Page 262 - Numerical Methods for Chemical Engineering

Optimal control 251

We thus have the following PDE problem,
∂ϕ C U 2 2 ∂ϕ
= [u(τ, x) − u set ] + [x − x set ] + [−(x − 1) + u(τ, x)]
∂τ 2 ∂x (5.167)
initial condition ϕ(0, x) = C H [x(t H ) − x set ] 2

To solve this problem numerically, we use the method of finite differences, explained in
further detail in Chapter 6. We restrict the x-domain to x lo ≤ x ≤ x hi , where the limits are
chosen to be larger than any conceivable x-value that could be encountered in practice.
Then, we place a grid of N points, uniformly-spaced, in this domain,
x hi − x lo
x k = x lo + (k − 1)( x) x = (5.168)
N − 1
At each x k , we use a finite difference approximation to estimate ∂ϕ/∂x,

1
∂ϕ
= [A lo ϕ k−1 + A mid ϕ k + A hi ϕ k+1 ] A lo + A mid + A hi = 0 (5.169)
∂x x
x j
where ϕ k (τ) ≡ ϕ(τ, x k ). For reasons that will become clear in our discussion of convection
in Chapter 6, we choose here the set of one-sided differences,

if f (x, u) ≤ 0 A lo =−1 A mid =+1 A hi = 0
(5.170)
else A lo = 0 A mid =−1 A hi =+1

Note that if x hi is large enough, f (t, x, u) =−(x − 1) + u < 0, and we have the one-sided
difference pointing “into” the grid, and we have no problem applying (5.169). Similar
reasoning holds at the lower boundary.
We now solve the HJB equation numerically by integrating the set of ODEs

dϕ k C U 2 2 ∂ϕ
= [u k (τ) − u set ] + [x k − x set ] + f (x k , u k (τ))
dτ 2 ∂x
x k
f (x k , u k (τ)) =−(x k − 1) + u k (τ) ϕ k (0) = C H [x k (t H ) − x set ] 2 (5.171)

−1 ∂ϕ
u k (τ) = u set − C U
∂x
x k
The feedback control law u con (x) is then
u con (x k ) = u k (τ = t H ) (5.172)

The optimal control at any point may be computed from (5.166). control 1D HJB.m solves
this HJB equation for specified t H , C U , C H .For t H = 10, C U = 1, and C H = 10, the resulting
feedback control law is shown in Figure 5.15. For this simple linear system and quadratic
cost functional, the optimal control law is a simple proportional controller with a gain of
K =−0.732. The advantage of this approach is that it can be extended (though at perhaps
great numerical cost) to nonlinear systems and to systems involving input constraints.