Michael G. Safonov

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. >

Exact calculation of the multiloop stability margin

A mapping theorem by L.A. Zadeh and C.A. Desoer (1963) serves as the basis for an algorithm that computes the stability margin k/sub m/ of diagonally perturbed multivariable feedback systems without conservatism. The stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs. The essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images. The technique is illustrated in an example with a plant model having three uncertainties. In general, the method is applicable to either SISO or MIMO LTI systems whose uncertainties can be modeled with noninteracting coefficients in the open-loop transfer function representations. >

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...

Synthesis of positive real multivariable feedback systems

This is a tutorial paper describing the application of infinity-norm optimal control theory to the synthesis of sector bounded and positive-real multivariable feedback systems. The paper gives a concise treatment of the practical aspects of the theory, describing how it has been implemented via state-space-oriented computer algorithms. Included is a concise summary of the state-space approach to infinity norm control developed by Doyle and by Safonov and Verma, based on the Hankel model reduction theory and incorporating the multivariable Hankel results of Glover.

A global optimization approach for the BMI problem

The biaffine matrix inequality (BMI) is a potentially very flexible new framework for approaching complex robust control system synthesis problems with multiple plants, multiple objectives and controller order constraints. The BMI problem may be viewed as the nondifferentiable biconvex programming problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. The BMI problem is non-local-global in general, i.e. there may exist local minima which are not global minima. While local optimization techniques sometimes yield good results, global optimization procedures need to be considered for the complete solution of the BMI problem. In this paper, we present a global optimization algorithm for the BMI based on the branch and bound approach. A simple numerical example is included.<<ETX>>

Biaffine matrix inequality properties and computational methods

Many robust control synthesis problems, including /spl mu//k/sub m/-synthesis, have been shown to be reducible to the problem of finding a feasible point under a biaffine matrix inequality (BMI) constraint. The paper discusses the related problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices, a biconvex, nonsmooth optimization problem. Various properties of the problem are examined and several local optimization approaches are presented, although the problem requires a global optimization approach in general.

Design of strictly positive real systems using constant output feedback

The authors present a linear matrix inequality (LMI) approach to the strictly positive real (SPR) synthesis problem: find an output feedback K such that the closed loop system T(s) is SPR. The authors establish that if no such constant output feedback K exists, then no dynamic output feedback with a proper transfer matrix exists to make the closed-loop system SPR. The existence of K to guarantee the SPR property of the closed-loop system is used to develop an adaptive control scheme that can stabilize any system of arbitrary unknown order and unknown parameters.

Unfalsified Virtual Reference Adaptive Switching Control of Plants with Persistent Disturbances

Abstract This paper addresses virtual reference adaptive switching control whereby a data-driven supervisor aims at stabilizing an unknown time-invariant dynamic system by switching at any time in feedback with system one element from a finite family of candidate controllers. Under the only assumption of problem feasibility, viz. the controller family contains a stabilizing controller, the resulting switched system is shown to be stable against arbitrary exogenous persistent bounded disturbances.

Safe Adaptive Switching Control: Stability and Convergence

A formal theoretical explanation of the model-mismatch instability problem associated with certain adaptive control design schemes is proposed, and a solution is provided. To address the model-mismatch problem, a primary task of adaptive control is formulated as finding an asymptotically optimal, stabilizing controller, given the feasibility of adaptive control problem. A class of data-driven cost functions called cost-detectable is introduced that detect evidence of instability without reference to prior plant models or plant assumptions. The problem of designing adaptive systems that are robustly immune to mismatch instability problems is thus placed in a setting of a standard optimization problem. We call the result safe adaptive control because it robustly achieves adaptive stabilization goals whenever feasible, without prior assumptions on the plant model and, hence, without the risk of model-mismatch instability. The result improves the robustness of previous results in hysteresis switching control, both for discrete and for continuously-parameterized candidate controller sets. Examples are provided.

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. >