Optimizing the Placement and Number of Measurement Points in Heating Process Control

Abstract

In this paper, a heating process control law with steam supply is designed for a fluid in a heat exchanger. The process is described by a linear hyperbolic equation of the first order with a nonlocal boundary condition with a time-delayed argument. The temperature of the supplied steam is found as a linear dependence on fluid temperature values at measurement points in the heat exchanger. Explicit formulas are obtained for the gradient of the objective functional of the control problem in the space of the feedback coefficients (parameters) of this dependence. A numerical scheme is developed for determining the feedback parameters based on these formulas. Finally, an algorithm is proposed for determining the rational (optimal) number of measurement points.

APPENDIX

Proof of Theorem 1. Explicit formulas expressing the increment of the objective functional through the increments of its optimized arguments can be obtained using the well-known method [25]. The linear part of the increment of the functional in each argument is the required component of its gradient with respect to the corresponding argument.

First of all, note the following. The initial fluid temperature φ ⊂ Φ and the heat loss coefficient γ ∈ Γ are mutually independent and do not depend on the heating process in the heat exchanger. Hence, from (2.11) and (2.12) it follows that

$$\begin{aligned} \operatorname{grad} J({\mathbf{P}}) = \operatorname{grad} \int\limits_\Phi {\int\limits_\Gamma {I({\mathbf{P}};\,\,\varphi ,\,\,\gamma ){{\rho }_{\Gamma }}(\gamma ){{\rho }_{\Phi }}(\varphi )d\gamma } d\varphi } \\ = \int\limits_\Phi {\int\limits_\Gamma {\operatorname{grad} I({\mathbf{P}};\,\,\varphi ,\,\,\gamma ){{\rho }_{\Gamma }}(\gamma ){{\rho }_{\Phi }}(\varphi )d\gamma } d\varphi } , \\ \end{aligned} $$

(A.1)

where

$$I({\mathbf{P}};\,\,\varphi ,\,\,\gamma ) = \int\limits_{{{t}_{b}}}^{{{t}_{f}}} {{{{[T(L,\,\,t;\,\,{\mathbf{P}},\,\,\varphi ,\,\,\gamma ) - V]}}^{2}}dt} + \varepsilon {{\left\| {{\mathbf{P}} - {\mathbf{\tilde {P}}}} \right\|}_{{{{\mathbb{R}}^{{3N}}}}}}.$$

Therefore, the formula for grad I(P; φ, γ) will be derived under arbitrary admissible feedback parameters P, heat loss coefficient γ ∈ Γ, and initial condition T(x, t) = φ, t $\leqslant$ 0.

Let T(x, t; P, φ, γ) be the solution of the loaded initial boundary-value problem (2.12), (2.3), and (2.5) under arbitrarily chosen optimized parameter vector P = (μ, α, β)', initial condition φ ∈ Φ and heat loss coefficient γ ∈ Γ. For brevity, whenever no confusion occurs, the parameters P, φ, γ of the solution T(x, t; P, φ, γ) will be omitted.

Consider an admissible increment ΔP = (Δμ, Δα, Δβ)' of the parameters P = (μ, α, β)' and let $\tilde {T}$(x, t) = $\tilde {T}$(x, t; ${\mathbf{\tilde {P}}}$, φ, γ) = T(x, t) + ΔT(x, t) be the solution of problem (2.12), (2.3), and (2.5) that corresponds to the incremental argument vector ${\mathbf{\tilde {P}}}$ = P + ΔP.

Substituting the function $\tilde {T}$(x, t) into conditions (2.12), (2.3), and (2.5) gives the initial boundary-value problem

$$\begin{gathered} \Delta {{T}_{t}}(x,\,\,t) + \vartheta \Delta {{T}_{x}}(x,\,\,t) = \lambda \sum\limits_{i = 1}^N {\left[ {{{\alpha }_{i}}\Delta T({{\mu }_{i}},\,\,t) + {{\alpha }_{i}}{{T}_{x}}({{\mu }_{i}},\,\,t)\Delta {{\mu }_{i}}} \right.} \\ \left. { + \;(T({{\mu }_{i}},\,\,t) - {{\beta }_{i}})\Delta {{\alpha }_{i}} - {{\alpha }_{i}}\Delta {{\beta }_{i}}} \right] - \lambda \Delta T(x,\,\,t),\quad (x,\,\,t) \in \Omega , \\ \end{gathered} $$

(A.2)

$$\Delta T(x,\,\,0) = 0,\quad x \in [0,\,\,l],$$

(A.3)

$$\Delta T(0,\,\,t) = \left\{ {\begin{array}{*{20}{l}} {0,}&{t\;\leqslant \;\tau } \\ {(1 - \gamma )\Delta T(L,\,\,t - \tau ),\;\;}&{t\; \geqslant \;\tau ,} \end{array}} \right.$$

