem of output-feedback control for a class of semilinear unstable heat equations is addressed within the framework of dissipativity theory. It is shown that using a simple proportional output-feedback control in a collocated input-output pairing the closed-loop stability is ensured if the control gain is chosen sufficiently large and the nonlinearity satisfies a quadratic dissipativity condition. The approach is illustrated using numerical simulations.

The problem of state feedback control for a class of semilinear partial integro-differential equations is addressed within the framework of dissipativity theory. Adapting ideas from absolute stability theory the system is viewed as interconnection of a linear dynamic and a nonlinear static subsystem. Using this interpretation sufficient conditions for the stabilization of the zero solution by means of a backstepping state feedback controller are derived considering a class of nonlinearities satisfying a quadratic dissipativity condition. The approach is illustrated using numerical simulations for a representative case example.

The paper develops new sufficient conditions for synchronization of a network of $N$ nonlinearly coupled Chua oscillators interconnected via the first state coordinate only. The nonlinear coupling strength is governed by a function residing within a sector, i.e. it is bounded from above and below by linear functions. The derived sufficient conditions provide a trade-off between the characteristics of the sector and the interconnection topology of the network to guarantee the synchronization of the oscillators., Comment: 6 pages, 3 figures, submitted to IFAC Symposium on Nonlinear Control Systems (NOLCOS 2019)

The global-nonlinear stochastic dynamics of a class of two-state two-parameter continuous exothermic chemical reactors subjected to noise disturbances is studied with Fokker-Planck’s (FP’s) partial differential equation (PDE) theory. The dependency of the stochastic on deterministic dynamics is formally established, and the fundamental characteristics of the associated two-state probability density function (PDF) are described in analytic form, including: (i) stationary state PDF, (ii) convection, diffusion, and metastability time scales along which the state PDF transient occurs, as well as (iii) conditions for PDF metastability. The correspondence between the determination of stochastic stationary state PDF and the assessment of deterministic multiplicity via Van Heerden’s (VH) diagram is established. The developments and results are illustrated with numerical solution of FP’s PDE, and put in perspective with typical and atypical results drawn with Monte Carlo (MC) method.

The dissipativity-based observer design approach is extended to a class of coupled systems of 1-D semi-linear parabolic partial differential equations (PDEs) of diffusion–convection–reaction type with in-domain point measurements. This class of systems covers important application examples like tubular or catalytic reactors. By combining a dissipativity (sector) condition for the nonlinearity with a modal measurement injection for the linear differential operator sufficient conditions for the exponential convergence of the observer are derived in the form of a linear matrix inequality (LMI). The performance of the proposed approach is illustrated for an exothermic tubular reactor model.

In this paper the problem of observer design for a class of 1D scalar semi-linear parabolic distributed parameter systems with unknown boundary input is addressed. An unknown input observer is proposed in which the measurement information is imposed directly at the measurement point in form of an algebraic constraint, similar to the reduced order observer in a finite-dimensional set-up. Sufficient conditions for strong state detectability of the system and exponential convergence of the observer are drawn in terms of measurement locations and system parameters. The approach is illustrated with two case studies: (i) an unstable heat equation, and (ii) a diffusion-convection-reaction system with multiple steady-state attractor profiles.

The open and closed-loop global state probability density function behavior of a class of single-state nonlinear chemical processes subjected to additive and multiplicative white noises is characterized with analytic formulae through Fokker Planck (FP) theory, in terms of: (i) stationary state probability density function (PDF), (ii) PDF evolution along deterministic, diffusion, and escape time scales, (iii) conditions for PDF metastability along escape time scale, and (iv) dependency of PDF motion on deterministic dynamics. Comparing with noise additivity, multiplicativeness can yield similar or substantially different open-loop PDF evolution behavior. The application of control to an open-loop (possibly fragile and metastable) multimodal PDF yields a closed-loop robust monomodal PDF with mode regulation capability. The developments and findings are illustrated with numerical simulations of FP’s PDE equation.

Detectability is the key property which underlies the design of any convergent observer. In this paper a method for analyzing the detectability for a class of networks of systems with unknown inputs is presented, exploiting the definition of indistinguishable trajectories and the structure of the underlying network graph. This analysis yields sufficient conditions for the sensor location in order to ensure the system’s strong detectability in presence of unknown inputs. Based on this analysis a simple (reduced order) unknown-input observer with explicit convergence conditions in terms of system parameters and the sensor locations is proposed and tested in numerical simulations for an illustrative case example of a Markov-chain based epidemic spreading model in a scale-free network with N = 103 nodes.

