Fall 2001 16.3112-4 Theorem: An Lti system is controllable iff it has no uncontrollable states We normally just say that the pair(A, B) is controllable Pseudo-Proof: The theorem essentially follows by the definition of an uncontrollable state If you had an uncontrollable state x*, then it is orthogonal to the forced response state a(t), which means that the system cannot reach it in finite time the system would be uncontrollable o Theorem The vector is an uncontrollable state iff (x)[BABA2B…A-B] See page 81 Simple test: Necessary and sufficient condition for controllability is that ankM会rank[BABA2B…A2-B]=nFall 2001 16.31 12—4 • Theorem: An LTI system is controllable iff it has no uncontrollable states. — We normally just say that the pair (A,B) is controllable. Pseudo-Proof: The theorem essentially follows by the definition of an uncontrollable state. — If you had an uncontrollable state x? , then it is orthogonal to the forced response state x(t), which means that the system cannot reach it in finite time ; the system would be uncontrollable. • Theorem: The vector x? is an uncontrollable state iff (x? ) T £ B AB A2B ··· An−1B ¤ = 0 — See page 81. • Simple test: Necessary and sufficient condition for controllability is that rank Mc , rank £ B AB A2B · · · An−1B ¤ = n