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Discretize a continuous LTI system: zero-order holding


The continuous LTI system is


The corresponding discrete system is


What are F and G? Usually we have two methods: use zero-order hold method to discretize it; or use approximation to a rigorous discrete system. Today I will show they are the same to the first order !!! So in the future, we are confident to directly use zero-order holding method to discretize a system as long as you want the first order accuracy!!!

1) Zero-order holding method:

x_k=x_{k-1}+(Ax_{k-1}+Bu_k)\Delta t

where \Delta t=t_{k}-t_{k-1}. Then

x_k=(I+\Delta t A)x_{k-1}+\Delta t Bu_k


>>>F=I+\Delta t A

>>>G=\Delta t B

2) Rigorous discrete system


x(t)=e^{A(t-t_0)}x_0+\int_{t_0}^te^{A(t-\tau)} Bu(\tau) \mbox{d}\tau

it is easy to see

>>>F=e^{A\Delta t}=I+\Delta t A + \frac{1}{2}A^2\Delta t^2\cdots

>>>G=\left(\int_0^{\Delta t}e^{A\alpha}d\alpha\right)B=\left(\Delta t+\frac{1}{2}A\Delta t^2+\cdots\right)B

Conclusion: Obviously, method 1 and 2 are equivalent to the first order!!!

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