LMI stability conditions for uncertain rational nonlinear systems

Abstract

This paper presents LMI conditions for local, regional, and global robust asymptotic stability of rational uncertain nonlinear systems. The uncertainties are modeled as real time varying parameters with magnitude and rate of variation bounded by given polytopes and the system vector field is a rational function of the states and uncertain parameters. Sufficient LMI conditions for asymptotic stability of the origin are given through a rational Lyapunov function of the states and uncertain parameters. The case where the time derivative of the Lyapunov function is negative semidefinite is also considered and connections with the well known LaSalle's invariance conditions are established. In regional stability problems, an algorithm to maximize the estimate of the region of attraction is proposed. The algorithm consists of maximizing the estimate for a given target region of initial states. The size and shape of the target region are recursively modified in the directions where the estimate can be enlarged. The target region can be taken as a polytope (convex set) or union of polytopes (non-convex set). The estimates of the region of attraction are robust with respect to the uncertain parameters and their rate of change. The case of global and orthant stability problems are also considered. Connections with some results found in sum of squares based methods and other related methods found in the literature are established. The LMIs in this paper are obtained by using the Finsler's Lemma and the notion of annihilators. The LMIs are characterized by affine functions of the state and uncertain parameters, and they are tested at the vertices of a polytopic region. It is also shown that, with some additional conservatism, the use of the vertices can be avoided by modifying the LMIs with the S-Procedure. Several numerical examples found in the literature are used to compare the results and illustrate the advantages of the proposed method. STABILITY CONDITIONS FOR NONLINEAR SYSTEMS 3125 the region of attraction can be found in [6,7]. The stability conditions in both references are based on polynomial Lyapunov functions. In [7, 8], a particular system decomposition is adopted and the LMIs are obtained with the Finsler's Lemma. The stability conditions in [6] are transformed into LMIs by using sum of squares (SOS) relaxations and the S -Procedure. Analysis and design methods based on SOS polynomials has received much attention this last decade [9][10][11][12][13]. The idea is to express the Lyapunov conditions for stability as SOS polynomials. The interest of SOS polynomials is that the problem of testing if a polynomial is SOS can be recast as an LMI problem [10,11,14,15]. It is shown in [16] that any locally exponentially stable system with a thrice differentiable vector field will have a polynomial Lyapunov function, which decreases exponentially on that region. If in one hand, this shows that polynomial Lyapunov functions, not necessarily of SOS type, seems to be rich enough to tackle a wide class of stability problems, o...

Computational Stability Analysis of Lotka-Volterra Systems

“…These transformations also result in the dimension reduction of the optimization problem compared to the results presented in Ref. [4]. The representation in Eq.…”

“…By applying the LFT and the further algebraic model transformation steps, a model is obtained in the representation of Eq. (4), where the entries of π(x) are second order monomials of the state variables.…”

“…As a consequence, the LFT will produce a model Eq. (4), in which π will contain every possible second-order monomial. This means that the number of monomials (including the state variables) is m b = (n 2 + 3n) / 2 in the case of a "full" model matrix.…”

“…Recently, an optimization-based method for DOA estimation was published [4], where the authors use Finsler's lemma and affine parameter-dependent LMIs to compute rational Lyapunov functions for a wide class of locally asymptotically stable nonlinear systems. Based on these results an improved method was published [5,6], where the transformation of the model to the form required for optimization is done automatically using the linear fraction transformation (LFT) and further automatic model simplification steps, which results in the dimension reduction of the optimization task.…”

“…The main motivation of our work was to evaluate the applicability of the approach [4] on a general polynomial system class consisting of low-degree monomials, and to study the limits of the method as the number of dimensions of the state space increase.…”