Introduction

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:

Parameters

• 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} ]

Quadratic Cost Function

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 ).

Results

• Total Cost Value for LMPC (( J=10 )): 8.98516268901749
• Total Cost Value for H2: 18.775862036811883

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