Sliding Mode Control

Sliding Mode Control

• Assign a state error vector \(E(k)=X_k - X_k^d\)
• Define the sliding surface or sliding hyperplane. \(S_k=GE_k+K_I\varepsilon_k\), this is a PI-type sliding surface in order to speed up the system response, where \(G, K_I\) are the gain matrix and integration matrix.
• Considering the equivalent control \(u^{eq}\) is the solution of \(\Delta S=S_{k+1}-S_{k}=0\), therefore, the following equation must hold:

\[S_{k+1}=GE_{k+1}+K_I\varepsilon_{k+1}\\=(G+K_I)(AE_k+Bu_k+d_k)+K_I\varepsilon_k=S_k \]

In this equation, \(d_k\) can be replaced with its one-step delayed value \(d_{k-1}=E_k-A_E_{k-1}-Bu_k\)
For gain vectors and integration vector, they are chosen to guarantee the stability of the closed-loop system. That means substituting \(u^{eq}\) into error equation, which will results in the closed-loop state error dynamics convergence.

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