Model Predictive Control

208 11 Receding Horizon Control
Stability, 2-Norm case
Consider system (11.1)-(11.2), the RHC law (11.7)-(11.10), the cost function(11.9)
and the closed-loop system (11.11). A simple choice Xf is obtained by choosing Xf
as the maximal positive invariant set (see Section 10.1) for the closed-loop system
x(k + 1) = (A + BF∞)x(k) where F∞ is the associated unconstrained infinite-time
optimal controller (8.32). With this choice the assumption (A3) in Theorem 11.2
becomes
x ′ (A ′ (P − PB(B ′ PB + R) −1 BP)A + Q − P)x ≤ 0, ∀x ∈ Xf
(11.25)
which is satisfied as an equality if P is chosen as the solution P∞ of the algebraic
Riccati equation (8.31) for system (11.1) (see proof in Section 8.5).
If system (11.1) is asymptotically stable, then Xf can be alternatively chosen
as the positive invariant set of the autonomous system x(k + 1) = Ax(k) subject
to the state constraints x ∈ X. Therefore in Xf the input 0 is feasible and the
assumption (A3) in Theorem 11.2 becomes
− x ′ Px + x ′ A ′ PAx + x ′ Qx ≤ 0, ∀x ∈ Xf
(11.26)
which is satisfied if P solves x ′ (−P +A ′ PA+Q)x = 0, i.e. the standard Lyapunov
equation (7.42). In this case stability implies exponential stability. The argument
is simple. As the system is closed-loop stable it enters the terminal region in finite
time. If Xf is chose as suggested, the closed-loop system is unconstrained after
entering Xf. For an unconstrained linear system the convergence to the origin is
exponential.
Stability, 1, ∞-Norm case
Consider system (11.1)-(11.2), the RHC law (11.7)-(11.10), the cost function(11.8)
and the closed-loop system (11.11). Let p = 1 or p = ∞. If system (11.1) is
asymptotically stable, then Xf can be chosen as the positively invariant set of the
autonomous system x(k + 1) = Ax(k) subject to the state constraints x ∈ X.
Therefore in Xf the input 0 is feasible and the assumption (A3) in Theorem 11.2
becomes
− Pxp + PAxp + Qxp ≤ 0, ∀x ∈ Xf (11.27)
which is the corresponding Lyapunov inequality for the 1, ∞-norm case (7.48)
whose solution has been discussed in Section 7.5.3.
In general, if the unconstrained optimal controller (9.31) exists it is PPWA. In
this case the computation of the maximal invariant set Xf for the closed-loop PWA
system
x(k + 1) = (A + F i )x(k) if H i x ≤ 0, i = 1, . . .,N r
(11.28)
is more involved. However if such Xf can be computed it can be used as terminal
constraint in Theorem 11.2. With this choice the assumption (A3) in Theorem 11.2
is satisfied by the infinite-time unconstrained optimal cost P∞ in (9.32).