Abstract not found

Several problems arising in control system analysis and design, such as reduced order controller synthesis, involve minimizing the rank of a matrix variable subject to linear matrix inequality (LMI) constraints. Except in some special cases, solving this rank minimization problem (globally) is very difficult. One simple and surprisingly effective heuristic, applicable when the matrix variable is symmetric and positive semidefinite, is to minimize its trace in place of its rank. This results in a semidefinite program (SDP) which can be efficiently solved. In this paper we describe a generalization of the trace heuristic that applies to general nonsymmetric, even non-square, matrices, and reduces to the trace heuristic when the matrix is positive semidefinite. The heuristic is to replace the (nonconvex) rank objective with the sum of the singular values of the matrix, which is the dual of the spectral norm. We show that this problem can be reduced to an SDP, hence efficiently solved. To motivate the heuristic, we show that the dual spectral norm is the convex envelope of the rank on the set of matrices with norm less than one. We demonstrate the method on the problem of minimum order system approximation. 1

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

In this paper we present a path-following (homotopy) method for (locally) solving bilinear matrix inequality (BMI) prob- lems in control. The method is to linearize the BMI using a first order perturbation approximation, and then iteratively compute a perturbation that "slightly" improves the controller performance by solving a semidefinite program (SDP). This process is repeated un- til the desired performance is achieved, or the performance cannot be improved any further. While this is an approximate method for solving BMIs, we present several examples that illustrate the effectiveness of the approach.

Since 1984 there has been a concentrated e ort to develop e cient interior-point methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interior-point methods (beyond their e ciency for LP): they extend gracefully to nonlinear convex optimization problems. New interior-point algorithms for problem classes such as semide nite programming (SDP) or second-order cone programming (SOCP) are now approaching the extreme e ciency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to e ciently solve nonlinear convex optimization problems opens up new applications. In the rst example we show how SOCP can be used to solve robust open-loop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the set-point and feedback gains for a controller, and compare this method with the more standard approach. Our nal application concerns analysis and synthesis via linear matrix inequalities and SDP. Submitted to a special issue of Journal of Process Control, edited by Y. Arkun & S. Shah, for papers presented at the 1997 IFAC Conference onAdvanced Process Control, June 1997, Ban. This and related papers available via anonymous FTP at

In this paper, we introduce a new class of Lyapunov functionals for analyzing hybrid dynamical systems. This class can be thought of as a generalization of the Lyapunov functional introduced by Yakubovich for systems with hysteresis nonlinearities which incorporates path integrals that account for the energy loss or gain every time a hysteresis loop is traversed. Hence, these Lyapunov functionals capture the path-dependence of the “stored energy ” in hybrid dynamical systems and are therefore less conservative over previously published approaches in analyzing such systems. More importantly, we show that searching over the proposed class of Lyapunov functionals to prove some specification (e.g., stability) can be cast as a semidefinite program (SDP), which can then be efficiently solved (globally) using widely available software. Examples are presented to show the effectiveness of this class of Lyapunov functionals in analyzing hybrid dynamical systems.

This paper presents an iterative algorithm for solving the parametric robust H2 controller synthesis problem and analyzes the convergence properties of the algorithm on several examples. Iterative procedures are normally applied to a large class of robust control design problems in which the formulation naturally leads to bilinear matrix inequalities (BMIs). It is di cult to make concrete statements about the behavior of these iterative algorithms, except that it is often conjectured that the cost in each step of the solution procedure is reduced, which implies that the algorithms should converge to a local minimum. Similar di culties exist for the new LMIbased iterative algorithm that we haverecently proposed to solve the BMIs that occur in robust H2 control design. The e ectiveness of the new algorithm has already been demonstrated on several numerical examples. This paper adds an important component tothediscussion on the convergence of the new algorithm by verifying that it e ciently converges to the optimal solution. In the process, we provide some new key insights on the proposed design technique which indicate that it exhibits properties similar to the D{K iteration of the complex =Km-synthesis. 1