# On Application of the Ray-Shooting Method for LQR via Static-Output-Feedback
## 1 Introduction  
Let a continuous-time system be given by  
\[ \begin{cases}\hat{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{cases}  \tag{1}\]  
where \(A\in\mathbb{R}^{p\times p},B\in\mathbb{R}^{p\times q}\), \(C\in\mathbb{R}^{r\times p}\) and \(x,u,y\) are the state, the input and the measurement, respectively. We assume that \(\left(A,B\right)\) and \(\left(A^{T},C^{T}\right)\) are controllable (see [1] for a new reduction from the case where \(\left(A,B\right)\) and \(\left(A^{T},C^{T}\right)\) are only stabilizable). Let  
\[J\left(x_{0},u\right):=\int_{0}^{\infty}\left(x\left(t\right)^{T}Qx\left(t \right)+u\left(t\right)^{T}Ru\left(t\right)\right)dt, \tag{2}\]  
denote the cost functional, where \(Q>0\) and \(R\geq 0\). Assuming that \(x_{0}=x\left(0\right)\) is given, the LQR problem is to attenuate the disturbance \(x_{0}\) using minimal control cost, i.e., to design a regulation input \(u\left(t\right)\) that minimizes \(J\left(x_{0},u\right)\), subject to the constraints given by (1). Let \(u=-Ky\) be the static-output-feedback (SOF) with the closed-loop matrix \(A_{\mathcal{c}\ell}\left(K\right):=A-BKC\). Let \(\mathbb{C}_{-}\) denote the left-half plane, let \(\alpha>0\) and let \(\mathbb{C}_{\alpha}\) denote the set of all \(z\in\mathbb{C}\) with \(\Re\left(z\right)\leq-\alpha\), where \(\Re\left(z\right)\) is the real part of \(z\). Let \(\mathcal{S}^{q\times r}\) denote the set of all matrices \(K\in\mathbb{R}^{q\times r}\), such that \(A_{\mathcal{c}\ell}\) is stable, i.e., \(\sigma\left(A_{\mathcal{c}\ell}\right)\subset\mathbb{C}_{-}\) (where \(\sigma\left(A_{\mathcal{c}\ell}\right)\) is the spectrum of \(A_{\mathcal{c}\ell}\)). By \(\mathcal{S}_{\alpha}^{q\times r}\), we denote the set of all matrices \(K\in\mathbb{R}^{q\times r}\), such that \(\sigma\left(A_{\mathcal{c}\ell}\right)\subset\mathbb{C}_{\alpha}\). In this case, we say that \(A_{\mathcal{c}\ell}\) is \(\alpha\)-stable. Below, we will occasionally write \(\mathcal{S}_{\alpha}\) instead of \(\mathcal{S}_{\alpha}^{q\times r}\), when it is clear what the size of the related matrices is. Let \(K\in\mathcal{S}_{\alpha}^{q\times r}\) be given. Substitution of \(u=-Ky=-KCx\) into (2) gives  
\[J\left(x_{0},K\right):=\int_{0}^{\infty}x\left(t\right)^{T}\left(Q+C^{T}K^{T} RKC\right)x\left(t\right)dt. \tag{3}\]  
Since \(Q+C^{T}K^{T}RKC>0\) and since \(A_{\mathcal{c}\ell}\left(K\right)\) is stable, it follows that the Lyapunov equation\[A_{c\ell}\left(K\right)^{T}P+PA_{c\ell}\left(K\right)=-\left(Q+C^{T}K^{T}RKC\right) \tag{4}\]  
has a unique solution \(P>0\), given by  
\[P=-mat\left(\left(I_{p}\otimes A_{c\ell}\left(K\right)^{T}+A_{c\ell}\left(K \right)^{T}\otimes I_{p}\right)^{-1}\cdot vec\left(Q+C^{T}K^{T}RKC\right) \right), \tag{5}\]  
where \(vec\) puts all the columns of the given matrix in a single column and \(mat\) is the inverse of \(vec\). Let us denote the solution (5) as \(P\left(K\right)\). Substitution of (4) into (3) and noting that \(\dot{x}\left(t\right)=A_{c\ell}\left(K\right)x\left(t\right)\) with \(A_{c\ell}\left(K\right)\) stable, leads to  
\[J\left(x_{0},K\right) =-\int_{0}^{\infty}x\left(t\right)^{T}\left(A_{c\ell}\left(K \right)^{T}P\left(K\right)+P\left(K\right)A_{c\ell}\left(K\right)\right)x \left(t\right)dt\] \[=-\int_{0}^{\infty}\frac{d}{dt}\left(x\left(t\right)^{T}P\left(K \right)x\left(t\right)\right)dt=x_{0}^{T}P\left(K\right)x_{0}.\]  
Thus, we look for \(K\in\mathcal{S}_{\alpha}^{q\times r}\) that minimizes the functional \(J\left(x_{0},K\right)=x_{0}^{T}P\left(K\right)x_{0}\). When \(x_{0}\) is unknown, we seek \(K\in\mathcal{S}_{\alpha}^{q\times r}\) for which  
\[\sigma_{max}\left(K\right):=\max\left(\sigma\left(P\left(K\right)\right)\right) \tag{6}\]  
is minimal. In this case, we get a robust LQR via SOF, in the sense that it minimizes \(J\left(x_{0},K\right)\) for the worst possible (unknown) \(x_{0}\).