(A.4)

where the accuracy is within the terms of the first order of smallness with respect to the increment ΔT(x, t) of the state variable. Formula (A.2) involves the relation

$$T({{\mu }_{i}} + \Delta {{\mu }_{i}},\,\,t) = T({{\mu }_{i}},\,\,t) + {{T}_{x}}({{\mu }_{i}},\,\,t)\Delta {{\mu }_{i}} + o\left( {\left| {\Delta {{\mu }_{i}}} \right|} \right).$$

The increment of the functional (2.12) can be easily represented as

$$\Delta I({\mathbf{P}};\,\,\varphi ,\,\,\gamma ) = I({\mathbf{\tilde {P}}};\,\,\varphi ,\,\,\gamma ) - I({\mathbf{P}};\,\,\varphi ,\,\,\gamma ) = I({\mathbf{P}} + \Delta {\mathbf{P}};\,\,\varphi ,\,\,\gamma ) - I({\mathbf{P}};\,\,\varphi ,\,\,\gamma )$$

$$ = 2\int\limits_{{{t}_{b}}}^{{{t}_{f}}} {[T(L,\,\,t;\,\,{\mathbf{P}},\,\,\varphi ,\,\,\gamma ) - V]\Delta T(L,\,\,t)dt} + 2\sigma \sum\limits_{i = 1}^{3N} {({{{\mathbf{P}}}_{i}} - {{{{\mathbf{\tilde {P}}}}}_{i}})\Delta {{{\mathbf{P}}}_{i}},} $$

$$\sum\limits_{i = 1}^{3N} {({{{\mathbf{P}}}_{i}} - {{{{\mathbf{\tilde {P}}}}}_{i}})\Delta {{{\mathbf{P}}}_{i}}} = \sum\limits_{i = 1}^{3N} {\left[ {({{\mu }_{i}} - {{{\tilde {\mu }}}_{i}})\Delta {{\mu }_{i}} + ({{\alpha }_{i}} - {{{\tilde {\alpha }}}_{i}})\Delta {{\alpha }_{i}} + ({{\beta }_{i}} - {{{\tilde {\beta }}}_{i}})\Delta {{\beta }_{i}}} \right].} $$

Let ψ(x, t; P, φ, γ) be an arbitrary (so far) function that is continuous everywhere in Ω except the points x = μ_i, i = 1, 2, …, N, differentiable with respect to x for x ∈ (μ_i, μ_i + 1), i = 0, 1, …, N, μ₀ = 0, μ_N + 1 = L, and differentiable with respect to t for t ∈ (0, T). The arguments P, φ, and γ of the function ψ(x, t; P, φ, γ) indicate its possible change when varying the feedback parameter vector P, the initial temperature φ, and the heat loss coefficient γ. Whenever possible, P, φ, and γ will be omitted for the function ψ(x, t; P, φ, γ). Under the accepted assumptions and conditions (A.3) and (A.4), integrating Eq. (A.2) with the factor ψ(x, t) along the rectangle Ω gives

$$\int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\psi (x,\,\,t)\Delta {{T}_{t}}(x,\,\,t)dxdt} } + \vartheta \sum\limits_{i = 0}^N {\int\limits_{{{\mu }_{i}}}^{{{\mu }_{{i + 1}}}} {\int\limits_0^{{{t}_{f}}} {\psi (x,\,\,t)\Delta {{T}_{x}}(x,\,\,t)dtdx} } } $$

$$ - \;\lambda \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\psi (x,\,\,t)\sum\limits_{i = 1}^N {\left[ {{{\alpha }_{i}}\Delta T({{\mu }_{i}},\,\,t) + {{\alpha }_{i}}{{T}_{x}}({{\mu }_{i}},\,\,t)\Delta {{\mu }_{i}}} \right.} } } $$

(A.5)

$$\left. { + \;(T({{\mu }_{i}},\,\,t) - {{\beta }_{i}})\Delta {{\alpha }_{i}} - {{\alpha }_{i}}\Delta {{\beta }_{i}}} \right]dxdt + \lambda \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\psi (x,\,\,t)\Delta T(x,\,\,t)dxdt} } = 0.$$

In view of (A.3)–(A.5), we integrate by parts the first and second terms in (A.5) separately to get

$$\begin{gathered} \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\psi (x,\,\,t)\Delta {{T}_{t}}(x,\,\,t)dxdt} } = \int\limits_0^L {\psi (x,\,\,{{t}_{f}})\Delta T(x,\,\,{{t}_{f}})dx} \\ + \;\int\limits_0^L {\left[ {\psi (x,\,\,t_{b}^{ - }) - \psi (x,\,\,t_{b}^{ + })} \right]\Delta T(x,\,\,{{t}_{b}})dx} - \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {{{\psi }_{t}}(x,\,\,t)\Delta T(x,\,\,t)dxdt} } , \\ \end{gathered} $$

