Lattice Trajectory Piecewise Linear Method for the Simulation of Diode Circuits

Lattice piecewise affine approximation of explicit nonlinear model predictive control with application to trajectory tracking of mobile robot

“…Assume that the number of reference points is K . Thus, after obtaining all of the discrete-time linear systems (4), we can construct a lattice PWA model (5) to linearly approximate the original nonlinear system function in a piecewise linear fashion according to the algorithm proposed in [24].…”

“…Consider the MPC problem (22) of nominal system. Suppose that Assumption 4 holds, there exists 𝜆(⋅), 𝛼(⋅) satisfying the condition (24) for all x ∈ 𝕏 and u ∈ 𝕌. Define the rotated cost as follows: le (x, u) = l e (x, u) − l e (x s , u s ) + 𝜆(x) − 𝜆( f LTPWL (x, u)).…”

“…Assume that the number of reference points is K . Thus, after obtaining all of the discrete‐time linear systems (4), we can construct a lattice PWA model (5) to linearly approximate the original nonlinear system function in a piecewise linear fashion according to the algorithm proposed in [24]. $$$$\begin{eqnarray} \hat{f}&=&f_{\text{LPWA}}=\max _{i=0, \ldots , K-1}{\lbrace \min _{j \in I_{\ge , i}}{\lbrace l_{j}\rbrace} \rbrace} \nonumber \\ I_{\ge , i}&=&{\lbrace k \mid l_{k}{(\bm{x}_{i}, \bm{u}_{i})} \ge l_{\operatorname{act}(i)}{(\bm{x}_{i}, \bm{u}_{i})}\rbrace} , \end{eqnarray}$$$$where $$l_{j}$$ is the affine function defined in (4).…”