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Suppose an LTI discrete-time system is given by the equations $$ x_{k+1} = Ax_k + Bu_k,\\ y_{k} = Cx_k + Du_k $$ with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\rho(A) < 1$ and $x_0 = 0$. Assume $r_k\in\mathbb{R}^{n}$ is a bounded reference for the output sequence. Define the total tracking error as $E(u) := \sum_{k=0}^{\infty}{||r_k - y_k||^2}$ for an input sequence $u := \{u_k\}_{k=0}^{\infty}$.

Question: What are the necessary and sufficient conditions for the existence of a bounded $u^*$ that minimizes $E$? Is output-controllability a sufficient condition?

Benjamin Tennyson
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  • @SampleTime, does there exist a system $(A, B, C, D)$ such that the condition you wrote is satisfied and it is still output controllable? – Benjamin Tennyson Mar 31 '24 at 19:00
  • I accidentally deleted my comment... I think $A=\begin{pmatrix}0.5&1\0&0.5\end{pmatrix},B=\begin{pmatrix}0\1\end{pmatrix},C=\begin{pmatrix}1&0\end{pmatrix},D=0$ is an example of such a system. – SampleTime Mar 31 '24 at 19:56
  • @SampleTime, I see your point for a causal system. But if I let the system be acausal, then one can provide the input from how many time-steps back (one in this case) one may require it to track the future reference, right? Ignoring the initial errors, can this procedure not yield a perfect tracking in this case? – Benjamin Tennyson Apr 01 '24 at 09:25
  • @SampleTime, assuming a causal system, what more conditions do you think will be necessary for a sufficient condition for the existence of such a $u^*$ along with output-controllability? – Benjamin Tennyson Apr 01 '24 at 09:41
  • Sorry, my last comment was incorrect, with an acausal system, perfect tracking is possible for this example, at least as long as the control signal does not saturate. If you require however, for example, that $|u|\leq 0.5$ and $r=1$, then its not anymore possible and with a toggling reference (for example between $0$ and $1$), $E\rightarrow\infty$ as $k\rightarrow\infty$. So there is at least a dependency on the reference signal in combination with the control signal limits. So output-controllability alone seems not like a sufficient condition. – SampleTime Apr 01 '24 at 15:54
  • @SampleTime, yes that is true. However, I am currently not considering such a restriction for $u$. The only restriction for $u$ is that given a reference $r$ one can always choose a $u$ such that there exists $M>0$ with $||u_k||<M$ for all $k\in\mathbb{N}\cup{0}$. But I am not placing any bounds on $M$ and the inputs need not be bounded uniformly by the same $M$ for any given reference. Do you think output-controllability is still not sufficient in this case? If yes, what extra conditions do you think would be required? – Benjamin Tennyson Apr 02 '24 at 10:26

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