Maurício C. de Oliveira

LMI characterization of structural and robust stability

Abstract This paper introduces several stability conditions for a given class of matrices expressed in terms of Linear Matrix Inequalities (LMI), being thus simply and efficiently computable. Diagonal and simultaneous stability, both characterized by polytopes of matrices, are addressed. Using this approach a method particularly attractive to test a given matrix for D-stability is proposed. Lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions. The key idea is to relate Hurwitz stability with a positive realness condition.

H 2 and H ∞ robust filtering for convex bounded uncertain systems

This paper investigates robust filtering design problems in H/sub 2/ and H/sub /spl infin// spaces for continuous-time systems subjected to parameter uncertainty belonging to a convex bounded polyhedral domain. It is shown that, by a suitable change of variables, both designs can be converted into convex programming problems written as linear matrix inequalities. All system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured. Assuming the order of the uncertain system is known, the optimal guaranteed performance H/sub 2/ and H/sub /spl infin// filters are proven to be of the same order as the order of the system. A numerical example illustrate the theoretical results.

Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions

Abstract This paper investigates the problems of checking robust stability and evaluating robust ℋ 2 performance of uncertain continuous-time linear systems with time-invariant parameters lying in polytopic domains. The novelty is the ability to check robust stability by constructing a particular parameter-dependent Lyapunov function which is a polynomial function of the uncertain system matrices, as opposed to a general polynomial function of the uncertain parameter. The degree of the polynomial is tied to a certain integer κ . The existence of such Lyapunov function can be proved by solving parameter-dependent Linear Matrix Inequalities (LMIs), which are guaranteed to be solvable for a sufficiently large yet finite value of κ whenever the system is robustly stable. Extensions to guaranteed ℋ 2 cost computation are also provided. Numerical aspects concerning the programming and the evaluations of the proposed tests are discussed and illustrated by examples.

Robust state feedback LMI methods for continuous-time linear systems: Discussions, extensions and numerical comparisons

This paper provides a brief survey on the subject of LMI (Linear Matrix Inequality) methods for robust state feedback control design. The focus is on continuous-time linear systems with time-invariant uncertain parameters belonging to a polytope. Several LMI conditions from the literature are reviewed and discussed. The relationship between quadratic stabilizability (i.e. constant Lyapunov matrix) and LMI conditions based on parameter-dependent Lyapunov functions is highlighted. As a contribution, a generalization of a family of parameter-dependent conditions is proposed. Discussions, possible extensions and interpretations are provided along the presentation. Finally, the numerical efficacy of the LMI conditions in finding robust controllers when one stabilizing gain is known to exist is investigated. The methods have been tested against a set of hard uncertain systems that are guaranteed to be stabilized by some robust state feedback controller, including a large subset of problems which are known to be stabilized by some robust controller but not to be quadratically stabilizable by any controller.

Synthesis of non-rational controllers for linear delay systems

This paper addresses the control of linear delay systems using non-rational controllers. The structure of the controller is chosen so as to copy the structure of the plant, reproducing the delays in the state and in the output. The resulting stabilization and performance design problems are entirely expressed as linear matrix inequalities. Although the design inequalities are based on delay independent stability conditions, the overall design is delay dependent, in the sense that the controller makes use of the delay parameter of the plant. This parameter is assumed to be constant yet arbitrary. Using non-rational controllers we overcome the main difficulty faced when designing rational controllers for linear delay systems, which is to incorporate in the design problem the matrix multiplier used to prove stability with respect to the delayed part of the system. We illustrate the paper with several examples and provide extensive comparisons with existent results.

Investigating duality on stability conditions

This paper is devoted to investigate the role played by duality in stability analysis of linear time-invariant systems. We seek for a dual statement of a recently developed method for generating stability conditions, which combines Lyapunov stability theory with Finslers Lemma. This method, developed in the time domain, is able to generate a set of (primal) equivalent stability tests involving extra multipliers. The resulting tests have very attractive properties. Stability is characterized via linear matrix inequalities and we use optimization theory to obtain the duals. The dual problems are given a frequency domain interpretation.

Engineering Systems and Free Semi-Algebraic Geometry

This article sketches a few of the developments in the recently emerging area of real algebraic geometry (in short RAG) in a free* algebra, in particular on “noncommutative inequalities”. Also we sketch the engineering problems which both motivated them and are expected to provide directions for future developments. The free* algebra is forced on us when we want to manipulate expressions where the unknowns enter naturally as matrices. Conditions requiring positive definite matrices force one to noncommutative inequalities. The theory developed to treat such situations has two main parts, one parallels classical semialgebraic geometry with sums of squares representations (Positivstellensatze) and the other has a new flavor focusing on how noncommutative convexity (similarly, a variety with positive curvature) is very constrained, so few actually exist.

Brief paper: Stability independent of delay using rational functions

This paper is concerned with the problem of assessing the stability of linear systems with a single time-delay. Stability analysis of linear systems with time-delays is complicated by the need to locate the roots of a transcendental characteristic equation. In this paper we show that a linear system with a single time-delay is stable independent of delay if and only if a certain rational function parameterized by an integer k and a positive real number T has only stable roots for any finite T>=0 and any k>=2. We then show how this stability result can be further simplified by analyzing the roots of an associated polynomial parameterized by a real number @d in the open interval (0,1). The paper is closed by showing counterexamples where stability of the roots of the rational function when k=1 is not sufficient for stability of the associated linear system with time-delay. We also introduce a variation of an existing frequency-sweeping necessary and sufficient condition for stability independent of delay which resembles the form of a generalized Nyquist criterion. The results are illustrated by numerical examples.

DuCTT: A tensegrity robot for exploring duct systems

A robot with the ability to traverse complex duct systems requires a large range of controllable motions as well as the ability to grip the duct walls in vertical shafts. We present a tensegrity robot with two linked tetrahedral frames, each containing a linear actuator, connected by a system of eight actuated cables. The robot climbs by alternately wedging each tetrahedron within the duct and moving one tetrahedron relative to the other. We first introduce our physical prototype, called DuCTT (Duct Climbing Tetrahedral Tensegrity). We next discuss the inverse kinematic control strategy used to actuate the robot and analyze the controllers capabilities within a physics simulation. Finally, we discuss the hardware prototype and compare its performance with simulation.

Optimal tensegrity structures in bending: The discrete Michell truss

Abstract This paper provides the closed form analytical solution to the problem of minimizing the material volume required to support a given set of bending loads with a given number of discrete structural members, subject to material yield constraints. The solution is expressed in terms of two variables, the aspect ratio, ρ - 1 , and complexity of the structure, q (the total number of members of the structure is equal to q ( q + 1 ) ). The minimal material volume (normalized) is also given in closed form by a simple function of ρ and q, namely, V = q ( ρ - 1 / q - ρ 1 / q ) . The forces for this nonlinear problem are shown to satisfy a linear recursive equation, from node-to-node of the structure. All member lengths are specified by a linear recursive equation, dependent only on the initial conditions involving a user specified length of the structure. The final optimal design is a class 2 tensegrity structure. Our results generate the 1904 results of Michell in the special case when the selected complexity q approaches infinity. Providing the optimum in terms of a given complexity has the obvious advantage of relating complexity q to other criteria, such as costs, fabrication issues, and control. If the structure is manufactured with perfect joints (no glue, welding material, etc.), the minimal mass complexity is infinite. But in the presence of any joint mass, the optimal structural complexity is finite, and indeed quite small. Hence, only simple structures (low complexity q) are needed for practical design.