An electro-hydraulic positioning system is shown in Figure P11.38. Use the following values.

K_a = 10 V/A             Ki = 10–² in./V 

K₂ = 3 × 10⁵ sec^–3  K₃ = 20 V/in. 

z = 0.8                       w_n = 100 rad/sec t = 0.01 sec

a. Develop a state-variable model of the plant with the controller current i_c as the input and the displacement y as the output.

b. Assuming that proportional control is used so that G_c(s) = K_p, develop a state model of the system with \y as the input and y as the output. Draw the root locus and use it to determine whether or not the system can be made stable with an appropriate choice for the value of K_P.

Figure P11.38

Load DR Position sensor Power Controller amplifier Command potentiometer Y,(s) K7Ys) K. Ge(s) Ka ControllerPowerMotor/ge ervomotor/load amplifier Sensor