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I have a discrete system of the form

$x(k+k_o) = Ax(k+k_o-1) + Bx(k)$

where $A$ and $B$ are $n\times n$ matrices ($k_o>0$). I want to know about the stability of the system when

  1. both A and B have eigen values inside the unit circle
  2. Eigen values of B are inside the unit circle while largest eigen value of A lies exactly on the unit circle.
peterh
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faisal
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2 Answers2

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In general, it is difficult to conclude on the stability of the system. In other words, the eigenvalues of the matrices A and B does not provide more information on the stability of the system.

For example, we take $k_0 = 1$ and $A = B = 3/4$. In this case, the matrices A and B have eigen values inside the unit circle but the system is unstable.

Similary, if take $A = B = 1/3$, the system is stability. In the same way, we can take some examples in case 2.

MichaelCarrick
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By defining a new state as

$$ z(k) = \begin{bmatrix} x(k) \\ x(k-1) \\ \vdots \\ x(k-k_o+1) \end{bmatrix}, $$

then the difference equation can be written as

$$ z(k+1) = \bar{A}\,z(k), $$

with

$$ \bar{A} = \begin{bmatrix} A & 0 & \cdots & 0 & B \\ I & 0 & \cdots & 0 & 0 \\ 0 & I & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 \\ 0 & \cdots & 0 & I & 0 \end{bmatrix}. $$

So stability can be checked by evaluating the eigenvalues of $\bar{A}$ (which must be Schur).

Kwin van der Veen
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  • wow, that was quite helpful, thanks buddy. – faisal Jul 20 '18 at 12:29
  • how can i prove the convergence/stability if there is also a constant present in that equation. x(k+ko)=Ax(k+ko−1)+Bx(k)+C – faisal Jul 23 '18 at 08:27
  • @faisal You can either try to find a coordinate transformation $x(k)=\hat{x}(k)+a$ such that $a=A,a+B,a+C$. This will mean that if the system is stable the state will asymptotically go to $a$. Or extend the state space $z(k)$ by one and add $\begin{bmatrix}C^\top & 0 & \cdots & 0 & 0 & 1\end{bmatrix}^\top$ as a new column to the right of $\bar{A}$ (the remainder of the bottom row should be zeros). The desired dynamics can be obtained by giving this new state an initial condition of one. This will always add an eigenvalue of one to your system. – Kwin van der Veen Jul 23 '18 at 12:36
  • I actually believe that $a$ can be found by looking at the eigenvector corresponding to this new eigenvalue of one (except its last element). – Kwin van der Veen Jul 23 '18 at 12:38