R. Y. Chiang

A Schur method for balanced-truncation model reduction

It is shown that a not-necessarily-balanced state-space realization of the Moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product PQ of the reachability and controllability Grammians. Two specific methods for computing these bases are proposed, one based on the ordered Schur decomposition of PQ and the other based on the Cholesky factors of P and Q. The algorithms perform reliably even for nonminimal models. >

Simplifying the H∞ theory via loop-shifting, matrix-pencil and descriptor concepts

The 2-Riccati H∞ controller formulae and derivations are simplified via various ‘loop-shifting’ transformations that are naturally expressed in terms of a degree-one polynomial system matrix (PSM) closely related to the Luenberger descriptor form of a system. The technique enables one without loss of generality to restrict attention to the simple case in which D 11 =0, D 22 = 0, D T 12= [0 l], D 21 = [0 l],D l 12 C1=0 and B 1 D T 21 =0. Matrix-fraction descriptions (MFDs) for the algebraic Riccati equation solutions afford another change of variables, which brings the 2-Riccati H∞ controller formulae into a cleaner, more symmetric descriptor form having the important practical advantage that it eliminates the numerical difficulties that can occur in cases where one or both of the Riccati solutions, Pi and Q, blow up and in cases where l— QP is nearly singular. Numerical difficulties previously associated with verifying the existence conditions P ≥ 0, Q ≥ 0, and λmaxlpar;QP) < 1 are largely eliminated by e...

Optimal Hankel model reduction for nonminimal systems

A basis-free descriptor system representation is shown to facilitate the computation of all minimum-degree and optimal kth-order all-pass extensions and Hankel-norm approximants. The descriptor representation has the same simple form for both the optimal and suboptimal cases. The method makes Hankel model reduction practical for nonminimal and nearly nonminimal systems by eliminating the ill-conditioned calculation of a minimal balanced realization. A simple, numerically sound method based on singular-value decomposition enables the results to be expressed in state-space form. >

CACSD using the state-space L/sup infinity / theory-a design example

The L/sup infinity / optimal control theory is used to achieve singular value loop-shaping design objectives in a multivariable aircraft pitch-axis control design example involving an unstable nonminimum phase plant. Comparison to the frequency-weighted LQG (Linear quadratic Gaussian) design shows that the L/sup infinity / theory easily produced a superior design having both higher bandwidth and greater stability margin, with no need for designer interaction for iterative adjustment of weights, than LQG. The design example provides a vivid illustration of how the all-pass property of the L/sup infinity / theory enables precise shaping of closed-loop singular-value Bode plots to conform pointwise to prespecified weighting functions. >

Model Reduction for Robust Control: A Schur Relative-Error Method

A numerically robust Relative Error Method (REM) for state-space model order reduction is described. Our algorithm is based on Desais Balanced Stochastic Truncation (BST) technique for which M. Green has obtained an L∞ relative-error bound. However, unlike previous methods, our Schur method completely circumvents the numerically delicate initial step of obtaining a minimal balanced stochastic realiztion (BSR) of the the power spectrum matrix G(s)GT(-s).

Real/complex multivariable stability margin computation via generalized Popov multiplier-LMI approach

Using classical results in spectral factorization and generalized positive real matrix theories, the computation of real/complex structural singular value using generalized Popov multipliers is formulated as a convex optimization problem involving linear matrix inequalities (LMIs). Efficient numerical algorithms exist for solving LMIs.

Convexity property of the one-sided multivariable stability margin

In evaluating the stability robustness of multivariable control systems having one-sided parameter uncertainty, a problem that naturally arises is the minimization over diagonal matrices D of the greatest eigenvalue of (e/sup D/Ae/sup -D/+(e/sup D/Ae/sup -D/)*)/2. The minimization is proved to be convex, thus guaranteeing that every local minimum is also a global minimum and, in theory, guaranteeing the global convergence of generalized gradient nonlinear programming algorithms for computing the minimizing D. >

Multiplier K/sub m///spl mu/-analysis - LMI approach

The theory of K/sub m///spl mu/-synthesis and analysis introduced by Safonov (1983) and Doyle (1983) essentially performs the following D-K iteration steps: 1) design a /spl Hscr//sub /spl infin//-controller K(s) such as to maximize the multivariable stabilty margin K/sub m/; and 2) for discrete frequencies, find a diagonal scaling frequency response matrix D(jw) as to maximize K/sub m/ for a fixed K(s). Step 1 can be cast as a linear matrix inequalities (LMI) optimization problem for full order controller. The K/sub m/ analysis in step 2 has been shown to be a LMI optimization problem. The paper reviews the fixed order optimal multiplier K/sub m/ analysis theory and presents state space parameterization for the set of multipliers subject to mixed dynamic and real parametric uncertainties. A numerical example involving the robustness analysis of the Cassini spacecraft thrust vector control system is also included.