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

Chesi, Graziano; Garulli, Andrea; Tesi, A.; Vicino, Antonio

doi:10.1109/tac.2005.843848

Cited by 233 publications

(160 citation statements)

References 14 publications

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“…Let us consider firstly the CT case. This supposition implies that Re(λ) < −0.5η * for all λ ∈ spc(A(p)) for all p ∈ P. In fact, from (6)- (8), (18) and Lemma 3 in [3], the first two constraints in (12) imply…”

Section: Theorem 1 [18] the Set A Is Stable If And Only If There Exismentioning

confidence: 95%

“…The following theorem is given in [3,18,12] and investigates robust stability of (1) via a HPD-QLF of degree m, i.e. via a Lyapunov function of the form x ′ P (p, β)x.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let us consider the stability statement, and let us observe that K in (11) is a full column rank (see e.g. [3]), which directly implies from (10) that…”

Section: Theorem 1 [18] the Set A Is Stable If And Only If There Exismentioning

confidence: 99%

“…Generally, these methods exploit parameter-dependent Lyapunov functions and LMIs, see e.g. [1] which considers Lyapunov functions with linear dependence, [2] which proposes Lyapunov functions with polynomial dependence, [3] which introduces the class of HPD-QLFs, [4] which proposes a general framework for LMI relaxations, [5] where homogeneous solutions are characterized, [6] which addresses the case of semi-algebraic sets, and [7,8] where matrix-dilation approaches are considered.…”

Section: Introductionmentioning

confidence: 99%

“…It is shown that a sufficient condition for establishing either stability or instability can be obtained by solving a generalized eigenvalue problem, and that this condition is also necessary by using a sufficiently large degree of the HPD-QLF. The idea behind this condition is to exploit the LMI relaxation introduced in [3] via the square matricial representation (SMR) 1 in order to characterize the instability via the presence of suitable vectors in certain eigenspaces.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

Chesi

2010

Automatica

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This paper investigates linear systems with polynomial dependence on time-invariant uncertainties constrained in the simplex via homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs). 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. Moreover, this condition is also necessary by using a sufficiently large degree of the HPD-QLF.

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Section: Theorem 1 [18] the Set A Is Stable If And Only If There Exismentioning

confidence: 95%

“…The following theorem is given in [3,18,12] and investigates robust stability of (1) via a HPD-QLF of degree m, i.e. via a Lyapunov function of the form x ′ P (p, β)x.…”

Section: Preliminariesmentioning

confidence: 99%

“…Let us consider the stability statement, and let us observe that K in (11) is a full column rank (see e.g. [3]), which directly implies from (10) that…”

Section: Theorem 1 [18] the Set A Is Stable If And Only If There Exismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

Chesi

2010

Automatica

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Consensus in Multi‐agent Systems

Wen

Chen

et al. 2016

Distributed Cooperative Control of Multi‐agent Systems

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A Carlos Elías y a Marina. AbstractDuring the last decade, the problem of consensus in multiagents systems has been studied with special emphasis on graph theoretical methods. Consensus can be regarded as a control objective in which is sought that all systems, or agents, in a network have an equivalent output value. This is achieved through a given control strategy usually referred to as consensus algorithm. The motivation to study such an objective comes from different areas, such as engineering, and social and natural sciences. In the control engineering field, important application examples are formation control of swarms of mobile robots and distributed electric generation. Most of the work in this area is done for agents with single or double integrator dynamics and algorithms derived as the Laplacian matrix of undirected graphs. That is, consensus is often studied as a property of particular networks and particular algorithms.In this thesis, we tackle the problem from a Control Theory perspective in an attempt to augment the class of systems that can be studied. For that we translate the consensus problem from its classical formulation for integrator systems into a general continuous time stability problem. From here, different algorithmic strategies under several dynamical assumptions of the agents can be studied through well known control theoretical tools -as Lyapunov's theory, linear matrix inequalities (LMI), or robust control -along with graph theoretical concepts. In particular, in this work we study consensus of agents with arbitrary linear dynamics under the influence of linear algorithms not necessarily derived from graphs. Furthermore, we include the possibility that the agents are disturbed by several factors as external signals, parameter uncertainties, switching dynamics, or communication failure. The theoretical analysis is also applied to the problem of power sharing in electric grids and to the analysis of distributed formation control.

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A convex optimization approach to adaptive stabilization of discrete‐time LTI systems with polytopic uncertainties

Lee

Joo

Tak

2014

Adaptive Control & Signal

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This paper suggests a simple convex optimization approach to state-feedback adaptive stabilization problem for a class of discrete-time LTI systems subject to polytopic uncertainties. The proposed method relies on estimating the uncertain parameters by solving an online optimization at each time step, such as a linear or quadratic programming, and then, on tuning the control law with that information, which can be conceptually viewed as a kind of gain-scheduling or indirect adaptive control. Specifically, an admissible domain of stabilizing state-feedback gain matrices is designed offline by means of linear matrix inequality problems, and based on the online estimation of the uncertain parameters, the state-feedback gain matrix is calculated over the set of stabilizing feedback gains. The proposed stabilization algorithm guarantees the asymptotic stability of the overall closed-loop control system. An example is given to show the effectiveness of the proposed approach.A CONVEX OPTIMIZATION APPROACH TO ADAPTIVE STABILIZATION 1117 deal of research has been dedicated to developing more general types of gain-scheduling controllers, for instance, path-dependent control laws [20] and polynomially parameter-dependent control laws [21,22].However, this is no longer the case for uncertain LTI systems in the sense that the uncertain parameters are literally assumed not to be measurable. For this reason, the development of less conservative design methods for the static state-feedback has become a fundamental and challenging problem. In the late 1990s, the so-called extended Schur complement and slack variable approaches were developed by the pioneering work [8,9], which played a significant role in the subsequent development of the LMI-based robust analysis and control approaches. More recently, an important advance on robust stabilization of discrete-time LTI systems was accomplished in [23,24], where the so-called periodically time-varying state-feedback controller was proposed and turned out to be effective in reducing the conservatism in the static state-feedback approaches.On the other hand, adaptive control technology [25-29] has a long history of interest in the control community [30][31][32][33] and has attracted considerable attention to date. One of the advantages of the adaptive control approach is the ability to adjust the controller so as to cope with complex, uncertain, and highly nonlinear nature of the plant especially in the absence of complete information on the system. The adaptive control strategies may be broadly classified into the following two categories: direct [29] and indirect [27,28] adaptive controls. The indirect adaptive control is a feedback control scheme in which the plant dynamics is estimated first, and the feedback controller is calculated by exploiting the estimation [34,35], while in the direct case, the ideal control law is approximated without the estimation of the plant.The philosophy of adaptive control and the gain-scheduling control leads us to the key motivation for the main re...

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

Cited by 233 publications

References 14 publications

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

Consensus in Multi‐agent Systems

A convex optimization approach to adaptive stabilization of discrete‐time LTI systems with polytopic uncertainties

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