The Cayley Transform as a Time Discretization Scheme

Abstract

We interpret the Cayley transform of linear (finite- or infinite-dimensional) state space systems as a numerical integration scheme of Crank–Nicolson type. The scheme is known as Tustin's method in the engineering literature, and it has the following important Hamiltonian integrator property: if Tustin's method is applied to a conservative (continuous time) linear system, then the resulting (discrete time) linear system is conservative in the discrete time sense. The purpose of this paper is to study the convergence of this integration scheme from the input/output point of view.

Keywords:

• Cayley–Tustin transform
• Conservative system
• Crank–Nicolson scheme

AMS Subject Classification:

• 47A48
• 65J10
• 93C25
• 34G10
• 47N70
• 65L70

ACKNOWLEDGMENT

We would like to thank several anonymous reviewers for valuable comments.

This work was supported by the European Commission's 5th Framework Programme: Smart Systems; New materials, adaptive systems and their nonlinearities (HPRN-CT-2002-00284).

Notes

¹To state this claim rigorously, we should define the sampling and interpolating operators T _2/h and . This is postponed to Section 2.2. Also note that we do not consider the approximation of x(·) in this paper but we restrict ourselves to the input/ouput framework.

²The rest of this section serves only as a motivation and background. An already well-motivated reader may skip to Section 2 without any loss to read the rest of this paper.

³We shall use the notation for X × Y.

⁴Note that by Proposition 2.1 and equality (2.4), we see that each L _σ: L ²(ℝ₊) → H ²(ℂ₊) is a co-isometry. The Laplace transform is a unitary mapping between the same spaces. Hence, the convergence of L _σ → ℒ must be rather weak.