# On Application of the Ray-Shooting Method for LQR via Static-Output-Feedback
## 2 The Practical Algorithm for the Problem of LQR via SOF
We conclude that the suggested algorithm outperforms the MC Algorithm regarding the above-mentioned key of performance, although the MC Algorithm outperforms the suggested algorithm in terms of CPU time. The MC Algorithm seems to perform better locally, while the suggested algorithm seems to perform better globally. Thus, practically, the best approach would be to apply the suggested algorithm in order to find a close neighborhood of a global minimum and then to apply the MC Algorithm on the result, for the local optimization. ## 4 Concluding Remarks

For a discrete-time system

\[ \begin{cases}x_{k+1}=Ax_{k}+Bu_{k}\\ y_{k}=Cx_{k}\end{cases}  \tag{9}\]

and cost functional

\[J\left(x_{0},u\right):=\sum_{k=0}^{\infty}\left(x_{k}^{T}Qx_{k}+u_{k}^{T}Rx_{ k}\right), \tag{10}\]

let \(u_{k}=-Ky_{k}\) be the SOF, and let \(A_{c\ell}\left(K\right):=A-BKC\) be the closed-loop matrix. Let \(\mathbb{D}\) denote the open unit-disk, let \(0<\alpha<1\) and let \(\mathbb{D}_{\alpha}\) denote the set of all \(z



Finally, if \(\lambda\) is an eigenvalue of \(A_{\mathit{c}\ell}\left(K\right)\) and \(v\) is a corresponding eigenvector then (12) yields \(1-\left|\lambda\right|^{2}=\frac{v^{*}\left(Q+C^{T}K^{T}RKC\right)v}{v^{*}P(K)v} \geq\frac{v^{*}Qv}{v^{*}P(K)v}\). Therefore \(\left|\lambda\right|^{2}\leq 1-\frac{v^{*}Qv}{v^{*}P(K)v}\), and thus, minimizing \(\sigma_{max}\left(K\right)\) results in larger abscissa. Now, the suggested algorithm can be readily applied to discrete-time systems. As to the MC Algorithm, we are not aware of any discrete-time analogue of it. We conclude that the Ray-Shooting Method is a powerful tool, since it practically solves the problem of LQR via SOF, for real-life systems. The suggested algorithm has good performance, and is proved to converge in probability. The suggested method can similarly handle the problem of LQR via SOF for discrete-time systems. Obviously, this enlarges the scope and usability of the Ray-Shooting Method and we expect to receive more results in this direction. I wish to dedicate this article to the memory of my beloved father [PERSON] and to the memory of my beloved wife and friend [PERSON]. The author declares no conflict of interest. ## References

* [PERSON] (2012) [PERSON] A characterization of all the static stabilizing controllers for LTI systems. _Linear Algebra Its Appl._**2012**, _437_, 525-548. * [PERSON] and [PERSON] (1970) [PERSON]; [PERSON] On the design of optimal dynamic compensators for linear constant systems. _IEEE Trans. Autom. Control_**1970**, _15_, 658-660. * [PERSON] and [PERSON] (1970) [PERSON]; [PERSON] On the determination of the optimal constant output feedback gains for linear multivariables systems. _IEEE Trans. Autom. Control_**1970**, _15_, 44-48. * [PERSON] et al. (1971) [PERSON]; [PERSON]; [PERSON] Optimal limited state variable feedback controllers for linear systems. _IEEE Trans. Autom. Control_**1971**, _16_, 785-793. * [PERSON] and [PERSON] (1985) [PERSON]; [PERSON] Convergence of numerical algorithm for calculating optimal output feedback gains. _IEEE Trans. Autom. Control_**1985**, _30_, 900-903. * [PERSON] and [PERSON] (1994) [PERSON]; [PERSON] All controllers for the general \(H_{\infty}\) control problem: LMI existance conditions and stste space formulas. _Automatica_**1994**, _30_, 1307-1317. * [PERSON] and [PERSON] (1994) [PERSON]; [PERSON] Linear quadratic suboptimal control with static output feedback. _Syst. Control Lett._**1994**, _23_, 421-430. * [PERSON] et al. (1993) [PERSON]; [PERSON]; [PERSON] [PERSON] Optimal \(H_{2}\) control by output feedback. In Proceedings of the 32 nd IEEE Conference on Decision and Control, San Antonio, TX, USA, 15-17 December 1993; pp. 102-107. * [PERSON] et al. (1999) [PERSON]; [PERSON]; [PERSON]. Design of A Vehicular Suspension Controller by Static Output Feedback. In Proceedings of the American Control Conference, San Diego, CA, USA, 2-4 June 1999. * [PERSON] and [PERSON] (1997) [PERSON]; [PERSON] [PERSON] NP-hardness of some linear control design problems. _SIAM J.