l11 - and D, as follows 5 A block diagram representation is...

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16.06 Lecture 11 State Space John Deyst September 29, 2003 Today’s Topics 1. The concept of system state 2. State vector deﬁnition 3. State space representation of LTI systems 1

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The state space approach models systems using linear vector space methods. The concept of a state vector (array) is important because it com- pletely represents the current status (state) of a system. Example: A Capacitor Typically, there are many possible ways to deﬁne the state. Charge q or voltage v c can be states. If v c is chosen as the state then the constituitive relation for the capacitor characterizes the behavior of the state If we have the simple circuit system 2
We have two constitutive relations so the equation for the system is with the solution and v ( t ) is the system state. 3

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Furthermore, if we know the state at any particular time t 1 ,thenwe know the state for all future time. If there are no inputs (homogeneous system) then the state is suﬃ- cient to predict the entire future behavior of the system. Deﬁnition of the state vector for a system. 4
We deﬁne the state vector as a x ( t ) In general the system will also have inputs u ( t ) and outputs y ( t ) As we have said a number of times, in this course we will only study linear, time invariant systems (LTI). Any LTI system can be charac- terized with four constant matrices- A,B,C

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Unformatted text preview: and D, as follows 5 A block diagram representation is where Simple example of a second order system The externally applied force f is the input and the output is the displacement d of the mass away from its rest position ( d = 0) 6 diﬀerential equation for this systems is, using ( F = ma ) This is a second order system so we need two states. Choose We need the time derivatives of these two states Thus we can write the vector/matrix equations 7 Also, any non-singular transformation of the state of a system is also a state of the system. For example, suppose the ( n × n ) matrix T is a non-singular transformation of the state x to a new state x ± . Then so If we then deﬁne the equations for the new state are 8 Similarly so and hence so the input and output are the same as before but we now have a transformed state x ± . 9...
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This note was uploaded on 11/06/2011 for the course AERO 100 taught by Professor Willcox during the Fall '03 term at MIT.

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l11 - and D, as follows 5 A block diagram representation is...

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