EE-486 Project

An antenna angular position motor is described by the following block diagram

By substituting values for the system for R, L, Km, N, J, f and Kb we can simplify the plant. We also expand the transfer block containing J and f to an integrator feedback loop.

From this diagram, we can create the state variable description of the system.

Note: Below is supposed to be a 2 by 2 matrix, Word doesn’t seem to have a way to insert matrices

[X1’ ]=[0 1][x1]+[0]Va+[0]*Td

[X2’]=[ -3.5][x2]+[5]Va+[1]*Td

Y=[1 0]*[x1]

[x2]

Error Tracking Feedback Control Design Control design begins with the following equations from the state variable description

Error=[Thetaa(x1)-Theta command(the input)]

Z=[x1’] w=Va’

[x2’]

Below is supposed to be a 3 by 3 matrix:

[e1’]=[0 1 0]*[x1] + [e]

[z1’]=[0 0 0.1]*[x2]+[z1]*Va+[0]

[z2’]=[0 0 -3.5] +[z2] [1]

The desired poles for the controller are specified as P(ƛ) = ( ƛ+ 5 -2j)( ƛ+ 5 +2j)( ƛ+5) = ƛ’’’+15ƛ ‘’+79ƛ’+145

P(A) the matrix that will be used to find the necessary gains for the controller is:

P(A)=[145 79 1.15] and P(C)=[0 0 0.5]

[0 145 3.875] [0 0.5 -1.75]

[0 0 9.375] [5 -17.5 61.25]

The determinant of PC is -1.25, so the system is fully controllable. Using Ackermann’s Formula k= [0 0 1 ]P (A)*P(C)^-1 to find the gains for the controller

K=[0 0 1]*[0 0 0.5]^-1 *[145 79 1.15]

[0 0.5 -1.75] [0 145 3.875]=[290 158 2.3]

[5 -17.5 61.25 [0 0 9.375]

K1=290 K2=158 K3=2.8

The input for the controller w = –k1e –k2z1 - k2z2 = Va’By integrating Va’Va=-k1 ∫e-k2*θa –k3*Ωm= -290∫( θa- θc ) -158A -2.3Ωm

Observer Design Formulating the Observer begins with the Observability Matrix from the original state equations.

P(o)=[1 0]

[0 0.1]

The determinant of POS is 0.1 indicating the system is fully observable. The poles for the observer P(ƛ) = (ƛ +40 +2j)( ƛ +40 -2j) = ƛ^2+80ƛ+1604

P(A) the 2 by 2 matrix that will determine the observer gains is:P(A)=[1604 7.65]

[0 1336.25]

Block diagram of controller:

Block diagram of controller with the observer:

Top: Theta (purple) and Theta command (yellow) vs time Bottom: Error vs Time

Both plots are for the uncompensated controller with the disturbance=0

Desired output (Yellow) vs Actual Output (Purple) for controller that has had the Luenberger observer added

Top: Armature voltage vs time for controller Bottom: Armature voltage vs time for controller with observer

Top: Error vs time for controller/observer system with disturbance=0 Bottom: Error vs time for controller/observer system with step disturbance at 5 seconds