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An iterative scheme for finite horizon model reduction of continuous-time linear time-varying systems

Kasturi Das, Srinivasan Krishnaswamy, Somanath Majhi

Abstract

In this paper, we obtain the functional derivatives of a finite horizon error norm between a full-order and a reduced-order continuous-time linear time-varying (LTV) system. Based on the functional derivatives, first-order necessary conditions for optimality of the error norm are derived, and a projection-based iterative scheme for model reduction is proposed. The iterative scheme upon convergence produces reduced-order models satisfying the optimality conditions. Finally, through a numerical example, we demonstrate the better performance of the proposed model reduction scheme in comparison to the finite horizon balanced truncation algorithm for continuous-time LTV systems.

An iterative scheme for finite horizon model reduction of continuous-time linear time-varying systems

Abstract

In this paper, we obtain the functional derivatives of a finite horizon error norm between a full-order and a reduced-order continuous-time linear time-varying (LTV) system. Based on the functional derivatives, first-order necessary conditions for optimality of the error norm are derived, and a projection-based iterative scheme for model reduction is proposed. The iterative scheme upon convergence produces reduced-order models satisfying the optimality conditions. Finally, through a numerical example, we demonstrate the better performance of the proposed model reduction scheme in comparison to the finite horizon balanced truncation algorithm for continuous-time LTV systems.
Paper Structure (12 sections, 62 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 62 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Time-limited $H_2$ optimal model order reduction (MOR)
  4. Time-Limited Two-Sided Iterative Algorithm (TL-TSIA)
  5. Finite horizon model order reduction for continuous-time LTV systems
  6. The finite horizon $H_2$ error norm
  7. Functional derivatives of the finite horizon $H_2$ error norm
  8. A projection-based model reduction scheme
  9. Numerical examples
  10. Example 1:
  11. Example 2:
  12. Conclusion

Figures (5)

  • Figure 1: Hankel singular values of the second-order LTV model.
  • Figure 2: (I) Step responses for the second-order LTV model and the first-order LTV approximations achieved by FH BT and various iterations of FH TSIA, (II) Approximation errors between the step responses of the original and the reduced-order models.
  • Figure 3: Hankel singular values of the fourth-order LTV model.
  • Figure 4: (I) Step responses for the $4^{th}$-order LTV model and the first-order approximations obtained by FH BT and various iterations of FH TSIA. (II) Approximation errors between the step responses of the original and the reduced-order models.
  • Figure 5: (I) Step responses for the $4^{th}$-order LTV model and the second-order LTV approximations obtained by FH BT and various iterations of FH TSIA. (II) Absolute value of error between the step responses of the original and the reduced-order models.