Stability Analysis With Dissipation Inequalities and Integral Quadratic Constraints

Seiler, Peter

doi:10.1109/tac.2014.2361004

Cited by 142 publications

(188 citation statements)

References 18 publications

Supporting

Mentioning

182

Contrasting

Order By: Relevance

“…It is possible to show that LMI and the dissipativity framework can be used to guarantee input‐output stability as in this alternative stability definition. The required steps parallel the continuous‐time result in [, Lemma 1]. It is also noted that related work in incorporates the effect of initial conditions.…”

Section: Preliminariesmentioning

confidence: 71%

“…The upper value of the game is defined as

{\overset{J}{true}}_{normalΨ, M} (ψ_{0}) : = \inf_{v \in ℓ_{2}^{n_{v}}} \sup_{w \in ℓ_{2}^{n_{w}}} J_{normalΨ, M} (v, w, ψ_{0})

The lower value of the game is defined as

{\underset{J}{false}}_{normalΨ, M} (ψ_{0}) : = \sup_{w \in ℓ_{2}^{n_{w}}} \inf_{v \in ℓ_{2}^{n_{v}}} J_{normalΨ, M} (v, w, ψ_{0})

The next two lemmas relate the upper and lower values of this open‐loop game to the properties of the IQC factorization (Ψ, M ). The proofs are omitted as they are similar to those used in the continuous‐time counterparts [, Lemma 2, Lemma 3].…”

Section: J‐spectral Factorizations and Related Gamesmentioning

confidence: 99%

See 1 more Smart Citation

Robustness analysis of uncertain discrete‐time systems with dissipation inequalities and integral quadratic constraints

Lacerda

Seiler

2016

Intl J Robust & Nonlinear

Self Cite

View full text Add to dashboard Cite

Summary This paper presents a connection between dissipation inequalities and integral quadratic constraints (IQCs) for robustness analysis of uncertain discrete‐time systems. Traditional IQC results derived from homotopy methods emphasize an operator‐theoretic input–output viewpoint. In contrast, the dissipativity‐based IQC approach explicitly incorporates the internal states of the uncertain system, thus providing a more direct procedure to analyze uniform stability with non‐zero initial states. The standard dissipation inequality requires a non‐negative definite storage function and ‘hard’ IQCs. The term ‘hard’ means that the IQCs must hold over all finite time horizons. This paper presents a modified dissipation inequality that requires neither non‐negative definite storage functions nor hard IQCs. This approach leads to linear matrix inequality conditions that can provide less conservative results in terms of robustness analysis. The proof relies on a key J‐spectral factorization lemma for IQC multipliers. A simple numerical example is provided to demonstrate the utility of the modified dissipation inequality. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

Section: Preliminariesmentioning

confidence: 71%

“…The upper value of the game is defined as

{\overset{J}{true}}_{normalΨ, M} (ψ_{0}) : = \inf_{v \in ℓ_{2}^{n_{v}}} \sup_{w \in ℓ_{2}^{n_{w}}} J_{normalΨ, M} (v, w, ψ_{0})

The lower value of the game is defined as

{\underset{J}{false}}_{normalΨ, M} (ψ_{0}) : = \sup_{w \in ℓ_{2}^{n_{w}}} \inf_{v \in ℓ_{2}^{n_{v}}} J_{normalΨ, M} (v, w, ψ_{0})

Section: J‐spectral Factorizations and Related Gamesmentioning

confidence: 99%

Robustness analysis of uncertain discrete‐time systems with dissipation inequalities and integral quadratic constraints

Lacerda

Seiler

2016

Intl J Robust & Nonlinear

Self Cite

View full text Add to dashboard Cite

show abstract

“…The relation between dissipativity (and hence Lyapunov methods) and the IQC theorem is beginning to be understood (Willems and Takaba, 2007;Veenman and Scherer, 2013;Seiler, 2015).…”

