Model Predictive Control

248 12 Constrained Robust Optimal Control
is the predicted state of system (12.38) at time k obtained by starting from the state
x0 = x(0) and applying to system xk+1 = A(w p
k )xk + B(w p
k )uk + Ew a k the input
sequence u0, . . . , uk−1 and the disturbance sequences w p
0
, . . . , wp
k−1 , wa 0, . . . , w a k−1.
Analogously, u(k) is the input applied to system (12.38) at time k while uk is the
k-th optimization variable of the optimization problem (12.45).
Problem (12.41) looks for the worst value J(x0, U) of the performance index
and the corresponding worst sequences wp∗ , wa∗ as a function of x0 and U0. Problem
(12.44)–(12.45) minimizes such a worst performance subject to the constraint
that the input sequence must be feasible for all possible disturbance realizations.
In other words, the worst-case performance is minimized under constraint fulfillment
against all possible realizations of wa , wp . Note that worst sequences wa∗ ,
wp∗for the performance are not necessarily worst sequences in terms of constraints
satisfaction.
The min-max formulation (12.41)–(12.45) is based on an open-loop prediction
and thus referred to as Constrained Robust Optimal Control with openloop
predictions (CROC-OL). The optimal control problem (12.41)–(12.45) can
be viewed as a deterministic zero-sum dynamic game between two players: the
controller U and the disturbance W [22, pag. 266-272]. The player U plays
first. Given the initial state x(0), U chooses his action over the whole horizon
{u0, . . . , uN−1}, reveals his plan to the opponent W, who decides on his actions
next {wa 0 , wp 0 , . . . , wa N−1 , wp
N−1 }.
For this reason the player U has the duty of counteracting any feasible disturbance
realization with just one single sequence {u0, . . . , uN−1}. This prediction
model does not consider that at the next time step, the payer can measure the
state x(1) and “adjust” his input u(1) based on the current measured state. By
not considering this fact, the effect of the uncertainty may grow over the prediction
horizon and may easily lead to infeasibility of the min problem (12.41)–(12.45).
On the contrary, in the closed-loop prediction scheme presented next, the optimization
scheme predicts that the disturbance and the controller play one move
at a time.
Closed-Loop Predictions
The constrained robust optimal control problem based on closed-loop predictions
(CROC-CL) is defined as follows:
J ∗ j (xj) min
uj
Jj(xj, uj) (12.47)
subj. to
xj ∈ X, uj ∈ U
∀wa j ∈ Wa , w p
j ∈ Wp
A(w p
j )xj + B(w p
j )uj + Ew a j
∈ Xj+1
(12.48)
Jj(xj, uj) max
w a j ∈Wa , w p
j ∈Wp
q(xj, uj) + J ∗ j+1(A(w p
j )xj + B(w p
j )uj + Ew a j ) ,
(12.49)