A gram-SOS approach for robust stability analysis of discrete-time systems with time-varying uncertainty

This paper addresses the problem of establishing robust asymptotical stability of discrete-time systems affected by time-varying parametric uncertainty. Specifically, it is supposed that the coefficients of the system depend linearly on the uncertainty, and that the uncertainty is confined into a polytope. In the continuous-time case, the problem can be addressed by imposing that the system admits a common homogeneous polynomial Lyapunov function (HPLF) at the vertices of the polytope. Unfortunately, such a strategy cannot be used in the discrete-time case since the derivative of the HPLF is nonlinear in the uncertainty. The problem is addressed in this paper through linear matrix inequalities (LMIs) by proposing a novel method for establishing decrease of the HPLF. This method consists, firstly, of introducing a Gram matrix built with respect to the state and parametrized by an arbitrary vector function of the uncertainty, and secondly, of requiring that a transformation of the introduced Gram matrix is a sum of squares (SOS) of matrix polynomials. The proposed method provides a condition for robust asymptotical stability that is sufficient for any degree of the HPLF candidate and that includes quadratic robust stability as special case. © 2013 AACC American Automatic Control Council.published_or_final_versio

Rational Lyapunov Functions for Estimating and Controlling the Robust Domain of Attraction

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On the robust stability of continuous-time and discrete-time time-invariant uncertain systems with rational dependence on the uncertainty: A non-conservative condition

A key problem in automatic control consists of investigating robust stability of systems with uncertainty. This paper considers linear systems with rational dependence on time-invariant uncertainties constrained in the simplex. It is shown that a sufficient condition for establishing whether the system is either stable or unstable can be obtained by solving a generalized eigenvalue problem constructed through homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs). Moreover, it is shown that this condition is also necessary for establishing either stability or instability by using a sufficiently large degree of the HPD-QLF. Some numerical examples illustrate the use of the proposed approach in both cases of continuous-time and discrete-time uncertain systems. ©2010 IEEE.published_or_final_versio

Stabilization and entropy reduction via SDP-based design of fixed-order output feedback controllers and tuning parameters

This paper addresses the problem of designing fixed-order output feedback controllers and tuning parameters for reducing the instability of linear time-invariant (LTI) systems. Specifically, continuous-time (CT) and discrete-time (DT) LTI systems are considered, whose coefficients are rational functions of design parameters that are searched for in a given semi-algebraic set. Two instability measures are considered, the first defined as the spectral abscissa (CT case) or the spectral radius (DT case), and the second defined as the sum of the real parts of the unstable eigenvalues (CT case) or the product of the magnitudes of the unstable eigenvalues (DT case). Two sufficient conditions are given for establishing either the non-existence or the existence of design parameters that reduce the considered instability measure under a desired value. These conditions require to solve a semidefinite program (SDP), which is a convex optimization problem, and to find the roots of a multivariate polynomial, which is a difficult problem in general. To overcome this difficulty, a technique based on linear algebra operations is exploited, which easily provides the sought roots in common cases by taking into account the structure of the polynomial under consideration. Also, it is shown that these conditions are also necessary by increasing enough the size of the SDP under some mild assumptions. Lastly, it is explained how the proposed methodology can be used to search for design parameters that minimize a given cost function while reducing the instability.postprin

A new condition and equivalence results for robust stability analysis of rationally time-varying uncertain linear systems

Uncertain systems is a fundamental area of automatic control. This paper addresses robust stability of uncertain linear systems with rational dependence on unknown time-varying parameters constrained in a polytope. For this problem, a new sufficient condition based on the search for a common homogeneous polynomial Lyapunov function is proposed through a particular representation of parameter-dependent polynomials and LMIs. Relationships with existing conditions based on the same class of Lyapunov functions are hence investigated, showing that the proposed condition is either equivalent to or less conservative than existing ones. As a matter of fact, the proposed condition turns out to be also necessary for a class of systems. Some numerical examples illustrate the use of the proposed condition and its benefits. © 2011 IEEE.published_or_final_versionThe 2011 IEEE International Symposium on Computer-Aided Control System Design (CACSD), Denver, CO., 28-30 September 2011. In IEEE CACSD International Symposium Proceedings, 2011, p. 234-23

On the admissible equilibrium points of nonlinear dynamical systems affected by parametric uncertainty: characterization via LMIs

Proceedings of the IEEE International Symposium on Computer-Aided Control System Design, 2010, p. 351-356Part of 2010 IEEE Multi-Conference on Systems and ControlThis paper investigates the set of admissible equilibrium points of nonlinear dynamical systems affected by parametric uncertainty. As it is well-known, determining this set is a difficult problem since one should compute the solutions of a system of nonlinear equations for all the admissible values of the uncertainty, which typically amounts to an infinite number of times. In order to address this problem, this paper proposes a characterization of this set via convex optimization for the case of polynomial nonlinearities and uncertainty constrained in a polytope. Specifically, it is shown that an upper bound of the smallest outer estimate with a freely selectable fixed shape can be obtained by solving a linear matrix inequality (LMI) problem built through the square matrix representation (SMR). Then, a necessary and sufficient condition is provided for establishing the tightness of the found upper bound. The proposed methodology and its benefits are illustrated through several numerical examples. © 2010 IEEE.published_or_final_versionThe 2010 IEEE International Symposium on Computer-Aided Control System Design (CACSD), Yokohama, Japan, 8-10 September 2010. in Proceedings of CACSD, 2010, p. 351-35

Characterizing the positive polynomials which are not SOS

Several analysis and synthesis tools in control systems are based on polynomial sum of squares (SOS) relaxations. However, almost nothing is known about the gap existing between positive polynomials and SOS of polynomials. This paper investigates such a gap proposing a matrix characterization of PNS, that is homogeneous forms that are not SOS. In particular, it is shown that any PNS is the vertex of an unbounded cone of PNS. Moreover, a complete parameterization of the set of PNS is introduced. © 2005 IEEE.published_or_final_versio

On the robust H∞ norm of 2D mixed continuous-discrete-time systems with uncertainty

This paper addresses the problem of determining the robust H∞ norm of 2D mixed continuous-discrete-time systems affected by uncertainty. Specifically, it is supposed that the matrices of the model are polynomial functions of an unknown vector constrained into a semialgebraic set. It is shown that an upper bound of the robust H∞ norm can be obtained via a semidefinite program (SDP) by introducing complex Lyapunov functions candidates with rational dependence on a frequency and polynomial dependence on the uncertainty. A necessary and sufficient condition is also provided to establish whether the found upper bound is tight. Some numerical examples illustrate the proposed approach.published_or_final_versio