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N , and rearranging terms, we may rewrite Eq. (75) as follows: 22 ADAPTIVE CONTROL Using the above methodology, we have therefore transformed the system in Eq. (65) into the following one: The above system is feedback-linearizable if the following assumption holds. A1. γ 0 (z) + ϑT γ 1 (z) = 0 for all z. Note now that in the case where θ (and thus ϑ) is known, a controller that meets the control objective is the controller of the form Under the above control law, the closed-loop dynamics become It can be shown that the matrix A0 is a stability matrix, provided that ci > 2.
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. 11. K. J. Astrom, B. Wittenmark, Adaptive Control, Reading, MA: Addison-Wesley, 1989. 12. P. A. Ioannou, J. Sun, Robust Adaptive Control, Upper Saddle River, NJ: Prentice-Hall, 1996.
STABILIZATION I: CONSTANT CONTROLS This section will introduce an important engineering design goal, stability, and the beginning of a running discussion of stabilization. Stabilization is an active area of research, in an effort to ﬁnd good design principles. , R(λi ) < − < 0. Then x˙ = F x is said to be exponentially stable (ES); as time increases, all solutions are bounded and ||x(t)|| < ||x(0)||e−t . If even one eigenvalue lies in the right half plane, almost all solutions will grow unboundedly and the system is called unstable.