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Given is the following discrete system

$$\begin{align*} &x(k + 1) = Ax(k) + Bu(k)\\ &x(0) = x_0\;. \end{align*}$$

How do we prove that the explicit solution formula for $x(k)$ (analogously to the variation of constants formula in the continuous time case) is

$$A^kx_0+\sum_{j=1}^kA^{j-k}Bu(j-1)\;?$$

Thanks a lot!

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    The power of $A$ inside the summation should be $k-j$. It is a convolution.2012-11-19

1 Answers 1