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By John G. Webster (Editor)

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6. P. H. Phillipson, Design methods for model reference adaptive systems, Proc. Inst. Mech. , 183 (35):, 695–706, 1968. 7. R. V. Monopoli, Lyapunov’s method for adaptive control design, IEEE Trans. Autom. Control, 3: 1967. 8. I. D. Landau, Adaptive Control: The Model Reference Approach, New York: Marcel Dekker, 1979. ADAPTIVE CONTROL 27 9. K. S. Narendra, A. M. Annaswamy, Stable Adaptive Systems, Englewood Cliffs, NJ: Prentice-Hall, 1989. 10. S. Sastry, M. Bodson, Adaptive Control: Stability, Convergence and Robustness, Englewood Cliffs, NJ: Prentice-Hall, 1989.

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).

If A = 0 in equations 4 or 5, the BLS is called symmetric: rank(g, Fg, . . , F n−1 g) = n x˙ u jB jx j=1 (4) B jx (5’) j=1 As a control system, equation 4 is time-invariant, and we need to use controls that can start and stop when we wish. The use of PWC controls not only is appropriate for that reason but also allows us to consider switched linear systems as BLS, and in that case, only a discrete set of control values such as {−1, 1} or {0, 1} is used. Later we will be concerned with state-dependent feedback controls, u = u(x), which may have to satisfy conditions that guarantee that differential equations like x˙ = Ax + u(x)Bx have well-behaved solutions.

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

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