Model Predictive Control

142 9 1/∞ Norm Optimal Control
9.1 Solution via Batch Approach
First we write the equality constraints in (9.4) explicitly to express all future states
x1, x2, . . . as a function of the future inputs u1, u2, . . . and then we eliminate all
intermediate states by using
xk = A k x0 + k−1
j=0 Aj Buk−1−j
(9.5)
so that all future states are explicit functions of the present state x(0) and the
future inputs u0, u1, u2, . . . only.
The optimal control problem (9.4) with p = ∞ can be rewritten as a linear
program by using the following standard approach (see e.g. [68]). The sum of
components of any vector {εx 0 , . . .,εx N , εu0 , . . .,εu N−1 } that satisfies
−1nεx k ≤ Qxk, k = 0, 1, . . .,N − 1
−1nεx k ≤ −Qxk, k = 0, 1, . . ., N − 1
−1rεx N ≤ PxN,
−1rεx N ≤ −PxN,
−1mεu k ≤ Ruk, k = 0, 1, . . ., N − 1
−1mεu k ≤ −Ruk, k = 0, 1, . . .,N − 1
(9.6)
forms an upper bound on J0(x(0), U0), where 1k [1
.
. . 1
]
k
′ , and the inequalities
(9.6) hold componentwise. It is easy to prove that the vector
z0 {εx 0, . . . , εx N , εu0, . . . , εu N−1 , u′ 0, . . . , u ′ N−1 } ∈ Rs , s (m + 1)N + N + 1, that
satisfies equations (9.6) and simultaneously minimizes J(z0) = εx 0 +. . .+εx N +εu0 +
. . . + εu N−1 also solves the original problem (9.4), i.e., the same optimum J ∗ 0 (x(0))
is achieved [258, 68]. Therefore, problem (9.4) can be reformulated as the following
LP problem
min
z0
subj. to −1nε x ⎡
k ≤ ±Q ⎣A k k−1
x0 +
ε x 0 + . . . + εx N + εu 0 + . . . + εu N−1
−1rε x N
⎡
≤ ±P ⎣A N x0 +
j=0
N−1
j=0
A j Buk−1−j
⎤
A j BuN−1−j
(9.7a)
⎦, (9.7b)
⎤
⎦, (9.7c)
−1mε u k ≤ ±Ruk, (9.7d)
k = 0, . . .,N − 1
x0 = x(0) (9.7e)
where constraints (9.7b)–(9.7d) are componentwise, and ± means that the constraint
appears once with each sign, as in (9.6).
Remark 9.1 The cost function (9.2) with p = ∞ can be interpreted as a special case
of a cost function with 1-norm over time and ∞-norm over space. For instance, the