Canonical forms for polynomial and quadratic differential operators

Kojima, Chiaki; Rapisarda, Paolo; Takaba, Kiyotsugu

doi:10.1016/j.sysconle.2007.06.004

Cited by 22 publications

(18 citation statements)

References 15 publications

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“…and Q Δ ( ) ≥ 0, ∀ ∈ C ∞ (R n , C m ). Substituting ζ = − jω and η = jω into (25), we obtain the FFDI (22). The above inequality guarantees that the FFDI (17) holds.…”

Section: Main Theoremmentioning

confidence: 91%

“…A physical interpretation of the characterization is provided in Section 4.2. Finally, we give a characterization of the property in terms of B-canonical polynomial matrices [22] in Section 4.3.…”

Section: Characterization Of Finite Frequency Propertiesmentioning

confidence: 99%

“…For the difficulty, this paper clarifies that a compensation rate Q Γ ( ) is induced by a 2n-variable polynomial matrix Γ(ζ, η) expressed as a summation of 2n-variable polynomial matrices which are nonnegative definite on each frequency domain. This is described in (19) and (22).…”

Section: Answermentioning

confidence: 99%

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Characterization of Finite Frequency Properties for n-Dimensional Behaviors Using Quadratic Differential Forms

Kojima

Hara

2014

SICE Journal of Control, Measurement, and System Integration

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Many of practical design specifications are provided by finite frequency properties described by inequalities over restricted finite frequency intervals. In this paper, the authors consider a characterization of the finite frequency domain inequalities (FFDIs) for n-dimensional systems from a view point of a dissipation theory using quadratic differential forms (QDFs), which are useful algebraic tools for the dissipation theory based on the behavioral approach. The QDFs allow us to derive a clear characterization of the FFDIs using some inequality analogous to dissipation inequality with a compensation rate and an inequality of an integral of the supply rate with a matrix integral quadratic constraint as a main result. This characterization leads to a physical interpretation in terms of the dissipativity for subbehavior with some rate constraints. The authors also show how to resolve a difficulty on the expression of a compensation rate peculiar to n-dimensional systems. The results of this paper can be regarded as a finite frequency version for the characterizations of frequency properties over the entire frequency domain due to Pillai and Willems (2002). 1 We call a system that depends on n independent variables (n ≥ 2) as an n-dimensional system.

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“…and Q Δ ( ) ≥ 0, ∀ ∈ C ∞ (R n , C m ). Substituting ζ = − jω and η = jω into (25), we obtain the FFDI (22). The above inequality guarantees that the FFDI (17) holds.…”

Section: Main Theoremmentioning

confidence: 91%

Section: Characterization Of Finite Frequency Propertiesmentioning

confidence: 99%

See 1 more Smart Citation

Characterization of Finite Frequency Properties for n-Dimensional Behaviors Using Quadratic Differential Forms

Kojima

Hara

2014

SICE Journal of Control, Measurement, and System Integration

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“…IfΨ is the coefficient matrix of a two-variable polynomial matrix Ψ(ζ, η), then the coefficient matrix of (ζη − 1)Ψ(ζ, η) is Consequently, the left-hand side of (27) is the coefficient matrix of Φ(ζ, η) − (ζη − 1)Ψ(ζ, η) = ∆(ζ, η). As noticed in the discussion appearing after equation (12), there is a one-to-one correspondence between the factorization of a two-variable polynomial matrix and the factorization of its coefficient matrix. Consequently, a factorizationF F of the left-hand side of (27) .…”

Section: Lemma 14mentioning

confidence: 98%

Time-relevant stability of 2D systems

2011

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For many 2D systems, one of the independent variables plays a distinct role in the evolution of the trajectories; since often this special independent variable is time, we call such systems 'time-relevant'. In this paper, we introduce a stability notion for time-relevant systems described by higher-order difference equations. We give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system. We also give an LMI characterization of time-relevant stability involving only constant matrices.

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“…Background material. We give only the minimum amount of information needed; see [10,17,18] for more information, and [11,13,14,21,23] for important details and for applications of 2D bilinear-and quadratic differential forms.…”

mentioning

confidence: 99%

State-Space Modeling of Two-Dimensional Vector-Exponential Trajectories

Rapisarda¹,

Antoulas²

2016

SIAM J. Control Optim.

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Abstract. We solve two problems in modelling polynomial vector-exponential trajectories dependent on two independent variables. In the first one we assume that the data-generating system has no inputs, and we compute a state representation of the Most Powerful Unfalsified Model for this data. In the second instance we assume that the data-generating system is controllable and quarterplane causal, and we compute a Roesser i/s/o model. We provide procedures for solving these identification problems, both based on the factorization of constant matrices directly constructed from the data, from which state trajectories can be computed.Key words. multidimensional systems, most powerful unfalsified model, Roesser models, bilinear differential forms AMS subject classifications. 93A30, 93B15, 93B20, 93B30, 93C201. Introduction. We consider two problems in modelling two-dimensional (2D in the following) continuous trajectories from data. In both cases the data consistis of polynomial vector-exponential trajectories, and we seek state-space models explaining it, i.e. systems of partial differential equations of first order in an auxiliary, "state" variable, and zeroth-order in the measured, "external" variable. The two situations differ in the model class we assume the data-generating system belongs to: in the first case we seek an autonomous state model , i.e. a system without inputs; in the second one we assume that an input/output partition of the external variable is given, and we compute an input-state-output (i/s/o in the following) model.Modelling 2D polynomial vector-exponential trajectories with autonomous systems has been considered in [32,33], on whose results the first part of this paper on the computation of autonomous models heavily relies. Modelling vector-exponential trajectories with transfer-function (i.e. input-output) models is closely related to twovariable rational interpolation; the latter has been investigated in the SISO case in [2]. The approach taken in the present paper differs fundamentally from those: we use data to first compute state trajectories corresponding to it, and in a second stage we compute a state representation for the data and the identified state trajectories by solving linear equations in the unknown state-, input-and output matrices.Modelling methodologies where state-trajectories are computed from data and state-equations are subsequently computed are well-known in the 1D case as subspace identification methods (see e.g. [12]). Such ideas have been pursued much less frequently in the 2D case: see [7,19] for a pioneering subspace-identification approach to the computation of i/s/o representations of denominator-separable 2D discretesystems from data. Our modelling approach differs essentially from those, in that it does not exploit the shift-invariance properties of data trajectories, but rather the fact that external properties-i.e. properties at the level of external variables, in our case duality-are reflected into internal properties-i.e. at the level of state. To make the...

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Canonical forms for polynomial and quadratic differential operators

Cited by 22 publications

References 15 publications

Characterization of Finite Frequency Properties for n-Dimensional Behaviors Using Quadratic Differential Forms

Characterization of Finite Frequency Properties for n-Dimensional Behaviors Using Quadratic Differential Forms

Time-relevant stability of 2D systems

State-Space Modeling of Two-Dimensional Vector-Exponential Trajectories

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