Model Predictive Control System Design and Implementation Using ...

8.4 CMPC with Asymptotic Stability 277
for simplicity of the solution, we have assumed that the weight matrix R is a
diagonal matrix.
Defining the data matrices
Ω =
Ψ =
∫ Tp
0
∫ Tp
the quadratic cost function (8.10) becomes
0
φ(τ)Qφ(τ) T dτ + R L (8.11)
φ(τ)Qe Aατ dτ, (8.12)
J = η T Ωη +2η T Ψx(t i )+constant. (8.13)
The optimal solution that minimizes the above quadratic cost function is
η = −Ω −1 Ψx(t i ). (8.14)
Upon obtaining η, the exponentially weighted derivative of the control signal
˙u α (τ) is constructed through
˙u α (τ) = [ L 1 (τ) T L 2 (τ) T ... L m (τ) T ] η. (8.15)
From the receding horizon control, the optimal solution for the actual ˙u(0) is
˙u(0) = ˙u α (0) = [ L 1 (0) T L 2 (0) T ... L m (0) T ] η. (8.16)
Because the optimization is performed on the transformed variables x α (.) and
˙u α (.), when constraints are introduced, all the original constraints are required
to be transformed from the variables x(.) and ˙u(.) tox α (.) and ˙u α (.). Constrained
control will be discussed further in the later sections of the chapter.
8.4 CMPC with Asymptotic Stability
This section establishes equivalent results with LQR when exponential weighting
is used. The results are investigated through two different cost functions,
and we then establish that the optimal control results are identical. The results
are summarized in the theorem as follows.
Case A
Suppose that the optimal control ˙u 1 (τ) is obtained by minimizing cost function
J 1 with Q ≥ 0, and R>0
J 1 =
∫ ∞
0
[
x(ti + τ | t i ) T Qx(t i + τ | t i )+ ˙u(τ) T R ˙u(τ) ] dτ, (8.17)