Question: Greedy regulation policy. Consider a linear dynamical system given by \(x_{t+1}=A x_{t}+\) \(B u_{t}\), where the \(n\) -vector \(x_{t}\) is the state at time \(t\), and the \(m\) -vector \(u_{t}\) is the input at time \(t\). The goal in regulation is to choose the input so as to make the state small. (In applications, the state \(x_{t}=0\) corresponds to the desired operating point, so small \(x_{t}\) means the state is close to the desired operating point.) One way to achieve this goal is to choose \(u_{t}\) so as to minimize

where \(\rho\) is a (given) positive parameter that trades off using a small input versus making the (next) state small. Show that choosing \(u_{t}\) this way leads to a state feedback policy \(u_{t}=K x_{t}\), where \(K\) is an \(m \times n\) matrix. Give a formula for \(K\) (in terms of $A, B$, and

\(\rho)\). If an inverse appears in your formula, state the conditions under which the inverse exists.

Remark. This policy is called greedy or myopic since it does not take into account the effect of the input \(u_{t}\) on future states, beyond \(x_{t+1} .\) It can work very poorly in practice.

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