Positive Real Balanced Truncation Research Articles

A New Passivity Preserving Model Order Reduction Method: Conic Positive Real Balanced Truncation Method

This article is dedicated to model order reduction of linear time-invariant passive systems, i.e., positive real (PR) systems, based on the balanced truncation (BT) concept. The main feature of this article is that we have used the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , in model order reduction for the first time in the proposed method, which results in more accurate reduced-order models. It is worth noting that this hypothesis has been never made before in any other model order reduction techniques. Furthermore, new gramians are calculated corresponding to their associated new Riccati equations and Lur’e equations, which are in turn achieved by using Kalman–Yakubovic–Popov (KYP) lemma and linear matrix inequalities (LMI). Thereafter, a novel passivity preserving model order reduction algorithm, which takes the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , into account, is presented. It is demonstrated that the proposed method is a generalization of the positive real balanced truncation (PRBT) method, and it is perfectly capable of providing more accurate approximation error compared to PRBT. Finally, numerical examples are included to figure out the effectiveness of the presented method.

Phase Preserving Balanced Truncation for Order Reduction of Positive Real Systems

This paper presents a new passivity-preserving order reduction method for linear time-invariant passive systems, which are also called positive real (PR) systems, with the aid of the balanced truncation (BT) method. The proposed method stems from the conic positive real balanced truncation (CPRBT) method, which is a modification of the BT method for PR systems. CPRBT presents an algorithm in which the reduced models are obtained from some Riccati equations in which the phase angle of the transfer function has been taken into consideration. Although CPRBT is a powerful algorithm for obtaining accurate PR reduced-order models, it cannot guarantee that the phase diagram of the reduced model remains inside the same interval as that of the original full-order system. We aim to address such a problem by modifying CPRBT in the way that the phase angle of the reduced transfer function always remains inside the conic and homolographic phase interval of the original system. This is proven through some matrix manipulations, which has added mathematical value to the paper. Finally, in order to assess the efficacy of the proposed method, two numerical examples are simulated.

Model Order Reduction of Positive Real Systems Based on Mixed Gramian Balanced Truncation with Error Bounds

In this paper, we discuss the problem of model order reduction for positive real systems based on balancing methods. The mixed gramian balanced truncation (MGBT) method, which is a modification of the positive real balanced truncation (PRBT) method, focuses on solving one Lyapunov equation and one Riccati equation resulting in less computational effort compared to PRBT requiring solving two Riccati equations. One major disadvantage of MGBT is that it cannot provide an error bound in contrast to PRBT. To overcome this issue, we have developed some novel modifications to MGBT which not only work with one Lyapunov and one Riccati equations but also provide error bounds. Thus, we can say that the presented methods take the key features of both MGBT and PRBT. These algorithms are presented with the aid of the new gramians which are extracted from new Lyapunov equations. The second algorithm is the frequency weighted version of the first algorithm. Additionally, it is also observed that the proposed methods can provide better error bounds compared to PRBT. Finally, comprehensive numerical examples are included to figure out the effectiveness of the presented method.

Balanced Truncation Model Reduction for Symmetric Second Order Systems---A Passivity-Based Approach

We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally, we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.

Error bounds in the gap metric for dissipative balanced approximations

We derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation. We prove these results by working in the context of dissipative driving-variable systems, as in behavioral and state/signal systems theory. In such a framework no prior distinction is made between inputs and outputs. Dissipativity preserving balanced truncation of dissipative driving-variable systems is addressed and a gap metric error bound is obtained. Bounded real and positive real input–state–output systems are manifestations of a dissipative driving-variable system through particular decompositions of the signal space. Under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow.

Bounded real and positive real balanced truncation for infinite-dimensional systems

Bounded real balanced truncation for infinite-dimensional systems is considered. This provides reduced order finite-dimensional systems that retain bounded realness. We obtain an error bound analogous to the finite-dimensional case in terms of the bounded real singular values. By using the Cayley transform a gap metric error bound for positive real balanced truncation is subsequently obtained. For a class of systems with an analytic semigroup, we show rapid decay of the bounded real and positive real singular values. Together with the established error bounds, this proves rapid convergence of the bounded real and positive real balanced truncations.

A Counterexample to “ Positive Realness Preserving Model Reduction With ${\cal H}_{\infty}$ Norm Error Bounds”

We provide a counterexample to the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> error bound for the difference of a positive real transfer function and its positive real balanced truncation stated in “Positive realness preserving model reduction with H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm error bounds,” IEEE Trans. Circuits Syst, I, Fundam. Theory Appl., vol. 42, no. 1, pp. 23-29 (1995). The proof of the error bound is based on a lemma from an earlier paper, “A tighter relative-error bound for balanced stochastic truncation,” Syst. Control Lett., vol. 14, no. 4, 307-317 (1990), which we also demonstrate is false by our counterexample. The main result of this paper was already known in the literature to be false. We state a correct H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> error bound for the difference of a proper positive real transfer function and its positive real balanced truncation and also an error bound in the gap metric.

