04.Automatic Control by John G. Webster (Editor)

By John G. Webster (Editor)

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4 of Reference 3. Once having written that expression for x(t), it can be observed that must satisfy the matrix differential equation ˙ = (A + u(t)B) , (t0 , t0 ) = I It has the composition property, also called the semigroup property (t, t1 ) (t1 , t0 ) = (t, t0 ). However, as a control system, equation 4 is timeinvariant, by definition. Then the most convenient families of admissible controls for BLS are the PWC and other piecewise-defined controls; such a control can be specified by its values on an interval of definition of duration τ, for instance {u(t), t ∈ (t0 , t0 + τ)}.

From the ad-condition, trajectories cannot remain in the set {x Bx = 0|x = 0}, and V (x(t)) → 0 as t → ∞. However, looking at the onedimensional case, one sees that for the differential equation ξ˙ = −ξ 3 , the approach to the origin is of order t −1/2 , and that this is also true for the n-dimensional problem using quadratic feedback. Using the same hypotheses and nearly the same short proof as in Reference 25, the reader can verify that the bounded feedback u = −x Bx/x x provides exponential stability.

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.

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