(A.6)

$$\vartheta \sum\limits_{i = 0}^N {\int\limits_{{{\mu }_{i}}}^{{{\mu }_{{i + 1}}}} {\int\limits_0^{{{t}_{f}}} {\psi (x,\,\,t)\Delta {{T}_{x}}(x,\,\,t)dtdx} } } = \vartheta \int\limits_0^{{{t}_{f}}} {[\psi (l,\,\,t)\Delta T(L,\,\,t) - \psi (0,\,\,t)\Delta T(0,\,\,t)]dt} $$

$$ + \;\vartheta \sum\limits_{i = 1}^N {\int\limits_0^{{{t}_{f}}} {\left[ {\psi (\mu _{i}^{ - },\,\,t) - \psi (\mu _{i}^{ + },\,\,t)} \right]\Delta T({{\mu }_{i}},\,\,t)dt} } - \vartheta \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {{{\psi }_{x}}(x,\,\,t)\Delta T(x,\,\,t)dxdt} } $$

$$\begin{gathered} = \vartheta \int\limits_0^{{{t}_{f}}} {\psi (L,\,\,t)\Delta T(L,\,\,t)dt} - \vartheta (1 - \gamma )\int\limits_\tau ^{{{t}_{f}}} {\psi (0,\,\,t)\Delta T(L,\,\,t - \tau )dt} \\ + \;a\sum\limits_{i = 1}^N {\int\limits_0^{{{t}_{f}}} {\left[ {\psi (\mu _{i}^{ - },\,\,t) - \psi (\mu _{i}^{ + },\,\,t)} \right]\Delta T({{\mu }_{i}},\,\,t)dt} } - \vartheta \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {{{\psi }_{x}}(x,\,\,t)\Delta T(x,\,\,t)dxdt} } \\ \end{gathered} $$

(A.7)

$$ = \vartheta \int\limits_0^{{{t}_{f}}} {\psi (L,\,\,t)\Delta T(L,\,\,t)dt} - \vartheta (1 - \gamma )\int\limits_0^{{{t}_{f}} - \tau } {\psi (0,\,\,t + \tau )\Delta T(L,\,\,t)dt} $$

$$ + \;\vartheta \sum\limits_{i = 1}^N {\int\limits_0^{{{t}_{f}}} {\left[ {\psi (\mu _{i}^{ - },t) - \psi (\mu _{i}^{ + },t)} \right]\Delta T({{\mu }_{i}},t)dt} } - \vartheta \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {{{\psi }_{x}}(x,t)\Delta T(x,t)dxdt.} } $$

In these formulas,

$$\psi (\mu _{i}^{ - },\,\,t) = \psi ({{\mu }_{i}} - 0,\,\,t),\quad \psi (\mu _{i}^{ + },\,\,t) = \psi ({{\mu }_{i}} + 0,\,\,t).$$

Considering (A.5)–(A.7), the increment of the objective functional takes the form

$$\Delta I = \int\limits_0^L {\psi (x,\,\,{{t}_{f}})\Delta T(x,\,\,{{t}_{f}})dx} + \int\limits_{{{t}_{f}} - \tau }^{{{t}_{f}}} {[\vartheta \psi (L,\,\,t) + 2(T(L,\,\,t) - V)]\Delta T(L,\,\,t)dt} $$

$$ + \;\int\limits_{{{t}_{k}}}^{{{t}_{f}} - \tau } {[\vartheta \psi (L,\,\,t) + \lambda (1 - \gamma )\psi (0,\,\,t + \tau ) + 2(T(L,\,\,t) - V)]\Delta T(L,\,\,t)dt} $$

$$ + \;\int\limits_0^{{{t}_{b}}} {[\vartheta \psi (L,\,\,t) + \lambda (1 - \gamma )\psi (0,\,\,t + \tau )]\Delta T(L,\,\,t)dt} $$

$$ + \;\int\limits_0^{{{t}_{f}}} {\int\limits_0^L {[ - {{\psi }_{t}}(x,\,\,t) - \vartheta {{\psi }_{x}}(x,\,\,t) + \lambda \psi (x,\,\,t)]\Delta T(x,\,\,t)dxdt} } $$

