Dynamic Optimization

The model of the process is time dependent

• Often the case in control engineering For example:

$$x_{t+1}=A_t{x}_t+B_t{u}_t$$

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General Dynamic Optimization Problem

$$\underset{z}{\min }\sum _{t=0}^N{f}_t(x_{t+1},u_t)s.t.\{\begin{array}{ll}{x}_{t+1}=g(x_t,u_t),& t=0,\dots ,N-1\\ h(x_t,u_t)\le 0,& t=0,\dots N\\ x_0:\text{given},z=(u_0,x_1,u_1,x_2,\dots u_{N-1},x_N{)}^T& \end{array}$$

Where

• $N$: The prediction horizon
• $f_t$: The state cost
• $g_t$: The dynamic model
• $h_t$: State-and input constraints

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