Boundary control of an unstable heat equation via measurement of domain-averaged temperature

Abstract

In this note, a feedback boundary controller for an unstable heat equation is designed. The equation can be viewed as a model of a thin rod with not only the heat loss to a surrounding medium (stabilizing) but also the heat generation inside the rod (destabilizing). The heat generation adds a destabilizing linear term on the right-hand side of the equation. The boundary control law designed is in the form of an integral operator with a known, continuous kernel function but can be interpreted as a backstepping control law. This interpretation provides a Lyapunov function for proving stability of the system. The control is applied by insulating one end of the rod and applying either Dirichlet or Neumann boundary actuation on the other.

Stability Analysis of the Heat Equation with Time-Delayed Feedback

“…Note that Ω(−a 1 ) = 1 and Ω(a−a 0 ) > 1 for all parameters a − a 0 > −a 1 . Therefore, if a 1 τ < 1, then the system is stable for all a − a 0 > −a 1 .…”

“…In [9] the same problem is considered with time varying delay, and a sufficient condition for stability is derived in terms of a linear matrix inequality formed by the parameters a, a 0 , a 1 , τ and the upper bound of the derivative of τ . In [1,4] the feedback contains the non-delayed term only f (z) = a 0 z(x, t) and in [11] it takes the form f (z) = a 0 z(x, t) + ∂ ∂x z(x, t). Other, possibly nonlinear, feedback forms are also considered in the literature, see e.g.…”

When is the discretization of a spatially distributed system good enough for control?

“…Consider the following nondimensionalised heat equation in a medium of one spatial dimension (Boskovic, Krstic & Liu 2001) with a measurement of temperature gradient at one end:…”

Explicit state and output feedback boundary controllers for partial differential equations

“…The methods for the boundary control of (1)-(2) include pole placement [9], LQR [3], and finitedimensional backstepping [12]. In [13], two stabilizing controllers (backstepping and pole placement) were constructed in a closed form, but only for λ 0 /ε < 3π 2 /4, i.e., in case of one unstable eigenvalue. However, the explicit (closed-form) boundary stabilization result in the case of arbitrary ε, λ 0 is not available in the literature even for this benchmark constant coefficient case.…”