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91
Approximation metrics for discrete and continuous systems
 IEEE Transactions on Automatic Control
, 2005
"... Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In thi ..."
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Cited by 105 (16 self)
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Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In this paper, we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion, approximate simulation, and approximate bisimulation relations. We define a hierarchy of approximation pseudometrics between two systems that quantify the quality of the approximation, and capture the established exact relationships as zero sections. Our approximation framework is compositional for a synchronous composition operator. Algorithms are developed for computing the proposed pseudometrics, both exactly and approximately. The exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations, or dually, the solution of a static game whose cost is the socalled branching distance between the systems. Approximations for the pseudometrics can be obtained by considering Lyapunovlike functions called simulation and bisimulation functions. We illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems.
A collection of benchmark examples for model reduction of linear time invariant dynamical systems. SLICOT Working Note 20022. Available from
"... Summary. We present a benchmark collection containing some useful real world examples, which can be used to test and compare numerical methods for model reduction. All systems can be downloaded from the web and we describe here the relevant characteristics of the benchmark examples. 1 ..."
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Cited by 73 (10 self)
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Summary. We present a benchmark collection containing some useful real world examples, which can be used to test and compare numerical methods for model reduction. All systems can be downloaded from the web and we describe here the relevant characteristics of the benchmark examples. 1
On the decay rate of Hankel singular values and related issues
 Systems Control Lett
"... Abstract This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropria ..."
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Cited by 64 (6 self)
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Abstract This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modiÿed transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues, like the Hankel singular values, are rapidly decaying. This fact has motivated the development of several algorithms for computing lowrank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically lowrank Cauchy matrix determined by the poles of the system. Bounds explaining rapid decay rates are obtained under some mild conditions.
A modified lowrank Smith method for largescale Lyapunov Equations
, 2003
"... ... to solutions of Lyapunov equations arising from largescale dynamical systems. Unlike the original cyclic lowrank Smith method introduced by Penzl in [20], the number of columns required by the modified method in the approximate solution does not necessarily increase at each step and is usually ..."
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Cited by 55 (14 self)
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... to solutions of Lyapunov equations arising from largescale dynamical systems. Unlike the original cyclic lowrank Smith method introduced by Penzl in [20], the number of columns required by the modified method in the approximate solution does not necessarily increase at each step and is usually much lower than in the original cyclic lowrank Smith method. The modified method never requires more columns than the original one. Upper bounds are established for the errors of the lowrank approximate solutions and also for the errors in the resulting approximate Hankel singular values. Numerical results are given to verify the efficiency and accuracy of the new algorithm.
The Sylvester equation and approximate balanced reduction
, 2002
"... The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection me ..."
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Cited by 53 (4 self)
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The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection methods, namely PVL (Pad via Lanczos) and rationa Krylov, which do not satisfy these properties. Our approach is based on the computation and approximation of the cross gramtan of the system. The cross gramtan is the solution of a Sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights.
Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations
 SIAM Journal on Scientific Computing
"... Abstract. We discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete Oseen equations with timeindependent advection and the linearized Navier–Stokes equations, linearized around a steady state. The purpose of this paper is twof ..."
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Cited by 23 (1 self)
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Abstract. We discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete Oseen equations with timeindependent advection and the linearized Navier–Stokes equations, linearized around a steady state. The purpose of this paper is twofold. First, we show how to apply standard balanced truncation model reduction techniques, which apply to dynamical systems given by ordinary differential equations, to this class of descriptor systems. This is accomplished by eliminating the algebraic equation using a projection. The second objective of this paper is to demonstrate how the important class of ADI/Smithtype methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection. Instead, we utilize the solution of saddle point problems. We demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete Oseen equations.
Uncovering operational interactions in genetic networks using asynchronous boolean dynamics
 in "J. Theor. Biol
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Approximate bisimulation relations for constrained linear systems
 AUTOMATICA
, 2007
"... In this paper, we define the notion of approximate bisimulation relation between two systems, extending the well established exact bisimulation relations for discrete and continuous systems. Exact bisimulation requires that the observations of two systems are and remain identical, approximate bisi ..."
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Cited by 18 (5 self)
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In this paper, we define the notion of approximate bisimulation relation between two systems, extending the well established exact bisimulation relations for discrete and continuous systems. Exact bisimulation requires that the observations of two systems are and remain identical, approximate bisimulation allows the observation to be different provided they are and remain arbitrarily close. Approximate bisimulation relations are conveniently defined as level sets of a function called bisimulation function. For the class of linear systems with constrained initial states and constrained inputs, we develop effective characterizations for bisimulation functions that can be interpreted in terms of linear matrix inequalities, set inclusion and games. We derive a computationally effective algorithm to evaluate the precision of the approximate bisimulation between a constrained linear system and its projection. This algorithm has been implemented in a MATLAB toolbox: MATISSE. Two examples of use of the toolbox in the context of safety verification are shown.
Approximate bisimulations for constrained linear systems
, 2005
"... In this paper, inspired by exact notions of bisimulation equivalence for discreteevent and continuoustime systems, we establish approximate bisimulation equivalence for linear systems with internal but bounded disturbances. This is achieved by developing a theory of approximation for transition ..."
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Cited by 14 (5 self)
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In this paper, inspired by exact notions of bisimulation equivalence for discreteevent and continuoustime systems, we establish approximate bisimulation equivalence for linear systems with internal but bounded disturbances. This is achieved by developing a theory of approximation for transition systems with observation metrics, which require that the distance between system observations is and remains arbitrarily close in the presence of nondeterministic evolution. Our notion of approximate bisimulation naturally reduces to exact bisimulation when the distance between the observations is zero. Approximate bisimulation relations are then characterized by a class of Lyapunovlike functions which are called bisimulation functions. For the class of linear systems with constrained disturbances, we obtain computable characterizations of bisimulation functions in terms of linear matrix inequalities, set inclusions, and optimal values of static games. We illustrate our framework in the context of safety verification.