lab1 - University of California Santa Barbara Department of...

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University of California, Santa Barbara Department of Electrical & Computer Engineering ECE 147b: Digital Control Lab 1: Digital Control Design Overview This lab will look at various means of transforming continuous-time designs to discrete-time designs. The plant we will attempt to control is the linear position servo. We will design multiple controllers for use in the unity feedback loop shown in Figure 1. C(z) + Encoder D/A P(s) y r - Figure 1: Closed-Loop System Model Derivation The DC motor equations are given by: V = I m R m + K m K g ω g = I m R m + K m K g ˙ x r , where V is the voltage applied to the motor, I m (Amp) is the motor current, K m ( V rad / sec ) is the back EMF constant, K g is the gear ratio in the motor gearbox, ω g ( rad sec ) is the motor output angular velocity, ˙ x ( m sec ) is the cart velocity, and r (m) is the radius of the motor pinion that meshes with the track. The torque generated by the motor is: τ = K m K g I m ,
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2 ECE147b: Lab #1 which exerts a force, F , on the cart via the pinion, F = τ r . This force results in an acceleration of the cart governed by Newton’s second law: F = m c ¨ x, where m c is the mass of the cart. Combining these two equations yields: x V = K m K g r s ( m c R m r 2 s + K 2 m K 2 g ) . The values of the coefficients are tabulated below. These have been taken from the manufacturer’s manual and may not be exactly correct for your particular hardware. Parameter Value K m Back EMF constant 0.00767 V/(rad/sec) R m Motor resistance 2.6 Ohms K g gear ratio 3.7 N p Motor pinion teeth 24 r pinion radius 0.00635 m m c cart mass 0.455 kg Substituting the numerical values for the constants in the above equation, and defining P ( s ) = x/V , yields P ( s ) = 3 . 78 s ( s + 16 . 88) . (1) Equation 1 describes the linear position servo which we will control in this lab. Analog Design Approaches We can design a controller by root locus or Bode methods. A suitable controller is given by C 1 ( s ) = 100( s + 16 . 88) s + 30 . This controller should place the poles of the closed-loop transfer function from the reference to the output at - 15 . 0 ± j 12 . 4. Note that if no pole-zero cancellation is acchieved succesfully, we will have another pole. In the cases that you have more than two poles, assign the others
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Digital Control Design 3 in stable locations. One possible location is - 16 . 9 since the closed-loop transfer function from the disturbance to the output has a pole at that location. Check this. Substituting s = 1 - z - 1 T , (2) into the expression for C 1 ( s ) gives a discrete equivalent with the following pulse transfer function: C 1 ( z ) = b 0 z + b 1 z + a 1 = b 0 + b 1 z - 1 1 + a 1 z - 1 .
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