1. MESA LAB A Brief Introduction to MFD (Matrix Fraction Description) Zhuo Li MESA LAB 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. MESA LAB 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 Slide-2/1024 AFC Workshop Series @ MESALAB @ UCMerced 04/21/2014

3. MESA LAB Definition 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-3/1024

4. MESA LAB Example 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-4/1024

5. MESA LAB Property 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-5/1024

6. MESA LAB 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] 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-6/1024

7. MESA LAB Example 1 For the system on the right The left MFD is 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-7/1024

8. MESA LAB 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 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-8/1024

9. MESA LAB Example 2 Two-input-two-output system Direct controllable state-space realization 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-9/1024 The order is 12

10. MESA LAB Example 2 – cont’d Rewrite G(s) in the polynomial denominator form 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-10/1024 The order is 10

11. MESA LAB 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 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-11/1024 U(s) is called divisor

12. MESA LAB 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. 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-12/1024

13. MESA LAB 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. 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-13/1024