4. An ODE Observer for Lyapunov-Based Global Stabilization of a Bioreactor Nonlinear PDE

We solve the stabilization problem of a neutrally stable nonlinear hyperbolic PDE with in-domain actuation, which models the population dynamics in a bioreactor, where the “spatial variable” is not the physical space but the age of the microorganisms being grown. The control challenges arise from (1) the structure of the plant dynamics in which the full state of the system gets recirculated back from the PDE domain to the inlet (birth) boundary condition, (2) the fact that control (harvesting rate across the entire age range) multiplies the state, (3) the fact that the state (population density distributed by age) must be kept nonnegative, and (4) the fact that the renewal kernel (the birth rate at different ages) is unknown. We find a nonlinear infinite-dimensional transformation which reveals that the system’s relative degree is one and that its zero dynamics are autonomous and exponentially stable, which we prove using a Lyapunov–Krasovskii functional. We take advantage of this structure and achieve stabilization of a desired measured population density under a saturated harvesting input and using a finite-dimensional observer-based feedback, where the observer estimates the harvesting rate setpoint, which depends on the unknown renewal kernel (birth rate).