ECE/ME 2646: Linear System Theory (3 Credits, Fall 2016)
Lecture 7: Controllability October 25, 2016
Zhi-Hong Mao Associate Professor of ECE and Bioengineering University of Pittsburgh, Pittsburgh, PA 1
Outline of this lecture • Homework 4 (due Nov. 15—in three weeks) • Controllability
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Controllability • Definition – Controllability deals with whether or not the state of a state-space equation can be controlled from the input – Consider a state equation x Ax Bu. This state equation or the pair (A, B) is said to be controllable if for any initial state x(0) = x0 and any final state x1, there exists an input that transfer x0 to x1 in a finite time. Otherwise the state equation or (A, B) is said to be uncontrollable 3
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Controllability •
Definition
• Examples of controllable and uncontrollable systems
1W
+ u -
1W +x -
1W
1W
+ u -
+ x1
+ x2
1F - 1F 1W
1W
Question: Are the above systems controllable (in the left system, the state variable is x; in the right system, the state variables are x1 and x2?
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Controllability •
Definition
• Examples of controllable and uncontrollable systems
Heart rate control, yoga practitioner, and implanted heart 5
Controllability •
Definition
• Examples of controllable and uncontrollable systems
Yoked eye movements and independent eye movements (in chameleon and sandlance) 6
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Controllability • •
Definition Examples of controllable and uncontrollable systems
• Theorems The pair (A, B) is controllable if and only if the matrix t
t
0
0
Wc (t ) eA BB' eA ' d eA ( t - ) BB' eA '( t - ) d is nonsingular for any t > 0.
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Controllability • •
Definition Examples of controllable and uncontrollable systems
• Theorems
t
t
0
0
Wc (t ) eA BB' eA ' d eA ( t - ) BB' eA '( t - ) d
Remark: If the pair (A, B) is controllable, then for any x(0) = x0 and any x(t1) = x1, the input -1
u(t ) -B' eA '( t1 -t )Wc (t1 )[eAt1 x0 - x1 ] will transfer x0 to x1 at time t1. Question: Please verify that t1
x(t1 ) eAt1 x(0) eA ( t1 - ) Bu( )d x1. 0
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Controllability • •
Definition Examples of controllable and uncontrollable systems
• Theorems The n-dimensional pair (A, B), where A and B are n by n and n by p matrices respectively, is controllable if and only if the n by np controllability matrix
C [B AB A 2B A n-1B] has rank n (full row rank).
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Controllability • •
Definition Examples of controllable and uncontrollable systems
• Theorems Example: Is the following state equation controllable
- 0.5 0 0.5 x x u ? 0 - 1 1
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Controllability • •
Definition Examples of controllable and uncontrollable systems
• Theorems An n by m matrix H, with n < m, has rank n, if and only if the n by n matrix HH’ has rank n or det(HH’) is nonzero. The n-dimensional pair (A, B) is controllable if and only if the n by np controllability matrix
C [B AB A 2B A n-1B] has rank n (full row rank) or the n by n matrix CC’ is nonsingular. 11
Controllability • •
Definition Examples of controllable and uncontrollable systems
• Theorems The controllability property is invariant under any equivalence transformation. (Why?)
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References • • • • • • •
C.-T. Chen. Linear System Theory and Design, 4th Edition, Oxford University Press, 2013. T. Hu. Lecture Notes for 16.513 Control Systems. University of Massachusetts at Lowell, 2006. C. L. Phillips and R. D. Harbor. Feedback Control Systems, 4th Edition, Prentice Hall, 2000. http://www.cartoonstock.com/lowres/abr1536l.jpg http://www.nature.com/nature/journal/v399/n6734/images/399305aa.eps.0.gif http://www.uq.edu.au/nuq/jack/sandlance.html http://www.yoga-age.com/
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