LMI Techniques for Optimization Over Polynomials in Control: A Survey

Chesi, Graziano

doi:10.1109/tac.2010.2046926

Cited by 360 publications

(314 citation statements)

References 56 publications

(79 reference statements)

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“…Here, we provide a computational approach to this problem based on polynomial Lyapunov functions and sum of squares techniques (SOS) [4], [15], [22]- [24]. The main idea is that a polynomial p(x) that can be written as a sum of squares, i.e.…”

Section: B Stability Using Sos Techniquesmentioning

confidence: 99%

Stability analysis of networked control systems: A sum of squares approach

2012

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Abstract-This paper presents a sum of squares (SOS) approach to the stability analysis of networked control systems (NCSs) incorporating time-varying delays and time-varying transmission intervals. We will provide mathematical models that describe these NCSs and transform them into suitable hybrid systems formulations. Based on these hybrid systems formulations we construct Lyapunov functions using SOS techniques that can be solved using LMI-based computations. This leads to several advantages: (i) we can deal with nonlinear polynomial controllers and systems, (ii) we can allow for nonzero lower bounds on the delays and transmission intervals in contrast with various existing approaches, (iii) we allow more flexibility in the Lyapunov functions thereby possibly obtaining improved bounds for the delays and transmission intervals than existing results, and finally (iv) it provides an automated method to address stability analysis problems in NCS.

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Section: B Stability Using Sos Techniquesmentioning

confidence: 99%

Stability analysis of networked control systems: A sum of squares approach

2012

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“…Theorem 1: Define k as the largest order of the Maclaurin polynomial in (4). Provided that there exists a polynomial s(x) ∈ P SOS 0 and c k is the optimum of the following polynomial optimization…”

Section: A Fixed Lyapunov Functionmentioning

confidence: 99%

“…The dimensions of φ ∈ R l(n,dx) and δ ∈ R ϑ(n,dx) can be calculated analogously to [4]. An exemplary illustration of SMR is provided as follows.…”

Section: B Bound Computation By Square Matrix Representation (Smr)mentioning

confidence: 99%

“…We exploit a fixed polynomial Lyapunov function v 0 (x) = 2x 4 1 + 2x 2 1 x 2 2 +5x 2 1 +10x 1 x 2 +2x 1 x 3 +10x 2 2 +8x 4 2 +6x 2 x 3 +4x 2 3 + 3x 4 3 and set the truncation degree as k = 3. By solving (36), we search a polynomial controller with d u = 2 and establish the tightness of the lower boundsγ corresponding to this controllers as shown in Fig.…”

Section: B Examplementioning

confidence: 99%

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Control synthesis for non-polynomial systems: A domain of attraction perspective

Han

Althoff

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-This paper studies a control synthesis problem to enlarge the domain of attraction (DA) for non-polynomial systems by using polynomial Lyapunov functions. The basic idea is to formulate an uncertain polynomial system with parameter ranges obtained from the truncated Taylor expansion and the parameterizable remainder of the non-polynomial system. A strategy for searching a polynomial output feedback controller and estimating the lower bound of the largest DA is proposed via an optimization of linear matrix inequalities (LMIs). Furthermore, in order to check the tightness of the lower bound of the largest estimated DA, a necessary and sufficient condition is given for the proposed controller. Lastly, several methods are provided to show how the proposed strategy can be extended to the case of variable Lyapunov functions. The effectiveness of this approach is demonstrated by numerical examples.

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“…However, it is admitted that the conservatism arises between Theorem 1 and Theorem 3 because of the gap between positive polynomials and Sum-of-Square polynomials which relates to the Hilbert's 17th problem [40].…”

Section: Remark 5: It Is Useful To Note Thatmentioning

confidence: 99%

Robust Synchronization via Homogeneous Parameter-Dependent Polynomial Contraction Matrix

Han

Chesi

2014

IEEE Trans. Circuits Syst. I

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Abstract-Robust synchronization problem is a key issue in chaotic circuits and nonlinear systems. This paper is concerned with robust synchronization problem of polynomial nonlinear system affected by time-varying uncertainties on topology, i.e., structured uncertain parameters constrained in a boundedrate polytope. Via partial contraction analysis, novel conditions, both for robust exponential synchronization and for robust asymptotical synchronization, are proposed by using parameterdependent contraction matrices. In addition, for polynomial nonlinear system, this paper introduces a new class of contraction matrix, i.e., homogeneous parameter-dependent polynomial contraction matrix (HPD-PCM), by which tractable conditions of linear matrix inequalities (LMIs) are provided via affine space parametrizations. Furthermore, the variant rate margin for robust asymptotical synchronization is, for the first time, proposed and investigated via handling generalized eigenvalue problems (GEVPs). A set of representative examples demonstrate the effectiveness of proposed method.

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LMI Techniques for Optimization Over Polynomials in Control: A Survey

Cited by 360 publications

References 56 publications

Stability analysis of networked control systems: A sum of squares approach

Stability analysis of networked control systems: A sum of squares approach

Control synthesis for non-polynomial systems: A domain of attraction perspective

Robust Synchronization via Homogeneous Parameter-Dependent Polynomial Contraction Matrix

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