By A. Fradkov
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Read or Download Cybernetical Physics: From Control of Chaos to Quantum Control (Understanding Complex Systems) PDF
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Additional info for Cybernetical Physics: From Control of Chaos to Quantum Control (Understanding Complex Systems)
Sample text
Generation (excitation) of oscillations. The third class of control goals corresponds to the problems of excitation or generation of oscillations. Here, it is assumed that the system is initially at rest. ) In this case the goal trajectory of the state vector x∗ (t) is not prespecified. Moreover, the goal trajectory may be unknown, or may even be irrelevant to the achievement of the control goal. Such problems are well known in electrical, radio engineering, acoustics, laser, and vibrational technologies – wherever it is necessary to create an oscillatory mode for the system.
2 Control goals 25 E. Modification of the limit sets (attractors) of the systems. The last class of the control goals is related to modification of some quantitative characteristics of the limit behavior of the system. ); – changing the position or the type of the bifurcation point in the parameter space of the system. Investigation of the above problems started in the end of the 1980s with the works on bifurcation control [3, 442] and continued in the works on control of chaos. Ott, Grebogi, and Yorke [331] and their followers introduced a new class of control goals, not requiring any quantitative characteristic of the desired motion.
Zm ) sign z is defined as sign z = col (sign z1 , . . , sign zm ). In order to make a reasonable choice of the control algorithm parameters the applicability conditions should be verified. The main conditions are: convexity of the function ω(x, u, t) in u and existence of “ideal” control u∗ such that ω(x, u∗ , t) ≤ 0 for all x (attainability condition). More precise formulations and mathematical proofs can be found in [29, 134, 135, 157, 164]. The speed-gradient algorithms can be modified to take into account constraints.
