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For linear time-invariant (LTI) systems, the design of an optimal controlleris a commonly encountered problem in many applications. Among all theoptimization approaches available, the linear quadratic regulator (LQR)methodology certainly garners much attention and interest. As is well-known,standard numerical tools in linear algebra are readily available which enablethe determination of the optimal static LQR feedback gain matrix when all thesystem state variables are measurable. However, in various certain scenarioswhere some of the system state variables are not measurable, and consequentprescribed structural constraints on the controller structure arise, theoptimization problem can become intractable due to the non-convexitycharacteristics that can then be present. In such cases, there have been somefirst-order methods proposed to cater to these problems, but all of thesefirst-order optimization methods, if at all successful, are limited to onlylinear convergence. To speed up the convergence, a second-order approach in thematrix space is essential, with appropriate methodology to solve the linearequality constrained static output feedback (SOF) problem with a suitablydefined linear quadratic cost function. Thus along this line, in this work, anefficient method is proposed in the matrix space to calculate the Hessianmatrix by solving several Lyapunov equations. Then a new optimization techniqueis applied to deal with the indefiniteness of the Hessian matrix. Subsequently,through Newton's method with linear equality constraints, a second-orderoptimization algorithm is developed to effectively solve the constrained SOFLQR problem. Finally, two numerical examples are described which demonstratethe applicability and effectiveness of the proposed method.

Second-Order Non-Convex Optimization for Constrained Fixed-Structure Static Output Feedback Controller Synthesis