MPC theory

An equivalent quadratic programming (QP) formulation for constrained DMC (as given in LQG_MPC_notes by Prof Sachin Patwardhan) is given as follows

$$min_{U_{f}}\frac{1}{2}U_{f}(k)^{T}HU_{f}(k)+F^{T}U_{f}(k)AU_{f}(k) \leq b \intertext{where}$$ (10.1)

$$A = \left[ {\begin{array}{cc} I_{qm} \\ -I_{qm} \\ \end{array} } \right]$$

$$b = \left[ {\begin{array}{cc} U^{H} \\ -U_{L} \\ \end{array} } \right]$$

$$x(k+1)=\Phi x(k) + \Gamma(k) + w(k)$$ (10.2)

$$y(k)= Cx(k) + v(k)$$ (10.3)

$\phi$ is represented by matrix A in the code, $\Gamma$ is represented as matrix B and C is represented as C matrix in the code.