Dual strategy for 2‐dimensional PDE optimal control problem in the reheating furnace

Abstract

Summary In this paper, the dual strategy is used to solve the 2‐dimensional partial differential equation optimal control problem by introducing the adjoint problem approach to the optimization model. It is proved that the Fr échet gradient of the cost functional could be written via the weak solution of the adjoint problem, and then, Lipschitz continuity of the gradient is derived. An improved conjugate gradient algorithm is used to solve this problem. Then, the proposed method is applied to obtain the reference values of the optimal furnace zone temperatures and achieve the desired temperature for steel slabs in the reheating furnace. Model validation and comparison between the mathematical model and the experiment results indicate that the present heat transfer model works well for the prediction of thermal behavior about the slab in the reheating furnace. Results of some computational experiments in the simulations of the A3 slab are illustrated, which verifies the effectiveness of the proposed method.

A combined adaptive entropy‐TOPSIS and model predictive control strategy for mixed loading and delay operations in the reheating furnace

“…The walking-beam type reheating furnace is 42.7 m in length and 12.6 m in width. 34 The highest furnace roof is about 5.18 m and shrinks to 4.03 m in the final soaking zone. The effective length of the whole furnace is 40 m. The effective lengths of the convection zone, preheating zone, heating zone 1, heating zone 2, and soaking zone are 0.3, 12.7, 7.7, 8.1, and 10.4 m, respectively.…”

First-Optimize-Then-Discretize Strategy for the Parabolic PDE Constrained Optimization Problem With Application to the Reheating Furnace

“…Choose an initial point w 0 = {u (t) , b (t)}. 2: Solve the original parabolic PDE (5) and the adjoint PDE (9) based on the input value w k , then we can obtain the T k (y, t) and p k (y, t). 3: Calculate the gradient of the cost functional g k based on the (23),…”

“…For instance, Wang et al [8] use the FOTD approach to solve a two-dimensional parabolic PDE-constrained optimal control problem, which is applied for the cooling process of steel billets in continuous casting secondary cooling zones. Luo et al [9] use the FOTD strategy to solve the 2-dimensional PDE optimal control problem to obtain the reference values of the optimal furnace zone temperatures for the reheating furnace. Chen et al [4] follow the FOTD approach to solve a coupled system of PDEs, which is derived from the continuous first order optimality conditions.…”

“…2 , t dt (8) According to the book [21] by M. Hinze, namely such that the equation ∆L = 0 forT , this is nothing else but the adjoint equation. Thus, it is easy to obtain the following adjoint equation (9). ∂p(y,t;w) ∂t…”

Parallel Numerical Calculation on GPU for the 3-Dimensional Mathematical Model in the Walking Beam Reheating Furnace

“…Li et al [9] make the simultaneous estimation of heat fluxes through the upper, side and lower surface of a steel slab based dynamic matrix control (DMC) in a walking beam type reheating furnace. Luo et al [10] build the 2D temperature prediction model rather than a precise 1D model and introduce the first-optimizethen-discretize approach to solve the 2D optimal control problems. Model validation and comparison indicate that the present 2D model works well for the prediction of thermal behavior about the slab in the reheating furnace.…”