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The state variables for a given plant constitute a vector function depending on time x(t). Knowledge of an initial state x(0), of future external inputs, and the ﬁrstorder vector differential equation that describes the plant determine the trajectory {x(t), t ≥ 0}. For the moment, we will suppose that the plant elements, such as capacitors, inductors, and resistors, are linear (Ohm’s Law) and constant in value. ) As written, equation 1 has constant coefﬁcients, and such a control system is called time-invariant, which means that its behavior does not depend on where we choose the origin of the independent variable t; the system can be initialized and control v exerted at any time.

Sometimes the duty cycle of a switch is under control, as in motors and DC-to-DC power conversion systems. A simple example is a conventional ignition sparking system for automobiles, in which the primary circuit can be modeled by assigning x 1 to voltage across capacitor of value C, x 2 to current in the primary coil of inductance L. The control is a distributor or electronic switch, either open (inﬁnite resistance) or closed (small resistance R) with duty cycle speciﬁed by the crankshaft rotation and timing controls.

This description seems to be characteristic of several such examples of strange attractors and bounded chaos, including the R¨ossler attractor. A= Bilinear Systems In electrical engineering, BLS viewpoints can be used to discuss switched and clocked circuits, in which the control takes on only a discrete set of values like {0, 1} and the plant is linear in each switched condition. The solutions are then easy to compute numerically. Sometimes the duty cycle of a switch is under control, as in motors and DC-to-DC power conversion systems.