A. Vicino

Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach

In this note, robust stability of state-space models with respect to real parametric uncertainty is considered. Specifically, a new class of parameter-dependent quadratic Lyapunov functions for establishing stability of a polytope of matrices is introduced, i.e., the homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs). The choice of this class, which contains parameter-dependent quadratic Lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form, is motivated by the property that a polytope of matrices is stable if and only there exists an HPD-QLF. The main result of the note is a sufficient condition for determining the sought HPD-QLF, which amounts to solving linear matrix inequalities (LMIs) derived via the complete square matricial representation (CSMR) of homogeneous matricial forms and the Lyapunov matrix equation. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties

Consideration is given to the problem of robust stability analysis of linear dynamic systems with uncertain physical parameters entering as polynomials in the state equation matrices. A method is proposed giving necessary and sufficient conditions for computing the uncertain system stability margin in parameter space, which provides a measure of maximal parameter perturbations preserving stability of the perturbed system around a known, stable, nominal system. A globally convergent optimization algorithm that enables solutions to the problem to be obtained is presented. Using the polynomial structure of the problem, the algorithm generates a convergent sequence of interval estimates of the global extremum. These intervals provide a measure of the accuracy of the approximating solution achieved at each step of the iterative procedure. Some numerical examples are reported, showing attractive features of the algorithm from the point of view of computational burden and convergence behavior. >

Robustness of pole location in perturbed systems

Abstract In this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients. A geometric approach in coefficient space is exploited to derive maximal deviations (in a given class of admissible perturbations) of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane. It is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane. This last result can be considered an extension of the result of the well-known theorem of Kharitonov. It is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations.

An LMI-based approach for characterizing the solution set of polynomial systems

Considers the problem of solving certain classes of polynomial systems. This is a well known problem in control system analysis and design. A novel approach is developed as a possible alternative to the commonly employed algebraic geometry and homotopy methods. The first result of the paper shows that the solution set of the polynomial system belongs to the kernel of a symmetric matrix. Such a matrix is obtained via the solution of a suitable linear matrix inequality (LMI) involving the maximization of the minimum eigenvalue of an affine family of symmetric matrices. The second result concerns the computation of the solutions from the kernel of the obtained matrix. In particular, it is shown that the solutions can be recovered quite easily if the dimension of the kernel is smaller than the degree of the polynomial system. Finally, some application examples are illustrated to show the features of the approach and to make a brief comparison with the algebraic geometry techniques.

Clockwise property of the Nyquist plot with implications for absolute stability

Abstract In this paper the clockwise property of the Nyquist plot of stable transfer functions is addressed. The relationships between clockwise property and modulus and/or phase monotonicity are analyzed. Some necessary and sufficient conditions for a class of stable transfer functions to enjoy the clockwise property are obtained. These results are used to enlarge the class of systems for which the Kalman conjecture on absolute stability of nonlinear systems holds.

An algorithm for nonconservative stability bounds computation for systems with nonlinearly correlated parametric uncertainties

The problem of robust stability analysis of linear dynamic systems with uncertain physical parameters entering polynomially in the state equation matrices is considered. A method is proposed for solving both the problems of checking stability of a family of matrices generated by a box in parameter space and that of computing maximal box domains of given shape in parameter space generating only stable matrices. A globally convergent optimization algorithm is presented which allows the stability problems to be solved. The implemented algorithm provides a measure of the accuracy of the solution achieved at each step. A numerical example is given, showing attractive features of the algorithm from the point of view of computational burden and convergence behavior.<<ETX>>

Design of Optimally Robust Controllers with Few Degrees of Freedom

Abstract In this paper a first attempt is made to the design of robust controllers for families of plants with interval and linearly dependent coefficient perturbations. An optimally robust controller is defined as a controller of fixed order stabilizing the nominal plant and maximizing the closed loop stability margin in the controller parameter space. Compensators of fixed structure and with coefficients depending on one or two free design parameters are considered. An algorithm for determining optimally robust controllers is proposed and an application example is presented to illustrate performance of the algorithm

LMI-based techniques for solving quadratic distance problems

The computation of the minimum distance from a point to a surface in a finite dimensional space is a key issue in several system analysis and control problems. The paper presents a general framework in which some classes of minimum distance problems are tackled via LMI techniques. Exploiting a suitable representation of homogeneous forms, a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one-parameter family of LMI optimization problems. Several properties of the proposed technique are discussed. In particular, tightness of the lower bound is investigated, providing both a simple algorithmic procedure for a posteriori optimality testing and a structural condition on the related homogeneous form that ensures optimality a priori. Extensive numerical simulations are reported showing promising performances of the proposed method.

Set membership pose estimation of mobile robots based on angle measurements

Addresses the problem of estimating position and orientation of a mobile robot navigating in an environment for which a landmark-based map is available. A set theoretic approach to the problem is proposed. Estimates of robot position and heading are derived in terms of uncertainty regions, under the hypothesis that the errors affecting all sensors measurements are unknown but bounded. A recursive estimation procedure for localization based on angle measurements is presented. Simulation and experimental results prove the effectiveness of the proposed approach.

Stability margin and robust control of systems with parametric nonlinear perturbations

A report is presented of results obtained by applying an algorithm presented in Vicino, Tesi, and Milanese (1988, 1990) for computing stability margins of linear control systems whose state matrix entries are polynomial (or rational) functions of a vector of physical uncertain parameters. The algorithm has been used on some meaningful application examples, giving a positive answer to important questions about the actual effectiveness of the algorithm both from computing time and numerical precision points of view.<<ETX>>