Self-Tuning Regulators (STR) Adaptive control

1. Contents
1.1 Indirect Self-Tuning Regulators (STR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 MATLAB Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1

2. 1.1. INDIRECT SELF-TUNING REGULATORS (STR) Adaptive Control
1.1 Indirect Self-Tuning Regulators (STR)
STR is one of the adaptive control methods that automatically finds its parameters in the control law by solving the Diophantine
equation.
Diophantine Equation
G(z−1
) =
z−dB(z−1)
A(z−1)
Gc(z−1
) =
S(z−1)
R(z−1)
Gcl(z−1
) =
y
uc
=
z−dBS
AR + z−dBS
A(z−1
) = 1 + a1z−1
+ a2z−2
+ ... + anaz−na
B(z−1
) = b0 + b1z−1
+ b2z−2
+ ... + bnbz−nb
R(z−1
) = 1 + r1z−1
+ r2z−2
+ ... + rnrz−nr
S(z−1
) = s0 + s1z−1
+ s2z−2
+ ... + snsz−ns
α(z−1
) = 1 + α1z−1
+ α2z−2
+ ... + αnαz−nα
nr = nb + d − 1 , ns = na − 1 , nα = na + nb + d − 1 = nr + ns + 1
C/C Eqn. : AR + z−d
BS = AmA0 = αc
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1 0 0 0
a1 1 0 0
a2 a1 1 0
a3 a2 a1 1
a4 a3 a2 a1
. a4 a3 a2
ana . a4 a3
0 ana . a4
0 0 ana .
0 0 0 ana
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
nα x nr
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0 0 0 0
0 0 0 0 (d−1) x (ns+1)
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
b0 0 0 0
b1 b0 0 0
b2 b1 b0 0
. b2 b1 b0
bnb . b2 b1
0 bnb . b2
0 0 bnb .
0 0 0 bnb
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nα x nα
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r1
r2
.
.
.
rnr
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s0
s1
s2
.
.
.
sns
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=
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α1
α2
α3
α4
α5
α6
.
.
.
αnα
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−
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a1
a2
a3
.
.
.
ana
0
.
.
.
0
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STR Configurations
Feed forward Controller Feed forward and Feedback Controller
Mohamed Mohamed El-Sayed Atyya Page 2 of 18

3. 1.2. EXAMPLES Adaptive Control
R(z) Configurations
B = B−B+
B+ : monic polynomial whose zeros are stable and well damped that they can be canceled
B− : monic polynomial whose zeros are unstable poorly damped that they cannot be canceled or a simple constant b0
Indirect STR without zero cancellation
Use Basic Diophantine Equation
Indirect STR with zero cancellation
B+ = B/b0 , B− = b0
C/C Eqn. : AR + z−dB−S = αc
R = B+R
T(z) Configurations
1. T(z) = Simple constant, to improve DCgain
2. T(z) = A0
3. Choose T(z) to cancel error dynamics
C/C Eqn. : ML + z−dBT = AmA0 = αc
L(z−1) = 1 + l1z−1 + l2z−2 + ... + lnlz−nl
T(z−1) = t0 + t1z−1 + t2z−2 + ... + tntz−nt
Where uc =
N(z)
M(z)
, nl = nb + d − 1 , nt = nm − 1 , nα = nm + nb + d − 1 = nl + nt + 1
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1 0 0
m1 1 0
m2 m1 1
m3 m2 m1
m4 m3 m2
. m4 m3
mnm . m4
0 mnm .
0 0 mnm
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nα x nl
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0 0 0 0
0 0 0 0 (d−1) x (nt+1)
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
b0 0 0 0
b1 b0 0 0
b2 b1 b0 0
. b2 b1 b0
bnb . b2 b1
0 bnb . b2
0 0 bnb .
0 0 0 bnb
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nα x nα
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l1
l2
.
.
.
lnl
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t0
t1
t2
.
.
.
tnt
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=
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α1
α2
α3
α4
α5
α6
.
.
.
αnα
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−
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m1
m2
m3
.
.
.
mnm
0
.
.
.
0
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4. choose T(z) as T(z)B(z) = z−n
1.2 Examples
The first step of STR is to estimate the system parameters by one of the linear estimation methods [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], then
design STR controller, the results of some STR techniques is presented,
Mohamed Mohamed El-Sayed Atyya Page 3 of 18

4. 1.2. EXAMPLES Adaptive Control
1. RLS estimation and model following without zero cancellation controller
Mohamed Mohamed El-Sayed Atyya Page 4 of 18

5. 1.2. EXAMPLES Adaptive Control
Mohamed Mohamed El-Sayed Atyya Page 5 of 18

6. 1.2. EXAMPLES Adaptive Control
2. RLS estimation and model following with zero cancellation controller
Mohamed Mohamed El-Sayed Atyya Page 6 of 18

7. 1.2. EXAMPLES Adaptive Control
Mohamed Mohamed El-Sayed Atyya Page 7 of 18

8. 1.2. EXAMPLES Adaptive Control
3. RLS estimation and model following without zero cancellation and controller and T = A0
Mohamed Mohamed El-Sayed Atyya Page 8 of 18

9. 1.2. EXAMPLES Adaptive Control
Mohamed Mohamed El-Sayed Atyya Page 9 of 18

10. 1.2. EXAMPLES Adaptive Control
4. RLS estimation and model following with zero cancellation controller and T = A0
Mohamed Mohamed El-Sayed Atyya Page 10 of 18

11. 1.2. EXAMPLES Adaptive Control
Mohamed Mohamed El-Sayed Atyya Page 11 of 18

12. 1.2. EXAMPLES Adaptive Control
5. RLS estimation and model following without zero cancellation controller and T is selected
to cancel error dynamics
Mohamed Mohamed El-Sayed Atyya Page 12 of 18

13. 1.2. EXAMPLES Adaptive Control
Mohamed Mohamed El-Sayed Atyya Page 13 of 18

14. 1.2. EXAMPLES Adaptive Control
6. RLS estimation and model following with zero cancellation controller and T is selected to
cancel error dynamics
Mohamed Mohamed El-Sayed Atyya Page 14 of 18

15. 1.2. EXAMPLES Adaptive Control
Mohamed Mohamed El-Sayed Atyya Page 15 of 18

16. 1.2. EXAMPLES Adaptive Control
7. RLS estimation and model following controller and T(z)B(z) = z−n
Mohamed Mohamed El-Sayed Atyya Page 16 of 18

17. 1.2. EXAMPLES Adaptive Control
Mohamed Mohamed El-Sayed Atyya Page 17 of 18

18. 1.3. MATLAB CODES Adaptive Control
1.3 MATLAB Codes
1.1 http://goo.gl/9qweI1
1.4 References
1. http://goo.gl/e7J2kq
2. http://goo.gl/3q6Yc6
3. http://goo.gl/SCPvEW
4. http://goo.gl/JnrdNh
5. http://goo.gl/xjpHha
6. http://goo.gl/6wVeuW
7. http://goo.gl/vuKeaL
8. http://goo.gl/mL0RCz
9. http://goo.gl/vzViYE
10. Karl Johan Astrom, Adaptive Control, 2nd
Edition.
11. Leonid B. Freidovich, lecture 12.
1.5 Contacts
mohamed.atyya94@eng-st.cu.edu.eg
Mohamed Mohamed El-Sayed Atyya Page 18 of 18