Model Predictive Control

122 7 General Formulation and Discussion
In general, however, we may have to resort to a “brute force” approach to
and to solve the dynamic program. Let us
is known and discuss how to construct
construct the cost-to-go function J ∗ j−1→N
assume that at time j−1 the cost-to-go J ∗ j→N
an approximation of J ∗ j−1→N . With J ∗ j→N known, for a fixed xj−1 the optimization
problem (7.14) becomes a standard nonlinear programming problem. Thus, we
can define a grid in the set Xj−1→N of the state space and compute the optimal
cost-to-go function on each grid point. We can then define an approximate value
function ˜ J ∗ j−1→N (xj−1) at intermediate points via interpolation. The complexity
of constructing the cost-to-go function in this manner increases rapidly with the
dimension of the state space (“curse of dimensionality”).
The extra benefit of solving the optimal control problem via dynamic programming
is that we do not only obtain the vector of optimal inputs U ∗ 0→N for a
particular initial state x(0) as with the batch approach. At each time j the optimal
cost-to-go function defines implicitly a nonlinear feedback control law.
u ∗ j (xj) = arg min
uj
q(xj, uj) + J ∗ j+1→N (g(xj, uj))
subj. to h(xj, uj) ≤ 0,
g(xj, uj) ∈ Xj+1→N
(7.18)
For a fixed xj this nonlinear programming problem can be solved quite easily in
order to find u ∗ j (xj). Because the optimal cost-to-go function J ∗ j→N (xj) changes
with time j, the nonlinear feedback control law is time-varying.
7.4 Optimal Control Problem with Infinite Horizon
We are interested in the optimal control problem (7.3)–(7.5) as the horizon N
approaches infinity.
J ∗ 0→∞ (x0) = minu0,u1,...
∞
q(xk, uk)
k=0
subj. to xk+1 = g(xk, uk), k = 0, . . . , ∞
h(xk, uk) ≤ 0, k = 0, . . .,∞
x0 = x(0)
We define the set of initial conditions for which this problem has a solution.
(7.19)
X0→∞ = {x(0) ∈ R n : Problem (7.19) is feasible and J ∗ 0→∞ (x(0)) < +∞}.
(7.20)
For the value function J ∗ 0→∞(x0) to be finite it must hold that
lim
k→∞ q(xk, uk) = 0
and because q(xk, uk) > 0 for all (xk, uk) = 0
lim
k→∞ xk = 0