Model Predictive Control

7.5 Lyapunov Stability 131
An effective method to find both H and P was proposed by Christophersen in [78].
To prove the stability of receding horizon control, later in this book, we will
need to find a ˜ P such that
− ˜ Px∞ + ˜ PAx∞ + Qx∞ ≤ 0, ∀x ∈ R n . (7.48)
Once we have constructed a P and H to fulfill the conditions of Theorem 7.7 we
can easily find ˜ P to satisfy (7.48) according to the following proposition:
Proposition 7.1 Let P and H be matrices satisfying conditions (7.47), with P
full column rank. Let σ 1 − H∞, ρ QP # ∞, where P # (P T P) −1 P T is
the left pseudoinverse of P. Then, the square matrix
satisfies condition (7.48).
˜P = ρ
P (7.49)
σ
Proof: Since ˜ P satisfies ˜ PA = H ˜ P, we obtain − ˜ Px∞+ ˜ PAx∞+Qx∞ =
− ˜ Px∞ + H ˜ Px∞ + Qx∞ ≤ (H∞ − 1) ˜ Px∞ + Qx∞ ≤ (H∞ −
1) ˜ Px∞ + QP # ∞Px∞ = 0. Therefore, (7.48) is satisfied. ✷
Note that the inequality (7.48) is equivalent to the Lyapunov inequality (7.44)
when p = 1 or p = ∞.