Metzler Matrix Transform Determination using a Nonsmooth Optimization Technique with an Application to Interval Observers

Chambon, Emmanuel; Apkarian, Pierre; Burlion, Laurent

doi:10.1137/1.9781611974072.29

Cited by 11 publications

(9 citation statements)

References 12 publications

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“…We also give a necessary and sufficient condition guaranteeing that A is similar to a matrix whose off diagonal entries are positive, which can also be easily checked. Our result complement [6] which proposed a technique making possible for controllable pairs (A, C) to find matrices P , L or appropriate dimension such that P (A + LC)P −1 is Metzler.…”

Section: Introductionsupporting

confidence: 55%

When is a Matrix of Dimension Three Similar to a Metzler Matrix? Application to Interval Observer Design

Mazenc¹,

Bernard

2022

IEEE Trans. Automat. Contr.

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A simple necessary and sufficient condition ensuring that a real matrix of dimension 3 is similar to a Metzler matrix is exhibited. When this condition is satisfied, a construction of the transfer matrix is given. This construction is used to design an interval observer for a family of continuous-time systems. An example is provided with interval observer design for the love dynamics in the case of limit cycles.

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Section: Introductionsupporting

confidence: 55%

When is a Matrix of Dimension Three Similar to a Metzler Matrix? Application to Interval Observer Design

Mazenc¹,

Bernard

2022

IEEE Trans. Automat. Contr.

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“…Using such techniques, additional control constraints can be taken into account in the determination of the appropriate state-coordinate transformation. This article extends the preliminary results presented in [5]. This approach was motivated by the interval observers cooperativity requirement but it is expected to be generalizable to any problem requiring specific structures of the matrices of the studied system.…”

Section: Introductionmentioning

confidence: 64%

“…In this article, a new approach to compute an appropriate change of coordinates for a system to be cooperative in these new coordinates has been introduced following preliminary results in [5]. Both the continuous-and discrete-time cases have been considered.…”

Section: Discussionmentioning

confidence: 99%

“…In this section, the different problems which arise in the proposed approach are exposed. As far as the state coordinate transformation determination is concerned, a reformulation into a control law synthesis problem is proposed, as exposed in [5]. A numerical approach to find a solution simultaneously satisfying to these problems is then proposed in Section 5.…”

Section: Problems Statementmentioning

confidence: 99%

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Overview of linear time‐invariant interval observer design: towards a non‐smooth optimisation‐based approach

Chambon¹,

Burlion²,

Apkarian³

2016

IET Control Theory & Applications

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Some applications in control require the state vector of a system to be expressed in the appropriate coordinates so as to satisfy to some mathematical properties which constrain the studied system dynamics. This is the case with the theory of linear interval observers which are trivial to implement on cooperative systems, a rather limited class of control systems. The available literature shows how to enforce this limiting cooperativity condition for any considered system through a statecoordinate transformation. This article proposes an overview of the existing numerical techniques to determine such a transformation. It is shown that in spite of being practical, these techniques have some limitations. Consequently, a reformulation of the problem is proposed so as to apply non-smooth control design techniques. A solution is obtained in both the continuous-and discretetime frameworks. Interestingly, the new method allows to formulate additional control constraints. Simulations are performed on three examples.

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“…SIVIA). For that reason, these prediction-correction schemes are better suited for pure offline applications and for systems in a non-cooperative state-space representation, while real-time control scenarios as investigated in this paper usually make use of the observer-based framework, possibly after transforming the sets of state equations into a cooperative representation [3,17].…”

Section: Introductionmentioning

confidence: 99%

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

Rauh

Kersten

2020

Acta Cybern

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One of the most important advantages of interval observers and the associated trajectory computation is their capability to provide estimates for a given dynamic system model in terms of guaranteed state bounds which are compatible with measured data subject to bounded uncertainty. However, the inevitable requirement for being able to produce such verified bounds is the knowledge about a dynamic system model in which possible uncertainties and inaccuracies are themselves represented by guaranteed bounds. For that reason, classical point-valued parameter identification schemes are often not sufficient or should, at least, be handled with sufficient care if safety critical applications are of interest. This paper provides an application-oriented description of the major steps leading from a control-oriented system model with an associated interval-valued parameter and disturbance identification to a verified design of interval observers which provide the basis for the development and implementation of cooperativity-preserving feedback controllers. Such combined control and observer structures allow for forecasting guaranteed lower and upper state bounds that can be determined by solving initial value problems for crisp-parameter models. As such, they replace the significantly more demanding task of computing tubes of reachable states by means of general-purpose interval methods. The corresponding computational steps for the cooperativity-preserving control and observer synthesis are described and visualized for the temperature control of a laboratory-scale test rig available at the Chair of Mechatronics at the University of Rostock.

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Metzler Matrix Transform Determination using a Nonsmooth Optimization Technique with an Application to Interval Observers

Cited by 11 publications

References 12 publications

When is a Matrix of Dimension Three Similar to a Metzler Matrix? Application to Interval Observer Design

When is a Matrix of Dimension Three Similar to a Metzler Matrix? Application to Interval Observer Design

Overview of linear time‐invariant interval observer design: towards a non‐smooth optimisation‐based approach

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

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