Model Predictive Control

146 9 1/∞ Norm Optimal Control
where the polyhedral sets {H i k x ≤ 0}, i = 1, . . .,Nr k , are a partition of Rn . The
optimal cost-to-go starting from the measured state x(k) is
J ∗ k (x(k)) = Pkx(k)∞
(9.24)
Here we have introduced the notation Pk to express the optimal cost-to-go J ∗ k (x(k)) =
Pkx(k)∞ from time k to the end of the horizon N. We also remark that the rows
of Pk correspond to the different affine functions constituting J ∗ k
and thus their
number varies with the time index k. Clearly, we do not have a closed form as for
the 2-norm Riccati Difference Equation (8.23) linking cost and control law at time
k given their value at time k − 1.
9.3 Comparison Of The Two Approaches
We will compare the batch and the recursive dynamic programming approach in
terms of the results and the methods used to obtain the results. Most importantly
we observe that the results obtained by the two methods are fundamentally different.
The batch approach yields a formula for the sequence of inputs as a function
of the initial state.
U ∗ 0 = ¯ F i 0x(0) if ¯ H i 0x(0) ≤ 0, i = 1, . . ., ¯ N r 0 (9.25)
The recursive dynamic programming approach yields a feedback policy, i.e., a sequence
of feedback laws expressing at each time step the control action as a function
of the state at that time.
u ∗ (k) = F i kx(k) if H i kx(k) ≤ 0, i = 1, . . .,N r k for k = 0, . . .,N − 1 (9.26)
As this expression implies, we determine u(k) at each time k as a function of the
current state x(k) rather than use a u(k) precomputed at k = 0 as in the batch
method. If the state evolves exactly according to the linear model (9.3) then the
sequence of control actions u(k) obtained from the two approaches is identical. In
practice, the result of applying the sequence (9.25) in an open-loop fashion may
be rather different from applying the time-varying feedback law (9.26) because the
model (9.3) for predicting the system states may be inaccurate and the system may
be subject to disturbances not included in the model. We expect the application
of the feedback law to be more robust because at each time step the observed state
x(k) is used to determine the control action rather than the state xk predicted at
time t = 0.
We note that we can get the same feedback effect with the batch approach
if we recalculate the optimal open-loop sequence at each time step j with the
current measurement as initial condition. In this case we need to solve the following
optimization problem
J ∗ j (x(j)) = minuj,··· ,uN −1 PxN ∞ + N−1
k=j Qxk∞ + Ruk∞
subj. to xj = x(j)
(9.27)