Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data

Rapisarda, Paolo; Trentelman, Harry L.

doi:10.1016/j.automatica.2011.02.048

Cited by 36 publications

(34 citation statements)

References 23 publications

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“…These results could be used in data-driven control and system identification. For example, in the work of Rapisarda and Trentelman (2011) they assume that the given data is generated by a dissipative system with respect to the given supply rate, but with our result it is possible to test if the data is indeed generated by a dissipative system. Our results are based on the concepts of behavioral system theory (see Willems (1989Willems ( , 1991), hence we use mathematical tools such as quadratic difference forms (QdFs)(see Kaneko and Fujii (2000)).…”

Section: Introductionmentioning

confidence: 96%

“…For example, to develop tests in which we can directly use system data to determine whether a system is dissipative. In the literature, the concept of dissipativity and data has been explored in Rapisarda and Trentelman (2011). In this case, the authors illustrate how to find a state space representation of a dissipative system using noise-free observed trajectories.…”

Section: Introductionmentioning

confidence: 99%

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On Lyapunov functions and data-driven dissipativity

2017

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Our contribution in this paper is twofold. In the first part, we study Lyapunov functions when a plant is interconnected with a dissipative stabilizing controller. In the second, we present results on data-driven approach to dissipative systems. In particular, we provide conditions under which an observed trajectory can be used to determine whether a system is dissipative with respect to a given supply rate. Our results are based on linear difference systems for which the use of quadratic difference forms play a central role for dissipativity and Lyapunov theory.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

On Lyapunov functions and data-driven dissipativity

2017

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“…In general it is difficult to obtain data from the dual system, and only data coming from the primal one are available (unless of course the two systems coincide-see [22,24] for examples in the 1D case). We now describe the mirroring technique, already used in the 1D case (see [8,9,25]), to obtain dual data on the basis of primal ones.…”

Section: Remark 5 (Data Dualization Via Mirroring)mentioning

confidence: 99%

“…not polynomial vector-exponential) discrete data; cf. [22] for a BDF approach to such problem in the 1D case. On a longer horizon and a broader perspective, we want to investigate the application of our duality-based approach to model reduction.…”

Section: Remark 5 (Data Dualization Via Mirroring)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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“…Thus, practical engineering control strategies for dealing with high-dimensional observational data revolve around dimensionality-reduction techniques. Such methods, often based on the singular value decomposition (SVD) of the data, allow one to construct low-dimensional subspaces where computationally tractable controllers can be designed and implemented [35,24,20,40,39,58,18]. Balanced truncation is a classic method developed to specifically take advantage of underlying low-dimensional observable and controllable subspaces to create a balanced, reduced-order model [35].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Mode Decomposition with Control

Proctor¹,

Brunton²,

Kutz³

2016

SIAM J. Appl. Dyn. Syst.

814

515

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Abstract. We develop a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, dynamic mode decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations-only snapshots in time of observables and actuation data from historical, experimental, or black-box simulations. We demonstrate the method on high-dimensional dynamical systems, including a model with relevance to the analysis of infectious disease data with mass vaccination (actuation).

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Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data

Cited by 36 publications

References 23 publications

On Lyapunov functions and data-driven dissipativity

On Lyapunov functions and data-driven dissipativity

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

Dynamic Mode Decomposition with Control

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