Shumon Koga

Backstepping control of the one-phase stefan problem

In this paper, a backstepping control of the one-phase Stefan Problem, which is a 1-D diffusion Partial Differential Equation (PDE) defined on a time varying spatial domain described by an ordinary differential equation (ODE), is studied. A new nonlinear backstepping transformation for moving boundary problem is utilized to transform the original coupled PDE-ODE system into a target system whose exponential stability is proved. The full-state boundary feedback controller ensures the exponential stability of the moving interface to a reference setpoint and the ℋ1-norm of the distributed temperature by a choice of the setpint satisfying given explicit inequality between initial states that guarantees the physical constraints imposed by the melting process.

Output feedback control of the one-phase Stefan problem

In this paper, a backstepping observer and an output feedback control law are designed for the stabilization of the one-phase Stefan problem. The present result is an improvement of the recent full state feedback backstepping controller proposed in our previous contribution. The one-phase Stefan problem describes the time-evolution of a temperature profile in a liquid-solid material and its liquid-solid moving interface. This phase transition problem is mathematically formulated as a 1-D diffusion Partial Differential Equation (PDE) of the melting zone defined on a time-varying spatial domain described by an Ordinary Differential Equation (ODE). We propose a backstepping observer allowing to estimate the temperature profile along the melting zone based on the available measurement, namely, the solid phase length. The designed observer and the output feedback controller ensure the exponential stability of the estimation errors, the moving interface, and the ℋ1-norm of the distributed temperature while keeping physical constraints, which is shown with the restriction on the gain parameter of the observer and the setpoint.

Gradient extremum seeking for static maps with actuation dynamics governed by diffusion PDEs

We design and analyze the scalar gradient extremum seeking control feedback for static maps with actuation dynamics governed by diffusion PDEs. Conceptually, a non-model based online optimization control scheme is paired with actuation dynamics which occur in chemistry, biology and economics. A learning-based adaptive control approach with known actuation dynamics is considered in this paper. In the design part, we first compensate the actuation dynamics in the dither signals. Secondly, we introduce an average-based actuation dynamics compensation controller via a backstepping transformation, which is fed by the perturbation-based gradient and Hessian estimates of the static map. The stability analysis of the error-dynamics is based on using Lyapunov’s method and applying averaging for infinite-dimensional systems to capture the infinite-dimensional state of the actuator model. Local exponential convergence to a small neighborhood of the optimal point is proven and illustrated by numerical simulations.

Backstepping control of the Stefan problem with flowing liquid

This paper presents a backstepping control design for the one-phase Stefan problem with flowing liquid. The problem is modeled by a 1-D diffusion-advection Partial Differential Equation (PDE) defined on a time-varying spatial domain described by an ordinary differential equation (ODE). Based on our recent contribution, a nonlinear backstepping transformation for moving boundary problems is utilized, and the associated boundary feedback controller is designed. We show that for the closed-loop system a set of physical inequalities is guaranteed, provided that the setpoint and initial values verify some (reasonable) constraints. These constraints also ensure the exponential convergence of the moving interface to a setpoint and exponential stability (in the ℋ1 norm) of the temperature equilibrium profile. A numerical study shows the performance of the proposed method.

Arctic sea ice temperature profile estimation via backstepping observer design

Recent rapid loss of the Arctic sea ice motivates the study of the Arctic sea ice thickness. Global climate model that describe the ices thickness evolution, require an accurate temperature profile of the Arctic sea ice. However, measuring the complete temperature profile is not feasible due to a limited number of thermal sensors. Instead, measuring the ices thickness is doable based on the acquisition of data from submarine and satellite devices. In this paper, we develop a backstepping observer algorithm to estimate temperature profile for the Arctic sea ice model via available measurements of sea ice thickness, sea ice surface temperature, and temperature gradient at the ice-ocean interface. The observer is designed in a rigorous manner to drive the temperature profile estimation error to zero, for salinity-free sea ice model. Moreover, the proposed observer is used to estimate the temperature profile of the original sea ice model with salinity via numerical simulation. In comparison with the straightforward open-loop algorithm, the simulation results illustrate that our observer design achieves faster convergence of the estimated temperature.