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# EE 350 Control Systems Assignments

## Assignment 1: Modeling of a dynamic system (cont'd...)

1.14 Consider the servo system shown in following figure which is self explanatory. The output voltage of this amplifier is applied to the armature circuit of the dc motor. A fixed voltage is applied to the field winding. If an error exists, the motor develops a torque to rotate the output load in such a way as to reduce the error to zero. For constant field current, the torque developed by the motor is $T=K_2i_a$, where $K_2$ is the motor torque constant and $i_a$ is the armature current. When the armature is rotating, a voltage proportional to the product of the flux and angular velocity is induced in the armature. For a constant flux, the induced voltage (back emf) $e_b=K_3d\theta/dt$ $K_3$ is the back emf constant of the motor, and $\theta$ is the angular displacement of the motor shaft.
a. Obtain the transfer function between the motor shaft angular displacement $\theta$ and the error voltage $e_v$.
b. Draw block diagram for this system and a simplify the block diagram when $L_a$ is negligible.

Fig 1.13 Position control of a DC motor

1.15 Consider the liquid-level system shown in the following figure. All the bars in the figure represent steady state values. Determine the state space model of the system when $h_l$ and $h_2$ are the outputs and $q_{i1}$, and $q_{i2}$ are the inputs. You may assume $q_{il}$, $q_{i2}$, $h_l$, $h_2$, $q_1$, and $q_0$ are considered small.

Fig 1.14 Liquid level system

#### Reference:

(i) Katsuhiko Ogata, “Modern Control Engineering”, Prentice-hall Of India Pvt Ltd, 4th edition, 2007.
(ii) Roland S. Burns, “Advanced Control Engineering”, Reed educational and professional publishing Ltd., Oxford, First Edition 2001.
(iii) I.J.Nagrath, and M. Gopal, "Control Systems Engineering", New age international publishers, New Delhi, fifth edition, reprint, 2014.