Model Predictive Control

7.4 Optimal Control Problem with Infinite Horizon 125
From the principle of optimality we know that
J ∗ 0→N(x0) = min
u0
q(x0, u0) + J ∗ 1→N(x1). (7.26)
Assume that we are at x(0) at time 0 and implement the optimal u∗ 0 that takes us
to the next state x1 = g(x(0), u∗ 0 ). At this state at time 1 we postulate to use over
the next N steps the sequence of optimal moves determined at the previous step
followed by zero: u∗ 1 , . . . , u∗N−1 , 0. This sequence is not optimal but the associated
cost over the shifted horizon from 1 to N + 1 can be easily determined. It consists
of three parts: 1) the optimal cost J ∗ 0→N (x0) from time 0 to N computed at time 0,
minus 2) the stage cost q(x0, u0) at time 0 plus 3) the cost at time N +1. But this
last cost is zero because we imposed the terminal constraint xN = 0 and assumed
uN = 0. Thus the cost over the shifted horizon for the assumed sequence of control
moves is
J ∗ 0→N(x0) − q(x0, u0).
Because this postulated sequence of inputs is not optimal at time 1
J ∗ 1→N+1(x1) ≤ J ∗ 0→N(x0) − q(x0, u0).
Because the system and the objective are time invariant J ∗ 1→N+1 (x1) = J ∗ 0→N (x1)
so that
J ∗ 0→N (x1) ≤ J ∗ 0→N (x0) − q(x0, u0).
As q ≻ 0 for all (x, u) = (0, 0), the sequence of optimal costs J ∗ 0→N (x0), J ∗ 0→N (x1), . . .
is strictly decreasing for all (x, u) = (0, 0). Because the cost J ∗ 0→N ≥ 0 the sequence
J ∗ 0→N (x0), J ∗ 0→N (x1), . . . (and thus the sequence x0, x1,. . .) is converging. Thus we
have established the following important theorem.
Theorem 7.1 At time step j consider the cost function
and the CFTOC problem
Jj→j+N(xj, uj, uj+1, . . . , uj+N−1)
j+N
k=j
q(xk, uk), q ≻ 0 (7.27)
J ∗ j→j+N (xj) minuj,uj+1,...,uj+N −1 Jj→j+N(xj, uj, uj+1, . . .,uj+N−1)
subj. to xk+1 = g(xk, uk)
h(xk, uk) ≤ 0, k = j, . . . , j + N − 1
xN = 0
(7.28)
Assume that only the optimal u∗ j is implemented. At the next time step j + 1 the
CFTOC problem is solved again starting from the resulting state xj+1 = g(xj, u∗ j )
(Receding Horizon Control). Assume that the CFTOC problem has a solution for
every one of the sequence of states xj, xj+1, . . . resulting from the control policy.
Then the system will converge to the origin as j → ∞.
Thus we have established that a receding horizon controller with terminal constraint
xN = 0 has the same desirable convergence characteristics as the infinite