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Extra resources for 04.Automatic Control
Control, 6 (3): 1958. 2. D. McRuer, I. Ashkenas, D. Graham, Aircraft Dynamics and Automatic Control, Princeton, NJ: Princeton Univ. Press, 1973. 3. J. B. , System Sensitivity Analysis, Stroudsburg, PA: Dowden, Hutchinson & Ross, 1973. 4. L. P. Grayson, Design via Lyapunov’ second method, Proc. 4th JACC, 1963. 5. P. C. Parks, Lyapunov redesign of model reference control systems, IEEE Trans. Autom. Control, 11: 1966. 6. P. H. Phillipson, Design methods for model reference adaptive systems, Proc. Inst.
For n ≥ 3 the puncture has negligible effect. In some applications, other unusual state spaces may be appropriate. There is an n-dimensional generalization of our scalar example, the diagonalizable BLS, which are (up to similarity) given by x˙ i = ui xi , i = 1, . . , n. If the initial state is on a coordinate half-axis, a quadrant of a coordinate plane, . . , or on one of the 2n orthants, the state must stay there forever. Other BLS that live this way on positive orthants occur rather often, and their controllability properties will be discussed in the “Positive Systems” section below.
Many papers on controllability and stabilization cite Jurdjevic and Quinn (25). For BLS their condition specializes as follows, for piecewise continuous signals. If A has eigenvaiues that are purely imaginary and distinct and the Lie rank condition is satisﬁed, then x˙ = Ax + uBx is controllable. This extends to multiple-input BLS immediately. A stabilization result using the ad-condition is also obtained. For symmetric systems, the Lie rank condition is a sufﬁcient condition for controllability, which is connected to the fact that in that case the transition matrices constitute a Lie group (discussed below).
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