Abstract not found

. We consider least-squares problems where the coefficient matrices A; b are unknown-butbounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Least-squares, uncertainty, robustness, second-order cone...

this paper is organized as follows. First, we discuss the formulation of the semidefinite programming problem used by CSDP. We then describe the predictor corrector algorithm used by CSDP to solve the SDP. We discuss the storage requirements of the algorithm as well as its computational complexity. Finally, we present results from the solution of a number of test problems. 2 The SDP Problem We consider semidefinite programming problems of the form max tr (CX)

. A variety of analysis and design problems in control, communication and information theory, statistics, combinatorial optimization, computational geometry, circuit design, and other fields can be expressed as semidefinite programming problems (SDPs) or determinant maximization problems (max-det problems). These problems often have matrix structure, i.e., some of the optimization variables are matrices. This matrix structure has two important practical ramifications: first, it makes the job of translating the problem into a standard SDP or maxdet format tedious, and, second, it opens the possibility of exploiting the structure to speed up the computation. In this paper we describe the design and implementation of sdpsol, a parser/solver for SDPs and max-det problems. sdpsol allows problems with matrix structure to be described in a simple, natural, and convenient way. Although the current implementation of sdpsol does not exploit matrix structure in the solution algorithm, the languag...

Conventional methods for optimal sizing of wires and transistors use linear RC circuit models and the Elmore delay as a measure of signal delay. If the RC circuit has a tree topology the sizing problem reduces to a convex optimization problem which can be solved using geometric programming. The tree topology restriction precludes the use of these methods in several sizing problems of significant importance to high-performance deep submicron design including, for example, circuits with loops of resistors, e.g., clock distribution meshes, and circuits with coupling capacitors, e.g., buses with crosstalk between the lines. The paper proposes a new optimization method which can be used to address these problems. The method uses the dominant time constant as a measure of signal propagation delay in an RC circuit, instead of Elmore delay. Using this measure, sizing of any RC circuit can be cast as a convex optimization problem which can be solved using the recently developed efficient interi...

Abstract—This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of nth-order linear time-invariant (LTI) systems with nu (respectively, ny) independent inputs (respectively, outputs). The algorithm is based on a “cone complementarity ” formulation of the problem and is guaranteed to produce a stabilizing controller of order m n 0 max(nu;ny), matching a generic stabilizability result of Davison and Chatterjee [7]. Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura’s generic stabilizability result (m n0nu0ny+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis. Index Terms — Complementarity problem, linear matrix inequality, reduced-order stabilization, static output feedback. I.

Abstract — In this paper we consider dynamical systems which are driven by “events ” that occur asynchronously. It is assumed that the event rates are fixed, or at least they can be bounded on any time period of length T. Such systems are becoming increasingly important in control due to the very rapid advances in digital systems, communication systems, and data networks. Examples of such systems include, control systems in which signals are transmitted over an asynchronous network; distributed control systems in which each subsystem has its own objective, sensors, resources and level of decision making; parallelized numerical algorithms in which the algorithm is separated into several local algorithms operating concurrently at different processors; and queuing networks. We present a Lyapunov-based theory for asynchronous dynamical systems and show how Lyapunov functions and controllers can be constructed for such systems by solving linear matrix inequality (LMI) and bilinear matrix inequality (BMI) problems. Examples are also presented to demonstrate the effectiveness of the approach.

In this paper we address the problem of low-authority controller (LAC) design. The premise is that the actuators have limited authority, and hence cannot significantly shift the eigenvalues of the system. As a result, the closed-loop eigenvalues can be well approximated analytically using perturbation theory. These analytical approximations may suffice to predict the behavior of the closed-loop system in practical cases, and will provide at least a very strong rationale for the first step in the design iteration loop. We will show that LAC design can be cast as convex optimization problems that can be solved efficiently in practice using interior-point methods. Also, we will show that by optimizing the ℓ1 norm of the feedback gains, we can arrive at sparse designs, i.e., designs in which only a small number of the control gains are nonzero. Thus, in effect, we can also solve actuator/sensor placement or controller architecture design problems. Keywords: Low-authority control, actuator/sensor placement, linear operator perturbation theory, convex optimization, second-order cone programming, semi-definite programming, linear matrix inequality. 1

We propose to use the dominant time constant of a resistor-capacitor (RC) circuit as a measure of the signal propagation delay through the circuit. We show that the dominant time constant is a quasiconvex function of the conductances and capacitances, and use this property to cast several interesting design problems as convex optimization problems, specifically, semidefinite programs (SDPs). For example, assuming that the conductances and capacitances are affine functions of the design parameters (which is a common model in transistor or interconnect wire sizing), one can minimize the power consumption or the area subject to an upper bound on the dominant time constant, or compute the optimal tradeoff surface between power, dominant time constant, and area. We will also note that, to a certain extent, convex optimization can be used to design the topology of the interconnect wires. This approach has two advantages over methods based on Elmore delay optimization. First, it handles a far wider class of circuits, e.g., those with non-grounded capacitors. Second, it always results in convex optimization problems for which very efficient interiorpoint methods have recently been developed. We illustrate the method, and extensions, with several examples involving optimal wire and transistor sizing.

. This report describes SDPpack, a package of Matlab files designed to solve semidefinite programs (SDP). SDP is a generalization of linear programming to the space of block diagonal, symmetric, positive semidefinite matrices. The main routine implements a primal--dual Mehrotra predictor-- corrector scheme based on the XZ+ZX search direction. We also provide certain specialized routines, one to solve SDP's with only diagonal constraints, and one to compute the Lov'asz ` function of a graph, using the XZ search direction. Routines are also provided to determine whether an SDP is primal or dual degenerate, and to compute the condition number of an SDP. The code optionally uses MEX files for improved performance; binaries are available for several platforms. Benchmarks show that the codes provide highly accurate solutions to a wide variety of problems. Copyright c fl 1997. All rights are reserved by the authors; restrictions in the copyright notice in each release also apply. ...