Pradeep Pandey

A collection of robust control problems leading to LMIs

The authors consider four control-related problems, all of which involve reformulation into linear matrix inequalities (LMIs). The problems are: structured singular value ( mu ) upper bound synthesis for constant matrix problems; robust-state-feedback problem with quadratic stability criteria for uncertain systems; optimal, constant, block diagonal, similarity scaling for full information and state feedback H/sub infinity / problem; and a partial theory for optimal performance in systems which depend on several independent variables.<<ETX>>

Optimal, constant I/O similarity scaling for full-information and state-feedback control problems

Abstract In robust control problems, capturing all robustness and performance objectives in a single H∞ norm cost function is impossible. An alternative approach, still untilizing the H∞ norm, involves diagonal similarity scaling of certain closed loop transfer functions. The set of allowable diagonal scalings is problem dependent, and reflects assumptions about the uncertainty, and desired performance objectives. The scaling set considered here is a prescribed convex set of positive definite matrices. We consider the optimal constant scaling problem for the Full-Information H∞ control problem. The solution is obtained by transforming the original problem into a convex feasibility problem, specifically, a structured, linear matrix inequality. In special cases, solvability of the Full-Information problem is equivalent to solvability of the State-Feedback problem.

Continuity properties of the real/complex structured singular value

The structured singular-value function ( mu ) is defined with respect to a given uncertainty set. This function is continuous if the uncertainties are allowed to be complex. However, if some uncertainties are required to be real, then it can be discontinuous. It is shown that mu is always upper semi-continuous, and conditions are derived under which it is also lower semi-continuous. With these results, the real-parameter robustness problem is reexamined. A related (although not equivalent) problem is formulated, which is always continuous, and the relationship between the new problem and the original real- mu m problem is made explicit. A numerical example and results obtained via this related problem are presented. >

A PARALLEL ALGORITHM FOR THE MATRIX SIGN FUNCTION

We propose a new parallel algorithm for computing the sign function of a matrix. The algorithm is based on the Pade approximation of a certain hypergeometric function which in turn leads to a rational function approximation to the sign function. Parallelism is achieved by developing a partial fraction expansion of the rational function approximation since each fraction can be evaluated on a separate processor in parallel. For the sign function the partial fraction expansion is numerically attractive since the roots and the weights are known analytically and can be computed very accurately. We also present experimental results obtained on a Cray Y-MP.

Quasi-Newton Methods for Solving Algebraic Riccati Equations

Variants of Newtons method for solving an algebraic Riccati equation are presented with the aim of developing efficient and parallel algorithms. Sufficient conditions for the convergence of these algorithms are given.

A note on invariant subspaces of Hamiltonian matrices

Algorithms involving certain Riccati equations of the type arising in H/sub /spl infin// control design methods are considered. Techniques are presented that avoid explicitly forming the Riccati solution by instead working directly with invariant subspace of the associated Hamiltonian matrix. It is shown that these techniques can be advantageous, both numerically and computationally.<<ETX>>