# On Application of the Ray-Shooting Method for LQR via Static-Output-Feedback
## 1 Introduction
Note that  
\[x_{0}^{T}P\left(K\right)x_{0}=\left\|P\left(K\right)^{\frac{1}{2}}x_{0}\right\| ^{2}\leq\left\|P\left(K\right)^{\frac{1}{2}}\right\|^{2}\left\|x_{0}\right\|^ {2}=\left\|P\left(K\right)\right\|\left\|x_{0}\right\|^{2}=\sigma_{max}\left( K\right)\left\|x_{0}\right\|^{2},\]  
and that there exists \(x_{0}\
eq 0\) for which equality holds. Therefore \(\frac{J\left(x_{0},K\right)}{\left\|x_{0}\right\|^{2}}\leq\sigma_{max}\left( K\right)\), where equality holds in the worst case. Note that the functionals \(J\left(x_{0},K\right)\) and \(\sigma_{max}\left(K\right)\) are generally not convex since their domain of definition \(\mathcal{S}^{q\times r}\) (and therefore \(\mathcal{S}_{\alpha}^{q\times r}\)) is generally non-convex. Necessary conditions for optimality were given as three quadratic matrix equations in [2; 3; 4; 5]. Necessary and sufficient conditions for optimality, based on linear matrix inequalities (LMI), were given in [6; 7; 8]. However, algorithms based on these formulations are generally not guaranteed to converge, seemingly because of the non-convexity of the coupled matrix equations or inequalities, and when they converge, it is to a local optimum only. The application of SOFs in LQRs is appealing for several reasons: they are reliable and cheap, and their implementation is simple and direct. Moreover, the long-term memory of dynamic-feedbacks is useless for systems subject to random disturbances, to fast dynamic loadings or to impulses, and the application of state-feedbacks is not always possible, due to unavailability of full-state measurements (see [9], for example). On the other hand, in practical applications, the entries of the needed SOFs are bounded, and since the problem of SOFs with interval constrained entries is NP-hard (see [10; 11]), one cannot expect the existence of a deterministic polynomial-time algorithm to solve this problem. Randomized algorithms are thus natural solutions to this problem. The probabilistic and randomized methods for the constrained SOF problem and robust stabilization via SOFs (among other hard problems) are discussed in [12; 13; 14; 15]. The Ray-Shooting Method was recently introduced in [16], where it was utilized to derive the Ray-Shooting (RS) randomized algorithm for the minimal-gain SOF problem with regional pole-assignment, where the region can be non-convex and unconnected. For a survey of the SOF problem see [17] and for a recent survey of the robust SOF problem see [18]. The contribution of this research is as follows:  
1. The suggested algorithm is based on the Ray-Shooting Method (see [16]), which, as opposed to smooth optimization methods, has the potential of finding a global optimum of continuous functions over compact non-convex and unconnected regions. 2. The suggested algorithm has a proof of convergence (in probability) and explicit complexity. 3. Experience with the algorithm shows good quality of controllers, high percent of success and good run-time for real-life systems. Thus, the suggested practical algorithm efficiently solves the problem of LQR via SOF. 4. The algorithm does not need to solve any Riccati equations and thus can be applied to large systems. 5. The suggested algorithm is one of the few that deals with LQR via SOF and has the ability to deal with discrete-time systems under the same formulation. The reminder of the article is organized as follows:  
In Section 2, we introduce the practical randomized algorithm for the problem of LQR via SOF. In Section 3, we give the results of the algorithm for some real-life systems and we compare its performance with the performance of a well known algorithm that has a proof of convergence to local minimum (under some reasonable assumptions). Finally, in Section 4 we conclude with some remarks.