Jovan Stefanovski

The problem of obtaining a dynamic controller such that its poles and the modes of the closed-loop (CL) system are in specified regions ${\mathbb {D}}^{\prime}$ and ${\mathbb {D}}^{\prime \prime }$ in the open left complex half-plane, such that the CL transfer matrix has as small as possible norm over a prescribed frequency range ${\mathbb {B}}$ is considered. A novel optimization algorithm for controller design is presented. The algorithm is based on iterative solving linear matrix inequalities and finds in every iteration a minimal norm static output feedback control for a descriptor system. A problem solution is obtained in every iteration, while the performance index decreases. A formal proof that the algorithm is descent in respect to the performance index is given for some subproblems.

The optimal variant of one‐sided ‐problem is elaborated, and an LMI‐based algorithm is presented, which works in the generic case. An application in controlled asymptotic filtering is presented, in which the matrices of the ‐problem are affine functions of controller coefficients. These two results are illustrated with examples.

Necessary and sufficient condition for the newly introduced problem to find a strictly F-positive real (SF-PR) interpolant for a one-sided rational interpolation are given,as well as the set of all solutions. The SF-PR property of the interpolant H, for a rational matrix (RM) F to be found, implies that H is bistable and biproper (BiPS). Since such an interpolant is needed in the strong stabilization problem, the interpolation result is used to develop a control design algorithm such that a stable stabilizing controller of degree $n+\ell$ is constructed using the RMs F and H, where $\ell$ is the number of unstable plant zeros. Examples, including one with a chemical reactor plant, illustrate the control design algorithm.

A deflating subspace approach is applied to the unknown input observer (UIO) design. The associated matrix pencil (MP) to the transfer matrix from the unknown input to the measured output is arbitrary. A unique necessary and sufficient condition is given, in terms of a proper stable left-deflating subspace (PSL-DS) of the MP. For the class of plants such that UIO exists, a UIO-based controller design is presented, based on obtaining a constant coefficient matrix that is a part of the controller, instead of a rational controller matrix. Then the design is applied to a fault-tolerant control (FTC). Illustrative examples are presented.

Summary: An iterative algorithm with linear matrix inequalities (ILMI) is presented, for robust stabilization of linear time‐invariant systems, where the uncertainty is a structured rational matrix belonging to a positive‐real‐like set, determined by scalar . The algorithm searches for a maximal set, in respect to inclusion of sets, that is, for the minimal parameter . A property of the algorithm is that a suboptimal solution is found in each iteration of the algorithm, and decreases in each iteration. Robustness in respect to the change of controller coefficients is elaborated on, and it is shown by examples that it can be greater than the robustness of the ‐controller. The efficiency of the algorithm, as well as its conservativity/non‐conservativity, is tested on examples with random numbers.

A formulation of the problem of fault tolerant tracking (FTT) in presence of general additive faults and disturbances is given. It is proved that a necessary condition for the FTT problem is that the plant is minimum‐phase. Then the problem is solved for the minimum‐phase descriptor systems, but possibly with zeros at infinity. The proposed overall controller consists of a feedback controller, which is stabilizing, and a feedforward controller, whose role is to achieve the tracking of an arbitrary reference signal by the controlled output. The feedback controller consists of a possibly unstable, improper and non‐minimum‐phase nominal controller, a fault estimator, and a fault compensator. The role of the nominal controller is to achieve a nominal tracking behavior in absence of faults. The role of the fault compensator is to achieve a nominal tracking behavior in presence of faults, in case that the fault estimation is accurate. However, the fault estimation need not to be accurate, if a design parameter, depending on the nominal controller, can be chosen small. Illustrative examples are given.