Model order reduction with preservation of passivity, non-expansivity and Markov moments

A new model order reduction technique is presented which preserves passivity and non-expansivity. It is a projection-based method which exploits the solution of linear matrix inequalities to generate a descriptor state space format which preserves positive-realness and bounded-realness. In the case of both non-singular and singular systems, solving the linear matrix inequality can be replaced by equivalently solving an algebraic Riccati equation, which is known to be a more efficient approach. A new algebraic Riccati equation and a frequency inversion technique are also presented to specifically deal with the important singular case. The preservation of Markov moments is also guaranteed by the judicious choice of a projection matrix. Three pertinent examples comparing the present approach with positive-real balanced truncation show the strength and accuracy of the present approach.

Bounded real and positive real balanced truncation using [formula omitted]-normalised coprime factors

In this article, we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix by Meyer (1990) [3] to balanced truncation with preservation of half line dissipativity. Special cases are preservation of positive realness and bounded realness. We consider a half line dissipative input–output system, with quadratic supply rate given by a nonsingular symmetric matrix Σ with the property that its positive signature is equal to the number of input components of the system. The transfer matrix of such system allows a Σ -normalised right coprime factorisation. We associate with such factorisation two Lyapunov equations, one of which is nonstandard, involving the indefinite matrix Σ . Balancing will be based on making the unique solutions of these two Lyapunov equations equal and diagonal. The diagonal elements will be called the Hankel Σ -singular values, because their squares are the nonzero eigenvalues of the composition of the ‘graph’ Hankel operator, multiplication by Σ , and the adjoint graph Hankel operator. This method of balanced truncation will be shown to preserve stability, minimality, and half line dissipativeness. We will characterize the ‘classical’ positive real and bounded real characteristic values in terms of the new Hankel Σ -singular values. Finally, we will derive one-step error bounds for the special case of balanced truncation of bounded real systems.

Passivity-Preserving Model Reduction Using Dominant Spectral-Zero Interpolation

In this paper, the dominant spectral-zero method (dominant SZM) is presented, a new passivity-preserving model-reduction method for circuit simulation. Passivity is guaranteed via spectral-zero interpolation, and a dominance criterion is proposed for selecting spectral zeros. Dominant SZM is implemented as an iterative eigenvalue-approximation problem using the subspace-accelerated dominant-pole algorithm. Passive circuits are reduced automatically irrespective of how the original system equations are formulated (e.g., circuit models containing controlled sources or susceptance elements). Dominant SZM gives comparable and often more accurate reduced models than known techniques such as PRIMA, modal approximation, or positive real balanced truncation.

Efficient Positive-Real Balanced Truncation of Symmetric Systems Via Cross-Riccati Equations

We present a highly efficient approach for realizing a positive-real balanced truncation (PRBT) of symmetric systems. The solution of a pair of dual algebraic Riccati equations in conventional PRBT, whose cost constrains practical large-scale deployment, is reduced to the solution of one cross-Riccati equation (XRE). The cross-Riccatian nature of the solution then allows a simple construction of PRBT projection matrices, using a Schur decomposition, without actual balancing. An invariant subspace method and a modified quadratic alternating-direction-implicit iteration scheme are proposed to efficiently solve the XRE. A low-rank variant of the latter is shown to offer a remarkably fast PRBT speed over the conventional implementations. The XRE-based framework can be applied to a large class of linear passive networks, and its effectiveness is demonstrated through numerical examples.

Passivity‐preserving model reduction of differential‐algebraic equations in circuit simulation

AbstractWe present an extension of the positive real balanced truncation model reduction method for differential‐algebraic equations that arise in circuit simulation. This method is based on balancing the solutions of the projected generalized algebraic Riccati equations. Important properties of this method are that passivity is preserved in the reduced‐order model and that there exists an approximation error bound. Numerical solution of the projected Riccati equations using the special structure of circuit equations is also discussed. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Fast Positive-Real Balanced Truncation Via Quadratic Alternating Direction Implicit Iteration

Balanced truncation (BT), as applied to date in model order reduction (MOR), is known for its superior accuracy and computable error bounds. Positive-real BT (PRBT) is a particular BT procedure that preserves passivity and stability and imposes no structural constraints on the original state space. However, PRBT requires solving two algebraic Riccati equations (AREs), whose computational complexity limits its practical use in large-scale systems. This paper introduces a novel quadratic extension of the alternating direction implicit (ADI) iteration, which is called quadratic ADI (QADI), that efficiently solves an ARE. A Cholesky factor version of QADI, which is called CFQADI, exploits low-rank matrices and further accelerates PRBT.

New Schemes for Positive Real Truncation

Model reduction, based on balanced truncation, of stable and of positive real systems are considered. An overview over some of the already existing techniques are given: Lyapunov balancing and stochastic balancing, which includes Riccati balancing. A novel scheme for positive real balanced truncation is then proposed, which is a combination of the already existing Lyapunov balancing and Riccati balancing. Using Riccati balancing, the solution of two Riccati equations are needed to obtain positive real reduced order systems. For the suggested method, only one Lyapunov equation and one Riccati equation are solved in order to obtain positive real reduced order systems, which is less computationally demanding. Further it is shown, that in order to get positive real reduced order systems, only one Riccati equation needs to be solved. Finally, this is used to obtain positive real frequency weighted balanced truncation.