Low-order control design for LMI problems using alternating projection methods

A Newton-like method for solving rank constrained linear matrix inequalities

“…By choosing alg equal to the of Problem 13, an algorithm calculated solution to Problem 13 will satisfy (16), (17) and (18). In addition, the constraint (19) will be satisfied to a tolerance of 2 , that is, at least 2n−(n+n c ) eigenvalues of the matrix in (19) will have magnitude 2 or less.…”

“…We now consider a particular reduced order output feedback problem from [4] (see also [18], [37,Chapter 10] and [17]). The system considered is a two-mass-spring system with state space representation given by Given α > 0, we wish to find an order 2 dynamic controller that places the closed loop poles in (15).…”

“…Currently available algorithms for the rank constrained LMI problem are largely heuristic in nature and are not guaranteed to converge to a solution even if one exists. Solution methods for this problem, or certain specializations of the problem, include those based on modified interior point methods [8], [9]; linearization [11], [22], [25]; alternating projections [18], [4], [17]; trace minimization methods that try to solve the problem by solving a related convex problem [34], [29]; augmented Lagrangian methods [12], [2]; and sequential semidefinite programming [13]. Aside from [13], these methods do not have established superlinear convergence rates and the challenge remains to find numerical schemes with verifiable local quadratic convergence rates.…”

“…The method is closely related to existing alternating projection methods but is expected to have improved convergence properties due to a built-in Newton-type step. In [18] and [17] alternating projection algorithms are proposed that involve tangent-like ideas, similar to our approach. However, the implementation details are different and the connection to the Newton method is neither mentioned nor obvious.…”

Reduced-Order H∞ Controllers Based on Improved Genetic Algorithm

“…Therefore, reduced-order H ∞ controller is a non-linear and non-protruded optimization problem. At present, there are not ordinary methods to solve it except for numerical value methods, such as eliminate method and variable substitute method (Grigoriadis et al, 1995). But, many restriction conditions about object in eliminate method, and substitute formula is difficult to find in substitute method because of complex non-linear relation among H ∞ controller variables.…”