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2.1 A dynamical system may be described in various ways - with a transfer function, with a differential equation, and with a system of differential equations on state- space form. In this problem we transfer between the different representations for four examples of dynamical systems, from biology, mechanics, electronics, and economics. a. A model for bacterial growth in a bioreactor is given by 10 x = -1 (44) x+8) ² (₁0): y = where u is the inflow of glucose to the reactor, and y is the bio mass. Determine the transfer function from u to y, and a differential equation that determines the relation between the input and output signals of the system. b. A simple model of a telescope is given by Jd²y dy + D = u dt dt² where yis the angle of the telescope to the earth surface, and u is the torque from the motor that controls the telescope. Determine the transfer function from u to y and write the system on state-space form. c. An electronic low pass filter is used at recordings to attenuate high frequency noise. The input u is the original noisy signal, and the output y is the recorded signal. The filter is given on state-space form as dx 1 1 == -x+ -u k k dt y = x Determine the transfer function from u to y.