The problem of estimating the spatio-temporal temperature and concentration distribution in a tubular reactor with first order exothermic reaction and in-domain pointwise temperature measurements is considered. The problem is addressed using a simple pointwise innovation (PWI) structure, which imposes the measurement information at the sensor location in the simulator. Rigorous convergence conditions are presented in terms of the sensor location and reactor parameters. The approach is tested in numerical simulation studies and put in perspective with the (continuous time) extended Kalman filter (EKF) for a benchmark reactor with up to five steady-state profiles. In the simulations the proposed PWI scheme shows a good convergence behavior even in the presence of large initial errors, unmeasured feed perturbations, measurement noise and parameter mismatch.

A feedforward-feedback control strategy for trajectory tracking of microalgae cultures is developed and experimentally evaluated. On the basis of the Droop model an inversion-based feedforward control is determined, which steers the biomass from given initial to final values in a finite prescribed time. Feedback control is employed for disturbance resilience and improving the stability properties of the desired trajectory. The approach is tested using numerical simulations and experiments in a lab-scale photocatalytic reactor with Chlorella sorokiniana.

In the present paper a nonlinear feedback control design for a class of coupled 1-D semilinear partial differential equations with in-domain point measurements and in-domain averaged or homogeneously distributed actuator configurations is addressed. The objective is to stabilize a desired (potentially open-loop unstable) profile with a nonlinear controller that mitigates the effect of the nonlinearity at the most sensitive location, for this purpose the actuator and sensor characteristics are key design degrees of freedom. Stability of the zero profile for the closed-loop system is ensured by studying it as a Lur’e system and by applying modal techniques while its uniqueness is studied using a bifurcation analysis and aims to an auxiliary sensor location criterion. The performance of the proposed controller is illustrated using numerical simulations.

The problem of state estimation for a class of continuous bioreactors with sampled biased measurements with nonperiodic rates is addressed within the framework of impulsive dynamical systems. Explicit conditions on the maximum time between measurements are established which ensure input-to-state stability of the estimation error with respect to the measurement uncertainty and, additionally, finitetime practical stability of the estimation error. The theoretical results are illustrated on the case study of a reactor with nonmonotonic reaction rate for which the state is not observable from continuous time measurements but only detectable.

In this work, the problem of stabilizing, about of a (possibly open-loop unstable) steady-state profile and through feedback control, a class of semilinear parabolic partial differential equations with collocated sensor and actuator setup is addressed. First, a state-feedback controller is build through an extension to the infinite-dimensional case of the notion of passivity for finite-dimensional systems. The stability analysis of the zero dynamics is carried out using by Lyapunov and modal techniques. Then the output-feedback version of the controller is constructed by combining the state-feedback controller with a pointwise measurement injection observer. The performance of the approach is illustrated with numerical simulations.

The problem of state estimation of spreading phenomena in complex networks is considered on the basis of a detectability-based approach. Using a simple, reduced model based state distribution estimator, where the monitored nodes are driven directly by the measured data, asymptotic convergence conditions are provided in terms of the number and location of the required sensors on the basis of the network topology. The convergence of the estimator is established in terms of the largest eigenvalue of a reduced connectivity matrix which stems from removing the monitored nodes and their connections from the original graph. In the case of unit weights, this condition corresponds to measuring the nodes with highest degree. Numerical simulations for a complete and a scale-free network each of 500 nodes and randomly distributed and unit weights, respectively, illustrate the estimator functioning with 20 sensors for the complete, and 38 sensors for the scale-free network.

The problem of state estimation for microalgae growth processes in batch cultures based on an extended Droop model is addressed. Sufficient conditions for local strong observability are derived within an unknown input framework considering a biased optical density (OD) measurement and a (non-biased) substrate measurement. To account for discrete-time measurements with different sampling rates a continuous-discrete extended Kalman Filter is designed and tested as well as within numerical simulations as also with experimental data for Haematococcus pluvialis.

In this paper the problem of designing an output-feedback control for the stabilization of the extinction steady-state in a virus spreading process over a complex network with quarantine is considered. Sufficient conditions are established for the choice of those nodes for which sensor information is necessary and those which should be controlled using notions from constructive control theory. A simple output-feedback control is proposed which exponentially stabilizes the extinction state. Numerical simulation results are provided to illustrate the functioning of the proposed control scheme for a scale-free network of one million nodes.

In the present paper the dissipativity-based observer design approach is extended to a class of 1-D semi-linear parabolic PDE systems with pointwise in-domain measurement. On the basis of a modal innovation over the set of slow eigenvalues of the linear operator and the consideration of a dissipativity (sector) condition for the nonlinearity an exponential convergence condition for the estimation error is derived in form of a linear operator inequality, which is solved on the basis of an eigenfunction expansion of the linear operator. The approach is illustrated by numerical simulations for a coupled two-state semi-linear PDE system.