Section: The Iqc Frameworkmentioning

confidence: 99%

Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches

Carrasco

Turner

Heath

2015

2015 European Control Conference (ECC)

View full text Add to dashboard Cite

Absolute stability attracted much attention in the sixties. Several stability conditions for loops with slope-restricted nonlinearities were developed. Results such as the Circle Criterion and the Popov Criterion form part of the core curriculum for students of control. Moreover, the equivalence of results obtained by different techniques, specifically Lyapunov and Popov's stability theories, led to one of the most important results in control engineering: the KYP Lemma.For Lurye 1 systems this work culminated in the class of multipliers proposed by O'Shea in 1966 and formalised by Zames and Falb in 1968. The superiority of this class was quickly and widely accepted. Nevertheless the result was ahead of its time as graphical techniques were preferred in the absence of readily available computer optimization. Its first systematic use as a stability criterion came twenty years after the initial proposal of the class. A further twenty years have been required to develop a proper understanding of the different techniques that can be used. In this long gestation some significant knowledge has been overlooked or forgotten. Most significantly, O'Shea's contribution and insight is no longer acknowledged; his papers are barely cited despite his original parameterization of the class.This tutorial paper aims to provide a clear and comprehensive introduction to the topic from a user's viewpoint. We review the main results: the stability theory, the properties of the multipliers (including their phase properties, phase-equivalence results and the issues associated with causality), and convex searches. For clarity of exposition we restrict our attention to continuous time multipliers for single-input single-output results. Nevertheless we include several recent significant developments by the authors and others. We illustrate all these topics using an example proposed by O'Shea himself.

show abstract

“…Since this paper uses IQCs for the robustness analysis of uncertain grid‐based LPV systems, IQCs also need to be expressed in the time domain. A multiplier

normalΠ \in double-struckR {double-struckL}_{\infty}^{false(n_{v} + n_{w} false) \times false(n_{v} + n_{w} false)}

can be factorized as Π=Ψ ∼ M Ψ, where

M = M^{T} \in R^{n_{z} \times n_{z}}

and

normalΨ \in double-struckR {double-struckH}_{\infty}^{n_{z} \times false(n_{v} + n_{w} false)}

. The IQC given in Equation can be rewritten as

\int_{0}^{\infty} z {false(t false)}^{T} M z false(t false) d t \geq 0,

where

z : = normalΨ ([], \begin{matrix} array v \\ array w \end{matrix}) \in R^{n_{z}}

is the output of the linear system Ψ driven by the input signals v and w starting from zero initial conditions .…”

Section: Introductionmentioning

confidence: 99%

“…A multiplier

normalΠ \in double-struckR {double-struckL}_{\infty}^{false(n_{v} + n_{w} false) \times false(n_{v} + n_{w} false)}

can be factorized as Π=Ψ ∼ M Ψ, where

M = M^{T} \in R^{n_{z} \times n_{z}}

and

normalΨ \in double-struckR {double-struckH}_{\infty}^{n_{z} \times false(n_{v} + n_{w} false)}

. The IQC given in Equation can be rewritten as

\int_{0}^{\infty} z {false(t false)}^{T} M z false(t false) d t \geq 0,

where

z : = normalΨ ([], \begin{matrix} array v \\ array w \end{matrix}) \in R^{n_{z}}

is the output of the linear system Ψ driven by the input signals v and w starting from zero initial conditions . Let Ψ have a state‐space realization given by

([], \begin{matrix} array {\dot{x}}_{normalΨ} \\ array z \end{matrix}) = ([], \begin{matrix} array A_{normalΨ} & array B_{Ψ v} & array B_{Ψ w} \\ array C_{normalΨ} & array D_{Ψ v} & array D_{Ψ w} \end{matrix}) ([], \begin{matrix} array x_{normalΨ} \\ array v \\ array w \end{matrix}),

where

x_{Ψ} \in R^{n_{Ψ}}

and x Ψ (0)=0.…”

Section: Introductionmentioning

confidence: 99%

Convex LPV synthesis of estimators and feedforwards using duality and integral quadratic constraints

Venkataraman

Seiler

2017

Intl J Robust & Nonlinear

Self Cite

View full text Add to dashboard Cite

Summary A method is presented for synthesizing output estimators and disturbance feedforward controllers for continuous‐time, uncertain, gridded, linear parameter‐varying (LPV) systems. Integral quadratic constraints are used to describe the uncertainty. Since the gridded LPV systems do not have a valid frequency‐domain interpretation, the time domain, dissipation inequality approach is followed. There are 2 main contributions. The first contribution is that a notion of duality is developed for the worst‐case gain analysis of uncertain, gridded LPV systems. This includes notions of dual LPV systems and dual integral quadratic constraints. Furthermore, several technical results are developed to demonstrate that the sufficient conditions for bounding the worst‐case gain of the primal and dual uncertain LPV systems are equivalent. The second contribution is that the convex conditions are derived for the synthesis of robust output estimators for uncertain LPV systems. The estimator synthesis conditions, together with the duality results, enable the convex synthesis of robust disturbance feedforward controllers. The effectiveness of the proposed method is demonstrated using a numerical example.

show abstract

Stability Analysis With Dissipation Inequalities and Integral Quadratic Constraints

Cited by 142 publications

References 18 publications

Robustness analysis of uncertain discrete‐time systems with dissipation inequalities and integral quadratic constraints

Robustness analysis of uncertain discrete‐time systems with dissipation inequalities and integral quadratic constraints

Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches

Convex LPV synthesis of estimators and feedforwards using duality and integral quadratic constraints

Contact Info

Product

Resources

About