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l18 - 16.06 Lecture 18 Controllability Continued John Deyst...

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16.06 Lecture 18 Controllability Continued John Deyst October 16, 2003 Today’s Topics 1. Controllability for systems with multiple inputs 2. Example 1
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In situations when we have multiple inputs we obtain a similar result as for the scalar case. As before we want to be able to drive the state anywhere in its state space. Now the control is a multi-dimensional vector u ( t ). Controllability is achieved if u ( t ) can drive the integral. anywhere in the state space in the time T . Similar to last time we now define vectors. so the integral becomes 2
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Let’s look at the first term which is a linear combination of other columns of the B matrix, similarly, for the second term which is a linear combination of other columns of AB . 3
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Extending this analysis to the other terms we conclude that the n dimensional vector represented by the integral Is a linear combination of the columns of the matrices Hence if the system is controllable there must be n columns of B, AB, A 2 B, . . . which are linearly independent. Alternatively, con-
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