The Chaos Effect by Pierre Christin
By Pierre Christin
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The simplicity of ﬁnding the gains is one of the main beneﬁts of the backstepping method. The standard methods for control of PDE, which are “PDE extensions” of linear-quadratic-optimal (LQR) methods for ﬁnite-dimensional systems, obtain their gains through the solution of Riccati equations—quadratic operator-valued equations, which are, in general, hard to solve. However, the numerical advantages that backstepping oﬀers are just one of its appeals. The other is the conceptual elegance of using a feedback transformation that eliminates exactly the part of the PDE that is undesirable (or adds a part that is missing in a PDE, like damping), while leaving the PDE in a familiar form where physical intuition can be used in shaping the closed-loop dynamics.
133) for G can be integrated, yielding as integral equation for G G(δ, η) = −g(η) − η + 0 + 1 4 δ 1 4 1 τ dτ − 2 4 λ η G (η + τ, η − τ ) g(τ )dτ + δ η μ+η−s G(τ, s)f η 0 μ δ s+τ s−τ , 2 2 η f 0 η δ 1 4 η λ 0 η τ −s 2 τ −s τ +s ,μ− 2 2 dτ ds G(τ, s)dsdτ dτ dsdμ. 136). 2 Mathematical Preliminaries and Notation 27 and for n ≥ 1, η Gn (δ, η) = Gn−1 (δ, η) + + + 1 4 1 4 δ η δ μ+η−s μ η λ η Gn−1 (η + τ, η − τ ) g(τ )dτ Gn−1 (τ, s)f 0 η 0 0 τ −s 2 τ +s τ −s ,μ− 2 2 Gn−1 (τ, s)dsdτ. 138) Then it can be shown that G(δ, η) = lim Gn (δ, η).
5 Simulation Study 51 This establishes asymptotic stability for the plant in the z-, w-coordinates, when ∈ (0, ∗ ). 67) which are derived taking the norm in the respective deﬁnitions. 29) is exponentially stable at the origin in the L2 sense, that is, there exist positive constants M and α, independent of the initial conditions, such that R2 2π v 2 (t, s) + ≤ M e−αt τ 2 (t, s, φ)dφ sds 0 R1 R2 2π v 2 (0, s) + R1 τ 2 (0, s, φ)dφ sds. 8) and the form of the boundary conditions. 5 Simulation Study We show a prototypical simulation case.