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Neurological studies show that injured neurons can regain their functionality with therapeutics such as Chondroitinase ABC (ChABC). These therapeutics promote axon elongation by manipulating the injured neuron and its intercellular space to modify tubulin protein concentration. This fundamental protein is the source of axon elongation, and its spatial distribution is the state of the axon growth dynamics. Such dynamics often contain time delays because of biological processes. This work introduces an input delay compensation with state-feedback control law for axon elongation by regulating tubulin concentration. Axon growth dynamics with input delay is modeled as coupled parabolic diffusion-reaction-advection Partial Differential Equations (PDE) with a boundary governed by Ordinary Differential Equations (ODE), associated with a transport PDE. A novel feedback law is proposed by using backstepping method for input-delay compensation. The gain kernels are provided after transforming the interconnected PDE-ODE-PDE system to a target system. The stability analysis is presented by applying Lyapunov analysis to the target system in the spatial H1-norm, thereby the local exponential stability of the original error system is proved by using norm equivalence., Submitted and accepted to 17th IFAC Workshop on Time Delay Systems (TDS)

The Stefan PDE system is a representative model for thermal phase change phenomena, such as melting and solidification, arising in numerous science and engineering processes. The mathematical description is given by a Partial Differential Equation (PDE) of the temperature distribution defined on a spatial interval with a moving boundary, where the boundary represents the liquid-solid interface and its dynamics are governed by an Ordinary Differential Equation (ODE). The PDE-ODE coupling at the boundary is nonlinear and creates a significant challenge for state estimation with provable convergence and robustness. This tutorial article presents a state estimation method based on PDE backstepping for the Stefan system, using measurements only at the moving boundary. PDE backstepping observer design generates an observer gain by employing a Volterra transformation of the observer error state into a desirable target system, solving a Goursat-form PDE for the transformation's kernel, and performing a Lyapunov analysis of the target observer error system. The observer is applied to models of problems motivated by climate change and the need for renewable energy storage: a model of polar ice dynamics and a model of charging and discharging in lithium-ion batteries. The numerical results for polar ice demonstrate a robust performance of the designed estimator with respect to the unmodeled salinity effect in sea ice. The results for an electrochemical PDE model of a lithium-ion battery with a phase transition material show the elimination of more than 15 \% error in State-of-Charge estimate within 5 minutes even in the presence of sensor noise., 45 pages, 18 figures, submitted to Annual Reviews in Control

In this paper, we generalize our previous results of Newton-based extremum seeking of higher derivatives via predictors with averaging-based estimates for uncertain delays. We show necessary conditions for the robustness to small delay uncertainties of the proposed extremum seeking controller with predictor feedback. Local stability and exponential convergence to a small neighborhood of the unknown extremum can still be guaranteed. A numerical simulation example is presented to illustrate the performance of the predictor-based control scheme for uncertain time-delay compensation.

We present a generalized form of Halanay’s inequality, having a time varying gain multiplying the delayed term, and having a constant decay rate. Unlike the usual Halanay’s conditions where the decay rate is required to be strictly larger than an upper bound on the gain multiplying the delayed term, we provide less restrictive conditions that allow times when the decay rate can be strictly less than the gain. We include an application to continuous time systems with switched delay values. This illustrates the utility of our generalized Halanay’s inequality conditions for proving asymptotic stability in significant cases that violate the contraction condition that was needed to prove asymptotic stability in previous trajectory based results, and which are also not amenable to previous Lyapunov function constructions.

We study the finite-time stabilization of an ODE plant having diffusive infinite-dimensional actuator dynamics. We follow a predictor-feedback approach relying on time-varying backstepping, where the actuator utilizes time-dependent gains selected to stabilize the actuator dynamics as well as the plant within a finite time which can be prescribed independently of the system’s initial conditions. As the attenuative actuator dynamics cascade into the ODE, the time-varying feedback gains result in cascading backstepping kernel PDEs. We utilize differential flatness of the kernel PDE as well as the method of successive approximations in order to recover these kernels.

In this paper we design a partial differential equation (PDE) observer for state estimation of a fundamental component of traffic flow network: one incoming road segment and two outgoing road segment connected by a junction. Traffic state estimation refers to acquisition of traffic state information from partially observed traffic data. We propose a PDE model-based observer to estimate traffic density and velocity on the ”Y” shape traffic network from boundary sensing at the nodes including inlet, outlet and middle junction. The macroscopic traffic dynamics on each road segment are governed by Aw-Rascle-Zhang (ARZ) model, consisting of second-order nonlinear PDEs of traffic density and velocity. Using PDE backstepping method, we first construct a boundary observer from a copy of the plant and output injections from boundary measurement errors of the middle junction. The exponential stability of the estimation error system to zero in the L2 norm is achieved. For robustness of the observer design, we prove the Input-to-State Stability (ISS) of the estimation error system with respect to in-domain spatially distributed disturbances and measurement noise.

Background Acquired thrombotic thrombocytopenic purpura (aTTP) is a rare, life‐threatening autoimmune thrombotic microangiopathy. Current standard of care is therapeutic plasma exchange, immunosuppression, and caplacizumab, an anti‐von Willebrand factor nanobody, which is effective in treating aTTP episodes. Patients/Methods Here we report on seven episodes of aTTP treated without plasma exchange in six female patients in Germany and Austria. Two episodes were initial presentations of aTTP; in five instances, patients experienced a relapse. In four episodes, moderate to severe organ dysfunction was observed; three cases presented with a mild course. All patients received caplacizumab immediately once aTTP was suspected or diagnosed, and plasma exchange was omitted based on shared decision making between patient and the treating physicians. Results We observed a rapid and robust increase of platelet counts already after the first dose of caplacizumab, leading to a doubling of platelet counts within 17 hours (median), platelet counts normalized (>150 G/L) after median 84 hours. Lactate dehydrogenase, as a surrogate parameter of organ damage, improved in parallel to the platelet counts, indicating resolving microangiopathy. Conclusions In conclusion, in selected cases of acute bouts of aTTP, it seems feasible to delay or omit plasma exchange if platelet counts increase and organ function is stable after start of caplacizumab therapy.

We study state estimation of the linearized Schrodinger equation within a prescribed terminal time. We make use of a time–varying, complex–valued observer gain and boundary measurements to construct the observer, where the gain is designed such that the estimate error converges to zero within the terminal time. The observer gain proposed herein is developed via the backstepping method by selecting a target error equation that stabilizes to zero within the terminal time. Our time–varying observer gain diverges as time approaches the terminal time. Nevertheless, we can guarantee prescribed–time stabilization of the estimator error equation by characterizing the growth–in–time of the observer gain and comparing it to the stability of the target error equation. We develop the full–state feedback dual result, and we combine the boundary estimation and control results to develop prescribed–time output regulation.