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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.

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.

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.

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.

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.

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.

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.

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.

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.

The saturated Output-Feedback (OF) control problem for a class of four-state anaerobic digestors with volatile fatty acids (VFAs) measurement is addressed. The reactor must operate about an optimal steady-state, with maximum VFA consumption, that is locally stable but not structurally unstable. The problem is addressed as an interlaced control-observer design within a geometric control framework in the light of passivity, observability, and bifurcation properties. The result is a saturated linear PI OF controller with global robust stability conditions in terms of control gains and limits. The proposed approach is illustrated with a representative case example through numerical simulations.

The robust feedforward (FF) output-feedback (OF) control problem of (possibly open-loop unstable) continuous exothermic jacketed reactors with isotonic kinetics with respect to reactant concentration is considered. The volume, temperature, and concentration must be regulated by manipulating the feed, exit and coolant flowrates on the basis of volume and temperature measurements. The problem is addressed as an interlaced controller-observer design within a constructive control framework. The result is a quasi linear-decentralized (qLD) FF-OF dynamic controller which: (i) recovers (up to observer convergence) the behavior of a robust nonlinear FF-SF controller, (ii) has closed-loop stability conditions coupled with conventional-like tuning guidelines, and (iii) constitutes an add-on to the PI control scheme commonly employed in industrial reactors. The approach is tested with an open-loop unstable reactor example through simulations.

The problem of designing a saturated Output-Feedback (OF) controller for a three species continuous fermenter with substrate and product-inhibited kinetics is addressed. The reactor must be operated at maximum production rate by manipulating the feed rate on the basis of a (biomass, substrate or reaction rate) measurement. The combination of global dynamics concepts with State-Feedback (SF) and OF nonlinear control tools yields a linear dynamical OF saturated control design with: (i) robust stability conditions in terms of the controller gains and saturation limits, and (ii) simple tuning guidelines. The linear OF saturated controller: (i) recovers, in a practical stability sense, the behavior of a globally stabilizing nonlinear robust SF saturated controller, and (ii) becomes a linear PI controller in the absence of saturation. The proposed approach is illustrated and tested with a representative example through simulations.

The problem of regulating the biomass growth rate at its maximum value in an open-loop unstable, continuous, non-monotonic biological reactor using biomass measurement and manipulation of substrate feed rate is considered. An Output-Feedback controller is designed exploiting the reactor open-loop stability and inherent observability properties and the relative degree structure to obtain robust and non-wasteful closed-loop performance. The resulting controller has a high degree of independency with respect to the kinetic growth rate expression, thus ensuring robustness against typical model uncertainties. A representative case example with non-monotonic Haldane kinetics is employed to test the proposed controller, in absence and presence of modeling and measurement errors.

The paper considers the observer design problem for a class of scalar distributed transport-reaction systems with boundary measurements, mixed Dirichlet-Neumann boundary conditions, linear (diffusive-convective) transport and nonlinear conically bounded source. A direct Lyapunov method is developed for the exponential stability analysis of the observation error. The Lyapunov functional is chosen with a constant kernel for the small values of the so-called Peclet number (i.e. the quotient of diffusion to convection time) and with a spatial-dependent kernel for high Peclet numbers. The derived Lyapunov-based stability conditions correspond to the ones based on the frequency-domain analysis. A numerical example illustrates the efficiency of the method.

The problem of controlling a (possibly open-loop unstable) continuous exothermic reactor with non-monotonic kinetics, temperature measurements, reactant and heat exchange rates as manipulated inputs, and operation at maximum production rate is addressed within a robustness-oriented practical (input-to-state nonlocal) stability framework, according to the related detectability and relative degree structure. The result is a dynamical output-feedback (OF) controller made by the combination of a passive state-feedback (SF) nonlinear controller with a dissipative nonlinear observer. The proposed design methodology has: (i) solvability conditions with physical meaning, and (ii) a closed-loop stability criterion coupled with simple tuning guidelines. The proposed approach is tested with a representative case example through numerical simulations.

The problem of designing a globally convergent observer for a class of tubular reactors with boundary measurements is addressed. The problem is tackled by extending a dissipativity theory-based observer design for nonlinear finite-dimensional systems, which has been recently applied to a class of continuous stirred tank reactors. The underlying idea of the proposed observer approach consists in designing the tubular reactor data-assimilation scheme so that the estimation error dynamics are given by a two-dissipative system interconnection: one linear distributed dynamical system with the convective and diffusive mechanisms, and one nonlinear lumped static system with the reaction kinetics. The approach is applied to a tubular reactor with non-monotonic kinetics and boundary measurements, and the associated sufficient solvability condition was identified and interpreted in terms of dissipativity and dimensionless numbers with physical meaning.

The observability properties of a 3-D aerobic biological reactor are analyzed in a general global sense. Using basic definitions of indistinguishability observability and detectability a recently developed method is used to determine a dynamical representation of all possible indistinguishable trajectories of the model for the reactor class. This dynamics is analyzed for the representative case of a non-monotonic Haldane-type kinetics law. It is shown that the reactor model is not observable. Nevertheless, using the theory of asymptotic autonomous systems it is shown that the system is detectable under practically reasonable conditions.

The problem of controlling continuous exothermic jacketed reactors with non-monotonic kinetics as well as temperature, level and flow measurements is addressed. The application of a constructive control procedure, with emphasis on the attainment of linearity, decentralization, robustness and model independency features, yields: (i) a control scheme with PI volume and cascade temperature loops, (ii) a material balance concentration controller driven by the integral actions of the temperature controller, and (iii) a closed-loop nonlocal stability criterion coupled with conventional-like tuning guidelines. The approach is tested with a representative example through simulations.