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Forward Euler Method: This method is given by the equation: $$ x_{k+1} = x_k + h f_\text{continuous} (x_k, u_k) $$
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Stability of the Discrete-Time System: The stability can be analyzed by linearizing around the equilibrium point: $$ \frac{\partial x_N}{\partial x_0} = \frac{\partial f_{\rm d}}{\partial x} \frac{\partial f_{\rm d}}{\partial x} ... {\frac{\partial f_{\rm d}}{\partial x}}{x_0} = A_{\rm d}^N $$ The stability requirement is: $$ | \textrm{eig} ( A_{\rm d}) | < 1 $$
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Key Takeaways:
- 🚫 Always check the simulator energy.
- ❌ Avoid using the Forward Euler integration as it can lead to instability.
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4th-Order Runge-Kutta Method (RK4): This method is more accurate and is given by: $$ x_{k+1} = f_{RK4} (x_k) $$ Where: $$ \begin{align*} K_1 & = f(x_k) \ K_2 & = f(x_k + \frac{1}{2} h K_1) \ K_3 & = f(x_k + \frac{1}{2} h K_2) \ K_4 & = f(x_k + h K_3) \ x_{k+1} & = x_{k} + \frac{h}{6} (K_1 + 2K_2 + 2K_3 + K_4) \end{align*} $$
- Key Takeaways:
- ✅ Provides better accuracy compared to the Forward Euler method.
- Key Takeaways:
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The implicit form is represented as: $$ f_{\rm d} (x_{k+1}, x_{k}, u_k) = 0 $$
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Backward Euler Method: This method is given by: $$ x_{k+1} = x_{k} + h f(x_{k+1}) $$ This method results in energy damping.
- Key Takeaways:
- ✅ The implicit form is generally more stable.
- ⏳ It is computationally more expensive.
- 📌 In some "direct" trajectory optimization methods, they are not any more expensive to use!
- Key Takeaways:
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Zero-Order Control: This method holds the control constant over the interval: $$ u(t) = u_k \ \text{for} \ t_{k} \leq t \leq t_{k+1} $$
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First-Order Control (Linear Interpolation): This method linearly interpolates the control over the interval: $$ u(t) = u_{k} + \frac{u_{k+1}-u_k}{h} (t-t_n) $$