Digital Control Systems

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Digital Control Systems

• Lecture Set 4C
• Digital Control Design via Continuous Design -
  Case Studies
• F,PW Chapter 7.2

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Digital Control Design via Continuous Design -
Part C

• Antenna Design Example

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Antenna control system design FPW Examples 7.2,
7.3 7.4
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Step 1 Choose ws

• FPW choose T1 sec (ws/wb) 2pi/1.153 5.447)
• which is much too long for design by emulation.
  We should expect their results to be very poor.
• A conservative choice would be T0.2 sec which
  should yield good results using emulation.

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Step 1 (Contd)

• Design by Discrete Equivalent Method

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.
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Step 2 Find D(s) (Contd)
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Step 2 Find D(s) (Contd)
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Root Loci-Discrete Equivalent Compensation with
T1
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Step 3 Compute D(z) for T0.2 sec
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Step 3 Transform D(z) into difference equation
9.1544 (z-0.9802) 9.1544 -
8.9731z-1 U(z) D(z) -----------------
______________ ________
(z-0.8187) 1 - 0.8187 z-1
E(z)
uk 0.8187uk-1 9.1544ek-8.9731ek-1
Typical Code in High-Order Language Interrupt
at Sample Time Read (Channel 0,e) A/D call
u0.8187u_last9.1544e-8.9731e_last
Control Law Output (Channel1,u) D/A call
u_lastu Save last output e_laste Save
last error sample Return Return to main
program
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Step 4 Analyze Discrete System using MATLAB
Emulation_Antenna.m Continuous Plant
Model plant_czpk(,0,-0.1,0.1) D_czpk(-0.1,
-1,10) closed_loop_cfeedback(D_cplant_c,1) da
mp(closed_loop_c) Discrete Plant
Model Tsinput('Enter desired sample period, Ts
') D_dc2d(D_c,Ts,'matched') plant_dc2d(plant_c,T
s) closed_loop_dfeedback(D_dplant_d,1) damp(clo
sed_loop_d) Comparison of Step
Responses figure(1),step(closed_loop_c,'b',closed_
loop_d,'r') title('Step Response Comparison')
Comparison of Frequency figure(2),bode(closed_loop
_c,'b',closed_loop_d,'r') title('Frequency
Response Comparison') Response of Discrete
System to Ramp Input of 0.01t Integrator_dtf(Ts,
1 -1,Ts) figure(3),step(0.01Integrator_dclose
d_loop_d) title('Unit Ramp Response of Discrete
Model ')
See M-file Emulation_Antenna.m provided on
website.
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MATLAB results with Ts1.0
Zero/pole/gain 0.1 --------- s (s0.1)
Zero/pole/gain 10 (s0.1) ---------- (s1)

Eigenvalue Damping
Freq. (rad/s)
-1.00e-001
1.00e000 1.00e-001 -5.00e-001
8.66e-001i 5.00e-001 1.00e000
-5.00e-001 - 8.66e-001i 5.00e-001
1.00e000

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MATLAB results with Ts1.0
Enter desired sample period, Ts1 Ts 1
Zero/pole/gain 6.6425 (z-0.9048) ---------------
-- (z-0.3679) Sampling time 1
Zero/pole/gain 0.048374 (z0.9672) -------------
------ (z-1) (z-0.9048) Sampling time 1


Eigenvalue Magnitude Equiv. Damping
Equiv. Freq. (rad/s)

9.05e-001 9.05e-001
1.00e000 1.00e-001
5.23e-001 6.36e-001i 8.24e-001
2.14e-001 9.04e-001 5.23e-001
- 6.36e-001i 8.24e-001 2.14e-001
9.04e-001
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Step Response with Ts1.0
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Frequency Response with Ts1.0
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MATLAB results with Ts0.2
Enter desired sample period, Ts 0.2 Ts
0.2000 Zero/pole/gain 9.1544
(z-0.9802) ----------------- (z-0.8187)
Sampling time 0.2 Zero/pole/gain 0.0019867
(z0.9934) -------------------- (z-1)
(z-0.9802) Sampling time 0.2


Eigenvalue Magnitude Equiv. Damping
Equiv. Freq. (rad/s)

9.80e-001 9.80e-001
1.00e000 1.00e-001
9.00e-001 1.62e-001i 9.15e-001
4.47e-001 9.96e-001 9.00e-001
- 1.62e-001i 9.15e-001 4.47e-001
9.96e-001
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Step Response with Ts0.2
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Frequency Response with Ts0.2
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0.01t ramp response with Ts0.2
Steady-state error 0.01 as predicted for a
Velocity Error Coefficient, Kv, of 1.0. Error
Slope/Kv
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Digital Control Design via Continuous Design -
Part C

• Case Study Applications Technology Satellite 6
  (ATS-6) Product Upgrade