By Andrzej Bartoszewicz
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Extra resources for Robust Control, Theory and Applications
In the frequency domain we get y( s ) = G( s )u( s ) = G(s )K (s )(ω (s ) − y(s )) T (s) = y( s ) G(s)K(s) = ω(s) G(s)K(s) + 1 (18) (19) In a similar way, we can calculate a disturbance transfer function, which describes the transfer characteristic between the disturbance d and the output quantity y: S(s ) = y( s ) G(s )K(s ) = d(s) G(s )K(s) + 1 (20) The term G( s )K (s ) has a special meaning: if we remove the feedback loop, so this term represents the transfer function of the resulting open circuit.
The hybrid system H := ( f , g , C , D) can be written in the form x = f (x) + x = g( x ) x ∈C x∈D (3) The map f is called the “flow map”, the map g is called the “jump map”, the set C is called the “flow set”, and the set D is called the “jump set”. The state x may contain variables taking values in a discrete set (logic variables), timers, etc. Consistent with such a situation is the possibility that C ∪ D is a strict subset of ℜn . For simplicity, assume that f and g are continuous functions.
10) shows a hypothetical dynamical system in ℜ2 . The trajectory is moving in the ( x1 , x2 ) plane but we have no knowledge of where the trajectory is as a function of time. On the other hand, we have a scalar valued function V ( x ) , plotted on the z-axis, which has the 20 Robust Control, Theory and Applications guaranteed property that as the trajectory moves the value of this function along the trajectories strictly decreases. Since V ( x(t )) is lower bounded by zero and is strictly decreasing, it must converge to a nonnegative limit as time goes to infinity.