# A Randomized Algorithm for Optimal PID Controllers
## 1 Introduction
Another closely related approach is given in [13] where the 2-Degree of Freedom PID (2-DOF-PID) problem is reduced to the SOF problem, which in order to incorporate the optimization problem of \(\gamma\) suboptimal \(H_{\infty}\) gain, is reformulated as a set of BMI's, where the last is solved via standard BMI solvers. Many problems can be reduced to the SOF problem, imposing structural or other constraints on the controllers. These include the reduction of the minimal-degree dynamic-feedback problem and robust or decentralized stability via static-feedback (see [14; 15], resp.). The design problem of remote digital PID controllers formulated as robust SOF problem for discrete-time Networked Control Systems (NCS), subject to time-delays and randomly missing-data is considered in [16]. A formulation of the reduced-order \(H_{\infty}\) filter problem as constrained SOF problem, is considered in [17]. Unfortunately, the SOF problem with bounded entries or other structural constraints is known to be NP-hard (see [14; 18]). Therefore, minimal-gain SOF with bounded entries is obviously NP-hard problem. Exact Pole-placement and simultaneous and robust stabilization via SOF are also NP-hard (see [14; 19], resp.). Thus, practically, one should expect that only approximation or randomized algorithms will be able to cope with these problems. The article is organized as follows:  
In Section 2 we define the feasibility problem of PID controllers and the specific reduction to the SOF problem that will be used. In Section 3 we define the PID controller optimization problems where the optimality is with respect to the \(H_{\infty}\)-norm, the \(H_{2}\)-norm and the LQR functional, with possible regional pole-placement. In Section 4 we describe the suggested randomized algorithm and finally, in Section 5 we give a real-life example for each one of the above-mentioned optimization problems. The notions are standard. By \(\Re\left(z\right)\) and \(\Im\left(z\right)\) we denote the real and imaginary parts of \(z\in\mathbb{C}\), resp., where by \(\mathbb{C}_{-}\) we denote the left half-plane. For a matrix \(Z\) we denote by \(Z^{T}\) its transpose and by \(\overline{Z}\) we denote the conjugate matrix. By \(Z^{*}\) we denote conjugate-transpose of \(Z\). By \(Z^{+}\) we denote the Moore-Penrose pseudo-inverse of \(Z\), where by \(L_{Z}\) and \(R_{Z}\) we denote the left and right orthogonal projections \(I-Z^{+}Z\) and \(I-ZZ^{+}\), resp.. For a square matrix \(Z\), we denote by \(\sigma\left(Z\right)\) the spectrum of \(Z\) and by \(\mathit{trace}\left(Z\right)\) we denote the trace of \(Z\). For matrices \(Z,W\) we denote by \(Z\otimes W\) the Kronecker product. By \(\|Z\|\) we denote the spectral norm of \(Z\), i.e., the largest singular value of \(Z\), where by \(\|Z\|_{F}\) we denote its Frobenious norm, i.e., \(\left(\mathit{trace}\left(Z^{*}Z\right)\right)^{\frac{1}{2}}\). For matrices \(Z,W\) with the same size, we denote by \(\left\langle Z,W\right\rangle_{F}=\mathit{trace}\left(W^{*}Z\right)\) the Frobenious inner product. For a vector \(v\) we denote by \(\|v\|\) its Euclidean norm \(\left(\sum_{j}\left|v_{j}\right|^{2}\right)^{\frac{1}{2}}\). By \(\mathit{vec}\left(Z\right)\) we denote the vector obtained from the matrix \(Z\) by chaining all its columns to a single column. By \(\mathit{mat}\) we denote the inverse function of \(\mathit{vec}\). For a transfer function \(T\left(s\right)\) analytic in the open right half-plane, we denote by \(\|T\|_{H_{\infty}}\) its \(H_{\infty}\) norm where \(\|T\|_{H_{\infty}}=\mathit{ess}-sup_{\omega\in\mathbb{R}}\left\|T\left(j \omega\right)\right\|\) and by \(\|T\|_{H_{2}}\) we denote its \(H_{2}\) norm where \(\left\|T\right\|_{H_{2}}=\left(\frac{1}{2\pi}\int_{-\infty}^{+\infty}trace\left(T \left(j\omega\right)^{*}T\left(j\omega\right)\right)d\omega\right)^{\frac{1}{2}}\). For a set \(X\) in a topological space, we denote by \(\overline{X}\) its closure and by \(int\left(X\right)\) its interior.