The problem of estimating the spatio-temporal temperature and concentration distribution in a tubular reactor with first order exothermic reaction and in-domain pointwise temperature measurement is considered. The problem is addressed using a simple point innovation structure, which imposes the measurement information at the sensor location in the simulator. Rigorous convergence conditions are presented in dependence of the sensor location and reactor parameters in terms of the dominant eigenvalues of the modified linear transport operators and the Lipschitz constant of the reaction rate. Numerical simulation results show that the proposed approach yields good performance even in the presence of unknown feed temperature perturbations.

Motivated by problems in process design, monitoring and control, the effect of multiplicative stochastic uncertainty injection on the behavior of nonlinear dynamical system is studied with Fokker Planck theory for two rather simple case examples selected to draw rigorous results with analytic formulae. The effect of noise multiplicativeness on the shape of the stationary state PDF and of the transient one along deterministic, probability diffusion, and escape time scales is characterized. The findings are corroborated and illustrated with numerical simulations. The results provide insight to assist the improvement of numerical Monte Carlo and polynomial chaos methods for stochastic multi-state multi-noise dynamical chemical processes underline by complex nonlinear behavior.

In this work, an experimental implementation of a model based output feedback (OF) scheme applied to regulate the volatile fatty acids concentration of an anaerobic digestion pilot plant is presented. The OF scheme was developed within a global nonlinear dynamics framework, resulting in an interlaced control-observer design in the light of passivity, observability, and bifurcation properties. From a industrial control perspective, the resulting OF scheme is a saturated PI controller with: (i) systematic construction and tuning and (ii) nonlocal robust stability conditions in terms of control gains and limits. Experimental results demonstrate the applicability, robustness and reliability of the OF control scheme in a real environment under several load disturbances, severe uncertainties and typical operational failures.

The saturated OF control problem for a class of four-state anaerobic digesters with volatile fatty acids (VFAs) measurement is addressed. The reactor must operate about an optimal steady-state, with maximum VFA consumption that is locally open-loop stable but not necessarily structurally stable. The problem is addressed as an interlaced control-observer design in the light of passivity, observability, and bifurcation properties. The result is a saturated linear PI controller with: (i) systematic construction and tuning and (ii) nonlocal robust stability conditions in terms of control gains and limits. The proposed approach is illustrated with a representative case example through numerical simulations, including a comparison with the switching control approach.

The problem of stabilizing the spreading process to a prescribed probability distribution over a complex network is considered, where the dynamics of the nodes in the network is given by discrete-time Markov-chain processes. Conditions for the positioning and identification of actuators and sensors are provided, and sufficient conditions for the exponential stability of the desired distribution are derived. Simulations results for a network of N = 10 6 corroborate our theoretical findings.

In this study, the dynamics of decisions in complex networks subject to external fields are studied within a Markov process framework using nonlinear dynamical systems theory. A mathematical discrete-time model is derived using a set of basic assumptions regarding the convincement mechanisms associated with two competing opinions. The model is analyzed with respect to the multiplicity of critical points and the stability of extinction states. Sufficient conditions for extinction are derived in terms of the convincement probabilities and the maximum eigenvalues of the associated connectivity matrices. The influences of exogenous (e.g., mass media-based) effects on decision behavior are analyzed qualitatively. The current analysis predicts: (i) the presence of fixed-point multiplicity (with a maximum number of four different fixed points), multi-stability, and sensitivity with respect to the process parameters; and (ii) the bounded but significant impact of exogenous perturbations on the decision behavior. These predictions were verified using a set of numerical simulations based on a scale-free network topology.

In this paper a method for the design of state observers for a class of complex network systems with nonlinear dynamics is presented. Necessary conditions on the dynamics of the nodes are given, i.e. how the interconnection enters and which states have to be measured, such that the arbitrary complex interconnection can be dealt with as an unknown input. Sufficient existence conditions for an Unknown Input Observer (UIO) for this class of systems are stated in terms of an LMI representing the dissipativity properties of the system. The design method is applied to monitor an epidemic disease in a complex network.

In this work, we present a method for estimating the diffusion coefficient in isothermal drying processes in those cases in which the associated shrinkage cannot be neglected. We assume that the drying process is described by the diffusion equation and depart of a numerical solution of reduced order to the corresponding problem. The method chosen for identifying the parameter in question is based in a high gain filter. As an example for showing the proposed method we take a solid presented by Aguerre et al. (Latin American Applied Research 2008;38:345–9). First, we solve the direct problem for generating an average moisture content profile and the corresponding radius profile. Taking the radius evolution as known data and the average moisture evolution as the innovation term in the filter, we solve the inverse problem and as a result a moisture-dependent diffusion coefficient is obtained, which is refined in a posterior and final optimization step.