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Extra info for 04.Automatic Control
Also, for the case of PPF systems presented in the previous subsection, the adaptive backstepping techniques guarantee global stability only in the case where assumption A2 is valid. Such restrictions are made because the computation of the adaptive control law depends on the existence of the inverse of the matrix that consists of the estimated input vector fields (or the Lie derivatives of the output functions along those vector fields). Even in the case of known parameters where the inverse of the corresponding matrix exists (this is trivially satisfied for feedback-linearizable systems), the inverse of the estimate of this matrix might not exist at each time due to insufficiently rich regressor signals, large initial parameter estimation errors, and so on.
There are many useful formulas involving adA , such as etadA (B) = etA Be−tA Accessibility and the Lie Rank Condition. What “state space” is most appropriate for a bilinear system? For inhomogeneous systems, Rn is appropriate. However, for homogeneous BLS, a trajectory starting at 0 can never leave it, and 0 can never be reached in ﬁnite time; the state space may as well be punctured at 0. This punctured n-space is denoted by Rn \0 or Rn0 , and it is of interest in understanding controllability.
Controllability and noncontrollability results are established for several families of A, B pairs, especially for n = 2. Bacciotti (29) completed the study for n = 2. He assumes the same conditions: A > 0; B is diagonal and nonsingular, with no repeated eigenvalues; and β2 > 0 (if not, reverse the sign of the control). Then for the BLS x˙ = Ax + uBx: 1. If β1 > 0, the BLS is completely controllable on Rn+ . 2. If β1 < 0 but δ = (β2 a11 − β1 a22 )2 + 4β1 β2 a12 a21 > 0 and β1 a22 − β2 a11 > 0, then the BLS is completely controllable on Rn+ .