Tudor Corneliu Ionescu

Families of moment matching based, low order approximations for linear systems ☆

The problem of finding a framework for linear moment matching based approximation of linear systems is discussed. It is shown that it is possible to define five (equivalent) notions of moments, i.e., one from a complex “s-domain” point of view, two using Krylov projections and two from a time-domain perspective. Based on these notions, classes of parameterized reduced order models that achieve moment matching are obtained. We analyze the controllability and observability properties of the models belonging to each of the classes of reduced order models. We find the subclasses of models of orders lower than the number of matched moments, i.e., we find the sets of parameters that allow for pole–zero cancellations to occur. Furthermore we compute the (subclass of) minimal order model(s). The minimal order is equal to half of the number of matched moments, i.e., generically the largest number of possible pole–zero cancellations is half of the number of matched moments. Finally, we present classes of reduced order models that match larger numbers of moments, based on interconnections between reduced order models.

Singular Value Analysis Of Nonlinear Symmetric Systems

In this paper, we introduce the notions of state-space symmetry for nonlinear systems and of the cross operator as a nontrivial natural extension of the linear symmetric case, in terms of the controllability and observability operators associated to it. We give a characterization of a symmetric nonlinear system in terms of the cross operator and a coordinate transformation. Then we analyze the use of the cross operator for solving the Hankel singular value problem of the system. The result is a new and simpler characterization of the solutions of this problem in terms of the cross operator and a metric.

Families of reduced order models that achieve nonlinear moment matching

In this paper we present a time-domain notion of moments for a class of single-input, single-output nonlinear systems, affine in the input, in terms of the steady-state response of the output of a generalized signal generator driven by the nonlinear system. In addition, we define a new notion of moment matching and present the class of (nonlinear) parameterized reduced order models that achieve moment matching. Furthermore, we establish relations with existing notions of moment, showing that the families of reduced order models that achieve nonlinear moment matching are equivalent. Furthermore, we compute the reduced order model that matches moments at two sets of interpolation points, simultaneously, i.e., the number of interpolation points is twice the order of the model.

Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems

In this paper we propose a solution to the problem of moment matching with preservation of the port Hamiltonian structure, in the framework of time-domain moment matching. We characterize several families of parameterized port Hamiltonian models that match the moments of a given port Hamiltonian system, at a set of finite interpolation points. We also discuss the problem of Markov parameters matching for linear systems as a moment matching problem for descriptor representations associated with the given system, at zero interpolation points. Solving this problem yields families of parameterized reduced order models that achieve Markov parameter matching. Finally, we apply these results to the port Hamiltonian case, resulting in families of parameterized reduced order port Hamiltonian approximations.

Dissipativity preserving balancing for nonlinear systems - A Hankel operator approach

Abstract In this paper we present a version of balancing for nonlinear systems which is dissipative with respect to a general quadratic supply rate that depends on the input and the output of the system. We discuss an approach that allows us to apply the theory of balancing based upon Hankel singular value analysis. In order to do that we prove that the available storage and the required supply of the original system are the controllability and the observability functions of a modified, asymptotically stable, system. Then Hankel singular value theory can be applied and the axis singular value functions of the modified system equal the nonlinear extensions of “similarity invariants” obtained from the required supply and available storage of the original system. Furthermore, we also consider an extension of normalized comprime factorizations and relate the available storage and required supply with the controllability and observability functions of the factorizations. The obtained relations are used to perform model order reduction based on balanced truncation, yielding dissipative reduced order models for the original systems. A second order electrical circuit example is included to illustrate the results.

On moment matching with preservation of passivity and stability

In this paper, the problem of moment matching with (preservation of) certain properties is tackled. Based on a new approach to linear and nonlinear matching we discuss the choices of interpolation (matching) points such that the resulting families of models have certain properties, namely passivity and stability, useful for modeling and analysis.

Positive real balancing for nonlinear systems

We extend the positive real balancing procedure for passive linear systems to the nonlinear systems case. We show that, just like in the linear case, model reduction based on this technique preserves passivity.

Nonlinear cross Gramians and gradient systems

We study the notion of cross Gramians for nonlinear gradient systems, using the characterization in terms of prolongation and gradient extension associated to the system. The cross Gramian is given for the variational system associated to the original nonlinear gradient system. We obtain linearization results that precisely correspond to the notion of a cross Gramian for symmetric linear systems. Furthermore, first steps towards relations with the singular value functions of the nonlinear Hankel operator are studied and yield promising results.

Moment matching with prescribed poles and zeros for infinite-dimensional systems

In this paper we approach the problem of moment matching for a class of infinite-dimensional systems, based on the unique solution of an operator Sylvester equation. It results in a class of parameterized, finite-dimensional, reduced order models that match a set of prescribed moments of the given system. We show that, by properly choosing the free parameters, additional constraints are met, e.g., pole placement, preservation of zeros. To illustrate the proposed method, we apply it to the heat equation with mixed boundary conditions. We obtain a second order reduced model which approximates the original systems better (in terms of the infinity norm of the approximation error) than the fourth order reduced model obtained by modal truncation.

Nonlinear Moment Matching-Based Model Order Reduction

In this paper we present a time-domain notion of moments for a class of single-input, single-output nonlinear systems in terms of the evolution of the output of a generalized signal generator driven by the nonlinear system. We also define a new notion of moment matching and present a family of (nonlinear) parametrized reduced order models that achieve moment matching. We establish relations with existing notions of moment for nonlinear systems, showing that the newly derived and the existing families of reduced order models that achieve nonlinear moment matching, respectively, are equivalent. Furthermore, we compute the reduced order model that matches the moments at two chosen signal generators (one exciting the input of the system and another driven by the system), simultaneously. We also present a family of models computed on the basis of a nonlinear extension of the Petrov-Galerkin projection that achieve moment matching. Finally, we specialize the results to the case of nonlinear, input-affine systems.