mesa lab a brief introduction to mfd (matrix fraction description) zhuo li mesa lab mesa...
TRANSCRIPT
MESA LAB
A Brief Introduction to MFD (Matrix Fraction Description)
Zhuo LiMESA (Mechatronics, Embedded Systems and Automation)LAB
School of Engineering,University of California, Merced
E: [email protected]: CAS Eng 820 (T: 209-228-4398)
Jul 28, 2014. Monday 4:00-6:00 PMApplied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced
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 @ UCMerced04/21/2014
MESA LAB
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]
04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced
Slide-3/1024
MESA LAB
Example
• Given • Then,
04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced
Slide-4/1024
MESA LAB
Property
• For any matrix in standard form– i.e. irreducible– and compromise, and is monic
there always exist LMFDs and RMFDs.
04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced
Slide-5/1024
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
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
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
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
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
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
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
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