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Homework 1 Classical Control Review

1. Consider the following notch filter with center frequency ω_n :

a) How many states are required to implement this filter? Derive the state space matrices for this transfer function: (ABC D, , , )  

b) What is the primary purpose of this filter?  

c) What happens when ζ is small, ζ is large?  

2. The dynamic equations for a given DC servo system are:

Jθ&&=− − +TL bθ& K im

L dt^di + = −Ri v K_mθ 

In the first equation, θ represents the angular position of the output shaft, J J is the load inertia, T_L is the load torque, K_m is the motor torque constant, b is the viscous

friction coefficient, and i is the field current in the motor windings. In the second equation, L is the motor inductance, R is the resistance, and v is the applied voltage.

Define x =_θ θ _ =[T v_L ]^T , and let y =_{θ θ}^&_^T .  Rewrite these equations in ^& i^T , let u

standard state-space form, defining the matrices (A B C D, , , )  

3. Consider the following phase-lock-loop dc-motor speed control system

Reference speed command fr =120 pulses/sec 

Kp = 0.1 V/pulse, KA = K , Ke =1, N=1.  Use the following transfer functions for the motor and controller:

ΩEm_A = s(0.0510s+1)   Gc (s) = 11++1.4TsTs ,  =0.01T  

a) Write the loop transfer function

b) Sketch the root locus for this system for the amplifier gain K 

c) What is the value of K when the roots break away from the real axis and become complex?

d) For K₌5, compute the velocity constant Kv 

e) Sketch the Nyquist D-contour and Nyquist locus for this system.

4. a) Consider a unity feedback system with G s( )= 1s₂ .   

a.1) Sketch the root locus for this system

a.2) Sketch the Nyquist D-contour and entire Nyquist locus for this system.   

b)  

Illustrate on the figure the GM, -GM, and PM.  The red curve is a unit circle.

GM = ___________  -GM = ______________ PM = _____________

Describe/estimate below the transfer function for this system as seen from the above

Nyquist locus

c)  

Illustrate on the figure the GM, -GM, and PM.  The red curve is a unit circle.

GM = ___________  -GM = ______________ PM = _____________

Describe/estimate below the transfer function for this system as seen from the above Nyquist locus.

d)  

GM = ___________  -GM = ______________ PM = _____________

ω_c = ___________  ω_g = ______________  

Describe/estimate below the transfer function for this system as seen from the above Bode plot