30 research outputs found
Robust Stability Analysis of Nonlinear Hybrid Systems
We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems
LMI techniques for optimization over polynomials in control: A survey
Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Plya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers. © 2006 IEEE.published_or_final_versio
Exactness Verification of Sum-of-Squares Approximations to Robust Semidefinite Programs with Functional Variables
Abstract-Robust semidefinite programs (robust SDPs in short) with functional variables are revisited in this paper. We firstly consider the approximate approach suggested by Jennawasin and Oishi (in Proceedings of the 17th IFAC World Congress, Seoul, Korea, July 2008), and then provide a numerically computable condition to verify when the optimal value of an approximate problem is actually equal to that of the original robust SDP. The idea is based on capturing some special structure of a dual feasible solution of the approximate problem
Robust stability of time-varying uncertain systems with rational dependence on the uncertainty
Robust stability of time-varying uncertain systems is a key problem in automatic control. This note considers the case of linear systems with rational dependence on an uncertain time-varying vector constrained in a polytope, which is typically addressed in the literature by using the linear fractional representation (LFR). A novel sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI) feasibility test by exploiting homogeneous polynomial Lyapunov functions, the square matrix representation and an extended version of Polya's theorem which considers structured matrix polynomials. It is shown that this condition is also necessary for second-order systems, and that this condition is less conservative than existing LMI conditions based on the LFR for any order. © 2010 IEEE.published_or_final_versio
Primal robustness and semidefinite cones
This paper reformulates and streamlines the core tools of robust stability
and performance for LTI systems using now-standard methods in convex
optimization. In particular, robustness analysis can be formulated directly as
a primal convex (semidefinite program or SDP) optimization problem using sets
of gramians whose closure is a semidefinite cone. This allows various
constraints such as structured uncertainty to be included directly, and
worst-case disturbances and perturbations constructed directly from the primal
variables. Well known results such as the KYP lemma and various scaled small
gain tests can also be obtained directly through standard SDP duality. To
readers familiar with robustness and SDPs, the framework should appear obvious,
if only in retrospect. But this is also part of its appeal and should enhance
pedagogy, and we hope suggest new research. There is a key lemma proving
closure of a grammian that is also obvious but our current proof appears
unnecessarily cumbersome, and a final aim of this paper is to enlist the help
of experts in robust control and convex optimization in finding simpler
alternatives.Comment: A shorter version submitted to CDC 1
The Non-Strict Projection Lemma
The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation
Controller Design via Experimental Exploration with Robustness Guarantees
For a partially unknown linear systems, we present a systematic control
design approach based on generated data from measurements of closed-loop
experiments with suitable test controllers. These experiments are used to
improve the achieved performance and to reduce the uncertainty about the
unknown parts of the system. This is achieved through a parametrization of
auspicious controllers with convex relaxation techniques from robust control,
which guarantees that their implementation on the unknown plant is safe. This
approach permits to systematically incorporate available prior knowledge about
the system by employing the framework of linear fractional representations
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
Combining Prior Knowledge and Data for Robust Controller Design
We present a framework for systematically combining data of an unknown linear
time-invariant system with prior knowledge on the system matrices or on the
uncertainty for robust controller design. Our approach leads to linear matrix
inequality (LMI) based feasibility criteria which guarantee stability and
performance robustly for all closed-loop systems consistent with the prior
knowledge and the available data. The design procedures rely on a combination
of multipliers inferred via prior knowledge and learnt from measured data,
where for the latter a novel and unifying disturbance description is employed.
While large parts of the paper focus on linear systems and input-state
measurements, we also provide extensions to robust output-feedback design based
on noisy input-output data and against nonlinear uncertainties. We illustrate
through numerical examples that our approach provides a flexible framework for
simultaneously leveraging prior knowledge and data, thereby reducing
conservatism and improving performance significantly if compared to black-box
approaches to data-driven control