A B rief Introduction to MFD (Matrix Fraction Description)

1. A Brief Introduction to MFD (Matrix Fraction Description) Zhuo Li MESA (Mechatronics, Embedded Systems and Automation)Lab School of Engineering, University of California, Merced E: zli32@ucmerced.edu Lab: CAS Eng 820 (T: 209-228-4398) Jul 28, 2014. Monday 4:00-6:00 PM Applied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced

2. What is MFD • Matrix fraction descriptions (MFDs) • A convenient way of representing rational matrices as the “ratio” of two polynomial matrices. • Useful for multi-input/multi-output linear transformations AFC Workshop Series @ MESALAB @ UCMerced

3. Definition • A rational matrix () • , where • Assume the existence of a non-singular polynomial matrix , which • Then, is called the left MFD of . • Right MFD is similarly defined • Ref [1] AFC Workshop Series @ MESALAB @ UCMerced

4. Example • Given • Then, AFC Workshop Series @ MESALAB @ UCMerced

5. Property • For any matrix in standard form • i.e. irreducible • and compromise, and is monic there always exist LMFDs and RMFDs. AFC Workshop Series @ MESALAB @ UCMerced

6. The use in control systems • To extend the results of scalar systems to multivariable systems. • Such as the transfer function to state-space realization • The closest analogy with the scalar results can be achieved by using the MFDs. • Ref [2] AFC Workshop Series @ MESALAB @ UCMerced

7. Example 1 • For the system on the right • The left MFD is AFC Workshop Series @ MESALAB @ UCMerced

8. The use in control systems • For scalar systems, nice controllability/ observability properties and minimal orders can be achieved through canonical form realization • For multi-variable systems, these properties may be lost AFC Workshop Series @ MESALAB @ UCMerced

9. Example 2 • Two-input-two-output system • Direct controllable state-space realization The order is 12 AFC Workshop Series @ MESALAB @ UCMerced

10. Example 2 – cont’d • Rewrite G(s) in the polynomial denominator form The order is 10 AFC Workshop Series @ MESALAB @ UCMerced

11. Question • For multi-variable systems, what the minimal order of a realization can be? • Corresponded to the degree of the denominator • A minimum-degree right MFD can be obtained by extracting a greatest common right divisor U(s) is called divisor AFC Workshop Series @ MESALAB @ UCMerced

12. Conclusion • The transformation from MFDs to state-space motivated the introduction of several concepts and properties specific to polynomial matrices. • There exist extensions to the results • e.g. Descriptor state-space representation. AFC Workshop Series @ MESALAB @ UCMerced

13. Reference • E Rosenwasser and B Lampe, “Multivariable computer-controlled systems”, Springer, 2006 • Didier Henrion, and Michael Sebek, “Polynomial And Matrix Fraction Description”, Lecture notes. • Rgtnikant V. Patel, “Computation of Matrix Fraction Descriptions of Linear Time-invariant Systems, IEEE Transactions On Automatic Control, 1981. AFC Workshop Series @ MESALAB @ UCMerced