A novel unified approach to invariance conditions for a linear dynamical system

Horváth, Zoltán; Song, Yunfei; Terlaky, Tamás

doi:10.1016/j.amc.2016.10.007

Cited by 12 publications

(32 citation statements)

References 52 publications

(141 reference statements)

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“…Since int(P) is an open set, we have that the statement is true for x k ∈ int(P). For x k ∈ ∂ P, by Nagumo's Theorem, see e.g., Horváth et al (2013); Nagumo (1942), the statement is also true.…”

Section: Of 19mentioning

confidence: 92%

“…Proof. According to Horváth et al (2013), we have that E is an invariant set for the discrete and continuous systems if and only if A T QA − Q 0 and A T Q + QA 0, respectively. Then by Lemma 4.2, we have thatτ = ∞.…”

Section: Uniform Steplength Threshold For Linear Systemsmentioning

confidence: 99%

“…The popularity of these special sets is due to their nice properties and the fact that they are widely used in modeling important applications. A unified approach to derive sufficient and necessary conditions under which a set is a positively invariant set is presented in Horváth et al (2013).…”

Section: Introductionmentioning

confidence: 99%

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Invariance Preserving Discretization Methods of Dynamical Systems

2018

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In this paper, we consider local and uniform invariance preserving steplength thresholds on a set when a discretization method is applied to a linear or nonlinear dynamical system. For the forward or backward Euler method, the existence of local and uniform invariance preserving steplength thresholds is proved when the invariant sets are polyhedra, ellipsoids, or Lorenz cones. Further, we also quantify the steplength thresholds of the backward Euler methods on these sets for linear dynamical systems. Finally, we present our main results on the existence of uniform invariance preserving steplength threshold of general discretization methods on general convex sets, compact sets, and proper cones both for linear and nonlinear dynamical systems. ZOLTÁN HORVÁTH, YUNFEI SONG AND TAMÁS TERLAKY sponding discrete system which is obtained by using the discretization method. Such a discretization method is called invariance preserving for the dynamical system on the positively invariant set.In this paper, our focus is to find conditions, in particular steplength thresholds for the discretization methods, such that the considered discretization method is invariance preserving for the given linear or nonlinear dynamical system. This topic is of great interest in the fields of dynamical systems, partial differential equations, and control theory. A basic result is presented in Bolley & Crouzeix (1978), which considers linear problems and invariance preserving on the positive orthant from a perspective of numerical methods. For invariance preserving on the positive orthant or polyhedron for Runge-Kutta methods, the reader is refereed to Horváth (2004); Horváth (2005, 2006). A similar concept named strong stability preserving (SSP) used in numerical methods is studied in Gottlieb et al. (2011Gottlieb et al. ( , 2001; Shu & Osher (1988). These papers deal with invariance preserving of general sets and they usually use the assumption that the Euler methods are invariance preserving with a steplength threshold τ 0 . Then the uniform invariance preserving steplength threshold for other advanced numerical methods, e.g., Runge-Kutta methods, is derived in terms of τ 0 . Therefore, to make the results applicable to solve real world problems, this approach requires to check whether such a positive τ 0 exists for Euler methods. We note that quantifications of steplength thresholds for some classes discretization methods on a polyhedron are studied in Horváth et al. (2015).In this paper, basic concepts and theorems are introduced in Section 2. In Section 3 first we prove that for the forward Euler method, a local invariance preserving steplength threshold exists for a given polyhedron when a linear dynamical system is considered. For the backward Euler method we prove that a local steplength threshold exists for polyhedron, ellipsoid, and Lorenz cone. These proofs are using elementary concepts. We also quantify a valid local steplength threshold for the backward Euler method. Second, we prove that a uniform invariance preserving steplengt...

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Section: Of 19mentioning

confidence: 92%

Section: Uniform Steplength Threshold For Linear Systemsmentioning

confidence: 99%

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Invariance Preserving Discretization Methods of Dynamical Systems

2018

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“…Observe that parameter β presented in Corollary 3.8 can be eliminated. In fact, one can show that [11].…”

Section: Remark 32 Consider the Following M Optimization Problemsmentioning

confidence: 99%

“…If the system in Theorem 3.1 is a linear dynamical system, then we have the following corollary, which is an invariance condition of polyhedral sets for linear systems. Note that Corollary 3.3 can also be referred to [11]. An alternative proof for Corollary 3.3, using optimality conditions is presented in the Appendix.…”

Section: Remark 32 Consider the Following M Optimization Problemsmentioning

confidence: 99%

Invariance Conditions for Nonlinear Dynamical Systems

Horváth

Song

Terlaky

2016

Optimization and Its Applications in Control and Data Sciences

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Recently, Horváth, Song, and Terlaky [A novel unified approach to invariance condition of dynamical system, submitted to Applied Mathematics and Computation] proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems.In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the nonlinear Farkas lemma and the S -lemma, together with Nagumo's Theorem are utilized to derive invariance conditions for discrete and continuous systems. Only standard assumptions are needed to establish invariance of broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we establish an optimization framework to computationally verify the derived invariance conditions. Finally, we derive analogous invariance conditions without any conditions.

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Invariance under discretization for positive systems

Bartosiewicz

2021

Math. Control Signals Syst.

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Positive dynamical or control systems have all their variables nonnegative. Euler discretization transforms a continuous-time system into a system on a discrete time scale. Some structural properties of the system may be preserved by discretization, while other may be lost. Four fundamental properties of positive systems are studied in the context of discretization: positivity, positive stability, positive reachability and positive observability. Both linear and nonlinear systems are investigated.

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A novel unified approach to invariance conditions for a linear dynamical system

Cited by 12 publications

References 52 publications

Invariance Preserving Discretization Methods of Dynamical Systems

Invariance Preserving Discretization Methods of Dynamical Systems

Invariance Conditions for Nonlinear Dynamical Systems

Invariance under discretization for positive systems

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