For assessing the performances of LMPC sampled data control and H2 optimal control, we compared the quadratic costs of them. The constraints for the cost function include:
-
Total Time (( T )): ( T = 10 ) s
-
State Weight Matrix (( Q )): ( Q = \text{diag}([1, 1, 1]) )
-
Control Weight Matrix (( R )): ( R = \text{diag}([0.1, 0.1]) )
-
Sampling Time (( \gamma )): ( \gamma = 0.1 ) s
-
LMPC Control Input Limits:
- ( v_{\text{max}} = 0.3 ), ( v_{\text{min}} = 0.0 )
- ( \omega_{\text{max}} = \frac{\pi}{4} ), ( \omega_{\text{min}} = -\frac{\pi}{4} )
-
Gain Matrix (( k )):
[ k = \begin{bmatrix} 0.4662 & 0.2978 & 0.2067 \ 0.8341 & 1.2165 & 1.3492 \end{bmatrix} ]
The quadratic cost function for a single timestep is given by:
[ J_{\text{total}} = \sum_{k=1}^{N} (\mathbf{x}_k^T \mathbf{Q} \mathbf{x}_k + \mathbf{u}_k^T \mathbf{R} \mathbf{u}_k) ]
Where ( N = 2 ).
- Total Cost Value for LMPC (( J=10 )): 8.98516268901749
- Total Cost Value for H2: 18.775862036811883