Backstepping PDE-based adaptive observer for a Single Particle Model of Lithium-Ion Batteries

Abstract

Abstract-Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof-Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L 2 -norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs.

Backstepping PDE Design: A Convex Optimization Approach

“…∀ (x, y) ∈ Ω L . Due to the quadratic term K 2 in the cost function (33) and the product c(y)K(x, y) in (35), this optimization problem is nonlinear, and in general, non-convex.…”

“…where Γ 0 is selected to achieve a required precision in the approximate solution of the Kernel-PDE (34)- (35), Γ 1 > − π 2 /4, ∀ x ∈ Ω, is selected according to the threshold of stability for (14), σ 0 > 0, σ 1 > 0 and σ 2 > 0 are weigh factors in the cost function, and…”

“…Proof: The convex formulation of (42)-(50) follows similar arguments as the ones given in the proof of Proposition 2. The residual δ 2 in (50) is an equivalent formulation of (35) by means of substituting the boundary condition (34) into (35) and using the relation described in (37). The residual δ 3 in (51) sets forth the condition (36), which is a necessary condition to use the relation (37).…”

“…On the other hand, (55) constrains the reactivity term c = c(x) to satisfy the threshold of stability in (14), ∀ x ∈ Ω. Therefore, by means of the term 1 0 f (x)dx, the optimization objective (41) minimizes the L 2 -norm of K(1, y), for a selected precision Γ 0 in the approximate solution of the Kernel-PDE (34)- (35).…”

“…A preliminary version of this paper has been published in [33]. Its application to the adaptive observer design problem has been presented in [34], [35].…”