Alexander Schaum

Estimating the state probability distribution for epidemic spreading in complex networks

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.

A simple observer scheme for a class of 1-D semi-linear parabolic distributed parameter systems

In this paper, a simple observer design scheme is proposed for a class of 1-D semi-linear parabolic distributed parameter systems with in-domain measurement. The main idea of the design resides in imposing the measurement on the observer dynamics in a similar way as it is done in the finite-dimensional reduced order observer design. For the distributed system this is achieved through an additional algebraic constraint in the domain instead of an output injection mechanism. Correspondingly, the observer convergence is not induced by the correction scheme, but by the sensor location and the system parameters, and thus amounts on the underlying detectability property rather than the system observability. Possible applications to the case of unknown inputs are discussed. The proposed estimation approach is illustrated and tested through simulations with two representative examples: an unstable diffusive linear system with Dirichlet boundary conditions, and a stable isothermal single-species nonlinear tubular reactor with non-monotonic kinetics and Robin-Newman (i.e., Danckwerts) boundary conditions.

Output-Feedback Control for Discrete-Time Spreading Models in Complex Networks

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=106 corroborate our theoretical findings.

Inversion-based feedforward control design for the Droop model

Abstract An inversion-based feedforward control for the Droop model for microalgae growth in the chemostat is presented. The dilution rate is manipulated in such a way that a desired biomass trajectory for set point changes is tracked. For this purpose a stable inversion technique is employed exploiting the asymptotic stability of the internal dynamics due to mass-conservation. The impact of unknown feed substrate step changes on the controller performance is analytically delimited using the mass-conservation-based bounded input-bounded output stability property. Numerical results show that the transition time can be significantly reduced by the proposed approach in comparison to a simple step change in the feed flow showing the bounded downgrade due to unknown variations in the feed concentration.

Backstepping Control for the Schrödinger Equation with an Arbitrary Potential in a Confined Space

In this work the control design problem for the Schrodinger equation with an arbitrary potential is addressed. In particular a controller is designed which (i) for a space-dependent potential steers the state probability density function to a prescribed solution and (ii) for a space and state-dependent potential exponentially stabilizes the zero solution. The problem is addressed using a backstepping controller that steers to zero the deviation between the initial probability wave function and the target probability wave function. The exponential convergence property is rigorously established and the convergence behavior is illustrated using numerical simulations for the Morse and the Poschl-Teller potentials as well as the semilinear Schrodinger equation with cubic potential.

Output-Feedback Control of Virus Spreading in Complex Networks With Quarantine

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.

Dissipative observers for coupled diffusion–convection–reaction systems

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.

Quasi-unknown input based 2-DOF control for a class of flat nonlinear SISO systems

The property of quasi unknown-input observability is put in perspective for the design of adaptive feedforward-feedback control for flat SISO systems. It is well-known that feedforward-feedback control for flat systems is quite straightforward given that the (flat) output dynamics is completely determined by the input and succesive output derivatives, thus allowing a direct algebraic solution to the feedforward control problem in absence of unknown inputs. In the presence of unknown inputs, their effects on the dynamics has to be compensated by some adaptation scheme. In this paper such an adaptation scheme is proposed for the SISO case (with respect to output, control input, and unknown input) for piecewise constant unknown inputs with relative degree n. The underlying observability property is discussed and a rigorous assessment of sufficient conditions for closed-loop exponential stability is provided. A representative case example with bistability and double saddle-node bifurcation is used to illustrate the main features of the approach.

A nonlinear quasi-unknown input observer for the chemostat Droop model

Abstract An unknown input observer is designed for the Droop model for microalgae growth in the chemostat considering the feed concentration as unknown input. Based on a biomass measurement, the nutrient quota and substrate are estimated together with the unknown (piecewise constant) feed concentration using a nonlinear observation scheme with PI-like correction mechanism. The estimation performance is illustrated with numerical simulations.