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Unformatted text preview: Topic #6 16.31 Feedback Control Systems StateSpace Systems What are statespace models? Why should we use them? How are they related to the transfer functions used in classical control design and how do we develop a statespace model? What are the basic properties of a statespace model, and how do we analyze these? Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 61 SS Introduction State space model: a representation of the dynamics of an N th order system as a first order differential equation in an Nvector, which is called the state . Convert the N th order differential equation that governs the dynamics into N firstorder differential equations Classic example: second order massspring system m p + c p + kp = F Let x 1 = p , then x 2 = p = x 1 , and x 2 = p = ( F c p kp ) /m = ( F cx 2 kx 1 ) /m p p = 1 k/m c/m p p + 1 /m u Let u = F and introduce the state x = x 1 x 2 = p p x = A x + Bu If the measured output of the system is the position, then we have that y = p = 1 0 p p = 1 0 x 1 x 2 = c x September 17, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 62 Most general continuoustime linear dynamical system has form x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) where: t R denotes time x ( t ) R n is the state (vector) u ( t ) R m is the input or control y ( t ) R p is the output A ( t ) R n n is the dynamics matrix B ( t ) R n m is the input matrix C ( t ) R p n is the output or sensor matrix D ( t ) R p m is the feedthrough matrix Note that the plant dynamics can be timevarying. Also note that this is a multiinput / multioutput (MIMO) system. We will typically deal with the timeinvariant case Linear TimeInvariant (LTI) state dynamics x ( t ) = A x ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t ) so that now A,B,C,D are constant and do not depend on t . September 17, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 63 Basic Definitions Linearity What is a linear dynamical system? A system G is linear with respect to its inputs and output u ( t ) G ( s ) y ( t ) iff superposition holds: G ( 1 u 1 + 2 u 2 ) = 1 Gu 1 + 2 Gu 2 So if y 1 is the response of G to u 1 ( y 1 = Gu 1 ), and y 2 is the response of G to u 2 ( y 2 = Gu 2 ), then the response to...
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.
 Fall '07
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