(A.8)

$$ + \;a\sum\limits_{i = 1}^N {\int\limits_0^{{{t}_{f}}} {\left[ {\psi (\mu _{i}^{ - },\,\,t) - \psi (\mu _{i}^{ + },\,\,t) - \frac{\lambda }{\vartheta }{{\alpha }_{i}}\int\limits_0^L {\psi (x,\,\,t)dx} } \right]\Delta T({{\mu }_{i}},\,\,t)dt} } $$

$$ - \;\lambda \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\psi (x,\,\,t)\sum\limits_{i = 1}^N {[{{\alpha }_{i}}{{T}_{x}}({{\mu }_{i}},\,\,t)\Delta {{\mu }_{i}} + (T({{\mu }_{i}},\,\,t) - {{\beta }_{i}})\Delta {{\alpha }_{i}} - {{\alpha }_{i}}\Delta {{\beta }_{i}}]dxdt} } } $$

$$ + \;2\sigma \sum\limits_{i = 1}^N {\left[ {({{\mu }_{i}} - {{{\tilde {\mu }}}_{i}})\Delta {{\xi }_{i}} - ({{\alpha }_{i}} - {{{\tilde {\alpha }}}_{i}})\Delta {{\alpha }_{i}} + ({{\beta }_{i}} - {{{\tilde {\beta }}}_{i}})\Delta {{\beta }_{i}}} \right].} $$

Since the function ψ(x, t) is arbitrary, let it be the solution of the initial boundary-value problem (3.5)–(3.9) almost everywhere.

Recall that the components of the gradient of the functional are determined by the linear part of its increment with respect to the increments of the corresponding arguments. Consequently,

$$\begin{aligned} {{\operatorname{grad} }_{{{{\mu }_{i}}}}}I = - \lambda {{\alpha }_{i}}\int\limits_0^{{{t}_{f}}} {\left( {\int\limits_0^L {\psi (x,\,\,t)dx} } \right){{T}_{x}}({{\mu }_{i}},\,\,t)dt} + 2\sigma ({{\mu }_{i}} - {{{\tilde {\mu }}}_{i}}), \\ i = 1,\,\,2,\,\, \ldots ,\,\,N, \\ \end{aligned} $$

(A.9)

$$\begin{aligned} {{\operatorname{grad} }_{{{{\alpha }_{i}}}}}I = - \lambda \int\limits_0^{{{t}_{b}}} {(T({{\mu }_{i}},\,\,t) - {{\beta }_{i}})\left( {\int\limits_0^L {\psi (x,\,\,t)dx} } \right)dt} + 2\sigma ({{\alpha }_{i}} - {{{\tilde {\alpha }}}_{i}}), \\ i = 1,\,\,2,\,\, \ldots ,\,\,N, \\ \end{aligned} $$

(A.10)

$${{\operatorname{grad} }_{{{{\beta }_{i}}}}}I = \lambda {{\alpha }_{i}}\int\limits_0^L {\psi (x,\,\,t)dx} + 2\sigma ({{\beta }_{i}} - {{\tilde {\beta }}_{i}}),\quad i = 1,\,\,2,\,\, \ldots ,\,\,N.$$

(A.11)

The proof of this theorem is complete.

It remains to obtain the adjoint initial boundary-value problem in the form (3.10) equivalent to (3.5)–(3.9) but without the jump conditions (3.9). Based on the property of the δ-function, the third term in (A.5) can be written as

$$\lambda \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\psi (x,\,\,t)\sum\limits_{i = 1}^N {({{\alpha }_{i}}\Delta T({{\mu }_{i}},\,\,t) + {{\alpha }_{i}}{{T}_{x}}({{\mu }_{i}},\,\,t)\Delta {{\mu }_{i}} + (T({{\mu }_{i}},\,\,t) - {{\beta }_{i}})\Delta {{\alpha }_{i}} - {{\alpha }_{i}}\Delta {{\beta }_{i}})dxdt} } } $$

$$ = \lambda \sum\limits_{i = 1}^N {{{\alpha }_{i}}\int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\int\limits_0^L {\psi (\zeta ,t)\delta (\zeta - {{\mu }_{i}})\Delta T(\zeta ,t)d\zeta dxdt} } } } $$

$$ + \;\lambda \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\psi (x,t)\sum\limits_{i = 1}^N {({{\alpha }_{i}}{{T}_{x}}({{\mu }_{i}},t)\Delta {{\mu }_{i}} + (T({{\mu }_{i}},t) - {{\beta }_{i}})\Delta {{\alpha }_{i}} - {{\alpha }_{i}}\Delta {{\beta }_{i}})dxdt.} } } $$

Now we change the order of integration over ζ and x in the first triple integral and swap the names of these variables to obtain

$$\int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\int\limits_0^L {\psi (\zeta ,\,\,t)\delta (\zeta - {{\mu }_{i}})\Delta T(\zeta ,\,\,t)d\zeta dxdt} } } = \int\limits_0^{{{t}_{f}}} {\int\limits_0^L {\left( {\int\limits_0^L {\psi (\zeta ,\,\,t)d\zeta } } \right)\delta (x - {{\mu }_{i}})\Delta T(x,\,\,t)dxdt.} } $$

(A.12)

In formula (A.5), we regroup the terms and, using (A.12), finally arrive from (3.5) and (3.8) at the integro-differential Eq. (3.12) for the adjoint problem.