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$$x(t+1) =Ax(t)+Bu(t)=\left(\begin{matrix}0&0&0.8\\1&0&0\\0&1&0 \end{matrix}\right)x(t)+\left(\begin{matrix}1\\0\\0 \end{matrix}\right)u(t)$$ Basically the excercise is to find the minimum value of $N$ such that $$z^T\left(\displaystyle\sum_{\tau=0}^{N-1}A^\tau B B^T \left(A^T\right)^\tau \right)^{-1}z ≤ 1.1z^T\lim_{t \rightarrow \infty}\left(\displaystyle\sum_{\tau=0}^{t-1}A^\tau B B^T \left(A^T\right)^\tau \right)^{-1}z $$ for all $z$.

There is a hint: the matlab command P=dlyap(A,W) computes the solution of the Lyapunov equation $AP A^T + W = P,$ but I can't see how that helps.

Karlis Olte
  • 637

2 Answers2

1

It is a little unclear, but judging by the hint:

For minimum energy control $$ x_0^tP^{-1}x_0=\min_{u_n} \sum_{n=1}^\infty u_n^t u_n $$ where $P$ is the solution to the discrete time Lyapunov equation.

riboch
  • 402
1

Consider

$$A \lim_{t\to\infty} \left( \sum_{\tau=0}^{t-1} A^\tau B B^T \left( A^T \right)^\tau \right) A^T - \lim_{t\to\infty} \left( \sum_{\tau=0}^{t-1} A^\tau B B^T \left( A^T \right)^\tau \right) \\ = ABB^TA^T+A^2 B B^T \left( A^T \right)^2+\dots-BB^T-ABB^TA^T-A^2 B B^T \left( A^T \right)^2-\dots \\ = -BB^T$$

assuming $\lim_{t\to\infty} A^t = 0$, which means the system is asymptotically stable. Therefore

$$P = \lim_{t\to\infty} \left( \sum_{\tau=0}^{t-1} A^\tau B B^T \left( A^T \right)^\tau \right)$$

is a (unique) solution to the equation

$$APA^T-P+BB^T=0$$

which is known as the Lyapunov equation for discrete time systems. Now you can use the Matlab command to find $P$ and calculate the minimum value of $N$ by trying.

obareey
  • 5,797