Time Variant Balancing and Nonlinear Balanced Realizations
Balancing for linear time varying systems and its application to model reduction via projection of dynamics (POD) are briefly reviewed. We argue that a generalization for balancing nonlinear systems may be expected to be based upon three sound principles:
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on Reduction techniques are routinely used to replace the relevant discretized PDE’s to ODE models of much smaller dimension. Most existing methods pertain to linear models and dynamics and fail to correctly model the nonlinear couplings and dynamics. Balancing for linear time invariant systems has been applied to problems of model reduction (via the Projection of Dynamics (POD), also called “balanced truncation”), in parameterization, sensitivity analysis and system identification. With these initial successes, extensions a balanced realization to other classes of systems soon followed. Balancing for linear time-varying systems [SSV83, VK83, SR02, VH98], and an alternative form of balancing, suitable for controller reduction, called LQG-balancing [Ve81a, Ve81b, JS83] were the first generalizations. This chapter presents a possible approach towards generalizing balanced realizations to nonlinear systems. In a series of papers [VG00, VG01b, Ve04, VG04] we extended the linear balancing method to a class of nonlinear systems. While sharing many similarities with the method first proposed by Scherpen [Sch94], there are some fundamental differences. Whereas Scherpen took the notions of observability and
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reachability1 functions as a starting point, our method is rooted in earlier work for linear time-varying systems. In particular, the Sliding-Interval-Balancing (SIB), proposed in [Ve80, VK83], is here used as a stepping stone to the nonlinear case. In Section 2, the principles behind balancing are motivated. Section 3 reviews SIB, and some new results are presented. The proposed approach to (local) nonlinear balancing is given in Section 4, which forms the main part of the chapter. The extension to global balancing is given in Section 5, where also an ‘obstruction’ is encountered. In Section 6 we introduce Mayer-Lie interpolation as a way around this obstruction. Application to nonlinear model reduction is briefly discussed in Section 7. Final comments regarding the proposed solution are formulated in Section 8. Some of this work was performed and evolved over many years in collaboration with Professor W. Steven Gray from the Old Dominion University.
2 Time Varying Linear Systems 2.1 Finite Time Gramians For general time varying linear systems, we define the reachability and observability map, and use adjoint operator techniques to solve various problems related to energy, ambiguity and uncertainty. This sheds light on the role played by the Gramian matrices (reachability and observability Gramian), and their subsequent importance in model reduction. It is assumed that there are n states, m inputs and p outputs. x(t) ˙ = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t)
(1) (2)
∂ Φ(t, τ ) = A(t) Let Φ(t, τ ) be the transition matrix, satisfying for all t and τ , ∂t Φ(t, τ ) with Φ(τ, τ ) = I. The finite time reachability and observability Gramians (for an interval of length δ > 0) are defined by t Φ(t, τ )B(τ )B(τ )T Φ(t, τ )T dτ (3) R(t, δ) = t−δ t+δ
O(t, δ) =
Φ(τ, t)T C(τ )T C(τ )Φ(τ, t) dτ.
(4)
t
In genera
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